%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%   The Abdus Salam International Centre for Theoretical Physics        %
%%%                                                                       %
%%%  `Preparing ICTP Lecture Notes for publication'                       %
%%%        Suggested Stencil version 1.3, April 1999                      %
%%%									  %
%%%   All the LaTeX constructs used in this stencil are documented in     %
%%%   detail in the Lamport's book and also in                            %
%%%   http://www.ictp.trieste.it/texi/teTeX/latex/latex2e-html            %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% DO NOT CHANGE THE FOLLOWING LINES

\documentclass[12pt,twoside]{article}
\usepackage{psfig}
\bibliographystyle{plain}
\input epsf
\topmargin      -0.3in  % distance to headers 
\headheight      0.2in  % height of header box 
\headsep         0.3in  % distance to top line 
\textheight      8.9in  % height of text 
\footskip        0.3in  % distance from bottom line 
\oddsidemargin   0.0in  % Horizontal alignment 
\evensidemargin  0.0in  % Horizontal alignment 
\textwidth       6.5in  % Horizontal alignment 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%    Volume-related  entities (reserved for publication office)         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ang{\,{\rm\AA}}
\def\flux{\,{\rm erg\,cm^{-2}\,arcsec^{-2}\,\AA^{-1}\,s^{-1}}}
\def\GeV{\,{\rm GeV}}
\def\TeV{\,{\rm TeV}}
\def\k{\bf k}
\def\x{\bf x}
\def\gev{\,{\rm GeV}}
\def\H{\,{\cal  H}}
\def\keV{\,{\rm keV}}
\def\MeV{\,{\rm MeV}}
\def\sec{\,{\rm sec}}
\newcommand{\Aa}{\ensuremath{\frac{a^{\prime}}{a}}}
\newcommand{\Ab}{\ensuremath{\Big(\frac{a^{\prime}}{a}\Big)^2}}
\newcommand{\Ac}{\ensuremath{\frac{a^{\prime\prime}}{a}}}
\newcommand{\Dp}{\ensuremath{\delta^{(1)}}}
\newcommand{\Ds}{\ensuremath{\delta^{(2)}}}
\newcommand{\La}{\ensuremath{\partial_i \,\partial^i}}
\newcommand{\deu}[1]{\ensuremath{{\delta #1}^{}}}
\newcommand{\ded}[1]{\ensuremath{{\delta #1}^{(2)}}}
\newcommand{\ze}[1]{\ensuremath{#1}^{(0)}}
\def\Gyr{\,{\rm Gyr}}
\def\yr{\,{\rm yr}}
\def\rcm{\,{\rm cm}}
\def\pc{\,{\rm pc}}
\def\kpc{\,{\rm kpc}}
\def\Mpc{\,{\rm Mpc}}
\def\mpc{\,{\rm Mpc}}
\def\eV{{\,\rm eV}}
\def\ev{{\,\rm eV}}
\def\erg{{\,\rm erg}}
\def\cmm2{{\,\rm cm^{-2}}}
\def\cm2{{\,{\rm cm}^2}}
\def\cmm3{{\,{\rm cm}^{-3}}}
\def\gcmm3{{\,{\rm g\,cm^{-3}}}}
\def\kms{\,{\rm km\,s^{-1}}}
\def\HO{{100h\,{\rm km\,sec^{-1}\,Mpc^{-1}}}}
\def\mpl{{m_{\rm Pl}}}
\def\pl{{\rm Pl}}
\def\mpp{{m_{\rm Pl,0}}}
\def\trh{T_{\rm RH}}
\def\g{\tilde g}
\def\R{{\cal R}}
\def\km{\rm \,km}
\def\yrs{\rm \,yrs}
\def\trh{T_{\rm RH}}

\def\baselinestretch{1.4}
\def\VEV#1{\left\langle #1\right\rangle}
\def\la{\mathrel{\mathpalette\fun <}}
\def\ga{\mathrel{\mathpalette\fun >}}
\def\fun#1#2{\lower3.6pt\vbox{\baselineskip0pt\lineskip.9pt
  \ialign{$\mathsurround=0pt#1\hfil##\hfil$\crcr#2\crcr\sim\crcr}}}
\newcommand{\VolumeHeader}{}
\newcommand{\VolumeSerial}{}


%%% YOUR CHANGES BELOW THIS LINE
%%% If you do not know the full official name and exact dates of the activity
%%% REPLACE the following with BLANKS, the secretariat will take care

\newcommand{\ActivityName}{ {\normalsize {\it 
Summer School on 
}}\\
{\normalsize {\it 
Astroparticle Physics and Cosmology
}}}
\newcommand{\ActivityDate}{ {\normalsize {\it
Trieste, 17 June - 5 July 2002
}}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%    Put your definitions here. For example                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
%\newcommand{\ra}{\rightarrow}
%\newcommand{\cB}{{\cal B}}

%%% THE FOLLOWING IS REQUIRED
%%% A short title for your page header

\newcommand{\LectureHeader}{Preparing...}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% add words to Tex's hyphenation list                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\hyphenation{re-commend-ed}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%            Automatic heading generation                               %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% DO NOT CHANGE THE FOLLOWING LINES

\begin{document}
\pagestyle{myheadings}
%\markboth{\LectureHeader}{\VolumeHeader}
\markright{\VolumeHeader}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%            Title page starts here                                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{titlepage}

%%% YOUR CHANGES BELOW THIS LINE

\title{Inflation and
the Theory of Cosmological Perturbations} 

\author{A. Riotto$^\dagger$\thanks{antonio.riotto@pd.infn.it}
\\[1cm]
{\normalsize
{\it $^\dagger$ INFN, Sezione di Padova, 
via Marzolo 8, I-35131, Padova, Italy.}}
\\[10cm]
%%% FOR FURTHER AUTHORS SEE WHAT IT IS WRITTEN IN THE ABSTRACT 
%%% DO NOT CHANGE THE FOLLOWING LINES
{\normalsize {\it Lectures given at the: }}
\\
\ActivityName 
\\
\ActivityDate 
\\[1cm]
{\small \VolumeSerial} 
}
\date{}
\maketitle
\thispagestyle{empty}
\end{titlepage}

\baselineskip=14pt
%\newpage
\thispagestyle{empty}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%            Abstract page starts here                                  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}

%%% YOUR CHANGES BELOW THIS LINE
These lectures provide a pedagogical introduction to inflation and the 
theory of cosmological perturbations generated during inflation
which  are  thought to be 
the origin of structure in the universe.



\end{abstract}

\vspace{6cm}

{\it Keywords:} Early Universe Physics and Particle Physics.

{\it PACS numbers:}
98.80.Cq

DFPD-TH/02/22

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%       Automatic TOC and your Text starts here                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
\thispagestyle{empty}
\tableofcontents

\newpage
\setcounter{page}{1}

\section{Introduction}

One of the basic ideas of modern cosmology is 
that there was an epoch early in the
history of the universe when potential, or vacuum, energy dominated
other forms of energy density such as matter or radiation. During the
vacuum-dominated era the scale factor grew exponentially (or nearly
exponentially) in time.  In this phase, known as inflation, 
a small, smooth spatial region of size less than the Hubble radius at
that time can grow so large as to easily encompass the comoving volume
of the entire presently observable universe.
If the early universe underwent this period of rapid expansion, then
one can understand why 
the observed universe is  so homogeneous and isotropic to high accuracy.
All  these virtues of inflation were noted when it was first proposed
by Guth in 1981 \cite{guth}. A more dramatic consequence of the inflationary
paradigm was noticed soon after 
\cite{hawking,starob82,guthpi}.
Starting with
a universe which is absolutely homogeneous and isotropic at the classical
level, the inflationary expansion of the universe will
`freeze in' the vacuum fluctuation of the inflaton field so that 
it becomes an essentially classical quantity. 
On each comoving scale, this happens soon after horizon exit.
Associated with this vacuum fluctuation is
a primordial energy density perturbation, which survives after inflation and 
may be the origin of all structure in the universe. In particular,
it may be responsible for the observed
cosmic microwave background (CMB) anisotropy and for the large-scale
distribution of galaxies and dark matter. 
Inflation also generates
primordial gravitational waves as a vacuum fluctuation, which may contribute
to the low multipoles of the CMB anisotropy.
Therefore, a prediction of inflation is that 
all of the
structure we see in the universe is a result of quantum-mechanical
fluctuations during the inflationary epoch. 


The goal of these lectures is to provide a pedagogical introduction
to the inflationary paradigm and to the theory of cosmological
perturbations generated during inflation. 


The lectures are organized as follows. In section 2 we give a brief
review of the Big-Bang theory. In section 3 we describe the shortcomings of the
Big-Bang theory leading to the idea of inflation which is addressed in section
4. In section 5 we present the idea of quantum fluctuations
which are studied in sections 6 and 7. Section 8 presents some details about
the post-inflationary evolution of the cosmological 
perturbations. Finally, section 8 contains some conclusions.


Since this lectures were devoted to a  a school, 
we will not provide an exhaustive list of
references to original material, but refer to several basic papers
(including several review papers) where students can find the
references to the original material.  The list of references include
Refs. \cite{abook,kolbbook,Bardeen,Stewart,Mukhanov,Liddle,Lidsey,lr,llbook}.

Finally, we warn the reader that references are listed in alphabetical 
order.




\section {Basics of the Big-Bang Model}

The standard cosmology is based upon the maximally spatially symmetric
Friedmann-Robertson-Walker (FRW) line element
\begin{equation}
ds^2 = -dt^2 +a(t)^2\left[ {dr^2\over 1-kr^2} +r^2
        (d\theta^2 + \sin^2\theta\,d\phi^2 ) \right];
\label{metric}
\end{equation}
where $a(t)$ is the cosmic-scale factor, $R_{\rm curv}\equiv
a(t)|k|^{-1/2}$ is the curvature radius, and $k= -1,
0, 1$ is the curvature signature.  All three models are
without boundary:  the positively curved model is finite
and ``curves'' back on itself; the negatively curved
and flat models are infinite in extent.  The Robertson-Walker
metric embodies the observed isotropy and homogeneity of
the universe.  It is interesting to note
that this form of the line element was originally introduced
for sake of mathematical simplicity; we now know that
it is well justified at early times or today on large
scales ($\gg 10\Mpc$), at least within our visible patch.

The coordinates, $r$, $\theta$, and $\phi$, are referred
to as {\it comoving} coordinates:  A particle at rest in these
coordinates remains at rest, {\it i.e.}, constant $r$, $\theta$,
and $\phi$.  A freely moving particle eventually comes
to rest these coordinates, as its momentum is red shifted
by the expansion, $p \propto a^{-1}$.
Motion with respect to the comoving coordinates (or cosmic
rest frame) is referred to as peculiar velocity; unless
``supported'' by the inhomogeneous distribution of matter
peculiar velocities decay away as $a^{-1}$.  Thus the
measurement of peculiar velocities, which is not easy
as it requires independent measures of both the distance
and velocity of an object, can be used to probe the
distribution of mass in the universe.

Physical separations  between freely moving particles
scale as $a(t)$; or said another way the physical separation
between two points is simply $a(t)$ times the coordinate
separation.  The momenta of freely propagating particles
decrease, or ``red shift,'' as $a(t)^{-1}$, and thus the
wavelength of a photon stretches as $a(t)$, which is
the origin of the cosmological red shift.  The red shift
suffered by a photon emitted from a distant galaxy
$1+z = a_0/a(t)$; that is, a galaxy whose light is
red shifted by $1+z$, emitted that light when the universe
was a factor of $(1+z)^{-1}$ smaller.  Thus, when the
light from the most distant quasar yet seen ($z=4.9$) was emitted
the universe was a factor of almost six smaller; when
CMB photons last scattered the universe was about $1100$ times 
smaller.

\subsection{Friedmann equations}

The evolution of the scale factor $a(t)$ is governed by Einstein equations
\begin{equation}
R_{\mu\nu}-\frac{1}{2}\,R\,g_{\mu\nu}\equiv G_{\mu\nu}=8\pi G\,
T_{\mu\nu}
\end{equation}
where $R_{\mu\nu}$ $(\mu,\nu=0,\cdots 3)$ 
is the Riemann tensor and $R$ is the Ricci scalar
constructed via the metric (\ref{metric}) \cite{kolbbook} and  
$T_{\mu\nu}$ is the energy-momentum tensor. Under the hypothesis
of homogeneity and isotropy, we can always write the energy-momentum
tensor under the form $T_{\mu\nu}={\rm diag}\left(\rho,p,p,p\right)$
where $\rho$ is the energy density of the system and $p$ its pressure.
They are functions of time.




The evolution of the cosmic-scale factor is governed
by the Friedmann equation
\begin{equation}
H^2 \equiv \left({\dot a \over a}\right)^2 =
        {8\pi G \rho \over 3} - {k\over a^2};
\label{fr1}
\end{equation}
where $\rho$ is the total energy density of the
universe, matter, radiation, vacuum energy, and so on.

Differentiating wrt to time both members of Eq. (\ref{f1}) and using the 
the mass conservation equation
\begin{equation}
\label{mass}
\dot{\rho} +3H(\rho +p) =0,
\end{equation}
we find the equation for the acceleration of the scale-factor
\begin{equation}
\label{acc}
\frac{\ddot{a}}{a} = - \frac{4\pi G}{3}
(\rho +3p).
\label{fr2}
\end{equation}
Combining Eqs. (\ref{fr1}) and (\ref{fr2}) we find
\begin{equation}
\dot H=-4\pi G\left(\rho+p\right).
\label{ll}
\end{equation}
The evolution of the energy
density of the universe is governed by
\begin{equation}
d(\rho a^3) = -p d\left(a^3\right);
\end{equation}
which is the First Law of Thermodynamics for
a fluid in the expanding universe.
(In the case that the stress energy of the universe is comprised
of several, noninteracting components, this relation applies
to each separately; {\it e.g.}, to the matter and radiation separately
today.)  For $p=\rho /3$, ultra-relativistic matter,
$\rho \propto a^{-4}$ and $a\sim t^{\frac{1}{2}}$; 
for $p=0$, very nonrelativistic
matter, $\rho \propto a^{-3}$ and $a\sim t^{\frac{2}{3}}$;
 and for $p=-\rho$, vacuum
energy, $\rho = \,$const.  If the rhs of the Friedmann
equation is dominated by a fluid
with equation of state $p = \gamma \rho$, it follows
that $\rho \propto a^{-3(1+\gamma )}$
and $a\propto t^{2/3(1+\gamma )}$.

We can use the Friedmann equation to relate the
curvature of the universe to the energy density and
expansion rate:
\begin{equation}
 \Omega -1={k \over a^2H^2}; \qquad
\Omega = {\rho\over \rho_{\rm crit}};
\label{curvature}
\end{equation}
and the critical density today $\rho_{\rm crit}
= 3H^2 /8\pi G = 1.88h^2\gcmm3 \simeq 1.05\times 10^{4}
\eV \cmm3$.  There is a one to one correspondence
between $\Omega$ and the spatial curvature of the universe:
positively curved, $\Omega_0 >1$; negatively curved, $\Omega_0
<1$; and flat ($\Omega_0 = 1$).  Further, the ``fate of the
universe'' is determined by the curvature:  model universes
with $k\le 0$ expand forever, while those with $k>0$ necessarily
recollapse.  The curvature radius of the universe is related
to the Hubble radius and $\Omega$ by
\begin{equation}
R_{\rm curv} = {H^{-1}\over |\Omega -1|^{1/2}}.
\label{curv}
\end{equation}
In physical terms, the curvature radius sets the scale for
the size of spatial separations where
the effects of curved space become
``pronounced.''  And in the case of the positively curved
model it is just the radius of the 3-sphere.

The energy content of the universe consists of matter
and radiation (today, photons and neutrinos).  Since
the photon temperature is accurately known,
$T_0=2.73\pm 0.01\,$K, the
fraction of critical density contributed by radiation
is also accurately known:  $\Omega_{R}h^2 = 4.2 \times
10^{-5}$, where $h=0.72\pm 0.07$ is the present Hubble rate in units of 
$100$ km 
${\rm sec}^{-1}$ ${\Mpc}^{-1}$ \cite{fr}.  The remaining content of the 
universe is 
another 
matter. Rapid progress has been made recently toward the measurement of 
cosmological parameters \cite{triangle}. Over the past three years 
the basic features of our universe have been determined. The universe  
is spatially flat; 
accelerating; comprised of one third of  dark matter
and two third a new 
form of dark energy. The measurements of the cosmic microwave background
 anisotropies at different angular scales performed by 
Boomerang, Maxima, DASI, CBI and VSA have recently significantly increase 
the case for  accelerated expansion in the early universe (the 
inflationary paradigm) 
and     at the current epoch (dark energy dominance), especially when 
combined with data on high redshift supernovae (SN1) and large scale 
structure (LSS) \cite{triangle}. A recent analysis \cite{bond} shows that
the CMB$+$LSS$+$SN1 data give 

$$\Omega_{0} 
=1.00^{+0.07}_{-0.03},
$$
meaning tha the present universe is spatially flat (or at least very close
to being flat). Restricting to    $\Omega_0=1$, the dark matter density
is given by \cite{bond}

$$ 
\Omega_{\rm DM}h^2 
=0.12^{+0.01}_{-0.01},
$$
and a baryon density

$$
\Omega_B h^2 = 0.022^{+0.003}_{-0.002},
$$
while the Big Bang nucleosynthesis estimate is $\Omega_B  
h^2=0.019\pm
     0.002.$ Substantial dark (unclustered) energy is inferred, 

$$
\Omega_Q 
\approx 0.68 \pm 0.05,
$$ 
compatible with the independent SN1 estimates! What is most
relevant for us, this universe is apparently born from a burst of rapid 
expansion, inflation, during which quantum noise was stretched to 
astrophysical size 
seeding cosmic structure. This is exactly the phenomena we want to address 
in these lectures.



\subsection{Some conformalities}
Before launching ourselves into the description of the early universe,
we would like to introduce the concept of conformal time which will
be useful in the next sections.
The conformal time $\tau$ is defined through the following relation
\begin{equation}
d\tau=\frac{dt}{a}.
\label{jj}
\end{equation}
The metric (\ref{metric}) then becomes
\begin{equation}
ds^2 = -a^2(\tau)\left[d\tau^2 - {dr^2\over 1-kr^2} -r^2
        (d\theta^2 + \sin^2\theta\,d\phi^2 ) \right].
\label{metricconf}
\end{equation}
The reason why $\tau$ is called conformal is manisfest from Eq. 
(\ref{metricconf}): the corresponding FRW line element is conformal
to the Minkowski line element describing a static four dimensional
hypersurface.

Any function $f(t)$ satisfies the rule
\bea
\dot{f}(t)&=&\frac{f^\prime(\tau)}{a(\tau)},\\
\ddot{f}(t)&=&\frac{f^{\prime\prime}(\tau)}{a^2(\tau)}-
\H\frac{f^\prime(\tau)}{a^2(\tau)},
\label{rule1}
\eea
where a prime now indicates differentation wrt to the conformal time
$\tau$ and
\be
\H=\frac{a^\prime}{a}.
\ee
In particular we can set the following rules
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
\bea
H&=&\frac{\dot a}{a}=\frac{a^\prime}{a^2}=\frac{\H}{a},\nonumber\\
\ddot a&=&\frac{a^{\prime\prime}}{a^2}-\frac{\H^2}{a},\nonumber\\
\dot H&=&\frac{\H^\prime}{a^2}-\frac{\H^2}{a^2},\nonumber\\
H^2&=&{8\pi G \rho \over 3} - {k\over a^2}\Longrightarrow
\H^2={8\pi G \rho a^2 \over 3}  -k\,\nonumber\\
\dot H&=&-4\pi G\left(\rho+p\right)\Longrightarrow 
\H^\prime=-\frac{4\pi G}{3}\left(\rho+3p\right)a^2,\nonumber\\
\dot{\rho} &+&3H(\rho +p) =0\Longrightarrow \rho^\prime +3\H(\rho +p)=0
\nonumber
\label{rules}
\eea
\\
\hline
\end{tabular}
\end{center}
Finally, 
if the scale factor $a(t)$ scales like $a\sim t^n$, solving the relation
(\ref{jj}) we find
\be
a\sim t^n\Longrightarrow a(\tau)\sim \tau^{\frac{n}{1-n}}.
\label{rule}
\ee


\subsection{The early, radiation-dominated universe}

In any case, at present, matter outweighs radiation
by a wide margin.  However, since the energy density
in matter decreases as $a^{-3}$, and that
in radiation as $a^{-4}$ (the extra factor due
to the red shifting of the energy of relativistic
particles), at early times the
universe was radiation dominated---indeed the calculations
of primordial nucleosynthesis provide excellent evidence for this.
Denoting the epoch of matter-radiation equality
by subscript `EQ,' and using $T_0=2.73\,$K, it follows that
\begin{equation}
a_{\rm EQ} = 4.18\times 10^{-5}\,(\Omega_0 h^2)^{-1};\qquad
T_{\rm EQ} = 5.62 (\Omega_0 h^2)\eV;
\end{equation}
\begin{equation}
t_{\rm EQ} = 4.17 \times 10^{10}(\Omega_0 h^2)^{-2}\sec .
\end{equation}
At early times the expansion rate and age of the universe were
determined by the temperature of the universe and
the number of relativistic degrees of freedom:
\begin{equation}
\rho_{\rm rad} = g_*(T){\pi^2T^4 \over 30}; \qquad
H\simeq 1.67g_*^{1/2} T^2 /\mpl ;
\end{equation}
\begin{equation}
\Rightarrow a\propto t^{1/2}; \qquad
t \simeq 2.42\times 10^{-6} g_*^{-1/2}(T/\GeV )^{-2}\,\sec ;
\end{equation}
where $g_* (T)$ counts the number of ultra-relativistic
degrees of freedom ($\approx$ the sum of the internal
degrees of freedom of particle species much less massive
than the temperature) and $\mpl \equiv G^{-1/2} = 1.22
\times 10^{19}\GeV$ is the Planck mass.  For example,
at the epoch of nucleosynthesis, $g_* = 10.75$ assuming
three, light ($\ll \MeV$) neutrino species; taking into
account all the species in the standard model,
$g_* = 106.75$ at temperatures much greater than $300\GeV$.


A quantity of importance related to $g_*$ is the
entropy density in relativistic particles,
$$s= {\rho +p \over T} = {2\pi^2\over 45}g_* T^3 ,$$
and the entropy per comoving volume,
$$S \ \  \propto\ \  a^3 s\ \  \propto\ \   g_*a^3T^3 .$$
By a wide margin most of the entropy
in the universe exists in the radiation bath.
The entropy density is proportional
to the number density of relativistic particles.
At present, the relativistic particle species
are the photons and neutrinos, and the
entropy density is a factor
of 7.04 times the photon-number density:
$n_\gamma =413 \cmm3$ and $s=2905 \cmm3$.

In thermal equilibrium---which provides a good description
of most of the history of the universe---the entropy per comoving
volume $S$ remains constant.  This fact is very useful.
First, it implies that the temperature and scale
factor are related by
\begin{equation}
T\propto g_*^{-1/3}a^{-1},
\end{equation}
which for $g_*=\,$const leads to the familiar $T\propto a^{-1}$.

Second, it provides a way of quantifying the net baryon
number (or any other particle number) per comoving volume:
\begin{equation}
N_B \equiv R^3n_B = {n_B\over s} \simeq (4-7)\times 10^{-11}.
\end{equation}
The baryon number of the universe tells us two things:
(1) the entropy per particle in the universe
is extremely high, about
$10^{10}$ or so compared to about $10^{-2}$ in the sun
and a few in the core of a newly formed neutron star.
(2)  The asymmetry between matter and antimatter is
very small, about $10^{-10}$, since at early times
quarks and antiquarks were roughly as abundant as
photons.  One of the great successes of particle
cosmology is baryogenesis, the idea that $B$, $C$, and
$CP$ violating interactions occurring out-of-equilibrium
early on allow the
universe to develop a net baryon number of this magnitude
\cite{baryo1,baryo2}.

Finally, the constancy of the entropy per comoving
volume allows us to characterize the size of comoving
volume corresponding to our present Hubble volume in
a very physical way:  by the entropy it contains,
\begin{equation}
S_{U} = {4\pi\over 3}H_0^{-3}s \simeq 10^{90}.
\end{equation}



The standard cosmology is tested back to times as early
as about 0.01 sec; it is only natural to ask how far back
one can sensibly extrapolate.  Since the fundamental
particles of Nature are point-like quarks and leptons
whose interactions are perturbatively weak at energies much greater
than $1\GeV$, one can imagine extrapolating
as far back as the epoch where general relativity
becomes suspect, i.e., where quantum gravitational
effects are likely to be important:  the Planck epoch,
$t\sim 10^{-43}\sec$ and $T\sim 10^{19}\GeV$.
Of course, at present, our firm understanding
of the elementary particles and their interactions
only extends to energies of the order of $100\GeV$,
which corresponds to a time of the order of
$10^{-11}\sec$ or so.  We can be relatively certain
that at a temperature of $100\MeV -200\MeV$ ($t\sim 10^{-5}\sec$)
there was a transition (likely a second-order
phase transition) from quark/gluon
plasma to very hot hadronic matter, and that some
kind of phase transition associated
with the symmetry breakdown of the electroweak theory
took place at a temperature of the order of $300\GeV$
($t\sim 10^{-11}\sec$).


\subsection{The concept of particle horizon}

In spite of the fact that the universe was
vanishingly small at early times, the rapid expansion
precluded causal contact from being established throughout.
Photons travel on null paths characterized
by $dr=dt/a(t)$; the physical distance that a photon
could have traveled since the bang until time $t$, the
distance to the particle horizon, is
\begin{eqnarray}
R_H(t) & = & a(t)\int_0^t {dt^\prime\over a(t^\prime )} \nonumber\\
       & = & \frac{t}{(1-n)} = n\,\frac{H^{-1}}{(1-n)}\sim
H^{-1} \qquad {\rm for}\ a(t)
        \propto t^n, \ \ n<1 .
\end{eqnarray}
Using the conformal time, the particle horizon becomes
\be
R_H(t)=a(\tau)\int_{\tau_0}^\tau\, d\tau,
\ee
where $\tau_0$ indicates the conformal time corresponding to $t=0$.
Note, in the standard cosmology the distance to the
horizon is finite, and up to numerical factors,
equal to the age of the universe or the Hubble radius, $H^{-1}$.
For this reason, we will use horizon and Hubble radius
interchangeably.\footnote{As we shall see, in inflationary models
the horizon and Hubble radius are not roughly equal
as the horizon distance grows exponentially relative
to the Hubble radius; in fact, at the end of inflation
they differ by $e^N$, where $N$ is the number of
e-folds of inflation.  However, we will slip and use
``horizon'' and ``Hubble radius'' interchangeably, though
we will always mean Hubble radius.}

Note also that a physical length scale $\lambda$ is within the horizon
if $\lambda<R_{H}\sim H^{-1}$. Since we can identify the length scale
$\lambda$ with its wavenumber $k$, $\lambda=2\pi a/k$, 
we will have the following
rule
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
\bea
\frac{k}{aH}&\ll& 1 \Longrightarrow {\rm SCALE}~~\lambda~~
{\rm OUTSIDE}~~{\rm THE}~~{\rm HORIZON}\nonumber\\
\frac{k}{aH}&\gg& 1 \Longrightarrow {\rm SCALE}~~\lambda~~
{\rm WITHIN}~~{\rm THE}~~{\rm HORIZON}\nonumber
\eea
%\\
\\
\hline
\end{tabular}
\end{center}
An important quantity is
the entropy within a horizon volume:  $S_{\rm HOR}
\sim H^{-3}T^3$; during the radiation-dominated epoch
$H\sim T^2/\mpl$, so that
\begin{equation}
S_{\rm HOR} \sim \left( {\mpl\over T} \right)^3;
\end{equation}
from this we will shortly conclude that at early times the comoving
volume that encompasses all that we can see today
(characterized by an entropy of about $10^{90}$) was comprised
of a very large number of causally disconnected regions.



\section{The shortcomings of the Standard Big-Bang Theory}




By now the shortcomings of the standard cosmology are
well appreciated:  the horizon or large-scale smoothness problem;
the small-scale inhomogeneity problem (origin of density perturbations);
and the flatness or oldness problem.
We will only briefly review them here.  They do not indicate any
logical inconsistencies of the standard cosmology; rather,
that very special initial data seem to be required for evolution
to a universe that is qualitatively similar to ours today.
Nor is inflation the first attempt to address these shortcomings:
over the past two decades cosmologists have pondered this
question and proposed alternative  solutions.
Inflation is a solution based upon well-defined,
albeit speculative, early universe microphysics describing
the post-Planck epoch.

\subsection{The Flatness Problem}


Let us make a tremendous extrapolation and assume  that Einstein
equations are valid until the Plank era, when the temperature of
the universe is $T_\pl\sim \mpl\sim 10^{19}$ GeV.
From Eq. (\ref{curvature}), we read that 
if the universe is perfectly flat, then $(\Omega=1)$ at all times.
On the other hand, if there is even a small curvature term, the time
dependence of $\left(\Omega-1\right)$ is quite different.

During a radiation-dominated period, we have that $H^2\propto
\rho_{R}\propto a^{-4}$ and
\begin{equation}
\Omega -1 \propto \frac{1}{a^2 a^{-4}}\propto a^2.
\end{equation}
During Matter Domination, $\rho_{M}\propto a^{-3}$ and   
\begin{equation}
\Omega -1 \propto \frac{1}{a^2 a^{-3}}\propto a.
\end{equation}
In both cases $(\Omega-1)$ decreases going backwards with time. 
Since we know that today $(\Omega_0-1)$ is of order unity at present, 
we can deduce its value at $t_{\pl}$ (the time at which the temperature
of the universe is $T_{\pl}\sim 10^{19}$ GeV)
\begin{equation}
\frac{\mid \Omega -1 \mid_{T=T_{\pl}}}{\mid \Omega -1
\mid_{T=T_{0}}} \approx \left(\frac{a^2_{\pl}}{a_{0}^2}\right) \approx
\left(\frac{T_0^2}{T^2_{\pl}}\right)  \approx \mathcal{O}(10^{-64}).
\end{equation}
where  $0$ stands for the present epoch, and $T_0\sim
10^{-13}$ GeV is the present-day
temperature of the CMB radiation. If we are not so brave and 
go back simply to the epoch of
nucleosynthesis when light elements abundances were formed, at
$T_N\sim$ 1 MeV,
we get
\begin{equation}
\frac{\mid \Omega -1 \mid_{T=T_N}}{\mid \Omega -1
\mid_{T=T_{0}}} \approx \left(\frac{a^2_{N}}{a_{0}^2}\right) \approx
\left(\frac{T_0^2}{T^2_N}\right)  \approx \mathcal{O}(10^{-16}).
\end{equation}
In order to get the correct value of $(\Omega_0-1)\sim 1$ at present,
the value of $(\Omega-1)$ at early times have to be fine-tuned to values
amazingly close to zero, but without being exactly zero. This is the reason why
the flatness problem is also dubbed the `fine-tuning problem'.\\

\subsection{The Entropy Problem}

Let us now see how the hypothesis of adiabatic expansion of the
universe is connected with the flatness problem. 
From the Friedman equation (\ref{f1}) we know that  during a 
radiation-dominated period
\begin{equation}
H^2\simeq \rho_{R}\simeq \frac{T^4}{\mpl^2},
\end{equation}
from which we deduce
\begin{equation}
\Omega-1=\frac{k \mpl^2}{a^4 T^4}=\frac{k \mpl^2}{S^{\frac{2}{3}} T^2}.
\end{equation}
Under the hypothesis of adiabaticity, $S$ is constant over the
evolution of the universe and therefore
\be
\left|\Omega -1\right|_{t=t_{\pl}}
=\frac{\mpl^2}{T_\pl^2}\frac{1}{S_U^{2/3}}=\frac{1}{S_U^{2/3}}\approx
10^{-60}.
\label{con}
\ee
We have discovered that  $(\Omega-1)$ is so close to
zero at early epochs because the total entropy of our universe
is so incredibly large.
The flatness problem is therefore a problem of understanding why the 
(classical) initial conditions corresponded to a universe that was so close 
to spatial flatness. In a sense, the problem is one of fine--tuning and 
although such a balance is possible in principle, one nevertheless feels 
that it is unlikely. On the other hand, the flatness problem arises because 
the entropy in a comoving volume is conserved. It is possible, therefore,  
that the problem could be resolved if the cosmic expansion was 
non--adiabatic for some finite time interval 
during the early history of the universe. \\

\subsection{The horizon problem}



According to the standard cosmology,
photons decoupled from the rest of the components (electrons and baryons)
 at a temperature of the order of 0.3 eV. This  corresponds to the
so-called  surface of `last-scattering' at a red shift of about $1100$ and
an age of about $180,000\,(\Omega_0 h^2)^{-1/2}\yrs$.
\begin{figure}[htb]
\centerline{\hbox{ \psfig{figure=spectrum.ps} }}
\caption{\footnotesize  The black body
spectrum of the cosmic background radiation  \label{wavelength}}
\end{figure}
From the epoch of last-scattering onwards, photons free-stream
and reach us basically untouched. Detecting primordial photons
is therefore equivalent to take a picture of the universe when the
latter was about 300,000 $\yrs$ old.
The spectrum of the cosmic background radiation (CBR) is consistent
that of a black body at temperature 2.73 K over more than three
decades in wavelength; see Fig. \ref{wavelength}.

The most accurate measurement of the temperature
and spectrum is that by the FIRAS instrument on the
COBE satellite which determined its temperature to be
$2.726\pm 0.01\,$K \cite{FIRAS}.  
The length corresponding to our present Hubble radius (which is
approximately the radius of our observable universe) at the time
of last-scattering was
$$
\lambda_H(t_{\rm LS})=R_H(t_0) \left(\frac{a_{\rm LS}}{a_0}
\right)=R_H(t_0) \left(\frac{T_{0}}{T_{\rm LS}}\right).
$$
On the other hand, during the matter-dominated period, 
the Hubble length has decreased with a different law
$$
H^2\propto \rho_{M} \propto a^{-3} \propto T^{3}.
$$
At last-scattering
$$
H_{LS}^{-1}=R_H(t_0)\left( \frac{T_{LS}}{T_0} \right)^{-3/2}\ll R_H(t_0).
$$
The length corresponding to our present Hubble radius was much
larger that the horizon at that time. This can be shown comparing
the volumes corresponding to these two scales

\begin{equation}
\frac{\lambda^3_{H}(T_{LS})}{H_{LS}^{-3}}=
\left(\frac{T_0}{T_{LS}}\right)^{-\frac{3}{2}}\approx 10^6.
\label{pp}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=3.5in \epsfbox{normal.eps}}
\caption{The horizon scale (green line) and a physical scale $\lambda$
(red line) as function of the scale factor $a$. From Ref. \cite{kolbreview}.
 \label{normal}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There were $\sim 10^6$ casually
disconnected regions within the volume that now corresponds to our
horizon! 
It is difficult to
come up with a process other
than an early hot and dense phase in the history
of the universe that would lead to a precise
black body \cite{dnsnature} for a bath of photons which
were causally disconnected the last time  they interacted with the
surrounding plasma.

The horizon problem is well represented by Fig. \ref{normal}
where the green line indicates the horizon scale and the red line any
generic physical length scale $\lambda$. Suppose, indeed that $\lambda$
indicates the distance between two photons we detect today. From
Eq. (\ref{pp}) we discover that at the time of emission (last-scattering)
the two photons could not talk to each other, the red line is above the
green line.
%\begin{figure}[htb]
%\centerline{\hbox{ \psfig{figure=normal.ps} }}
%\caption{\footnotesize Explanation \label{normal}}
%\end{figure}
There is another aspect of the horizon problem which is related to the
problem of initial conditions for the cosmological perturbations.
We have every indication that the universe at early
times, say $t\ll 300,000\yrs$, was very homogeneous;
however, today inhomogeneity (or structure) is ubiquitous:
stars ($\delta\rho /\rho \sim 10^{30}$),
galaxies ($\delta\rho /\rho \sim 10^{5}$),
clusters of galaxies ($\delta\rho /\rho \sim 10-10^{3}$),
superclusters, or ``clusters of clusters''
($\delta\rho /\rho \sim 1$), voids ($\delta\rho /\rho
\sim -1$), great walls, and so on.
For some twenty-five  years the standard cosmology has provided
a general framework for understanding this picture.
Once the universe becomes matter dominated (around 1000 yrs
after the bang) primeval density
inhomogeneities ($\delta\rho /\rho \sim 10^{-5}$)
are amplified by gravity and
grow into the structure we see today \cite{sf}.
The existence of density inhomogeneities has another important
consequence:  fluctuations in the temperature of the CMB radiation  of
a similar amplitude.  The temperature difference measured
between two points separated by a large angle ($\ga 1^\circ$)
arises due to a very simple physical effect:
the difference in the gravitational potential between the two points on
the last-scattering surface, which in turn is related
to the density perturbation, determines
the temperature anisotropy on the angular scale subtended
by that length scale,
\begin{equation}
\left({\delta T \over T}\right)_\theta 
\approx
\left( {\delta\rho\over \rho}\right)_{\lambda},
\end{equation}
where the scale $\lambda \sim 100h^{-1}\Mpc(\theta /{\rm deg})$
subtends an angle $\theta$ on the last-scattering
surface.  This is known as the Sachs-Wolfe effect \cite{SW}.
The  CMB experiments looking for the
tiny anisotropies are  of three kinds: satellite
experiments, balloon experiments, and ground based experiments.
The technical and economical advantages of ground based
experiments are evident, but their main problem is atmospheric
fluctuations. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=4.5in \epsfbox{cobe.eps}}
\caption{The CMBR anisotropy as function of $\ell$. From Ref. 
\cite{homepageteg}.
\label{figcobe}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The problem can be limited choosing a very high and
cold site, or
working on small scales (as the Dasi experiment \cite{dasi}).
Balloon based experiments limit the atmospheric problems, but have
to face the following problems: they must be limited in weight,
they can not be manipulated during the  flight, they have a rather short
duration (and they have to be recovered intact). Maxima \cite{maxima},
and Boomerang \cite{boomerang} are experiments of this kind.\\
At present, there is a satellite experiment -- MAP (Microwave
Anisotropy Probe) sponsored by  NASA mission, which is taking
data \cite{map}.
Finally, a satellite mission PLANCK is planned by ESA to be
launched in 2007 \cite{planck}.
The
temperature anisotropy is commonly expanded  in spherical harmonics
\begin{equation}
\frac{\Delta T}{T}(x_0,\tau_0,{\bf n})=\sum_{\ell m}
a_{\ell,m}(x_0)Y_{\ell m}({\bf n}),
\end{equation}
where $x_0$ and $\tau_0$ are our position and the preset time, respectively,
 ${\bf n}$ is the
direction of observation, $\ell'$s are the
different multipoles  and\footnote{An alternative definition is $C_\ell=
\langle \left|a_{\ell m}\right|^2\rangle=\frac{1}{2\ell +1}
\sum_{m=-\ell}^{\ell}
\left|a_{\ell m}\right|^2$.}
\begin{equation}
\langle a_{\ell m}a^*_{\ell'm'}\rangle=\delta_{\ell,\ell'}\delta_{m,m'} C_\ell,
\end{equation}
where the deltas are due to the fact that the process that created
the anisotropy is statistically isotropic. 
%\begin{figure}[htb]
%\centerline{\hbox{ \psfig{figure=cobe.eps} }}
%\caption{\footnotesize A plot for illustration.  \label{figcobe}}
%\end{figure}
The $C_\ell$ are the so-called CMB power spectrum.
For homogeneity and isotropy, the $C_\ell$'s are neither a function
of $x_0$, nor of $m$.
The two-point-correlation function is related to the $C_l$'s in
the following way
\begin{eqnarray}
\Big<\frac{\delta T({\bf n})}{T}\frac{\delta T({\bf n}')}{T}\Big>&=&
\sum_{\ell\ell'mm'}\langle a_{\ell m}a^*_{\ell'm'}\rangle
Y_{\ell m}({\bf n})Y^*_{\ell'm'}({\bf n}')\nonumber\\
&=&\sum_\ell C_\ell \sum_m
Y_{\ell m}({\bf n})Y^*_{\ell m}({\bf n}')=\frac{1}{4\pi}\sum_\ell (2\ell+1) 
C_\ell
P_\ell(\mu={\bf n}\cdot{\bf n}')
\label{j}
\end{eqnarray}
where we have used the addition theorem for the spherical
harmonics, and $P_\ell$ is the Legendre polynom of order $\ell$. 
In expression (\ref{j}) the expectation value is an ensamble average. It 
can be regarded as an average over the possible observer positions, but
not in general as an average over the single sky we observe, because of the
cosmic variance\footnote{The usual hypothesis is that we observe a typical
realization of the ensamble. This means that we expect  the difference 
between the observed values $|a_{\ell m}|^2$ and the 
ensamble averages $C_\ell$ to be of the order of the mean-square deviation
of  $|a_{\ell m}|^2$ from $C_\ell$. The latter is called
cosmic variance and, because we are dealing with a gaussian distribution, it is
equal to $2C_\ell$ for each multipole $\ell$. For a single $\ell$, averaging 
over the $(2\ell +1)$ values of $m$ reduces the cosmic variance
by a factor $(2\ell +1)$, but it remains a serious limitation for low 
multipoles.}.


Let us now consider the last-scattering surface. In comoving coordinates
the latter is `far' from us a distance equal to
\be
\int_{t_{\rm LS}}^{t_0}\,\frac{dt}{a}=\int_{\tau_{\rm LS}}^{\tau_0}\,
d\tau=\left(\tau_0-\tau_{\rm LS}\right).
\ee
A given comoving 
scale $\lambda$  is therefore projected on the last-scattering surface
sky on an angular scale
\begin{equation}
\theta \simeq \frac{\lambda}{\left(\tau_0-\tau_{\rm LS}\right)},
\end{equation}
where we have neglected tiny curvature effects.
Consider now
that the scale $\lambda$ is of the order of the comoving sound horizon at the 
time of last-scattering, $\lambda\sim c_s\tau_{\rm LS}$, where
$c_s\simeq 1/\sqrt{3}$ is the sound velocity at which photons
propagate in the plasma at the last-scattering.
This corresponds
to an angle 
\be
\theta\simeq c_s\frac{\tau_{\rm LS}}{\left(\tau_0-\tau_{\rm LS}\right)}\simeq
c_s\frac{\tau_{\rm LS}}{\tau_0},
\ee
where the last passage has been performed knowing that $\tau_0\gg
\tau_{\rm LS}$. Since the universe is matter-dominated from the
time of last-scattering onwards, the scale factor has the following
behaviour:  $a\sim T^{-1}\sim 
t^{2/3}\sim \tau^2$, where we have made use of the relation (\ref{rule}).
The angle $\theta_{\rm HOR}$ 
subtended by the sound horizon on the last-scattering
surface then becomes
\be
\theta_{\rm HOR}
\simeq c_s\left(\frac{T_0}{T_{\rm LS}}\right)^{1/2}\sim 1^\circ,
\ee
where we have used $T_{\rm LS}\simeq 0.3$ eV and $T_0\sim 10^{-13}$ GeV.
This corresponds to a multipole $\ell_{\rm HOR}$
 
\begin{equation}
\ell_{\rm HOR}=\frac{\pi}{\theta_{\rm HOR}}\simeq 200.
\end{equation}
From these estimates we conclude that  
two photons which on the last-scattering surface were separated
by an angle larger than $\theta_{\rm HOR}$, corresponding to
multipoles smaller than $\ell_{\rm HOR}\sim 200$ were not in causal
contact. 
On the other hand, 
from Fig. (\ref{figcobe}) it is clear that small anisotropies, of the 
{\it same} order of magnitude $\delta T/T\sim 10^{-5}$ are present at $\ell\ll
200$. We conclude that one of the striking features of the CMB 
fluctuations is that they appear to be noncausal.
Photons at the last-scattering surface which were causally disconnected
have the same small anisotropies!
The existence of particle
horizons in the standard cosmology precludes explaining
the smoothness as a result of microphysical events:  the
horizon at decoupling, the last time one could imagine
temperature fluctuations being smoothed by particle interactions,
corresponds to an angular scale on the sky of about
$1^\circ$, which precludes temperature variations on larger scales
from being erased.  

To account for the small-scale lumpiness of the
universe today, density perturbations with horizon-crossing
amplitudes of $10^{-5}$ on scales of $1\Mpc$ to $10^4\Mpc$ or so
are required.  As can be seen in Fig. \ref{normal}, in the
standard cosmology the physical
size of a perturbation, which grows as the scale factor,
begins larger than the horizon and relatively late
in the history of the universe crosses inside the horizon.
This precludes a causal microphysical explanation for
the origin of the required density perturbations.

From the considerations made so far, it appears that solving the
shortcomings of the standard Big Bang theory requires two basic
modifications of the assumptions made so far:

\begin{itemize}

\item { The universe has to go through a non-adiabatic period.
This is necessary to solve the entropy and the flatness problem. A 
non-adiabatic phase may give rise to the large entropy $S_U$ we observe
today.}

\item{The universe has to go through a primordial period during which
the physical scales $\lambda$ evolve faster than the horizon scale $H^{-1}$.}
\end{itemize}
The second  condition is obvious from Fig. \ref{inflation}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=4.5in \epsfbox{inflation.eps}}
\caption{The behaviour of a generic scale $\lambda$ and the horizon scale
$H^{-1}$ in the standard inflationary model. From Ref. \cite{kolbreview}.
\label{inflation}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If there is period during which physical length scales grow faster
than $H^{-1}$, length scales $\lambda$ 
which are within the horizon today, $\lambda<H^{-1}$ (such
as the distance between two detected photons) and 
were outside the
horizon for some period,  $\lambda>H^{-1}$ (for 
istance at the time of last-scattering when the
two photons were emitted), had a chance to be within the horizon
at some primordial epoch,  $\lambda<H^{-1}$ again. If this happens,
the homogeneity and the isotropy of the CMB can be easily explained:
photons that we receive today and were emitted from the last-scattering 
surface from causally disconnected regions have the same temperature
because they had a chance to `talk' to each other at some primordial
stage of the evolution of the universe.

The second condition can be easily expressed as a condition on the
 scale factor $a$. Since a given scale $\lambda$ scales like
$\lambda\sim a$ and $H^{-1}=a/\dot a$, we need to impose
that there is a period during which
$$
\left(\frac{\lambda}{H^{-1}}\right)^{\cdot}=\ddot a>0.
\label{fund}
$$ 
We can therefore introduced
the following rigorous definition: an  inflationary stage \cite{guth} is 
a period of the universe during which the latter accelerates
\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
{\rm INFLATION}~~~\Longleftrightarrow~~~\ddot a>0.
$$
%\\
\\
\hline
\end{tabular}
\end{center}
\vskip 0.2cm

{\it \underline{Comment}:}Let us stress that during such a accelerating  phase 
the universe expands {\it adiabatically}. This means that during inflation
one can exploit the usual FRW equations (\ref{fr1}) and (\ref{fr2}). 
It must be 
clear therefore that the non-adiabaticity condition is satisfied not during 
inflation, but during the phase transition between the end of inflation
and the beginning of the radiation-dominated phase. At this transition phase
a large entropy is generated under the form of relativistic degrees of freedom:
the Big Bange has taken place.









\section{The standard inflationary universe}

From the previous section we have learned that 
an accelerating stage during the primordial phases  of the evolution of
the universe
might be able to solve the horizon problem. From Eq. (\ref{fr2})
we learn that 
$$
\ddot a>0 \Longleftrightarrow (\rho +3p)<0.
\label{gg}
$$
An accelerating period is obtainable only if the overall pressure $p$ 
of the universe is negative: $p<-\rho/3$. Neither 
a radiation-dominated
phase nor a matter-dominated phase (for which $p=\rho/3$ and $p=0$, 
respectively) satisfy such a condition. Let us postpone for the time being
the problem of finding a `candidate' able to provide the condition
$p<-\rho/3$. For sure, 
inflation  is a phase of the history of the universe
occurring before the era of nucleosynthesis ($t \approx 1$ sec, $T
\approx 1$ MeV) during which the light elements abundances were
formed. This is because nucleosynthesis 
is the earliest epoch  we have
experimental data from and they are in agreement
with the predictions of the standard Big-Bang theory. However,  
the thermal history of the universe before 
the epoch of nucleosynthesis is unknown. 

In order to study the properties of the period of inflation, we assume the
extreme condition $p=-\rho$ which considerably simplifies the
analysis. A period of the universe during which
$p=-\rho$ is called  {\it de Sitter} stage. 
By inspecting Eqs. (\ref{fr1}) and 
(\ref{mass}),
we learn that during the de Sitter phase 
\bea
\rho&=&~~{\rm constant},\nonumber\\
H_I&=&~~{\rm constant},\nonumber
\eea
where we have indicated by $H_I$ the value of the Hubble rate during inflation.
Correspondingly, solving Eq. (\ref{fr1}) gives
\be
a=a_i\, e^{H_I(t-t_i)},
\ee
where $t_i$ denotes the time at which inflation starts.
Let us now see how such a period of exponential expansion
takes care of the shortcomings of the standard Big Bang Theory.\footnote{
Despite the fact that the growth of the scale factor is exponential
and the expansion is {\it superluminal}, this is not
in contradiction with what dictated by relativity. Indeed, it is the
spacetime itself which is progating so fast and not a light signal in it.}

\subsection{Inflation and the horizon Problem}

During the  inflationary (de Sitter) epoch the horizon scale $H^{-1}$  is
constant. If inflation lasts long enough, all the physical scales
that have left  the horizon during the radiation-dominated or 
matter-dominated phase can
re-enter the horizon in the past: this is 
because such scales  are exponentially reduced.
As we have seen in the previous section, 
this explains both the problem of the homogeneity of CMB
and the initial condition problem of small
cosmological perturbations.
Once the physical length is within the horizon,
microphysics can act, the universe can be made
approximately homogeneous and the primaeval inhomogeneities can
be created. 

Let us see how long inflation must
be sustained in order to solve the horizon problem.
Let $t_i$ and $t_f$ be, respectively, the time of beginning and
end of inflation. We can define the corresponding number of e-foldings $N$
\be
N={\rm ln}\left[H_I(t_e-t_i)\right].
\ee
A necessary condition to solve the horizon problem is that the
largest scale we observe today, the present horizon $H_0^{-1}$, was
reduced  during inflation to a value $\lambda_{H_0}(t_{i})$ 
smaller than the value of
horizon length $H_I^{-1}$ during inflation.
This gives
$$
\lambda_{H_0}(t_{i})=H^{-1}_0 \left(\frac{a_{t_f}}{a_{t_0}}
\right) \left(\frac{a_{t_i}}{a_{t_f}}\right)=
H_0^{-1} \left(\frac{T_0}{T_f}
\right) e^{-N}\la H_I^{-1},
$$
where we have neglected for simplicity the short period of matter-domination
and we have called $T_f$ the temperature at the end of inflation (to be 
indentified with  the reheating temperature $T_{RH}$ at the beginning of 
the radiation-dominated phase after inflation, see later).  
We get 
$$
N \ga \ln\left( \frac{T_0}{H_0} \right) - \ln\left( \frac{T_f}{H_I}
\right) \approx 67 + \ln\left( \frac{T_f}{H_I} \right).
$$
Apart from the logarithmic dependence, we obtain  $N \ga 70$.

\subsection{Inflation and the flateness problem}

Inflation solves elegantly the flatness problem. Since during inflation the
Hubble rate is constant
$$
\Omega -1 = \frac{k}{a^2H^2}\propto \frac{1}{a^2}.
$$
On the other end the condition (\ref{con}) tells us that
to reproduce a value of $(\Omega_0-1)$ of order of unity today
the initial value of $(\Omega-1)$ at the beginning of the
radiation-dominated phase must be $\left|\Omega-1\right|\sim 10^{-60}$.
Since we identify the beginning of the radiation-dominated phase
with the beginning of inflation, we require
$$
\left|\Omega -1\right|_{t=t_{f}}\sim 10^{-60}.
$$
During inflation
\begin{equation}
\frac{\left|\Omega -1\right|_{t=t_{f}}}{\left|
\Omega -1\right|_{t=t_{i}}}= \left(\frac{a_i}{a_f}
   \right)^2 = e^{-2N}.
\label{fold}
  \end{equation}
Taking $\left|
\Omega -1\right|_{t=t_{i}}$ of order unity, 
it is enough to
require that $N \approx 70$ to solve the flatness problem.

{\it 1. \underline{Comment}:} In the previous section we have written that
the flateness problem can be also seen as a fine-tuning problem of one
part over $10^{60}$. Inflation ameliorates this fine-tuning problem, by
explaining a tiny number $\sim 10^{-60}$ with a number $N$ of the order
70.  


{\it 2. \underline{Comment}:} The number $N\simeq 70$ 
has been obtained requiring that
the present-day value of 
$(\Omega_0-1)$ is of order unity. For the expression (\ref{fold}), it
is clear that --if the period of inflation lasts longer than 70 e-foldings
the present-day value of $\Omega_0$ will be equal to unity with a great
precision. One can say that a generic prediction of inflation is that
\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
{\rm INFLATION}~~~\Longrightarrow~~~\Omega_0=1.
$$
%\\
\\
\hline\end{tabular}
\end{center}
This statement, however, must be taken {\it cum grano salis} and
properly specified. Inflation does not change the global geometric properties
of the spacetime. If the universe is open or closed, it will
always remain flat or closed, independently from inflation. 
What inflation does is to magnify the radius of curvature 
$R_{\rm curv}$ defined in Eq. (\ref{curv}) 
so that locally
the universe is flat with a great precision. 
%This is well depicted in
%Fig. \ref{formica}. 
%\begin{figure}[htb]
%\centerline{\hbox{ \psfig{figure=FG27_013.ps} }}
%\caption{\footnotesize  Inflation predicts an almost
%spatially flat universe and  $\Omega_0=1$.  \label{formica}}
%\end{figure}
As we have seen in section 2, the current data on the
CMB anisotropies confirm this prediction!

\subsection{Inflation and the entropy problem}

In the previous section, we have seen that the  
flatness problem arises because 
the entropy in a comoving volume is conserved. It is possible, therefore,  
that the problem could be resolved if the cosmic expansion was 
non-adiabatic for some finite time interval 
during the early history of the universe. We need to produce
a large amount of entropy $S_U\sim 10^{90}$. Let us  
postulate that the entropy changed by an amount
\begin{equation}
S_f=Z^3\,S_i
\end{equation}
from the beginning to the end of the inflationary 
period, where $Z$ is a numerical factor.
It is very natural to assume that the total entropy of the universe
at the beginning of inflation was of order unity, one particle
per horizon. Since, from the end of inflation onwards, the universe expands
adiabatically, we have $S_f=S_U$. This gives $Z\sim 10^{30}$. On the other
hand, since $S_f\sim \left(a_f  T_f\right)^3$ and 
$S_i\sim \left(a_i  T_i\right)^3$, where $T_f$ and $T_i$ are the temperatures
of the universe at the end and at the beginning of inflation,
we get
\be
\left(\frac{a_f}{a_i}\right)=e^N\approx 10^{30}\left(\frac{T_i}{T_f}\right),
\ee
which gives again $N\sim 70$ up to the logarithmic factor ${\rm ln}
\left(\frac{T_i}{T_f}\right)$.
We stress again that such a large amount of entopy is not
produced during inflation, but  during the non-adiabatic 
phase transition which gives rise to the usual radiation-dominated phase.


\subsection{Inflation and the inflaton}

In the previous subsections we have described the various
adavantages of having a period of accelerating phase. The latter
required $p<-\rho/3$. Now, we would like to show that this condition
can be attained by means of  a simple 
scalar field. We shall call  this field the {\it  inflaton} $\phi$.

The action of the inflaton field reads
\begin{equation}
S=\int d^4x\, \sqrt{-g}\,\mathcal{L}=\int\, d^4x\, \sqrt{-g}\,
\left[\frac{1}{2}
\partial_{\mu}\phi
\partial^{\mu}\phi +V(\phi)\right],
\end{equation}
where $\sqrt{-g}=a^3$ for the FRW metric (\ref{metric}).
From the Eulero-Lagrange equations
\begin{equation}
\partial^{\mu}\frac{\delta(\sqrt{-g}\mathcal{L})}{\delta\,
\partial^{\mu}\phi}- \frac{\delta(\sqrt{-g}\mathcal{L})}{\delta
\phi}=0, 
\end{equation}
we obtain
\begin{equation}
\ddot{\phi}+ 3H\dot{\phi}-\frac{\nabla^2\phi}{a^2}+V'(\phi)=0,
\label{nabla}
\end{equation}
where $V'(\phi)=\left(dV(\phi)/d\phi\right)$. Note, in particular, the
appearance of the friction term $3H\dot{\phi}$: a scalar field
rolling down its potential suffers a friction due to the
expansion of the universe.


We can write the energy-momentum tensor of the scalar field
$$
T_{\mu\nu}=\partial_{\mu}\phi \partial_{\nu}\phi
-g_{\mu\nu}\, \mathcal{L}.
$$
The corresponding energy density $\rho_\phi$ and pressure density $p_\phi$ 
are
\begin{eqnarray}
T_{00}=\rho_{\phi}=\frac{\dot{\phi}^2}{2} + V(\phi)+ 
\frac{(\nabla \phi)^2}{2a^2},  \\
T_{ii}=p_{\phi}=\frac{\dot{\phi}^2}{2} - V(\phi)- \frac{(\nabla
\phi)^2}{6a^2}.
\end{eqnarray}
Notice that, if
the gradient term were dominant, we would obtain
$p_\phi=-\frac{\rho_\phi}{3}$, not enough to drive inflation. 
We can now split the inflaton field in 
$$
\phi(t)=\phi_{0}(t)+\delta\phi({\bf x},t),
$$
where $\phi_{0}$ is the `classical' (infinite wavelength) field, that is 
the expectation value of the inflaton field 
on the initial isotropic and
homogeneous state, while $\delta\phi({\bf x},t)$ represents the quantum
fluctuations around $\phi_{0}$.
In this section, we will be only concerned with the evolution of the
classical field $\phi_0$. The next section will be devoted to the
crucial issue of the evolution of quantum perturbations during inflation.
This separation is justified by the fact that quantum fluctuations are much
smaller than the classical value and therefore negligible when looking at the 
classical evolution. To not be overwhelmed by the notation, we will
keep indicating
from now on the classical value of the inflaton field by $\phi$.
The energy-momentum tensor becomes
\begin{eqnarray}
T_{00}=\rho_{\phi}=\frac{\dot{\phi}^2}{2} + V(\phi)\\
T_{ii}=p_{\phi}=\frac{\dot{\phi}^2}{2} - V(\phi).
\end{eqnarray}
If
$$
V(\phi) \gg \dot{\phi}^2
$$
we obtain the following condition
$$
p_\phi\simeq -\rho_\phi
$$
From this simple calculation, 
we realize that a scalar field whose energy is dominant
in the universe and whose potential energy  
dominates over the kinetic term gives inflation! Inflation
is driven by the vacuum energy of the inflaton field.

\subsection{Slow-roll conditions}

Let us now quantify  better under which circumstances a scalar field
may give rise to a period of inflation. 
The equation of motion of the field is
 \begin{equation}
 \ddot{\phi}+3H\dot{\phi}+V'(\phi)=0
\label{poi}
 \end{equation}
If we require that $\dot{\phi}^2\ll V(\phi)$, the scalar field 
is slowly rolling down
its potential. This is the reason why such  a period is called {\it slow-roll}.
We may also expect that -- being the potential flat -- 
$\ddot{\phi}$ is negligible as well. We
 will assume that this is true and we will quantify this condition soon.
The FRW equation (\ref{fr1}) becomes
\be
H^2\simeq \frac{8\pi G}{3}\,V(\phi),
\ee
where we have assumed that the inflaton field dominates the
energy density of the universe.
The new equation of motion becomes
\be
 3H\dot{\phi}=-V'(\phi)
\label{friction}
\ee
which gives $\dot{\phi}$ as a function of $V'(\phi)$.
Using Eq. (\ref{friction}) slow-roll  conditions then require
$$
\dot\phi^2 \ll  V(\phi)   \\  \Longrightarrow  \\  \frac{(V')^2}{V} \ll
H^2 \label{slowroll1}
$$
and
$$
\ddot{\phi} \ll 3H\dot{\phi} \\  \Longrightarrow  \\  V'' \ll H^2.  
\label{slowroll2}
$$
It is now useful to define the  slow-roll 
parameters, $\epsilon$ and $\eta$ in the following way
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
\begin{eqnarray}
\epsilon&=&-\frac{\dot{H}}{H^2}=4\pi G\frac{\dot{\phi}^2}{H^2}
=\frac{1}{16\pi
G}\left(\frac{V'}{V}\right)^2, \label{epsilon slow roll}\nonumber\\
\eta&=&\frac{1}{8\pi G} \left(\frac{V''}{V}\right)=\frac{1}{3}
\frac{V''}{H^2},\nonumber\\
\delta&=&\eta-\epsilon=-\frac{\ddot{\phi}}{H \dot{\phi}}\, 
.\nonumber
\end{eqnarray}
\\
\hline
\end{tabular}
\end{center}
It might be useful to have the same parameters expressed in terms of
conformal time
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
\begin{eqnarray}
\epsilon&=&1-\frac{\H^\prime}{\H^2}=4\pi G\frac{\phi{^\prime}^2}{\H^2}
\label{epsilon slow roll conf}\nonumber\\
\delta&=&\eta-\epsilon=1-\frac{\phi^{\prime\prime}}{\H \phi^\prime}
\, .\nonumber
\end{eqnarray}
%\\
\\
\hline
\end{tabular}
\end{center}
The parameter $\epsilon$ quantifies how much the
Hubble rate $H$ changes with time during inflation. Notice that, since
$$
\frac{\ddot a}{a}=\dot H+H^2=\left(1-\epsilon\right)H^2,
$$
inflation can be attained only if $\epsilon<1$:
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
{\rm INFLATION}~~~\Longleftrightarrow ~~~\epsilon <1.
$$
%\\
\\
\hline
\end{tabular}
\end{center}
As soon as this condition fails, inflation ends. In general, slow-roll 
inflation
is attained if $\epsilon\ll 1$ and $|\eta|\ll 1$. During inflation
the  slow-roll parameters $\epsilon$ and $\eta$ can be considered
to be approximately constant  since the potential $V(\phi)$
is very flat.

\vskip 0.2cm

{\it \underline{Comment}:} In the following, we will work at {\it
first-order} perturbation in the slow-roll parameters, that is we will
take only the first power of them. Since, using their definition, it is
easy to see that $\dot\epsilon,\dot\eta={\cal O}
\left(\epsilon^2,\eta^2\right)$, this amounts to saying that we will
trat the slow-roll parameters as constant in time.
\vskip 0.2cm

Within these approximations, it is easy to compute the number of
e-foldings between the beginning and the end of inflation.
If we indicate by $\phi_i$ and  $\phi_f$ the values of the inflaton
field at the beginning and at the end of inflation, respectively,
we have that the {\it total} number of e-foldings is
\begin{eqnarray}
N&\equiv&\int_{t_i}^{t_f}\,H\,dt\nonumber\\
&\simeq& H\int^{\phi_f}_{\phi_i}
\frac{d\phi}{\dot{\phi}}\nonumber\\
&\simeq&-3 H^2\int^{\phi_f}_{\phi_i}
\frac{d\phi}{V'}\nonumber\\
&\simeq& -8\pi G \int^{\phi_f}_{\phi_i} \frac{V}{V'}\,d\phi.
 \end{eqnarray}

We may also compute the number of e-foldings $\Delta N$ which are left to go
to the end of inflation

\be
\label{togo}
\Delta N\simeq  8\pi G \int^{\phi_{\Delta N}}_{\phi_f}
\frac{V}{V'}\,d\phi,
\ee
where $\phi_{\Delta N}$ is the value of the inflaton field
when there are $\Delta N$ e-foldings to the end of inflation.



{\it 1. \underline{Comment}:} According to the criterion 
given in subsection 2.4, a given scale
length $\lambda=a/k$ leaves the horizon when $k=aH_k$
where
$H_k$ is the 
the value of the Hubble rate at that time. One can compute easily
the rate of change of $H^2_k$ as a function of $k$
\be
\frac{d {\rm ln} \,H_k^2}{d {\rm ln} \,k}=
\left(\frac{d {\rm ln} \,H_k^2}{dt}\right)\left(\frac{dt}{d {\rm ln} \,a}
\right)\left( \frac{d {\rm ln} \,a}{d {\rm ln} \,k}\right)=
2\frac{\dot H}{H}
\times \frac{1}{H}\times 
1=2\frac{\dot H}{H^2}=-2\epsilon.
\label{z}
\ee


{\it 2. \underline{Comment}:} Take a  given physical scale $\lambda$ today 
which crossed the horizon scale during inflation. This happened when
$$
\lambda\left(\frac{a_f}{a_0}\right)e^{-\Delta N_\lambda}=\lambda
\left(\frac{T_0}{T_f}\right)e^{-\Delta N_\lambda}=H_I^{-1}
$$
where $\Delta N_\lambda$ indicates the number of e-foldings from the time the
scale crossed the horizon during inflation and the end of inflation.
This relation gives a way to determine the number of e-foldings 
to the end of inflation corresponding to a given scale
$$
\Delta N_\lambda\simeq 65 +{\rm ln}\left(\frac{\lambda}{3000\,\,{\rm Mpc}}
\right)+2\,{\rm ln}\left(\frac{V^{1/4}}{10^{14}\,\,{\rm GeV}}
\right)-{\rm ln}\left(\frac{T_f}{10^{10}\,\,{\rm GeV}}
\right).
$$
Scales relevant  for the CMB anisotropies 
correspond  to $\Delta N\sim $60.



\subsection{The last stage of inflation and reheating}


Inflation ended when the potential energy associated with the inflaton
field became smaller than the kinetic energy of the field.  By that
time, any pre-inflation entropy in the universe had been inflated
away, and the energy of the universe was entirely in the form of
coherent oscillations of the inflaton condensate around the minimum of
its potential.  The universe may be said to be frozen after the end of
inflation. We know that somehow the low-entropy cold universe
dominated by the energy of coherent motion of the $\phi$ field must be
transformed into a high-entropy hot universe dominated by
radiation. The process by which the energy of the inflaton field is
transferred from the inflaton field to radiation has been dubbed
{\it reheating}. In the old theory of reheating \cite{dolgov,abbot}, 
the simplest way to envision this process is if the comoving energy
density in the zero mode of the inflaton decays into normal particles,
which then scatter and thermalize to form a thermal background.  It is
usually assumed that the decay width of this process is the same as
the decay width of a free inflaton field.

Of particular interest is a quantity known usually 
as the reheat temperature,
denoted as $T_{RH}$\footnote{So far, we have indicated it with
$T_f$.}. The reheat temperature is calculated by assuming 
an instantaneous conversion of the energy density in the inflaton 
field into radiation when the decay width of the inflaton energy,
$\Gamma_\phi$, is equal to $H$, the expansion rate of the universe. 

The reheat temperature is calculated quite easily.   After inflation
the inflaton field executes coherent oscillations about the minimum
of the potential.  Averaged over several oscillations, the coherent
oscillation energy density redshifts as matter: $\rho_\phi \propto
a^{-3}$, where $a$ is the Robertson--Walker scale factor.  If we
denote as $\rho_I$ and $a_I$ the total inflaton energy density 
and the scale factor at the initiation of coherent oscillations,
then the Hubble expansion rate as a function of $a$ is 
\begin{equation}
H^2(a) = \frac{8\pi}{3}\frac{\rho_I}{\mpl^2}
	\left( \frac{a_I}{a} \right)^3.
\end{equation}
Equating $H(a)$ and $\Gamma_\phi$ leads to an expression for $a_I/a$.
Now if we assume that all available coherent energy density is
instantaneously converted into radiation at this value of $a_I/a$, we
can find the reheat temperature by setting the coherent energy
density, $\rho_\phi=\rho_I(a_I/a)^3$, equal to the radiation energy
density, $\rho_R=(\pi^2/30)g_*T_{RH}^4$, where $g_*$ is the effective
number of relativistic degrees of freedom at temperature $T_{RH}$.
The result is
\begin{equation}
\label{eq:TRH}
T_{RH} = \left( \frac{90}{8\pi^3g_*} \right)^{1/4}
	\sqrt{ \Gamma_\phi \mpl } \
       = 0.2 \left(\frac{200}{g_*}\right)^{1/4}
      \sqrt{ \Gamma_\phi \mpl } \ .
\end{equation}
In some models of inflation reheating can be anticipated by a period of
preheating \cite{preheating} when the 
the classical inflaton field  very rapidly (explosively) 
decays into $\phi$-particles or 
into other bosons due to broad parametric resonance. 
This stage cannot be described by the standard elementary approach to 
reheating based on perturbation theory. The bosons produced at 
this stage  further decay into other particles, 
which eventually become thermalized. 

\subsection{A brief survey of inflationary models}

Even restricting ourselves to a simple single-field inflation scenario, the
number of models available to choose from is large \cite{lr}.
 It is convenient to define a general classification
 scheme, or ``zoology'' for 
models of inflation. We divide models into three general types
\cite{dodelson97}: {\it 
large-field}, {\it small-field}, and {\it hybrid},  with a fourth 
classification. A generic single-field potential  can be characterized by two 
independent mass scales: a ``height'' $\Lambda^4$, corresponding to the vacuum 
energy density during inflation, and a ``width'' $\mu$, corresponding to the 
change in the field value $\Delta \phi$ during inflation:
\begin{equation}
V\left(\phi\right) = \Lambda^4 f\left({\phi \over \mu}\right).
\end{equation}
Different models have different forms for the function $f$.
Let us now briefly describe the different class of models.







\subsubsection{Large-field models}

Large-field models are potentials typical of the ``chaotic'' 
inflation 
scenario\cite{linde83}, 
in which the scalar field is displaced from the minimum 
of the potential by an amount usually of order the Planck mass. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=3.5in \epsfbox{large_field.eps}}
\caption{Large field models of inflation. From Ref. \cite{kolbreview}.
 \label{large}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Such models are 
characterized by  $V''\left(\phi\right) > 0$, and $-\epsilon < \delta \leq 
\epsilon$. The generic large-field potentials we consider are polynomial 
potentials $V\left(\phi\right) = \Lambda^4
\left({\phi / \mu}\right)^p$,
and exponential potentials, $V\left(\phi\right) = \Lambda^4 \exp\left({\phi / 
\mu}\right)$. 
In the chaotic inflation scenario, it is assumed that the universe emerged 
from a quantum gravitational state with an energy density comparable to that 
of the Planck density. This implies that $V (\phi ) \approx \mpl^4$ 
and results in a large friction term in the Friedmann equation (\ref{f2}). 
Consequently, the inflaton will slowly roll  down its potential.
The condition 
for inflation is therefore satisfied and the scale factor grows as 
\begin{equation}
a(t) =a_{i} e^{\left( \int^t_{t_{i}} dt' H(t') \right)}.
\end{equation}
The simplest chaotic inflation model is that of a free field with a 
quadratic potential, $V(\phi) =m^2 \phi^2/2$, where $m$ represents the mass 
of the inflaton. During inflation the scale factor grows as 
\begin{equation}
a(t) = a_{i} e^{2\pi (\phi^2_{i} - \phi^2 (t))}
\end{equation}
and inflation ends when $\phi = {\cal{O}} (1)$ $ \mpl$. If inflation 
begins when $V(\phi_{\rm i} ) \approx \mpl^4$, the scale factor grows 
by a factor $\exp( 4\pi \mpl^2/m^2)$ before the inflaton reaches the 
minimum of its potential. We will later  show that the mass 
of the field should be $m \approx 10^{-6}\mpl$ if the microwave 
background constraints are to be satisfied. This implies that the volume of 
the universe will increase by a factor of $Z^3 \approx 10^{3 \times 
10^{12}}$ and this is more than enough inflation to solve the problems of 
the hot big bang model.

In the chaotic inflationary scenarios, the present-day universe is only
a small portion of the universe which suffered inflation!
Notice also that the typical values of the inflaton field
during inflation are of the order of $\mpl$, giving rise to the
possibility of testing planckian physics \cite{ckrt}.










\subsubsection{Small-field models}

Small-field models are the type of potentials that arise naturally
 from spontaneous symmetry breaking (such as the original models of ``new'' 
inflation \cite{linde82,albrecht82}) and from pseudo Nambu-Goldstone modes 
(natural inflation\cite{freese90}). The field starts from near an
 unstable equilibrium (taken to be at the origin) and rolls 
down the potential to a stable minimum. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=3.5in \epsfbox{small_field.eps}}
\caption{Small field models of inflation. From Ref. \cite{kolbreview}.
 \label{small}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Small-field models are 
characterized by $V''\left(\phi\right) < 0$ and $\eta < -\epsilon$. Typically 
$\epsilon$ is close to zero. 
The generic small-field potentials we consider are of the form 
$V\left(\phi\right) = \Lambda^4 \left[1 - \left({\phi / \mu}\right)^p\right]$,
 which can be viewed as a lowest-order Taylor expansion of an arbitrary
 potential about the origin. See, for instance, Ref. \cite{Dine:1997kf}.


\subsubsection{Hybrid models}

The hybrid scenario\cite{linde91,linde94,copeland94} frequently appears in 
models which incorporate inflation into supersymmetry \cite{Riotto:1997iv}
and supergravity \cite{Linde:1997sj}. In a typical hybrid 
inflation model, the scalar field responsible
for inflation evolves toward a minimum with nonzero vacuum energy. The end of 
inflation arises as a
 result of instability in a second field. Such models are 
characterized by $V''\left(\phi\right) > 0$ and $0 < \epsilon < \delta$. We 
consider generic potentials for hybrid inflation of the form 
$V\left(\phi\right) 
= \Lambda^4 \left[1 + \left({\phi / \mu}\right)^p\right].$ The field value at 
the end of inflation is determined by some other physics, so there is a second 
free parameter characterizing the models. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=3.5in \epsfbox{hybrid.eps}}
\caption{Hybrid field models of inflation. From Ref. \cite{kolbreview}.
 \label{hybrid}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This enumeration of models is certainly not exhaustive. There are a number of
single-field models that do not fit well into this scheme, for example
logarithmic potentials $V\left(\phi\right) \propto
\ln\left(\phi\right)$ typical of supersymmetry
\cite{lr,Halyo:1996pp,Binetruy:1996xj,Dvali:1997mh,
Lyth:1997pf,Riotto:1997wy,Espinosa:1998ks,King:1998uv}. 
Another example is potentials
with negative powers of the scalar field $V\left(\phi\right) \propto
\phi^{-p}$ used in intermediate inflation 
\cite{barrow93} and dynamical
supersymmetric inflation \cite{kinney97,kinney98}. 
Both of these cases require 
and auxilliary field to end inflation and are more properly categorized as 
hybrid models, but fall into the small-field class. 
However, the three classes
categorized by the relationship between the slow-roll parameters 
as $-\epsilon < 
\delta \leq \epsilon$ (large-field), $\delta 
\leq -\epsilon$ (small-field) 
and $0 < \epsilon < \delta$ (hybrid) seems to be good enough for
comparing theoretical expectations with experimental data.


\section{Inflation and the cosmological perturbations}

As we have seen in the previous section, 
the early universe was made very nearly uniform by a primordial
inflationary stage.  However, the important
caveat in that statement is the word `nearly'.  Our current
understanding of the origin of structure in the universe is that it
originated from small `seed' perturbations, which over time grew to
become all of the structure we observe.  
Once the universe becomes matter dominated (around 1000 yrs
after the bang) primeval density
inhomogeneities ($\delta\rho /\rho \sim 10^{-5}$)
are amplified by gravity and
grow into the structure we see today \cite{sf}.
The fact that a fluid of
self-gravitating particles is unstable to the growth of
small inhomogeneities was first pointed out by Jeans
and is known as the Jeans instability.  Furthermore, 
the existence of these inhomogeneities was confirmed
by the COBE  discovery of CMB anisotropies;   the temperature
anisotropies detected almost certainly owe their
existence to primeval density inhomogeneities, since, as we have seen, 
causality precludes microphysical processes
from producing anisotropies on angular scales larger
than about $1^\circ$, the angular size of the horizon
at last-scattering.

The growth of small matter inhomogeneities of wavelength
smaller than the Hubble scale ($\lambda \la
H^{-1}$) is governed by a Newtonian equation:
\begin{equation}
{\ddot\delta}_{\bf k} + 2H{\dot\delta}_{\bf k} +v_s^2\frac{k^2}{a^2}
\delta_{\bf k} 
= 4\pi G\rho_M \delta_{\bf k} ,
\label{newt}
\end{equation}
where $v_s^2 = \partial p /\partial \rho_M$ is the square of the sound
speed and we have expanded 
the perturbation to the matter density in  plane waves
\begin{equation}
{\delta\rho_M ({\bf x}, t)\over\rho_M} = {1\over (2\pi )^3}\int d^3k\,
	\delta_{\bf k}(t) e^{-i{\bf k}
	\cdot {\bf x}}.
\end{equation} 
Competition between the pressure term and
the gravity term on the rhs of Eq. (\ref{newt}) determines whether or
not pressure can counteract gravity:
perturbations with wavenumber larger than the Jeans wavenumber,
$k_J^2 = 4\pi Ga^2 \rho_M /v_s^2$, are Jeans stable
and just oscillate; perturbations with smaller
wavenumber are Jeans unstable and can grow.

Let us discuss solutions to this equation under
different circumstances.  First, consider the Jeans
problem, evolution of perturbations in a static fluid,
{\it i.e.}, $H=0$.  In this case Jeans unstable
perturbations grow exponentially,
$\delta_{\bf k} \propto \exp (t/\tau )$ where $\tau = 1/\sqrt{4G\pi\rho_M}$.
Next, consider the growth of Jeans unstable perturbations in a
matter-dominated universe,
{\it i.e.}, $H^2=8\pi G\rho_M/3$ and $a\propto t^{2/3}$.  Because
the expansion tends to ``pull particles away from
one another,'' the growth is only power law,
$\delta_{\bf k} \propto t^{2/3}$; {\it i.e.}, at the same rate as the
scale factor.  Finally, consider a radiation-dominated 
universe. In this case, the expansion
is so rapid that matter perturbations grow very slowly,
as $\ln a$ in radiation-dominated epoch. Therefore, perturbations
may grow only in a matter-dominated period. 
Once a perturbation reaches an overdensity of order unity
or larger it ``separates'' from the expansion --{\it i.e.}, becomes
its own self-gravitating system and ceases to expand
any further.  In the process of virial
relaxation, its size decreases by a factor of two---density
increases by a factor of 8; thereafter, its density contrast
grows as $a^3$ since the average
matter density is decreasing as $a^{-3}$, though smaller
scales could become Jeans unstable and collapse further to form
smaller objects of higher density.



In order for structure formation to occur via gravitational
instability, there must have been small preexisting fluctuations on
physical length scales when they crossed the Hubble radius in the 
radiation-dominated 
and matter-dominated 
 eras.  In the standard Big-Bang model these small perturbations
have to be put in by hand, because it is impossible to produce
fluctuations on any length scale while it is larger than the horizon.  Since
the goal of cosmology is to understand the universe on the basis of
physical laws, this appeal to initial conditions is unsatisfactory.
The challenge is therefore to give an explanation to the small seed
perturbations which allow  the gravitational growth of the
matter perturbations.

Our  best guess for the origin
of these perturbations is quantum fluctuations during an inflationary
era in the early universe.
Although originally introduced as a
possible solution to the  cosmological conundrums such as the
horizon, flatness and entopy problems, by far the most useful
property of inflation is that it generates spectra of both density
perturbations 
and gravitational waves. These perturbations extend from extremely
short scales to scales considerably in excess of the size of the
observable universe. 

During inflation the scale factor grows
quasi-exponentially, while the Hubble radius remains almost constant.
Consequently the wavelength of a quantum fluctuation -- either in the
scalar field whose potential energy drives inflation or in the
graviton field -- soon exceeds the Hubble radius.  The amplitude of
the fluctuation therefore becomes `frozen in'. This is
quantum mechanics in action at macroscopic scales!

According to quantum field theory, empty space is 
not entirely empty. It is filled with quantum fluctuations 
of all types of physical fields. The fluctuations can 
be regarded as waves of physical fields
with all possible wavelenghts, moving in all possible directions.
If the values of these fields, averaged over some macroscopically
large time, vanish then the space filled with these fields seems to us 
empty and can be called the vacuum.

In the exponentially expanding universe the 
vacuum structure is much more complicated. The wavelenghts of all 
vacuum fluctuations of the inflaton field $\phi$ grow exponentially in the
expnading universe. When the wavelength of any particular fluctuation
becomes greater than $H^{-1}$, this fluctuation stops propagating, and 
its amplitude freezes at some nonzero value $\delta\phi$ because of the 
large
friction term $3H\dot\phi$ i the equation of motion of the
field $\phi$. The amplitude of this fluctuation then remains 
almost unchanged for a very long time, whereas its wavelength grows 
exponentially. Therefore, the appearance of such frozen fluctuation
is equivalent to the appearance of a classical field
$\delta\phi$ that does not vanish after having averaged over some 
macroscopic interval of time. Because the vacuum contains fluctuations
of all possible wavelength, inflation leads to the creation of
more and more new perturbations of the classical field with
wavelength larger than the horizon scale.



Once inflation has ended,
however, the Hubble radius increases faster than the scale factor, so
the fluctuations eventually reenter the Hubble radius during the
radiation- or matter-dominated eras. The fluctuations that exit around
60 $e$-foldings or so before reheating reenter with physical
wavelengths in the range accessible to cosmological observations.
These spectra provide a distinctive signature of inflation. They can
be measured in a variety of different ways including the analysis of
microwave background anisotropies. 

The physical processes which give rise to the structures we observe today
are well-explained in Fig. \ref{figpert}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=3.5in \epsfbox{fig4.eps}}
\caption{The horizon scale (green line) and a physical scale $\lambda$
(red line) as function of the scale factor $a$. From Ref. \cite{kolbreview}.
\label{figpert}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Quantum fluctuations of the inflaton field are generated during inflation.
Since gravity talks to any component of the universe, small fluctuations
of the inflaton field are intimately related to fluctuations of the
spacetime metric, giving rise to perturbations of the curvature
${\cal R}$ (which will be  defined in the following; the reader
may loosely think of it as a gravitational potential). The
wavelenghts $\lambda$ of these  
perturbations grow exponentially and leave soon the
horizon when $\lambda>R_H$. On superhorizon scales, curvature 
fluctuations are frozen in and may be considered as classical. Finally, 
when the
wavelength of these fluctuations reenters the horizon, at some
radiation- or matter-dominated epoch, the curvature (gravitational
potential) perturbations
of the spacetime give rise to matter (and temperature) 
perturbations $\delta\rho$
via the Poisson equation. These fluctuations will then start growing
giving rise to the structures we observe today.

In summary, two are the key ingredients for understanding 
the observed structures in the universe within the
inflationary scenario:

\begin{itemize}

\item Quantum fluctuations of the inflaton field 
are excited during inflation and stretched to cosmological scales. At the
same time, being the inflaton fluctuations connected
to the metric perturbations through  Einstein's equations, 
ripples on the metric are also excited and stretched to cosmological 
scales.

\item Gravity acts a   messanger since it communicates to baryons and photons
the small seed perturbations once a given wavelength becomes smaller
than the horizon scale after inflation.

\end{itemize}

Let us know see how quantum fluctuations are generated during inflation.
We will proceed by steps. First, we will consider the simplest
problem of studying the quantum fluctuations of a generic scalar field
during inflation: we will
learn how perturbations evolve as a function of time and compute their
spectrum. Then -- since a satisfactory description of the generation of
quantum fluctuations have to take both the inflaton and the metric
perturbations into account -- we will study the system composed by 
quantum
fluctuations of the inflaton field  and  quantum fluctuations
of the metric.











\section{Quantum fluctuations of a generic massless scalar field during 
inflation}





Let us first see how the
fluctuations  of a generic scalar field $\chi$, which is {\it not}
the inflaton field, behave during inflation. To warm up we first
consider a de Sitter epoch during which the Hubble rate is
constant.

\subsection{Quantum fluctuations of a generic massless scalar field during 
a de Sitter stage}

We assume this field to be massless. The massive case will be analyzed in 
the
next subsection.

Expanding the scalar field $\chi$ in Fourier modes
$$
\delta\chi({\bf x},t)=\int\,\frac{d^3{\bf k}}{(2\pi)^{3/2}}\,e^{i{\bf k}
\cdot{\bf x}}\,
\delta\chi_{{\bf k}}(t),
$$
we can write the equation for the
fluctuations as 
\be
\delta\ddot{\chi}_{\bf  k}
+3H\,\delta\dot{\chi}_{\bf k}+
\frac{k^2}{a^2}\,\delta\chi_{\bf k}=0.
\label{quantum}
\ee
Let us study the qualitative behaviour of the solution to Eq. (\ref{quantum}).

\begin{itemize}

\item For wavelengths within the horizon, $\lambda\ll H^{-1}$, the 
corresponding wavenumber satisfies the relation $k\gg a\,H$. In this
regime, we can neglect the friction term $3H\,\delta\dot{\chi}_{\bf k}$
and 
Eq. (\ref{quantum}) reduces to
\be
\delta\ddot{\chi}_{\bf k}+
\frac{k^2}{a^2}\,\delta\chi_{\bf k}=0,
\ee
which is -- basically -- the equation of motion of an harmonic oscillator.
Of course, the frequency term $k^2/a^2$ depends upon time because
the scale factor $a$ grows exponentially. On the qualitative level,
however, one expects that when the wavelength of the fluctuation is within
the horizon, the fluctuation oscillates.




\item For wavelengths above the horizon, $\lambda\gg H^{-1}$, the 
corresponding wavenumber satisfies the relation $k\ll aH$
and the term $k^2/a^2$ can be safely neglected. Eq. (\ref{quantum}) reduces to
\be
\delta\ddot{\chi}_{\bf  k}
+3H\,\delta\dot{\chi}_{\bf k}=0,
\ee
which tells us that on superhorizon scales $\delta\chi_{\bf k}$ remains 
constant.

\end{itemize}

We have therefore the following picture: take a given fluctuation
whose initial wavelength $\lambda\sim a/k$ is within the horizon. The
fluctuations oscillates till the wavelength becomes of the order
of the horizon scale. When the wavelength crosses the horizon, the
fluctuation ceases to oscillate and gets frozen in.

Let us know study the evolution of the fluctuation is a more quantitative
way. To do so, we perform the following redefinition
$$
\delta\chi_{\bf k}=\frac{\delta\sigma_{\bf k}}{a}
$$
and we work in conformal time $d\tau=dt/a$. For the time
being, we solve the problem for a pure de Sitter expansion
and we take the scale 
factor
 exponentially growing as $a\sim e^{Ht}$; 
the corresponding
conformal factor reads (after choosing properly the integration constants) 
$$
a(\tau)=-\frac{1}{H\tau}\,\,\,\,(\tau<0).
$$
In the following we will also solve the problem in the
case of quasi de Sitter expansion.
The beginning of inflation coincides with some initial time $\tau_i\ll 0$.
Using the set of rules (\ref{rules}), we find that Eq. (\ref{quantum})
becomes
\be
\delta\sigma^{\prime\prime}_{\bf k}+
\left(k^2-\frac{a^{\prime\prime}}{a}\right)\delta\sigma_{\bf k}=0.
\label{qq}
\ee
We obtain an equation which is very `close' to the equation for a 
Klein-Gordon scalar field in flat spacetime, the only difference being
a negative time-dependent mass term $-a^{\prime\prime}/a=-2/\tau^2$.
Eq. (\ref{qq}) can be obtained from an action of the type
\be
\delta S_{\bf k}=\int\,d\tau\,\left[\frac{1}{2}
\delta\sigma^{\prime 2}_{\bf k}-\frac{1}{2}
\left(k^2-\frac{a^{\prime\prime}}{a}\right)\delta\sigma^2_{\bf k}
\right],
\label{action}
\ee
which is the canonical action for a simple harmonic oscillator with
canonical commutation relations $\delta\sigma^*_{\bf 
k}\delta\sigma^\prime_{\bf k}
-\delta\sigma_{\bf k}\delta\sigma^{*\prime}_{\bf k}=-i$.

Let us study the behaviour of this equation on subhorizon and superhorizon
scales. Since 
$$
\frac{k}{aH}=-k\,\tau,
$$
on subhorizon scales $k^2\gg a^{\prime\prime}/a$   Eq. (\ref{qq})
reduces to 
$$
\delta\sigma^{\prime\prime}_{\bf k}+
k^2\,\delta\sigma_{\bf k}=0,
$$
whose solution is a plane wave
\be
\delta\sigma_{\bf k}=\frac{e^{-ik\tau}}{\sqrt{2k}}\,\,\,\,(k\gg aH).
\label{q1}
\ee
We find again that  fluctuations with wavelength within the horizon
oscillate exactly like in flat spacetime. 
This does not come as a surprise. In the 
ultraviolet regime, that is for wavelengths much smaller than the horizon
scale, one expects that approximating the spacetime as flat
is a good approximation.


On superhorizon scales, 
$k^2\ll a^{\prime\prime}/a$ Eq. (\ref{qq})
reduces to 
$$
\delta\sigma^{\prime\prime}_{\bf k}-
\frac{a^{\prime\prime}}{a}\delta\sigma_{\bf k}=0,
$$
which is satisfied by 
\be
\delta\sigma_{\bf k}=B(k)\,a \,\,\,\,(k\ll aH).
\label{x2}
\ee
where $B(k)$ is a constant of integration. Roughly matching  the (absolute
values of the) solutions
$(\ref{q1})$ and $(\ref{x2})$ at $k=aH$ ($-k\tau=1$), we can determine the
(absolute value of the) constant $B(k)$
$$
\left|B(k)\right|a=\frac{1}{\sqrt{2k}}\Longrightarrow
\left|B(k)\right|=\frac{1}{a\sqrt{2k}}=\frac{H}{\sqrt{2k^3}}.
$$
Going back to the original variable 
$\delta\chi_{\bf k}$, we obtain that the quantum fluctuation of the 
$\chi$
field on superhorizon scales is constant and approximately
equal to
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
\left|\delta\chi_{\bf k}\right|\simeq \frac{H}{\sqrt{2k^3}}\,\,\,\,
({\rm ON}\,\,{\rm SUPERHORIZON}\,\,{\rm SCALES})
$$
%\\
\\
\hline
\end{tabular}
\end{center}
In fact we can do much better, since Eq. (\ref{qq}) has an {\it exact} 
solution:
\be
\label{sigma}
\delta\sigma_{\bf k}=\frac{e^{-ik\tau}}{\sqrt{2k}}\left(
1+\frac{i}{k\tau}\right).
\ee
This solution reproduces all what we have found by qualitative arguments
in the two extreme regimes $k\ll aH$ and $k\gg aH$. The reason why we have
performed the matching procedure is to show that the latter can be
very useful to determine the behaviour of the solution on superhorizon scales
when the exact solution is not known.

 







\subsection{Quantum fluctuations of a generic massive scalar field during
a de Sitter stage}

So far, we have solved the equation for the quantum perturbations of 
a generic massless  field, that is  neglecting the mass squared term 
$m_\chi^2$. 
Let us know discuss
the solution when such a mass term is present. Eq. (\ref{qq})
becomes
\be
\delta\sigma^{\prime\prime}_{\bf k}+
\left[k^2+M^2(\tau)\right]\delta\sigma_{\bf k}=0,
\label{qqq}
\ee
where
$$
M^2(\tau)=\left(m_\chi^2-2H^2\right)a^2(\tau)=\frac{1}{\tau^2}
\left(\frac{m^2}{H^2}-2\right).
$$
Eq. (\ref{qqq}) can be recast in the form
\be
\delta\sigma^{\prime\prime}_{\bf k}+
\left[k^2-\frac{1}{\tau^2}\left(\nu_\chi^2-\frac{1}{4}\right)
\right]\delta\sigma_{\bf k}=0,
\label{qqqq}
\ee
where 
\be
\nu_\chi^2=\left(\frac{9}{4}-\frac{m_\chi^2}{H^2}\right).
\label{zz}
\ee
The generic
solution to Eq. (\ref{qqq}) for $\nu_\chi$ {\it real} is
$$
\delta\sigma_{\bf k}=\sqrt{-\tau}\left[c_1(k)\,H_{\nu_\chi}^{(1)}(-k\tau)+
c_2(k)\,H_{\nu_\chi}^{(2)}(-k\tau)\right],
$$
where $H_{\nu_\chi}^{(1)}$ and $H_{\nu_\chi}^{(2)}$ are the Hankel's 
functions of the
first and second kind, respectively.
If we impose that in the ultraviolet regime $k\gg aH$  $(-k\tau\gg 1$)
the solution matches the plane-wave solution $e^{-ik\tau}/\sqrt{2k}$
that we expect in flat spacetime and knowing that
$$
H_{\nu_\chi}^{(1)}(x\gg 1)\sim \sqrt{\frac{2}{\pi x}}\,e^{i\left(x-
\frac{\pi}{2}\nu_\chi-\frac{\pi}{4}\right)}\,\,\,\, ,
H_{\nu_\chi}^{(2)}(x\gg 1)\sim \sqrt{\frac{2}{\pi x}}\,
e^{-i\left(x-
\frac{\pi}{2}\nu_\chi-\frac{\pi}{4}\right)},
$$ 
we set $c_2(k)=0$ and $c_1(k)=
\frac{\sqrt{\pi}}{2}\,e^{i\left(\nu_\chi+\frac{1}{2}\right)\frac{\pi}{2}}$. 
The exact solution becomes
\be
\delta\sigma_{\bf k}=\frac{\sqrt{\pi}}{2}\,
e^{i\left(\nu_\chi+\frac{1}{2}\right)\frac{\pi}{2}}\,
\sqrt{-\tau}\,H_{\nu_\chi}^{(1)}(-k\tau).
\label{exactm}
\ee
On superhorizon scales, since $H_{\nu_\chi}^{(1)}(x\ll 1)\sim
\sqrt{2/\pi}\, e^{-i\frac{\pi}{2}}\,2^{\nu_\chi-\frac{3}{2}}\,
(\Gamma(\nu_\chi)/\Gamma(3/2))\, x^{-\nu_\chi}$, 
the fluctuation (\ref{exactm}) becomes
$$
\delta\sigma_{\bf k}=e^{i\left(\nu_\chi-\frac{1}{2}\right)\frac{\pi}{2}}
2^{\left(\nu_\chi-\frac{3}{2}\right)}\frac{\Gamma(\nu_\chi)}{\Gamma(3/2)}
\frac{1}{\sqrt{2k}}\,(-k\tau)^{\frac{1}{2}-\nu_\chi}.
$$
Going back to the old variable $\delta\chi_{\bf k}$, we find that
on superhorizon scales, the fluctuation with nonvanishing mass
is not exactly constant, but
it acquires a tiny dependence upon the time
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
\left|\delta\chi_{\bf k}\right|\simeq \frac{H}{\sqrt{2k^3}}
\left(\frac{k}{aH}\right)^{\frac{3}{2}-\nu_\chi}
\,\,\,\,
({\rm ON}\,\,{\rm SUPERHORIZON}\,\,{\rm SCALES})
$$
%\\
\\
\hline
\end{tabular}
\end{center}
If we now define, in analogy with the definition of the slow roll
parameters $\eta$ and $\epsilon$ for the 
inflaton field,  the 
parameter 
$\eta_\chi=(m_\chi^2/3H^2)\ll 1 $, one finds
\be
\frac{3}{2}-\nu_\chi\simeq \eta_\chi.
\label{nu}
\ee
It is instructive  to analyze the case in which $\nu_\chi$
is {\it imaginary}, that is $m_\chi/H>3/2$. In such a case, 
we define $\widetilde{\nu}=i\nu$. In superhorizon scales, performing the
same steps we have done for the case of $\nu_\chi$ real, we find

\begin{eqnarray}
&\left|{\delta \chi}_{\bf k}\right|^2=\frac{\pi}{4}
\frac{e^{-\pi\widetilde{\nu}}}{a^2} \frac{1}{aH}\Big[
\frac{(1+{\rm cot}(\pi\widetilde{\nu})^2){\rm sinh}(\pi\widetilde{\nu})}
{\pi\widetilde\nu}
+\frac{\widetilde{\nu}}{\pi
{\rm sinh}(\pi\widetilde{\nu})}+\\ \nonumber
&2\,{\rm Re}\Big(i\left({\rm cos}\,(2\,\widetilde{\nu}\,
{\rm ln}(\frac{k\eta}{2}))+
i\,
{\rm sin}\,(2\,\widetilde{\nu}\,{\rm ln}(\frac{k\tau}{2}))\right)
\frac{1-{\rm cot}(\pi
\widetilde{\nu})}{\Gamma(1+i\widetilde{\nu})}
\frac{\Gamma(-i\widetilde{\nu})}{\pi}\Big)
\Big]\Big(\frac{k}{aH}\Big)^{n-1}.
\end{eqnarray}
In the limit of long wavelengths, the highly oscillating term which
appears in the real part can be neglected  because its
average on $k$ is 0.
The resulting power spectrum is the following
\begin{eqnarray}
&{\cal P}_{\delta \chi}(k)=\frac{\pi}{4} e^{-\pi\widetilde{\nu}}
\Big(\frac{H}{2\pi}\Big)^2 \Big(
\frac{(1+{\rm cot}(\pi\tilde{\nu})^2){\rm sinh}(\pi\widetilde{\nu})}
{\pi\widetilde\nu}+\frac{\widetilde{\nu}}{\pi
{\rm sinh}(\pi\tilde{\nu})}\Big)\\
\nonumber &\times\Big(\frac{k}{a H}\Big)^{n-1} \simeq
\left(\frac{H}{2\pi}\right)^2\left(\frac{H}{m_\chi}\right)
 \Big(\frac{k}{aH}\Big)^3. \label{PS nu
imaginary}
\end{eqnarray}
Therefore, for very massive scalar fields, $m_\chi>3H/2$, the power
spectrum has an amplitude which is suppressed by the ratio $(H/m_\chi)$
and the spectrum falls down rapidly al large wavelengths $k^{-1}$ as
$k^{3}$.


\subsection{Quantum to classical transition}

We have previously said that the quantum flactuations can be regarded
as classical when their corresponding wavelengths cross the 
horizon. To  better motivate this statement, we should compute the
number of particles $n_{\bf k}$ per wavenumber ${\bf k}$ on superhorizon
scales and check that it is indeed much larger than unity, $n_{\bf k}\gg 1$
(in this limit one can neglect the ``quantum" factor $1/2$ in the 
Hamiltonian $H_{\bf k}=\omega_{\bf k}\left(n_{\bf k}+\frac{1}{2}\right)$
where $\omega_{\bf k}$ is the energy eigenvalue).
If so, the fluctuation  can be regarded as classical. The number of
particles $n_{\bf k}$ can be estimated to be of the order of $H_{\bf 
k}/\omega_{\bf k}$,
where $H_{\bf k}$ is the Hamiltonian corresponding to the action
\be
\delta S_{\bf k}=\int\,d\tau\,\left[\frac{1}{2}
\delta\sigma^{\prime 2}_{\bf k}+\frac{1}{2}
\left(k^2-M^2(\tau)\right)\delta\sigma^2_{\bf k}
\right],
\ee
One obtains on superhorizon scales
$$
n_{\bf k}\simeq \frac{M^2(\tau)\left|
\delta\chi_{\bf k}\right|^2}{\omega_{\bf k}}\sim 
\left(\frac{k}{aH}\right)^{-3}\gg 1,
$$
which confirms that fluctuations on superhorizon scales may be indeed 
considered as classical.









\subsection{The power spectrum}

Let us define now the power spectrum, a useful quantity to
characterize the properties of the perturbations. 
For a generic quantity $g({\bf x},t)$, which can expanded in
Fourier space as
$$
g({\bf x},t)=\int\,\frac{d^3{\bf k}}{(2\pi)^{3/2}}\,e^{i{\bf k}
\cdot{\bf x}}\,
g_{{\bf k}}(t),
$$
the power spectrum can be defined as
\be
\langle 0|g^{*}_{{\bf k}_1}g_{{\bf k}_2}|0\rangle
\equiv\delta^{(3)}\left({\bf k}_1-{\bf k}_2\right)\,\frac{2\pi^2}{k^3}\,
{\cal P}_{g}(k),
\ee
where $\left|0\right.\rangle$ is the vacuum quantum state of the system. This
definition leads to the usual relation
\be
\langle 0|g^2({\bf x},t)|0\rangle=\int\,\frac{dk}{k}\,
{\cal P}_{g}(k).
\ee


\subsection{Quantum fluctuations of a generic scalar field in a quasi de 
Sitter stage}

So far, we have computed the time evolution and the spectrum of the 
quantum flutuations of a generic scalar field $\chi$ 
supposing that the 
scale factor evolves
like in a pure de Sitter expansion, $a(\tau)=-1/(H\tau)$. However, 
during
inflation the Hubble rate is not exactly constant, but changes with time
as $\dot H=-\epsilon\,H^2$ (quasi de Sitter expansion),
In this subsection, we will solve for the perturbations
in a quasi de Sitter expansion. Using the definition of the
conformal time, one can show that the scale factor for small values
of $\epsilon$
becomes
$$
a(\tau)=-\frac{1}{H}\frac{1}{\tau(1-\epsilon)}.
$$
Eq. (\ref{qqq}) has now a squared mass term
$$
M^2(\tau)=m_\chi^2a^2-\frac{a^{\prime\prime}}{a},
$$
where
\bea
\label{ap}
\frac{a^{\prime\prime}}{a}&=&a^2\left(\frac{\ddot{a}}{a}+H^2\right)=
a^2\left(\dot H+2\,H^2\right)\nonumber\\
&=&a^2\left(2-\epsilon\right)H^2=
\frac{\left(2-\epsilon\right)}{\tau^2\left(1-\epsilon\right)^2}\nonumber\\
&\simeq& \frac{1}{\tau^2}\left(2+3\epsilon\right).
\eea
Taking $m_\chi^2/H^2=3\eta_\chi$ and expanding for small values
of $\epsilon$ and $\eta$ we get Eq. 
(\ref{qqqq}) with
\be
\label{vv}
\nu_\chi\simeq \frac{3}{2}+\epsilon-\eta_\chi.
\ee
Armed with these results, we may  compute the
variance of the perturbations of the generic $\chi$ field
\bea
\langle 0|\left(\delta\chi({\bf x},t)\right)^2|0\rangle&=&
\int\,\frac{d^3k}{(2\pi)^3}\,\left|\delta\chi_{\bf k}\right|^2\nonumber\\
&=&\int\,\frac{dk}{k}\,\frac{k^3}{2\pi^2}
\,\left|\delta\chi_{\bf k}\right|^2\nonumber\\
&=& \int\,\frac{dk}{k}\,{\cal P}_{\delta\chi}(k),
\eea
which defines the power spectrum of the
fluctuations of the scalar field $\chi$
\be
{\cal P}_{\delta\chi}(k)\equiv\frac{k^3}{2\pi^2}
\,\left|\delta\chi_{\bf k}\right|^2.
\label{spectrum}
\ee
Since we have seen that fluctuations are (nearly) 
frozen in on superhorizon scales,
a way of characterizing the perturbations is to compute
the spectrum on scales larger than the horizon. For a massive scalar 
field, we obtain
\be
{\cal P}_{\delta\chi}(k)=\left(\frac{H}{2\pi}\right)^2
\left(\frac{k}{aH}\right)^{3-2\nu_\chi}.
\label{fff}
\ee
We may also define the {\it spectral index} $n_{\delta\chi}$ 
of the fluctuations
as
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
n_{\delta\chi}-1=
\frac{d {\rm ln} \,{\cal P}_{\delta\phi}}{d {\rm ln} \,k}=3-2\nu_\chi=
2\eta_\chi-2\epsilon.
$$
\\
\hline
\end{tabular}
\end{center}
The power spectrum of fluctuations of the scalar
field $\chi$  is therefore
{\it nearly flat}, that is is nearly independent from the wavelength
$\lambda=\pi/k$: the amplitude of the 
fluctuation on superhorizon scales does not (almost) depend upon the 
time at which the fluctuations crosses the horizon and becomes frozen
in. The small tilt of the power spectrum arises from the fact that
the scalar field $\chi$ is massive and because 
during inflation the Hubble rate is not exactly constant, but
nearly constant, where `nearly' is quantified by the slow-roll
parameters $\epsilon$. Adopting the  traditional 
terminology,
we may say that the spectrum of perturbations is blue if 
$n_{\delta\chi}>1$
(more power in the ultraviolet)
and red if $n_{\delta\chi}<1$ (more power in the infrared).
The power spectrum of the perturbations
of a generic scalar field $\chi$ generated during a period
of slow roll inflation may be either blue or red. This
depends upon the relative magnitude between $\eta_\chi$ and $\epsilon$.
For instance, in chaotic inflation with a quadric potential
$V(\phi)=\frac{m_\phi^2\phi^2}{2}$, one can easily
compute 

$$
n_{\delta\chi}-1=
2\eta_\chi-2\epsilon=\frac{2}{3H^2}\left(m_\chi^2-m_\phi^2\right),
$$
which tells us that the spectrum is blue (red) if $m_\chi^2>m_\phi^2$
($m_\chi^2>m_\phi^2$). 

\vskip 0.2cm

{\it \underline{Comment}:} We might have computed the
spectral index of the spectrum ${\cal P}_{\delta\chi}(k)$ by first
solving the equation for the perturbations of the field $\chi$
in a di Sitter stage, with $H=$ constant and therefore $\epsilon=0$, and
then taking into account the time-evolution of the Hubble rate
introducing  the subscript in $H_k$ whose time variation is determined 
by  Eq. (\ref{z}). Correspondingly, $H_k$ 
is the value of the Hubble rate  when a given wavelength $\sim 
k^{-1}$ crosses
the horizon (from that point on the fluctuations remains
frozen in). The power spectrum in such an approach would read

\be
{\cal P}_{\delta\chi}(k)=\left(\frac{H_k}{2\pi}\right)^2
\left(\frac{k}{aH}\right)^{3-2\nu_\chi}
\label{bb}
\ee
with $3-2\nu_\chi\simeq \eta_\chi$. Using Eq. (\ref{z}), one finds

$$
n_{\delta\chi}-1=
\frac{d {\rm ln} \,{\cal P}_{\delta\phi}}{d {\rm ln} \,k}=
\frac{d {\rm ln} \,H_k^2}{d {\rm ln} \, k}+3-2\nu_\chi=
2\eta_\chi-2\epsilon
$$
which 
reproduces our previous findings.

\vskip 0.2cm   

{\it \underline{Comment}:} Since on superhorizon scales 

$$
\delta\chi_{\bf k}\simeq \frac{H}{\sqrt{2k^3}}
\left(\frac{k}{aH}\right)^{\eta_\chi-\epsilon}\simeq
\frac{H}{\sqrt{2k^3}}\left[1+\left(\eta_\chi-\epsilon\right)
{\rm ln}\,\left(\frac{k}{aH}\right)\right],
$$
we discover that 

\begin{equation}
\label{e}
\left|\delta\dot{\chi}_{\bf k}\right|\simeq \left|
H\left(\eta_\chi-\epsilon\right)
\,\delta\chi_{\bf k}\right|\ll \left|H\,\delta\chi_{\bf k}\right|,
\end{equation}
that is on superhorizon scales the time variation of the 
perturbations can be safely neglected.


\section{Quantum fluctuations during inflation}

As we have mentioned in the previous section, the linear theory of the
cosmological perturbations represent a cornerstone of modern cosmology
and is used to describe the formation and evolution of structures
in the universe as well as the anisotrpies of the CMB. The seeds
for these inhomegeneities were generated during inflation and
stretched over astronomical scales because of the rapid superluminal
expansion of the universe during the (quasi) de Sitter epoch.


In the previous section we have already seen
that pertubations of a generic scalar field $\chi$ are generated
during a (quasi) de Sitter expansion. The inflaton
field is a scalar field and, as such, we conclude that 
inflaton fluctuations will be generated as well. However, 
the inflaton is special from the point of view
of perturbations. The reason is very simple. By assumption, the
inflaton field dominates the energy density of the universe during
inflation. Any perturbation in the inflaton field means a perturbation
of the stress energy-momentum tensor

$$
\delta\phi\Longrightarrow \delta T_{\mu\nu}.
$$
A perturbation in the stress energy-momentum tensor implies,
through Einstein's equations of motion, a perturbation of the metric

$$
\delta T_{\mu\nu}\Longrightarrow \left[
\delta R_{\mu\nu}-\frac{1}{2}\delta\left(g_{\mu\nu}R\right)\right]
=8\pi G\delta T_{\mu\nu}\Longrightarrow \delta g_{\mu\nu}.
$$
On the other hand, a pertubation of the metric induces a backreaction
on the evolution of the inflaton perturbation through the 
perturbed Klein-Gordon equation of the inflaton field

$$
\delta g_{\mu\nu}\Longrightarrow \delta\left(\partial_\mu\partial^\mu\phi+
\frac{\partial V}{\partial\phi}\right)=0\Longrightarrow\delta\phi.
$$
This logic chain makes us conclude that the perturbations of the
inflaton field and of the metric are tightly coupled to each other
and have to be studied together


\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
\delta\phi\Longleftrightarrow\delta g_{\mu\nu}
$$
%\\
\\
\hline
\end{tabular}
\end{center}
As we will see shortly, this
relation is stronger than one might thought because of the issue
of gauge invariance.





Before launching ourselves into the problem of finding the evolution
of the quantum perturbations of  the inflaton field when they are coupled 
to gravity, 
let us give  a heuristic 
explanation of why we expect that during inflation such fluctuations are
indeed present.

If we take Eq. (\ref{nabla}) and split the  inflaton field as
its classical value $\phi_0$ plus the quantum flucutation $\delta\phi$,  
$\phi({\bf x},t)=\phi_{0}(t)+\delta\phi({\bf x},t)$, the quantum perturbation
$\delta\phi$ satisfies the equation of motion
\begin{equation}
\delta\ddot{\phi}+3H\,\delta{\dot\phi}-\frac{\nabla^2\delta\phi}{a^2}
+V''\,
\delta\phi=0.
\label{aa}
\end{equation}
Differentiating  Eq. (\ref{poi}) wrt time and taking $H$ constant
(de Sitter expansion)  we find
\begin{equation}
({\phi}_0)^{\cdot\cdot\cdot}+3H\ddot{\phi}_0 +V''\,\dot{\phi}_0=0.
\end{equation}
Let us consider for simplicity the limit ${\bf k}^2/a^2\ll 1$ 
and let us disregard
the gradient term. Under this condition we see that
$\dot{\phi}_0$ and $\delta\phi$ solve the same equation. The solutions
have therefore to be related to each other by a constant of proportionality
which depends upon time , that is 

\be
\label{old}
\delta\phi=-\dot{\phi}_0\,\delta t({\bf x}).
\ee

This tells us that $\phi({\bf x},t)$ will have the form
$$
\phi({\bf x},t)=\phi_0\left({\bf x},t-\delta t({\bf x})\right).
$$

This equation indicates that the inflaton field does not acquire
the same value at a given time $t$ in all the space. On the contrary,
when the inflaton field is rolling down its potential, it acquires
different values from one spatial point ${\bf x}$ to the other. The inflaton
field is not homogeneous and fluctuations are present. These fluctuations, 
in turn, will induce fluctuations in the metric.



\subsection{The metric fluctuations}


The mathematical tool do describe the linear evolution
of the cosmological perturbations is obtained  by 
perturbing at  the first-order the 
FRW metric $g^{(0)}_{\mu\nu}$, see Eq. (\ref{metric})

\begin{equation}
g_{\mu\nu}\quad = \quad  g^{(0)}_{\mu\nu}(t) \,+\, 
g_{\mu\nu}(\mathbf{x},t)\,; \qquad  g_{\mu\nu} \,\ll
\,g^{(0)}_{\mu\nu}\,.
\end{equation}
The metric perturbations can be decomposed according to their spin
with respect to a local rotation of the spatial
coordinates on hypersurfaces of constant time. This leads to

\begin{itemize}

\item {\it scalar perturbations}

\item {\it vector perturbations}

\item{\it tensor perturbations}

\end{itemize}

Tensor perturbations or gravitational waves 
have spin 2 and are the ``true'' degrees of 
freedom of the gravitational fields  in the sense that they can
exist even in the vacuum. Vector perturbations are spin 1 modes arising from
rotational velocity fields and are also called vorticity modes. Finally,
scalar perturbations have spin 0. 

Let us make  a simple exercise to count how many scalar degrees of freedom
are present. Take a spacetime of dimensions $D=n+1$, of which $n$
coordinates are spatial coordinates. The symmetric metric tensor $g_{\mu\nu}$
has $\frac{1}{2}(n+2)(n+1)$ degrees of freedom. We can perform $(n+1)$
coordinate transformations in order to eliminate $(n+1)$ degrees of freedom,
this leaves us with $\frac{1}{2}n(n+1)$ degrees of freedom.
These $\frac{1}{2}n(n+1)$ degrees of freedom contain scalar, vector
and tensor modes. According to Helmholtz's theorem we can always decompose
a vector $u_i$ $(i=1,\cdots,n)$ as $u_i=\partial_i v +v_i$, where
$v$ is a scalar (usually called potential flow) which is curl-free,
$v_{[i,j]}=0$, 
and $v_i$ is a real vector (usually called vorticity) which is divergence-free,
$\nabla\cdot v=0$. This means that the real vector (vorticity) modes
are $(n-1)$. Furthermore, a generic traceless tensor $\Pi_{ij}$
can always be decomposed as $\Pi_{ij} =\Pi^S_{ij}+\Pi_{ij}^V+
\Pi_{ij}^T$, where $\Pi^S_{ij}=\left(-\frac{k_i k_j}{k^2}+
\frac{1}{3}\delta_{ij}\right)\Pi$, $\Pi^V_{ij}=(-i/2k)\left(k_i\Pi_j
+k_j\Pi_i\right)$ $(k_i\Pi_i=0)$  and $k_i\Pi^T_{ij}=0$. This means that the
true symmetric, traceless and transverse tensor degreees of freedom
are $\frac{1}{2}(n-2)(n+1)$. 

The number of
scalar degrees of freedom are therefore

$$
\frac{1}{2}n(n+1)-(n-1)-\frac{1}{2}(n-2)(n+1)=2,
$$
while the degrees of freedom  
of true vector modes are $(n-1)$ and the number of degrees of freedom of 
true
tensor modes (gravitational waves) are $\frac{1}{2}(n-2)(n+1)$. In four
dimensions $n=3$, meaning that one expects 2 scalar degrees of freedom,
2 vector degrees of freedom and 2 tensor degrees of freedom.
As we shall see, to the 2 scalar degrees of freedom from the 
metric, one has to add an another
one, the inflaton field perturbation $\delta\phi$. However, since
Einstein's equations will tell us that the two scalar degrees of freedom
from the metric are equal during inflation, we expect a total number
of scalar degrees of freedom equal to 2.

At the linear order, the scalar, vector and tensor perturbations evolve
independently (they decouple) and it is therefore possible to analyze
them separately. Vector perturbations are not excited
during inflation because there are no rotational velocity fields
during the inflationary stage. We will analyze the generation
of tensor modes (gravitational waves) in the following. For the
time being we want to focus on the scalar degrees of freedom of the metric.



Considering only the scalar degrees of freedom of the perturbed
metric, the most generic perturbed metric reads

\begin{equation}
g_{\mu\nu}\,=\, a^2 \left(
\begin{array}{c c}
- 1 \,-\, 2\,A & \partial_i B \\
\partial_i B & \left( 1 \,-\, 2\,\psi\right)\delta_{ij} \,+\, D_{ij}
E \\
\end{array}
\right),
\end{equation}
while the line-element can be written as 
\begin{equation}
ds^2 \,=\, a^2 \big( ( - 1 - 2\,A)d\tau^2 \,+\, 2 \,\partial_i B
\,d\tau\,dx^i \,+\, \left((1 - 2\,\psi)\delta_{ij} \,+\,
D_{ij}E\right) \,dx^i\,dx^j \big).
\end{equation}
Here $D_{ij}\,=\left(\partial_i
\partial_j \,-\, \frac{1}{3}\,\delta_{ij}\,\nabla^2\right)$.

We now want to determine the inverse $g^{\mu\nu}$ of the metric
at the linear order 
\begin{equation}
\label{inv} g^{\mu\alpha}\,g_{\alpha\nu}\,=\,
\delta^{\mu}_{\nu}.
\end{equation}
We have therefore to solve the equations
\begin{equation}
\label{metricper}
 \left( g^{\mu\alpha}_{(0)} \,+\, 
{g^{\mu\alpha}} \right)\left( g^{(0)}_{\alpha\nu} \,+\, 
{g_{\alpha\nu}} \right) \,=\, \delta^{\mu}_{\nu}\,,
\end{equation}
where $g^{\mu\alpha}_{(0)}$ is simply the unperturbed FRW metric
(\ref{metric}). 
Since 
\begin{equation}
g_{(0)}^{\mu\nu}\,=\, \frac{1}{a^2}\left(
\begin{array}{c c}
-1 & 0 \\
0 & \delta^{ij}
\end{array}
\right),
\end{equation}
we can write in general
\begin{eqnarray}
g^{00}\,&=& \,\frac{1}{a^2} \left( \,-1 \,+\, X \right)\,;\nonumber \\
 g^{0i}\,&=& \,\frac{1}{a^2} \,\partial^i Y \,;\nonumber\\
g^{ij}\,&=& \,\frac{1}{a^2}\,\left( \left( 1 \,+\, 2\,Z
\right)\delta^{ij} \,+
 \,D^{ij} K \right).
 \end{eqnarray}
Plugging these expressions into 
Eq. (\ref{metricper}) we find for   $\mu = \nu = 0$ 
\begin{equation}
( - 1 \,+\, X )( -1 \,- \,2\,A ) \,+\, \partial^i Y \,\partial_i B
\,=\, 1.
\end{equation}
Neglecting the terms $ - \,2\,A \cdot X$ e $\partial^i Y \cdot
\partial_i B$ because they are second-order in the 
perturbations, we find
\begin{equation}
1 \,-\,X \,+\, 2\,A \,=\, 1 \qquad \Rightarrow \qquad X\,=\,
2\,A\,.
\end{equation}
Analogously, 
the components $\mu = 0 ,\, \nu = i$ of Eq. 
(\ref{metricper}) give
\begin{equation}
( - 1 \,+\,2\,A)( \partial_i B ) \,+\, \partial^j Y \left[ ( 1
\,-\, 2\,\psi ) \delta_{ji} \,+\, D_{ji} E \right] \,=\, 0.
\end{equation}
At the first-order, we obtain 
\begin{equation}
- \partial_i B \,+\, \partial_i Y \,=\, 0 \qquad \Rightarrow
\qquad Y = B \,.
\end{equation}
Finally, the components
$\mu = i$, \, $\nu = j$ give
\begin{equation}
\partial^i B \,\partial_j B \,+\, \left( ( 1 \,+\,2\,Z )\delta^{ik}
\,+\, D^{ik} K \right)\,\left( ( 1 - 2\,\psi) \delta_{kj} \,+\,
D_{kj} E \right) \,=\, \delta^i_j.
\end{equation}
Neglecting the second-order terms, we obtain
\begin{equation}
 ( 1 \,-\, 2\,\psi \,+\, 2\,Z )\delta^i_j \,+\, D^i_j E \,+\,
D^i_j K \,=\, \delta^i_j 
 \Rightarrow Z \,=\,\psi \,; \qquad K\,= \,-\,E\,.
\end{equation}
The metric $g^{\mu\nu}$ finally reads
\begin{equation}
g^{\mu\nu}\,=\, \frac{1}{a^2} \left(
\begin{array}{c c}
-1 \,+\, 2\,A & \partial^i B \\
\partial^i B & ( 1 \,+\, 2\,\psi )\delta^{ij} \,-\,D^{ij}E
\end{array}
\right).
\end{equation}





\subsection{Perturbed affine connections and Einstein's tensor}

In this subsection we provide the reader with the perturbed affine connections
and Einstein's tensor. 

First, let us list the unperturbed affine connections
\begin{equation}
\Gamma^0_{00}\,=\, \Aa \,; \qquad
\Gamma^i_{0j}\,=\,\Aa\,\delta^i_j\,; \qquad
\Gamma^0_{ij}\,=\,\Aa\,\delta_{ij}\,;
\end{equation}
\begin{equation}
\Gamma^i_{00}\quad=\quad\Gamma^0_{0i}\quad=\quad\Gamma^i_{jk}\quad=\quad
0 \,.
\end{equation}
The expression for the affine connections in terms
of the metric is 
\begin{equation}
\label{conness} \Gamma^\alpha_{\beta\gamma}\,=\,
\frac{1}{2}\,g^{\alpha\rho}\left( \frac{\partial
g_{\rho\gamma}}{\partial x^{\beta}} \,+\, \frac{\partial
g_{\beta\rho}}{\partial x^{\gamma}} \,-\, \frac{\partial
g_{\beta\gamma}}{\partial x^{\rho}}\right)
\end{equation}
which implies
\begin{eqnarray}
\deu{\Gamma^\alpha_{\beta\gamma}}& \,=\ &
\frac{1}{2}\,\deu{g^{\alpha\rho}}\left( \frac{\partial
g_{\rho\gamma}}{\partial x^{\beta}} \,+\, \frac{\partial
g_{\beta\rho}}{\partial x^{\gamma}} \,-\, \frac{\partial
g_{\beta\gamma}}{\partial x^{\rho}}\right) \nonumber\\
& + &\frac{1}{2}\,g^{\alpha\rho}\left( \frac{\partial
\deu{g_{\rho\gamma}}}{\partial x^{\beta}} \,+\, \frac{\partial
\deu{g_{\beta\rho}}}{\partial x^{\gamma}} \,-\, \frac{\partial
\deu{g_{\beta\gamma}}}{\partial x^{\rho}}\right),
\end{eqnarray}
or in components
\begin{equation}
\deu{\Gamma^0_{00}} \,=\, A^{\prime} \ ;
\end{equation}
\begin{equation}
\deu{\Gamma^0_{0i}} \,=\, \partial_i\, A \,+\,
\frac{a^{\prime}}{a}\partial_i\,B \, ;
\end{equation}
\begin{equation}
\deu{\Gamma^i_{00}} \,=\, \frac{a^{\prime}}{a}\,\partial^i B \,+\,
\partial^i B^{\prime} \,+\, \partial^i A \, ;
\end{equation}
\begin{eqnarray}
\deu{\Gamma^0_{ij}}  &\,=\,&
-\,2\,\frac{a^{\prime}}{a}\,A\,\delta_{ij} \,-\,
\partial_i \partial_j B \,-\,
2\,\frac{a^{\prime}}{a}\,\psi\,\delta_{ij} \nonumber\\
& -\, \psi^{\prime}\,\delta_{ij} \,-\,
\frac{a^{\prime}}{a}\,D_{ij} E \,+\, \frac{1}{2}\,D_{ij}E^{\prime}
\, ;
\end{eqnarray}
\begin{equation}
\deu{\Gamma^i_{0j}} \,=\, - \,\psi^{\prime}\delta_{ij} \,+\,
\frac{1}{2}\,D_{ij}E^{\prime} \ ;
\end{equation}
\begin{eqnarray}
\deu{\Gamma^i_{jk}} &\,=\,& \partial_j \psi \,\delta_k^i \,-\,
\partial_k \psi\, \delta_j^i \,+\, \partial^i \psi \,\delta_{jk}
\,-\, \frac{a^{\prime}}{a}\,\partial^i B
\,\delta_{jk}\nonumber\\
   & +&\, \frac{1}{2}\,\partial_j D^i_k E \,+\, \frac{1}{2}\,\partial_k D^i_j E
  \,-\,
  \frac{1}{2}\,\partial^i D_{jk} E \, .
\end{eqnarray}\\
We may now compite the Ricci scalar
defines as 
\begin{equation}
R_{\mu\nu}\,=\, \partial_\alpha\,\Gamma^\alpha_{\mu\nu} \,-\,
\partial_{\mu}\,\Gamma^\alpha_{\nu\alpha} \,+\,
\Gamma^\alpha_{\sigma\alpha}\,\Gamma^\sigma_{\mu\nu} \,-\,
\Gamma^\alpha_{\sigma\nu} \,\Gamma^\sigma_{\mu\alpha}\,.
\end{equation}
Its variation at the first-order reads
\begin{eqnarray}
\deu{R_{\mu\nu}}\,&
=&\,\partial_\alpha\,\deu{\Gamma^\alpha_{\mu\nu}} \,-\,
\partial_{\mu}\,\deu{\Gamma^\alpha_{\nu\alpha}} \,+\,
\deu{\Gamma^\alpha_{\sigma\alpha}}\,\Gamma^\sigma_{\mu\nu}
\,+\,\Gamma^\alpha_{\sigma\alpha}\,\deu{\Gamma^\sigma_{\mu\nu}} \nonumber\\
&  -&\, \deu{\Gamma^\alpha_{\sigma\nu}}
\,\Gamma^\sigma_{\mu\alpha}\,-\,\Gamma^\alpha_{\sigma\nu}
\,\deu{\Gamma^\sigma_{\mu\alpha}}\,.
\end{eqnarray}
The background values are given by 
\begin{equation}
R_{00}\,=\,-\,3\,\Ac \,+\,3\,\Ab \,; \qquad R_{0i}\,=\,0\,;
\end{equation}
\begin{equation}
R_{ij}\,=\,\Big( \Ac \,+\,\Ab \Big)\,\delta_{ij}\,
\end{equation}
which give
\begin{equation}
\deu {R_{00}} = \frac{a^{\prime}}{a}\partial_i\partial^i B +
\partial_i\partial^i B^{\prime} + \partial_i\partial^i A +
3\psi^{\prime\prime} + 3\frac{a^{\prime}}{a}\psi^{\prime} +
3\frac{a^{\prime}}{a}A^{\prime} \ ;
\end{equation}
\begin{equation}
\deu {R_{0i}} = \frac{a^{\prime\prime}}{a}\partial_i B +
\left(\frac{a^{\prime}}{a}\right)^2\partial_i B +
2\partial_i\psi^{\prime} + 2\frac{a^{\prime}}{a}\partial_i A +
\frac{1}{2}\partial_k D^k_i E^{\prime} \ ;
\end{equation}
\begin{eqnarray}
\deu {R_{ij}} &=& \Big(-\frac{a^{\prime}}{a}A^{\prime} -
5\frac{a^{\prime}}{a}\psi^{\prime} - 2\frac{a^{\prime\prime}}{a}A
-2\left(\frac{a^{\prime}}{a}\right)^2A \nonumber\\
& -&2\frac{a^{\prime\prime}}{a}\psi -
2\left(\frac{a^{\prime}}{a}\right)^2\psi - \psi^{\prime\prime} +
\partial_k\partial^k\psi -
\frac{a^{\prime}}{a}\partial_k\partial^k B \Big) \delta_{ij}\nonumber\\
& -&\partial_i\partial_j B^{\prime} +
\frac{a^{\prime}}{a}D_{ij}E^{\prime} +
\frac{a^{\prime\prime}}{a}D_{ij}E +
\left(\frac{a^{\prime}}{a}\right)^2 D_{ij}E \nonumber\\
& +& \frac{1}{2}D_{ij}E^{\prime\prime} + \partial_i\partial_j \psi
-
\partial_i\partial_j A - 2\frac{a^{\prime}}{a}\partial_i\partial_j
B\nonumber\\
& +& \frac{1}{2}\partial_k\partial_iD^k_j E +
\frac{1}{2}\partial_k\partial_j D^k_i E -
\frac{1}{2}\partial_k\partial^k D_{ij} E \ ;
\end{eqnarray}
The perturbation of the scalar curvature
\begin{equation}
\label{curvat} R \,=\, g^{\mu\alpha}\,R_{\alpha\mu}\,,
\end{equation}
for which the first-order perturbation is
\begin{equation}
\label{r} \deu R \,=\, \deu{g^{\mu\alpha}}\,R_{\alpha\mu} \,+\,
g^{\mu\alpha}\,\deu R_{\alpha\mu}\,.
\end{equation}
The background value is 
\begin{equation}
R \,=\, \frac{6}{a^2}\,\Ac\,
\end{equation}
while from Eq.  (\ref{r}) one finds
\begin{eqnarray}
\deu R &=& \frac{1}{a^2} \Big( 
-6\frac{a^{\prime}}{a}\partial_i\partial^i B -
2\partial_i\partial^i B^{\prime} - 2\partial_i\partial^i A
-6\psi^{\prime\prime}\nonumber\\& -& 6\frac{a^{\prime}}{a}A^{\prime}
-18\frac{a^{\prime}}{a}\psi^{\prime} -
12\frac{a^{\prime\prime}}{a}A +4\partial_i\partial^i\psi +
\partial_k\partial^iD^k_i E \Big).
\end{eqnarray}
Finally, we may compute the 
perturbations of the Einstein tensor
\begin{equation}
G_{\mu\nu} \,=\, R_{\mu\nu} \,-\, \frac{1}{2}\,g_{\mu\nu}\,R\,,
\end{equation}
whose background components are  
\begin{equation}
G_{00}\,=\,3\,\Ab\,; \qquad G_{0i}\,=\,0\,;\qquad G_{ij}\,=\,\Big(
-\,2\,\Ac \,+\, \Ab \Big)\,\delta_{ij}\,.
\end{equation}
At first-order, one finds
\begin{equation}
\deu{G_{\mu\nu}}\,=\, \deu{R_{\mu\nu}} \,-\,
\frac{1}{2}\,\deu{g_{\mu\nu}}\,R \,-\,
\frac{1}{2}\,g_{\mu\nu}\,\deu R \,,
\end{equation}
or in components
\begin{equation}
\deu{G_{00}} =
-2\frac{a^{\prime}}{a}\thinspace\partial_i\partial^i\thinspace B -
6\frac{a^{\prime}}{a}\thinspace\psi^{\prime} +
2\thinspace\partial_i\partial^i\thinspace\psi +
\frac{1}{2}\thinspace\partial_k\partial^i D^k_i E \,;\\
\end{equation}
\\
\begin{equation}
\deu {G_{0i}} =  -2\frac{a^{\prime\prime}}{a}\thinspace\partial_i
B + \left(\frac{a^{\prime}}{a}\right)^2\thinspace\partial_i B +
2\partial_i\thinspace\psi^{\prime} +
\frac{1}{2}\partial_k\thinspace D^k_i E^{\prime} +
2\thinspace\frac{a^{\prime}}{a}\thinspace\partial_i A \,; \\
\end{equation}
\\
\begin{eqnarray}
\deu {G_{ij}} &= & \Bigg( 2\frac{a^{\prime}}{a}\thinspace
A^{\prime} + 4\frac{a^{\prime}}{a}\thinspace \psi^{\prime} +
4\frac{a^{\prime\prime}}{a}\thinspace A
-2\left(\frac{a^{\prime}}{a}\thinspace\right)^2 A \nonumber\\
& +& 4\frac{a^{\prime\prime}}{a}\thinspace\psi
-2\left(\frac{a^{\prime}}{a}\thinspace\right)^2 \psi +
2\psi^{\prime\prime} - \partial_k\partial^k\thinspace\psi \nonumber\\
& + &2\frac{a^{\prime}}{a}\thinspace\partial_k\partial^k B +
\partial_k\partial^k B^{\prime} + \partial_k\partial^k A
+ \frac{1}{2}\partial_k\partial^m\thinspace D^k_m\, E\Bigg)\delta_{ij}
\nonumber\\
& -&\partial_i\partial_j B^{\prime} +\partial_i\partial_j \psi
 - \partial_i\partial_j A + \frac{a^{\prime}}{a}\thinspace
 D_{ij}E^{\prime}
 - 2\,\frac{a^{\prime\prime}}{a}\thinspace D_{ij}E\nonumber\\
 & + &\left(\frac{a^{\prime}}{a}\thinspace\right)^2 D_{ij}E +
\frac{1}{2}D_{ij}E^{\prime\prime}+ \frac{1}{2}\partial_k\partial_i
D^k_j E\nonumber\\
&+& \frac{1}{2}\partial^k\partial_j D_{ik}E
-\frac{1}{2}\partial_k\partial^k D_{ij}E - 2 \,\Aa
\partial_i\partial_j B \,.
\end{eqnarray}
For convenience, we also give the expressions for the pertubations
with one index up and one index down

\begin{eqnarray}
\deu {G^\mu_\nu}\,& =&\,\deu{(g^{\mu\alpha \, G_{\alpha\nu}})} \,
\nonumber\\
&=&
\deu{g^{\mu\alpha}}\,G_{\alpha\nu}\,+\,g^{\mu\alpha}\,\deu{G_{\alpha\nu}}\,,
\end{eqnarray}
or in components
\begin{equation}
 \deu {G_0^0} \,=\,6\,\Ab A \,+\,6\,\Aa\psi^{\prime}\,+\, 2\,\Aa \La
 B\,-\, 2\,\La \psi \,-\, \frac{1}{2}\,\partial_k \partial^i
 \,D^k_i E \,.
\label{00}
 \end{equation}
 \begin{equation}
 \deu {G^0_i} \,=\, -\,2\,\Aa \partial_i A \,-\, 2\,\partial_i
 \psi^{\prime} \,-\, \frac{1}{2}\,\partial_k D^k_i E^{\prime} \,.
\label{0i} 
\end{equation}
  \begin{eqnarray}
\deu {G^i_j} &=& \Bigg( 2\,\Aa A^{\prime} \,+\, 4\,\Ac A \,-\,
2\,\Ab A \,+\, \La A \,+\, 4\,\Aa \psi^{\prime} 
\,+\, 2\,\psi^{\prime\prime}  \nonumber\\
& -&\, \La \psi \,+\, 2\,\Aa \La B \,+\, \La B^{\prime} \,+\,
\frac{1}{2}\partial_k \partial^m D^k_m
E \Bigg)\delta_j^i \nonumber\\
& -& \partial^i\partial_j A \,+\, \partial^i\partial_j \psi \,-\,
2\,\Aa \partial^i\partial_j B \,-\, \partial^i\partial_j
B^{\prime} \,+\, \Aa D^i_j E^{\prime} \,+\, \frac{1}{2}\,D^i_j
E^{\prime\prime}
\nonumber\\
& +&\, \frac{1}{2}\,\partial_k\partial^i \,D^k_j E \,+\,
\frac{1}{2}\,\partial_k\partial_j \,D^{ik} E \,-\,
\frac{1}{2}\,\partial_k\partial^k \,D^i_j E \,. 
\label{ij}
\end{eqnarray}


\subsection{Perturbed  stress energy-momentum
tensor}

As we have seen previously, the perturbations of the metric
are induced by the perturbations of the  stress energy-momentum
tensor of the inflaton field
\begin{equation}
T_{\mu\nu}\,=\,\partial_\mu \phi\,\partial_\nu \phi \,-\,
g_{\mu\nu}\left( \frac{1}{2}\,g^{\alpha\beta}\,\partial_\alpha
\phi\,\partial_\beta \phi \,-\, V(\phi)\right)\,,
\end{equation}
whose background values are 
\begin{eqnarray}
 T_{00}&=&\frac{1}{2}\,{\phi^{\prime}}^2 \,+\, V(\phi)\,a^2\,\nonumber\\
 T_{0i}&=&\,0\,; \nonumber\\
T_{ij}&=&\left( \frac{1}{2}\,{\phi^{\prime}}^2 \,-\,
V(\phi)\,a^2\right)\,\delta_{ij}\,.
\end{eqnarray}
The perturbed stress energy-momentum tensor reads
\begin{eqnarray}
\deu{T_{\mu\nu}} \,&=&\,\partial_\mu \deu\phi \,\partial_\nu \phi
\,+\,\partial_\mu \phi \,\partial_\nu \deu\phi \,-\,
\deu{g_{\mu\nu}}\left(
\frac{1}{2}\,g^{\alpha\beta}\,\partial_\alpha \phi\,\partial_\beta
\phi \,+\, V(\phi)\right)\nonumber\\
& -&\,g_{\mu\nu}\left(
\frac{1}{2}\deu{g^{\alpha\beta}}\,\partial_\alpha
\phi\,\partial_\beta \phi \,+\, g^{\alpha\beta}\,\partial_\alpha
\deu\phi\,\partial_\beta \phi \,+\, \frac{\partial V}{\partial
\phi}\,\deu\phi \,+\, \frac{\partial V}{\partial
\phi}\deu{\phi}\right)\,.
\end{eqnarray}
In components 
we have
\begin{equation}
\deu {T_{00}} \,=\, \deu{\phi^{\prime}}\thinspace\phi^{\prime}
\,+\, 2\,A\,V(\phi)\,a^2 \,+\, a^2\thinspace\frac{\partial
V}{\partial \phi}\thinspace\deu\phi \, ;
\end{equation}
\begin{equation}
\deu T_{0i} \,=\, \partial_i \thinspace\deu\phi \, \phi^{\prime}
\,+\, \frac{1}{2}\,\partial_i B \,{\phi^{\prime}}^2
 \,-\, \partial_i B \,V(\phi)\,a^2 \,;
\end{equation}
\begin{eqnarray}
\deu T_{ij}  \,&=&\,  \left(\deu{\phi^{\prime}}\,\phi^{\prime}
\,-\, A\thinspace{\phi^{\prime}}^2 \,-\,
a^2\thinspace\frac{\partial
V}{\partial\phi}\thinspace\delta^{(1)}\phi
 \,-\, \psi \,{\phi^{\prime}}^2 \,+\, 2\,\psi\,V(\phi)\,a^2 
\right)\delta_{ij}
 \nonumber\\
\,&+&\, \frac{1}{2}\,D_{ij}E\,{\phi^{\prime}}^2 \,-\,
 D_{ij}E\,V(\phi)\,a^2
 \ .
\end{eqnarray}
For covenience, we list the mixed components
\begin{eqnarray}
\deu{T^\mu_\nu}\,& =&\, \deu{(g^{\mu\alpha}\,T_{\alpha\nu})} \nonumber\\
& =&\, \deu{g^{\mu\alpha}}\,T_{\alpha\nu} \,+\,
g^{\mu\alpha}\,\deu{T_{\alpha\nu}}
\end{eqnarray}
or
\begin{eqnarray}
  \deu {T^0_0} &=& A\,{\phi^{\prime}}^2 \,-\,
  \deu{\phi^{\prime}}\,\phi^{\prime} 
\,-\, \deu \phi\, \frac{\partial V}{\partial
  \phi}\,a^2 \,;\nonumber\\
  \deu {T^i_0} &=& \partial^i B\,{\phi^{\prime}}^2 \,+\,
  \partial^i \deu\phi\,\phi^{\prime} \,; \nonumber\\
  \deu {T^0_i} &=& -\,\partial^i \deu{\phi}\,\phi^{\prime} \,;
\nonumber\\
  \deu {T^i_j} &=& \left( -\, A\,{\phi^{\prime}}^2 \,+\, \deu{
  \phi^{\prime}}\,\phi^{\prime} \,-\, \deu\phi\, \frac{\partial V}{\partial
  \phi}\,a^2 \right) \delta^i_j \,. 
  \end{eqnarray}

\subsection{Perturbed Klein-Gordon equation}

The inflaton equation of motion is the Klein-Gordon equation of
a scalar field under the action of its potential $V(\phi)$.
The equation to
perturb is therefore 
\begin{eqnarray}
\label{kg}
 \partial^\mu\partial_\mu
 \,\phi \,&=&\, \frac{\partial V}{\partial \phi}\,; \nonumber\\
 \partial_\mu\partial^\mu\phi \,&=&\,
 \frac{1}{\sqrt {- g}}\,\partial_\nu \left( \sqrt{-
g}\,g^{\mu\nu}\,\partial_\nu \phi\right)\,;
\end{eqnarray}
which at the zero-th order gives the inflaton equation
of motion 
\begin{equation}
\label{kg_espl} \phi^{\prime\prime}\,+\,
2\,\Aa\,\phi^{\prime}\,=\, -\,\frac{\partial V}{\partial
\phi}\,a^2 \,.
\end{equation}
The variation of Eq. (\ref{kg}) is the sum of four different
contributions corresponding to the variations of
$\frac{1}{\sqrt{- g}}$, $\sqrt{- g}$, $g^{\mu\nu}$ and $\phi$.
For the variation of $g$ we have 
\begin{equation}
\label{gi} \deu g \,=\,g\,g^{\mu\nu}\deu{g_{\nu\mu}}\,=\, dg\,=\,
g^{\mu\nu}dg_{\mu\nu}
\end{equation}
which give at the linear order
\begin{eqnarray}
 \deu {\sqrt{-g}}\,&=&\, -\, \frac{\deu g}{2\,\sqrt{- g}}\,;\nonumber \\
 \deu {\frac{1}{\sqrt{- g}}} \,&=&\, \frac{\deu{\sqrt{- g}}}{g}\,.
\end{eqnarray}
Plugging these results into 
the expression for the variation of 
Eq. (\ref{kg_espl}) 
\begin{eqnarray}
\label{mongo}
\deu \partial_\mu\partial^\mu
\,\phi \,& =& \,-\,\deu{\phi^{\prime\prime}} \,-\, 2\,\Aa
{\deu \phi}^{\prime} \,+\, \La \deu \phi \,+\,
2\,A\,\phi^{\prime\prime}
\,+\, 4\,\Aa A\,\phi^{\prime} \,+\, A^{\prime}\phi^{\prime} \nonumber\\
& +&\, 3\,\psi^{\prime}\phi^{\prime} \,+\, \La B \,\phi^{\prime} \,\nonumber
\\
& =&\,\deu \phi \,\frac{\partial^2 V}{\partial \phi^2}\,a^2 \,.
\end{eqnarray}
Using Eq.  (\ref{kg_espl}) to write 
\begin{equation}
2\,A\,\phi^{\prime\prime}\,+\, 4 \,\Aa \phi^{\prime}\,=\,
2\,A\,\frac{\partial V}{\partial \phi}\,,
\end{equation}
Eq. (\ref{mongo}) becomes
\begin{eqnarray}
\label{kgg}
\,{\deu\phi}^{\prime\prime} \,+\, 2\,\Aa {\deu \phi}^{\prime}
\,&-&\, \La \deu \phi \,-\, A^{\prime}\phi^{\prime} \,-\,
 3\,\psi^{\prime}\phi^{\prime} \,-\, \La B \,\phi^{\prime} \,\nonumber\\
& =&\,-\deu \phi \,\frac{\partial^2 V}{\partial \phi^2}\,a^2 \,-\,
2\,A\,\frac{\partial V}{\partial \phi}.
\end{eqnarray}
After having computed the perturbations at the linear order
of the Einstein's tensor and of the stress energy-momentum
tensor, we are ready to solve the perturbed Einstein's equations
in order to quantify the inflaton and the metric fluctuations.
We pause, however, for a moment in order to deal with the problem
of gauge invariance.

\subsection{The issue of gauge invariance}

When studying the cosmological density perturbations, 
what we are interested in is following the evolution of a spacetime which
is neither homogeneous nor isotropic.  This is done by following
the evolution of the differences between the actual spacetime and a
well understood reference spacetime.  So we will consider small
perturbations away from the homogeneous, isotropic spacetime (see
Fig.\ \ref{tau}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{\epsfxsize=3.5in \epsfbox{fig3.eps}}
\caption{In the reference unperturbed universe, constant-time
surfaces have constant spatial curvature (zero for a flat FRW model).
In the actual perturbed universe, constant-time surfaces have
spatially varying spatial curvature. From Ref. \cite{kolbreview}.
\label{tau}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The reference system in our case is the
spatially flat Friedmann--Robertson--Walker (FRW) spacetime, with line
element $ds^2 = a^2(\tau) \left\{ d\tau^2 - \delta_{ij} dx^idx^j
\right\}$.
Now, the key issue is that general relativity is a gauge theory where
the gauge transformations are the generic coordinate transformations
from a local reference frame to another. 

When we compute the perturbation of a given quantity, this is defined
to be the difference between the value that this quantity assumes on the
real physical  spacetime and the value it assumes on the unperturbed 
background. Nonetheless, to perform a comparison between these two
values, it is necessary to compute the at the same spacetime point.
Since the two values ``live'' on two different geometries, it is necessary to
specify a map which allows to link univocally the
same point on the two different spacetimes. This correspondance is called
a gauge choice and changing the map means performing a gauge transformation.


Fixing a gauge in general relativity implies choosing
a coordinate system. A choice of coordinates 
defines a {\it threading} of spacetime into 
lines (corresponding to fixed spatial coordinates  ${\bf x}$) and a 
{\it slicing} into hypersurfaces (corresponding to fixed time $\tau$).
A choice of coordinates is is called a {\it gauge} and there is no unique
preferred gauge
\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
{\rm GAUGE ~~CHOICE}~~~\Longleftrightarrow~~~{\rm SLICING ~~AND ~~THREADING}
$$
%\\
\\
\hline
\end{tabular}
\end{center}
From a more formal point of view, operating  an infinitesimal 
 gauge tranformation on the coordinates
\begin{equation}
\label{gauge} \widetilde{x^\mu}\,=\, x^\mu \,+\, \delta x^\mu
\end{equation}
implies on a generic quantity $Q$ a tranformation on its
perturbation
\begin{equation}
\widetilde{\deu Q}\,=\, \deu Q \,+\,\pounds_{\delta x} \,Q_0\,
\label{formal}
\end{equation}
where $Q_0$ is the value assumed by the quantity $Q$ on the background
and $\pounds_{\delta x}$ is the Lie-derivative of $Q$ along
the vector $\delta x^\mu$.

Decomposing in the
usual manner the vector $\delta x^\mu$ 
\begin{eqnarray}
 \delta x^0 \,&=&\, \xi^0(x^\mu)\,; \nonumber\\
\delta x^i \,&=&\, \partial^i \beta(x^\mu) \,+\, v^i(x^\mu) 
\,; \qquad \partial_i v^i
\,=\,0\,,
\end{eqnarray}
we can easily deduce the transformation
law of a scalar quantity $f$ (like the inflaton scalar field $\phi$
and energy density $\rho$). Instead of applying the formal definition
(\ref{formal}), we find the transformation law in an alternative (and 
more pedagogical) way.
We first write $\delta f(x)=f(x)-f_0(x)$,
where $f_0(x)$ is the background value. Under a gauge transformation
we have $\widetilde{\delta f}(\widetilde{x^\mu})=
\widetilde{f}(\widetilde{x^\mu})-\widetilde{f}_0(\widetilde{x^\mu})$.
Since $f$ is a scalar we can write $f(\widetilde{x^\mu})=f(x^\mu)$
(the value of the scalar function in a given physical point is
the same in all the coordinate system). On the other side, on the
unperturbed background hypersurface $\widetilde{f}_0=
f_0$. We have therefore

\begin{eqnarray}
\widetilde{\delta f}(\widetilde{x^\mu})&=&
\widetilde{f}(\widetilde{x^\mu})-\widetilde{f}_0(\widetilde{x^\mu})\nonumber\\
&=& f(x^\mu)-f_0(\widetilde{x^\mu})\nonumber\\
&=&f\left(\widetilde{x^\mu}\right)-f_0(\widetilde{x^\mu})\nonumber\\
&=&f(\widetilde{x^\mu})-\delta x^\mu\,\frac{\partial f}{\partial x^\mu}
(\widetilde{x})-f_0(\widetilde{x^\mu}),\nonumber\\
\end{eqnarray}
from which we finally deduce, being $f_0=f_0(x^0)$, 

\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
$$
\widetilde{\delta f}=\delta f-f^\prime\,\xi^0
$$
\\
\hline
\end{tabular}
\end{center}
For the spin zero perturbations of  the metric, we can proceed analogously.
We use the following trick. Upon a coordinate transformation
$x^\mu\rightarrow \widetilde{x^\mu}=x^\mu+\delta x^\mu$, the line
element is left invariant, $ds^2=\widetilde{ds^2}$. This implies, for instance,
that $a^2(\widetilde{x^0})\left(1+\widetilde{A}\right)\left(
d\widetilde{x^0}\right)^2=
a^2(x^0)\left(1+A\right)(dx^0)^2$. Since 
$a^2(\widetilde{x^0})\simeq a^2(x^0)+2 a\, a^\prime\,\xi^0$ and
$d\widetilde{x^0}=\left(1+\xi^{0\prime}\right)dx^0+
\frac{\partial x^0}{\partial x^i}\,d x^i$, we obtain 
$1+2 A=1+2\widetilde{A}+2\H\xi^0+2\xi^{0\prime}$. A similar procedure
leads to the following transformation laws

\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
\begin{eqnarray}
\label{trasf}
\widetilde{A}\,&=&\, A \,-\, \xi^{0\prime} \,-\, \Aa \xi^0\,;\nonumber\\
 \widetilde{B} \,&=&\, B \,+\, \xi^0 \,+\, \beta^{\prime} \nonumber\\
 \widetilde{\psi} \,&=&\, \psi \,-\, \frac{1}{3}\,\nabla^2 \beta \,+\,
\Aa \xi^0 \,;\nonumber\\
  \widetilde{E}\,&=&\, E \,+\, 2\,\beta\,\nonumber.
\end{eqnarray}
\\
\hline
\end{tabular}
\end{center}
The gauge problem stems from the fact that a change of the map (a change
of the coordinate system) implies the variation of the perturbation
of a given quantity which may therefore assume different values (all of
them on a equal footing!) according to the gauge choice. 
To eliminate this ambiguity, one has therefore
a double choice: 

\begin{itemize}

\item Indentify those combinations
representing gauge invariant quantities; 

\item choose a given gauge
and perform the calculations in that gauge. 

\end{itemize}
Both options have  advantages and drawbacks. Choosing a gauge may 
render the computation technically simpler with the danger, however, 
of including gauge artifacts, {\it i.e.} gauge freedoms which are
not physical. Performing a gauge-invariant computation may be
technically more involved, but has the advantage of treating only physical
quantities. 

Let us first indicate some gauge-invariant quantities which have
been introduced first in Ref. \cite{Bardeen}. They are the so-called
gauge invariant potentials or Bardeen's potentials
\begin{equation}
\Phi\,=\, -\, A \,+\, \frac{1}{a}\,\left[ \left( -\, B \,+\,
\frac{E^{\prime}}{2}\right) a \right]^{\prime}\,,
\label{bar1}
\end{equation}
\begin{equation}
\Psi\,=\, -\,\psi \,-\, \frac{1}{6}\,\nabla^2\,E \,+\, \Aa \left(
B \,-\, \frac{E^{\prime}}{2}\right)\,.
\label{bar2}
\end{equation}
Analogously, one can define a gauge invariant quantity for the
perturbation of the inflaton field. Since $\phi$ is a scalar field
$\widetilde{\delta \phi}=\left(\delta \phi-\phi^\prime\,\xi^0\right)
$ and therefore 
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
$$
\deu{\phi^{({\rm GI})}} \,=\,  \,-\deu{\phi} \,+\,
\phi^{\prime}\left(\frac{E^{\prime}}{2} \,-\, B \right)
$$
\\
\hline
\end{tabular}
\end{center}
is gauge-invariant.

Analogously, one can define a gauge-invariant energy-density perturbation

\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
$$
\deu{\rho^{({\rm GI})}} \,=\,  \,-\deu{\rho} \,+\,
\rho^{\prime}\left(\frac{E^{\prime}}{2} \,-\, B \right)
$$
\\
\hline
\end{tabular}
\end{center}







We now want to pause to introduce in details 
some gauge-invariant quantities
which play  a major role in the computation of the
density perturbations. 
In the following we will be interested only in the 
coordinate  transformations on constant time hypersurfaces
and therefore gauge invariance will be equivalent to independent of the 
slicing.

\subsection{The comoving curvature perturbation}

The intrinsic spatial curvature on hypersurfaces on constant
conformal  time $\tau$ and for a flat universe is given by

$$
^{(3)}R=\frac{4}{a^2}\nabla^2\,\psi.
$$
The quantity $\psi$ is usually referred to as the {\it curvature perturbation}.
We have seen, however, that the the curvature potential
$\psi$ is {\it not} gauge invariant, but is defined only on a 
given slicing.
Under a transformation on constant time hypersurfaces
$t\rightarrow t+\delta \tau$ (change of the slicing)

$$
\psi\rightarrow \psi+\H\,\delta\tau.
$$
We now consider the {\it comoving slicing} which is defined to be the
slicing orthogonal to the worldlines of 
comoving observers. The latter are    are free-falling
and the expansion defined by them is isotropic. In practice, what
this means is that there is no flux of energy measured by these
observers, that is $T_{0i}=0$. During inflation this means
that these observers measure $\delta\phi_{\rm com}=0$ since 
$T_{0i}$ goes like $\partial_i\delta\phi({\bf x},\tau)\phi^\prime(\tau)$.

Since $\delta\phi\rightarrow \delta\phi-\phi^\prime\delta\tau$ for a 
transformation on constant time hypersurfaces, this means that

$$
\delta\phi\rightarrow\delta\phi_{\rm com}=\delta\phi-\phi^\prime\,\delta\tau=0
\Longrightarrow \delta\tau=\frac{\delta\phi}{\phi^\prime},
$$
that is $\delta\tau=\frac{\delta\phi}{\phi^\prime}$ is the time-displacement
needed to go from a generic slicing with generic $\delta\phi$ to the
comoving slicing where $\delta\phi_{\rm com}=0$.
At the same time the curvature pertubation $\psi$ transforms
into

$$
\psi\rightarrow\psi_{\rm com}= \psi+\H\,\delta\tau=\psi+
\H\frac{\delta\phi}{\phi^\prime}.
$$
The quantity

\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
{\cal R}=\psi+
\H\frac{\delta\phi}{\phi^\prime}=\psi+H\frac{\delta\phi}{\dot\phi}
$$
%\\
\\
\hline
\end{tabular}
\end{center}
is the {\it comoving curvature perturbation}. This quantity is gauge invariant
by construction and is related to the gauge-dependent
curvature perturbation $\psi$ on a generic slicing to the inflaton
perturbation $\delta\phi$ in that gauge. By construction, the meaning of
${\cal R}$ is that it represents the gravitational potential on 
comoving hypersurfaces where $\delta\phi=0$

$$
{\cal R}=\left.\psi\right|_{\delta\phi=0}.
$$




\subsection{The curvature perturbation on spatial slices of uniform energy 
density}


We now consider the {\it slicing of uniform energy density } which 
is defined to be the
the slicing where there is no perturbation in the energy density,
 $\delta\rho=0$.

Since $\delta\rho\rightarrow \delta\rho-\rho^\prime\,\delta\tau$ for a 
transformation on constant time hypersurfaces, this means that

$$
\delta\rho\rightarrow\delta\rho_{\rm unif}=\delta\rho-\rho^\prime\,\delta\tau=0
\Longrightarrow \delta\tau=\frac{\delta\rho}{\rho^\prime},
$$
that is $\delta\tau=\frac{\delta\rho}{\rho^\prime}$ is the time-displacement
needed to go from a generic slicing with generic $\delta\rho$ to the
slicing of uniform energy density where $\delta\rho_{\rm unif}=0$.
At the same time the curvature pertubation $\psi$ transforms
into

$$
\psi\rightarrow\psi_{\rm unif}= \psi+\H\,\delta\tau=\psi+
\H\frac{\delta\rho}{\rho^\prime}.
$$
The quantity

\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
\zeta=\psi+
\H\frac{\delta\rho}{\rho^\prime}=\psi+H\frac{\delta\rho}{\dot\rho}
$$
%\\
\\
\hline
\end{tabular}
\end{center}
is the {\it curvature perturbation on slices of uniform energy density}. 
This quantity is gauge invariant
by construction and is related to the gauge-dependent
curvature perturbation $\psi$ on a generic slicing and to the energy density
perturbation $\delta\rho$ in that gauge. By construction, the meaning of
$\zeta$ is that it represents the gravitational potential on 
slices of uniform energy density

$$
\zeta=\left.\psi\right|_{\delta\rho=0}.
$$
Notice that, using the energy-conservation equation
$\rho^\prime+3\H(\rho+p)=0$, the  curvature perturbation on 
slices of uniform energy density
can be also written as

$$
\zeta=\psi
-\frac{\delta\rho}{3(\rho+p)}.
$$
During inflation $\rho+p=\dot{\phi}^2$. Furthermore, on superhorizon scales
from what we have learned in the previous section (and will be rigously
shown in the following) the inflaton fluctuation $\delta\phi$ is
frozen in and $\delta\dot\phi=($slow roll parameters$)\times H\,\delta\phi$.
This implies that 
$\delta\rho=\dot\phi\delta\dot\phi+V^\prime\delta\phi\simeq
V^\prime\delta\phi\simeq -3H
\dot\phi$, leading to 

$$
\zeta\simeq 
\psi+\frac{3H\dot\phi}{3\dot{\phi}^2}=\psi+H\frac{\delta\phi}{\dot\phi}
{\cal R}~~~~~({\rm ON 
~SUPERHORIZON~SCALES})
$$
The comoving curvature pertubation and the curvature perturbation
on uniform energy density slices are equal on superhorizon scales.

\subsection{Scalar field perturbations in the spatially flat
gauge}


We now consider the {\it spatially flat gauge}  which 
is defined to be the
the slicing where there is no curvature $\psi_{\rm flat}=0$.

Since $\psi\rightarrow \psi+\H\,\delta\tau$ for a 
transformation on constant time hypersurfaces, this means that

$$
\psi\rightarrow\psi_{\rm flat}= \psi+\H\,\delta\tau=0\Longrightarrow
\delta\tau=-\frac{\psi}{\H},
$$
that is $\delta\tau=-\psi/\H$ is the time-displacement
needed to go from a generic slicing with generic $\psi$ to the
spatially flat gauge  where $\psi_{\rm flat}=0$.
At the same time the fluctuation of the inflaton field  transforms
a

$$
\delta\phi\rightarrow\delta\phi -\phi^\prime\,\delta\tau=\delta\phi+
\frac{\phi^\prime}{\H}\,\psi.
$$
The quantity

\begin{center}
\begin{tabular}{|p{13.0 cm}|}
\hline
%\\
$$
Q=\delta\phi+
\frac{\phi^\prime}{\H}\,\,\psi=\delta\phi+\frac{\dot\phi}{H}\psi\equiv
\frac{\dot\phi}{H}\,{\cal R}
$$
%\\
\\
\hline
\end{tabular}
\end{center}
is the inflaton perturbation on spatially flat gauges. 
This quantity is gauge invariant
by construction and is related to the inflaton perturbation
$\delta\phi$ on a generic slicing and to to the curvature perturbation
$\psi$ in that gauge. By construction, the meaning of
$Q$ is that it represents the inflaton potential on spatially flat 
slices 

$$
Q=\left.\delta\phi\right|_{\delta\psi=0}.
$$
This quantity is often referred to as the Sasaki or Mukhanov
variable \cite{q1,q2}.

Notice that $\delta\phi=-\phi^\prime\delta\tau=-\dot\phi\delta t$
on flat slices, where $\delta t$ is the time displacement going from
flat to comoving slices. This relation makes somehow rigorous the expression 
(\ref{old}). Analogously, going from flat to comoving slices one has
${\cal R}=H\,\delta t$.

\subsection{Adiabatic and isocurvature perturbations}

Arbitrary cosmological perturbations can be decomposed into:

\begin{itemize} 

\item {\it adiabatic or curvature perturbations} which perturb the solution
along the same trajectory in phase-space as the 
as the background solution. The perturbations in any scalar
quantity $X$ can be described 
by a unique perturbation in expansion with respect
to the background

$$
H\,\delta t=H\,\frac{\delta X}{\dot X}~~~~~{\rm FOR ~ EVERY~} X.
$$
In particular, this holds for the energy density and the pressure

$$
\frac{\delta\rho}{\dot \rho}=\frac{\delta p}{\dot p}
$$
which implies that $p=p(\rho)$. This explains why 
they are called adiabatic. They are called curvature perturbations because
a given time displacement $\delta t $ causes the same relative
change $\delta X/\dot X$ for all quantities. In other words the perturbations
is democratically shared by all components of the universe.

\item {\it isocurvature perturbations} which perturb the solution off the
background solution

$$
\frac{\delta X}{\dot X}\neq\frac{\delta Y}{\dot Y}~~{\rm FOR ~ SOME~}
X~{\rm AND}~ Y.
$$
One way of specifying a generic isocurvature perturbation $\delta X$ is to
give its value on uniform-density slices, related to its value on a different
slicing by the gauge-invariant definition 

$$
H\,\left.\frac{\delta X}{\dot X}\right|_{\delta\rho=0}=
H\left(\frac{\delta X}{\dot X}-\frac{\delta \rho}{\dot \rho}\right).
$$
For a set of fluids with energy density $\rho_i$, the isocurvature 
perturbations are conventionally defined by the gauge invariant quantities

$$
S_{ij}=3H\left(\frac{\delta \rho_i}{\dot{\rho}_i}-
\frac{\delta \rho_j}{\dot{\rho}_j}\right)=3\left(\zeta_i-\zeta_j\right).
$$
One simple example of isocurvature perturbations is the baryon-to-photon
ratio $S=\delta(n_B/n_\gamma)=(\delta n_B/n_B)-(\delta n_\gamma/n_\gamma)$.

\vskip 0.2cm

{\it 1. \underline{Comment}:}

From the definitions above, it follows that
the cosmological perturbations generated during inflation are of the adiabatic
type {\it if} the inflaton field is the only fiels driving inflation.
However, if inflation is driven by more than one field, isocurvature 
perturbations are expected to be generated (and they
might even be cross-correlated to the adiabatic ones \cite{b1,b2,b3}). 
In the following, however, we will keep
focussing -- as done so far --  on
the one-single field case, that is we will be dealing only with
adiabatic/curvature perturbations.


{\it 2. \underline{Comment}:} The perturbations generated during inflation
are  {\it gaussian}, {\it i.e.} the two-point correlation functions
(like the power spectrum) suffice to define all the higher-order
even correlation fucntions, while the odd correlation functions (such
as the three-point correlation function) vanish. This conclusion is drawn 
by the very same fact that cosmological perturbations are studied {\it 
linearizing} Einstein's and Klein-Gordon equations. This turns out
to be a good approximation because we know that the inflaton potential
needs to be very flat in order to drive inflation and the interaction 
terms in the inflaton potential might be present, but they are small. 
Non-gaussian features are therefore suppressed since the non-linearities 
of the inflaton potential are suppressed too. The same argument applies
to the metric perturbations; non-linearities appear only at the 
second-order in deviations from the homogeneous background solution and 
are therefore small. This expectation has been recently confirmed
by the  first computation of the cosmological perturbations 
generated during inflation up to second-order in deviations from the
     homogeneous background solution which fully account for 
the inflaton self-interactions as well as for the second-order 
fluctuations of
the background metric \cite{ac}.

\end{itemize}

\subsection{The next steps}

After all these technicalities, it is useful to rest for a moment and to
go back to  physics. Up to now we have learned that during inflation
quantum fluctuations of the inflaton field are generated and their wavelengths
are stretched on large scales by the rapid expansion of the universe.
We have also seen that the quantum fluctuations of the inflaton field
are in fact impossible to disantagle from the metric perturbations.
This happens not only because they are tightly coupled 
to each other through Einstein's equations, but also because
of the issue of gauge invariance. Take, for instance, the
gauge invariant quantity $
Q=\delta\phi+
\frac{\phi^\prime}{\H}\,\,\psi$. We can always go to a gauge where
the fluctuation is entirely in the curvature potential $\psi$, $Q=
\frac{\phi^\prime}{\H}\,\,\psi$, or entirely in the inflaton
field, $Q=\delta\phi$. However, as we have stressed at the end
of the previous section, once ripples in the curvature 
are frozen in
on superhorizon scales during inflation, it is in fact gravity that acts
as a messanger communicating to baryons and photons the small
seeds of perturbations once a given scale reenters the horizon
after inflation. This happens thanks to Newtonian physics; 
a small perturbation in the gravitational potential $\psi$ induces
a small perturbation of the energy density $\rho$ through Poisson's
equiation $\nabla^2\psi=4\pi G\delta\rho$. Similarly, 
if perturbations
are adiabatic/curvature  perturbations and, as such, treat democratically
all the components, a ripple in the curvature is communicated
to photons as well, giving rise to a nonvanishing $\delta T/T$.

These considerations make it clear that the next steps of these lectures will
be

\begin{itemize}

\item Compute the curvature perturbation generated during inflation
on superhorizon scales. As we have seen we can either compute the comoving
curvature perturbation  ${\cal R}$ 
or the curvature on uniform energy density hypersurfaces $\zeta$. They
will tell us about the fluctuations of the gravitational potential.

\item See how the fluctuations of the gravitational
potential are transmitted to baryons and photons.

\end{itemize}

We now intend to address the first point. As stressed previously, we are 
free
to follow two alternative roads: either pick up  a gauge and compute
the gauge-invariant curvature in that gauge or perform
a gauge-invariant calculation. We take
both options.

\subsection{Computation of the  curvature perturbation using 
the longitudinal gauge}

The longitudinal (or conformal newtonian) 
gauge is a convenient  gauge to compute the
cosmological perturbations. It is defined by performing a coordinate 
transformation such that $B=E=0$. This leaves behind two 
degrees of freedom in the scalar perturbations, $A$ and $\psi$.
As we have previously seen in subsection 7.1, these two degrees of
freedom fully account for the scalar perturbations in the metric.

First of all, we take the non-diagonal part ($i\neq j$) of Eq. (\ref{ij}).
Since the stress energy-momentum tensor does not have
any non-diagonal component (no stress), we have

$$
\partial_i\partial_j\left(\psi-A\right)=0\Longrightarrow \psi=A
$$
and we can now work only with one variable, let it be $\psi$.

Eq. (\ref{0i}) gives (in cosmic time)

\begin{equation}
\dot\psi+H\,\psi=4\pi G\dot\phi\,\delta\phi=\epsilon\,H
\frac{\delta\phi}{\dot\phi},
\label{oo}
\end{equation}
while Eq. (\ref{00}) and the diagonal part of (\ref{ij}) ($i=j$)
give respectively

\begin{eqnarray}
\label{v}
-3H\left(\dot\psi+H \psi\right)+\frac{\nabla^2\psi}{a^2}&=&
4\pi G\left(\dot\phi\delta\dot\phi-\dot{\phi}^2\psi+
V^\prime\delta\phi\right),\\
\label{vvv}
-\left(2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2\right)\psi-
3H\dot\psi-\ddot\psi&=&-\left(\dot\phi\delta\dot\phi-\dot{\phi}^2\psi
-V^\prime\delta\phi\right),
\end{eqnarray}
If we now use the fact that $\dot H=4\pi G\dot{\phi}^2$, sum
Eqs. (\ref{v}) and {\ref{vvv}) and 
use  the background Klein-Gordon
equation to eliminate $V^\prime$, we arrive at the equation for the
gravitational potential

\begin{equation}
\label{master}
\ddot{\psi}_{\bf k}+\left(H-2\frac{\ddot\phi}{\dot\phi}\right)
\dot{\psi}_{\bf k}+
2\left(\dot H-H\frac{\ddot\phi}{\dot\phi}\right)\psi_{\bf k}+\frac{k^2}{a^2}
\psi_{\bf k}=0
\end{equation}
We may  rewrite it in conformal time

\begin{equation}
\label{masterconf}
{\psi}^{\prime\prime}_{\bf k}+2\left(\H-
\frac{\phi^{\prime\prime}}{\phi^\prime}\right)
\psi^\prime_{\bf k}+
2\left(\H^\prime-\H\frac{\phi^{\prime\prime}}{\phi^\prime}\right)
\psi_{\bf k}+k^2\,
\psi_{\bf k}=0
\end{equation}
and in terms of the slow-roll parameters $\epsilon$ and $\eta$

\begin{equation}
\label{masterslowroll}
{\psi}^{\prime\prime}_{\bf k}+2\H\left(\eta-\epsilon\right)
\psi^\prime_{\bf k}+
2\H^2\left(\eta-2\epsilon\right)
\psi_{\bf k}+k^2\,
\psi_{\bf k}=0.
\end{equation}
Using the same logic leading to Eq. (\ref{e}), 
from Eq. (\ref{master}) we can infer that on superhorizon scales
the gravitational potential $\psi$ is nearly constant (up to a  mild 
logarithmic
time-dependence proportional to slow-roll parameters), that is
$\dot{\psi}_{\bf k}\sim($slow-roll parameters$)\times$$\psi_{\bf k}$.
This is hardly 
surprising, we know that fluctuations are frozen in on superhorizon scales.

Using Eq. (\ref{oo}), we can therefore relate the fluctuation 
of the gravitational potential $\psi$ to the
fluctuation of the inflaton field $\delta\phi$ on superhorizon scales

\begin{equation}
\label{ee}
\psi_{\bf k}\simeq \epsilon\,H\frac{\delta\phi_{\bf k}}{\dot\phi}~~~
({\rm  ON ~ SUPERHORIZON~ SCALES})
\end{equation}
This gives us the chance to compute the gauge-invariant
comoving curvature perturbation ${\cal R}_{\bf k}$ 


\begin{equation}
{\cal R}_{\bf k}=\psi_{\bf k}+H\,\frac{\delta\phi_{\bf k}}{\dot\phi}
=\left(1+\epsilon\right)
\frac{\delta\phi_{\bf k}}{\dot\phi}\simeq \frac{\delta\phi_{\bf k}}{\dot\phi}.
\end{equation}
The power spectrum of the the comoving curvature perturbation 
${\cal R}_{\bf k}$ then reads on superhorizon scales

$$
{\cal P}_{{\cal R}}=\frac{k^3}{2\pi^2}\frac{H^2}{\dot{\phi}^2}\left|
\delta\phi_{\bf k}\right|^2=\frac{k^3}{4\mpl^2\epsilon\,\pi^2}
\left|
\delta\phi_{\bf k}\right|^2.
$$
What is left to evaluate is the time evolution of $\delta\phi_{\bf k}$.
To do so, we consider the  perturbed Klein-Gordon equation (\ref{kgg})
in the longitudinal gauge (in cosmic time)

$$
\delta\ddot{\phi}_{\bf k}+3H\delta\dot{\phi}_{\bf k}+\frac{k^2}{a^2}
\delta\phi_{\bf k}+V^{\prime\prime}\delta\phi_{\bf k}=-2\psi_{\bf k}V^\prime
+4\dot{\psi}_{\bf k}\dot{\phi}.
$$
Since on superhorizon scales $\left|4\dot{\psi}_{\bf k}\dot{\phi}\right|\ll
\left|\psi_{\bf k}V^\prime\right|$, using Eq. (\ref{ee}) and
the relation $V^\prime\simeq
-3H\dot{\phi}$, 
we can rewrite the
perturbed Klein-Gordon equation on superhorizon scales as

$$
\delta\ddot{\phi}_{\bf k}+3H\delta\dot{\phi}_{\bf k}+\left(V^{\prime\prime}
+6\epsilon H^2\right)\delta\phi_{\bf k}=0.
$$
We now introduce as usual the field $\delta\chi_{\bf k}=\delta\phi_{\bf k}/a$
and go to conformal time $\tau$. The perturbed Klein-Gordon equation on 
superhorizon scales becomes, using Eq. (\ref{ap}),  

\bea
\delta\chi_{\bf 
k}^{\prime\prime}&-&\frac{1}{\tau^2}\left(\nu^2-\frac{1}{4}
\right)\delta\chi_{\bf k}=0,\nonumber\\
\nu^2&=&\frac{9}{4}+9\epsilon-3\eta.
\eea
Using what we have learned in the previous section, we conclude that

$$
\left|\delta\phi_{\bf k}\right|\simeq \frac{H}{\sqrt{2k^3}}
\left(\frac{k}{aH}\right)^{\frac{3}{2}-\nu}
\,\,\,\,
({\rm ON}\,\,{\rm SUPERHORIZON}\,\,{\rm SCALES})
$$
which justifies our initial assumption that both the inflaton
perturbation and the gravitational potential are nearly constant
on superhorizon scale. 

We may now compute the power spectrum of the comoving curvature
perturbation on superhorizon scales

\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
{\cal P}_{{\cal 
R}}(k)=\frac{1}{2\mpl^2\epsilon}\left(\frac{H}{2\pi}\right)^2
\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}\equiv A^2_{\cal R}
\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}
$$
\\
\hline
\end{tabular}
\end{center}

where we have defined the {\it spectral index} $n_{{\cal R}}$ of the comoving
curvature perturbation
as
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
n_{{\cal R}}-1=
\frac{d {\rm ln} \,{\cal P}_{{\cal R}}}{d {\rm ln} \,k}=3-2\nu=
2\eta-6\epsilon.
$$
\\
\hline
\end{tabular}
\end{center}
\vskip 0.2cm
We conclude that inflation is responsible for the generation of 
adiabatic/curvature perturbations with an almost scale-independent
spectrum.



From the curvature perturbation we can easily deduce the
behaviour of the gravitational potential $\psi_{\bf k}$ from Eq. (\ref{oo}).
The latter is solved by 

$$
\psi_{\bf k}=\frac{A(k)}{a}+\frac{4\pi G}{a}\,\int^t\,dt^\prime\,a(t^\prime)
\,\dot\phi(t^\prime)\,
\delta\phi_{\bf k}(t^\prime)\simeq\frac{A(k)}{a}+\epsilon\,{\cal R}_{\bf k}.
$$
We find that during inflation and on superhorizon scales the gravitational
potential is the sum of a decreasing function plus a nearly constant in
time piece
proportional to the curvature perturbation. Notice in particular that
in an exact de Sitter stage, that is $\epsilon=0$, the gravitational
potential is not sourced and any initial condition  in the
gravitational potential is washed out as $a^{-1}$ during the inflationary
stage.



{\it \underline{Comment}:} We might have computed the
spectral index of the spectrum ${\cal P}_{{\cal R}}(k)$ by first
solving the equation for the perturbation $\delta\phi_{\bf k}$
in a di Sitter stage, with $H=$ constant ($\epsilon=\eta=0$),
whose solution is Eq. (\ref{sigma}) and
then taking into account the time-evolution of the Hubble rate
and of $\phi$ introducing  the subscript in $H_k$ and $\dot{\phi}_k$. The 
time variation of the latter  is determined 
by 
\be
\frac{d {\rm ln} \,\dot{\phi}_k}{d {\rm ln} \,k}=
\left(\frac{d {\rm ln} \,\dot{\phi}_k}{dt}\right)
\left(\frac{dt}{d {\rm ln} \,a}
\right)\left(\frac{d {\rm ln} \,a}{d {\rm ln} \,k}\right)=
\frac{\ddot{\phi}_k}{\dot\phi_k}
\times \frac{1}{H}\times 
1=-\delta=\epsilon-\eta.
\label{zzz}
\ee
Correspondingly, $\dot{\phi}_k$ 
is the value of the time derivative of the inflaton field
  when a given wavelength $\sim 
k^{-1}$ crosses
the horizon (from that point on the fluctuations remains
frozen in). The curvature 
perturbation in such an approach would read

$$
{\cal R}_{\bf k}\simeq \frac{H_k}{\dot\phi_k}\,\delta\phi_{\bf k}\simeq
\frac{1}{2\pi}\left(\frac{H_k^2}{\dot\phi_k}\right).
$$
Correspondigly

$$
n_{{\cal R}}-1=
\frac{d {\rm ln} \,{\cal P}_{{\cal R}}}{d {\rm ln} \,k}=
\frac{d {\rm ln} \,H_k^4}{d {\rm ln} \, k}-
\frac{d {\rm ln} \,\dot{\phi}_k^2}{d {\rm ln} \, k}=
-4\epsilon+(2\eta-2\epsilon)=2\eta-6\epsilon
$$
which 
reproduces our previous findings. 

During inflation the curvature
perturbation is generated on superhorizon scales with a spectrum which
is nearly scale invariant,  that is is nearly independent from the wavelength
$\lambda=\pi/k$: the amplitude of the 
fluctuation on superhorizon scales does not (almost) depend upon the 
time at which the fluctuations crosses the horizon and becomes frozen
in. The small tilt of the power spectrum arises from the fact that
the inflaton field  is massive, giving rise to a nonvanishing $\eta$
 and because 
during inflation the Hubble rate is not exactly constant, but
nearly constant, where `nearly' is quantified by the slow-roll
parameters $\epsilon$. 

\vskip 0.2cm

{\it \underline{Comment}:} From what found so far, we may conclude that
on superhorizon scales the comoving curvature perturbation ${\cal R}$
and the uniform-density gauge curvature $\zeta$ satisfy on superhorizon scales
the relation

$$
\dot{\cal R}_{\bf k}\simeq\dot{\zeta}_{\bf k}\simeq 0.
$$
An independent argument of the fact that they are nearly constant
on superhorizon scales is given in the Appendix A.



\subsection{Gauge-invariant computation of the curvature
perturbation}

In this subsection we would like to show how the computation of the
curvature perturbation can be performed in a gauge-invariant way. 
We first rewrite Einstein's equations in terms of Bardeen's potentials
(\ref{bar1}) and (\ref{bar2})


\begin{eqnarray}
 \deu{G^0_0}\,&=&\,\frac{2}{a^2}\Bigg( -
3\,\mathcal{H}\left(\mathcal{H}\,\Phi \,+\, \Psi^{\prime}\right)
\,+\, \nabla^2 \Psi \,+\, 3\,\mathcal{H}\left( -
\mathcal{H}^{\prime} \,+\,\mathcal{H}^2
\right)\left(\frac{E^{\prime}}{2}\,-\, B \right)\Bigg)\,,\\
 \deu{G^0_i}\,&=&\, \frac{2}{a^2}\,\partial_i \,\Bigg(
\mathcal{H}\,\Phi \,+\, \Psi^{\prime} \,+\, \left(
\mathcal{H}^{\prime} \,-\, \mathcal{H}^2\right)\left(
\frac{E^{\prime}}{2}\,-\, B\right)\Bigg)\,, \\
 \deu{G^i_j} \,&=&\, -\, \frac{2}{a^2}\,\Bigg( \left( \left(
2\,\mathcal{H}^{\prime} \,+\, 2\,\mathcal{H}^2\right)\Phi \,+\,
\mathcal{H}\,\Phi^{\prime} \,+\, \Psi^{\prime\prime} \,+\,
2\,\mathcal{H}\,\Psi^{\prime} \,+\, \frac{1}{2}\,\nabla^2
D \right)\delta^i_j \nonumber\\
&+&\, \left(\mathcal{H}^{\prime\prime} \,-\,
\mathcal{H}\,\mathcal{H}^{\prime} \,-\,
\mathcal{H}^3\right)\left(\frac{E^{\prime}}{2}\,-\,
B\right)\delta^i_j \,-\, \frac{1}{2}\,\partial^i\partial_j D
\Bigg),
\end{eqnarray}
with $D \,=\Phi- \Psi$. 
These quantities are not gauge-invariant, but using the 
gauge transformations described in subsection 7.6, we can easily
generalize them to gauge-invariant quantities


\begin{eqnarray}
 \deu{G^{({\rm GI})0}_0}&=&\, \deu{G^0_0} \,+\, (G^0_0)^{\prime}\left(
\frac{E^{\prime}}{2} \,-\, B\right)\,, \\
 \deu{G^{({\rm GI})0}_i}&=&\, \deu{G^0_i} \,+\, \left(G^0_i \,-\, \frac{1}{3}
\,T^k_k \right)\partial_i\, \left( \frac{E^{\prime}}{2} \,-\,
B\right)\,,\\
\deu{G^{({\rm GI})i}_j}&=&\, \deu{G^i_j} \,+\, (G^i_j)^{\prime}\left(
\frac{E^{\prime}}{2} \,-\, B\right)\,
\label{f1}
\end{eqnarray}
and

\begin{eqnarray}
 \deu{T^{({\rm GI})0}_0}&=&\, \deu{T^0_0} \,+\, (T^0_0)^{\prime}\left(
\frac{E^{\prime}}{2} \,-\, B\right)=-\delta\rho^{({\rm GI})}\,, \\
 \deu{T^{({\rm GI})0}_i}&=&\, \deu{T^0_i} \,+\, \left(T^0_i \,-\, \frac{1}{3}
\,T^k_k \right)\partial_i\, \left( \frac{E^{\prime}}{2} \,-\,
B\right)=\left(\rho+p\right)a^{-1}\delta v_i^{({\rm GI})}\,,\\
\deu{T^{({\rm GI})i}_j}&=&\, \deu{T^i_j} \,+\, (T^i_j)^{\prime}\left(
\frac{E^{\prime}}{2} \,-\, B\right)=\delta p^{({\rm GI})}\,
\label{f2}
\end{eqnarray}
where we have written the stress energy-momentum tensor as $T^{\mu\nu}=
\left(\rho+p\right)u^\mu u^\nu +p\eta^{\mu\nu}$ with $u^\mu=(1,v^i)$.



Einstein's equations can now be written in a gauge-invariant
way

\begin{eqnarray}
\label{equ}
 &-& 3\,\mathcal{H}\left( \mathcal{H}\,\Phi \,+\,
\Psi^{\prime}\right) \,+\, \nabla^2 \,\Psi  \\
& =&\, 4\,\pi\,G \left( - \Phi \,{\phi^{\prime}}^2 \,+\,
\deu{\phi^{({\rm GI})}}\,\phi^{\prime} \,+\,
\deu{\phi^{({\rm GI})}}
\,\frac{\partial V}{\partial \phi}\,a^2\right)\,, \nonumber\\
&& \partial_i \,\left( \mathcal{H}\,\Phi \,+\,
\Psi^{\prime}\right)\,=\,4\,\pi\,G
\left(\partial_i\,\deu{\phi^{({\rm GI})}}\,\phi^{\prime}\right)\,,\nonumber\\
&& \left(\left(2\,\mathcal{H}^{\prime}\,+\,\mathcal{H}^2\right)\Phi
\,+\, \mathcal{H}\,\Phi^{\prime} \,+\, \Psi^{\prime\prime} \,+\,
2\,\mathcal{H}\,\Psi^{\prime} \,+\, \frac{1}{2}\,\nabla^2 D
\right)\delta^i_j \,-\, \frac{1}{2}\partial^i\partial_j D , \nonumber\\
& =&\, -\,4\,\pi\,G \left( \Phi\,{\phi^{\prime}}^2 \,-\,
\deu{\phi^{({\rm GI})}}\,\phi^{\prime} \,+\,
\deu{\phi^{({\rm GI})}}\,\frac{\partial V}{\partial
\phi}\,a^2\right)\delta^i_j\,.
\label{sys}
\end{eqnarray}
Taking $i\neq j$ from the third equation, we find $D=0$, that is $\Psi=\Phi$
and from now on we can work with only the variable $\Phi$. Using
the background relation

\begin{equation}
2 \Ab \,-\, \Ac \,=\, 4\,\pi\,G\,{\phi^{\prime}}^2
\end{equation}
we can rewrite the system of Eqs. (\ref{sys})
in the form 

\begin{eqnarray}
\label{pippo}
 \nabla^2\,\Phi \,-\, 3\,\mathcal{H}\,\Phi^{\prime}
\,-\,\left(\mathcal{H}^{\prime} \,+\, 2\,\mathcal{H}^2\right)
\,&=&\,4\,\pi\,G \left( \deu{\phi^{({\rm GI})}}\,\phi^{\prime} \,+\,
\deu{\phi^{({\rm GI})}}\,\frac{\partial V}{\partial
\phi}\,a^2\right)\,;\nonumber\\
 \Phi^{\prime} \,+\, \mathcal{H}\,\Phi \,&=&\, \,4\,\pi\,G
\left(\deu{\phi^{({\rm GI})}}\,\phi^{\prime}\right)\,;\nonumber\\
 \Phi^{\prime\prime} \,+\, 3\,\mathcal{H}\,\Phi^{\prime} \,+\,
\left( \mathcal{H}^{\prime} \,+\, 2\,\mathcal{H}^2\right)\Phi
\,&=&\,4\,\pi\,G \left(\deu{\phi^{({\rm GI})}}\,\phi^{\prime} \,-\,
\deu{\phi^{({\rm GI})}}\,\frac{\partial V}{\partial \phi}\,a^2\right)\,.
\end{eqnarray}
Substracting the first equation from the third, using the second 
equation to express $\deu{\phi^{({\rm GI})}}$ as a function
of $\Phi$ and $\Phi^\prime$ and using the Klein-Gordon equation one finally
finds the 

\begin{equation}
\Phi^{\prime\prime}\,+\, 2\left(\mathcal{H}\,-\,
\frac{\phi^{\prime\prime}}{\phi^{\prime}}\right)\Phi^{\prime}
\,-\, \nabla^2 \,\Phi \,+\, 2\left(\mathcal{H}^{\prime} \,-\,
\mathcal{H}\,\frac{\phi^{\prime\prime}}{\phi^{\prime}}\right)\Phi
\,=\, 0 \,,
\label{nn}
\end{equation}
for the gauge-invariant potential $\Phi$.

We now introduce the gauge-invariant quantity


\begin{eqnarray}
u \, &\equiv& \,a\,\deu{\phi^{({\rm GI})}} \,+\, z\,\Psi \,,\\
z \, &\equiv& a\,\frac{\phi^{\prime}}{\mathcal{H}}=
a\frac{\dot\phi}{H}.
 \end{eqnarray}
Notice that the variable $u$ is equal to 
$-a\,Q$, the gauge-invariant inflaton perturbation on 
spatially flat gauges. 

Eq. (\ref{nn}) becomes

\begin{equation}
 \label{muk}
 u^{\prime\prime} \,-\, \nabla^2 \,u \,-\,
 \frac{z^{\prime\prime}}{z}\,u \,=\,0\,,
\label{fundu}
 \end{equation}
while the two remaining equations of the system (\ref{pippo})
can be written as

\begin{eqnarray}
\label{rest1}
  \nabla^2 \,\Phi \,&=&\, 4\,\pi\,G\,\frac{\mathcal{H}}{a^2}\left(
 z\,u^{\prime} \,-\, z^{\prime}u \right)\,,\\
  \left( \frac{a^2
 \,\Phi}{\mathcal{H}}\right)^{\prime}\,&=&\,4\,\pi\,G\,z\,u\,,
\label{rest2} 
\end{eqnarray}
which allow to determine the variables $\Phi$ and $\deu{\phi^{({\rm GI})}}$.

We have now to solve Eq. (\ref{fundu}). First, we have to evaluate
$ \frac{z^{\prime\prime}}{z}$ in terms of the slow-roll
parameters

$$
\frac{z^\prime}{\H z}=\frac{a^\prime}{\H a}+
\frac{\phi^{\prime\prime}}{\H \phi^\prime}-\frac{\H^\prime}{\H^2}=
\epsilon+\frac{\phi^{\prime\prime}}{\H \phi^\prime}.
$$
We then deduce that

$$
\delta\equiv 1-
\frac{\phi^{\prime\prime}}{\H \phi^\prime}=1+\epsilon
-\frac{z^\prime}{\H z}.
$$
Keeping the slow-roll parameters constat in time (as we have mentioned,
this corresponds to expand all quantities to first-order in the
slow-roll parameters), we find

$$
0\simeq \delta^\prime=\epsilon^\prime(\simeq 0)-
\frac{z^{\prime\prime}}{\H z}+\frac{z^\prime\H^\prime}{z\H^2}+
\frac{\left(z^\prime\right)^2}{\H z^2},
$$
from which we deduce

$$
\frac{z^{\prime\prime}}{z}\simeq 
\frac{z^\prime\H^\prime}{z\H}+\frac{\left(z^\prime\right)^2}{z^2}.
$$
Expanding in slow-roll parameters we find

$$
\frac{z^{\prime\prime}}{z}\simeq \left(1+\epsilon-\delta\right)
\left(1-\epsilon\right)\H^2+\left(1+\epsilon-\delta\right)^2\H^2\simeq
\H^2\left(2+2\epsilon-3\delta\right).
$$
If we set

$$
\frac{z^{\prime\prime}}{z}=\frac{1}{\tau^2}\left(\nu^2-\frac{1}{4}\right),
$$
this corresponds to

$$
\nu\simeq \frac{1}{2}\left[1+4
\frac{\left(1+\epsilon-\delta\right)(2-\delta)}{(1-\epsilon)^2}\right]^{1/2}
\simeq\frac{3}{2}+\left(2\epsilon-\delta\right)
\simeq\frac{3}{2}+3\epsilon-\eta.
$$
On subhorizon scales $(k\gg aH)$, the solution of equation (\ref{fundu})
is obviously $u_{\bf k}\simeq e^{-ik\tau}/\sqrt{2k}$. Rewriting
Eq. (\ref{rest2}) as

$$
\Phi_{\bf k}=-\frac{4\pi G a^2}{k^2}\frac{\dot{\phi}^2}{H}
\left(\frac{H}{a\dot\phi}u_{\bf k}\right)^{\cdot},
$$
we infer that on subhorizon scales

$$
\Phi_{\bf k}\simeq i\,\frac{4\pi G\dot\phi}{\sqrt{2 k^3}}\,
e^{-i\frac{k}{a}}.
$$
On superhorizon scales $(k\ll aH)$, one obvious solution to Eq. (\ref{fundu}) 
is $u_{\bf k}
\propto z$. To find the other solution, we may set $u_{\bf k}=z\,
\widetilde{u}_{\bf k}$, which satisfies the equation

$$
\frac{\widetilde{u}_{\bf k}^{\prime\prime}}{
\widetilde{u}_{\bf k}^{\prime}}=-2\frac{z^\prime}{z},
$$
which gives

$$
\widetilde{u}_{\bf k}=\int^\tau\,\frac{d\tau^\prime}{z^2(\tau^\prime)}.
$$
On superhorizon scales  therefore we find

$$
u_{\bf k}=c_1(k)\frac{a\dot\phi}{H}+c_2(k)\frac{a\dot\phi}{H}
\int^t\,dt^\prime\,\frac{H^2}{a^3\dot{\phi}^2}.
\simeq c_1(k)
\frac{a\dot\phi}{H}-c_2(k)\frac{1}{3 a^2\dot\phi},
$$
where the last passage has been performed supposing a de Sitter
epoch, $H=$ constant. 
The first piece is the constant mode
$c_1(k)z$, while the second is the
decreasing mode. To find the constant $c_1(k)$, we 
apply what we have learned in subsection 6.5. We know that
on superhorizon scales  
the exact solution of equation (\ref{fundu}) is
\be
u_{\bf k}=\frac{\sqrt{\pi}}{2}\,
e^{i\left(\nu+\frac{1}{2}\right)\frac{\pi}{2}}\,
\sqrt{-\tau}\,H_{\nu}^{(1)}(-k\tau).
\label{exactfund}
\ee
On superhorizon scales, since $H_{\nu}^{(1)}(x\ll 1)\sim
\sqrt{2/\pi}\, e^{-i\frac{\pi}{2}}\,2^{\nu-\frac{3}{2}}\,
(\Gamma(\nu_\chi)/\Gamma(3/2))\, x^{-\nu}$, 
the fluctuation (\ref{exactfund}) becomes
$$
u_{\bf k}=e^{i\left(\nu-\frac{1}{2}\right)\frac{\pi}{2}}
2^{\left(\nu-\frac{3}{2}\right)}\frac{\Gamma(\nu)}{\Gamma(3/2)}
\frac{1}{\sqrt{2k}}\,(-k\tau)^{\frac{1}{2}-\nu}.
$$

Therefore

\be
c_1(k)={\rm lim}_{k\rightarrow 0}\left|\frac{u_{\bf k}}{z}\right|=
\frac{H}{a\dot\phi}\frac{1}{\sqrt{2k}}\left(
\frac{k}{aH}\right)^{\frac{1}{2}-\nu}=
\frac{H}{\dot\phi}\frac{1}{\sqrt{2k^3}}\left(
\frac{k}{aH}\right)^{\eta-3\epsilon}
\label{c1}
\ee

The last steps consist in relating the variable $u$ to the comoving
curvature ${\cal R}$ and to the gravitational potential $\Phi$. The comoving
curvature takes the form    

\begin{equation}
\label{curva} \mathcal{R}\,\equiv\,-\,\Psi \,-\,
\frac{H}{\phi^{\prime}}\,\delta\phi^{({\rm GI})}=-\frac{u}{z}.
\end{equation}
Since $z=a\dot\phi/H=a\sqrt{2\epsilon}\mpl$, 
the power spectrum of the comoving curvature can be expressed on
superhorizon scales as 
\begin{equation}
\mathcal{P}_\mathcal{R}(k)\,=\,
\frac{k^3}{2\,\pi^2}\,\left| \frac{u_{\bf k}}{z}\right|^2=
\frac{1}{2\mpl^2\epsilon}\left(\frac{H}{2\pi}\right)^2
\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}\equiv A^2_{\cal R}
\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}
\end{equation}
with 

\be
n_{{\cal R}}-1
=3-2\nu=
2\eta-6\epsilon.
\ee
These results reproduce those found in the previous subsection.

The last step is to find the behaviour of the gauge-invariant potential
$\Phi$ on superhorizon scales. If we  recast equation (\ref{rest2}) in the form

\be
u_{\bf k}=\frac{1}{4\pi G}\frac{H}{\dot\phi}
\left(\frac{a}{H}\Phi_{\bf k}\right)^{\cdot},
\label{jjj}
\ee
we can infer that on superhorizon scales the nearly constant mode
of the gravitational potential during inflation reads

\be
\Phi_{\bf k}=c_1(k)\left[1-\frac{H}{a}\,\int^t\,dt^\prime\,
a\left(t^\prime\right)\right]\simeq -c_1(k)\frac{\dot H}{H^2}=\epsilon\,
c_1(k)\simeq \epsilon\frac{u_{\bf k}}{z}\simeq -\epsilon\,{\cal R}_{\bf k}.
\label{c2}
\ee
Indeed, plugging this solution into Eq. (\ref{jjj}), one reproduces
$u_{\bf k}=c_1(k)\frac{a\dot\phi}{H}$.




\subsection{Gravitational waves}

Quantum fluctuations in the gravitational fields are generated 
in a similar fashion of that of the scalar perturbations 
discussed so far. A gravitational wave
may be viewed as a ripple of spacetime in the FRW background metric
(\ref{metric}) and in general the linear tensor perturbations
may be written as 

$$
g_{\mu\nu}=a^2(\tau)\left[-d\tau^2+\left(\delta_{ij}+h_{ij}\right)
dx^i dx^j\right],
$$
where $\left|h_{ij}\right|\ll 1$. The tensor $h_{ij}$ has six degrees of 
freedom, but, as we studied in subsection 7.1,
 the tensor perturbations are traceless, $\delta^{ij}h_{ij}=0$,
and transverse $\partial^i h_{ij}=0$ $(i=1,2,3)$. With these
4 constraints, there remain 2 physical degrees of freedom, or 
polarizations, 
 which are
usually indicated 
$\lambda=+,\times$. More precisely, we can write

$$
h_{ij}=h_+\,e_{ij}^+ +h_\times\,e_{ij}^\times,
$$
where $e^+$ and $e^\times$ are the polarization tensors which 
have the following properties

$$
e_{ij}=e_{ji},~~~ k^i e_{ij}=0,~~~,e_{ii}=0,
$$
$$
e_{ij}(-{\bf k},\lambda)=e^*_{ij}({\bf k},\lambda),~~~
\sum_\lambda\,e^*_{ij}({\bf k},\lambda)e^{ij}({\bf k},\lambda)=4.
$$
Notice also that
the tensors $h_{ij}$ are gauge-invariant and therefore represent physical
degrees of freedom.

If the stress-energy momentum tensor is diagonal, as the one 
provided by the inflaton potential $T_{\mu\nu}=\partial_\mu\phi
\partial_\nu\phi-g_{\mu\nu}{\cal L}$, the tensor modes do not have any
source in their equation and their action can be written as 

$$
\frac{\mpl^2}{2}\,\int\,d^4x\,\sqrt{-g}\, \frac{1}{2}\partial_\sigma
h_{ij}\,\partial^\sigma h_{ij},
$$
that is the action of four independent massless scalar fields. The 
gauge-invariant tensor amplitude

$$
v_{\bf k}=a\mpl\frac{1}{\sqrt{2}}\, h_{\bf k},
$$
satisfies therefore the equation

$$
v_{\bf k}^{\prime\prime}+\left(k^2-\frac{a^{\prime\prime}}{a}\right)
v_{\bf k}=0,
$$
which is the equation of motion of a massless scalar field
in a quasi-de Sitter epoch. We can therefore make use of the results
present in subsection 6.5 and Eq. (\ref{vv}) to  conclude that
on superhorizon scales the tensor modes scale like

$$
\left|v_{\bf k}\right|=\left(\frac{H}{2\pi}\right)\left(\frac{k}{aH}\right)
^{\frac{3}{2}-\nu_T},
$$
where 

$$
\nu_T\simeq \frac{3}{2}-\epsilon.
$$
Since fluctuations are (nearly)
frozen in on superhorizon scales,
a way of characterizing the tensor pertubations is to compute
the spectrum on scales larger than the horizon

\be   
{\cal P}_{T}(k)=\frac{k^3}{2\pi^2}\sum_{\lambda}\left|
h_{\bf k}
\right|^2=4\times 2\frac{k^3}{2\pi^2}\left|v_{\bf k}\right|^2.
\ee
This gives the power spectrum on superhorizon scales

\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
{\cal P}_{T}(k)=\frac{8}{\mpl^2}\left(\frac{H}{2\pi}\right)^2
\left(\frac{k}{aH}\right)^{n_T}\equiv A^2_{T}
\left(\frac{k}{aH}\right)^{n_T}
\label{ttt}
$$
\\
\hline
\end{tabular}
\end{center}
where we have defined the {\it spectral index} $n_{T}$ of the tensor 
perturbations 
as
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\   
$$
n_T=
\frac{d {\rm ln} \,{\cal P}_{T}}{d {\rm ln} \,k}=3-2\nu_T=
-2\epsilon.
$$
\\
\hline
\end{tabular}
\end{center}
The tensor perturbation is almost scale-invariant. Notice that
the amplitude of the tensor modes depends only on the value
of the Hubble rate during inflation. This amounts
to saying that it depends only on the energy scale $V^{1/4}$
associated to the inflaton potential. A detection of gravitational
waves from inflation will be therefore a direct measurement
of the energy scale associated to inflation.


\subsection{The consistency relation}

The results obtained so far for the scalar and 
tensor perturbations allow to predict a {\it consistency relation}
which holds for the models of inflation addressed in these
lectures, {\it i.e.} the models of inflation driven by
one-single field $\phi$. We define tensor-to-scalar amplitude ratio to be

$$
r=\frac{\frac{1}{100} A_T^2}{\frac{4}{25}A_{\cal R}^2}=
\frac{\frac{1}{100} 8 \left(\frac{H}{2\, 
\pi\,\mpl}\right)^2}{\frac{4}{25}(2\epsilon)^{-1}\left(
\frac{H}{2\,\pi\,\mpl}\right)^2}=\epsilon.
$$
This means that 

\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
r=-\frac{n_T}{2}
$$
\\
\hline
\end{tabular}
\end{center}
One-single models of inflation predict that during inflation
driven by a single scalar field, the ratio between the
amplitude of the tensor modes and that of the curvature perturbations
is equal to minus one-half of the tilt of the spectrum of tensor modes.
If this relation turns out to be
falsified by the future measurements of the CMB anisotropies, this
does not mean that inflation is wrong, but only that
inflation has not been driven by
only one field. Generalizations to two-field models of inflation
can be found for instance in Refs. \cite{b2,b3}.



\section{The post-inflationary evolution of the cosmological
perturbations}

So far, we have computed the evolution of the cosmological
perturbations within the horizon and outside
the horizon during inflation. However, what we are really interested
in is their evolution after inflation and to compute the amplitude
of perturbations when they re-enter the horizon during radiation- or 
matter-domination.

To this purpouse, we use
the following procedure. We use Eqs. (\ref{f1}) and (\ref{f2}) to write



\begin{eqnarray}
\label{pippo1}
 \nabla^2\,\Phi \,-\, 3\,\mathcal{H}\,\Phi^{\prime}
\,-\,\left(\mathcal{H}^{\prime} \,+\, 2\,\mathcal{H}^2\right)
\,&=&\,4\,\pi\,G \,a^2\,\delta\rho^{({\rm GI})}\,;\\
 \Phi^{\prime} \,+\, \mathcal{H}\,\Phi \,&=&\, 
\,4\,\pi\,G\,a^2\left(\rho+p\right)
\delta v^{({\rm GI})}\,;\\
 \Phi^{\prime\prime} \,+\, 3\,\mathcal{H}\,\Phi^{\prime} \,+\,
\left( \mathcal{H}^{\prime} \,+\, 2\,\mathcal{H}^2\right)\Phi
\,&=&\,4\,\pi\,G\,a^2 \delta p^{({\rm GI})}\,.
\end{eqnarray}
Combining these equations one finds

\be
\label{hj}
\Phi^{\prime\prime}+3\H\left(1+c_s^2\right)\Phi^\prime-
c_s^2\nabla^2\Phi+\left[2\H^\prime+\left(1+c_s^2\right)\H^2\right]\Phi=0,
\ee
where $c_s^2=\dot p/\dot\rho$. This equation can be rewritten as

$$
\dot{{\cal R}}_{\bf k}=0
$$
where we have set 

$$
{\cal R}_{\bf k}=-\Phi-\frac{2}{3}
\frac{\H^{-1}\dot\Phi+\Phi}{\left(1+w\right)}-\Phi.
$$
Here $w=p/\rho$. Notice that during inflation, when $p=\frac{1}{2}\dot{\phi}^2-
V$ and $\rho=\frac{1}{2}\dot{\phi}^2+V$, ${\cal R}_{\bf k}$ takes the form



\be
{\cal R}_{\bf k}=-\Phi_{\bf k}-\frac{1}{\epsilon \H}\left(
\phi^\prime_{\bf k}+\H\Phi_{\bf k}\right),
\label{rr}
\ee
which using the equation

$$
\Phi^{\prime} \,+\, \mathcal{H}\,\Phi \,=\, \,4\,\pi\,G
\left(\deu{\Phi^{({\rm GI})}}\,\phi^{\prime}\right)\,
$$
reduces to the comoving curvature perturbation (\ref{curva}).

Eq. (\ref{hj}) is solved by

$$
\Phi_{\bf k}=c_1(k)\left(1-\frac{\H}{a^2}\,\int^\tau\,d\tau^\prime
a^2(\tau^\prime)\right)+c_2(k)\frac{\H}{a^2}~~{\rm (ON~SUPERHORIZON ~
SCALES)}
$$
This nearly constant solution can be rewritten in cosmic time as

$$
\Phi_{\bf k}=c_1(k)\left(1-\frac{H}{a}\,\int^t\,dt^\prime
a(t^\prime)\right).
$$
which is the same form of solution we found during inflation, see
Eq. (\ref{c2}). This explains why we can choose the constant
$c_1(k)$ to be the one given by Eq. (\ref{c1}), $c_1(k)=|u_{\bf k}/z|$
for superhorizon scales.





 


Since during radiation-domination $a\sim t^{n}$ with $n=1/2$ and during
matter-domination $a\sim t^{n}$ with $n=2/3$, we can write

$$
\Phi_{\bf k}=c_1(k)\left(1-\frac{H}{a}\,\int^t\,dt^\prime
a(t^\prime)\right)=c_1(k)\left(1-\frac{n}{n+1}\right)=
\frac{c_1(k)}{n+1}={\cal R}_{\bf k}\left\{
\begin{array}{cc}
\frac{2}{3} & {\rm RD}\\
\frac{3}{5} & {\rm MD}
\end{array}\right.
$$
This relation tells us that, after a gravitational perturbation with a given
wavelength
is generated during inflation, it evolves on superhorizon scales
after inflation simply
slightly rescaling its amplitude. When the given wavelength re-enters the 
horizon, the amplitude of the gravitational potential depends upon
the time of re-enter. If the perturbation re-enters the horizon
when the universe is still dominated by radiation, then 
$\Phi_{\bf k}=\frac{2}{3}{\cal R}_{\bf k}$; if 
the perturbation re-enters the horizon
when the universe is  dominated by matter, then 
$\Phi_{\bf k}=\frac{3}{5}{\cal R}_{\bf k}$. For instance, 
the power spectrum of the gravitational perturbations
during matter-domination reads


$$
{\cal P}_{\Phi}=\left(\frac{3}{5}\right)^2\,{\cal P}_{{\cal R}}
=\left(\frac{3}{5}\right)^2\,
\frac{1}{2\mpl^2\epsilon}\left(\frac{H}{2\pi}\right)^2
\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}
$$

\subsection{From the inflationary seeds to the matter power spectrum}


As the curvature perturbations enter the causal horizon during radiation-
or matter-domination, they create density fluctuations $\delta\rho_{\bf k}$ 
via gravitational
attractions of the potential wells. The density contrast $\delta_{\bf k}=
\frac{\delta\rho_{\bf k}}{\overline\rho}$
can be deduced from Poisson equation

$$
\frac{k^2}{a^2}=-4\pi G\,\delta\rho_{\bf k}=
-4\pi G\,\frac{\delta\rho_{\bf k}}{\overline\rho}\,\overline\rho=
\frac{3}{2}\,H^2\,\frac{\delta\rho_{\bf k}}{\overline\rho}
$$
where $\overline\rho$ is the background average energy density.
This means that

$$
\delta_{\bf k}=\frac{2}{3}\,\left(\frac{k}{a H}\right)^2\,\Phi_{\bf k}.
$$
From this expression we can compute the power spectrum 
of matter density perturbations induced by inflation when they
re-enter the horizon during matter-domination

$$
{\cal P}_{\delta\rho}=\langle\left|\delta_{\bf k}\right|^2\rangle=
A\,\left(\frac{k}{a H}\right)^n=\frac{2\pi^2}{k^3}\left(\frac{2}{5}\right)^2
 A^2_{\cal R}\left(\frac{k}{aH}\right)^4\,
\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}
$$
from which we deduce that matter perturbations scale linearly
with the wavenumber and have a scalar tilt

$$
n=n_{{\cal R}}=1+2\eta-6\epsilon.
$$

The primordial spectrum ${\cal P}_{\delta\rho}$ is of course
reprocessed by gravitational
instabilities after the universe becomes matter-dominated. Indeed, 
as we have seen in section 6, 
perturbations evolve after entering the horizon and the power spectrum will
not remain constant. To see how the density contrast is reprocessed
we have first to analyze how it evolves
on superhorizon scales before horizon-crossing. 
We use the following trick. Consider 
a flat universe with average energy density $\overline\rho$. The corresponding
Hubble rate is 

$$
H^2=\frac{8\pi G}{3}\,\overline\rho.
$$

A small
positive fluctuation $\delta\rho$ will cause the universe to be
closed

$$
H^2=\frac{8\pi G}{3}\left(\overline\rho+\delta\rho\right)-
\frac{k}{a^2}.
$$
Substracting the two equations we find

$$
\frac{\delta\rho}{\rho}=\frac{3}{8\pi G}\frac{k}{a^2\rho}\sim
\left\{\begin{array}{cc}
a^2 &{\rm RD}\\
a & {\rm MD}\end{array}\right.
$$
Notice that $\Phi_{\bf k}\sim \delta\rho a^2/k^2\sim(\delta\rho/\rho)\rho
a^2/k^2=$ constant for both RD and MD which confirms our previous findings.


When the matter densities enter the horizon, they 
do not increase appreciably before 
matter-domination because before matter-domination pressure
is too large and does not allow the matter inhomogeneities to grow.
On the other hand, the suppression of growth due to radiation
is restricted to scales smaller than the horizon, while large-scale
perturbations remain unaffected. This is the reason why
the horizon size at equality sets an important scale for structure growth

$$
k_{\rm EQ}=H^{-1}\left(a_{\rm EQ}\right)\simeq
0.08\,h\,{\rm Mpc}^{-1}.
$$
Therefore, perturbations with $k\gg k_{\rm EQ}$ are perturbations
which have entered the horizon before matter-domination and have remained
nearly constant till equality. This means that they are suppressed
with respect to those perturbations having $k\ll k_{\rm EQ}$ by a factor
$(a_{\rm ENT}/a_{\rm EQ})^2=(k_{\rm EQ}/k)^2$. 
If we define the 
 transfer function $T(k)$ by the relation
${\cal R}_{\rm final}=T(k)\,{\cal R}_{\rm initial}$ we find therefore
that roughly speaking
$$
T(k)=\left\{\begin{array}{cc}
1 &k\ll k_{\rm EQ},\\
(k_{\rm EQ}/k)^2 &k\gg k_{\rm EQ}. \end{array}\right.
$$
The corresponding power spectrum will be

$$
{\cal P}_{\delta\rho}(k)\sim\left\{\begin{array}{cc}
\left(\frac{k}{aH}\right) & k\ll k_{\rm EQ},\\
 \left(\frac{k}{aH}\right)^{-3} & k\gg k_{\rm EQ}. \end{array}\right.
$$
Of course, a more careful computation needs to include many other effects
such as neutrino free-streeming, photon diffusion and the diffusion of baryons
along with photons. It is encouraging however that this rough estimate
turns out to be confirmed by present data on large scale structures \cite{hu}.


\subsection{From inflation to large-angle CMB anisotropy}

Temperature fluctuations in the CMB arise due to five distinct
physical effects: our peculiar velocity with respect to
the cosmic rest frame; fluctuations in the gravitational potential
on the last scattering surface; fluctuations intrinsic to the radiation
field itself on the last-scattering surface; the peculiar velocity of the
last-scattering surface and damping of anisotropies if the universe
should be re-ionized after decoupling. The first effect gives rise to
the dipole anisotropy.
The second effect, known as the Sachs-Wolfe effect is the dominat contribution
to the anisotropy on large-angular scales, $\theta\gg\theta_{\rm HOR}\sim
 1^\circ$. The last three
effects provide the dominant contributions to the anisotropy on small
angular scales, $\theta\ll 1^\circ$.

We consider here the temperature fluctuations on large-angular scales
that arise due to the Sachs-Wolfe effect. These anisotropies
probe length scales that were superhorizon sized at photon decoupling and
therefore insensitive to microphysical processes. On the contrary, they provide
a probe of the original spectrum of primeveal fluctuations produced during 
inflation.

To proceed, we consider the CMB anisotropy measured at positions other than
our own and at earlier times. This is called the brightness
function $\Theta(t,{\bf x},{\bf n})\equiv \delta T(t,{\bf x},{\bf n})/T(t)$.
The photons with momentum ${\bf p}$ in a given range $d^3p$ have intensity
$I$ proportional to $T^4(t,{\bf x},{\bf n})$ and therefore
$\delta I/I=4\Theta$. The brightness function
depends upon the direction ${\bf n}$ of the photon momentum or, equivalently,
on the direction of observation ${\bf e}=-{\bf n}$.
Because the CMB travels freely from the last-scattering, we can write

$$
\frac{\delta T}{T}=\Theta\left(t_{\rm LS},{\bf x}_{\rm LS},{\bf n}\right)+
\left(\frac{\delta T}{T}\right)_{*},
$$
where ${\bf x}_{\rm LS}=-x_{\rm LS}{\bf n}$ is the point of the origin 
of the photon
coming from the direction ${\bf e}$. The comoving distance of the 
last-scattering distance is $x_{\rm LS}=2/H_0$. The first term corresponds 
to the anisotropy already present at last scattering and the second term
 is the additional anisotropy acquired during the travel towards us, equal to
minus the fractional pertubation
in the redshift  of the radiation.
Notice that the separation between each term depends on the slicing, but
the sum is not. 

Consider the redshift perturbation on comoving slicing.
We imagine the 
universe populated by comoving observers along the line of sight. The relative
velocity of adjacent comoving observers is equal to their distance times the 
velocity gradient measured along ${\bf n}$ of the photon. In the
unperturbed universe, we have ${\bf u}=H{\bf r}$, leading to
the velocity gradient $u_{ij}=\partial u_i/\partial r_j=
u_{ij}=H(t)\delta_{ij}$ with zero vorticity and shear. Including a peculiar
velocity field as 
perturbation, ${\bf u}=H{\bf r}+{\bf v}$ and $u_{ij}=H(t)\delta_{ij}+
\frac{1}{a}\frac{\partial v_i}{\partial v_j}$. The corresponding
Doppler shift is

$$
\frac{d\lambda}{\lambda}=\frac{da}{a}+n_i n_j\frac{\partial v_i}{\partial x_j}
dx.
$$

The perturbed FRW equation is 

$$
\delta H=\frac{1}{3}\nabla\cdot{\bf v},
$$
while

$$
(\delta\rho)^\cdot=-3\rho\delta H-3H\delta\rho.
$$ 
Instead of $\delta\rho$, let
us work with the density contrast $\delta=\delta\rho/\rho$. Remembering
that $\rho\sim a^{-3}$, we find that $\dot\delta=-3\delta H$, which give

$$
\nabla\cdot {\bf v}=-\dot{\delta}_{\bf k}.
$$

From Euler equation $\dot{\bf u}=-\rho^{-1}\nabla p-\nabla\Phi$,
we deduce $\dot{\bf v}+H{\bf v}=-\nabla\Phi-\rho^{-1}\nabla p$.
Therefore, for $a\sim t^{2/3}$ and negligible 
pressure gradient, since the gravitational potential is constant, we find

$$
{\bf v}=-t\nabla\Phi
$$
leading to

\be
\label{sw1}
\left(\frac{\delta T}{T}\right)_{*}=\int_0^{x_{\rm LS}}\,\frac{t}{a}
\frac{d^2\Phi}{dx^2}\,dx.
\ee
The photon trajectory is $a d{\bf x}/dt={\bf n}$. Using $a\sim t^{2/3}$ gives

$$
x(t)=\int_t^{t_0}\frac{dt^\prime}{a}=3\left(\frac{a_0}{t_0}-
\frac{t}{a}\right).
$$
Integrating by parts Eq. (\ref{sw1}), we finally find

$$
\left(\frac{\delta T}{T}\right)_{*}=\frac{1}{3}\left[
\Phi({\bf x}_{\rm LS})-
\Phi(0)\right]+{\bf e}\cdot\left[{\bf v}(0,t_0)-{\bf v}(
{\bf x}_{\rm LS},t_{\rm LS})\right].
$$
Te potential at our position contributes only to
the unobservable monopole and can be dropped. On scales outside the horizon,
${\bf v}=-t\nabla\Phi\sim 0$. The remaining term is the
Sachs-Wolfe effect

$$
\frac{\delta T({\bf e})}{T}=\frac{1}{3}\Phi({\bf x}_{\rm LS})=
\frac{1}{5}{\cal R}({\bf x}_{\rm LS}).
$$
Therefore, at large angular scales, the theory of cosmological perturbations
predicts a remarkable simple formula relating the CMB anisotropy to
the curvature perturbation generated during inflation.

In section 3, we have seen that 
the 
temperature anisotropy is commonly expanded  in spherical harmonics
$
\frac{\Delta T}{T}(x_0,\tau_0,{\bf n})=\sum_{\ell m}
a_{\ell,m}(x_0)Y_{\ell m}({\bf n}),
$
where $x_0$ and $\tau_0$ are our position and the preset time, respectively,
 ${\bf n}$ is the
direction of observation, $\ell'$s are the
different multipoles  and
$
\langle a_{\ell m}a^*_{\ell'm'}\rangle=\delta_{\ell,\ell'}\delta_{m,m'} C_\ell
$,
where the deltas are due to the fact that the process that created
the anisotropy is statistically isotropic. 
The $C_\ell$ are the so-called CMB power spectrum.
For homogeneity and isotropy, the $C_\ell$'s are neither a function
of $x_0$, nor of $m$.
The two-point-correlation function is related to the $C_l$'s
according to Eq. (\ref{j}).

For adiabatic perturbations we have seen that on large scales,
larger than the horizon on the last-scattering surface (corresponding
to angles larger than $\theta_{\rm HOR}\sim 1^\circ$)
$\delta T/T=
\frac{1}{3}\Phi({\bf x}_{\rm LS})$
In Fourier transform
\begin{equation}
\frac{\delta T({\bf k},\tau_0,{\bf n})}{T}=
\frac{1}{3}\Phi_{{\bf k}}\,e^{i\,{\bf k}\cdot{\bf n}(\tau_0-
\tau_{{\rm LS}})}
\end{equation}
Using the decomposition
\begin{equation}
\exp(i\,{\bf k}\cdot{\bf n}(\tau_0-\tau_{{\rm LS}}))=
\sum_{\ell=0}^\infty (2\ell+1) i^\ell
j_\ell(k(\tau_0-\tau_{{\rm LS}})) P_{\ell}({\bf k}\cdot{\bf n})
\end{equation}
where $j_\ell$ is the spherical Bessel function of order $\ell$ and 
substituting, we get
\begin{eqnarray}
&&\Big<\frac{\delta T(x_0,\tau_0,{\bf n})}{T}\frac{\delta
T(x_0,\tau_0,{\bf n'})}{T}\Big>=\\ \nonumber &&=\frac{1}{V}\int
d^3x \Big<\frac{\delta T(x_0,\tau_0,{\bf n})}{T}\frac{\delta
T(x_0,\tau_0,{\bf n}')}{T}\Big>=\\ \nonumber
&&=\frac{1}{(2\pi)^3}\int d^3k \Big<\frac{\delta
T({\bf k},\tau_0,{\bf n})}{T}\left(\frac{\delta
T({\bf k},\tau_0,{\bf n}')}{T}\right)^*\Big>=\\ \nonumber
&&=\frac{1}{(2\pi)^3}\int d^3k
\Big(\Big<\frac{1}{3}|\Phi|^2\Big>
\sum_{\ell,\ell'=0}^{\infty}
(2\ell+1)(2\ell'+1)j_\ell(k(\tau_0-\tau_{\rm LS}))\nonumber\\
&&j_{\ell'}(k(\tau_0-\tau_{{\rm LS}}))
P_\ell({\bf k}\cdot{\bf n}) P_{\ell'}({\bf k}'\cdot{\bf n}')\Big)
\end{eqnarray}
Inserting $P_\ell({\bf k}\cdot{\bf n})=\frac{4\pi}{2\ell+1}\sum_m
Y^*_{lm}({\bf k})Y_{\ell m}({\bf n})$ and analogously for
$P_\ell({\bf k}'\cdot{\bf n}')$, integrating over the directions
$d\Omega_k$ generates $\delta_{\ell\ell'}\delta_{mm'}\sum_m
Y^*_{\ell m}({\bf n})Y_{\ell m}({\bf n}')$. Using as well $\sum_m
Y^*_{\ell m}({\bf n})Y_{\ell m}({\bf n}')=\frac{2\ell+1}{4\pi}
P_\ell({\bf n}\cdot {\bf n}')$, we
get
\begin{eqnarray}
&&\Big<\frac{\delta T(x_0,\tau_0,{\bf n})}{T}\frac{\delta
T(x_0,\tau_0,{\bf n}')}{T}\Big>\\ \nonumber &&=\Sigma_\ell
\frac{2\ell+1}{4\pi}P_\ell({\bf n}\cdot{\bf n}') 
\frac{2}{\pi}\int \frac{dk}{k}
\Big<\frac{1}{9}|\Phi|^2\Big> k^3 j^2_\ell(k(\tau_0-\tau_{\rm LS})).
\end{eqnarray}
Comparing this expression with that for the $C_\ell$, we get the
expression for the $C^{{\rm AD}}_\ell$, where the suffix ``AD'' stays for
adiabatic 
\begin{equation}
C^{\rm AD}_\ell=\frac{2}{\pi}\int \frac{dk}{k} \Big<\frac{1}{9}\left|\Phi
\right|^2\Big> k^3
j^2_\ell(k(\tau_0-\tau_{\rm LS}))
\end{equation}
which is valid for $2\leq \ell\ll
(\tau_0-\tau_{\rm LS})/\tau_{\rm LS}\sim 100$.

If we generically indicate by 
$\langle|\Phi_{\bf k}|^2\rangle k^3=A^2\,(k\tau_0)^{n-1}$, 
we can perform the integration and
get
\begin{equation}
\frac{\ell(\ell+1)C^{\rm AD}_\ell}{2\pi}
=\left[
\frac{\sqrt{\pi}}{2}\ell(\ell+1)\frac{\Gamma(\frac{3-n}{2})
\Gamma(\ell+\frac{n-)}{2})}{\Gamma\left(\frac{4-n}{2}\right)\Gamma
\left(\ell+
\frac{5-n}{2}\right)}\right]
\frac{A^2}{9}\left(\frac{H_0}{2}\right)^{n-1}
\end{equation}
For $n\simeq 1$ and $\ell\gg 1$, we can approximate this expression to

\begin{equation}
\frac{\ell(\ell+1)C^{\rm AD}_l}{2\pi}=\frac{A^2}{9}.
\label{Clad}
\end{equation}
This result shows that inflation predicts
a very flat spectrum  for  low $\ell$. This prediction has
been confirmed by the COBE satellite \cite{FIRAS}.
Furthermore, 
since inflation predicts $\Phi_{\bf k}=\frac{3}{5}{\cal R}_{\bf k}$, we find
that 

\begin{equation}
\pi\,\ell(\ell+1)C^{\rm AD}_l=\frac{A_{\cal R}^2}{25}=
\frac{1}{25}\frac{1}{2\,\mpl^2\,\epsilon}\left(\frac{H}{2\pi}\right)^2.
\end{equation}

COBE data imply that $\frac{\ell(\ell+1)C^{\rm AD}_l}{2\pi}\simeq 10^{-10}$
or
\begin{center}
\begin{tabular}{|p{13 cm}|}
\hline
%\\
$$
\left(\frac{V}{\epsilon}\right)^{1/4}\simeq 6.7\times 10^{16}\,{\rm GeV}
$$
\\
\hline
\end{tabular}
\end{center}
Take for instance a model of chaotic inflation with quadratic 
potential $V(\phi)=\frac{1}{2}m_\phi^2\phi^2$. Using Eq. (\ref{togo})
one easily computes that when there are $\Delta N$ e-foldings to go,
the value of the inflaton field is $\phi_{\Delta N}^2=(\Delta N/2\pi G)$
and the corresponding value of $\epsilon$ is $1/(2 \Delta N)$. Taking
$\Delta N\simeq 60$ (corresponding to large-angle CMB anisotropies), one finds
that COBE normalization imposes $m_\phi\simeq 10^{13}$ GeV.


\section{Conclusions}

Along these lectures, we have learned that a stage
of inflation during the early epochs of the evolution of the
universe solves many drawbacks of the standard Big-Bang cosmology, such
as the flatness or entropy problem and the horizon problem. Luckily,
despite inflation occurs after a tiny bit after the bang, 
it leaves behind some observable predictions:

\begin{itemize}

\item {\it The universe should be flat}, that is the total density
of all components of matter should sum to the critical energy
density and $\Omega_0=1$. The current data on the CMB anisotropies
offer a spectacular confirmation of such a prediction. The universe
appears indeed to be spatially flat.

\item {\it Primordial perturbations are adiabatic}. Inflation provides the 
seeds for the cosmological perturbations. In one-single field
models of inflation, the perturbations  are {\it adiabatic} or curvature
pertrubations, {\it i.e.} they are fluctuations in the total energy density
of the universe or, equivalently, scalar perturbations to the
spacetime metric. Adiabaticity implies that the spatial
distribution of each species in the universe is the same, that is the ratio
of number densities of any two of these species is everywhere
the same. Adiabatic perturbations predict a contribution to the
CMB anisotropy which is related to the curvature perturbation ${\cal R}$ 
on large angles,
$\delta T/T=\frac{1}{5}\,{\cal R}$, and are in excellent agreeement
with the CMB data.
Adiabatic perturbations can be
contrasted to isocurvature perturbations which are fluctuations in the
ratios between the various species in the universe. Isocurvature perturbations
predict that on large angles $\delta T/T=-2\Phi$ and are presently ruled
out, even though a certain amount of isocurvature perturbations,
possibly correlated with the adiabatic fluctuations,  cannot
be excluded by present CMB data \cite{Amendola:2001ni}. 

\item {\it Primordial perturbations are almost scale-independent}.
The primordial power spectrum predicted by inflation has a characteristic
feature, it is almost scale-independent, that is the spectral index
$n_{\cal R}$ is very close to unity. Possible deviations from exact
scale-independence arise because during inflation the inflaton is not
massless and the Hubble rate is not exactly constant.
A recent analysis \cite{bond}
shows that $n_{\cal R}=0.97^{+0.08}_{-0.05}$, again in agreement
with the theoretical prediction.

\item {\it Primordial perturbations are nearly gaussian}. The fact that
cosmological perturbations are tiny allow their analysis in terms of 
linear perturbation theory. 
Non-gaussian features are therefore suppressed since the non-linearities 
of the inflaton potential and of the metric perturbations are suppressed.
Non-gaussian features are indeed present, but 
may appear only at the 
second-order in deviations from the homogeneous background solution and 
are therefore small \cite{ac}. This is rigously true only for 
one-single field models of inflation. Many-field models of inflation
may give rise to some level of non-gaussianity \cite{b4}.
If the next generation of satellites will detect a non-negligible
amount of non-gaussianity in the CMB anisotropy, this will rule
out one-single field models of inflation.


\item {\it Production of gravitational waves}. A stochastic
background of gravitational waves is produced during inflation
 in the very same way classical perturbations to the inflaton
field are generated. The spectrum of such gravitational waves is
again flat, {\it i.e.} scale-independent and the tensor-to-scalar
amplitude ratio $r$ is, 
at least in one-single
models of inflation, related to the  spectral index $n_T$  by the 
consistency relation $r=-n_T/2$. A confirmation of such a relation
would be a spectacular proof  of one-single field models
of inflation. The detection of the primordial stochastic
background of gravitational waves from inflation is challenging, but
would not only set the energy scale of inflation, but would also give the
opportunity of discriminating among different models of inflation
\cite{kmr,bacci}.

\end{itemize}

\section*{Acknowledgments}

The author would like to thank the organizers of the School,
G. Dvali, A. Perez-Lorenzana, G. Senjanovic, G. Thompson and F. Vissani
for providing
the students and the lecturers with such an excellent and stimulating
environment. He also thanks the students for their questions
and enthusiasm.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%            Appendices (if any) start here                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% IF YOU DO NOT HAVE ANY APPENDIX, 
%%% REMOVE EVERYTHING TILL "End of Appendixes"

\newpage
\appendix

\section{Evolution of the curvature perturbation on superhorizon scales}
\def\theequation{\thesection.\arabic{equation}}
\setcounter{equation}{0}
In this appendix, we repeat the general arguments following 
from energy-momentum conservation
given in Ref. \cite{mal}
to show that the curvature perturbation on constant-time hypersurfaces
$\psi$ is constant on superhorizon scales if perturbations are
adiabatic.

The constant-time hypersurfaces are orthogonal to the unit time-like 
vector field $n^\mu=(1-A,-\partial^i B/2)$. Local conservation of 
the energy-momentum tensor tells us that $T^\mu_{\nu;\mu}=0$. The energy
conservation equation $n^\nu T^\mu_{\nu;\mu}=0$ for first-order
density perturbations and on superhorizon scales give

$$
\delta\dot\rho=-3H\left(\delta\rho+\delta p\right)-3\dot\psi
\left(\rho+p\right).
$$
We
write $\delta p=\delta p_{\rm nad}+c_s^2\delta\rho$, where 
$\delta p_{\rm nad}$ is the non-adiabatic component
of the pressure perturbation and $c_s^2=\delta p_{\rm ad}/\delta\rho$
is the adiabatic one. In the uniform-density gauge 
$\psi=\zeta$ and $\delta\rho=0$ and therefore $\delta p_{\rm ad}=0$. 
The energy conservation
equation implies

$$
\dot\zeta=-\frac{H}{p+\rho}\,\delta p_{\rm nad}.
$$
If perturbations are adiabatic, the curvature on  uniform-density gauge 
is constant on superhorizon scales. The same holds for
the comoving curvature ${\cal R}$ as the latter and $\zeta$ are
equal on superhorizon scales, see section 7.


%%%           End of Appendixes						  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%           References starts here                                      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
\addcontentsline{toc}{section}{References}
\begin{thebibliography}{99}


\bibitem{abbot}  L.~F.~Abbott, E.~Farhi and M.~B.~Wise,
Phys.\ Lett.\ B {\bf 117}, 29 (1982).

\bibitem{ac} V.~Acquaviva, N.~Bartolo, S.~Matarrese and A.~Riotto,
arXiv:.

\bibitem{albrecht82} A. Albrecht and P. J. Steinhardt,
 Phys. Rev. Lett {\bf48}, 
1220 (1982).



\cite{Amendola:2001ni}
\bibitem{Amendola:2001ni}
L.~Amendola, C.~Gordon, D.~Wands and M.~Sasaki,
Phys.\ Rev.\ Lett.\  {\bf 88}, 211302 (2002).


\bibitem{bacci} 
C.~Baccigalupi, A.~Balbi, S.~Matarrese, F.~Perrotta and N.~Vittorio,
Phys.\ Rev.\ D {\bf 65}, 063520 (2002).



\bibitem{triangle} For a review, see, for instance, 
N.~A.~Bahcall, J.~P.~Ostriker, S.~Perlmutter and P.~J.~Steinhardt,
Science {\bf 284}, 1481 (1999).


\bibitem{Bardeen} J.M. Bardeen, Phys. Rev. D{\bf 22}, 1882 (1980);
J.M. Bardeen, P. J. Steinhardt and M. S. Turner,  
	Phys. Rev. D{\bf 28}, 679 (1983).



\bibitem{barrow93} J. D. Barrow and A. R. Liddle, Phys. Rev. D {\bf 47}, R5219 
(1993).

\bibitem{b1} N.~Bartolo, S.~Matarrese and A.~Riotto,
Phys.\ Rev.\ D {\bf 64}, 083514 (2001).

\bibitem{b2} 
N.~Bartolo, S.~Matarrese and A.~Riotto,
Phys.\ Rev.\ D {\bf 64}, 123504 (2001).

\bibitem{b4} N.~Bartolo, S.~Matarrese and A.~Riotto,
Phys.\ Rev.\ D {\bf 65}, 103505 (2002).




\bibitem{b3} N.~Bartolo, S.~Matarrese, A.~Riotto and D. Wands,
Phys.\ Rev.\ D {\bf 66}, 043520 (2002).





\bibitem{Binetruy:1996xj}
P.~Binetruy and G.~R.~Dvali,
Phys.\ Lett.\ B {\bf 388}, 241 (1996).




\bibitem{bond} J.R. Bond et {\it al.}, .


\bibitem{boomerang} See {\tt http://www.physics.ucsb.edu/~boomerang}.




\bibitem{copeland94} E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart, 
and D. Wands, Phys. Rev. D {\bf 49}, 6410 (1994).

\bibitem{ckrt} 
D.~J.~Chung, E.~W.~Kolb, A.~Riotto and I.~I.~Tkachev,
Phys.\ Rev.\ D {\bf 62}, 043508 (2000).



\bibitem{dasi} See {\tt http://astro.uchicago.edu/dasi}.


\bibitem{Dine:1997kf}
M.~Dine and A.~Riotto,
Phys.\ Rev.\ Lett.\  {\bf 79}, 2632 (1997).



\bibitem{dodelson97} S. Dodelson, 
W. H. Kinney and E. W. Kolb, Phys. Rev. D {\bf 
56} 3207 (1997).


\bibitem{dolgov} A.~D.~Dolgov,
Phys.\ Rept.\  {\bf 222}, 309 (1992).

\bibitem{Dvali:1997mh}
G.~R.~Dvali and A.~Riotto,
Phys.\ Lett.\ B {\bf 417}, 20 (1998).




\bibitem{Espinosa:1998ks}
J.~R.~Espinosa, A.~Riotto and G.~G.~Ross,
Nucl.\ Phys.\ B {\bf 531}, 461 (1998).




\bibitem{freese90} K. Freese, 
J. Frieman and A. Olinto, Phys. Rev. Lett. {\bf 
65}, 3233 (1990).

\bibitem{fr} W. Friedmann, plenary talk given at {\it COSMO02},  
Chicago, Illinois, USA,
             September 18-21, 2002.

\bibitem{guth} A.~H.~Guth,
Phys.\ Rev.\ D {\bf 23}, 347 (1981).



\bibitem{guthpi}
A. H. Guth and S.-Y. Pi, {Phys. Rev. Lett.} {\bf 49} 1110 (1982).

\bibitem{Halyo:1996pp}
E.~Halyo,
Phys.\ Lett.\ B {\bf 387}, 43 (1996).

\bibitem{hawking}
S. W. Hawking, {Phys. Lett.} {\bf B115} 295 (1982) . 

\bibitem{hu} W. Hu, this series of lectures. 

\bibitem{kinney97} W. H. Kinney and A. Riotto, Astropart. Phys. {\bf 10}, 387 
(1999), .
\bibitem{kinney98} W. H. Kinney and A. Riotto, Phys. Lett.{\bf 435B}, 272 
(1998).

\bibitem{kmr} W.~H.~Kinney, A.~Melchiorri and A.~Riotto,
Phys.\ Rev.\ D {\bf 63}, 023505 (2001).

\bibitem{King:1998uv}
S.~F.~King and A.~Riotto,
Phys.\ Lett.\ B {\bf 442}, 68 (1998).

\bibitem{preheating}
L.~Kofman, A.~D.~Linde and A.~A.~Starobinsky,
Phys.\ Rev.\ Lett.\  {\bf 73}, 3195 (1994).




\bibitem{kolbbook} E.~W.~Kolb and M.~S.~Turner,
``The Early universe,''
{\it  Redwood City, USA: 
Addison-Wesley (1990) 547 p. (Frontiers in physics, 69)}.


\bibitem{kolbreview} E.~W.~Kolb,
arXiv:.



\bibitem{Liddle} A.R. Liddle and D. H. Lyth, 1993, Phys. Rept. {\bf 231}, 1
(1993).

\bibitem{llbook} A.R. Liddle and D.H. Lyth, 
{\it COSMOLOGICAL INFLATION AND LARGE-SCALE STRUCTURE},
Cambridge Univ. Pr. (2000). 


\bibitem{Lidsey} .~E.~Lidsey, A.~R.~Liddle, E.~W.~Kolb, E.~J.~Copeland, 
T.~Barreiro and M.~Abney,
Rev.\ Mod.\ Phys.\  {\bf 69}, 373 (1997).

\bibitem{abook} A. D. Linde, {\em Particle Physics and Inflationary Cosmology},
        Harwood Academic, Switzerland (1990).


\bibitem{linde82} A.~D.~Linde, Phys.\ Lett.\  {\bf B108} 389, 1982.
\bibitem{linde83} A.~D.~Linde, Phys.\ Lett.\  {\bf B129}, 177 (1983). 
\bibitem{linde91} A.~Linde, Phys.\ Lett.\  {\bf B259}, 38 (1991).
\bibitem{linde94} A.~Linde, Phys.\ Rev.\  D {\bf 49}, 748 (1994).

\bibitem{Linde:1997sj}
A.~D.~Linde and A.~Riotto,
Phys.\ Rev.\ D {\bf 56}, 1841 (1997).




\bibitem{Lyth:1997pf}
D.~H.~Lyth and A.~Riotto,
Phys.\ Lett.\ B {\bf 412}, 28 (1997).

\bibitem{lr}
D.~H.~Lyth and A.~Riotto,
Phys.\ Rept.\  {\bf 314}, 1 (1999).


\bibitem{FIRAS} J.~Mather et al., {\it Astrophys. J.}, in press (1993).

\bibitem{map} See {\tt http://map.gsfc.nasa.gov}.
\bibitem{maxima} See {\tt http://cosmology.berkeley.edu/group/cmb}.

\bibitem{q2} V.~F.~Mukhanov,
Sov.\ Phys.\ JETP {\bf 67} (1988) 1297
[Zh.\ Eksp.\ Teor.\ Fiz.\  {\bf 94N7} (1988\ ZETFA,94,1-11.1988) 1].



\bibitem{Mukhanov} V.F. Mukhanov,  
H.A. Feldman and R.H. Brandenberger, 
	Phys. Rept. {\bf 215}, 203 (1992).


\bibitem{sf}  For a more complete pedagogical discussion
of structure formation see e.g. P.J.E.~Peebles,
{\it The Large-scale Structure of the universe}
(Princeton Univ. Press, Princeton, 1980)


\bibitem{dnsnature} P.J.E.~Peebles, D.N.~Schramm,
E.~Turner, and R.~Kron, {\it Nature} {\bf 352}, 769 (1991).


\bibitem{planck} See 
{\tt http://astro.estec.esa.nl/SA-general/Projects/Planck}.

\bibitem{Riotto:1997iv}
A.~Riotto,
Nucl.\ Phys.\ B {\bf 515}, 413 (1998).




\bibitem{Riotto:1997wy}
A.~Riotto,
arXiv:.


\bibitem{baryo1} For a review, see 
A.~Riotto,
 lectures delivered at the 
{\it ICTP Summer School in High-Energy Physics and Cosmology},
Miramare, Trieste, Italy, 29 Jun - 17 Jul 1998. 




\bibitem{baryo2} A.~Riotto and M.~Trodden,
Ann.\ Rev.\ Nucl.\ Part.\ Sci.\  {\bf 49}, 35 (1999).




\bibitem{SW} R.K.~Sachs and A.M.~Wolfe, {\it Astrophys. J.}
{\bf 147}, 73 (1967).

\bibitem{q1} M.~Sasaki,
Prog.\ Theor.\ Phys.\  {\bf 76}, 1036 (1986).

\bibitem{starob82}
A. A. Starobinsky, {Phys. Lett.} {\bf B117} 175 (1982) .


\bibitem{Stewart}  J.M. Stewart, Class. Quant. Grav. {\bf 7}, 1169 (1990).

\bibitem{homepageteg} See {\tt http://www.hep.upenn.edu/~max}.

\bibitem{mal} D.~Wands, K.~A.~Malik, D.~H.~Lyth and A.~R.~Liddle,
Phys.\ Rev.\ D {\bf 62}, 043527 (2000).








\end{thebibliography}

\end{document}

%%% End Of Stencil

