\documentstyle[11pt,epsfig]{article}

\def\double{\baselineskip 24pt \lineskip 10pt}

%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}%}

%\renewcommand{\thefigure}{\arabic{section}.\arabic{figure}}



\textheight 8.5in

\textwidth 6in

\oddsidemargin 8pt

\topmargin -30pt



\def\baselinestretch{1.2}

\parskip 0.2cm



%minore o circa uguale

\def\laq{~\raise 0.4ex\hbox{$<$}\kern -0.8em\lower 0.62

ex\hbox{$\sim$}~}

%maggiore o circa uguale

\def\gaq{~\raise 0.4ex\hbox{$>$}\kern -0.7em\lower 0.62

ex\hbox{$\sim$}~}



\def\beq{\begin{equation}}

\def\eeq{\end{equation}}

\def\bea{\begin{eqnarray}}

\def\eea{\end{eqnarray}}

\def\bean{\begin{eqnarray*}}

\def\eean{\end{eqnarray*}}



\def\vp{\varphi}

\def \vpb {{\overline {\vp}}}

\def \vpbp {{\dot{\vpb}}}

\def \rb {\overline \rho}

\def \pb {\overline p}

\def \sb {\overline \sg}

\def \H {{a^\prime \over a}}

\def \fp {{\phi^\prime}}

\def \hp {\dot{h}}

\def \hpp {\ddot{h}}



\def \pa {\partial}

\def \ra {\rightarrow}

\def \fp {{\phi^\prime}}

\def \ti {\widetilde}

\def \la {\lambda}

\def \ls {\lambda_{\rm s}}

\def \lp {\lambda_{\rm P}}

\def \Ms {M_{\rm s}}

\def \Mp {M_{\rm P}}

\def \La {\Lambda}

\def \Da {\Delta}

\def \da {\delta}

\def \b {\beta}

\def \a {\alpha}

\def \ap {\alpha^{\prime}}

\def \ka {\kappa}

\def \Ga {\Gamma}

\def \ga {\gamma}

\def \sg {\sigma}

\def \da {\delta}

\def \ep {\epsilon}

\def \r {\rho}

\def \om {\omega}

\def \Om {\Omega}

\def \noi {\noindent}

\def \lap {\triangle}



\begin{document}



\begin{titlepage}



\begin{flushright}

BA-TH/03-463\\

CERN-TH/2003-128\\

hep-th/yymmnnn

\end{flushright}



\vspace*{1.8 cm}



\begin{center}



\huge

{Perturbations in a non-singular bouncing Universe}



\vspace{1cm}



\large{M. Gasperini$^{1}$, M. Giovannini$^2$ and G. Veneziano$^2$}



\normalsize

\vspace{.2in}



{\sl $^1$Dipartimento di Fisica, Universit\`a di Bari, \\ 

Via G. Amendola 173, 70126 Bari, Italy\\

and \\

Istituto Nazionale di Fisica Nucleare, Sezione di Bari\\

Via G. Amendola 173, 70126 Bari, Italy}



\vspace{.2in}



{\sl $^2$Theoretical Physics Division, CERN, \\

CH-1211 Geneva 23, Switzerland }



\vspace*{1.5cm}



\begin{abstract}

We complement the low-energy gravi-dilaton 

effective action of string theory  with a non-local, general-covariant

dilaton potential, and  obtain homogeneous solutions describing 

a non-singular (bouncing-curvature) cosmology.  We then compute, both

analytically and numerically, the spectrum of  amplified scalar and

tensor perturbations, and draw some  general lessons on

how to extract observable consequences from  pre-big bang and

ekpyrotic scenarios. 



\end{abstract}

\end{center}



\end{titlepage}

\newpage





In recent years string/M-theory has inspired new cosmological 

scenarios in which a long period of accelerated (growing-curvature)

evolution, emerging from an almost trivial initial geometry, turns into a

standard (decreasing-curvature)  FRW-type cosmology, after going smoothly

through a big bang-like event. There are by now several variations on

this pre-big  bang theme. Besides the original pre-big bang (PBB)

scenario \cite{1,2,3}, based on the duality symmetries of string

cosmology,  new models incorporating brane and M-theory ideas

have been proposed under the generic name of ekpyrotic (EKP)  scenarios

\cite{4}.

While different proposals differ in the way the scale factor behaves 

during the growing-curvature phase, they all share the  feature

of describing a bounce in $|H|$ (the absolute value

of the Hubble parameter) or, in more geometrical terms, in the  space-time

curvature. The latter always starts vanishingly

small, grows to a maximum at the would-be big bang, and then decreases again in

the FRW phase. A common theoretical challenge to all these models is

that of being able to describe the transition between the two regimes.



At a more phenomenological level, instead, the challenge is to compute,

in a reliable way, the final spectrum of amplified quantum fluctuations

to be compared with present data on CMB radiation and large-scale

structure.

In the PBB case it was admitted early on \cite{5,6} that adiabatic- 

curvature perturbations had too large a spectral index to be of any

relevance at cosmologically interesting scales (while being possibly

important for gravitational waves searches \cite{7,8}). Isocurvature

perturbations (related to the Kalb--Ramond two-form)  can instead  be

produced with an interestingly flat spectrum \cite{9}, but have to be

converted into adiabatic-curvature perturbations through the so-called

curvaton mechanism \cite{10} before they can provide a viable

scenario for large scale-anisotropies \cite{11}.

Proponents of the ekpyrotic scenario, while agreeing with the PBB 

result of a steep spectrum of tensor perturbations, have also

repeatedly claimed \cite{4,12} to obtain ``naturally" an almost

scale-invariant spectrum of adiabatic-curvature perturbations, very

much as in ordinary models of slow-roll inflation. These claims have

generated a lot of heated discussion (see

for instance \cite{13}), with many arguments given  in favour or against

the phenomenological viability of  EKP scenarios  in the absence of a

curvaton's help. The reasons for the disagreement  can be ultimately

traced back to the fact that the curvature bounce (hereafter simply

referred to as the bounce) is put in by hand, rather than being derived

from an underlying action. This leaves different authors to make

different assumptions on how to smoothly connect  perturbations across

the bounce itself, and this often results in completely different physical

predictions.



Our aim is to present a class of models where a regular 

bounce is derived from the field equations of a 

general-covariant, albeit non-local,  action, 

and where perturbations can be

studied from beginning to end.  We are thus able to test, at least within

the model, the assumptions made by different groups about how

different perturbations should or should not behave across the bounce.

In this letter we only consider a simple 

gravi-dilaton model and present  the main results,

working in the string frame and skipping their detailed derivation. 

In a longer, forthcoming paper \cite{16} we shall give

full details of the calculations in both the string and Einstein 

frames (with identical physical conclusions), and generalize the class of models 

to include a fluid component.

We first  present the action, the

background-field equations, and a class of smooth, bouncing solutions.

We then discuss, successively, the amplification of tensor and scalar 

perturbations from an initial spectrum of vacuum

quantum fluctuations, and  compare analytical and

numerical results. We finally summarize our results and draw some conclusions. 





We recall that the homogeneous cosmological solutions derived from the tree-level, low-energy

string effective action exhibit in general a curvature singularity,

disconnecting the pre-big bang branch from the post-big bang one \cite{BV}. 

Such a singularity is expected to

be removed by higher-loops {\em and} higher-curvature $\ap$

corrections (see for instance \cite{3}), but it is known \cite{14} that non-singular

solutions can be explicitly obtained already at low curvatures \cite{2}

in the presence of an appropriate {\em non-local} effective potential.

An example of such a possibility can be illustrated by  using a

$(d+1)$-dimensional, general covariant  gravi-dilaton

effective action, which,  in the string frame, reads: 

\beq

S= -{1\over 2\ls^{d-1}} \int d^{d+1}x \sqrt{|g|}~

e^{-\vp}\left[R+ \left(\nabla \vp\right)^2+V(\vpb)\right]

\label{action}

\eeq

(metric conventions: $+--- \dots$).  Here $\ls= \sqrt{\ap} = \Ms^{-1}$ is the string

length scale, and the potential $V(\vpb(x))$, a local function of $\vpb$, is instead a

non-local function (yet a scalar under general coordinate transformations)

of the dilaton owing to the definitions:

\beq 

V= V(e^{-\vpb}),  ~~~~~~

e^{-\vpb(x)} =    \int {d^{d+1}y \over \ls^d}\sqrt{|g(y)|}~ e^{-\vp(y)}

\sqrt{\pa_\mu \vp (y) \pa^\mu \vp(y)}

~\delta\left(\vp (x) -   \vp (y)\right) .

\label{21}

\eeq

Note that $e^{\vpb}$ plays the role of a ``reduced" coupling constant from $d+1$ to $0+1$ space-time dimensions.

 We may thus expect $V(e^{-\vpb})$ to go like some inverse power of its argument as this becomes large (i.e. in perturbation theory). 

We shall discuss in \cite{16} how such a non-local potential may be

induced by loop corrections in (higher-dimensional) manifolds

with compact spatial sections; here, we take  this simply as a  toy model that avoids the singularity, 

while staying all the time at low energy/curvature.

For a background manifold isometric with respect to $d$ spatial

translations, it is known that  action (\ref{action}) leads to field equations that are

covariant under scale-factor duality \cite{1} and, in  the presence of the two-form background $B_{\mu\nu}$,  also under global $O(d,d)$ transformations \cite{14}. 



Variation of the above action with respect to $g_{\mu\nu}$ and

$\vp$, though somewhat unusual, is straightforward and leads to the following field equations  (in units of

$2\ls^{d-1}=1$):

\beq

G_{\mu\nu} + \nabla_{\mu} \nabla_{\nu} \vp + \frac{1}{2} g_{\mu\nu}  

\left[ (\nabla \vp)^2 - 2 \nabla^2 \vp - V\right] -  \frac{1}{2} e^{-\vp} 

\sqrt{(\partial \vp)^2} ~\ga_{\mu\nu} I_1 = 0,

\label{genback1}

\eeq

\beq

R + 2 \nabla^2 \vp - (\nabla \vp)^2 + V - {\pa V\over \pa{\bar{\vp}}} 

+  e^{-\vp}  \frac{\hat{\nabla}^2 \vp}{\sqrt{(\partial \vp)^2}}  I_1 - 

e^{-\vp}  V'   I_2 = 0,

\label{genback2}

\eeq

where (with a prime denoting differentiation with respect to the argument)

\begin{eqnarray}

&&

I_1 = {1\over \ls^d}\int d^{d + 1}y \sqrt{|g(y)|}~V'(e^{-\vpb(y)}) 

~\delta\left(\vp (x) -   \vp (y)\right) , 

\nonumber\\

&&

I_2 = {1\over \ls^d} \int d^{d + 1} y \sqrt{|g(y)|}~ 

\sqrt{\pa_\mu \vp (y) \pa^\mu \vp(y)}

~\delta'\left(\vp (x) -   \vp (y)\right) \; ,

\end{eqnarray} 

and we have also  introduced the induced metric and Laplacian:

\beq

\gamma_{\mu\nu} = g_{\mu\nu} - \frac{\partial_{\mu}\vp \partial_{\nu}\vp}{(\partial \vp)^2},  

~~~~~~~~~~  \hat{\nabla}^2 \vp = \ga_{\mu\nu} \nabla^\mu\nabla^\nu \vp.

\eeq

For a homogeneous, isotropic and spatially flat background, we can set 

$g_{00}=1$, $g_{ij}=-a^2(t)\da_{ij}$, $\vp=\vp(t)$, and  obtain $e^{-\vpb}= e^{-\vp}a^d$,

where we have absorbed into $\vp$ the dimensionless constant $-\ln 

(\int d^dy/\la_s^d)$, associated with the (finite) comoving spatial volume. 

In such a case, the time and space components of Eq.(\ref{genback1}), and the dilaton equation (\ref{genback2}), lead  to a set of equations already studied in \cite{14,2,5}:

\begin{eqnarray} 

&&

\dot{\overline{\vp}}^2 - d H^2 -V=0, ~~~~~~~~~~~~~~~~~~~~~

\dot H -H\vpbp=0,

\label{29}\\

&&

2 \ddot \vpb - \vpbp^2 -d H^2 +V -{\pa V\over \pa \vpb} =0,

\label{210}

\end{eqnarray}

where the  third (dilaton) equation follows from  the first two, {\em provided} $\vpbp\not=0$. 



As noticed in \cite{14}, the above set of equations admits, quite generically, non-singular solutions. To recall this point and give an explicit analytical  example we note that the two equations (\ref{29}) can be reduced to quadratures, i.e.  

\beq

H = m e^\vpb, ~~~~~~~~~

t= \int^\la d\la' \left[d+\la'^2 V(m\la')\right]^{-1/2} \;, 

\eeq

where $m$ is an integration constant, and $\la=m^{-1} e^{-\vpb}$. If the function appearing above between square

brackets has a simple zero, we obtain a regular bouncing solution \cite{14}. Furthermore, if $V\la^2 \ra 0$

at large $|\la|$ (corresponding to a potential generated beyond two loops), the asymptotic solutions approach the

usual vacuum solutions at large $|t|$. 



 Consider, in particular, the class of  potentials

\beq

V(m\la)= \la^{-2}\left( \left[\a -(m\la)^{-2n}\right]^{2-1/n} - d \right),

\label{212}

\eeq

parametrized by the dimensionless coefficients $\a$ and by the ``loop-counting" parameter $n$. For $\a >0$ and $n>0$, Eq. (\ref{212}) leads  to the general exact solution

\beq

H =m e^\vpb =m\left[\a \over 1+(\a mt)^{2n}   \right]^{1/2n}.

\label{213}

\eeq

As $|t| \ra \infty$, the Hubble parameter behaves like $H|t| \sim \a^{(1-2n)/2n}$, so that the minimal pre-big bang solutions dominated by the dilaton kinetic energy are only recovered for $ \a^{(2-1/n)}=d$ (see for instance \cite{3}). If this condition is not satisfied, the background may still be non-singular, but  the dilaton potential cannot be neglected, even  asymptotically.





 For the perturbation analysis of this paper it will be sufficient to use as a toy model the simple regular background  associated to the four-loop potential 

$V(\vpb)=-V_0e^{4 \vpb}$. In this case the general solution, in $d=3$ spatial dimensions, can be written as:

\beq

a(\tau) =  \biggl[ \tau + \sqrt{\tau^2 + 1}\biggr]^{1/\sqrt{3}},

~~~~~~~~~~

 \overline{\vp} = - \frac{1}{2} \ln{ ( 1 + \tau^2)} + \vp_{0},  

~~~~~~~~~~ \tau=t/t_0, 

\label{214}

\eeq

where $\vp_0$ is an integration constant and $t_0^{-1} = e^{{\vp}_0} \sqrt{V_0}$. The solution is thus characterized 

by two relevant parameters,  $t_0^{-1}$ and $e^{{\vp}_0}$, corresponding, up to some numerical factors, to the

Hubble parameter and string coupling at the bounce, respectively (without loss of generality we have set 

$a(0) =1$ with $t=0$, the time at which the bounce occurs).

We stress that the possibility of regular backgrounds is not limited to the class of potentials illustrated in Eq. (\ref{212}), and that regular bouncing solutions can also be obtained by adding to the action (\ref{action}) fluid matter sources, as already pointed out in \cite{2} and  illustrated in  \cite{16}. 







The evolution equation of tensor (transverse and traceless) 

metric perturbations   can be obtained by perturbing 

to first order 

the $(i,j)$ component of Eq. (\ref{genback1}). For 

each polarization we obtain, in Fourier space,

\begin{equation}

\ddot{h}_{k} - \dot{\overline{\vp}} \dot{h}_k + \om ^2 h_k =0 \;  ,  

~~~~~~~~~~\om \equiv {k}/{a}.

\label{GW1}

\end{equation}

We shall concentrate our attention on the case of regular 

solutions approaching asymptotically  the well- 

known minimal gravi-dilaton model with negligible potential.

The gravitational wave spectra arising in this  case

 are characterized, in general,   by a steep  spectrum ($n_T >1$), up to

 logarithmic corrections. In fact the asymptotic 

solution of Eq. (\ref{GW1}) for large wavelengths can be written as

\begin{equation}

h_{k} = A_{k} + B_{k} \int_{t_{\rm ex}}^{t} 

\frac{e^{\vp}}{a^{3}} dt,

\label{outhor}

\end{equation}

where $t_{\rm ex} \sim \om^{-1}$ denotes the time at which the perturbation 

exits the horizon. Since in the case under consideration

 $e^{\vp}a^{-3} \sim  |t|^{-1}$ for $t \to -\infty$,

we must expect 

 a $\ln{\om t}$  growth of $h_{k}$,  leading,  ultimately, to

$\ln{k}$  corrections in the power spectrum.



For an accurate camparison of analytical and numerical 

results, we shall restrict our attention to the 

specific regular bouncing 

 solution (\ref{214}). In this case, the evolution 

equation for the canonical normal mode 

$\mu_{k} = a e^{-\vp/2} h_{k}$ has asymptotic solutions with 

normalization to an initial vacuum fluctuation spectrum

given in terms of Hankel functions of index zero. The application 

of the standard matching procedure leads to  a Bogoliubov 

coefficient (see below for its exact definition) given by 

\begin{equation}

|\beta_{k}|^2 =\epsilon_{1} + \epsilon_{2}\ln^2({k_1/k}),

\label{exptensor}

\end{equation}

where $\epsilon_{1}$ and $\epsilon_2$ are numerical factors 

of order $1$,  and $k_1$ is a characteristic 

momentum scale. The associated power spectrum is 

\begin{equation}

|\delta h_{k}|^2 = k^3 |h_{k}|^2 \sim\left(H(0)\over \Mp(0)\right)^2

 \left(k\over k_1\right)^2 \ln^2\left(k\over k_1\right),

\label{dhnorm}

\end{equation}

where the normalization $H(0)/M_{P}(0) \sim (\lambda_{s} /t_{0})~

 e^{\vp(0)/2}$  is controlled by the ratio between the 

curvature scale and the Planck mass at the bounce $t =0$. 



For the numerical calculation, it is convenient 

to work with the rescaled variable $\hat{h} = e^{-\vp/2} a^{3/2} h =

\sqrt{a} \mu $. In order to compute the amplification, we impose 

for $t \to -\infty$ the quantum mechanical initial conditions 

\begin{equation}

\mu_{k} = \frac{1}{\sqrt{2 k} } e^{- i k \eta}~~~~~\rightarrow~~~~~

\hat{h}_{k} = \frac{1}{\sqrt{2 \omega} } e^{ - i \int \omega dt},

\end{equation}

where we recall that we are using units in which

$\Ms^2 = 2$. The action of the gravi-dilaton background 

will  produce a mixing in the positive and negative frequency modes so

that,  for $t \to +\infty$, the solution  can be parametrized as 

\begin{equation}

\hat{h}_{k} =\frac{1}{\sqrt{2 \omega}} \left(\alpha_{k} e^{- i\int 

\omega dt} + \beta_{k} e^{i \int \omega dt}\right).

\label{18}

\end{equation}

In a second quantized approach $|\beta_{k}|^2$ is nothing but the 

number of produced gravitons of given momentum $k$. The energy 

density of the produced gravitons will then be, up to numerical factors, 

$\int d \ln{k}~ k^4 |\beta_{k}|^2$. Thus, using the 

theoretical expectations given above, this 

corresponds to  an  

energy  spectrum going as $k^4$ (up to logarithmic corrections).

\begin{figure}[!ht]

\centerline{\epsfxsize = 11cm  \epsffile{FIG1.eps}}

\vskip 3mm

\caption[a]{Behaviour of the tensor power  spectrum $|\delta h_{k}|$, for different values of $\kappa = k t_0$, 

in the region around  the bounce. In the present and in the following figures the $\log$ denotes the 

logarithm in basis ten.}

\label{FIGURE1} 

\end{figure}

The numerical integration in terms shows that 

both  $\hat{h}$ and $h$ pass regularly through the bounce; this 

is  illustrated in Fig. \ref{FIGURE1}, 

where we report the tensor power spectrum, $|\delta h_{k}|$, 

in the region around $t=0$, clearly in qualitative agreement with 

Eq. (\ref{outhor}).The  absolute normalization 

of the tensor power spectrum has been fixed by imposing $H(0)/M_{P}(0)=10^{-2}$.



The mixing 

coefficients of Eq. (\ref{18}) can be obtained in terms 

of the asymptotic  ($\tau \to +\infty$) values of the real and imaginary parts of $\hat{h}$,  via the expressions:

\begin{eqnarray}

&& |\alpha_{k}|^2 + |\beta_{k}|^2 = \left( \omega |\hat{h}_{k}|^2  

+ \frac{1}{\omega}\left| \dot{\hat{h}}_{k} - \frac{H}{2} \hat{h}_{k}\right|^2

 \right),

\label{sum}\\

&& |\alpha_{k}|^2 - |\beta_{k}|^2= i \biggl( \dot{\hat{h}}_{k} \hat{h}_{k}^{\ast} 

-{\dot{\hat{h}}}^{\ast}_{k} \hat{h}_{k}\biggr). 

\label{difference}

\end{eqnarray}

We expect Eq. (\ref{difference}) to be identically 

equal to $1$ thanks to the conservation of the Wronskian.  On the other hand, Eq. (\ref{sum}) 

should  approach a ($k$-dependent) constant only at late 

enough times.

These two expectations are perfectly fulfilled, as  illustrated in Fig.

\ref{FIGURE2}, which thus represents 

a highly non-trivial consistency check of our numerical

procedure.



\begin{figure}

\centerline{\epsfxsize = 11cm  \epsffile{FIG2.eps}}

\vskip 3mm

\caption[a]{Asymptotic behaviours  of the mixing coefficients given 

in Eqs. (\ref{sum}) and (\ref{difference}) for different values of 

$\kappa =k t_{0}$.}

\label{FIGURE2} 

\end{figure}



Clearly,  smaller $k$ re-enter the horizon (and thus reach their asymptotic value) later.

The asymptotic value gives, for each $k$, the sought  after Bogoliubov  coefficient. The results are plotted in Fig.

\ref{FIGURE3} (crosses) and fitted to the theoretical expectation (\ref{exptensor}).

\begin{figure}

\centerline{\epsfxsize = 11cm  \epsffile{FIG3.eps}}

\vskip 3mm

\caption[a]{ The numerically determined values of  $|\beta_k|^2$  are fitted to theoretical expectations for tensor (crosses) and scalar (stars) perturbations.}

\label{FIGURE3} 

\end{figure}

The fit is very good and the parameters $\epsilon_{1} , \epsilon_{2}, k_1$ are determined according to:

\begin{equation}

|\beta_{k}|^2 = 0.46 \ln^2(k/k_1) -4.84, ~~~~~~~~~~~~~ k_1 = 

6.68/t_{0}.

\end{equation}

We have not yet attempted to make a theoretical estimate of  $\epsilon_{2}$.



 Unlike tensor perturbations, scalar fluctuations are, in general, neither gauge- nor frame-independent.

Gauge-invariant perturbations can be defined, as usual, in each frame. Among these,  the spatial curvature perturbation  on uniform dilaton 

hypersurfaces ${\cal R}$, and the Bardeen potentials $\Psi$ and $\Phi$ are   particularly important for CMB

phenomenology. Denoting by $\delta \varphi = \chi$ the dilaton perturbation, and introducing  scalar metric perturbations in conformal time via the standard expressions 

\begin{equation} 

 \delta g_{00} = 2 a^2 \phi,~~~~~~~\delta g_{i j} 

= 2 a^2 ( \psi \delta_{ij} - \partial_{i}\partial_{j } E), ~~~~~~~ \delta g_{i 0} = a^2 \partial_{i} B  \; ,

\label{linepert}

\end{equation}

we may express  ${\cal R}$,  $\Psi$ and $\Phi$ as:

\begin{equation}

{\cal R} = - \psi - \frac{{\cal H} }{\varphi'} \chi,~~~

 \Psi = \psi + {\cal H} (E' - B),  ~~~ \Phi = \phi - {\cal H} (E' - B) - (E' - B)',

\end{equation}

where primes denote derivatives with respect to conformal time and 

${\cal H}=a'/a$.

Introducing the Einstein frame, with scale factor $a_E=a e^{-\vp/2}$, 

we have $\phi_E = \phi_s - \frac{1}{2} \chi$,  $\psi_E = \psi_s  + \frac{1}{2} \chi$ (while $B$ and $E$ coincide in the two frames), and  we can easily prove that

${\cal R}$ and  $\Psi +  \Phi$ are frame-independent,  while $(\Psi -  \Phi)_s = (\Psi -  \Phi)_E - \chi + \vp' E'$.



We shall work in the convenient 

 uniform dilaton gauge $\chi = 0$, and also set $B=0$, so that

${\cal R} = - \psi$ and $\Psi_E = \psi + {\cal H}_{E} E'$, where

${\cal H}_{E}= {\cal H} - {\vp'}/{2}$. 

This gauge is particularly useful since all the perturbations variables in (\ref{linepert}) coincide in the two frames. Another 

advantage is that, by perturbing the background equation (\ref{genback1}), we find in this gauge $\delta \gamma_{\mu}^{\nu}=0$,

$\delta e^{-\vpb}=0$, $\delta I_1 =0$.

The perturbations of the background equations 

thus simply lead to the 

following set of evolution equations:

\begin{eqnarray}

&& 3 (\vp' - 2 {\cal H}) \psi' + ( 6 {\cal H} \vp' - {\vp'}^2 - 6 {\cal H}^2) 

\phi + 2 \nabla^2 \psi - (\vp' - 2 {\cal H}) \nabla^2 E' =0, ~~~~~

\label{00P1}\\

&& (\vp' - 2 {\cal H}) \phi = 2 \psi',~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~

\label{i0P1}\\

&& E'' + (2{\cal H} - \vp') E' + \psi - \phi =0, ~~~~~~~~~~~~~~~~~~~

~~~~~~~~~

\label{ineqjP1}

\end{eqnarray}

following,  respectively, from the 

$(00)$, $(0i)$ and  $(i\neq j)$ components of the  perturbed 

Eq. (\ref{genback1}).  Note that the last equation, upon using ${\cal H}_{E}= {\cal H} - {\vp'}/{2}$,  is nothing but $(\Psi -  \Phi)_E =0$, and that

the additional equations obtained from the $(i=j)$ component 

and from the dilaton equation (\ref{genback2}) are redundant.



We may combine the above system to get the interesting equation 

\begin{equation}

\psi' = \frac{( \vp' -2 {\cal H})}{{\varphi'}^2} \nabla^2 

 \left[ ( \vp' -2 {\cal H})E' - 2 \psi \right],

\label{pspr}

\end{equation}

which  turns out 

to be  exactly the standard evolution equation for

${\cal R}$:

\begin{equation}

{\cal R}' = - \frac{ 4 {\cal H}_{e}}{{\vp'}^2} \nabla^2  \Psi_E.

\label{R}

\end{equation}

Although this equation  is {\em formally } the same 

as the standard equation  of ordinary

cosmological perturbation theory, the relation between

$\varphi'^2$ to $({\cal H}_{E}^2 - {\cal H}_{E}')$ 

is different in our case. 

Indeed, it can be easily  checked, by transforming the background equations (\ref{29}), (\ref{210}) to the Einstein frame, that:

\begin{equation}

4 ({\cal H}_{E}' - {\cal H}_{E}^2) + {\vp '}^2 + 

\frac{\partial V}{\partial \overline{\vp}} a_{e}^2 e^{\vp}=0.

\end{equation}

This shows that the usual identity 

$ {\vp '}^2 \propto  ({\cal H}' - {\cal H}^2)$ is not satisfied

in our case. And indeed, in our 

class of bouncing solutions

$\varphi'$, unlike ${\cal H}_E' - {\cal H}_E^2$,  {\em never } vanishes.

As a consequence, the pre-factor on the right-hand side 

of Eq. (\ref{R}) does not diverge during the bounce; we can thus infer, already at this 

stage, that ${\cal R}$ is likely to be conserved on super-horizon scales.



Let us now turn to the specific study 

of the background model already analyzed in the case of

the tensor perturbations.

On the basis of simple estimates based 

on the asymptotic behaviour of the solutions prior to the bounce and after, 

we expect the amplification coefficients for the 

scalar and the  tensor modes of the geometry to be very similar, both

qualitatively and quantitatively. Indeed, after some lengthy 

but trivial algebra, the following decoupled evolution equation for 

${\cal R}$ 

can be obtained from Eqs.(\ref{00P1})--(\ref{ineqjP1}):

\begin{equation}

{\cal R}_{k}'' + 2 \frac{z'}{z} {\cal R}_{k}' + k^2 C_{s}^2 {\cal R}_{k} =0,

\end{equation}

where  

\begin{equation}

z = \frac{\varphi'}{{\cal H}_e} a e^{-\vp/2},~~~~~

~~~~

C_{s}^2= \left( 1 + \frac{\partial V}{\partial \vpb} \frac{a^2}{{\vp'}^2} 

\right).

\end{equation} 

On the other hand, in order to enforce the correct quantum normalization, we note 

that ${\cal R}$ is related to the canonical normal mode of scalar perturbations, 

$v_{k} = z {\cal R}_{k}$. For large $|t|$, $V\to 0$, $z \propto 

a e^{- \vp/2}$ and we are led exactly to the same 

equations as for tensor perturbations, the same asymptotic solutions, and thus 

the same spectrum for $\delta R_{k}$.



These expectations will now be checked  numerically. 

We first notice that it is impossible to follow the evolution of perturbations directly through the canonical variable

$v$ (or the other often used variable $u$), since these become singular at the (Einstein-frame) bounce. This is not a problem, however, since $v$ is only needed at very early or very late times, for normalization purposes.

We can instead integrate directly a first-order system of differential equations,  

involving $\psi =- \cal{R}$ and $E'$, which is completely regular throughout. Going over to (string-frame) cosmic time, and using 

the constraint (\ref{i0P1}) into Eqs. (\ref{00P1}) and 

(\ref{ineqjP1}), we can eliminate $\phi$. We then obtain:

\begin{equation}

 \dot{\psi}_{k} = A_{k}(t) \psi_{k} + B_{k}(t) {\cal E}_{k}, ~~~~~~~~~

 \dot{{\cal E}}_{k} = C_{k}(t) \psi_{k} + D_{k}(t) {\cal E}_{k},

\label{seconeq}

\end{equation}

where ${\cal E}_{k}= a^2 \dot{E}_{k}$, and 

\begin{eqnarray}

A_{k}(t) = \frac{2 (\dot{\vp} - 2  H)}{\dot{\vp} }\omega^2, ~~ &&~~ B_{k}(t) =-\biggl(\frac{\dot{\vp} - 2  H}{\dot{\vp}}\biggr)^2 

\omega^2,

\nonumber\\

C_{k}(t) = \frac{4\omega^2}{ \dot{\vp}^2} -1 ,~~ &&~~ 

D_{k}(t) = \dot{\varphi} -  H  - A_{k}(t).

\label{ABCD}

\end{eqnarray}

 This system can be solved by imposing

quantum mechanical initial conditions for  the fluctuations of the

curvature. Using $v_{k} = z {\cal R}_{k}$ we expect, 

asymptotically 

\begin{equation}

{\cal R}_{k} \sim \frac{1}{w} \frac{1}{\sqrt{2 \omega} } \biggl( \alpha_{k} e^{- i \int \omega dt} + 

\beta_{k} e^{ i \int \omega dt} \biggr),

\end{equation}

where $w= \sqrt{a} z$.



\begin{figure}

\centerline{\epsfxsize = 11cm  \epsffile{FIG4.eps}}

\vskip 3mm

\caption[a]{Behaviour of the power spectrum 

of ${\cal R}$, for different values of $\kappa = k t_{0}$.}

\label{FIGURE4} 

\end{figure}



Following the same lines the discussion as were developed in the case of  the

tensor modes of the  geometry, we can obtain, in the asymptotic region

after the bounce,   the appropriate combinations that  determine

the mixing coefficients: 

\begin{eqnarray}

|\alpha_{k}|^2 - |\beta_{k}|^2 &=&  i w^2 \left(  {\cal R}_{k}^* \dot{\cal R}_{k} - {\cal R}_{k} \dot{\cal R}_{k}^* \right) ,

\nonumber\\

 |\alpha_{k}|^2 + |\beta_{k}|^2 &= & w^2 \left( \omega  |{\cal R}_{k}|^2 

+\frac{1}{\omega}  |Y {\cal R}_{k} +\dot{\cal R}_{k}|^2\right)\; , 

\label{MI}

\end{eqnarray}

where $Y = \dot z/z$. 

The numerically computed  power spectrum of ${\cal R}$ is reported in Fig. \ref{FIGURE4}. 

In full analogy with the procedure discussed before, we have also  plotted the quantities appearing 

in Eqs. (\ref{MI}), checking that the first one gives identically $1$, and that the second approaches a limiting value at late times.

 We can thus extract the corresponding value of $|\beta_{k}|^2$  and fit it to the theoretical

expectation.

The results of this analysis  

are  reported in Fig. \ref{FIGURE3},  together with the already discussed results for the 

tensors. The fit for the scalar case (starred points) gives:

\begin{equation}

|\beta_{k}|^2 = 0.46 \ln^2(k/k_{2}) - 2.22,~~~~~~~~~~~~~~k_{2} 

= 2.2 /t_{0}.

\end{equation}

This formula is very similar to the one obtained in 

the case of tensors perturbations.



\begin{figure}

\centerline{\epsfxsize = 10cm  \epsffile{FIG5.eps}}

\vskip 3mm

\caption[a]{Time evolution of the power spectrum of the 

(Einstein-frame) Bardeen potential  $\Psi_E$ (thin and dashed curves).

 The bold curve represents the behaviour of an (appropriately rescaled) $\cal{R}$ mode.} 

\label{FIGURE5} 

\end{figure}



We now turn our attention to the behaviour of the (Einstein-frame) Bardeen potential. 

Its time evolution is shown in Fig. \ref{FIGURE5}, where it is clearly visible that  $\Psi$ 

becomes much larger than one for small $\kappa \eta$,  signalling a breakdown of perturbation theory in the 

longitudinal gauge (a point already emphasized by the first paper quoted in Ref. \cite{13}). 

The two Bardeen potentials in the string frame, though unequal, exhibit similar pathological behaviours.

We also see that, although all variables are regular across the bounce, the values of $\Psi$ and $\Psi'$, unlike those of $\cal{R}$,

change drastically across the bounce. Thus,

assuming continuity of $\Psi$ and of its derivative in simplistic models for the bounce is very dangerous.



Let us finally turn to another popular variable, the spatial-curvature perturbation on constant-density hypersurfaces, $\zeta$. 

It can be shown from its definition that it is not frame-independent, i.e. that

\begin{equation}

\zeta_s = - \psi_s  - \frac{{\cal H}_{s} }{\rho'_s} {\delta \rho_s}\not=

\zeta_E = - \psi_E  - \frac{{\cal H}_{E} }{\rho'_E} {\delta \rho_E}. 

\end{equation}

In any case, in our gauge, $\zeta_E$ and  $\zeta_s$  differ from $\cal{R}$ by terms proportional to $\phi$.

 The latter, however, is very suppressed at large scales (cf. Eqs. (\ref{i0P1}) and (\ref{pspr})). We conclude that

also $\zeta_s$ and $\zeta_E$ are well-behaved across the bounce. The fact that the power spectra of  both $\zeta_E$,  $\zeta_s$ and $\cal{R}$

stay small across the bounce implies that perturbation theory remains valid at all times in our gauge,  provided

$H(0)/M_P(0) < 1$.



We now summarize our results and draw some conclusions:

\begin{itemize}

\item Having defined a non-singular  bouncing cosmology, we are able to

follow the behaviour of the various perturbation variables from

beginning to end. \item Tensor perturbations behave as expected, with

$h$ becoming constant on superhorizon scales. Its power spectrum can

be computed by studying the associated canonical variable $\mu$ that

turns out to be also regular throughout the bounce. The numerically

computed power spectrum is in perfect agreement with the analytic 

expectations (including log corrections). 

\item For scalar perturbations, the Bardeen

potential $\Psi$,  the curvature perturbation on uniform dilaton

hypersurfaces $\cal{R}$, and the curvature perturbation on constant

 density hypersurfaces $\zeta$, all  go smoothly through the

bounce.  

\item Both $\cal{R}$ and $\zeta$ stay constant on superhorizon

scales, in agreement with general arguments \cite{w}, while $\Psi$ does not.

\item Provided that the ratio of the Hubble parameter to the effective Planck

mass  at the bounce ($t=0$) is small,  $H(0)/M_P(0) < 1$,  $\cal{R}$ and

$\zeta$ remain sufficiently small {\it at all scales} for perturbation

theory to remain valid at all times in the comoving gauge.

\item By

contrast, $\Psi$ becomes so large, at large scales, near the bounce that

perturbation theory breaks down in the longitudinal gauge.

\item The canonical variable $v$ of scalar perturbations 

(as well as another often discussed

variable, $u$) exhibits singularities at the bounce. This is not a problem

since the only use of $v$ is that of giving the initial normalization of the

fluctuations and the final Bogolubov coefficient, and $v$ is well behaved

at sufficiently early or late times.

\item We are thus able to compute

numerically the scalar perturbation power spectrum. We find that it is

very similar to that of tensor perturbations and in agreement with the

analytic results that follow by assuming a smooth behaviour of 

$\cal{R}$ (or $\zeta$) through the bounce (even when the background

itself has discontinuous derivatives). 

\end{itemize} 

In conclusion, at

least within the class of models considered in this paper, the procedure

used to argue  that, in bouncing Universes, the spectrum of adiabatic

scalar perturbations can be much flatter than the one 

of tensor perturbations, appears to be unjustified.



We are grateful to Valerio Bozza for help during the early stages of 

this work.





\begin{thebibliography}{999}

\newcommand{\bb}{\bibitem}



\bb{1}G.  Veneziano, Phys. Lett. {\bf B265}, 287  (1991).



\bb{2}M. Gasperini and G. Veneziano, Astropart. Phys. 

{\bf 1}, 317 (1993).



\bibitem{3} M. Gasperini and G. Veneziano, Phys. Rep. {\bf 373},  1

(2003).



\bb{4}J. Khouri, B. A. Ovrut, P. J. Steinhardt and N. Turok, 

Phys. Rev. {\bf D64}, 123522 (2001); 

J. Khouri, B. A. Ovrut, N. Seiberg, P. J. Steinhardt,  and N. Turok, 

Phys. Rev. {\bf D65}, 086007 (2002).



\bb{5}M. Gasperini and G. Veneziano, 

Phys. Rev. {\bf D50}, 2519 (1994).



\bb{6}R. Brustein, M. Gasperini, M. Giovannini,

V. Mukhanov and G. Veneziano, Phys. Rev. {\bf D51}, 6744 (1995).



\bb{7}M. Gasperini and M. Giovannini, Phys. Lett. {\bf B282},

36 (1992); M. Gasperini and M. Giovannini, Phys. Rev.  {\bf D47}, 1519

(1993).  



\bb{8}R. Brustein, M. Gasperini, M. Giovannini and G.

Veneziano, Phys. Lett. {\bf B361}, 45 (1995).



\bb{9}E. J. Copeland, R. Easther and D. Wands, Phys. Rev. {\bf D56}, 874

(1997); E. J. Copeland, J. E. Lidsey and D. Wands,

Nucl. Phys. {\bf B506}, 407 (1997).



\bb{10}K. Enqvist and M. S. Sloth, Nucl. Phys. {\bf B626}, 395

(2002); D. H. Lyth and D. Wands, Phys. Lett. {\bf B524}, 5 (2002); T. Moroi

and T. Takahashi, Phys. Lett. {\bf B522}, 215 (2001).



\bibitem{11}V. Bozza, M. Gasperini, M. Giovannini and G.

Veneziano, Phys. Lett.  {\bf B543}, 14 (2002); Phys. Rev. D {\bf 67}

063514, (2003).



\bb{12}J. Khouri, B. A. Ovrut, P. J. Steinhardt and N. Turok, 

Phys. Rev. {\bf D66}, 046005 (2002);

S. Gratton, J. Khoury, P. J. Steinhardt, and  N. Turok, 

{\it Conditions for generating scale-invariant density perturbations}

astro-ph/0301395. 



\bb{13} D. H. Lyth, Phys. Lett. B {\bf 526}, 173 (2002); 

Phys.Lett. B {\bf 524}, 1 (2002); 

R. Brandenberger and F. Finelli, JHEP {\bf 11}, 056 (2001);

 S. Tsujikawa, R. Brandenberger, and

F. Finelli, Phys.Rev.D {\bf 66}, 083513 (2002); 

 J. Martin, P. Peter, N. Pinto-Neto, and D. J. Schwarz 

 Phys.Rev.D {\bf 65} 123513 (2002);

 R. Durrer and F. Vernizzi, 

Phys. Rev. D {\bf 66}, 083503 (2002); C. Cartier, R. Durrer and E.

J. Copeland, Phys. Rev. D {\bf 67}, 103517 (2003). 



\bb{16}M. Gasperini, M. Giovannini and G. Veneziano, to appear. 



\bb{BV} R. Brustein and G. Veneziano, Phys. Lett.  {\bf B329},  429 (1994). 



\bb{14}K. A. Meissner and G. Veneziano, Mod. Phys. Lett. 

{\bf A6}, 3397 (1991); Phys. Lett.  {\bf B267},  33 (1991). 



\bibitem{w} S. Weinberg, {\it Adiabatic modes in cosmology}, UTTG-12-02,

 astro-ph/0302326.



\end{thebibliography}

\end{document}

