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\begin{document}                                                 
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\def\RN{Reis\-sner-Nord\-str\"{o}m }
\def\rc{\rho_{\rm crit}}
\def\rl{\rho_\Lambda}
\def\rt{\rho_{\rm tot}}
\def\ie{{\it i.e.\;}}
\def\lp{\ell_{\rm Pl}}
\def\mp{m_{\rm Pl}}
\def\tp{t_{\rm Pl}}
\def\tf{t_{\rm FP}}
%\def\om{\Omega_{\rm M}}
\def\om{\Omega_{\phi}}
\def\oa{\Omega_{\Lambda}}
\def\ot{\Omega_{\rm tot}}
\def\tla{\widetilde{\lambda}_\ast}
\def\tom{\widetilde{\omega}_\ast}\
%\def\luv{\lambda_\ast^{\rm UV}}
%\def\guv{g_\ast^{\rm UV}}
\def\luv{\lambda_\ast}
\def\guv{g_\ast}
\def\lir{\lambda_\ast^{\rm IR}}
\def\gir{g_\ast^{\rm IR}}
\def\lir{\lambda_\ast^{\rm IR}}
\def\gir{g_\ast^{\rm IR}}
\preprint{DSF 2003/5}
\title{A Class of Renormalization Group Invariant\\
 Scalar Field Cosmologies}
\author{Alfio Bonanno}
\email{Electronic address: abo@ct.astro.it}
\affiliation{INAF, Osservatorio Astrofisico di Catania, Via S.Sofia 78,
I-95123 Catania, Italy}
\altaffiliation[Also at ]{INFN, Sezione di Catania}
\author{Giampiero Esposito}
\email{giampiero.esposito@na.infn.it}
\affiliation{INFN, Sezione di Napoli, and Dipartimento di 
Scienze Fisiche\\
Complesso Universitario di Monte S. Angelo, Via Cintia,
Edificio N', 80126 Napoli, Italy}
\author{Claudio Rubano}
\email{Electronic address: claudio.rubano@na.infn.it}
\affiliation{Dipartimento di Scienze Fisiche and INFN,
Sezione di Napoli\\
Complesso Universitario di Monte S. Angelo,\\
Via Cintia, Edificio N', 80126 Napoli, Italy}
\altaffiliation[Also at ]{INFN, Sezione di Napoli}
\begin{abstract}
We present a class of scalar field cosmologies with a dynamically 
evolving Newton parameter $G$ and cosmological term $\Lambda$.
In particular, we discuss a class of solutions which are consistent 
with a renormalization group scaling
for $G$ and $\Lambda$  near a fixed point.
Moreover, we propose a modified action for gravity which includes the 
effective running of $G$ and $\Lambda$ near the fixed point.
\end{abstract}

\maketitle

\section{introduction}
The recent discovery that Einstein gravity is most probably renormalizable
at a non-perturbative level \cite{ol1,ol2,ol3,frank}
has triggered many investigations on the 
possible consequences of these findings
in cosmology. In \cite{br1}, a cosmology of the Planck Era, 
valid immediately after the initial singularity, 
was discussed. In this model the Newton constant $G$ 
and the cosmological constant $\Lambda$ are dynamically  
coupled to the geometry by ``improving'' 
the Einstein equations with the renormalization group (hereafter RG) 
equations for Quantum Einstein Gravity \cite{mr}.
This modified Einstein theory is not affected by the 
horizon and flatness problems of the cosmological standard model. 

In \cite{br2}, a similar framework has been extended 
to the study of the large scale dynamics of the Universe. 
In this case a solution of the ``cosmic coincidence problem'' 
\cite{coscon2} arises naturally without 
the introduction of a {\rm quintessence} field, because
the vacuum energy density $\rho_\Lambda\equiv\Lambda/8\pi G$
is automatically adjusted so as to equal the matter energy density, 
{\it i.e.} $\oa=\Omega_{\rm matter} =1/2$ \cite{br2}. 
We shall term the models discussed in \cite{br1,br2} as {\it fixed point} 
(hereafter FP) cosmologies, or equivalently, 
RG-invariant cosmologies. 

In a nutshell, the {\it renormalization group improvement}  
consists in the modified Einstein equations
\be\label{ein}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = -g_{\mu\nu}\Lambda(k)+
8\pi G(k) T_{\mu\nu}
\ee
where the Newton parameter $G$ and cosmological term $\Lambda$ 
are now dependent on the scale $k$, $k$ being 
the running cut-off of the renormalization group equation \cite{mr}.
This framework has been also applied in General Relativity in \cite{bh1},
in the dynamical context of a gravitational collapse,
and in \cite{bh2} for a Schwarzschild black hole. 
 
In cosmology, the dynamical evolution is instead determined by a set of 
renormalization group equations by means of the cut-off identification $k = k (t)$ which 
relates the energy scale of the running 
cutoff $k$ of the renormalization group, with the cosmic time $t$. 
In \cite{br2} it has been shown that, in a cosmological setting, the correct
cutoff identification is $k \propto t^{-1}$;
it is thus possible to determine $G(k(t))$ and $\Lambda(k(t))$ in
Eq.(\ref{ein}) once a RG trajectory is available. 
The aim of this paper is to extend the results discussed in \cite{br1,br2}
to the case of a scalar field coupled to gravity.

Let us in fact assume that, besides the non-Gaussian fixed point discovered
in \cite{ol1} for pure gravity, the standard Gaussian fixed point is accessible in
perturbation theory in the scalar sector (this is actually the case
for a free scalar field \cite{perc}, and it also emerges from the analysis 
of \cite{perc2} for a self interacting scalar theory). 
Then, a solution which is compatible with a possible RG trajectory for the scalar
sector must predict a simple renormalizable potential for spin-0 
particles. We thus show that there exists a class of solutions 
for the well-known renormalizable $\phi^4$ potential. 

In addition, we also discuss a possible renormalization group improvement
at the level of the Einstein-Hilbert Lagrangian itself. In this
case, solutions for a class of power law self-interaction potentials are available only
for some specific values of the quartic self-interaction coupling
constants.

The plan of this work is the following: in Sec.II we introduce 
the basic equations and present the scalar
field solution. In Sec.III we discuss the RG improvement
in the Einstein-Hilbert Lagrangian. Sec.IV is devoted to the conclusions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{the model}
We now introduce the basic equations of the FP cosmologies for a scalar 
field matter component. 
%The scalar field Lagrangian reads
%\be\label{lag}
%{\cal L} = - (\partial_\mu \phi \partial^\mu \phi + V(\phi))
%\ee
Let us recall that the effective energy density and pressure of a generic 
scalar field read:
\ba\label{eq:dens}
&&\rho_\phi = {1\over 2}\dot{\phi}^2+V(\phi), \\[2mm]
&& p_\phi   = {1\over 2}\dot{\phi}^2-V(\phi),
\ea
respectively. In term of $\rho_\phi$ and 
$p_\phi$ the coupled system of RG improved evolution equations 
read
\begin{subequations}
\label{eq:system}
\ba
&&\left(\frac{\dot{a}}{a}\right)^2+\frac{K}{a^2}=\frac{1}{3}\Lambda+
\frac{8\pi}{3}G\rho_\phi, \label{1a}\\
&&\ddot{\phi}+3\frac{\dot{a}}{a}\dot\phi+V'(\phi)=0, \label{1b}\\
&&\dot{\Lambda}+8 \pi \dot G \rho_\phi=0, \label{1c}\\
&&G(t) \equiv G(k(t)), \;\; \Lambda(t) \equiv \Lambda(k(t)), \label{1d}
\ea
\end{subequations}
Eq.(\ref{1a}) is the improved Friedman equation, 
Eq.(\ref{1b}) is the Klein-Gordon equation,
Eq.(\ref{1c}) follows from the Bianchi identities, 
and Eqs.(\ref{1d}) are determined from the 
renormalization group equations once the cutoff 
identification $k=k(t)$ is given.
%Let us also recall that the effective energy density and pressure of the 
%scalar field read
%\ba\label{eq:dens}
%&&\rho_\phi = {1\over 2}\dot{\phi}^2+V(\phi), \\[2mm]
%&& p_\phi   = {1\over 2}\dot{\phi}^2-V(\phi),
%\ea
%respectively. 
We define the vacuum energy density $\rl$, the total energy density
$\rt$ and the critical energy density $\rc$ according to
\ba
&&\rho_{\Lambda}(t) \equiv \frac{\Lambda(t)}{8 \pi G(t)}, \\
&&\rho_{\rm{tot}}(t) \equiv \rho_\phi + \rl , \\
\label{eq:cdens}
&&\rho_{\rm{crit}}(t) \equiv \frac{3} {8 \pi G(t)} \left(\frac{\dot{a}}{a}
\right)^{2} ,
\ea
with $H \equiv \dot a / a$.
Hence we may rewrite the improved Friedman equation (\ref{1a}) in the form
\be
\label{eq:IFE2}
\frac{\dot a^2+K}{a^2} = \frac{8 \pi}{3} G(t) \rt .
\ee
We refer the various energy densities to the critical density
(\ref{eq:cdens}):
\ba
&&\Omega_{\phi} \equiv \frac{\rho}{\rho_{\rm crit}}, 
\quad \Omega_{\Lambda} \equiv 
\frac{\rho_{\Lambda}}{\rho_{\rm crit}} , \\
&&\Omega_{\rm tot}= \Omega_{\phi} + 
\Omega_{\Lambda} \equiv \frac{\rho_{\rm tot}}{\rho_{\rm crit}} .
\ea
It follows from these definitions that
the Friedman equation (\ref{eq:IFE2}) becomes
\be
\label{eq:kol}
K=\dot a^2 \; [\Omega_{\rm tot}-1].
\ee
For a spatially flat universe ($K=0$) we need $\rt=\rc$, as in
standard cosmology. In the following we shall discuss only the $K=0$ case. 
In order to solve the system (\ref{eq:system}) 
we consider the first three equations
in (\ref{eq:system}) without the RG equations (\ref{1d}). 
While in general (\ref{eq:system}) can be solved once $V(\phi)$ is given, 
we shall see  that the {\it perfect fluid} 
ansatz $p_\phi = w \rho_\phi$, $w$ being a constant, 
is equivalent to assume a class of 
power-law potential $V(\phi) \propto \phi^m$.

We first consider the first three equations
in (\ref{eq:system}) without the RG equations (\ref{1d}), and then we determine
the solutions consistent with a given RG trajectory.
The potential can be written as
\be\label{eq:pot}
V(\phi) = \frac{1}{2}\dot{\phi}^2 \Big ( \frac{1-w}{1+w} \Big ),
\ee
which shows that the value $w=-1$ should be ruled out, as
we will do from now on. By
substitution in the Klein-Gordon equation (\ref{1b}) we readily obtain
\be\label{eq:eu}
\rho_\phi = \frac{1}{1+w}\dot{\phi}^{2}
\equiv \frac{\cal M}{8\pi a^{3(1+w)}},
\ee
where ${\cal M}$ is an integration constant. 
By substituting into Eq.(\ref{1a}) we derive the following power-law 
solutions:
\begin{subequations}
\label{sol}
\ba\label{23a}
&&a(t) = \Big [\frac{3(1+w)^2}{2(n+2)}\;{\cal M}\; C\Big ]^{1/(3+3w)}
\;t^{(n+2)/(3+3w)}, \\[2mm]
&&\phi(t) = \Big(\frac{4(n+2)}{12\pi(1+w)Cn^2}\Big)^{1/2} 
t^{-n/2}, \label{23b}\\[2mm] 
&&G(t) = C\; t^{n} , \label{23c}\\[2mm]
&&\Lambda(t) = \frac{n(n+2)}{3(1+w)^2}\;\frac{1}{t^2}, \label{23d}
\ea
\end{subequations}
where $C$ is a constant 
and $n$ is a positive integer. 
For example, writing 
$a(t)=A \; t^{\alpha}, \Lambda=B \; t^{-2}$ and expressing $G$ as
in (\ref{23c}), Eq.(\ref{1a}) yields, for $K=0$, a first-order algebraic
equation for $\alpha$, which is solved by $\alpha={(n+2)\over 3(1+w)}$.
Eq.(\ref{eq:eu}) is then integrated to get the result (\ref{23b}).
As anticipated, the potential is also a power law, i.e.
\be\label{potential}
V(\phi) = \frac{1-w}{2+2w}
\Big ( \frac{12\pi(w+1)C}{n+2} \Big)^{\frac{2}{n}} 
(\frac{n}{2})^{\frac{2(n+2)}{n}} 
\;\;\;\phi^{\frac{2(n+2)}{n}}.
\ee

The RG equations (\ref{1d}) have not been used so far. What is the correct 
RG trajectory for a self-interacting scalar field coupled with gravity? 
Let us consider the RG-trajectory discussed in the introduction, where
in the deep UV region we must have the non-Gaussian fixed point 
\cite{ol1} in the gravitational
sector, and  the Gaussian one in the scalar field sector.
In this case, the renormalized trajectory for the dimensionful quantities
reads:
\be\label{rg}
G(k) = {\guv}/{k^2}, \;\;\;\; \Lambda(k) = \luv \;k^2 , 
\ee
where $\guv$, $\luv$ are the dimensionless couplings $g(k)$ and $\lambda(k)$,
respectively, at the ultraviolet non-Gaussian fixed point $k\rightarrow \infty$.
The numerical values have been obtained in the analysis of \cite{perc,perc2}
and read  $\guv \approx 0.32$, $\luv \approx 0.36$ approximately. 

The next step is to determine $k$ as a function of $t$. 
In \cite{br1} it was shown that the 
correct cutoff identification is given by 
\be\label{cuti}
k(t) = {\xi}/{t} .
\ee
Therefore, we see from (\ref{rg}) that we are led to set
$n=2$ in (\ref{sol}) and $\xi^2 = 8/3(1+w)^2\luv$ in (\ref{cuti}). At last,
the following {\it renormalization group invariant} 
(or fixed-point) solution is obtained:
\begin{subequations}
\label{sol2}
\ba\label{24a}
&&a(t) = \Big [\Big (\frac{3}{8}\Big )^2 (1+w)^4 \guv \luv \;{\cal M} 
\Big ]^{1/(3+3w)}\;t^{4/(3+3w)}, \\[2mm]
&&\phi(t) = \Big(\frac{8}{9\pi(1+w)^3\guv\luv}\Big)^{1/2} 
\; \frac{1}{t}, \label{24b}\\[2mm] 
&&G(t) = \frac{3}{8}(1+w)^2\guv\luv t^{2}, \label{24c}\\[2mm]
&&\Lambda(t) = \frac{8}{3(1+w)^2}\;\frac{1}{t^2}. \label{24d}
\ea
\end{subequations}
The solution, as far as $a(t)$, $G(t)$ and $\Lambda(t)$ are concerned, is 
basically the same as what already discussed in \cite{br1,br2} but in this case the
potential reads
\be\label{pot2}
V(\phi) = \frac{9 \pi}{16}(1-w)(1+w)^2\guv\luv \; \phi^{4},
\ee
which is the standard renormalizable quartic self-interacting potential for 
a massless scalar theory. The role of $w$ is now clear: it allows a 
convenient parametrization of the solution in terms of the parameter $w$ 
instead of the self-interaction coupling constant in the potential.
It in fact measures the self-coupling strength $9(1-w)(1+w)^2\guv\luv/16$:
for $w=1$ ({\it stiff matter} equation of state) $V=0$ and 
$\phi$ is a free field, while for $0<w<1$, $\phi$ is an interacting field.  
For $w>1$ the theory is not bounded from below. 

Other properties of the solution (\ref{sol2}) have extensively been discussed
in \cite{br1,br2} and we shall not repeat that discussion here.
We simply observe that for the solution (\ref{sol2}), 
we have $\Omega_\phi=\Omega_\Lambda=1/2$ at any time.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%
\section{improving the action}
One of the striking properties of the renormalization group 
trajectory (\ref{rg}) is that the following 
relation holds:
\be\label{rela}
\Lambda = \frac{\guv\luv}{G} .
\ee
This fact has a deep meaning and is related to the possibility of reducing 
the number of coupling constants in a RG-invariant theory \cite{zimmer}.
What happens in our case is that near the
fixed point it is always possible to consider 
$\Lambda=\Lambda(G)$ and the effective
scaling is ruled only by $G$, for instance.
This fact suggests that a more fundamental approach 
should consider $\Lambda$ as a function of $G$ from the beginning, 
perhaps at the level of the action itself.

Let us then consider the action
\be\label{action}
S = \int d^4x\;\sqrt{-g}\;\Big (\frac{R}{G}-\frac{2\Lambda(G)}{G}\Big )+S_{\rm m},
\ee
where $S_{\rm m}$ is the action for the matter field.
In this case, variation with respect to $g_{\mu\nu}$ 
leads to Eqs.(\ref{eq:system})
and variation with respect to $G$ gives an additional constraint equation 
(see also \cite{krori}):
\be\label{con}
-\frac{R}{G}+\frac{2\Lambda}{G}-2\frac{d\Lambda}{d G} =0.
\ee
This equation, jointly with Eq.({\ref{1c}) and the field equations yields
\be\label{con2}
2\Lambda = 8\pi G (\rho_\phi+3p_\phi) = 8\pi G\rho_\phi(1+3 w).
\ee
By inserting the general solution (\ref{sol}) in (\ref{con2}) we have
\be\label{con3}
n = (1+3w).
\ee
In particular, for the case of interest $n=2$, and hence $w=1/3$, leading
in turn to the renormalizable interaction
\be\label{pot3}
V(\phi) =\frac{2 \pi}{3}\guv \luv \phi^{4}. 
\ee
The relevant property of this solution is that the effective 
strength of the interaction self-coupling is determined entirely by
the fixed point values $\guv$ and $\luv$. 
For a free scalar field $\guv \luv \approx 0.11$ \cite{perc} and this value does not
change in a significant way in the interacting case \cite{perc2}.
Loop corrections are then expected to be small and the leading tree
level form of the potential (\ref{pot3}) holds.
We can thus regard the cosmology (\ref{sol2}) with $w=1/3$
as an exact solution of the modified Einstein-Lagrangian (\ref{action}) which 
is consistent with a RG flow near the non-Gaussian fixed point in the gravitational sector
and the Gaussian one in the matter sector.
\section{conclusion}
We have presented a class of power-law cosmologies 
with variable $G$ and $\Lambda$ in the case
of a scalar field matter component, Eq.(\ref{sol}). 
We have then extended the FP cosmology 
presented in \cite{br1} by including the 
RG evolution Eq.(\ref{rg}) in the general solution (\ref{sol}). 
Last, we have presented a new RG-improvement at the level of the action
which picks out a specific self-interaction strength 
value for the scalar field potential.   
The scalar solution (\ref{sol2}) with $w=1/3$ can, at best, be considered 
only a toy model of the initial state of the Universe. 
However, it may be helpful in understanding a more 
complete framework where the dynamical evolution of the gravitational field 
and the matter field near the initial singularity 
is consistent with RG scaling law of the renormalized 
theory near a fixed point.   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
The authors are indebted to the INFN and Dipartimento di Scienze Fisiche
of Naples University for financial support.
We thank M. Reuter, R.Percacci and D.Perini for useful discussions. 
A.B. also acknowledges the warm 
hospitality of the University of Naples where part of this work was written.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}



