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\begin{document}
\begin{titlepage}
\def\baselinestretch{1.1}
\begin{flushright}
BA-TH/02-451\\
CERN-TH/2002-352 \\
hep-ph/0212112
\end{flushright}
\vspace*{1cm}
\begin{center}
{\large{\bf Constraints on pre-big bang parameter space\\
from CMBR anisotropies}}
\vspace*{1cm}
{\sl V. Bozza$^{a,b,c}$, M. Gasperini$^{d,e}$,
M. Giovannini$^{c}$ and G. Veneziano$^c$}
{\sl $^a$ Dipartimento di Fisica
``E. R. Caianiello", Universit\`a di
Salerno, 84081 Baronissi, Italy}
{\sl $^b$ INFN, Sezione di
Napoli, Gruppo Collegato di Salerno, Salerno, Italy}
{\sl $^c$ Theoretical Physics Division, CERN, CH-1211
Geneva 23, Switzerland}
{\sl $^d$ Dipartimento di Fisica,
Universit\`a di Bari, Via G. Amendola 173, 70126 Bari, Italy}
{\sl $^e$ INFN, Sezione di Bari, Bari, Italy}
\begin{abstract}
The so-called curvaton mechanism --a way to convert isocurvature
perturbations into adiabatic ones-- is investigated
both analytically and numerically in a
pre-big bang scenario where
the r\^ole of the
curvaton is played by a sufficiently massive Kalb--Ramond axion
of superstring theory.
When combined with observations of CMBR anisotropies
at large and moderate angular scales,
the present analysis allows us to
constrain quite considerably the parameter space of
the model: in particular, the initial displacement of the axion from the
minimum of its potential and the rate of evolution of the
compactification volume during pre-big bang inflation. The combination
of theoretical and experimental constraints favours a slightly blue
spectrum of scalar perturbations, and/or a value of the string scale
in the vicinity of the SUSY-GUT scale.
\end{abstract}
\end{center}
\end{titlepage}
\noindent
\renewcommand{\theequation}{1.\arabic{equation}}
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\section{Introduction}
\label{Sec1}
At present, the largest-scale temperature fluctuations
of the Cosmic Microwave Background radiation (CMBR)
are consistent with a (quasi) scale-invariant spectrum of Gaussian
primordial curvature fluctuations \cite{cobe,cob2,silk}.
The analysis
of the first acoustic oscillations occurring on shorter angular scales
adds the information that such
curvature fluctuations should be predominantly adiabatic
\cite{boom,dasi,maxima,archeops}.
Although sufficiently small amounts of non-Gaussianity and/or
isocurvature perturbations are not excluded,
the above-mentioned observational features represent
an important constraint for any scenario trying to model
the initial stages of our Universe.
In a previous Letter \cite{us} we tried to confront the pre-big
bang scenario \cite{g1,lwc,g2}
with these constraints.
The present paper contains a full description of that analysis and
completes it.
Let us recall that, during the pre-big bang phase,
the quantum fluctuations of all the
light modes present in the low energy effective action are
parametrically amplified. Nonetheless, sizeable large-scale
adiabatic fluctuations are not easily produced from the initial
vacuum through the usual mechanism of parametric amplification.
In particular, both tensor and scalar-metric
fluctuations are amplified with very steep spectra \cite{GV94,bmggv},
resulting in adiabatic modes which are
far too small to explain the observed level of large-scale
CMBR anisotropies \cite{smoo1}.
However, not all the primordial spectra of pre-big bang cosmology are
blue. For instance,
in a pure gravi-dilaton background, the pseudo-scalar
supersymmetric partner of the
dilaton in the dimensionally-reduced string effective action,
the so-called Kalb--Ramond axion, emerges from the pre-big bang
phase with a fluctuation spectrum whose tilt depends on the rate of
change of the compactification
volume \cite{cew,clw}.
Depending on this, the axion tilt can be negative (red spectrum),
positive (blue spectrum), or zero (scale-invariant spectrum).
However, since the homogeneous (background) component of the
axion is trivial, such a spectrum does not affect directly the metric,
hence no curvature perturbation
is generated at this primordial level. In other words, the axion
perturbations are entropic, isocurvature perturbations. This feature
persists, unfortunately,
under $S$-duality
transformations \cite{cew,clw}.
If such an axion is massless, or at least light enough not to have
decayed yet , the induced CMBR fluctuations on large scales
can fit the COBE normalization \cite{dgs1,dgs2,dgs3,mas1},
but, being neither adiabatic nor Gaussian
\cite{mvdv}, they are not able to fit the observed structure of the
first few acoustic peaks.
A possible way out of this problem \cite{us,lwc,es,lw,Moroi} is offered
by the alternative scenario of a massive axion, initially displaced
from the minimum of its non-perturbative potential. In
that case axion perturbations couple to scalar metric perturbations
through the non-vanishing axion's VEV. Eventually, the axion
relaxes toward the minimum of the potential
and then, if heavy enough, decays prior to nucleosynthesis.
During the relaxation
process the dominant source of energy undergoes a drastic change:
it consists of
the radiation produced at the end of the pre-big bang
evolution, and later becomes the pressureless fluid corresponding
to the damped coherent oscillations of the axion. This non-trivial
evolution
results in a non-adiabatic
pressure perturbation which, in turn, is well known \cite{ks,mfb} to
induce curvature perturbations on constant energy (or comoving)
hypersurfaces even on superhorizon scales.
The interplay of such different sources of inhomogeneity,
throughout the different stages of the background evolution,
eventually determines the spectral amplitude
of scalar curvature perturbations right after matter-radiation
equality, when all the
scales of interest for the CMBR data are still outside
the horizon. This
conversion of isocurvature into adiabatic perturbations,
originally suggested in a different context by Mollerach \cite{mol},
also applies to more general cases \cite{lw,Moroi}.
Depending upon the initial value $\sigma_{\rm i}$ of the Kalb--Ramond
background, different
post-big bang histories are possible. If
$\bar{\sigma} < \sigma_{\rm i} \ll 1$ in Planck units
(see below, Eq. (\ref{c1}), for the definition of
$\bar{\sigma}$),
the axion oscillates for a long time before becoming
dominant and eventually decays. For $ \sigma_{\rm i} < \bar{\sigma}$
it may never fully dominate
the energy density before decaying. If, instead, $\sg_{\rm i} \gg 1$ the axion
will dominate before oscillating and a slow-roll (low-scale)
inflationary phase could take place in that epoch. As we
shall see, of all these possibilities CMBR observations seem
to favour the ``natural" one, $\sigma_{\rm i} \sim 1$. In any case, even if
different post-big bang histories will lead
to different spectral amplitudes of the Bardeen potential,
adiabatic scalar metric perturbations will
always be present at some level outside the horizon, prior to
decoupling.
The purpose of the present paper is to report on the calculation of
the spectral amplitude of the induced adiabatic metric
perturbations, and on the comparison of the predictions
of the pre-big bang scenario with the observations coming from the
physics of the CMBR anisotropies. In order to achieve this goal
it is mandatory to have a good understanding both of the
axion relaxation mechanism and of the evolution of the
inhomogeneities. Hence analytical results will be supported with
numerical examples and vice versa. We will present, in particular, a
full derivation of the results for the final adiabatic spectrum of the
Bardeen potential (some of these results have been
summarized already in \cite{us}).
The paper is organized as follows.
In Section II the basic equations describing the
post-big bang evolution of the inhomogeneities
and of the background geometry will be introduced.
In Section III the physics of the different post-big bang histories
will be analysed.
In Section IV the evolution of the background and of its perturbations
will be discussed for the case in which the amplitude of the initial axion
background is
smaller than $1$ in Planck units, $\sg_{\rm i}<1$. In Section V we will
discuss the evolution of the system in the complementary
case $\sg_{\rm i}>1$. Section VI is devoted to the phenomenological
implications of the large-scale
adiabatic perturbations produced through the relaxation
of the axionic background. The obtained results will be compared with
observations. Constraints on the pre-big bang parameters
will be derived.
Section VII contains our concluding remarks while, in the
Appendix, a self-contained derivation of the axionic
spectra produced by the pre-big bang evolution has been included.
\renewcommand{\theequation}{2.\arabic{equation}}
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\section{Background and perturbation equations}
As already mentioned, we shall start our analysis at some
time $\eta_{\rm i}$ in the post-big bang epoch, assuming that
the axion field has inherited from the preceding epoch
appreciable large--scale
fluctuations, while other sources of energy as well as the metric
are exactly homogeneous. It will also be assumed that, initially,
the dominant source of energy is in the form of radiation.
The post-big bang dynamics takes place, in the present analysis,
when the curvature scale has fallen to a sufficiently
small value (in string units) so that the use of the low-energy
effective action
is appropriate. Furthermore, for $\eta >\eta_{\rm i}$
the dilaton is assumed to be frozen already at its present value.
Under these assumptions,
the evolution of the geometry is determined by the Einstein
equations, supplemented by the conservation
equations determining the dynamics of the
sources\footnote{Gravitational units
$16 \pi G= 1$
will be used throughout. When explicitely written in the formulae,
$M_{\rm P} = (16 \pi G)^{-1/2} = 1.72\times 10^{18}~{\rm GeV}$. In these
units,
$\sigma$ is the canonically
normalized axion field.}:
\begin{eqnarray}
&& R_{\alpha}^{\beta} - \frac{1}{2} \delta_{\alpha}^{\beta} R =
\frac{1}{2} \biggl( T_{\alpha}^{\beta}(\sigma)
+ {\cal T}_{\alpha}^{\beta} \biggr),
\label{g1}\\
&& g^{\alpha\beta} \nabla_{\alpha} \nabla_{\beta} \sigma +
\frac{\partial V}{\partial\sigma} =0,
\label{g2}
\end{eqnarray}
where $T_{\alpha}^{\beta}(\sigma)$ and ${\cal T}_{\alpha}^{\beta}$ are,
respectively, the energy-momentum tensors of the axionic background
and of the matter fluid. Notice that the covariant conservation of
$T_{\alpha}^{\beta}(\sigma)$ is dynamically
equivalent to the evolution equation of the axionic field, i.e.
Eq. (\ref{g2}), and implies, through the contracted Bianchi identities:
\beq
\nabla_{\alpha} {\cal T}_{\beta}^{\alpha} =0.
\label{g3}
\eeq
In a conformally flat background geometry,
\begin{equation}
ds^2 = a^2(\eta) [ d\eta^2 - d\vec{x}^2],
\label{lel}
\end{equation}
Eqs. (\ref{g1})--(\ref{g3}) lead to a set of three independent
equations, whose specific form is dictated by the fluid content
of the primordial plasma. In the case of
a radiation fluid we have
\begin{equation}
{\cal T}_{0}^{0} = \rho_{\rm r}, ~~~~~ {\cal T}_{i}^{j}
= - p_{\rm r} \delta_{i}^{j}, ~~~~~p_{\rm r}
= \frac{\rho_{\rm r}}{3},
\end{equation}
and Eqs. (\ref{g1})--(\ref{g3}) lead to
\begin{eqnarray}
&& {\cal H}' = - \frac{a^2}{6} \biggl[ \rho_{\rm r}
+ \frac{ {\sigma'}^2}{a^2} - V\biggr],
\label{dyn}\\
&& \sigma'' + 2 {\cal H} \sigma' + a^2 \frac{\partial V}{\partial\sigma}
=0, \label{kg}\\
&& \rho_{\rm r}' + 4 {\cal H} \rho_{\rm r} =0.
\label{contrad}
\end{eqnarray}
Here the prime denotes the derivation with
respect to the conformal time coordinate $\eta$, and
${\cal H}= (\ln{a})'$. For future convenience we also
recall that the connection between ${\cal H}$ and the
Hubble parameter is $ H= {\cal H}/a$.
The
effective energy and pressure densities of $\sigma$ will be given by
\begin{equation}
\rho_{\sigma} = \frac{{\sigma'}^2}{2 a^2} + V, ~~~~~~~~
p_{\sigma} = \frac{{\sigma'}^2}{2 a^2} - V.
\label{backdef}
\end{equation}
The set of dynamical equations (\ref{dyn})--(\ref{contrad}) is
supplemented by the Hamiltonian constraint
\begin{equation}
{\cal H}^2 = \frac{a^2}{6} \biggl[ \rho_{\rm r} + \frac{{\sigma'}^2}{2 a^2}
+ V\biggr],
\label{ham}
\end{equation}
which imposes a specific relation on the set of initial
data and is required, in particular, for the numerical integration of the
background evolution.
During the post big-bang phase, the first order perturbation of Eqs.
(\ref{g1})--(\ref{g3}) provides the linear (coupled) system of evolution
equations of the inhomogeneities. To first order in the scalar
metric fluctuations, the line element (\ref{lel}) can be written as
\cite{mfb} \begin{equation}
ds^2 = a^2(\eta) \{( 1 + 2 \phi) d\eta^2 - 2 \partial_{i}B dx^{i} d\eta^2 -
[ ( 1 - 2 \psi) \delta_{i j} + 2 \partial_{i} \partial_{j} E ] dx^i dx^j \}.
\label{pertlel}
\end{equation}
Since there are two gauge transformations preserving the
scalar nature of the above metric fluctuations $(\psi, \phi,E,B)$,
two gauge-invariant
(Bardeen) potentials can be defined \cite{mfb,bar}:
\begin{eqnarray}
&&\Phi = \phi + \frac{1}{a} [ ( B- E') a]',
\label{b1}\\
&& \Psi = \psi - {\cal H} ( B - E').
\label{b2}
\end{eqnarray}
Appropriate gauge-invariant variables
can also be defined
for the perturbations of the sources, in such a way that
\begin{eqnarray}
&&\chi^{(\rm gi)} = \delta\sigma + \sigma' ( B - E') ,
\label{chidef}\\
&& \delta\rho^{(\rm gi)} _{\rm r} = \delta\rho_{\rm r} +
\rho_{\rm r}' ( B - E'),
\label{rhrdef}\\
&& v_{\rm r}^{(\rm gi)} = v_{\rm r} + (B -E'),
\label{potdef}
\end{eqnarray}
whose physical interpretation is particularly simple in the
so-called longitudinal gauge \cite{mfb} in which $E=0=B$. Here
$\delta {\cal T}_{0}^{0} = \delta \rho_{\rm r}$, and the velocity
potential is defined by the off-diagonal fluctuations of the radiation
energy-momentum tensor as \begin{equation}
\delta {\cal T}_{i}^{0} = (p_{\rm r} + \rho_{\rm r}) u^{0} \delta u_{i},
\end{equation}
where $ u^{0} = 1/a$ and, in the longitudinal gauge,
$ \delta u_{i} = a \partial_{i} v_{\rm r}$.
By perturbing the diagonal components of $T_{\mu}^{\nu}(\sigma)$, and
using
Eqs. (\ref{b1}) and (\ref{chidef}),
the fluctuations of the axionic energy and pressure
densities can be expressed in a fully gauge-invariant way as follows
\footnote{ In the
following, since we will be dealing only with gauge-invariant quantities,
the superscript ``$({\rm gi})$'' can be consistently dropped
without confusion. }:
\begin{eqnarray}
&& \delta \rho_{\sigma} = \frac{1}{a^2} \biggl[ - \Phi {\sigma'}^2 + \sigma'
\chi' + \frac{\partial V}{\partial\sigma} a^2\chi \biggr],
\label{deltarhosigma}\\
&& \delta p_{\sigma} = \frac{1}{a^2} \biggl[ - \Phi {\sigma'}^2 +
\sigma' \chi' - \frac{\partial V}{\partial\sigma} a^2 \chi\biggr].
\label{deltapsigma}
\end{eqnarray}
The variables characterizing the gauge-invariant
fluctuations of the sources can be defined in different, but equivalent,
ways \cite{ks,dur}.
For instance, it is sometimes useful (especially in the case
of fluids with constant speed of sound) to write
equations for the combination $(\delta \r_r/\r_r- 4 \Phi)$, whose
evolution greatly simplifies at large scales.
The fluctuations of the off-diagonal (space-like) components of Eq.
(\ref{g1}) imply that $\Phi = \Psi$. Hence, in terms of the variables
defined in Eqs. (\ref{b1})--(\ref{deltapsigma}), the ($00$) and ($0i$)
components of the perturbed Einstein equations (acting as Hamiltonian
and momentum constraints for the evolution of the Bardeen potential)
can be written in terms of the gauge-invariant velocity potentials
$v_{\rm r}$, $v_{\sigma}$, and of the radiation and axion density
contrasts $ \delta_{\rm r}=\delta\rho_{\rm r}/\rho_{\rm r}$,
$\delta_{\sigma} = \delta \rho_{\sigma}/\rho_{\sigma} $, as follows:
\begin{eqnarray}
&& \nabla^2 \Phi - 3 {\cal H} ( {\cal H} \Phi + \Phi') =
\frac{a^2}{4} \biggl( \rho_{{\rm r}} \delta_{\rm r} + \rho_{\sigma}
\delta_{\sigma} \biggr),
\label{hamp}\\
&& {\cal H} \Phi + \Phi' = \frac{a^2}{4}\biggl[ (\rho_{\rm r} + p_{\rm r} )
v_{\rm r} + (\rho_{\sigma} + p_{\sigma}) v_{\sigma} \biggr],
\label{momp}
\end{eqnarray}
Here the
axion velocity potential, $v_{\sigma}$, is defined by
\begin{equation}
v_{\sigma} = \frac{ \chi}{a \sqrt{ p_{\sigma} + \rho_{\sigma}}}
\label{usigma}
\end{equation}
and is the axionic counterpart of
the velocity potential introduced for the radiation fluid.
The constraints (\ref{hamp}) and (\ref{momp})
are to be supplemented by the dynamical equations coming from the
perturbation of the ($ii$) components of Einstein's
equations (\ref{g1}), of the axion equation (\ref{g2}) and of the
continuity equation (\ref{g3}). For the
gauge-invariant quantities defined above, such dynamical equations
are, respectively:
\begin{eqnarray} && \Phi'' + 3 {\cal H} \Phi' + ( {\cal
H}^2 + 2 {\cal H}') \Phi = \frac{a^2}{12} \rho_{\rm r} \delta_{\rm r} +
\frac{a^2}{4} \delta p_{\sigma} , \label{ij}\\
&& \chi'' + 2 {\cal H} \chi' - \nabla^2 \chi +
\frac{\partial^2 V}{\partial\sigma^2} a^2 \chi - 4 \sigma' \Phi' + 2
\frac{\partial V}{\partial \sigma }a^2 \Phi =0,
\label{chired} \\
&& \delta_{\rm r}' - 4 \Phi' - \frac{4}{3} \nabla^2 v_{\rm r} =0,
\label{deltar}\\
&& v_{\rm r}' - \frac{1}{4} \delta_{\rm r} - \Phi =0.
\label{ureq}
\end{eqnarray}
Finally, the perturbation of the covariant conservation of the axionic
energy-momentum tensor leads to two useful equations:
\begin{eqnarray}
&& \rho_{\sigma} \delta_{\sigma}' - ( p_{\sigma} + \rho_{\sigma})
\nabla^2 v_{\sigma} - 3 {\cal H} p_{\sigma} \delta_{\sigma}
- 3 \Phi' ( p_{\sigma} + \rho_{\sigma} ) + 3 {\cal H} \delta p_{\sigma} =0,
\label{contsig}\\
&& v_{\sigma}' + \biggl( 4 {\cal H} +
\frac{ p_{\sigma}' + \rho_{\sigma}'}{p_{\sigma} +
\rho_{\sigma}}\biggr) v_{\sigma} - \frac{\delta p_{\sigma}}{p_{\sigma} +
\rho_{\sigma} } - \Phi =0,
\label{usigeq}
\end{eqnarray}
which are implied, as it should be, by Eqs. (\ref{ij})--(\ref{ureq}) when
the background equations (\ref{dyn})--(\ref{kg}) are used.
It is also useful to notice that, by combining Eqs. (\ref{ham}) and
(\ref{ij}), we can eliminate the fluid variables, and we obtain
\begin{equation}
\Phi'' + 4 {\cal H} \Phi' + 2 ( {\cal H}^2 + {\cal H}') \Phi - \frac{1}{3}
\nabla^2 \Phi = - \frac{{\sigma'}^2}{6} \Phi + \frac{\sigma'}{6} \chi'
- \frac{1}{3} \frac{\partial V}{\partial \sigma} a^2 \chi,
\label{phred}
\end{equation}
which, together with Eq. (\ref{chired}), provides a closed
system of equations for $\Phi$ and $\chi$. Of course, the velocity
potential and the density contrast of the fluid do not disappear
from the physics of our problem, and have to be
directly computed using the Hamiltonian and momentum
constraints of Eqs. (\ref{hamp}) and (\ref{momp}).
\subsection{Curvature perturbations from non adiabaticity}
Given the system of Eqs. (\ref{ij})--(\ref{ureq}), supplemented
by the constraints (\ref{hamp})--(\ref{momp}), it is
sometimes appropriate to select variables obeying simple
evolution equations in the long-wavelength limit, in which
the spatial gradients are negligible. For this purpose,
a particular combination of Eqs. (\ref{hamp})
and (\ref{ij}) will be considered, and
the fluctuations in the total energy and pressure densities will be
defined:
\beq
\delta \rho_{\rm tot} = \delta \rho_{\sigma}+ \delta \rho_{\rm r},
~~~~~~~~~~~
\delta p_{\rm tot} = \delta p_{\sigma}+ \delta p_{\rm r}.
\label{tot}
\eeq
In terms of the quantities
defined in Eq. (\ref{tot}),
the evolution of the Bardeen potential can be formally
written in terms of a single equation
\begin{equation}
\Phi'' + 3 {\cal H} ( 1 + c_{s}^2) \Phi' +
[ 2 {\cal H}' + {\cal H}^2 ( 1 + 3 c_{s}^2)]\Phi - c_s^2 \nabla^2 \Phi=
\frac{a^2}{4} [ \delta p_{\rm tot} - c_{s}^2 \delta \rho_{\rm tot}],
\label{source}
\end{equation}
where $c_s$ is the speed of sound for the total system, defined by
\begin{equation}
c_{s}^2 = \frac{ p_{\rm tot}'}{\rho_{\rm tot}'} \equiv
\frac{ p_{\sigma}' + p_{\rm r}'}{\rho_{\sigma}' + \rho_{\rm r}'},
\end{equation}
or, using the explicit form of the background equations,
\begin{equation}
c_{s}^2 =
\frac{1}{3}\biggl\{ \frac{\rho_{\rm r}
+ \frac{9}{4} ( p_{\sigma} + \rho_{\sigma}) +
\frac{3}{2} \frac{\sigma'}{{\cal H}} V_{,\sigma} }{\rho_{\rm r} +
\frac{3}{4} ( p_{\sigma} + \rho_{\sigma})}\biggr\},
\label{sps}
\end{equation}
where $V_{,\sigma} \equiv \pa V/\pa \sg$.
The left-hand side of Eq. (\ref{source}) (except for the Laplacian term)
can now be expressed as the time derivative of a single
gauge-invariant function $\zeta$, namely
\begin{equation}
\zeta = - \biggl[ \Phi + \frac{4{\cal H}}{a^2}\biggl( \frac{ {\cal H} \Phi
+ \Phi'}{\rho_{\rm tot} + p_{\rm tot}}
\biggr) \biggr] \equiv - \biggl( \Phi + {\cal H} \frac{ {\cal H} \Phi
+ \Phi'}{{\cal H}^2 - {\cal H}'}\biggr),
\label{zeta1}
\end{equation}
where the second equality follows by using the background
equations of motion (\ref{dyn})--(\ref{ham}). By using this variable,
Eq. (\ref{source}) can be written as
\begin{equation}
\frac{ d \zeta}{d\eta}
= - \frac{{\cal H}}{p_{\rm tot} + \rho_{\rm tot}} \delta p_{\rm nad}
- \frac{ 4 {\cal H} c_{s}^2}{a^2 (\rho_{\rm tot} + p_{\rm tot})}
\nabla^2 \Phi.
\label{zetapr}
\end{equation}
where we have defined
\begin{equation}
\delta p_{\rm nad} = \delta p_{\rm tot} - c_s^2 \delta \rho_{\rm tot}.
\label{nad}
\end{equation}
As noticed long ago \cite{ks,bar,lyt}, the variable $\zeta$ represents
the inhomogeneities in the spatial part of the space-time curvature,
measured with respect to comoving hypersurfaces ($\sigma= {\rm
constant}$).
Using Eq. (\ref{momp}), the variable $\zeta$ can also be usefully
related to the total velocity potential as
\begin{equation}
\zeta = - ( \Phi + {\cal H} v_{\rm tot}),
\label{zeta}
\end{equation}
where
\begin{equation}
(p_{\rm tot} + \rho_{\rm tot}) v_{\rm tot} = ( p_{\rm r} + \rho_{\rm r})
v_{\rm r} + ( p_{\sigma} + \rho_{\sigma} )v_{\sigma}.
\label{utot}
\end{equation}
In our specific case, using the full set of background and perturbation
equations in the long-wavelength limit, where
$ \delta_{\rm r} \sim 4 \Phi$ according to Eq. (\ref{deltar}),
the expression for $\zeta$ can be written in the following convenient
form
\begin{equation}
\zeta= - \frac{ \frac{3}{4} (p _{\sigma} + \rho_{\sigma}) \delta_{\rm r}
- \rho_{\sigma} \delta_{\sigma} }{4 \rho_{\rm r} + 3
( p_{\sigma} + \rho_{\sigma}) }.
\label{zetaex}
\end{equation}
As we will discuss in detail in Sect. V, in the absence of a dominant
radiation fluid $\delta p_{\rm nad}$ is zero at
large scales, i.e. up to terms containing the Laplacian of $\Phi$.
However, in a radiation dominated regime,
$\delta p_{\rm nad} \neq 0$ and Eq.
(\ref{zetapr}) implies $\zeta' \neq 0$ even in the long-wavelength
limit. Let us then compute the
general form of $\delta p_{\rm nad}$, for the full system of axion plus
fluid perturbations. By using the previous definitions we obtain
\begin{equation} \delta p_{\rm nad} = \rho_{\rm r} \biggl(\frac{1}{3} -
c_{s}^2\biggr) \delta_{\rm r}
+ \Phi ( c_{s}^2 -1) ( p_{\sigma} + \rho_{\sigma}) +
\frac{\sigma' \chi'}{a^2} ( 1 - c_{s}^2) -
\frac{\partial V}{\partial\sigma} \chi ( 1 + c_s^2).
\label{dpnadex}
\end{equation}
On the other hand, using Eqs. (\ref{hamp})
and (\ref{momp}), we can write
\begin{equation}
\Phi(p_{\sigma} + \rho_{\sigma}) = - \frac{4}{a^2} \nabla^2 \Phi
+ 3 {\cal H} \frac{ \sigma' \chi}{a^2} + 4 {\cal H} \rho_{\rm r} v_{\rm r}
+ \rho_{\rm r} \delta_{\rm r} +
\biggl[ \frac{\sigma' \chi'}{a^2} + \frac{\partial V}{\partial\sigma}
\chi\biggr].
\label{intermed}
\end{equation}
Thus (neglecting
the spatial gradient of $\Phi$) we get
\begin{equation}
\delta p_{\rm nad} = - \frac{2}{3} \rho_{\rm r} \delta_{\rm r}
- 2 \frac{\partial V}{\partial\sigma} \chi + 4 {\cal H} \rho_{\rm r}
v_{\rm r}
(c_s^2 -1)
+3 {\cal H} (c_s^2 -1) \frac{ \sigma' \chi}{a^2}.
\label{dpnad2}
\end{equation}
The above equations are useful to compute, in some specific phase of
the dynamical evolution, the source term of Eq. (\ref{zetapr}) whose
integration allows us to obtain the explicit time dependence of $\zeta$.
In \cite{us} we have determined the evolution of the fluctuations by
following the $\zeta$ variable. In the present investigation we will
solve the perturbation equations both in terms of $\Phi$ and
$\zeta$, checking numerically the consistency of the two
approaches.
\renewcommand{\theequation}{3.\arabic{equation}}
\setcounter{equation}{0}
\section{Post big-bang histories}
At the beginning of the post-big bang evolution the background is
characterized by a ``maximal" curvature scale $H_1$, whose finite
value regularizes the big bang singularity of the standard cosmological
scenario, and provides a natural cutoff for the spectrum of
quantum fluctuations amplified by the phase of pre-big bang inflation
(see below, in particular Section VI). In string cosmology models such
an initial curvature scale is at most of the order of the string mass
scale, i.e. $H_1 \laq M_{\rm s} \sim 10^{17}$ GeV.
The Kalb--Ramond axion
has gravitational coupling to photons and to the QCD
topological current but it is not necessarily identified
with the invisible axion \cite{kim} usually
invoked in the explanation of the strong CP problem via
an initial misalignment of the QCD vacuum angle $\vartheta$ \cite{pre}.
The potential of Kalb--Ramond axion is, strictly speaking,
periodic. The periodicity of the potential occurs whenever
a Peccei--Quinn symmetry is spontaneously broken down to
a discrete symmetry corresponding to shifting the
$\vartheta$ angle by multiples
of $2\pi$ (see, for instance, \cite{dvv}).
However, close to the
minimum of the potential (i.e. sufficiently late in the process of
relaxation) the potential can be assumed to be quadratic.
Such an approximation is expected to
be realistic for values of $\sigma$ that are small compared to its
periodicity.
Unfortunately, translating periodicity in $\vartheta$ into periodicity in $\sigma$
involves a normalization factor that is unknown in the strong-coupling region
where the dilaton is supposed to be frozen at
late times. For this reason, we shall keep
the initial dispacement in Planck units,
$\sigma_{\rm i}$, as a free parameter.
We start our study of the background and perturbation evolution at
an initial curvature scale $H_{\rm i}\leq H_1$, when the energy density
of the background is mainly stored in the radiation fluid, while the
energy density of the axion is dominated by the potential:
\begin{equation} \rho_{\rm r}(\eta_{\rm i}) \gg
\rho_{\sigma}(\eta_{\rm i})\simeq V(\eta_{\rm i}).
\end{equation}
During the first stages of the evolution $\sigma$ remains
approximately fixed at the initial value $\sigma_{\rm i}$ up to
corrections ${\cal O}(V_{,\sigma})$. In the course of such a
``slow-roll" phase, the curvature scale of the background decreases,
until it becomes comparable with the curvature of the potential.
The axion background will then start oscillating,
at a typical scale
\begin{equation}
H_{\rm osc} \sim m
\label{osc}
\end{equation}
(note that, as already mentioned, we are assuming
that the potential is quadratic).
At the curvature scale
\begin{equation}
H_{\sigma} \sim m \sigma(t),
\label{dominance}
\end{equation}
the axion field will dominate the background.
The specific value of the scale $H_{\sigma}$ depends upon $\sigma_{\rm i}$
and also upon the evolution after $\eta_{\rm i}$.
In fact, during the oscillatory phase the axionic energy density
decreases, on the average, as $a^{-3}$, i.e. slower than the energy of
the radiation background, $\rho_{\rm r}
\simeq a^{-4}$.
From Eqs. (\ref{osc}) and (\ref{dominance})
it is then clear
that, depending on the initial value of $\sigma$, the
oscillations of the axionic background may arise either before
or after the phase of $\sigma$-dominance.
Irrespectively of its initial value,
the coupling of $\sigma$ to photons is gravitational, i.e.
suppressed by the Planck mass.
The decay takes place when the curvature
scale is of the same order as the decay rate, namely when
\begin{equation}
H\sim H_{\rm d} \sim \frac{m^3}{M_{\rm P}^2}.
\label{decb}
\end{equation}
The late decay of $\sigma$ is in general associated to a
significant entropy release, which has to be carefully
constrained \cite{ell1,ell2,g3} not to spoil the light nuclei abundances
and the baryon asymmetry generated, respectively, by
primordial nucleosynthesis and baryogenesis.
In our context, for typical values of $H_1$, and for a
realistic scenario, the decay of $\sigma$ is constrained to occur
prior to nucleosynthesis, i.e. at a scale
$H_{\rm d}>H_{\rm N} \sim
(1 {\rm MeV})^2/M_{\rm P}$, which implies $m \gaq 10$ TeV.
The lower bound on the axion mass is even larger if we
require that the decay occurs prior to baryogenesis at the
electroweak scale (characterized by a temperature of the
cosmological plasma of order $0.1$ TeV), which implies
$m \gaq 10^4$ TeV. If, on the contrary, baryogenesis occurs at a
large enough scale preceeding the phase of axion
dominance and decay, then the minimal value of $m$
allowed by the entropy constraints \cite{ell1,ell2,g3} is, in general,
$\sg_{\rm i}$-dependent. In that case, however, the resulting
lower bound is strongly dependent on the given model of
baryogenesis, and can be somewhat relaxed by various
mechanisms. In the rest of this paper we will thus adopt a
conservative approach, by taking the nucleosynthesis bound
$m \gaq 10^{-14} M_{\rm P}$ as a typical reference value.
\subsection{Late dominance of the axion: $\sigma_{\rm i} <1$}
If $\sigma(\eta_{\rm i}) = \sigma_{\rm i} < 1$,
then
the axionic background first experiences
a phase of radiation-dominated oscillations,
from $H_{\rm osc}$ down to $H_{\sigma}$.
The duration of this phase depends upon $\sigma_{\rm i}$, since
$(a_{\rm osc}/a_{\sigma}) \sim \sigma_{\rm i}^2$, and it may be rather
long, if $\sigma_{\rm i} \ll 1$.
During this phase the axion potential energy decreases as
$a^{-3}$. Consequently,
the typical scale of axion dominance is, from Eq. (\ref{dominance}),
\begin{equation}
H_{\sigma} \simeq m \sigma_{\rm i}^4.
\end{equation}
From $H_{\sigma}$ down to $H_{\rm d}$, i.e. inside the regime
of axion-dominated oscillations,
the effective equation of state of the gravitational sources, averaged
over one oscillation, mimics that of
dusty matter, with $\langle p_{\sigma} \rangle =0$.
During this regime the scale factor and the Hubble
parameter also have oscillating corrections, which vanish on the
average, and decay away for large times. It should be stressed,
however, that the effective equation of state of the axion background,
for curvature scales smaller than $H_{\sigma}$,
depends upon the curvature of the potential
around the minimum. If, for instance, the potential
is not quadratic, but quartic, the coherent
oscillations will lead to an effective equation of state
that simulates a radiation fluid, i.e.
$3 \langle p_{\sigma}\rangle = \langle \rho_{\sigma}\rangle$ \cite{tur}.
The occurrence of the axion-dominated phase requires
\begin{equation}
H_{\rm d} < H_{\sigma},
\label{h1}
\end{equation}
which imposes a lower bound on the
initial axionic amplitude, namely
\begin{equation}
1>\sigma_{\rm i} > \sqrt{{m}/{M_{P}}}\equiv \bar{\sigma}.
\label{c1}
\end{equation}
This constraint,
however, is not so demanding, given the generous lower
bound on $m$ (in Planck units) allowed by nucleosynthesis and
baryogenesis. Finally, after the axion decay, the Universe enters a
subsequent
radiation-dominated epoch. From this moment on, the evolution
of the background fields is standard.
\subsection{Early dominance of the axion: $\sigma_{\rm i} >1$}
If $\sigma(\eta_{\rm i}) =\sigma_{\rm i} >1$,
then the axion, right after
the onset of the radiation-dominated epoch, starts again rolling down
its potential. This initial part of the
evolution is completely analogous to that of the $\sigma_{\rm i} < 1$
case.
However, for $\sigma_{\rm i} >1$, the axion dominance
will occur before the onset of the axion oscillating phase,
i.e.
\begin{equation}
H_{\rm osc} < H_{\sigma},
\label{h2}
\end{equation}
where, for a generic potential,
\begin{equation}
H_{\sigma} \sim \sqrt{V(\sigma_{\rm i})}
\label{dom2}
\end{equation}
(since the kinetic energy of the axion
is negligible during the slow-roll evolution).
At $H=H_{\sigma}$ the Universe enters a phase of accelerated
expansion (slow-roll inflation) whose duration, for a quadratic
potential, is given by
\begin{equation}
Z_{\sigma} = \frac{a_{\rm final}}{a_{\rm initial}}=
\exp\left[{ \frac{1}{8} ( \sigma_{\rm initial}^2 - \sigma_{\rm
final}^2)}\right].
\label{srdurtaion}
\end{equation}
This
inflationary phase will last until $H=H_{\rm osc}\sim m$,
$\sigma_{\rm final}
\simeq 1$
(if we assume, again, that close to its minimum the potential is
quadratic). For $H 1.
\label{c2}
\end{equation}
As in the case of Eq. (\ref{c1}), this bound is not so restrictive,
given the limits on the axion mass. Indeed, in the case
$\sigma_{\rm i} >1$, the most stringent constraints are coming
not from the
evolution of the background geometry but, as we shall see, from the
evolution of the fluctuations that forbid too large
values of $\sigma_{\rm i}$. One is then left with a situation where
$\sigma_{\rm i} \simeq 1$ and $H_{\sigma} \simeq H_{\rm osc}$. In
such a case, the phase of axion-dominated oscillations will take place
right after the radiation-dominated period of slow-roll, without
a long intermediate epoch of inflation.
\subsection{Initial conditions for the fluctuations}
Given the coupled system of gauge-invariant perturbation equations,
the initial conditions for the Bardeen potential, for the perturbed
radiation density and for the radiation velocity field, will be imposed as
follows
\begin{equation}
\Phi_{k}(\eta_{\rm i}) =0,~~~~~\delta_{\rm r}(\eta_{\rm i}, k) =0,
~~~~v_{\rm r}(\eta_{\rm i},k) =0,
\label{incon3}
\end{equation}
assuming that no appreciable amount of adiabatic metric perturbations
has been directly generated (on large scales) by the
pre-big bang dynamics. The only non-vanishing initial
fluctuations are the (isocurvature) axionic seeds, amplified from
the vacuum during the pre-big bang evolution:
\begin{equation}
\chi_{k}(\eta_{\rm i}) = \chi_{\rm i} (k)\neq 0,
\end{equation}
so that, from Eq. (\ref{deltarhosigma}),
$\delta_{\sigma} (\eta_{\rm i}, \vec{x} ) \neq 0$.
In the present analysis we shall assume that the amplitude of the
initial axion fluctuations is smaller (for all modes)
than $\sigma_{\rm i}$, i.e.
\begin{equation}
k^{3/2}|\chi_{\rm i}(k)| < \sigma_{\rm i}.
\label{incon3a}
\end{equation}
In the opposite case, $k^{3/2}|\chi_{\rm i}(k)| > \sigma_{\rm i}$,
we are led to the case already analysed in
\cite{dgs1,dgs2,dgs3,mvdv} where $\sigma_{\rm i}$ was assumed
to be zero, and the obtained
large-scale fluctuations are of the isocurvature
type, and strongly non-Gaussian.
If $\sigma_{\rm i} >\bar{\sigma}$, the non-Gaussianity is rather
small, but can be enhanced if the axion does not dominate
at decay ($\sigma_{\rm i} < \bar{\sigma}$) \cite{lw,luw}.
\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
\section{Background and perturbation evolution for
$\sigma_{\i} < 1$}
In view
of the forthcoming phenomenological applications,
the main quantity that we need to compute is the
spectral amplitude of the Bardeen potential after axion
decay, during the subsequent radiation-dominated evolution, as a
function of the spectral
amplitude of the axion fluctuations amplified by the phase of pre-big
bang inflation. It is important, for this purpose, to have a reasonably
accurate control on the evolution of the background and of the
fluctuations. Using different approximations, motivated by the
hierarchy of scales discussed in the previous section, we will first
analytically determine the evolution
of the system through the different cosmological stages.
Numerical integrations will then be used in order to check
the analytical results in the cross-over
regimes connecting the different phases of the background evolution.
\subsection{The radiation-dominated slow-roll regime}
During the first stage of evolution,
$\rho_{\rm r}(\eta_{\rm i}) \gg V(\eta_{\rm i})$. In this limit, Eqs.
(\ref{dyn})--(\ref{contrad}) and (\ref{ham}) simplify
to:
\begin{eqnarray}
&&{\cal H}^2 = \frac{a^2}{6} \rho_{\rm r},
\label{slowrolla}\\
&& \rho_{\rm r}' + 4 {\cal H} \rho_{\rm r} =0,
\label{slowrollb}\\
&& \sigma'' + 2 {\cal H} \sigma' + a^2 \frac{\partial V}{\partial\sigma}
=0.
\label{slowroll}
\end{eqnarray}
Equations (\ref{slowrolla}) and (\ref{slowrollb})
imply that ${\cal H} a$ is constant. Furthermore,
since the kinetic energy of $\sigma$ is subleading with respect to
the potential, the axionic field
slowly rolls down the potential.
In such a situation a systematic expansion
in the gradient of the potential, $V_{,\sigma}$,
can be developed, and the background evolution
is adequately described by the following approximate
equation
\begin{equation}
\sigma' = - \frac{1}{5} \frac{a^2}{\cal H} V_{,\sigma},
\label{sre}
\end{equation}
which can be integrated to give
\begin{equation}
\sigma \simeq \sigma_{\rm i} - \frac{1}{20}
\biggl(\frac{V_{,\sigma}}{H^2}-
\left. \frac{V_{,\sigma}}{H^2} \right|_{\rm i}\biggr),
\label{sre2}
\end{equation}
i.e. $\sigma$ is approximately constant up to
corrections that depend upon the specific form of the
potential, and which induce a slight decrease of the
axion background.
In order to solve
the Hamiltonian constraint (\ref{hamp})
it is now convenient to work in terms of the Fourier components of the
perturbation variables, $\Phi_k, \delta_{\rm r}(k)$, and so on. Since we are
interested in large scale inhomogeneities we first obtain,
from Eq. (\ref{deltar}),
\begin{equation}
\delta_{\rm r} (k) \simeq 4 \Phi_{k} ,
\label{drph}
\end{equation}
where the integration constants vanish because of Eq. (\ref{incon3}).
Consequently, using Eq. (\ref{deltarhosigma}), the Hamiltonian
constraint (\ref{hamp}) can be written as
\begin{equation}
- 3 {\cal H} ( {\cal H} \Phi_{k} + \Phi_{k}' ) -
\Phi_{k}\biggl[ a^2 \rho_{\rm r} - \frac{{\sigma '}^2}{4}\biggr]
= \frac{1}{4} \biggl[ \sigma' \chi_{k}' + V_{,\sigma}
a^2 \chi_{k}\biggr],
\label{hamint}
\end{equation}
where the spatial gradients have been consistently
neglected.
Using Eq. (\ref{slowrolla}),
\begin{equation}
\Phi_{k}' + 3 {\cal H}\Phi_{k}
\simeq - \frac{1}{12 {\cal H} }\biggl[ \sigma' \chi_{k}'
+ \frac{\partial V}{\partial\sigma} a^2 \chi_{k}\biggr].
\label{app}
\end{equation}
On the other hand, from Eq. (\ref{chired}), the evolution of $\chi_{k}$
is approximately given by
\begin{equation}
\chi_{k}' \simeq - \frac{1}{5} V_{,\sigma\sigma}
\frac{a^2}{{\cal H}} \chi_{k}.
\label{chev}
\end{equation}
The first term on the r.h.s. of Eq. (\ref{app}) thus
contains three derivatives of the potential, and it is
subleading with respect to the second term.
Direct integration of Eq. (\ref{app}) then gives
\begin{equation}
\Phi_{k}(\eta) = - \frac{1}{84} \frac{a^2}{{\cal H}^2}
V_{,\sigma} \chi_{k} + {\cal O}( V_{,\sigma}^2) \simeq
- \frac{1}{14~\rho_{\rm r}} V_{,\sigma} \chi_{k}
+ {\cal O}( V_{,\sigma}^2).
\label{phgen}
\end{equation}
As a consequence, from Eqs. (\ref{deltar}) and (\ref{ureq}) we can
determine $\delta_{\rm r}$ and $v_{\rm r}$ as
\beq
\delta_{\rm r}(k,\eta) = - \frac{1}{21} \frac{a^2}{{\cal H}^2}
V_{,\sigma} \chi_{k} + {\cal O}( V_{,\sigma}^2),
~~~~
v_{\rm r}(k,\eta) = - \frac{1}{210} \frac{a^2}{{\cal H}^3}
V_{,\sigma} \chi_{k} + {\cal O}( V_{,\sigma}^2).
\label{flsol}
\eeq
Inserting now the obtained solutions in the remaining equations
(\ref{momp}) and (\ref{ij})
(not used for the above derivation), we can see that they are satisfied
with the same accuracy.
The time evolution
of $\zeta_{k}$ in the radiation-dominated, slow-roll regime can
finally be determined from Eq. (\ref{zeta1}):
\begin{equation}
\zeta_{k}(\eta) \simeq \frac{1}{4 \rho_{\rm r}}
\frac{\partial V}{\partial\sigma}
\chi_{k} + {\cal O}( V_{,\sigma}^2),
\label{zet2}
\end{equation}
so that $\Phi_{k}$ and $\zeta_{k}$ obey the
following simple relation
\begin{equation}
\Phi_{k}(\eta) \simeq -(2/7) \zeta_{k}(\eta) + {\cal O}( V_{,\sigma}^2).
\end{equation}
It should be appreciated that Eq.
(\ref{zet2}) can also be obtained by direct integration of Eq.
(\ref{zetapr}).
In the limit $(p_{\sigma} + \rho_{\sigma}) \ll \rho_{\rm r}$,
Eq. (\ref{sps}) implies indeed
\begin{equation}
c_{s}^2 \simeq \frac{1}{3} + \frac{1}{2\rho_{\rm r}}
\frac{\sigma'}{{\cal H}} V_{,\sigma}.
\label{sps1}
\end{equation}
On the other hand, from Eqs. (\ref{dpnadex}) and
(\ref{drph}),
the approximate form of $\delta p_{\rm nad}(k)$ is
\begin{equation}
\delta p_{\rm nad} (k) \simeq - \frac{4}{3} V_{,\sigma} \chi_{k} +
{\cal O}( V_{,\sigma}^2)
\label{dpnadsr}
\end{equation}
(again, terms with more than one derivative of the potential
have been neglected). By
inserting this result into Eq. (\ref{zetapr})
we are led to the equation
\begin{equation}
\frac{d \zeta_{k}}{d \ln{a}} = \frac{V_{,\sigma} \chi_{k}}{\rho_{\rm r}},
\end{equation}
whose direct integration (recall that $\rho_{\rm r} \sim a^{-4}$)
leads, as expected, to Eq. (\ref{zet2}).
The above approximate results for $\Phi_{k}$ and $\zeta_{k}$
hold for a generic (flat enough)
potential. However, in order to check the correctness of
our approximations numerically, it is useful to consider the
simple case of a quadratic potential:
\begin{equation}
V(\sigma) = \frac{m^2}{2} \sigma^2.
\end{equation}
In that case, Eqs. (\ref{sre}),
(\ref{chev}), (\ref{phgen}), (\ref{zet2}) lead to
\begin{eqnarray}
&& \sigma(\tau) \simeq \sigma_{\rm i} \left[ 1 - \frac{\mu^2}{20}
\left(\tau^4 -1\right) + {\cal O}( \mu^4 \tau^8) \right],
\label{sigsr}\\
&& \chi_{k}(\tau) \simeq \chi_{\rm i}(k) \left[ 1 - \frac{\mu^2}{20}
\left(\tau^4 -1\right) + {\cal O}( \mu^4 \tau^8) \right],
\label{chsr}\\
&&
\Phi_{k}(\tau) \simeq - \frac{\sigma_{\rm i} \chi_{\rm i}(k)}{84}\left[
\mu^2 \left(\tau^4 -1\right) + {\cal O}( \mu^4 \tau^8)\right],
\label{phsr}\\
&&\zeta(\tau) \simeq
\frac{\sigma_{\rm i} \chi_{\rm i}(k) }{24 } \left[\mu^2
\left(\tau^4 -1\right)
+ {\cal O}( \mu^4 \tau^8)\right],
\label{zetsr}
\end{eqnarray}
where $\chi_{\rm i}(k)$ is the initial spectrum of the axionic
fluctuations, and we have defined the (dimensionless) rescaled mass
and conformal time coordinate:
\begin{equation}
\tau= \frac{\eta}{\eta_{\rm i}}, ~~~~~\mu = m\,\eta_{\rm i}\,a_{\rm i}=m/H_{\rm i}.
\label{resc}
\end{equation}
The time $\eta_{\rm i}$ is the initial
integration time and $a_{\rm i}$ the initial normalization
of the scale factor.
These rescalings are useful in order to compare the numerical results
with the analytical calculations.
We have performed a numerical integration by choosing
initial conditions at sub-Planckian curvature scales, i.e.
\begin{equation}
H_{\rm i} = \frac{{\cal H}_{\rm i}}{a_{\rm i}} \ll 1, ~~~~~~~\eta_{\rm i} \gg 1
\end{equation}
(in Planck units), and setting $a_{\rm i} =1$.
Given a value of $\sigma_{\rm i}$ compatible, for a given mass,
with Eq. (\ref{c1}), the constraint (\ref{ham})
fixes
the initial radiation background
$\rho_{\rm r}(\eta_{\rm i}) $ to a value
much larger than the axionic potential.
Similarly,
the initial values of the derivatives of $\Phi_{k}$ and $\chi_{k}$ are
obtained by imposing, on the initial data
(\ref{incon3})--(\ref{incon3a}), the Hamiltonian
and momentum constraints of Eqs. (\ref{hamp}) and (\ref{momp}).
It has been checked
that all the constraints (both for the background and for the
fluctuations) are satisfied at every time
all along the numerical integration. The system describing
the evolution of the fluctuations, in particular, can be integrated
in two different (and complementary) ways. We could either use
Eqs. (\ref{ij})--(\ref{ureq}) and follow the evolution of all variables,
or use Eqs. (\ref{chired})--(\ref{phred}) and integrate the system
only in terms of $\Phi_{k}$ and $\chi_{k}$. We have performed the
numerical integration in both ways, and checked that the results are
the same.
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{SR1.eps}}
\vskip 3mm
\caption[a]{The full curve shows the result of a numerical
integration for
the case ${\cal H}_{\rm i} = 0.01$, and for the set of parameters
reported in the figure. The dashed curve shows the approximate
analytical solution based on Eqs. (\ref{phgen}) and (\ref{phsr}).}
\label{SR1}
\end{figure}
In Figs. \ref{SR1}, \ref{SR2} and \ref{SR2a} we report, as full
curves, the results of the numerical integration for a quadratic
potential. The analytical results of
Eqs. (\ref{sigsr})--(\ref{zetsr}), based on the slow-roll
approximation, are also illustrated, for comparison, by the dashed
curves.
\begin{figure}
\centerline{\epsfxsize = 11 cm \epsffile{SR2.eps}}
\vskip 3mm
\caption[a]{Evolution of $\chi_{k}$, reported for the same
set of parameters as in Fig. \ref{SR1}. The dashed curve
corresponds to the approximate analytical result
of Eq. (\ref{chsr}). }
\label{SR2}
\end{figure}
\begin{figure}
\centerline{\epsfxsize = 11 cm \epsffile{SR2a.eps}}
\vskip 3mm
\caption[a]{Evolution of $\zeta_k$, for the same set of parameters as in
Figs. \ref{SR1} and \ref{SR2}. The numerical result (full curve) is
compared with the approximate analytical result of Eq. (\ref{zetsr})
(dashed curve).}
\label{SR2a}
\end{figure}
\subsection{Radiation-dominated oscillations}
During the radiation-dominated regime, and for a quadratic potential,
the axion evolution equation (\ref{kg}) can be written as
\begin{equation}
\frac{ d^2 g}{d \tau^2} + \mu^2 \tau^2 g = 0, ~~~~~g = \sigma a,
\label{f}
\end{equation}
and its exact solution can be given in terms of Bessel functions
\cite{abr} as
\begin{equation}
g(\tau) = \sqrt{\tau} {\cal C}_{1/4} \bigg( \frac{\mu \tau^2}{2}\biggr),
\end{equation}
where ${\cal C}_{1/4}$ is a linear combination (with
constant coefficients) of Bessel functions of
order $1/4$ and $\mu\tau^2/2\sim m(t -t_{\rm i})$.
By imposing the correct boundary conditions and
normalization, in such a way that
$\sigma(\tau) \to \sigma_{\rm i}$ for $ \tau \to 1$, we obtain
\begin{equation}
\sigma(\tau) = \frac{\sigma_{\rm i} }{\sqrt{\tau}}
\frac{1}{J_{1/4} (\mu/2) } J_{1/4}\biggl( \frac{\mu}{2} \tau^2\biggr),
~~~~ \eta_{\rm i}<\eta<\eta_\sg,
\label{exsig}
\end{equation}
where $J_{1/4}$ is the first-kind Bessel function.
Notice that the small argument limit of this equation, for $\mu\ll 1 $
and $\tau \ra 1$, exactly gives the result (\ref{sigsr}), obtained in
the slow-roll approximation. This exact
analytical solution can also be
used as a consistency check of the quadratic approximations
when the potential, during slow-roll, has a more complicated
analytical form.
The onset of the axion oscillations can be determined
from the first zero of $J_{1/4}(\mu \tau^2/2)$, which
occurs for
\begin{equation}
\frac{\mu \tau^2}{2} \simeq 2.78,
\end{equation}
namely for
\begin{equation}
\tau_{\rm m} = \frac{\epsilon_1}{\sqrt{\mu}}, ~~~~~\epsilon_{1}
\simeq 2.35.
\label{430}
\end{equation}
Different definitions of the oscillation starting time, for instance
related to the breakdown of the slow-roll approximation,
would lead to similar numerical values, i.e. to Eq. (\ref{430})
with $\epsilon_1 \simeq (12)^{1/4}$. In the large argument limit
($\mu/2 \tau^2 \gg 1$) of Eq. (\ref{exsig}) the solution finally describes
the oscillating regime,
\begin{equation}
\sigma(\tau) \simeq \frac{ 2 \sigma_{\rm i} }{\tau^{3/2}
\sqrt{\pi \mu}
J_{1/4}(\mu/2) } \cos{\biggl( \frac{\mu \tau^2}{2} -
\frac{3}{8} \pi\biggr)}, ~~~~ \eta_{\rm osc} <\eta<\eta_\sg,
\label{wkbsig}
\end{equation}
where the phase and amplitude of oscillations are fixed by the
initial conditions.
An approximate solution of Eq. (\ref{chired}), similar to Eq.
(\ref{exsig}), holds for the axion fluctuations, namely
\begin{equation}
\chi_{k}(\tau) \simeq
\frac{\chi_{\rm i}(k) }{\sqrt{\tau}} \frac{1}{J_{1/4}
(\mu/2) } J_{1/4}\biggl( \frac{\mu}{2} \tau^2\biggr),
~~~~~~ \eta_{\rm i}< \eta<\eta_\sg,
\label{exch}
\end{equation}
whose small and large argument limits lead, respectively, to
\begin{eqnarray}
&& \chi_{k}(\tau) \simeq \chi_{\rm i}(k) \left[ 1 - \frac{\mu^2}{20}
\left(\tau^4 -1\right)\right],
\label{chsmall}\\
&&\chi_{k}(\tau) \simeq \frac{ 2 \chi_{\rm i}(k) }{ \tau^{3/2}
\sqrt{\pi \mu}
J_{1/4}(\mu/2) } \cos{\biggl( \frac{\mu \tau^2}{2} -
\frac{3}{8} \pi\biggr)}.
\label{chlarge}
\end{eqnarray}
We notice that Eq. (\ref{exch})
is obtained by solving the {\em approximate}
evolution equation
\begin{equation}
\chi_{k}'' +
2 {\cal H} \chi_{k}' + m^2 a^2 \chi_{k}
\simeq 0,
\label{approx}
\end{equation}
i.e. neglecting the terms containing the Bardeen potentials in
Eq. (\ref{chired}).
In the slow-roll approximation, as previously stressed,
these terms can be neglected for a {\em generic} potential term.
However, they can also be neglected in the oscillating phase,
provided the potential is well approximated by a quadratic form.
We have explicitly checked that the exact analytical solutions
(\ref{exsig}) and (\ref{exch}) are in perfect agreement with the results
of a numerical integration performed with a quadratic potential.
Thus, for a potential which is generic during the slow-roll phase (but
still quadratic during the oscillating regime) it will be sufficient to
work out the slow-roll solutions specific to that potential, from Eqs.
(\ref{sre}), (\ref{chev}), and match them (with their first
derivative) to the WKB solutions of Eqs. (\ref{f}) and (\ref{approx}),
namely \begin{eqnarray}
&&\sigma(\tau) \simeq \frac{\sigma_{2}}{\sqrt{2 a_{\rm i}^2 \mu \tau^3}}
\cos{\biggl(
\frac{\mu \tau^2}{2} + \beta\biggr)},
\label{solsig2}\\
&&
\chi_{k}(\tau) \simeq \frac{\chi_{2}(k)}{\sqrt{2 a_{\rm i}^2 \mu \tau^3}} \cos{\biggl(
\frac{\mu \tau^2}{2} + \gamma\biggr)}.
\label{solchi2}
\end{eqnarray}
The matching will allow a determination of the precise amplitudes and
phases of $\sigma$ (and $\chi_{k}$) in terms of $\sigma_{\rm i}$ (and
$\chi_{\rm i}(k)$).
\begin{figure}[t]
\centerline{\epsfxsize = 12 cm \epsffile{SR5.eps}}
\vskip 3 mm
\caption[a]{ The exact, numerical evolution of $\sigma$ and $\chi_{k}$
(full curves) is compared with the interpolating solution (dashed curves)
obtained by matching the slow-roll solutions (\ref{sigsr}) and (\ref{chsr})
with the WKB approximated solutions (\ref{solsig2}) and (\ref{solchi2}),
valid during the radiation-dominated oscillations of the axion.}
\label{SR5}
\end{figure}
As an application of this technique let us consider the
example of the quadratic potential, using the slow-roll solutions
for $\tau < \tau_{\rm m}$.
The result of this exercise is reported in Fig. \ref{SR5} where, with the
full curves, we illustrate the numerical results (coinciding exactly
with the analytical solutions). With the dashed curves we show the
interpolating solutions obtained by matching Eqs.
(\ref{sigsr}) and (\ref{chsr}) (obtained in the slow-roll approximation)
with the WKB solutions (\ref{solsig2}) and (\ref{solchi2}), valid in the
oscillating regime.
The time evolution of $\sigma(\eta)$ and
$\chi_{k}(\eta)$ explains why,
for $\tau \geq \tau_{\rm m}$ (i.e. after the slow-roll regime where
$\Phi_k \sim a^4$ according to Eq. (\ref{phgen})), the Bardeen potential
enters a phase of linear evolution (in conformal time).
This feature is illustrated in Fig. \ref{SR6a},
where we report the numerical results for the evolution of the
Bardeen potential, computed for different values of the axion mass.
An analytical estimate of
the slope of the linear regression for $\Phi_{k}$, after
the end of the radiation-dominated slow roll,
can be obtained from the Hamiltonian
constraint (\ref{hamp}),
which can be recast in the following form:
\begin{equation}
\frac{ \partial}{\partial \tau} ( \tau^3 \Phi_{k}) = - \frac{\tau^4}{12}
\biggl[
\frac{\partial\sigma}{\partial\tau} \frac{\partial\chi_{k}}{\partial\tau}
+ \mu^2 \tau^2 \sigma \chi_{k} \biggr],
\label{hamred}
\end{equation}
where we only assumed a quadratic form for the axion potential. By
using the WKB solutions (\ref{solsig2}) and (\ref{solchi2})
we obtain
\begin{equation}
\Phi_{k} = - \frac{ \sigma_2 \chi_{2}(k)}{96 a_{\rm i}^2} ~\mu \tau -
\frac{ \sigma_2 \chi_2(k)}{384 a_{\rm i}^2\mu\tau^3}
\biggl[ - 12 \cos{( 2 \gamma + \mu\tau^2)} + 18 \int^{\mu\tau^2/2}
\frac{dx}{x} \cos{(x + \gamma)}\biggr],
\label{integr}
\end{equation}
where the integral can be expressed in terms of ${\rm Ci}(w) =
-\int_{w}^{\infty} \frac{\cos{x}}{x} dx$ and
${\rm Si}(w) = \int_{0}^{w} \frac{\sin{x}}{x} dx$, and we have assumed
$\beta=\gamma$.
The oscillating terms are suppressed by $\tau^{-3}$ and can be
neglected (in agreement with the numerical results of Fig. \ref{SR6a}),
since we are considering the regime $\tau > \tau_{\rm m} \gg 1$. On top
of the oscillating terms, the amplitude of the term responsible
for the linear growth can be extracted from the numerical solutions by
fitting their asymptotic behaviour with the line
\begin{equation}
\Phi_{k}(\eta) = \sigma_{\rm i} \chi_{\rm i} [ - \epsilon_2 - \epsilon_3
\sqrt{\mu} \tau],~~~~~~ \tau > \tau_{\rm m}.
\label{regr}
\end{equation}
In the case of a quadratic
potential we obtain
\begin{equation}
\epsilon_2 \simeq 0.001, ~~~~~~~~~~~\epsilon_3 \simeq 0.0437.
\label{est}
\end{equation}
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{SR6a.eps}}
\vskip 3 mm
\caption[a]{The results of the numerical integrations for the Bardeen
potential are illustrated by the full curves for a quadratic potential and
for different values of the mass. The dashed lines represent the linear
fit of Eq. (\ref{regr}).}
\label{SR6a}
\end{figure}
The accuracy of this result can be appreciated from Fig. \ref{SR6a},
where the dashed lines (barely distinguishable from the numerical
solutions) are plotted according to Eqs. (\ref{regr}), (\ref{est}).
A different form of the potential will not affect the angular coefficient
of the regression (which is determined by the phase of the
radiation-dominated oscillations), but only the constant $\epsilon_2$.
It may be interesting to look also at the analytical estimate of
$\epsilon_{3}$, for a quadratic potential.
The initial amplitudes
$\sigma_2$ and $\chi_{2}(k)$ of Eq. (\ref{integr}) are determined
by the large argument limit of the exact solutions, Eqs.
(\ref{wkbsig}) and (\ref{chlarge}). In this case we get
\begin{equation}
\epsilon_{3}^{\rm th} = \frac{\Gamma^2(5/4)}{6 \pi} = 0.0435,
\label{est2}
\end{equation}
in excellent agreement with (\ref{est}). For a generic potential,
we could also determine $\epsilon_{3}$ by adopting an approximate
procedure, i.e. by taking the slow-roll solutions for $\chi_{k}$ and
$\sigma$ from Eqs. (\ref{sre}) and (\ref{chev}), and matching them
in $\tau_{\rm m}$ to the WKB solutions
(\ref{solsig2}) and (\ref{solchi2}), in order to determine amplitude and
phase.
The linear growth of the Bardeen potential continues until
$\rho_{\sigma}$, decreasing as $a^{-3}$, equals $\rho_{\rm r}$. This
happens at a time $\tau_\sg$ such
that
\begin{equation}
12 {\cal H}^2 \simeq a^2 m^2 \sigma^2.
\label{eq2}
\end{equation}
The expansion of Eq. (\ref{wkbsig}) for $\mu \ll 1$ then gives
\begin{equation}
\tau_{\sigma} = \frac{\epsilon_4}{\sigma_{\rm i}^2~ \sqrt{\mu}},
~~~~~~~~~~~~
\epsilon_4 = \frac{3 \pi }{2 \Gamma^2(5/4)}\simeq 5.74.
\label{tsig2}
\end{equation}
From Eq. (\ref{regr}) we can then finally obtain the value of the Bardeen
potential, at the onset of the phase of $\sigma$-dominated
oscillations. By using the value of $\tau_\sg$ given by Eq. (\ref{tsig2}),
the result is
\begin{equation}
\Phi_{k}(\eta_{\sigma}) \simeq \epsilon_{5}
\frac{ \chi_{\rm i}(k)}{\sigma_{\rm i}},
~~~~~ \epsilon_{5} = \epsilon_4 \epsilon_3 \simeq 0.25.
\label{finph}
\end{equation}
\subsection{The axion-dominated oscillations}
Using standard techniques suitable for the
oscillating regime \cite{tur,mfb},
Eqs. (\ref{ham}) and (\ref{kg}) can be solved and,
in the case of a quadratic potential, the oscillating
terms lead
to a geometry that reproduces (but only on the average) a
matter-dominated Universe. The oscillating corrections
will be suppressed for large (cosmic or conformal) times, and can be
easily computed in the cosmic-time gauge, where:
\begin{eqnarray}
&&\dot{H} = - \frac{1}{4} \dot{\sigma}^2,
\nonumber\\
&& H^2 = \frac{1}{12} \biggl[ \dot{\sigma}^2 + m^2 \sigma^2 \biggr],
\nonumber\\
&& \ddot{\sigma} + 3 H \dot{\sigma} + m^2 \sigma =0.
\end{eqnarray}
Using the auxiliary variable $Z= d \sg/d\ln{a}$, which satisfies
\begin{equation}
\frac{d Z}{d\sigma} = -
3 \bigg( 1 - \frac{Z^2}{12}\biggr)\biggl( 1 + 4 \frac{Z}{\sigma}\biggr),
\label{red}
\end{equation}
and defining two angular variables $(r, \theta)$,
\begin{equation}
Z = \sqrt{12} \sin{\theta} ,~~~~\sigma= r\cos{\theta},
\end{equation}
(such that $H=mr/\sqrt{12}$),
the following two equations are obtained:
\beq
\dot{\theta} = - \frac{\sqrt{3}}{4} m r \sin{2\theta} - m,
~~~~~~~~~~~
\dot{r} = -\frac{\sqrt{3}}{2} m r^2 \sin^2{\theta}.
\eeq
They can be solved, and the solutions expanded for large times at any
order in $1/t$.
A similar procedure can be carried out in conformal time.
Equations (\ref{kg})--(\ref{ham}) are equivalent to the following set of
equations
\beq
r' = - \frac{\sqrt{3}}{2} m a_{\rm f} \eta^2 r^2
\sin^2{\biggl(\frac{m a_f \eta^3}{3}\biggr)},
~~~~~~~~~
\sigma = r \cos{\biggl(\frac{m a_f \eta^3}{3}\biggr)}
\eeq
(where $a_f$ is an appropriate dimensionful integration constant), and
their solution leads to the expansion
\begin{eqnarray}
&& a(\eta) \simeq a_{\rm f} \biggl[
\eta^2 - \frac{3}{2 a_{\rm f}^2 m^2 \eta^4}
\cos{\biggl(\frac{2~m a_{\rm f} \eta^3}{3}\biggr)} +
{\cal O}(\frac{1}{\eta^5})\biggr],
\nonumber\\
&& {\cal H}(\eta) \simeq
\biggl[ \frac{2}{\eta} + \frac{3}{ m a_{\rm f} \eta^{4}}
\sin{\biggl(\frac{2~m a_{\rm f} \eta^3}{3}\biggr)}
+ {\cal O}(\frac{1}{\eta^5}) \biggr],
\nonumber\\
&& \sigma(\eta) \simeq \frac{4 \sqrt{3}}{m a_{\rm f} \eta^3}
\cos{\biggl(\frac{m a_{\rm f} \eta^3}{3}\biggr)} + {\cal O}(\frac{1}{\eta^5}).
\label{oscback}
\end{eqnarray}
A posteriori, as a cross-check,
Eqs. (\ref{oscback}) can be inserted into
Eqs. (\ref{dyn})--(\ref{ham}), to see that all terms
up to ${\cal O}(1/\eta^5)$ cancel, as expected.
In the phase dominated by the oscillating axion the effective
gravitational source is pressureless, on the average. By inserting the
condition $\langle p_{\sigma}\rangle =0 $ into the background and
perturbation equations, we get
\begin{equation}
\langle \delta_{\sigma}(k) \rangle \sim - 2 \langle \Phi_{k}\rangle,
~~~~~~~~~~~
\langle \delta_{\rm r}(k)
\rangle \sim - 2 \langle \delta_{\sigma}(k) \rangle,
\label{relph}
\end{equation}
which can be inserted into Eq. (\ref{zetaex}),
obtaining
\begin{equation}
\langle \zeta_{k} \rangle
\simeq \frac{5}{6} \langle \delta_{\sigma}(k) \rangle \simeq -
\frac{5}{3} \langle
\Phi_{k}\rangle.
\end{equation}
when $\langle p_{\sigma}\rangle \sim 0$ and $\rho_{\sigma}
\gg \rho_{\rm r} $.
Since $p_{\sigma} $ vanishes only on the average, more accurate
solutions have to be supplemented by oscillating corrections.
Using Eqs. (\ref{oscback}) an approximate
form of the perturbations in the oscillating
regime can be obtained.
One finds that $\delta_{\sigma}(k)$ and $\Phi_k$ are almost constant
(up to oscillations), i.e.
\begin{eqnarray}
&&\delta_{\sigma}(\eta) \simeq -\frac{1}{2} \Phi_0(k)\biggl[1
- \cos{\biggl(\frac{2 a_{\rm f} m \eta^3}{3}\biggr)} \biggr],
\label{dsosc}\\
&&\Phi_{k}(\eta) \simeq \Phi_0(k)\biggl[ 1 -
\frac{1}{ m a_{\rm f} \eta^3} \sin{\biggl(\frac{2}{3}
m a_{\rm f} \eta^3\biggr)} -
\frac{4 }{(m a_{\rm f} \eta^3)^2 } \cos{
\biggl(2\frac{m a_f \eta^3}{3}\biggr)} \biggr],
\label{phiosc}\\
&& \chi_{k}(\eta)
\simeq \chi_{0}(k)\biggl[
\sin{\biggl(\frac{m a_{\rm f} \eta^3}{3}\biggr)} +
\frac{ 3 }{ m a_{\rm f} \eta^3}
\cos{\biggl(\frac{m a_{\rm f} \eta^3}{3}\biggr)}\biggr],
\label{chosc}
\end{eqnarray}
where
\begin{equation}
\Phi_0(k) = \langle \Phi_{k}(\eta) \rangle,~~~~~~~~~\eta_\sg<\eta <
\eta_{\rm d},
\end{equation}
and
\begin{equation}
\chi_{0}(k) = - \frac{4}{\sqrt{3}} \Phi_{0}(k).
\end{equation}
The above solutions
satisfy the evolution equations of the fluctuations up to
${\cal O}(\eta^{-5})$.
\subsection{The axion decay and the subsequent radiation-dominated
phase}
When the decay rate of the axion equals the cosmological expansion
rate, energy is transferred from the coherent oscillations of $\sigma$
to the radiation produced by the axion decay. The radiation produced thanks
to the decay of the axion will quickly dominate the expansion and the second
radiation-dominated phase will take place.
The Bardeen potential prior to decay is given by Eq.
(\ref{phiosc}) while, after the decay, its evolution equation
(\ref{phred}) reduces to
\begin{equation}
\Phi'' + 4 {\cal H} \Phi' + 2 ( {\cal H}^2 + {\cal H}') \Phi -
\frac{1}{3} \nabla^2 \Phi = 0, ~~~~~~ \eta >\eta_{\rm d},
\label{radbard}
\end{equation}
and the corresponding exact solution can be expressed
as \cite{mfb}
\begin{equation}
\Phi_{k}(\eta) = \frac{1}{\eta^3} \biggl[ B_{1}(k) ( \omega\eta
\cos{\omega\eta} - \sin{\omega\eta} ) + B_{2}(k)
( \omega\eta \sin{\omega\eta} + \cos{\omega\eta} )\biggr],
~~~ \eta>\eta_{\rm d},
\label{f1}
\end{equation}
where $\omega = k/\sqrt{3}$. In the sudden approximation, the
two (dimensional) arbitrary constants $B_1(k)$ and $B_2(k)$ can be
uniquely fixed by matching, at the decay time $\eta_{\rm d}$, the solutions
(\ref{phiosc}) and (\ref{f1}) together with their first derivatives.
The terms containing
$(k\eta_{\rm d})$ are small and negligible for
modes that are outside the horizon at the time of the axion decay,
i.e. for the ones relevant to the physics of the observed CMBR
anisotropies. Furthermore, terms proportional to inverse powers of
$m/H_{\rm d}\sim M_{\rm P}^2/m^2$ are
also small and can be consistently neglected.
Hence, up to subleading terms, the final value of
the Bardeen potential can be written as
\begin{equation}
\Phi_{k}(\eta)
= \Phi_{0}(k) \left[ 2\cos{\left(\frac{2 \b}{3}\right)} - 3\right] \biggl[
\frac{\cos{\omega \eta}}{(\omega \eta)^2} -
\frac{\sin{\omega\eta}}{(\omega\eta)^3} \biggr],
\label{bardint2}
\end{equation}
where $\b = m \eta_{\rm d} a(\eta_{\rm d})
\sim m/H_{\rm d} \sim M_{\rm P}^2/m^2$.
The $\b$-dependent prefactor is a consequence of the
approximation of sudden decay where the axion field
is assumed to decay at a specific time $\eta_{\rm d}$. This
sudden approximation also neglects the possible (exponential) damping
of the oscillations in $\Phi_{k}$ arising in
Eq. (\ref{phiosc}).
It will now be shown
that the $\beta$-dependent prefactor is an artefact of the sudden
approximation.
In a realistic model of decay, in fact, the
energy-momentum tensors of the radiation fluid and of the
axion will not be separately conserved, because of their relative
coupling induced by the friction term $\Ga (\sg'/a)^2$, which leads, in
cosmic time, to the generalized conservation equations:
\begin{eqnarray}
&&\dot{\rho}_{\sigma} + (3 H + \Gamma)
(\rho_{\sigma} + p_{\sigma})=0, \nonumber\\ && \dot{\rho}_{\rm r} + 4
H \rho_{\rm r} - \Gamma(\rho_{\sigma} + p_{\sigma}) =0.
\label{dec2}
\end{eqnarray}
The fluctuations $\chi_{k}$ will experience
a similar damping,
\begin{equation}
\chi'' + (2 {\cal H}+\Gamma a) \chi' - \nabla^2 \chi +
\frac{\partial^2 V}{\partial\sigma^2} a^2 \chi - 4 \sigma' \Phi' + 2
\frac{\partial V}{\partial \sigma }a^2 \Phi =0,
\label{chiGamma}
\end{equation}
while the $\Phi_{k}$ evolution will still be described by Eq. (\ref{phred}).
This treatment of the damping of the fluctuations
was suggested in \cite{us} (see also \cite{mwu}).
The effect of $\Ga$ is, primarily, to induce a
damping in the oscillations
of the background and of the axion fluctuations according to Eqs.
(\ref{dec2}) and (\ref{chiGamma}).
Moreover, the (damped) fluctuations of the axionic field will also
influence the dynamics of $\Phi_{k}$ according to Eq. (\ref{phred}).
The time-dependent oscillations of Eq. (\ref{phiosc}) (occurring
in the absence
of friction) will then be further suppressed if $\Gamma \neq 0$
(more details
will be given in the following section).
As a consequence, the $\b$-dependent correction tends to
disappear from Eq. (\ref{bardint2}), leading to the final result
\begin{equation}
\Phi_{k}(\eta) = 3 \Phi_{0}(k)
\biggl[
\frac{\sin{\omega\eta}}{(\omega\eta)^3}
- \frac{\cos{\omega \eta}}{(\omega \eta)^2} \biggr].
\label{bardint3}
\end{equation}
In the equation for $\Phi_{k}$
the effect of the finite duration $\Gamma^{-1}$
is then equivalent to averaging
over the decay time.
At the end of the following section, numerical examples of the decay
will be discussed in detail.
\renewcommand{\theequation}{5.\arabic{equation}}
\setcounter{equation}{0}
\section{Background and perturbation equations for
$\sigma_{\i} > 1$}
If $\sigma_{\rm i}>1$, the epoch
of axion domination precedes the oscillation epoch. The previous
solutions for the slow-roll regime are still valid, and the axion starts
dominating when
\begin{equation}
\rho_{\rm r} a^2 = 6 {\cal H}^2
\simeq V a^2,
\label{dom}
\end{equation}
i.e., for a quadratic potential, when
\begin{equation}
\tau=\tau_{\sigma} \simeq \frac{(12)^{1/4}}{\sqrt{\mu\sigma_{\rm i}}}.
\label{tsig}
\end{equation}
If $\sigma_{\rm i} \simeq 1$, then
\begin{equation}
\tau_{\rm m} \simeq \tau_{\sigma}
,\label{hier1}
\end{equation}
the axion oscillates almost immediately after becoming dominant, and
the amplitude of the Bardeen potential
at the onset of the oscillatory phase
is obtained from Eq. (\ref{phsr}) as
\begin{equation}
\Phi_{k} ( \eta_{\rm m}) \simeq - \frac{\mu^2}{84} \chi_{\rm i}(k)
\sigma_{\rm i} \tau_{\rm m}^4 = - {1\over 7} {\chi_{\rm i}(k) \over
\sigma_{\rm i}} \simeq - {1\over 7} \chi_{\rm i}(k) .
\label{mean1}
\end{equation}
If the initial value $\sg_{\rm i}$ is larger than $1$, but not too large, then
Eqs. (\ref{hier1}) and (\ref{phsr}) are still valid, but Eq. (\ref{mean1}) is
to be multiplied by the factor $\sg_{\rm i}^2$, arising from a short period of
axion dominance toward the end of the slow-roll evolution (see
below). This effect, for moderate values of $\sg_{\rm i}$, is illustrated in
Fig \ref{SR3a} where, for the given parameters of the plot, the
final amplitude of the Bardeen potential is estimated as
\begin{equation}
\Phi_{k} ( \eta_{\rm m}) \simeq - \epsilon_{6} \chi_{\rm i}(k)
\sigma_{\rm i}, ~~~~~~~~~~~
\epsilon_6 = 0.143
\label{57}
\end{equation}
still in good agreement with the approximate value $1/7$ of Eq.
(\ref{mean1}).
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{SR3a.eps}}
\vskip 3 mm
\caption[a]{The full curves are the results of numerical integrations
for the case $\sigma_{\rm i} >1$.
The dashed curves correspond to the approximated results of
Eqs. (\ref{phsr}) and (\ref{57}).}
\label{SR3a}
\end{figure}
If $\sigma_{\rm i} \gg 1$, then $\tau_{\rm m} \gg \tau_{\sigma}$,
and a phase of inflationary expansion dominated by the axion potential
will take place between $\tau_{\sg}$ and
$\tau_{\rm m}$. During this phase the axion slowly rolls, the radiation
energy-density is quickly diluted as $\rho_{\rm r} \sim a^{-4}$, and
the time evolution of the fluctuations is correspondingly
modified. The Hamiltonian and momentum constraints
(\ref{hamp}) and (\ref{momp}) can now be combined to give
\begin{equation}
4\nabla^2 \Phi_{k} = 3 {\cal H} \sigma' \chi_{k} - \Phi_{k}
{\sigma'}^2 + \sigma' \chi_{k}' +
V_{,\sigma} a^2 \chi_{k},
\label{comb}
\end{equation}
and the speed of sound of Eq. (\ref{sps}) becomes
\begin{equation}
c_s^2 \simeq 1 + \frac{2 a^2}{3 {\cal H} \sigma'} V_{,\sigma},
\label{sps2}
\end{equation}
from which, using Eq. (\ref{dpnadex}),
\begin{equation}
\delta p_{\rm nad} = \frac{2 V_{\sigma}}{3 {\cal H} \sigma'}
\biggl[ \Phi_{k}
{\sigma'}^2 + \sigma' \chi_{k}' - 3 {\cal H} \sigma' \chi_{k} - a^2
V_{,\sigma}
\chi_{k}
\biggr].
\label{dpnadinf}
\end{equation}
The combination of Eqs. (\ref{comb}) and (\ref{dpnadinf}) leads to
\begin{equation}
\delta p_{\rm nad} = \frac{ 8 V_{,\sigma}}{3 {\cal H} \sigma'} \nabla^2\Phi_{k}.
\end{equation}
Hence, from Eq. (\ref{zetapr}), we get that $\zeta_{k} ' \sim 0$ at large
scales.
According to its definition, on the other hand, the constancy of
$\zeta_{k}$ implies, in cosmic time, that
\begin{equation}
\Phi_{k} \biggl( \frac{ 2 + \alpha_1}{ 1 + \alpha_1}\biggr) +
\frac{\dot{\Phi}_{k}}{H}
\frac{1}{1 + \alpha_1}
\label{constant}
\end{equation}
is also a constant, where
\begin{equation}
\alpha_{1} = - {\dot{H}}/{H^2} .
\end{equation}
It follows that, during inflation, we can parametrize the evolution of
$\Phi_k$, to lowest order, as
\begin{equation}
\Phi_{k} = A \biggl({\dot{H}}/{H^2}\biggr),
\label{Phinf}
\end{equation}
where $A$ is a constant controlled by the value of the Bardeen
potential at the beginning of inflation. Assuming a quadratic potential
for $\tau < \tau_{\sigma}$ we have, from Eq. (\ref{mean1}), $A
\simeq -(1/7)
(\chi_{\rm i}(k)/\sigma_{\rm i})(H^2/\dot H)_{\tau_\sg}$. By using the
dynamics of slow-roll inflation,
\begin{eqnarray}
&& H^2 \simeq \frac{m^2}{12} \sigma^2,
\label{srin1}\\
&& \dot{\sigma} \simeq - \frac{m^2}{3 H} \sigma,
\label{srin2}
\end{eqnarray}
we can deduce that
\begin{equation}
\frac{\dot{H}}{H^2} \simeq -\frac{4}{\sigma^2}.
\label{hdoh}
\end{equation}
Using Eq. (\ref{hdoh})
we finally obtain the Bardeen potential at the onset of the
phase of $\sigma$-dominated oscillations:
\begin{equation}
\Phi_{k}(\tau_{\rm m}) \sim \sigma_{\rm i} \chi_{\rm i}(k),
\label{515}
\end{equation}
where we used the expression for $A$ given above and the fact that
$\sigma(\tau_{\sigma}) = \sigma_{\rm i}$
and $\sigma(\tau_{\rm m}) \sim {\cal O}(1)$.
For $\tau>\tau_{\rm m}$ the axion eventually oscillates, and the subsequent
evolution has the same features as already discussed in the previous
section, for the case $\sigma_{\rm i}<1$.
In the following, the dynamics of the decay will be investigated
numerically, and Eq. (\ref{phred}) will be solved together
with Eqs. (\ref{dec2}) and (\ref{chiGamma}).
In order to illustrate the results let us recall
that, in the absence of friction ($\Gamma =0$ in Eqs. (\ref{chiGamma})),
the evolution of $\chi_{k}$ and $\Phi_{k}$, during the
axion-dominated oscillations, is given by Eqs. (\ref{phiosc}) and
(\ref{chosc}). In particular,
\begin{equation}
\Phi_{k}(\eta) \simeq \Phi_{0}(k) + \delta \Phi_{k}(\eta),
\label{phdecdef}
\end{equation}
where $\delta\Phi_{k}(\eta)$ is an oscillating function\footnote{In
order to avoid confusion we note that $\delta\Phi_{k}$ and, in the
following, $\delta\chi_{k}$, are not the power spectra of $\Phi$ and
$\chi$.} decaying as $\eta^{-3} \sim t^{-1}$. The frequency of
oscillation of $\delta\Phi_{k}$ is controlled by the axion mass. In
analogy with Eq. (\ref{phdecdef}) we can also define $\delta \chi_{k}$
which, for $\Gamma =0$, corresponds to the oscillating function
appearing in Eq. (\ref{chosc}). The evolution of $t\delta\Phi_{k}$ and of
$\delta\chi_{k}$,
for $\Gamma=0$, is represented by the full bold curves of
Figs. \ref{Fdec1} and \ref{Fdec2}. Notice that $\delta\Phi_{k}$ and
$\delta\chi_{k}$ oscillate very fast and that, for our illustrative
purpose, we have plotted their amplitudes calculated as the
average of the semi-difference
between the maximum and the minimum of each oscillation, and the
semi-difference between the successive maximum and the same
minimum.
If the oscillations
of $\delta\Phi_{k}$ are only suppressed by a power-law
function of time, we have seen
that there are mass-dependent terms that appear in the amplitude of
the Bardeen potential after the decay.
The integration of Eqs. (\ref{phred}) and of
(\ref{dec2}) and (\ref{chiGamma}) shows however that, with the inclusion
of the appropriate friction terms (due to the decay) into the
energy-momentum conservation equations,
the oscillations in $\delta\Phi_{k}$ and $\delta\chi_{k}$ are
exponentially suppressed and, in such a case, no mass-dependent
correction is left in the amplitude of the Bardeen potential (the
asymptotic, constant value of $\Phi_{k}$ is however unaffected by
such a damping mechanism). The damping of the oscillations, on a time
scale of order $\Gamma^{-1}$, is illustrated by the dashed curves of
Figs. \ref{Fdec1} and \ref{Fdec2}.
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{valpsi.eps}}
\vskip 3 mm
\caption[a]{Time evolution of the amplitude of the $\delta\Phi_k$
oscillations (multiplied by $t$ in cosmic time), with and without the
damping term due to the axion decay.}
\label{Fdec1}
\end{figure}
If we compare the sudden-decay approximation, discussed
in the previous section,
with the numerical results of Figs. \ref{Fdec1} and \ref{Fdec2}, we
see that the finite duration of the decay process can be physically
represented as a dynamical average to zero of the oscillatory terms in
the evolution of $\Phi_k$. In view of these results, when matching
$\Phi_k$ to the post-decay phase, we should take into account the fact
that all
the derivatives of $\Phi_k$ are exponentially suppressed with respect
to $\Phi_k/t_{\rm d}$, and thus can be safely neglected. This
leads to the result reported in Eq. (\ref{bardint3}).
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{valchi.eps}}
\vskip 3 mm
\caption[a]{Time evolution of the amplitude of
the $\delta\chi_{k}$ oscillations, with and without the
damping term due to the axion decay.}
\label{Fdec2}
\end{figure}
\renewcommand{\theequation}{6.\arabic{equation}}
\setcounter{equation}{0}
\section{Large-scale adiabatic fluctuations}
In order to discuss the direct impact of our results on the possible
generation of the observed CMBR anisotropies, the
evolution of the large-scale metric fluctuations should be followed
down to the matter-dominated phase, for all times $\eta>\eta_{\rm
eq}$. In particular, the phase and the amplitude of the Bardeen
potential prior to $\eta_{\rm eq}$ will fix the initial conditions for the
subsequent evolution of the inhomogeneities, and will be crucial to
determine whether they are of adiabatic or isocurvature nature.
We recall that, after the axion decay, the amplitude of the Bardeen
potential has been computed as
\begin{equation}
\Phi_{k}(\eta) = 3 \Phi_{0}(k)
\biggl[
\frac{\sin{\omega\eta}}{(\omega\eta)^3}-
\frac{\cos{\omega \eta}}{(\omega \eta)^2} \biggr],
~~~~~\eta \leq \eta_{\rm eq},
\label{bardint4}
\end{equation}
where, as in the previous section, $\omega= k/\sqrt{3}$.
For $\eta > \eta_{\rm eq}$, matter domination sets in, the background
satisfies $ 2 {\cal H}' + {\cal H}^2=0$, so that the evolution of the
Bardeen potential (outside the horizon) is described by
\begin{equation}
\Phi_{k}'' + 3 {\cal H}\Phi_{k}' =0,~~~~~{\cal H} =
\frac{2}{\eta}, \label{mdeq}
\end{equation}
whose solution can be written as
\begin{equation}
\Phi_{k}(\eta) = A(k) + \frac{B(k)}{\eta^{5}},~~~~\eta\geq \eta_{\rm eq}.
\label{mdsol}
\end{equation}
Imposing the continuity of the solutions (\ref{bardint4}) and (\ref{mdsol})
and of their first derivatives at $\eta= \eta_{\rm eq}$ one obtains:
\begin{eqnarray}
&& A(k) = \frac{3 \Phi_{0}(k)}{5 x_{\rm eq}^3}[ - 2 x_{\rm eq}
\cos{x_{\rm eq}} +
(x_{\rm eq}^2 + 2) \sin{x_{\rm eq}}],
\label{Ak}\\
&&B(k) = \frac{3 \Phi_{0}(k) \eta_{\rm eq}^5}{5 x_{\rm eq}^3}
[ 3 x_{\rm eq} \cos{x_{\rm eq} } +
(x_{\rm eq}^2 - 3) \sin{ x_{\rm eq}}],
\label{Bk}
\end{eqnarray}
where $x_{\rm eq} = \omega \eta_{\rm eq} \equiv k
\eta_{\rm eq} /\sqrt{3}$.
For scales that are outside the horizon prior to decoupling,
$x_{\rm eq} \ll 1$, and Eq. (\ref{mdsol})
becomes
\begin{equation}
\Phi_{k}(\eta) = \Phi_{0}(k)\biggl[1 + \frac{ (k\eta_{\rm eq})^2 }{75}
\biggl( \frac{\eta_{\rm eq}}{\eta}\biggr)^{5}\biggr],
~~~~~~~\eta> \eta_{\rm eq}.
\label{MD}
\end{equation}
For $\eta> \eta_{\rm eq}$ the decaying mode
is highly suppressed, and we are then in the situation of constant
Bardeen potential right after equality, with an amplitude $\Phi_{0}(k)$,
which (recalling the previous results (\ref{finph}), (\ref{mean1}),
(\ref{515})) is completely determined by the axion spectrum
and by the initial conditions of the axion background. More precisely,
the final amplitude can be parametrized as follows
\begin{equation}
\Phi_{0}(k) \equiv \Phi_{k}(\eta_{\rm d}) \equiv -
f(\sigma_{\rm i}) \chi_{\rm i}(k),
\label{phif}
\end{equation}
where
\begin{equation}
f(\sigma_{\rm i}) = c_1 \sigma_{\rm i} +\frac{c_2}{\sigma_{\rm i}} -
c_3,
\label{fsig}
\end{equation}
and
\beq
c_1 \simeq 0.13,
~~~~~~~
c_2 \simeq 0.25,
~~~~~~~
c_3 \simeq 0.01.
\label{c123}
\eeq
The above coefficients $c_{i}$ have been obtained
by integrating numerically the evolution equations of the background
and of the fluctuations for different values of $\sigma_{\rm i}$ (both
larger and smaller than $1$). Then, following the
hint of the analytical results obtained by solving the evolution piece-wise,
the final value of $\Phi_{k}(\eta)$ has been fitted with Eq. (\ref{fsig}),
and the values reported in Eq. (\ref{c123}) have been determined.
The value (\ref{fsig}) of the Bardeen potential provides the initial
condition for the subsequent hydrodynamical evolution.
Such evolution will allow us to determine, in turn, the precise value of
the temperature fluctuations through the Sachs--Wolfe effect. In
particular, the modes that are outside the horizon for $\eta_{\rm
eq}< \eta <\eta_{\rm dec} $ will determine the large-scale temperature
fluctuations relevant to the COBE observations.
By perturbing the corresponding conservation equations on a
matter-dominated background, we obtain:
\begin{eqnarray}
&& \delta_{\rm r}' - \frac{4}{3} \nabla^2 v_{\rm r} - 4 \Phi' =0,
\label{dr1}\\
&& v_{\rm r}' - \frac{1}{4} \delta_{\rm r} - \Phi = 0,
\label{ur1}\\
&& \delta_{\rm m}' - \nabla^2 v_{\rm m} - 3 \Phi' =0,
\label{dm1}\\
&& v_{\rm m}' + {\cal H} v_{\rm m} - \Phi =0,
\label{udm}
\end{eqnarray}
where $\delta_{\rm m}=\da \r_{\rm m}/\r_{\rm m}$ and $v_{\rm m}$,
following the notation of the previous sections, are
the gauge-invariant density contrast and velocity potential of the
matter fluctuations. Also, in the above equations,
\begin{equation} {\cal H} = \frac{2}{\eta},~~~~~\rho_{\rm m} a^2 =
\frac{24}{\eta^2}.
\label{backdm}
\end{equation}
As already stressed at the beginning
of this section, $\Phi$ is constant during the matter-dominated phase.
Using this property we can now work out the specific relations
between the different fluid variables, for
modes that are outside the horizon right after equality, so as to
explicitly check the adiabaticity of the fluid perturbations.
The system of Eqs. (\ref{dr1})--(\ref{udm})
can be easily solved by going to Fourier space. For $v_{\rm m}$ we have
\begin{equation}
k v_{\rm m}(k) \simeq \frac{k\eta}{3} \Phi_0(k),~~~~~~~k\eta \ll 1.
\end{equation}
Since $\vec{\nabla} v_{\rm m}$ (evaluated outside the horizon) contributes
directly to the Sachs--Wolfe effect, it is important to notice
that this term is subleading with respect to the other contributions arising
in the case of adiabatic fluctuations.
We will indeed show that, unlike $\vec{\nabla} v_{\rm m}$, which
is suppressed, the contrast
$\delta_{\rm r}$ is instead constant outside the horizon, and
proportional to $\Phi_{0}(k)$.
Inserting
$v_{\rm r}$ from Eq. (\ref{dr1}) into Eq. (\ref{ur1}) we get a
decoupled equation for $\delta_{\rm r}$, namely,
\begin{equation}
\delta_{\rm r}'' + \frac{k^2}{3} \delta_{\rm r} = - \frac{4}{3} k^2
\Phi_{0}(k).
\label{decdr}
\end{equation}
The general solution is
\begin{equation}
\delta_{\rm r}(k \eta) = A_1 \cos{\omega \eta} + B_1\sin{\omega\eta} +
4 \Phi_{0}(k) [\cos{\omega\eta} - 1],
\end{equation}
and the constants $A_{1}$ and $B_{1}$
can be determined by consistency with the other
equations and with the Hamiltonian constraint (\ref{hamp}) written in the case
of a matter-radiation fluid. The final result is:
\begin{eqnarray}
&& \delta_{\rm r}(k,\eta) = \frac{4}{3}\Phi_{0}(k) \biggl[ \cos{\omega\eta} - 3\biggr]
\label{delrel}\\
&& k v_{\rm r}(k,\eta) = \frac{\Phi_0(k)}{\sqrt{3}} \sin{\omega\eta},
\label{urel}\\
&& \delta_{\rm m}(k,\eta) = - 2 \Phi_{0}(k)-\frac{\Phi_{0}(k)}{6} (k\eta)^2,
\label{dmrel}\\
&& k v_{\rm m}(k,\eta) = \frac{ (k\eta)}{3} \Phi_{0}(k).
\label{uma}
\end{eqnarray}
Notice that, outside the horizon, $ k v_{\rm m} \equiv k v_{\rm r}$ as required by
local thermodynamical equilibrium. Furthermore, for $k\eta \ll 1$, the velocities
of the two fluids are proportional to $(k\eta)$. When the modes are outside the horizon,
Eqs. (\ref{delrel}) and (\ref{dmrel}) imply that the density
contrasts $\delta_{\rm r}$ and $\delta_{\rm m}$ are both
constant and proportional according to
\begin{equation}
\delta_{\rm r} \simeq (4/3) \delta_{\rm m}.
\label{dmrad}
\end{equation}
This result has a simple physical interpretation, and
implies the adiabaticity of the fluid perturbations.
The entropy per matter particle is indeed proportional to
$S=T^3/n_{\rm m}$, where $n_{\rm m}$ is the number density of
matter particles and $T$ is the radiation temperature.
The associated entropy fluctuation, $\da S$, satisfies
\begin{equation}
\frac{\delta S}{S} = \frac{3}{4} \delta_{\rm r} - \delta_{\rm m},
\end{equation}
where we used the fact that $\rho_{\rm r} \sim T^4 $ and that
$\rho_{\rm m} = m n_{\rm m}$, where $m$ is the typical mass
of the particles in the matter fluid. Equation (\ref{dmrad}) thus implies $\delta S/S
=0$, in agreement with the adiabaticity of the fluctuations.
\subsection{Sachs-Wolfe effect and COBE scales}
The fluctuations of the Bardeen potential and of the
radiation density contrast are sources of a slight temperature
difference between photons coming from different sky directions. This is the
essence of the Sachs--Wolfe effect \cite{sw}. In terms of the
gauge-invariant variables introduced in the present analysis, the
various contributions to the Sachs--Wolfe effect, along the $\vec{n}$
direction, can be written as \cite{mfb,dur}
\begin{equation}
\frac{\Delta T}{T}(\vec{n},\eta_0,x_0) = \biggl[ \frac{\delta_{\rm r}}{4}
+ \vec{n} \cdot \vec{\nabla} v_{\rm b}
+ \Phi\biggr](\eta_{\rm dec}, \vec{x}(\eta_{\rm dec}))
- \int_{\eta_0}^{\eta_{\rm dec}} (\Phi' + \Psi')(\eta , \vec{x}(\eta))
d\eta,
\label{SW}
\end{equation}
where $\eta_0$ is the present time, and $\vec{x}(\eta)=\vec{x}_0-
\vec{n}(\eta-\eta_0)$ is the
unperturbed photon position
at the time $\eta$ for an observer
in $\vec{x}_0$. The term $\vec{v}_{\rm b}$ is the peculiar velocity
of the baryonic matter component.
We are preliminarily interested in the effects of scales still outside the
horizon at the time of the matter-radiation equality, which are the
scales relevant to the observations of the COBE-DMR experiment
\cite{smoo1,smoo2}.
In order to correctly take into account the constraints
imposed by the COBE normalization on the
spectral amplitude of the Bardeen potential, let us compare the
relative weight of the different terms appearing in the Sachs-Wolfe
formula (\ref{SW}).
From Eq. (\ref{urel}) we can see that, for our adiabatic initial
conditions, the fluctuation
in the matter velocity potential is
subleading for superhorizon scales, suppressed by the term $k\eta
\ll 1$ with respect to the constant values of $\delta_{\rm
r}$ and $\Phi_{k}$. Furthermore, since $\Phi'\simeq 0$ and $\Psi =
\Phi$, the integrated Sachs--Wolfe effect can also be neglected. By
inserting Eq. (\ref{delrel}) into Eq. (\ref{SW}) we thus obtain the usual
result for adiabatic fluctuations, namely
\begin{equation}
\frac{\Delta T}{T} (\vec{n},\eta_0,x_0) = \frac{1}{3}
\Phi (\eta_{\rm dec}, \vec{x}(\eta_{\rm dec})),
\label{ff}
\end{equation}
to be used for the comparison of our theoretical predictions
with the COBE normalization.
On the other hand, by taking the Legendre transform at the present
time $\eta_0$, the temperature fluctuations of Eq. (\ref{SW}) can be
generally expanded into spherical harmonic functions, $Y_{\ell m}$, as
\begin{equation}
\frac{\Delta T}{T}(\vec{x}_0,\vec{n},\eta_0)
=\sum_{\ell,m} a_{\ell m}(\vec{x}_0) Y_{\ell m}(\vec{n}),
\end{equation}
where the coefficients $a_{\ell m}$ define the angular power spectrum
$C_\ell$ by
\begin{equation}
\biggl\langle a_{\ell m}\cdot a_{\ell'm'}^*\biggr\rangle
= \delta_{\ell\ell'}\delta_{mm'}C_\ell,
\label{anps}
\end{equation}
and determine the
two-point correlation function of the temperature fluctuations,
namely
\begin{eqnarray}
\left\langle{\delta T\over T}(\vec{n}){\delta T\over T}(\vec{n}') \right\rangle_{{~}_{\!\!(\vec{n}\cdot
\vec{n}'=\cos\vartheta)}}&=&
\sum_{\ell \ell' m m'}\biggl\langle a_{\ell m}
a_{\ell'm'}^*\biggr\rangle Y_{\ell m}(\vec{n}) Y_{\ell' m'}^*(\vec{n'})
\nonumber \\
& =& {1\over 4\pi}\sum_\ell(2\ell+1)C_\ell P_\ell(\cos\vartheta).
\end{eqnarray}
These coefficients $C_{\ell}$, in
turn, are related through Eq. (\ref{ff}), to the power spectrum of
$\Phi_{0}(k)$, and for $2 \leq \ell \ll 100$ they can be expressed as
\cite{dur2}
\begin{equation}
C_\ell \simeq
\frac{2}{9 \pi} \int_0^\infty \frac{dk}{k}
\biggl\langle |\Phi_{0}(k)|^2
\biggr\rangle k^3 j_\ell^2 [k(\eta_0-\eta_{\rm dec})].
\label{CL1}
\end{equation}
As already stressed, the spectrum of the Bardeen potential is fully
determined, in our context, by the initial spectrum of axionic
fluctuations amplified by the pre-big bang
dynamics. A self-contained derivation of such a spectrum, including
the mass contribution, is presented in Appendix A.
Consider first the case of minimal pre-big bang models, whose related
spectrum is reported in Eq. (\ref{estimate}). The spectrum of curvature
perturbations will then be, at large scales,
\begin{equation}
k^3 \left| \Phi_0(k)\right|^2 = f^2(\sigma_{\rm i}) k^3
\left|\chi_k\right|^2 = f^2(\sigma_{\rm i}) \biggl(\frac{H_1}{M_{\rm P}}
\biggr)^2
\left( k\over k_1\right)^{n-1}, ~~~~~~k1.
\label{ga2}
\end{eqnarray}
(we have rescaled $\om_1$ taking into account the kinematics of the
various cosmological phases from $t_1$ down to $t_0$).
The factor
$Z_\sigma= (a_{\rm osc}/a_\sigma)$ denotes the amplification of the
scale factor during the phase of axion-dominated, slow-roll inflation,
for the case $\sg_{\rm i}>1$.
Notice that $\omega_1(t_0)$
depends on the mass, on the initial amplitude of the
axion background and on the axion decay rate.
If the axion decays at a typical scale fixed by
Eq. (\ref{decb}), Eqs. (\ref{ga1}) and (\ref{ga2}) lead to
\begin{eqnarray}
\omega_1(t_0) &\simeq & 10^{29} \om_0
\left(H_1\over M_{\rm P}\right)^{1/2}
\left(m\over \sigma_{\rm i}^2 M_{\rm P}\right)^{
1/3},~~~~~~~~~~~~~~~~~~~\sigma_{\rm i} < 1,
\label{om1a}\\
&\simeq & 10^{29}\om_0
\left(\sigma_{\rm i} H_1\over M_{\rm P}\right)^{
1/2} \left(m\over \ M_{\rm P}\right)^{
1/3}Z_\sigma^{-1},~~~~~~~~~~~~~~\sigma_{\rm i} > 1
\label{om2a}
\end{eqnarray}
(we have used $H_0 \simeq 10^{-6} H_{\rm eq} \simeq 10^{-60}M_{\rm P}$).
Hence, in spite of the fact that the initial axionic spectrum does not have
any mass dependence, the mass appears again when computing the
amplitude of the spectrum at the present horizon scale $\omega_0$.
The amplitude of the Bardeen potential, on the other hand, is
constrained by the COBE normalization of the quadrupole coefficient
$C_2$, which in our case is given by
\begin{equation}
C_2=
\alpha^2_n f^2(\sigma_{\rm i})
\left(H_1\over M_{\rm P}\right)^2\left(\omega_0\over
\omega_1\right)^{n-1},
\label{C2}
\end{equation}
where
\begin{equation}
\alpha_n^2 = \frac{2^{n}}{72}{\Gamma(3-n) \Gamma\left({3+n\over
2}\right) \over
\Gamma^2\left({4-n\over 2}\right) \Gamma\left({9-n\over 2}\right)}.
\label{alpha}
\end{equation}
Using the experimental result \cite{ban}
\begin{equation}
C_2=(1.9\pm 0.23)\times 10^{-10},
\label{expnorm}
\end{equation}
we are thus led to the bounds
\begin{eqnarray}
&&
\alpha^2_n f^2(\sg_{\rm i})\sigma_{\rm i}^{2(n-1)/3}
\left(H_1\over M_{\rm P}\right)^{(5-n)/2}
\left(m\over M_{\rm P}\right)^{-(n-1)/3} 10^{-
29(n-1)} \simeq 1.9 \times 10^{-10},
~~~~\sigma_{\rm i} < 1,
\label{C2a}\\
&&
\alpha^2_n f^2(\sg_{\rm i})Z_\sigma^{n-1} \sigma_{\rm i}^{(1-n)/2}
\left(H_1\over M_{\rm P}\right)^{(5-n)/2}
\left(m\over M_{\rm P}\right)^{-(n-1)/3} 10^{-
29(n-1)} \simeq 1.9 \times 10^{-10},\,~~\sigma_{\rm i} >1.
\label{C2b}
\end{eqnarray}
These constraints, imposed by the COBE normalization,
will be discussed at the end of the
present section, and combined
with other theoretical constraints
pertaining to the various models of background evolution.
\subsection{Acoustic peak region}
In the previous discussion of the modes that are outside the horizon
before decoupling, we have completely neglected the
possible scattering of radiation with baryons. In fact, if we move to
smaller angular scales (i.e. typically to $\ell \gaq 100$), the main contribution
to the CMBR temperature fluctuations comes from the oscillations of the various
plasma quantities, the so-called Sakharov oscillations \cite{sak}.
A correct approach to this
problem is then to perturb consistently the Boltzmann equations for the
different species of the plasma \cite{ks2,hs1,hs2}. Furthermore
it can be relevant
to discuss the case of a smooth transition between radiation
and matter dominated epochs. In such a context it
becomes difficult to provide an analytical description of the system and,
in order to compute the patterns of the acoustic oscillations,
we will indeed present some numerical examples in the third part of the
present section.
It is however useful to emphasize that the phases of the Bardeen
potential for the adiabatic mode of Eq. (\ref{bardint4}) determine
not only the relative weight of the Sachs--Wolfe contributions, but also
the specific phase of the oscillatory patterns at small scales in the
temperature fluctuations.
For scales $\ell \gaq 100$ the contribution to the temperature perturbations
given in Eq. (\ref{SW}) is dominated by acoustic oscillations.
This aspect can be appreciated by looking at Eqs. (\ref{delrel})--(\ref{uma})
in the limit $k\eta >1$, where the peculiar velocity
of baryonic matter does not oscillate. Instead, from Eq. (\ref{delrel}),
we find that the terms $\delta_{\rm r}/4$ and $
\Phi$, appearing in Eq. (\ref{SW}),
combine to give a single term oscillating like a cosine:
\begin{equation}
\frac{\Delta T}{T}(k,\eta_{0},\eta_{\rm dec}) \simeq
\frac{1}{4}\delta_{\rm r}(k,\eta_{\rm dec}) + \Phi_{0}(k) \sim
\frac{\Phi_{0}(k)}{3} \cos{\omega\eta_{\rm dec}}.
\end{equation}
In this argument the interactions of baryons
with the radiation fluid have been neglected.
The dynamics of $(\Delta T/T)_{k}$ can be obtained from an exact
Boltzmann equation with source term provided by Compton scattering coupled
to the continuity
and Euler equations for the fluid variables.
Before recombination,
Compton scattering is very rapid and therefore
the Boltzmann, Euler and continuity equations for the
photon--baryon system can be expanded in powers of the Compton
scattering time \cite{hs1,hs2}.
Within this approximation the baryon velocity
field is damped and $(\Delta T/T)_{k}$ oscillates
as a cosine for adiabatic initial conditions.
In the approximation
of \cite{hs1,hs2}, the oscillations
in $(\Delta T/T)_{k}$ have an amplitude
proportional to $(1 + R)^{-1/4}$
where $R(\eta) = 3 \rho_{\rm b}/( 4 \rho_{\rm r})$. This
result simply tells that the baryonic content
of the plasma determines the height of the first peak.
Notice that this is in sharp contrast with what happens in the
case of light axions \cite{dgs1,dgs2}, where the Bardeen potential is
quadratic in the axion fluctuations, and the initial
conditions for the hydrodynamical evolution are of the isocurvature type.
This implies, in particular, that the oscillatory patterns of the CMBR
anisotropies will be shifted by $\pi/2$ if compared with the case
discussed in the present paper.
\subsection{Constraints on pre-big bang models}
In this subsection we will discuss the bounds imposed by the COBE
normalization, together with other constraints following from the
evolution of the background geometry. Let us start with the axion
spectrum of minimal pre-big bang models, Eq. (\ref{norm}). In such a
case, and for a flat Harrison--Zeldovich spectrum (i.e.
$n=1$), the COBE normalization is inconsistent with a cut-off $H_1$ at
the standard value $M_{\rm s} \sim 10^{-1} ~M_{\rm P}$ of the string mass
scale \cite{kap}. By using $n=1$, and taking for $\sg_{\rm i}$ the value
minimizing $f(\sg_{\rm i})$,
\begin{equation}
\sigma_{\rm i}^{\rm min} = \sqrt{\frac{c_2}{c_1}}\simeq 1.38,~~~~~~~~
f(\sigma_{\rm i}^{\min}) \simeq 0.34,
\end{equation}
we have indeed, from Eqs. (\ref{C2})--(\ref{expnorm}),
\begin{equation}
H_1 \simeq 5.2 \times 10^{-4}~M_{\rm P}.
\end{equation}
However, the precise value of $H_1$ is one of the main uncertainties
of pre-big bang models. As we shall see in a
moment, the value $H_1 = M_{\rm s}$ (or $H_1 = M_{\rm GUT}$) may become consistent with the
COBE normalization for non-flat (blue) spectra, and even for a strictly
flat spectrum in the case of non-minimal implementations of the pre-big
bang scenarios.
Let us first recall the various constraints to be imposed on the
spectrum. The condition (\ref{C2a}) is to be combined with the constraint
(\ref{c1}), the condition (\ref{C2b}) with the constraint (\ref{c2}), which
are required for the consistency of the corresponding classes of
background evolution. Both conditions are to be intersected with
the experimentally allowed range of the spectral index. We will use (as
a reference value) the generous upper bound \cite{cobe}, $ n \laq
1.4$. Also, for our illustrative purpose, we will take the
maximally extended range of allowed values of the axion
mass, satisfying the nucleosynthesis constraint $m \gaq 10$
TeV.
We will assume, finally, that in the case $\sg_{\rm i} >1$ the axion-driven
inflation is short enough, to avoid a possible contribution to $C_{\ell}$
arising from the metric fluctuations directly amplified from the
vacuum during such a phase of axionic inflation. This requires that the
smallest amplified frequency mode $\om_\sg$, crossing the horizon at
the beginning of inflation, at decoupling be still larger than the Hubble
horizon at the corresponding epoch. This imposes the condition
$\om_\sg(t_0) =H_\sg
(a_\sg/a_0)>\om_{\rm dec}(t_0)=H_{\rm dec}(a_{\rm dec}/a_0)$ ,
namely
\begin{equation} Z_\sg \laq 10^{27} \sg_{\rm i} \left(m\over
M_{\rm P}\right)^{5/6},
\label{648}
\end{equation}
to be added to the constraint (\ref{c2}) for $\sg_{\rm i} >1$.
The allowed region in the plane $\{\log \sg_{\rm i}, \log (m/M_{\rm P})\}$ is
illustrated in Fig. \ref{f9} for $H_1=10^{-2}M_{\rm P}$,
using for the inflation
factor the parametrization $Z_\sg =
\exp((\sg_{\rm i}^2-1)/8)$. Along the thin full curves the parameters satisfy
the COBE normalization, for fixed values of $n$, ranging from $1.1$ to
$1.4$ (the condition (\ref{648}), in this case, is always automatically
satisfied). A growing (``blue") spectrum is thus allowed even if
$H_1\sim M_{\rm s}$, for a wide range of axion masses, and for a (narrower)
range of values of $\sg_{\rm i}$. In particular, for the case $H_1=10^{-2}
M_{\rm P}$, we find $1 \gaq \sg_{\rm i} \gaq 10^{-4}$, for $\sg_{\rm i}<1$. For
$\sg_{\rm i} >1$ the results are complementary for the spectral index, but
there are much more stringent bounds on $\sg_{\rm i}$, because the
inflationary red-shift factor $Z_\sg$ grows exponentially with
$\sg_{\rm i}^2$. As a consequence, the allowed region for
$\sg_{\rm i}>1$ is distorted and compressed, as illustrated in Fig. \ref{f9}.
\begin{figure}
\centerline{\epsfxsize = 11 cm \epsffile{f9.eps}}
\vskip 3mm
\caption{ Allowed values of $\sg_{\rm i}$ and $m$ (in Planck units)
according to Eqs. (\ref{C2a}), (\ref{C2b}), with $H_1=10^{-2}M_{\rm P}$.
The allowed region (within the thick lines) is bounded by the condition
$n<1.4$ (left and right bold lines), by the nuclesosynthesis constraint
$m>10$ TeV (lower bold line), by the condition (\ref{c1}) (upper left bold
line) and (\ref{c2}) (upper right bold line).}
\label{f9}
\end{figure}
The allowed region may be further extended if the inflation scale $H_1$
is lowered (see for instance \cite{sc}), and a flat ($n=1$) or
almost flat spectrum may become possible if $c_2\a_1 H_1 \laq
10^{-5}M_{\rm P} \sg_{\rm i}$, for $\sg_{\rm i}<1$, and if $c_1\a_1H_1 \laq
10^{-5}M_{\rm P} /\sg_{\rm i}$, for $\sg_{\rm i}>1$ (see Eqs. (\ref{C2a}),
(\ref{C2b})). The corresponding allowed values of $H_1$ and $\sg_{\rm i}$ are
illustrated in Fig. \ref{f9a} for $m=10^{-9}M_{\rm P}$, and for three
different values of $n$ around $1$.
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{f9a.ps}}
\vskip 3mm
\caption{Allowed values of $H_1$ as a function of $\sigma_{\rm i}$
for different values of the spectral index and for $m = 10^{-9} M_{\rm P}$. }
\label{f9a}
\end{figure}
However, a flat spectrum
may be allowed even keeping pre-big bang inflation at the string scale
($H_1 \sim M_{\rm s}$), provided we consider a non-minimal pre-big bang
scenario. In that context, in fact, the high-frequency branch of the
axionic spectrum may be modified, getting steeper enough to match the
string-scale normalization at the end-point of the spectrum, while
the low-frequency branch remains flat (or
quasi-flat, see Appendix), to agree with large-scale observations. Examples of
realistic pre-big bang backgrounds producing such an axion spectrum
have been presented already in \cite{dgs3}.
A non-minimal spectrum can be parametrized by the Bogoliubov
coefficients (which will be given in Eq. (\ref{delta})), in terms of a generic
break-scale $k_s$ and of the high-frequency slope parameter $\delta$.
In that case, for a long and/or steep enough high-frequency branch
of the spectrum, the large-scale amplitude may be suppressed
sufficiently to allow flat (or even red) spectra at the COBE scale.
In fact, for the non-minimal spectrum (\ref{delta}), the normalization
condition (\ref{C2}) becomes
\begin{equation}
C_2=
\alpha^2_n f^2(\sigma_{\rm i})
\left(H_1\over M_{\rm P}\right)^2\left(\omega_0\over
\omega_1\right)^{n-1}\left(\om_s\over \om_1\right)^\da.
\label{649}
\end{equation}
Flat or red spectra ($n \leq 1$) are thus possible even for $H_1 \gaq
10^{-2}M_{\rm P}$, provided
\begin{equation}
\alpha^2_1 f^2(\sigma_{\rm i})
\left(\om_s\over \om_1\right)^\da \laq 10^{-6}.
\label{649a}
\end{equation}
In order to illustrate this possibility we will choose a specific model of
background by identifying $k_s$ with the equilibrium scale $k_{\rm
eq}$, in such a way that $n$ corresponds to the spectral index of all
scales relevant to the CMBR anisotropies, while $n+\da$ provides the
average spectral index for all other scales, up to $k_1$. We will also
assume for the axion background the ``natural" initial value $\sg_{\rm i}=1$,
so that
\begin{equation}
\frac{\om_1}{\om_s}=\frac{\om_1}{\om_{\rm eq}}\simeq 10^{27}
\biggl(\frac{H_1}{M_{\rm P}}\biggr)^{1/2}
\biggl(\frac{m}{M_{\rm P}}\biggr)^{1/3}.
\label{650}
\end{equation}
The COBE normalization can then be written explicitly as
\begin{equation}
C_2 = \alpha_{n}^2 f^2(1)
\biggl(\frac{H_1}{M_{P}}\biggr)^{(5-n - \delta)/2}
\biggl(\frac{m}{M_{P}}\biggr)^{- ( n -1 + \delta)/3} 10^{-[27\delta +
29(n-1)]}.
\label{C2break}
\end{equation}
By using the experimental value of $C_2$ given in Eq. (\ref{expnorm})
we can now obtain a relation between the high-frequency
slope parameter $\da$ and the spectral index $n$ at the COBE scale, for
any given value of $H_1$ and $m$. In Fig. \ref{f10} we illustrate such a
relation for different (realistic) values of $H_1$, and for a typical axion
mass $m=10^{-9}M_{\rm P}$. It should be stressed that, for $n \simeq
1$, and $\sg_{\rm i}$ of order 1 (i.e. near the minimum of $f(\sg_{\rm i})$),
the curves at constant $H_1$ are almost insensitive to the values of
$m$, and remain stable even if we change $m$ by various orders of
magnitude, as illustrated in Fig. \ref{f11}.
We have also reported, in Fig. \ref{f10}, the (present) most stringent
bounds on $n$, obtained by a recent analysis of the CMBR anisotropies
and large-scale structures \cite{ber,wtz}, i.e. $0.87 \leq n\leq 1.06$.
They are all compatible with $H_1 \simeq M_{\rm s}$, provided we allow for a
small break of the minimal spectrum, with $\da \simeq 0.2$--$ 0.3$. On
the other hand, as already stressed, no break at all is needed (i.e. $\da
=0$) if, for some dynamical mechanism (see for instance \cite{sc}) the
string mass is lowered down to the GUT scale, i.e. $H_1 \simeq
10^{-3}M_{\rm P}$.
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{f10.ps}}
\vskip 3mm
\caption[a]{Relation between $\da$ and $n$ for different values of
$H_1$ (in Planck units), for $m=10^{-9}M_{\rm P}$, and for $\sg_{\rm i}=1$.
The vertical dashed lines denote the experimentally allowed range
$0.87 \leq n\leq 1.06$.}
\label{f10}
\end{figure}
In Fig. \ref{f11} we have plotted the same curves of Fig. \ref{f10} for
two, very different values of the axion mass, $10^{-9}M_{\rm P}$ (bold
curves) and $10^{-14}M_{\rm P}$ (thin dashed curves). As clearly
illustrated by the figure, the dependence on the mass is very mild, and
it becomes practically inappreciable (for the given range of parameters)
when $H_1$ approaches $M_{\rm GUT}$.
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{f11.ps}}
\vskip 3mm
\caption[a]{Stability of the curves of Fig. 10 for two different choices
of the axion mass, $10^{-9}M_{\rm P}$ (bold curves) and
$10^{-14}M_{\rm P}$
(thin dashed curves).}
\label{f11}
\end{figure}
Having discussed the constraints imposed by the
COBE normalization we can present now the plots of the angular
coefficients $C_{\ell}$ for the scalar spectrum, in the case of a spatially
flat background and for few relevant choices of
the spectral index. In Figs. \ref{SR12a} and \ref{SR13a}
the (scalar) angular power spectrum defined in Eq. (\ref{anps}) is reported
for a flat, slightly red and slightly blue spectrum.
In order to obtain the results of Figs. \ref{SR12a} and \ref{SR13a}
we used the latest release of CMBFAST, relaxing the strict
COBE normalization at $\ell =10$, in favour of a better general agreement of the
overall fit.
In Figs. \ref{SR12a} and \ref{SR13a} we used the following
values of the cosmological parameters, selected according to fits of
CMBR anisotropy experiments \cite{wtz}: $h_{0} = 0.65$,
$\Omega_{\Lambda}= 0.7$, and $ h_0^2\Omega_{b} = 0.02$. The selected
value of $h_{0}^2\Omega_{b}$ is rather robust, even if no final
consensus has been reached on the second significant figure beyond
$0.02$. We have also assumed the simplest scenario for the
late-time cosmological evolution, with no significant effects of
reionization.
\begin{figure}[ht]
\centerline{\epsfxsize = 12 cm \epsffile{CL1.eps}}
\vskip 3mm
\caption[a]{The spectrum of $C_{\ell}$ is illustrated
for a fiducial set of parameters ($h_0 =0.65$, $\Omega_{\rm b}=
0.04733$, $\Omega_{\Lambda} = 0.7$, $\Omega_{\rm m} = 0.25267$)
and for flat (full line, $n =1$), slightly red (dashed line, $n =0.9$)
and slightly blue (dotted line, $n =1.02$) spectral indices.}
\label{SR12a}
\end{figure}
In Fig. \ref{SR12a} the $C_{\ell}$ are plotted on a linear
scale, whereas in Fig. \ref{SR13a} we present the same plot with
a semi-logarithmic scale, in such a way that the region relevant to
the COBE observations is less compressed.
We recall, finally, that the flat, red and blue
spectral indices may correspond to particular combinations of the
parameters $H_1,m,\da$ and $\sg_{\rm i}$, chosen in such a way as to satisfy
the COBE normalization, Eq. (\ref{649}).
\begin{figure}
\centerline{\epsfxsize = 12 cm \epsffile{CL2.eps}}
\vskip 3mm
\caption[a]{The same plot as in Fig. \ref{SR12a}, but
on a semi-logarithmic scale.}
\label{SR13a}
\end{figure}
The data points reported in Figs. \ref{SR12a} and \ref{SR13a}
are those from COBE \cite{cobe,cob2}, BOOMERANG \cite{boom},
DASI \cite{dasi}, MAXIMA \cite{maxima} and ARCHEOPS \cite{archeops}.
Notice that the data reported in \cite{archeops} fill
the ``gap'' between the last COBE points and the points of
\cite{boom,dasi,maxima}. Therefore, one could think of normalizing the
spectra not to COBE but directly to ARCHEOPS.
In spite of this possibility, the forthcoming MAP data
will give even more accurate determination of the $C_{\ell}$ spectra. It
will then be interesting to use these data in order to make
more consistent and accurate determinations of the pre-big bang
parameter space.
\renewcommand{\theequation}{7.\arabic{equation}}
\setcounter{equation}{0}
\section{Concluding remarks}
In the present paper the possible conversion of isocurvature,
primordial axionic fluctuations into adiabatic, large-scale metric
perturbations has been discussed in the context of the pre-big bang
scenario. Depending upon the specific relaxation of the axionic
background toward the minimum of the potential, a constant (and
large enough) mode in the Bardeen potential can be generated, for
scales that are still outside the horizon right after matter-radiation
equality.
After analysing the dynamics of the background and of its fluctuations,
the final amplitude and spectrum of the Bardeen potential
has been related to the initial axion spectrum
directly arising from the vacuum fluctuations amplified during the
pre-big bang epoch. Our goal has been to include, with reasonable
accuracy, the details of the post-big bang evolution, in such a way that
the pre-big bang parameters could be directly constrained
by the COBE normalization, and by the analysis of the Doppler-peak
structure. All the theoretical uncertainty reflects in our lack
of knowledge of $H_1$ which determines the end point of the primordal
axion spectrum.
The main conclusion of this work is that a phenomenologically appealing
spectrum of
adiabatic scalar perturbations can naturally emerge from the simplest pre-big
bang
scenario through a conversion of the initial isocurvature perturbations of the
Kalb-Ramond axion.
Since, at the large scales tested by CMBR experiments, the above conversion
preserves
the scale-dependence of the original spectrum, it is important for the latter
to be
quasi-scale-invariant at large scales. This can be achieved, for instance, if
the very
early stages of pre-big bang cosmology at weak coupling involve a symmetric
evolution
of all $9$ spatial dimensions (modulo $T$-duality).
Since the constant mode of the curvature fluctuations leads to
adiabatic initial conditions for the fluid evolution after matter-radiation
equality, the location of the Doppler peak is correctly reproduced.
On the other hand, the absolute normalization of fluctuations at large scales
(say those
relevant for COBE)
depend on several details of the model. Indeed, the axion spectrum is naturally
normalized at its end-point, given by our parameter $H_1$.
If one takes, naively, $H_1 \sim M_{\rm s} \sim 10^{17} ~{\rm GeV}$ and assumes a flat
spectrum
one finds values of $\Delta T/T$ that are a couple of orders of magnitude too
large
when compared with COBE's data.
However, one can think of many (individual or combined) effects that
can bring down our normalization to agree with the data,
e.g.
\begin{itemize}
\item A slight (blue) tilt to the spectrum;
\item A blue spectrum just at high frequency
(i.e. for scales that exit late, during the strongly coupled regime);
\item A lower $H_1/ M_{\rm s}$ ratio;
\item A lower $M_{\rm s}/ M_{\rm P}$ ratio.
\end{itemize}
In the near future we hope to extend the present discussion to
forthcoming CMBR anisotropy data at even smaller angular scales. It would be interesting
to see if a combined analysis of the experimental data
may give further useful hints on the parameter space of the scenario
explored in the present investigation.
\section*{Acknowledgements}
V. Bozza would like to thank the ``Museo Storico della
Fisica e Centro Studi e Ricerche E. Fermi'' for financial support.
G. Veneziano wishes to acknowledge the support of a ``Chaire Internationale
Blaise Pascal'', administred by the ``Fondation de l'Ecole Normale
Sup\'erieure''. M. Giovannini is indebted to the
``Institut de Physique Th\'eorique''
of the ``Universit\'e de Lausanne'' for partial support.
\newpage
\begin{appendix}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\section{Axionic spectra}
During the pre-big bang phase the quantum mechanical
fluctuations of the axionic field will be amplified from the initial
vacuum state. The obtained spectrum provides the initial
condition for the evolution of the axion fluctuations in the post-big
bang phase. At very large-scales, such a spectrum will not depend so
much upon the details of the pre-big bang evolution. At smaller scales,
however, it can be strongly affected by specific dynamics of the strong
coupling and high-curvature regime. In
spite of the fact that the spectral slope at large scales is not affected
by high energy corrections, the large scale amplitude is affected and,
in particular, a steeper slope at small scales has important
consequences for the
normalization of the low-frequency branch of the spectrum. In this
appendix we will consider, separately, the axion spectrum obtained
in the case of minimal and non-minimal pre-big bang models.
\subsection{Minimal pre-big bang models}
The linearized evolution of massive axion
inhomogeneities $\chi_k$, neglecting their coupling to scalar metric
perturbations, in a spatially flat cosmological
background, is described in general by the equation
\begin{equation}
\psi_{k}'' + \biggl[ k^2 + m^2 a^2 - \frac{z''}{z}\biggr] \psi_{k} =0,
\label{flucchi}
\end{equation}
where
\begin{equation}
z = a~e^{\varphi/2}, ~~~~\psi_{k} = z\chi_{k}.
\label{pump}
\end{equation}
In the pre-big bang phase ($\eta < \eta_1$)
the axion is massless. In the post-big bang,
radiation-dominated phase, taking place for $\eta> \eta_1$,
the gauge coupling freezes ($\varphi=$ const) and
the axion acquires a mass. The produced
axion spectrum, in principle, has a relativistic and a
non-relativistic branch: this is because, in the radiation
era, the proper momentum is red-shifted
with respect to the rest mass, and the whole spectrum, mode by mode,
tends to become non-relativistic. The spectral slope of the relativistic
and non-relativistic branches of the spectrum are in general different.
However, if the axion modes, as in the present case, become
non-relativistic when they are still outside the horizon,
the solution is then
exactly the same as in the relativistic limit.
Consider first the relativistic branch of the spectrum.
For $\eta <\eta_1$ the solution of Eq. (\ref{flucchi}) can be expressed
in terms of the
second-kind Hankel functions \cite{abr} as:
\begin{equation}
\psi_k(\eta)=\eta^{1/2}H_\mu^{(2)} (k\eta),
\label{psirel}
\end{equation}
where $\mu$ depends on the parameters controlling the kinematics of
the pre-big bang background (a specific example
will be given below, see Eqs. (\ref{ckappa}) and (\ref{index})). In the
radiation era, $\eta>\eta_1$, one has $z''/z=0$, and the evolution
equation of $\psi_{k}$ acquires a massive correction:
\begin{equation}
\psi_k'' +\left(k^2 + m^2 a^2\right)\psi_k=0,
\label{psinr}
\end{equation}
Assuming that the axion mass is negligible at the transition epoch
$\eta_1$, the solution (\ref{psirel}) can be matched to the
plane-wave solution
\begin{equation}
\psi_k= {1\over \sqrt k}\left[c_+(k) e^{-ik\eta}+
c_-(k) e^{ik\eta}\right],
\end{equation}
and the final result for $\chi_{k}$ is
\begin{equation}
\chi_{k}( \eta) = {c(k)\over a \sqrt k}\sin (k\eta),
\label{chires}
\end{equation}
where
\begin{equation}
c(k) \simeq \biggl(\frac{k}{k_1}\biggr)^{\frac{n - 5}{2}},
\label{ckappa}
\end{equation}
with $n = 4 - 2|\mu|$ . Note that
the expression of the Bogoliubov coefficient $c(k)$
and of the mean number of produced axions,
$\overline{n}_{k} = |c(k)|^2$,
contains different numerical factors of order $1$. At the same
time the maximal amplified momentum $k_1$ can be defined in different
ways, all equivalent up to numerical factors.
In the present analysis we will define the maximal scale $k_1$ as the
energy scale where one axion is produced per unit volume of phase-space.
Consider now the non--relativistic spectrum in the
case when the mode becomes non--relativistic
while it is still {\em outside the horizon}.
Defining
as $k_m$ the limiting comoving frequency of a mode that becomes
non-relativistic ($k_m=ma_m$) at the time it re-enters the horizon
($k_m=H_ma_m$), we find, in the radiation era \cite{dgs2,dgs3},
\begin{equation}
k_m= k_1 \left(m\over H_1\right)^{1/2}.
\label{c11}
\end{equation}
We are thus considering modes with $k\ll k_m$.
In order to estimate the spectrum, in this limit,
let us write Eq. (\ref{psinr}) in a form
suitable for comparison with known results
of parabolic cylinder equations:
\begin{equation}
{d^2\psi_k\over dx^2}+\left({x^2\over 4} -b\right)\psi_k=0,
~~~~~ x=\eta (2 \alpha)^{1/2}, ~~~~ -b= k^2/2\alpha ,
\label{parab}
\end{equation}
where
\begin{equation}
m^2a^2 =\alpha^2\eta^2, ~~~~~~~~~~~~~~~~
\alpha= m H_1a_1^2,
\end{equation}
and where $a \sim \eta$ has been assumed. The corresponding
general solution can be written as
\begin{equation}
\psi=A y_1(b,x)+B y_2(b,x)~,
\label{seceq}
\end{equation}
where $y_1$ and $y_2$ are the even and odd parts of the parabolic
cylinder functions \cite{abr}. The normalization to Eq. (\ref{chires})
in the relativistic limit (i.e. $x \to 0$) gives $A=0$ and
\begin{equation}
\psi_k \simeq c(k)\left(k\over 2\alpha \right)^{1/2} y_2(b,x) .
\end{equation}
Outside the horizon, $k\eta \ll1$, and for non-relativistic modes, $k \ll
ma$, we take (respectively) the limits $-b x^2 \ll1$ and $-b \ll x^2$,
the solution can be expanded as $y_2 \sim x\sim \eta \sqrt{2\a}$, so
that the mass disappears from the amplitude:
\begin{equation}
|\chi_{k}| \simeq {|c(k)|k^{1/2} \over a_1\eta_1}.
\end{equation}
The insertion of the spectrum (\ref{ckappa}), using $k_1=a_1 H_1$,
leads to the final result
\begin{equation}
k^{3/2} |\chi_{k}| \simeq H_{1}\biggl(\frac{k}{k_1}
\biggr)^{\frac{n -1 }{2}}.
\label{estimate}
\end{equation}
\subsection{Non-minimal pre-big bang evolution
and spectral breaks}
Equation (\ref{estimate}) holds in the case of minimal
pre-big bang models, where the dynamical evolution
of the dilaton field is dictated by the solution
of the low-energy equations of motion. However, when the dilaton
enters the strong coupling regime, different types of
scenarios may emerge. In particular, relation (\ref{pump})
defining the form of the axion pump field, may change in the infinite
bare string-coupling limit, as suggested by the arguments recently
developed in \cite{sc}. In the framework of \cite{sc}
the axion coupling function, as well as the other coupling functions
pertaining to fields of different spin, may have a finite limit for
infinite bare string coupling. Hence, toward the end
of the pre-big bang phase (i.e. when strong coupling is presumably
reached), \begin{equation}
z \sim a [c_{z} + {\cal O}(e^{- \varphi/2})],
\end{equation}
where $c_{z}$ is a constant. Since the axionic pump field now depends only on
the scale factor, it will naturally be steeper for small length
scales. A complementary possibility, discussed in
\cite{dgs3}, is the presence of an intermediate high-energy phase,
which precedes the standard radiation era, and which is still part of the
accelerated pre-big bang regime, but in which the kinematics of the
(usual) canonical pump field is significantly different from its
low-energy behaviour.
In all these cases the obtained spectra, at small scales, are possibly
steeper than in the case of minimal pre-big bang models.
In the simplest case the spectrum will have only one break,
at a momentum scale that will be conventionally denoted by $k_{s}$,
and the Bogoliubov coefficients can be written in the form
\begin{eqnarray}
|c_k|^2 &=& \left(k\over k_1\right)^{n- 5 + \delta},
~~~~~~~~~~~~~~~~~~~~~~k_s 0 $ parametrizes the slope of the break at high
frequency, while $n$ is the usual spectral index appearing at
large scales and computed on the basis of the perturbative
evolution of the dilaton field.
From Eq. (\ref{delta}) it can be argued that
the steeper and/or the longer the
high-frequency branch of the spectrum, the larger the suppression
at low-frequency scales.
\end{appendix}
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