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%preprint No.DFAQ-2001/04-TH
%\hfill HIP-2002-18/TH



\title{Post-inflationary thermalization and hadronization: QCD based approach}
\author{Prashanth Jaikumar and Anupam Mazumdar} 
\address{
Physics Department, McGill University, Montr\'eal, 3600,
Qu\'ebec, Canada H3A 2T8.}
%\date{\today}
\maketitle

\begin{abstract}

We study thermalization of the early Universe when the inflaton can
decay into the Standard Model quarks and gluons, using perturbative QCD 
arguments. We comment on the nature of the thermal plasma of soft gluons
and quarks that can be formed well before the completion of reheating.
We also discuss hadronization while thermalizing the decay products of
the inflaton. Hadronization becomes a part of thermalization especially
when the reheat temperature of the Universe is sufficiently low but 
above the temperature of the Big Bang nucleosynthesis. We discuss 
relevant interaction rates of leading order processes and their 
corresponding thermalization time scale. We will also highlight 
similarities and dissimilarities vis-a-vis collider based 
ultra-relativistic heavy-ion collisions, concentrating especially 
on hadronization of the inflaton decay products.
\end{abstract}

\vskip2pc]


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

After the end of inflation, which is required to solve some of the
outstanding issues of hot Big Bang cosmology \cite{linde90}, the
homogeneous inflaton starts oscillating around its minimum. During
this period the average oscillations of the inflaton mimics the
equation of state of a pressureless fluid. The inflaton oscillations 
persist until the inflaton decays and fragments in order to provide 
a bath of relativistic species in kinetic and chemical equilibrium. 
Often in the literature the largest temperature of the relativistic 
thermal bath acquired from the complete decay products of the inflaton 
is known as the {\it reheat temperature}: $T_{rh}$. The equilibration of 
thermal and chemical bath takes finite time and the whole process is 
known as {\it thermalization} of the inflaton decay products \cite{kolb90}. 
The time scale of thermalization crucially depends upon the inflaton 
coupling to the decay products, which is usually not known when the 
inflaton is considered to be a SM and SU(3) gauge singlet \footnote{
In order to keep the success of the slow rolling scalar field inflation, 
Standard Model (SM) scalar field such as Higgs cannot act as an inflaton. 
In fact numerous attempts failed to obtain a successful model for 
inflation which could be embedded within some gauge group beyond the SM, 
such as SU(5), see for example \cite{olive90}. Therefore, lacking hard 
evidence, it is hard to pin down the precise nature of the inflaton sector.}.


Note that it is important for Big Bang nucleosynthesis (BBN) that at the 
time when electroweak interaction between neutron and proton freezes out
(around the temperature of few MeV) the Universe must be dominated by the 
SM relativistic degrees of freedom. Therefore it becomes important that 
the inflaton must couple to the SM fields if not directly but via 
non-renormalizable interactions \footnote{This impass\'e can be solved 
elegantly if we cleverly separate the good and bad qualities of the 
inflaton sector. Recently it has been observed that thermalization, 
reheating, and curvature perturbations can naturally arise from the 
known sector of the minimal supersymmetric SM (MSSM) \cite{enqvist02c}, 
thereby making part of the observable predictions more robust, while 
keeping the inflaton sector as a bolt-on accessory for solving 
the homogeneity and the flatness problems. For a review on MSSM flat 
directions and cosmology, see \cite{enqvist02}.}. 



Although a thermal bath of temperature $\sim 1$~MeV is necessary, 
there is no direct evidence of thermal history beyond the BBN era.
Therefore reheat temperature could in principle lie anywhere between 
$m_{\phi}\geq T_{rh}\geq {\cal O}(1)$~MeV, where $m_{\phi}$ is the inflaton
mass. In supersymmetric (SUSY) theories there is an upper bound on reheat 
temperature arising from thermal production of the superpartner of the 
graviton: the gravitino. Gravitinos are trouble makers in minimal 
supergravity (mSUGRA), because their mass is around SUSY breaking 
scale $\sim 100$~GeV, and their coupling to the SM fields is Planck 
mass suppressed (here we adopt the reduced Planck mass: 
$M_{\rm P}=2.4\times 10^{18}$~GeV). The decay life time exceeds that 
of the BBN era. When gravitinos decay, they dump entropy which destroys 
the abundance of Helium and Deuterium \cite{sarkar96}. In order not to 
ruin the primordial abundance of the light nuclei, the entropy dumped 
via gravitino decay must be constrained. In terms of gravitino abundance: 
$n_{3/2}/s\leq 10^{10}$, where $s$ is the entropy density of the thermal bath
\cite{ellis84,sarkar96}. This restrains the temperature of the thermal bath, 
or, the reheat temperature  should be $T_{rh}\leq 10^{9}$~GeV \cite{ellis84} 
(assuming that the Universe expands adiabatically from $T_{rh}$ onwards). 
Late inflatino (superpartner of inflaton in a supersymmetric theory) 
\cite{nilles01}, and gravitino \cite{enqvist01} production have also 
been suggested, but their abundance is usually small enough to put 
any independent constraint. Moreover, there are decay channels of 
inflatino similar to the inflaton, see \cite{allahverdi01}.


The issue of thermalization has been addressed in many papers
\cite{ellis87,dodelson88,enqvist90,enqvist93,enqvist94,zimdahl97,mcdonald00,allahverdi00}
and very recently some concrete ideas have been put forward in
\cite{sarkar00,allahverdi02}. In earlier papers
\cite{ellis87,dodelson88,zimdahl97} elastic interactions, such as
$2\rightarrow 2$ scattering and annihilation have been considered,
while in \cite{enqvist90,enqvist93}, it was shown that elastic
collisions lead to kinetic equilibrium while redistributing the
energy density, and $2\rightarrow 3$ (particle number changing)
processes that lead to chemical equilibrium take a longer time. In
\cite{mcdonald00}, it was suggested that elastic scattering followed
by prompt decay might lead to rapid thermalization. The new idea
behind rapid thermalization was proposed in \cite{sarkar00}, where
inelastic scattering such as $2\rightarrow 3$ process have been
invoked from the very beginning.  It was found that thermalization
time scale is roughly given by the inverse of the inelastic scattering
rate $\Gamma_{inel}^{-1}$. In fact our analysis would come quite close
to their analysis, but our plan is to stretch the energy scales below
the QCD scale $\Lambda_{QCD}\sim 1$~GeV. We also highlight other
processes and compare the time scales which we believe will give some
more insight in understanding thermalization.


Especially when the inflaton scale is as low as $m_{\phi}\sim {\cal O}(1)$
TeV~\cite{randall95}, see also~\cite{mazumdar99}, and the reheat temperature 
is below the QCD scale $\sim 200$~MeV, then the importance of thermalization 
and hadronization of the quark gluon plasma (QGP) becomes an interesting issue.
Further note that the inflaton decay products still have energy ranging from
${\cal O}(m_{\phi})$ down to $T_{rh}$. The hard hitting quarks and gluons 
must lose their initial momentum towards the last stages of reheating.
Depending on whether there is a hadronic bath, or a soft bath of quarks and
gluons, the inflaton decay products will lose their energy differently.


In this paper our main goal will be to understand hadronization and
the process of thermalization of the inflaton decay products. In the
next section we discuss the interactions among the inflaton decay
products, where we point out the possibility of early hadronization
from quarks via fragmentation, and their subsequent break-up back into
elementary constituents.  We describe the formation of a thermal bath
of soft gluons and quarks prior to reheating via inelastic processes.
We suggest to use similar techniques as applied in understanding the
issue of thermalization in the context of heavy-ion collisions. Often
we will draw parallels between these two systems. In the last section
we will describe hadronization of the soft bath. However, it can
happen that the still decaying inflaton can produce energetic quarks
and gluons, which can break the hadrons via deep inelastic
scattering. We discuss the hadronization of the last decaying inflaton
products, where there are three competing processes; energy loss by
inelastic scattering of quarks producing soft gluons, fragmentation of
the pair of quarks, and energy loss via deep-inelastic scattering with
the hadronic bath and then subsequent hadronization.


 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{interaction among inflaton decay products}


After inflation has ceased, the inflaton field $\phi$ begins to
execute coherent oscillations about the minimum of its potential. 
The perturbative decay occurs over several such oscillations
\footnote{The initial stages of the inflaton oscillations could give 
rise to non-thermal production of bosons and fermions, usually known 
as preheating \cite{traschen90,dolgov90}. The large occupation number 
of particles can be built, but the effect of backreaction (in the 
bosonic preheating the scattering processes with the zero mode of the 
inflaton with the excited quanta) shuts off preheating. In many cases 
due to running coupling the inflaton condensate fragments and forms 
inflatonic $Q$-balls and reheating occurs only through surface evaporation 
\cite{enqvist02a,enqvist02b}. Nevertheless it is important to highlight 
that preheating does not give rise to thermalization. Preheating produces 
multi particle non-thermal spectrum at initial stages which still requires 
to be thermalized with chemical and kinetic equilibrium. Preheating is 
successful only in specific models with large coupling of the inflaton 
and large initial amplitude of the oscillations, which makes it a rather
model and parameter dependent issue.}. The initial stage of reheating begins 
right after $H_{inf}\simeq m_{\phi}$, where $H_{inf}$ denotes the Hubble 
expansion of the Universe towards the end of inflation, and $m_{\phi}$ 
denotes the mass of the inflaton. Note that the inflaton oscillations always 
dominate the energy density until thermalization is completed
\beq
H(a)\simeq\frac{\rho_{i}}{3 M_{\rm P}^2}\left(\frac{a_{in}}{a}\right)^3\sim
\Gamma_{\phi}\,,
\eeq 
where $\Gamma_{\phi}$ is the inflaton decay rate, and $\rho_{i}$ is the
energy density stored in the inflaton sector. The inflaton oscillations
dominate until $\tau\sim \Gamma^{-1}_{\inf}$. $a$ is the scale factor 
in the Friedman-Robertson-Walker (FRW) metric, where the subscript denotes 
the initial time. In the simplest scenario, when the inflaton completely 
decays, it releases its energy into a thermal bath of relativistic particles 
whose energy density is determined by the reheat temperature $T_{rh}$, 
given by 
\begin{equation}
\label{reheat}
T_{rh} \sim g_{\ast}^{-1/4}\left(\Gamma_{\phi}M_{\rm P}\right)^{1/2}\,,
\end{equation}
where $g_{\ast}$ is the relativistic degrees of freedom. The above
estimation is simple and it still stands as the only valid estimate 
of the reheat temperature. 



As $\phi$ is a gauge singlet with respect to all gauge symmetries
under consideration, it follows that for the case of QCD with quarks,
it can couple only to color singlets, and to SU(2) gauge singlets
constructed from the left-handed quark field which forms a doublet in
the standard model. In this paper we mainly concentrate upon 
interactions 
\beq {\cal L}_{\phi g g}=\frac{\phi}{M_{\rm P}}
F^{\mu\nu}F_{\mu\nu}
\eeq 
which describes the perturbative decay of the inflaton to gluons.
Inflaton could also decay into SM quarks through
\beq {\cal L}_{\phi q q}=\frac{\phi}{M_{\rm P}}
(H\bar{q}_L)q_R\,, 
\eeq 
where $H$ is the Higgs doublet, $q_{L}$ is the SU(2) doublet, and 
$q_{R}$ is singlet. Neutral Higgs scalar eventually decays into the SM 
quarks and leptons via the Yukawa coupling. In supersymmetric theories
there are more decay channels; inflaton can decay into sleptons 
and Higgsinos. The decay products carry an energy of the order of 
the mass of the inflaton $\sim m_{\phi}$, and could be extremely 
energetic depending on the mass of the inflaton.


Note that the inflaton always couples through non-renormalizable scale 
which could be taken here as the Planck mass. In order to set our 
notation we define the inflaton coupling to the SM fields as
$\alpha_{\phi}\ll(m_{\phi}/M_{\rm P})$. 


We can now estimate the number density of hard hitting pairs of quarks, 
and gluons at time: $\tau>\tau_{in}$, where $\tau_{in}$ is the 
reference time when the inflaton starts oscillating. The number 
density is given by 
\beq
n_{\chi}(\tau)=2n_{\phi}(a_{in})(1-{\rm e}^{-\Gamma_{\phi}(\tau-\tau_{in})})
\biggl(\frac{a_{in}}{a}\biggr)^3\,,
\label{ninf}
\eeq
where $n_{\phi}(a_{in})=\rho_{\phi}(a_{in})/m_{\phi}$, and we collectively 
designate hard hitting quarks and gluons by $\chi$. Note that we are
neglecting here the particle number changing processes in the above 
estimation.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Early Fragmentation}


In reality, we expect pair-production mechanisms or {\it fragmentation} 
processes from the inflaton decay products that would produce more 
quarks and gluons. From Eq.~(\ref{ninf}), it is clear that for very 
early times $\tau\simeq\tau_{in}$, there is a small but finite number
density of hard quarks and gluons.  The inverse spacing between two
typical quarks could be of order $\Lambda_{QCD}$, as these quarks rush
away from each other very fast due to their large kinetic energy,
especially if $m_{\phi}\gg\Lambda_{QCD}$. At the separation scale
$\Lambda_{QCD}^{-1}$, strong confining forces come into play, and the
string tension which was low when the quarks were close together, now
increases so that the color string is stretched and breaks at the
threshold to create a quark-anti-quark pair from the QCD vacuum. In
turn, there arise strong forces between these newly created quarks and
the original quarks which are separating rapidly, resulting in further
pair production. It is important to note that unlike for the original
fast moving quarks, the pair-produced quarks will immediately
hadronize with unit probability due to confinement. In fact, one can
estimate the hadron multiplicity at this very early stage due to such
a fragmentation process.


A simple parameterization of the non-perturbative aspects of these quark 
jets is given by the Feynman-Field approach~\cite{Feyn78}, which 
models the recursive principle described above. In their simplest 
form, the fragmentation functions $D_q^h(z)$ are often written 
as~\cite{halzen84} 
\beq 
\label{had}
D_q^h(z)=(n+1)<z>\frac{(1-z)^n}{z}\,, 
\eeq 
where $z=E_h/E_q$ is the fractional energy of the quark $q$ residing 
in the hadron $h$, and $n$ is a constant chosen as a best fit by 
examining the $e^+e^-\rightarrow h+X$ data. $<z>$ is the average 
fraction of quark energy carried by hadrons of type $h$ after 
fragmentation. The energy scale of these experiments is typically 
$E_q\sim 1$~TeV, while $E_q\sim 10^{10}$ TeV is possible for the 
quark produced as an inflaton decay product when
$m_{\phi}\sim 10^{13}$~GeV in the case of chaotic inflation
\cite{linde90}. At such large energies, perturbative QCD corrections
to scaling of the fragmentation functions will be important. Physically, 
these corrections come from including gluon emissions from the emitted 
quark (though they are not the only cause of scaling violations that 
can be imagined at low energies, where one has to consider the crossing 
of the threshold for heavy quark production). However a conservative 
estimate of the hadron multiplicity can be derived from the momentum 
and probability conservation requirements imposed on the fragmentation
functions~\cite{Feyn78} 
\beqy 
\sum_{h}\int_0^1zD_q^h(z)dz=1\,,\nonumber \\ 
\sum_{q}\int_{z_{min}}^1[D_q^h(z)+D_{{\bar q}}^h(z)]dz=n_h\,, 
\eeqy 
where $z_{min}=2m_h/E_q$ is the threshold energy for producing a 
hadron of mass $m_h$. In the above equation $n_h$ is the approximate 
hadron multiplicity 
\beq 
n_h\sim {\rm log}\biggl(\frac{m_{\phi}}{2m_h}\biggr)\,.  
\eeq 
The logarithmic growth with available energy comes from the fact that 
the soft part ($1/z$) in Eq.~(\ref{had}) dominates the fragmentation 
process. This particular scenario of early hadronization of fragmented 
quarks does not take place in heavy ion collisions due to the large 
number density of un-equilibrated partons (quarks and gluons), which 
causes them to remain at relatively close separation. It is only when this
close-packed debris of quarks and gluons becomes sufficiently dilute
that hadronization followed by chemical and thermal freeze out, in
that order, can occur. In ultra-relativistic heavy ion collisions it
is also believed that the quarks and gluons attain a thermalized state
before pressure-driven expansion takes over and pushes the
constituents of this high temperature plasma away from each other.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Break-up of hadrons at an early stage}

In our case also, unless the inflaton mass is small, fragmentation is 
unlikely to occur. Since by the time quarks attain a typical separation 
of $\Lambda_{QCD}^{-1}$, the inflaton decay products rapidly build up 
a large number density of hard quarks, which can screen strong 
forces. Although hadrons (we can take them to be dominated by the 
lightest modes, the pions) would be formed at such an early time,
they would soon break up as bombardment from the hard particles 
(quarks and gluons) becomes more frequent. This is because there 
is already a large number of inflaton quanta still decaying: 
$n_{\phi}=\rho_{\phi}/m_{\phi}\simeq m_{\phi}M_{\rm P}^2$. 



The break-up happens through the process of deep inelastic scattering, 
which involves a large momentum transfer. Note that the typical energies 
of these hard particles is well above the hadron binding energy of few 
hundred MeV. By the uncertainty principle, we can roughly estimate 
that a break up is likely to occur within a time scale 
\beq
\tau \approx \frac{\hbar}{Q}
\eeq
where $Q$ is the typical momentum transfer, which can be as large 
as $m_{\phi}$. At large values of $Q$ permissible in cosmological scenarios, 
the structure functions are unknown, but we estimate a bound $\tau_{max}$ 
within which the hadron is sure to break up. This corresponds to a 
momentum transfer of the order of the hadron binding energy $\sim 1$~GeV
\beq
\label{unc}
\tau_{max}\sim 10^{-24}{\rm s}\,, 
\eeq 
which is the typical strong interaction time scale. 



Another expression of the break-up time follows from the mean 
free path argument, which relates to the cross-section for deep 
inelastic scattering
\beq
\tau\sim (n_q \sigma_{DIS})^{-1}\,, 
\eeq 
where $n_q$ is the number density of quarks at early times (it is 
then proportional to the time $\tau-\tau_{in}$) and $\sigma_{DIS}$ 
is the integrated cross-section for deep inelastic scattering. 
For a virtual photon of invariant mass $t$, and for incident and 
scattered quark energies $E$ and $E^{\prime}$, we have for the 
differential cross-section~\cite{halzen84} 
\beq
\frac{d\sigma}{dE^{\prime}d\Omega}=\frac{\alpha_e^2}{t^2}
\frac{E^{\prime}}{E}(L^q)^{\mu\nu}W_{\mu\nu}\,,
\eeq 
where $\alpha_e=e^2/4\pi$, $L^{\mu\nu}$ is the standard leptonic
tensor, and $W_{\mu\nu}$ can be expressed in terms of hadron structure
functions~\cite{halzen84}. 
 


It is certain, however, that due to the rapidly increasing number of
hard particles with time, the hadrons will break up into
quarks and gluons once again. Thus, within a time of at most
$10^{-18}$s, (assuming a typical cross-section of a few micro barns),
the universe is filled with a large number density of quarks and
gluons, such that they are interacting weakly, and behaving almost as
free particles. For $m_{\phi}$ large (${\cal O}(10^{12})$GeV), this
happens much sooner and perhaps without any fragmentation process.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Formation of the initial thermal bath of soft quarks and gluons}

We may now ask the question as to how this dense system of hard quarks
and gluons will thermalize, and what the corresponding temperature
would be. In the context of high energy heavy-ion collisions of large
nuclei such as those studied at the Relativistic Heavy Ion Collider
(RHIC), thermalization is not guaranteed by perturbative QCD evolution
of the assumed initial conditions~\cite{wang96}. The initial parton
scattering and cascading processes do not offer a definite answer to
the question of an early thermalization (within 1fm/c). On the other
hand, parameters such as the elliptic flow $v_2$, which relates to
the spatial anisotropy of non-central collisions of nuclei~\cite{jyo92}, 
when calculated on the basis of a hydrodynamic evolution (which assumes 
a thermalized state is formed very early on), agree quantitatively with 
data collected at RHIC. This seems to indicate that thermalization does 
indeed occur very early on, during the longitudinal expansion of the gluon 
rich soup. 


To begin with, we note that the previous cosmological studies
\cite{sarkar00,allahverdi02} have addressed thermalization of
the inflaton decay products in some detail, albeit within the
framework of a generic and unspecified theory of fermions and 
gauge bosons. By adopting a particular theory, namely QCD, we 
are aiming at two points. The first one is to map the cosmological 
problem to the heavy-ion physics that is under active investigation 
in order to make the physics qualitatively definite. The second 
is to understand the resulting quantitative estimates of thermalization 
time and hadronization time for the early universe when it was initially
filled with quarks and gluons, and also the time to reach the final
reheat temperature $T_{rh}$. It is worthwhile at this stage to 
review and recast the arguments for rapid thermalization that appear 
in references \cite{sarkar00,allahverdi02} in a QCD based approach.



The mechanism therein involves creating a soft thermal bath of
fermions and gauge bosons from scattering and decay of hard particles
(in our case, these would be the quarks and gluons) which then quickly
thermalize the remaining hard constituents because of two reasons. 
Firstly, the energy transfer is more efficient in scattering hard 
particles off of soft particles compared to hard-hard scattering.
Secondly, the $2\rightarrow 2$ scattering is accompanied by the
emission of a soft gluon from one of the legs of the $2\rightarrow 2$ 
diagram. This is the gluon Bremsstrahlung. The emitted gluon can now
be involved in a Bremsstrahlung process with another quark (or gluon),
thereby leading to an exponential growth in the number density of soft
particles~\cite{sarkar00};
\beq
\frac{dn_g}{d\tau}\sim(\sigma_{qq\rightarrow qqg} n_q^2+
\sigma_{qg\rightarrow qgg} n_qn_g)~\quad \,.
\eeq 
Due to pair annihilation processes, the thermal bath in QCD will be 
composed of soft quarks as well as soft gluons. The soft quarks and 
gluons can reach a thermal plasma with an instantaneous temperature
\cite{kolb90} 
\beq
\label{inst}
T_{inst}\sim \biggl(g_*^{-1/2}H(\tau_{inst})\Gamma_{\phi} M_{\rm P}^2
\biggr)^{1/4}\,,
\eeq  
where $H(\tau_{inst})\geq \Gamma_{\phi}$ is the Hubble parameter at 
the time when the soft thermal bath is created. This instantaneous 
temperature reaches its maximum $T_{max}\leq m_{\phi}$ soon after 
the inflaton field starts to oscillate around the minimum of its potential. 
Also an important point to note here is that since the inflaton energy 
density is still dominating 
$\rho_{\phi}\approx H^2M_{\rm P}^2\sim a^{-3/2}$, the 
instantaneous temperature  falls as 
\begin{equation}
T_{inst}\sim a^{-3/8}\left(g_{\ast}^{-1/2}\frac{\Gamma_{\phi}}{H}\right)^{1/4}
\,,
\end{equation}
instead of $T_{inst}\sim a^{-1}$. Numerical simulations also support 
this argument, see \cite{chung99}. In the next section we will estimate 
$T_{max}$.


For elastic processes, one cannot reduce the average energy of the
system quickly since no new particles are created (we are ignoring
the red-shift of the particle momenta for the moment). The
Bremsstrahlung process redistributes the energy among softer
constituents and these soft particles act as a very efficient sink of
energy once the bath starts to form. An analogous mechanism for
heavy-ion collisions was suggested by Shuryak~\cite{shuryak92} that a
thermally equilibrated quark-gluon plasma (QGP) would be created in a
two-step process (``hot-glue scenario''), wherein the gluonic bath
would be created in a much shorter time than it took the quarks to
thermalize \footnote{More recently, a proposed method of thermalization 
in heavy-ion collisions at RHIC or the Linear Hadron Collider (LHC) 
mirrors that discussed in the cosmological context. In~\cite{baier00}, 
the authors advocate that an overwhelmingly large number of soft gluons can be 
produced in the early stages of the collision which then forms a bath 
that draws energy from the hard constituents. Full thermalization is 
defined to have occurred only when the primary hard gluons have lost 
almost all their energy. Their estimations of time scale for 
thermalization is presented in terms of the gluon saturation scale 
(beyond which the gluons inside ultra relativistic nuclei cannot be 
described as a classical coherent field) and the strong coupling 
constant (which is weak at the saturation scale).}.


We now proceed to make some quantitative estimates along the lines
of~\cite{sarkar00} for the cross-sections and scattering rates. We
note that these are order of magnitude estimates, and more accurate
methods such as using Boltzmann equations with collision terms taken
from perturbative QCD may be developed as has been done for the finite
size system in heavy-ion collisions~\cite{wong96}. However, in light
of the uncertainties in the cosmological context, it is adequate to
first obtain an order of magnitude estimates for a proposed physical
scenario.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$2\rightarrow 2$ processes}

We begin by considering the $2\rightarrow 2$ processes among quarks 
mediated by one gluon exchange. The rationale for calculating 
perturbative cross-sections is that the quarks and gluons have 
large momenta and large number densities. We take the dominant 
interaction between them to be one gluon exchange with a minimum 
momentum transfer that is characterized by the typical inter-particle 
separation (see below). 



In terms of the Mandelstam variables $t$, we may write the most
singular part of the differential cross-sections in elastic scattering
as~\cite{peskin} 
\beqy 
\frac{d\sigma^{qq\rightarrow
qq}}{dt}&\sim&\frac{8\pi\alpha_s^2}{9t^2}\,, \nonumber\\
\frac{d\sigma^{qg\rightarrow
qg}}{dt}&\sim&\frac{2\pi\alpha_s^2}{t^2}\,, \nonumber\\
\frac{d\sigma^{gg\rightarrow
gg}}{dt}&\sim&\frac{9\pi\alpha_s^2}{2t^2}\,, 
\eeqy 
where $\alpha_s^2=g^2/4\pi$, with $g$ the strong coupling constant, and we
have kept only the dominant kinematic contributions in the scattering
channels (the $t-$ channel). There are also annihilation processes
that can produce chemical equilibrium between quarks and gluons.
\beqy 
\frac{d\sigma^{q\bar{q}\rightarrow gg}}{dt}&\sim&\frac{32\pi\alpha_s^2}
{27s^2}\biggl[\frac{u}{t}+\frac{t}{u}\biggr]\,, \nonumber\\
\frac{d\sigma^{gg\rightarrow q\bar{q}}}{dt}&\sim&\frac{\pi\alpha_s^2}{6s^2}
\biggl[\frac{u}{t}+\frac{t}{u}\biggr]\,. 
\eeqy 
Note that the integrated cross-section goes as $\sim 1/t$ for the
scattering processes. On the other hand for the annihilation process 
the cross-section goes as $\sim {\rm log}(1/t)$. These potential divergences 
can be regulated by medium effects. 




When a plasma of soft quarks and gluons is eventually formed, exchanged 
gluons will be screened over a distance scale $\mu^{-1}\sim (1/gT)$ 
corresponding to the momentum scale $gT$~\footnote{The magneto static 
gluons are not screened perturbatively, and this scale does not apply to
them. If magnetic screening effect is also taken into account then 
$\mu^{-1}\sim (1/g^2T)$. However, for our estimation purposes, a starting 
point is to take the minimal form for the screening of the gluon propagator.}
\cite{lebellac}. In the case of quarks being exchanged in a plasma, we can 
use the cutoff given by the kinetic theory which gives the quarks an 
effective mass of ${\cal O}(gT)$. A more consistent perturbative approach 
in a thermal plasma would be to the Braaten-Pisarski re-summation 
scheme~\cite{braaten90}, which incorporates effective propagators and 
vertices for the quark and gluons, and amounts to a reordering of 
perturbation theory. Since we are interested only in order of magnitude 
estimations here, we choose the bare propagators and minimal expressions 
for the cutoff. 



For a sufficiently dilute system of quarks and gluons, the concept of
a thermal plasma does not make sense, in which case, we choose to
regulate in the infra-red by using, at any instant, the inverse of the
inter-particle separation of the decay products of the inflaton (hard
quarks and gluons which we denote by $q$). This is determined by
Eq.~(\ref{ninf}) at any instant of time. For $\Gamma_{\phi}\leq H(\tau)$, 
we expand the right hand side of the expression Eq.~(\ref{ninf}) in terms 
of $\Gamma_{\phi}/H$, and with the help of Eq.~(\ref{inst}), we immediately 
obtain the number density $n_{\chi}$ of hard quarks and gluons, and from it, 
the infra-red cutoff
\begin{equation}
n_{\chi}^{-1/3}\sim \left(\frac{T_{inst}^4}{m_{\phi}}\right)^{-1/3}\,,
\end{equation}
It should be noted that the expressions for energy transfer will be
less divergent than those for the absolute cross-section since the
energy transfer also tends to zero for very soft scattering. This
reduces the efficacy of soft processes for thermalization.


The rate for thermalization assuming only such $2\rightarrow 2$ elastic 
scattering processes can be estimated as in~\cite{sarkar00} by using 
Bjorken's estimate of the energy loss suffered by fast moving partons 
incident on a (as yet) un-thermalized system of quarks and gluons 
characterized by a (non-equilibrium) effective distribution function
$\rho(k)=(2/3)\rho_q+(3/2)\rho_g$, and a temperature $T_{inst}$.  
\beq
\frac{dE}{d\tau}=\pi\alpha_s^2\biggl(\frac{2}{3}\biggr)^{\pm 1}\int
\frac{d^3k}{k}\rho(k){\rm
log}\biggl(\frac{\nu_{max}}{\nu_{min}}\biggr)\,. 
\eeq 
In the above equation, $\nu=(E/s)|t|$. In our case 
$\nu_{max}\sim m_{\phi}$ and $\nu_{min}\sim T_{inst}$
\cite{sarkar00,bjorken82}. The factor $2/3$ applies to incident 
quarks while $3/2$ applies to incident gluons ($E$ is their incident 
energy). If the lower cutoff $\nu_{min}$ is taken to be 
$\sim \alpha_{s}^{1/2}T_{inst}$, this leads to an elastic scattering 
rate of
\beqy
\label{elas}
\Gamma_{\rm elas}&=&\frac{1}{E}\frac{dE}{d\tau}\sim
\frac{\alpha_s^2n_{\chi}}{m_{\phi}^2} {\rm
log}\biggl(\frac{t_{max}}{t_{min}}\biggr)\nonumber\\
&&\sim\frac{\alpha_s^2n_{\chi}}{m_{\phi}^2} {\rm
log}\biggl(\frac{m_{\phi}}{\alpha_{s}^{1/2}T_{inst}}\biggr)\,, 
\eeqy 
which is similar to the scattering rate obtained in~\cite{sarkar00}, 
except that $T_{rh}\rightarrow T_{inst}$, and there is another factor 
of $\alpha_{s}^{1/2}$ appearing in the denominator of the logarithm. 
The reason for choosing $\nu_{min}\sim T_{inst}$ is that the system 
is characterized by a temperature $T_{inst}$ rather than $T_{rh}$ at 
the time when the plasma of soft gluons and quarks are 
formed~\cite{allahverdi02}.


Note that this extra enhancement in the elastic cross section
compared to \cite{sarkar00,allahverdi02} does not significantly 
change the scattering rate, because the modification is inside the
logarithm. However, we will see that inclusion of minimal screening
can have a large effect for the case of inelastic $2\rightarrow 3$
processes, which we address in the next section.


As emerged in studies of thermal and chemical equilibration in 
heavy-ion collisions, inelastic processes ($2\rightarrow n$, where $n>2$) 
are likely to be the dominant mechanism for rapid thermalization and 
entropy generation. Within perturbative QCD itself, it has been shown 
that processes that are higher order in $\alpha_s$ can be more important 
for equilibration. Thus, gluon branching from a quark or gluon line must 
be taken into account, and we turn now to consideration of such 
$2\rightarrow n$ processes.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$2\rightarrow n$ processes}


To achieve thermalization at a temperature well below $m_{\phi}$, one
needs to generate many soft particles so that while the energy density
remains constant, the number density rapidly increases. Then the
average energy per particle also decreases. The creation of a soft
gluonic bath can proceed via Bremsstrahlung emission of a gluon from a
quark or gluon participating in a scattering process ($2\rightarrow 3$
process). The energy loss due to Bremsstrahlung can be mitigated by
the Landau-Pomeranchuk-Migdal (LPM) effect~\cite{lpmigdal} in a dense
medium. This effect has been studied in a QED~\cite{moore} 
as well as in a QCD plasma, with significant modifications in the energy 
loss profile of a parton jet traversing a QCD plasma~\cite{wang93}. If 
the mean free path of the incident parton is small compared to the 
formation time of the emitted gluon (which can become large for small 
angle scattering by uncertainty principle arguments), interference 
effects between multiple scattering has to be taken into account. 
This issue has been addressed in calculations of photon and gluon 
production rates from an equilibrated QGP, as well as in the phenomenon 
of jet quenching. The importance of this effect can be gauged as follows. 



Suppose the incident quark of energy $E_q$ emits at a small angle
$\theta$, a gluon ($k^{\mu}=(\omega,k_z,k_{\perp})$) with energy
$\omega_g\sim T_{inst}$, thereby itself losing an energy $\Delta E(k)$. 
The formation time of the emitted gluon would be given by the uncertainty 
principle as~\cite{wang93} 
\beq
\tau(k)\sim\frac{\hbar}{\Delta E(k)}\sim
\frac{2\omega_g}{k_{\perp}^2}\sim \frac{2}{\omega_g\theta^2}\,, 
\eeq
where $\theta\approx k_{\perp}/\omega_g$. The ratio of the mean free
path to the formation time is given by 
\beq
\frac{L}{\tau}\sim\frac{1}{n_{\chi}\sigma_{\rm inel}}
\frac{\omega_g\theta^2}{2}\,,
\label{lpm}
\eeq 
where $\sigma_{\rm inel}=\sigma_{\chi\chi\rightarrow \chi\chi g}$ is 
the inelastic cross-section. It has a logarithmic dependence on $t$ 
from the Bremsstrahlung emission, and a $1/t$ dependence from the exchanged 
momentum. The extra emitted gluon gives one extra power of $\alpha_s$ 
compared to the elastic case leading to   
\beq
\sigma_{\rm inel}\sim\frac{\alpha_s^3}{t_{\rm min}}{\rm log}
\biggl(\frac{m_{\phi}^2}{t_{\rm min}}\biggr)\,.  
\eeq 
Substituting the above equation into Eq.~(\ref{lpm}), and using the fact that
$t_{min}\sim \alpha_s T_{inst}^2$ and $n_{\chi}\sim T_{inst}^4/m_{\phi}$, 
we find 
\beq
\frac{L}{\tau}\sim\frac{m_{\phi}}{\alpha_s~T_{inst}~{\rm log}
\biggl(\frac{m_{\phi}^2}{\alpha_s T_{inst}^2}\biggr)}\,.
\label{landau}
\eeq 
If this ratio is much less than one, the LPM effect causes
suppression of energy loss, and consequently, the hard quarks (or
gluons) lose their energy at a slower rate. This can weaken the
functional dependence of $dE/d\tau$ on the incident energy~\cite{wang93}, 
leading to a suppression of energy loss, and an increase in thermalization 
time. From Eq.~(\ref{landau}), it is evident that for a given 
$\alpha_s\ll 1$, the assumption of additive energy loss from successive 
scattering applies, since the fraction $T_{inst}/m_{\phi}$ is much less 
than unity implying that $L/\tau\gg 1$ (the logarithm cannot counteract 
the dependence from the prefactor). 




Continuing with the inelastic process in the limit $L/\tau\gg 1$ 
(additive energy loss), the time scale for an inflaton decay product 
to lose an energy $\sim m_{\phi}$ by such processes turns out to be much 
smaller than in the elastic case, as was shown in~\cite{sarkar00}. 
The reason is that the soft gluon population (neglecting its 
annihilation to fermions) grows exponentially since each soft gluon 
produced radiatively also acts as a subsequent scattering center for 
the production of further soft gluons. The bath will be actually 
composed of a certain number of fermions as well, due to the forward 
and backward annihilation processes between quarks and gluons. However, 
for our estimation purposes, it only matters that hard quarks/gluons 
are scattering off soft particles, whose total number is important 
though not its species content. Due to the (almost) exponential 
increase in the number of soft particles, we estimate that the rate 
of energy loss is given by 
\beq
\frac{dE}{d\tau} \approx n_g\sigma_{\rm inel}T_{\rm inst}\,, 
\eeq 
where $n_{g}$ is the number density of the soft quarks (and gluons)
$n_{g}\sim g_{\ast}T_{inst}^3$. The inelastic scattering rate is 
then given by
\beq
\Gamma_{\rm inel}=\frac{1}{E}\frac{dE}{d\tau}\sim 
\frac{g_{\ast}\alpha_s^2T_{inst}^2}{m_{\phi}}~{\rm log}\biggl(
\frac{m_{\phi}^2}{\alpha_sT_{inst}^2}\biggr)\,.
\eeq
Note that if we were to consider higher order gluon emission 
(2 or more emitted gluons), $\Gamma_{\rm inel}$ would be suppressed 
by further powers of $\alpha_s$ and will not compete with the 
$2\rightarrow 3$ process. Furthermore, emission of gluons softer 
than $T_{inst}$, which would ultimately require handling collinear 
divergences are not considered since those ultra soft gluons would 
have to be re-scattered back up to a momentum of $T_{inst}$. As we 
are interested only in the time scale to create a thermalized plasma 
at roughly $T_{inst}$, this aspect will not concern us 
here~\cite{sarkar00}. We will return to the point about the 
$2\rightarrow n$ processes at the end of this section. Continuing 
with our estimate, the time to lose an energy $m_{\phi}$ is then  
\beq 
\tau_{\rm inel}=\Gamma_{\rm inel}^{-1}\sim \frac{m_{\phi}}{\frac{dE}{d\tau}}
\sim\frac{m_{\phi}}{g_{\ast}\alpha_s^2~T_{inst}^2~{\rm log}
\biggl(\frac{m_{\phi}^2}{\alpha_s T_{inst}^2}\biggr)}\,.\label{tinel}
\eeq
This can be re-expressed as
\beq
\tau_{\rm inel}\sim\biggl(\frac{T_{inst}^2}{m_{\phi}^2}\biggr)
\tau_{\rm elas}\,,
\eeq 
where $\tau_{\rm elas}=\Gamma_{\rm elas}^{-1}$. Note that this result is a
factor $\alpha_{s}$ larger than previous estimates \cite{sarkar00}, 
because the lower cutoff is taken as the screening mass rather than 
the typical temperature (which would be the energy of the individual 
constituents themselves, and would be ignorant of many-body effects). 
Further note that $\tau_{\rm inel}\ll \tau_{\rm el}$, because 
$T_{inst}\ll m_{\phi}$ at later times. 


Now we are able to estimate the largest temperature of the
instantaneous thermal bath before reheating. By using 
$H(t_{inst})\simeq \Gamma_{inel}$, Eq.~(\ref{inst}), and 
$\Gamma_{\phi}=g_{\ast}^{1/4}T_{rh}^2/M_{\rm P}$, we obtain the maximum
instantaneous temperature 
\begin{equation}
\frac{T_{max}}{T_{rh}} \sim \left(g_{\ast}^{3/4}\alpha_{s}^2\frac{M_{\rm P}}
{m_{\phi}}\right)^{1/2}\,.
\end{equation}
Note that $T_{max}$ is couple of magnitudes larger than the reheat 
temperature, but still smaller compared to the inflaton mass: 
$T_{max}\leq m_{\phi}$. 


It is sufficient for us to estimate the thermalization time by 
studying $2\rightarrow 3$ processes, since further gluon branching 
($2\rightarrow n, n>3$) will come with additional factors of 
$\alpha_s\ll 1$. The lower scattering cross-section for such 
processes renders them higher-order corrections to the inelastic 
scattering rate and the thermalization time scale. This conclusion 
can change if the mean free path of the hard particles becomes 
comparable to the formation time of the emitted gluon, i.e., if 
the LPM suppression is severe. 


We have also justifiably neglected the inverse reaction of quarks 
and gluons going into the inflaton, the reason being that 
$\alpha_{\phi}\ll \alpha_s$ and the large number of relativistic
degrees of freedom of quarks and gluons. It would be an interesting 
study however, to consider the inverse reactions amidst quarks and 
gluons themselves, to determine the evolution of the quark-gluon 
number densities within a Boltzmann equation approach. Numerical 
results have been obtained previously in the context of heavy-ion 
collisions~\cite{wong96}, and can be adapted to the cosmological 
problem, which we defer to future work.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Towards hadronization}


Thus far, we have described a generic scenario towards reheating the
Universe without bothering too much on the reheat temperature. An
implicit assumption was made that the reheat temperature was sufficiently 
above the QCD scale. However, a priori there is no reason that the 
Universe could not be reheated at a temperature lower than the QCD scale and
above the BBN temperature. Such a scenario is plausible if the inflaton 
scale is sufficiently low. Keeping the reheat temperature 
$T_{rh}< \Lambda_{QCD}\approx 200$~MeV, we will be concentrating upon the
thermalization issue. 


Naturally if the thermal bath of soft quarks and gluons is formed near 
$T_{\rm inst}\sim T_c$, where $T_c$ is the critical temperature 
($\sim 200$ MeV), the quarks and gluons will rapidly hadronize. 
However, if the thermal bath is initially formed at a much higher 
temperature than a GeV, it will hadronize at a later time when
the expansion of the universe has red-shifted the energy down to the
confinement scale~\footnote{Hadronization in heavy-ion collisions
follows the expansion of a relativistic fluid, and its subsequent
transition to a hadronic state as its energy density falls below the
critical energy density of a quark-hadron phase transition.
Three-dimensional expansion of the fluid implies that the time to
reach the critical temperature is given by
$\tau_c=({T(\tau_0)}/{T_c})\tau_0$, where
$\tau_0\simeq 1$ fm/c in heavy-ion collisions.}. This is particularly
important if thermalization occurs within a Hubble time $H^{-1}$.  



Since the inflaton decay rate $\Gamma_{\phi} \ll H$, the hard inflaton 
decay products will tend to destroy the hadronic state. In any case, 
the final state obtained must be a dense hadronic system with different
degrees of freedom and thermodynamics than an equilibrated plasma of
quarks and gluons. 


Let us imagine that there already exists a plasma of soft quarks and gluons.
Since the temperature of the plasma is inversely related to the scale factor 
of the FRW metric as $T\propto a^{-3/8}$. It follows that during the inflaton
oscillations dominated phase 
\beqy 
\frac{T(\tau)}{T(\tau_0)}&=&\biggl(\frac{\tau_0}{\tau}\biggr)^{1/4}\,,
\quad\nonumber \\
\frac{s(\tau)}{s(\tau_0)}&=&\frac{T^3(\tau)a^3(\tau)}{T^3(\tau_{0})
a^3(\tau_{0})}=\left(\frac{\tau}{\tau_{0}}\right)^{5/4}\,.
\label{entropy}
\eeqy 
where $\tau_{0}$ denotes any reference time.
It is interesting to mention here that similar relations are derived 
for the evolution of the relativistic plasma of quarks and gluons 
formed in heavy-ion collisions~\footnote{There, it is appropriate 
to adopt Bjorken's idealized hydrodynamical scenario of the quark-gluon 
plasma as an expanding relativistic fluid~\cite{bjorken83}.
Approximate longitudinal boost invariance and the ideal equation of 
state for a relativistic fluid imply that the temperature and entropy 
density depend on the proper time $\tau$ as 
${T(\tau)}/{T(\tau_0)}=({\tau_0}/{\tau})^{1/3}$, and
${s(\tau)}/{s(\tau_0)}=({\tau_0}/{\tau})$.}.



As we run the time until $\tau=\tau_c$ when $T=T_c\sim 200$~MeV, the
critical temperature for hadronization, the fluid is dilute enough
that strong non-perturbative forces recombine the quark-gluon soup
into colorless hadrons. This constitutes a phase transition whose
nature depends on the number of light quark flavors (from universality
arguments~\cite{wilczek84}) and the quark masses (which act as
external fields). The transition in the early universe occurs at
almost zero entropy per baryon, and assuming the strange quark to be
light, we expect a first order phase transition with release of latent
heat. As the strange quark mass is tuned to be heavy, the transition
changes to weakly first order (or analytic crossover) and then becomes
second order~\cite{rajagopal}. Given the uncertainty in the exact 
value of the strange quark mass, we assume it to be light on the 
$\Lambda_{QCD}$ scale. We can say that the hadronization proceeds 
by a first order phase transition with bubbles of hadronic matter 
appearing within the quark-gluon plasma, which grows in number till
the end of the phase transition~\cite{shuryakbook}. The time for this 
transition to occur is denoted by $\tau_h$; the hadronization time. 


It is possible to estimate $\tau_{h}$ as follows~\cite{CYKbook}. 
Throughout the phase transition, the temperature remains constant 
at $T_c$, while entropy is decreased in going from the quark-gluon 
to the hadronic phase. This means that Eq.~(\ref{entropy}) cannot be
applicable in describing the phase transition. It is helpful to imagine
the quark gluon plasma as a subsystem where the temperature remains
$T(\tau_0)=T(\tau)=T_{c}\sim 200$~MeV for the duration of the phase 
transition. Then, the ratio of the entropy densities is simply the ratio of
$\epsilon + p$ of the plasma, which scales as the fourth power of temperature. 
From Eq.~(\ref{entropy}), it follows that 
\begin{eqnarray}
\label{entropy1}
\frac{s(\tau)}{s(\tau_c)}&=&\frac{\epsilon(\tau)+p(\tau)}{T_{c}}\cdot
\frac{T_{c}}{\epsilon(\tau_{c})+p(\tau_{c})}\, \nonumber \\ &=&
\left(\frac{\tau_c}{\tau}\right)\,,\quad
\tau>\tau_c \,,
\end{eqnarray} 
In the case of heavy-ion collisions, a rerun of the above steps yields 
the relation ${s(\tau)}/{s(\tau_c)}=({\tau_c}/{\tau})^{4/3}$.



This means that the phase transition in the early universe occurs
at a slower rate than in heavy-ion collisions, since the
entropy/particle is some fixed quantity in the two different
phases. We will see that this fact is borne out by our final
expression for the hadronization time. During the transition, the
system can be described by a mixed phase, with a fraction $f(\tau)$ in
the quark-gluon phase ($1-f(\tau)$ in the hadronic phase). Since
entropy is additive, we find using Eq.~(\ref{entropy1}), 
\beqy
f(\tau)&=&\frac{1}{s_{qg}(T_c)-s_h(T_c)}\biggl(\{f(\tau_c)s_{qg}(T_c)+
[1-f(\tau_c)]\nonumber \\
&&s_h(T_c)\}\times\frac{\tau_c}{\tau}-s_h(T_c)\biggr) 
\eeqy
The entropy densities in the two phases can be traded for the respective 
degeneracies ($g_{qg}\sim 37$ and $g_h\sim 3$) to give 
\beq
f(\tau)=\frac{1}{g_{qg}-g_h}\left({f(\tau_c)g_{qg}+[1-f(\tau_c)]g_h}
\frac{\tau_c}{\tau}-g_h\right)\,.
\eeq 
At $\tau=\tau_h$, the quark-gluon phase is completely hadronized. Thus, 
it follows from the above equation that 
\beq
\tau_h=\left[\frac{g_{qg}}{g_h}f(\tau_c)+1-f(\tau_c)\right]\tau_c 
\label{hadtime}\,.
\eeq 
Starting with quark-gluon matter at $f(\tau_c)=1$, we find that
$\tau_h\sim 10\tau_c$. In heavy-ion collisions, this relation gives
$\tau_h\sim 6\tau_c$~\footnote{In the heavy-ion case, there is a power 
of $3/4$ for the term within square brackets in Eq.~(\ref{hadtime}), 
which follows from the different scaling law of entropy density with time. 
The fractional power implies a reduced hadronization time in heavy ion 
collisions.}. 




Coming back to Eq.~(\ref{entropy}), we observe that $\tau_c$ denotes the 
time for the plasma to cool down to the critical temperature for 
hadronization $T_c$. The origin of time is taken to be the moment 
when the inflaton begins its oscillations. Taking 
$\tau_{\rm inf}\sim m_{\phi}^{-1}$, we can estimate an upper bound 
on $\tau_c$ as 
\beq 
\tau_c\leq m_{\phi}^{-1}\alpha_{\phi}^{-1}\sim \Gamma_{\phi}^{-1}\,,
\eeq 
where $\alpha_{\phi}$ is the non-renormalizable coupling of the inflaton
to the SM quarks and leptons $\alpha_{\phi}\ll (m_{\phi}/M_{\rm P})$. It 
is known that this time in the lab frame must be on the $\mu$s time scale. 



Unlike for the case of heavy-ion collisions, where hadrons, once formed, 
decouple chemically and thermally from each other after a certain time 
and stream freely to the particle detectors, that may not be the case 
here. Since the hadronization time is short compared to the inflaton 
decay lifetime, it follows that the (hard) inflaton decay products will 
continue impinging on the newly-formed hadrons and cause break-up due 
to the large energy transfer. How efficient this process is depends 
on the number density of these hard particles. One also needs some 
estimate of the rate of energy lost by these hard particles 
traveling now in a hadronic phase rather than a soft gluonic bath. 
It is interesting to compare this mode of energy loss for the hard 
particles to the previously considered inelastic scattering process. 
We also expect fragmentation to be a competitive mode of energy loss. 
This is because color charge is now confined within a typical hadronic 
radius, and wavelengths at this scale are unlikely to see the screening 
effects of this color charge. We can estimate and compare the time scale 
for these late-era hard hitting particles to hadronize via these various 
mechanisms. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Late break-up of hadrons}
\subsection{Deep-inelastic scattering}

The break-up of hadrons by hard-hitting quarks can be assessed through
DIS. As mentioned previously, the structure functions are unknown for
energy transfers of the cosmological scale, but we may nevertheless
place an upper bound on the time scale by considering low momentum
transfers on the GeV scale (which are sufficient to break up the
hadron). From Eq.~(\ref{ninf}), we can conclude that the number
density of hard particles at temperature of order the hadronization
scale ($\sim 200$ MeV) is very low ($n_q\leq 10^{-6}$ fm$^{-3}$),
whereas the number of hadrons is $n_h\sim (m_hT_h)^{3/2}{\rm e}^{-m_h/T}$. 
The Boltzmann suppression is least for the lightest hadron, namely pions, 
whose mass $m_{\pi}=140$ MeV is very close to
$T_h$, where the hadronization temperature $T_h\simeq 150$
MeV. Therefore $n_h\sim T_h^3e^{-1}\sim 10^{-1}$ fm$^{-3}$.  This
implies that while not many hadrons break up, the hard quarks can lose
energy quite efficiently due to the large number of participants. The
mean free path of the hard quark is approximately
$l\sim(n_h~\sigma_{DIS})^{-1}$. Assuming a typical cross-section of
microbarns for the integrated deep-inelastic cross-section at a
momentum transfer $Q\sim 1$ GeV (which will give us an upper bound on
the break up time by the uncertainty principle), we find $l\sim 10^4$
fm. Since these quarks are relativistic, this implies an energy loss
of 1 GeV in a time $\sim 10^{-19}$s. Considering that these quarks
have energies of $10^{12}$ GeV (of order $m_{\phi}$), we obtain a
maximum time of $10^{-7}$s to lose their energy. This is an upper
bound, and for smaller $m_{\phi}$, the time scale would be much shorter.
Also, the possibility of very large momentum transfers also implies a
much shorter time scale. However, cross-sections at such high energies
are unknown, and in any case, large momentum transfers imply that the 
hadron breaks into very energetic particles which will themselves take 
a long time to reach the hadronization scale.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Inelastic processes}

The mechanism of energy loss by 2$\rightarrow$ 3 inelastic scattering
can still occur, though it is not correct to use one gluon exchange as
the only interaction when quarks are so dilute. If we continue to use
the perturbative estimate, at the temperature $T_h$, from
Eq.(~\ref{tinel}), we obtain $\tau_{inel}\sim 10^{-11}$s. However,
we believe that this would not be an accurate estimate because of the
strong non-perturbative forces active at this separation scale of the
quarks. In light of these strong non-perturbative forces, we may
consider the fragmentation process whereby the energy of the quark is
degraded by producing $q\bar{q}$ pairs in the medium.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fragmentation}

For purposes of estimation, we neglect the effect of the hadronic 
medium and use the expression for the formation time (in the lab frame) 
as obtained in a fragmentation model, popular in heavy-ion physics. To 
be consistent with the broad rapidity plateau observed in ultra-relativistic 
heavy-ion collisions, one can use the symmetric LUND model~\cite{lund}, 
which generates a probability distribution for the location of the 
vertices at which the string breaks. Since these vertices lie on a 
curve of constant proper time in the symmetric LUND model, the hadron 
spectrum is consistent with experimental hadronic yields in the central 
rapidity region. While in heavy-ion collisions, there is a case for 
longitudinal boost invariance, we cannot justify similar assumptions 
here, but for an order of magnitude estimate, we will utilize this model 
with the understanding that there is some uncertainty in the formation 
time which is not so serious as to affect our estimate. 

In terms of a typical proper time of formation
$\langle\tau\rangle\simeq 1-2$ fm/c, which depends on fragmentation 
parameters obtained from experiment, the time in the lab frame (time dilated)
is given by
\beq
t_{\rm frag}\sim \langle\tau\rangle\frac{E_q}{m_h}
\eeq    
Since the energy of the hard quark $E_h\sim m_{\phi}$ and the 
hadronic mass is about $0.1$~GeV, we find $t_{\rm frag}\sim 10^{-11}$s. 
Compared to the time scale for the DIS process, this is faster and
it is likely that the fragmentation process is most efficient 
in degrading the energy of the hard quark, and this happens in a 
time that is very short compared to the inflaton decay lifetime, so 
that complete hadronization time is achieved only when all inflatons
have decayed. It should be noted that hadronization happens very early 
on, and also towards the end, by fragmenting quarks. The reheat 
temperature $T_{\rm rh}$, when all inflatons have decayed and hadrons 
fill the universe will be close to $T_h\sim 0.1$ GeV, the hadronization 
temperature. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion}

In this article, we have studied the perturbative decay of the 
inflaton to quarks and gluons, focusing particularly on the aspect 
of thermalization and energy loss of these hard particles in an expanding 
universe. For a given inflaton mass $m_{\phi}$, the coupling has to be 
small enough in order that thermalization is achieved before inflaton 
decay is completed. The thermalization time is extremely short, driven 
principally by 2$\rightarrow$ 3 inelastic processes that are considerably 
more efficient than 2$\rightarrow$ 2 processes at degrading the energy of 
hard quarks and gluons. After the thermal plasma is formed, it continues 
to cool in the background of an expanding universe, until a critical 
temperature for hadronization is reached. The laws of expansion are 
different than those of a relativistic fluid expanding on account of 
it's own pressure, as is the case with heavy-ion collisions. 


We estimate that the time to reach the hadronization scale is about a few 
$\mu$s, based on the relationship between the maximum temperature of the 
plasma and the reheat temperature, which should be of the order of the 
$\Lambda_{QCD}$. This depends on an adjustment of the values of the 
inflaton mass and coupling. While we have not examined in detail the 
hadronization process, which occurs in a mixed phase at roughly 
a constant temperature and is accompanied by a reduction in entropy, 
we have estimated that the hadronization time is likely to be on the 
order of micro-seconds as well ($\sim 10\mu$s). 

This state of hadrons may not be immediately stable, however, 
on account of the small cool-down and thermalization time in comparison
to the inflaton decay lifetime. Hard inflaton decay products will 
impinge on this bath of hadrons, and cause some break up into quarks and 
gluons once again. One can also safely conclude, on account of the small 
hadronization time, that complete formation of a hadronic bath 
will be stable only when all inflatons have decayed and the number 
density of hard particles is negligible. The temperature characterizing 
that state of the thermal hadronic bath composed of non-relativistic 
particles can be considered to be the reheat temperature $T_{\rm rh}$. 
Since the number density of hard particles by the time of hadronization 
is actually quite small, we find that fragmentation processes will 
rapidly degrade the energy of the hard particles, and the reheat 
temperature is about the same or slightly lower than the critical 
temperature for hadronization. 



Throughout this paper, we have borrowed ideas and expressions from 
relativistic heavy-ion physics to discuss the thermalization of the 
inflaton decay products. This intermingling is natural especially 
near the hadronization scale where current experiments are active. 
However, at much higher energies, cosmology stands alone and it 
is not likely that such energy scales will be attained experimentally 
anytime soon. We point out the estimations of thermalization time 
scales in the two regimes, and the importance of inelastic processes. 
There are differences too, in the speed at which the phase boundary 
is crossed, and the break-up effects of the inflaton decay products 
at late times. We feel that many more aspects from high-energy heavy-ion 
collisions can be brought to bear on analyzing the problem of thermalizing 
and reheating the universe, and current estimates could be successively 
improved by a more rigorous treatment of the early non-equilibrium 
stages prior to thermalization.  

  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgment}

The authors are thankful to Rouzbeh Allahverdi, and Guy Moore for 
discussion. P. J acknowledges support from the Natural Science and
Engineering Council of Canada. A. M is a Cita-National fellow.

                       
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Note added}

During the completion of this work, \cite{fromerth02} appeared, 
which focused exclusively on the quark-hadron phase transition 
when the Universe was already dominated by the radiation era. It
was assumed that the Universe was reheated before the quark hadron 
phase transition.







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\end{document}


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