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% "CONSTRAINTS ON PRE-BIG BANG MODELS FOR SEEDING
% LARGE-SCALE ANISOTROPY BY MASSIVE
% KALB-RAMOND AXIONS"
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%minore o circa uguale
\def\laq{\raise 0.4ex\hbox{$<$}\kern -0.8em\lower 0.62
ex\hbox{$\sim$}}
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\def\gaq{\raise 0.4ex\hbox{$>$}\kern -0.7em\lower 0.62
ex\hbox{$\sim$}}
\def \pa {\partial}
\def \ra {\rightarrow}
\def \la {\lambda}
\def \La {\Lambda}
\def \Da {\Delta}
\def \b {\beta}
\def \a {\alpha}
\def \ap {\alpha^{\prime}}
\def \Ga {\Gamma}
\def \ga {\gamma}
\def \sg {\sigma}
\def \da {\delta}
\def \ep {\epsilon}
\def \r {\rho}
\def \om {\omega}
\def \Om {\Omega}
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\begin{document}
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{\large
\begin{flushright}
DFTT-31/98\\
CERN-TH/98-180\\
hep-ph/9806327
\end{flushright}
}
\vspace*{0.3truein}
\vskip 1.5 cm
{\Large\bf\centering\ignorespaces
Constraints on pre-big bang models for seeding large-scale \\
anisotropy by massive Kalb--Ramond axions
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{\large M. Gasperini}
\par}
{\large\it\centering\ignorespaces
Dipartimento di Fisica Teorica, Universit\`a di Torino,
Via P. Giuria 1, 10125 Turin, Italy \\
and Istituto Nazionale di Fisica Nucleare, Sezione di Torino,
Turin, Italy \\
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{\large G. Veneziano}
\par}
{\large\it\centering\ignorespaces
Theory Division, CERN, CH-1211 Geneva 23, Switzerland \\
\par}
%{\small\rm\centering(\ignorespaces May 1998\unskip)\par}
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\centerline{\Large Abstract}
\vskip 0.5 cm
\noi
{\large
We discuss the conditions under which pre-big bang models can
fit the observed large-scale anisotropy with a primordial
spectrum of massive (Kalb--Ramond) axion fluctuations.
The primordial spectrum must be sufficiently flat at
low frequency and sufficiently steeper at
high frequency. For a steep and/or long enough
high-frequency branch of the spectrum the
bounds imposed by COBE's normalization allow axion masses of the
typical order for a Peccei--Quinn--Weinberg--Wilczek axion. We
provide a particular example in which an appropriate axion
spectrum is obtained from a class of backgrounds satisfying the
low-energy string cosmology equations.
}
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{\large
\begin{flushleft}
CERN-TH/98-180\\
May 1998
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}
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\begin{flushright}
DFTT-31/98\\
CERN-TH/98-180\\
hep-ph/9806327\\
\end{flushright}
\vskip 0.5 true cm
{\large\bf\centering\ignorespaces
Constraints on pre-big bang models for seeding large-scale
anisotropy \\
by massive Kalb--Ramond axions
\vskip2.5pt}
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\nointerlineskip \rm\centering
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M. Gasperini
\par}
{\small\it\centering\ignorespaces
Dipartimento di Fisica Teorica, Universit\`a di Torino,
Via P. Giuria 1, 10125 Turin, Italy \\
and Istituto Nazionale di Fisica Nucleare, Sezione di Torino,
Turin, Italy \\
\par}
{\dimen0=-\prevdepth \advance\dimen0 by23pt
\nointerlineskip \rm\centering
\vrule height\dimen0 width0pt\relax\ignorespaces
G. Veneziano
\par}
{\small\it\centering\ignorespaces
Theory Division, CERN, CH-1211 Geneva 23, Switzerland \\
\par}
{\small\rm\centering(\ignorespaces May 1998\unskip)\par}
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%\begin{abstract}
We discuss the conditions under which pre-big bang models can
fit the observed large-scale anisotropy with a primordial
spectrum of massive (Kalb--Ramond) axion fluctuations.
The primordial spectrum must be sufficiently flat at
low frequency and sufficiently steeper at
high frequency. For a steep and/or long enough
high-frequency branch of the spectrum the
bounds imposed by COBE's normalization allow axion masses of the
typical order for a Peccei--Quinn--Weinberg--Wilczek axion. We
provide a particular example in which an appropriate axion
spectrum is obtained from a class of backgrounds satisfying the
low-energy string cosmology equations.
%\end{abstract}
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%\pacs{}
\section{INTRODUCTION}
\label{I}
It has recently been shown \cite{1,1a} that a stochastic
background of massless axions can induce large-scale anisotropies
of the Cosmic Microwave Background (CMB), in agreement with
present observations \cite{2,2a}, provided it is primordially
produced with a sufficiently flat spectrum. It has also been
shown that a massive axion background can
satisfy the same experimental constraints, with some
additional restrictions on the tilt of the spectrum, but only if the
axion mass lies inside an appropriate ultra-light mass window
\cite{1} having an upper limit of
$10^{-17}$ eV.
The cosmic axion background considered in \cite{1,1a} is obtained by
amplifying the vacuum fluctuations of the so-called universal axion
of string theory \cite{3}, i.e. the (four-dimensional) dual of the
Kalb--Ramond (KR) antisymmetric tensor field appearing in the
low-energy string effective action. The KR axion has interactions
of gravitational strength, hence its mass is not significantly
constrained by present tests of the equivalence principle for
polarized macroscopic bodies \cite{4}. Although (gravitationally)
coupled to the QCD topological current, the KR axion
is not to be necessarily identified with
the ``invisible" axion \cite{5} responsible for solving
the strong CP problem. Other, more strongly coupled
pseudoscalars, can play the traditional axion's role. In this
case, the standard Weinberg--Wilczek formula \cite{6} would give
the mass of the appropriate combination of pseudoscalars which is
coupled to the topological charge, while the KR axion would
mostly lie along the orthogonal combinations, which
remain (almost) massless.
In this context it is thus possible that
KR axions are neither produced from an initial
misalignment of the QCD vacuum angle \cite{7}, nor from the
decay of axionic cosmic strings \cite{8}, so that existing
cosmological bounds on the axion mass \cite{9} can be evaded.
Also, KR axions are treated in \cite{1,1a} as ``seeds", i.e. as
inhomogeneous perturbations of a background that is {\it not}
axion-dominated, so that the mechanism of anisotropy production
is different from previous computations of isocurvature axion
perturbations \cite{10}. The seed approximation is not inconsistent
either with the presence of mass or with the possible resonant
amplification of quantum fluctuations through oscillations in the full
axion potential \cite{kolb}.
In spite of all this, it is quite likely that the KR axion
will be heavier than $10^{-17}$ eV, in which
case, taking the results of \cite{1} at face value, KR axions
would be unable to seed the observed
CMB anisotropies. The main purpose of
this paper is to point out that this conclusion is not inescapable,
provided one is willing to add more structure to the primordial
axion spectrum through more complicated cosmological
backgrounds than the one considered in \cite{1}.
It turns out, in particular, that the bounds on the axion mass can
be relaxed, provided the axion spectrum, before non-relativistic
corrections, grows monotonically with frequency, but with a
frequency-dependent slope. At low frequencies the slope must be
small enough to reproduce the approximate scale-invariant
spectrum found by COBE, while at high frequencies the spectrum
has to be steeper. Since position and normalization of
the end point of the spectrum are basically
fixed in string cosmology, the steeper and/or longer
the high-frequency spectrum, the larger the
suppression of the amplitude at the low-frequency scales
relevant for COBE's observation.
On the other hand, the low-frequency amplitude is
proportional to a positive power of
the axion mass, in the non-relativistic regime. This
is the reason why, for a steep and/or long enough high-frequency
branch of the spectrum, it becomes possible to relax the bounds of
\cite{1} on the axion mass while remaining compatible
with COBE's data.
In the context of the pre-big bang scenario \cite{11} it is
known that relativistic axions with a nearly flat spectrum can be
produced in the transition from a dilaton-dominated,
higher-dimensional phase to the standard radiation-dominated
phase \cite{3}. A spectrum with an effective slope that grows
with frequency can be easily obtained if the phase of accelerated
pre-big bang evolution consists of (at least) two distinct regimes.
In that case the allowed mass window can be enlarged
to include a more conventional range of values.
Conversely, no significant
relaxation of the bounds given in \cite{1} seems to be possible
if the two
branches of the spectrum are obtained through a period of
decelerated post-big bang evolution, preceding the radiation era.
The rest of the paper is organized as follows. In Sections \ref{II}
and \ref{III} we generalize
the results of \cite{1,1a} for massive axions to an arbitrary
background of the pre-big bang type. In particular,
we will derive an equation expressing the predicted
CMB anisotropy in terms of the axion mass, of the string coupling,
and of the behaviour of the cosmological
background {\em before} the
radiation era. In Section \ref{IV} we will discuss in detail the
example of an axion spectrum
consisting of just a low- and a high-frequency branch. For this
case we will determine the
region in parameter space that gives consistency with COBE's data
without violating other important constraints. We will show
that backgrounds of this kind can emerge, for instance, from the
low-energy string cosmology equations in the presence of
classical string sources. Section \ref{V} is finally devoted to our
concluding remarks.
\section{MASSIVE AXION SPECTRA IN THE PRE-BIG
BANG SCENARIO}
\label{II}
We start by considering the dimensionally-reduced effective action
for Kalb--Ramond axion perturbations ($\sg$), to lowest order in
$\ap$ and in the string coupling parameter. For
a spatially flat background, in the string frame, we are led to the
four-dimensional action \cite{3}:
\beq
S={1\over 2} \int d^3x d\eta \left[ a^2 e^{\phi} \left(\sg^{\prime
2} + \sg\nabla^2 \sg\right) \right].
\label{21}
\eeq
Here a prime denotes differentiation with respect to conformal
time $\eta$, $\phi$ is the dimensionally reduced dilaton that
controls the effective four-dimensional gauge coupling
$g=e^{\phi/2}$, and $a$ is the scale factor of the external,
isotropic three-dimensional space, in the string frame.
Variation of the action (\ref{21}) leads to the canonical
perturbation equations, which can be written in terms of the pump
field $\xi$ and of the normal mode $\psi$ as:
\beq
\psi'' + \left(k^2-{\xi''\over \xi}\right)\psi=0,
~~~ \psi=\sg \xi, ~~~\xi=ae^{\phi/2}
\label{22}
\eeq
(we have implicitly assumed a Fourier expansion of perturbations,
by setting $\nabla^2 \psi=-k^2\psi$).
For any given model of background evolution,
$a(\eta)$, $\phi(\eta)$, the amplified axion spectrum
can be easily computed, starting with an initial vacuum
fluctuation spectrum and applying the standard formalism of
cosmological perturbation theory \cite{19}. One has to solve the
perturbation equation (\ref{22}) in the various cosmological
phases, and to match the solution at the
transition epochs. From the final axion
amplitude $\sg (\eta)$, $\eta \ra +\infty$, one then obtains the
so-called Bogoliubov coefficients which determine, in the free-field,
oscillating regime (i.e. well inside the horizon), the total number
distribution $n(\om)$ of the produced axions.
For the purpose of this paper it will be enough to consider the case,
appropriate to the pre-big bang scenario, in which the pump field
$\xi$ keeps growing during the whole pre-big bang
epoch, i.e. from $\eta=-\infty$ up to the final time $\eta=\eta_r$,
when the background enters the standard,
radiation-dominated regime with frozen dilaton.
For $\eta>\eta_r$ the pump
field is still growing, as the dilaton stays constant and $\xi(\eta)$
coincides with the expanding external scale factor $a(\eta)$.
With this model of background, it is convenient to refer the
spectrum to the maximal amplified frequency, $\om_r
=k_r/a\simeq H_r a_r/a$, where $H=a'/a^2$ is the Hubble
parameter, and $\om=k/a$ denotes proper frequency.
The axion energy distribution per logarithmic
interval of frequency, in this background, can
thus be written (in units of critical energy density $\r_c =
3H^2/8\pi G$) as:
\bea
\Om_\sg(\om, \eta)&= &{1\over \r_c}{d \r\over d\ln \om}=
{4G\over 3 \pi H^2}\om^4 n(\om)\nonumber \\
&=& g_r^2 \Om_\ga (\eta)
\left(\om \over \om_r\right)^4 n(\om).
\label{28}
\eea
We have denoted with $\Om_\ga(\eta)=(H_r/H)^2(a_r/a)^4$ the
time-dependent radiation energy density (in critical units), that
becomes dominant at $\eta=\eta_r$. Also, we have identified the
curvature scale at the inflation--radiation transition with the
string mass scale, $H_r \simeq M_s$, and we have denoted by
$g_r\equiv g(\eta_r)$ the final value of the string coupling
parameter, approaching the present value of the fundamental ratio
between string and Planck mass \cite{17}:
\beq
g_r =e^{\phi_r/2} \simeq M_s/M_p \sim 0.1 - 0.01 .
\label{29}
\eeq
Numerical coefficients of order $1$ have been absorbed into
$g_r^2$, also in view of the uncertainty with which we can identify
the transition scale and the string scale.
The number distribution $n(\om)$, appearing in eq. (\ref{28}),
is completely determined by the
background evolution, and can be estimated by truncating the solution
of the perturbation equation (\ref{22}), ouside the horizon, to the
frozen part of the axion field, i.e. $\sg (\eta)=$ const for
$|k\eta|\ll 1$. This is common practice in the context of the
standard inflationary scenario \cite{19}, where the non-frozen
part of the fluctuations quickly decays in time outside the horizon.
It can be shown, however, that such an estimate is
generally valid,
quite independently of the behaviour of $\sg$ outside the
horizon, provided the total energy density is correctly computed
by including in the Hamiltonian the contribution of the frozen
modes of the fluctuation and of its conjugate momentum
\cite{19a}.
For a monotonically growing pump field, as
in the case we are considering, we obtain the estimate
\beq
n(\om)\simeq {\xi^2_{re}(\om)\over \xi^2_{ex}(\om)}=
{\xi^2_{r}\over \xi^2_{ex}(\om)}{a^2_{re}(\om)\over a^2_{r}}.
\label{26}
\eeq
Here the label $r$ denotes, as before, the beginning of the
radiation phase;
the labels ``{\it ex}" and ``{\it re}" mean evaluation of the
fields at the times $|\eta|\simeq (a\om)^{-1}$ when a mode $\om$,
respectively, ``exits the horizon" during the pre-big bang epoch,
and ``re-enters the horizon" in the post-big bang epoch.
Let us now discuss how the above spectrum has to be modified
when axions become massive. We will first assume that, at the
beginning of the radiation era,
the axion field has already acquired a mass, but the mass is so
small that it does not affect, initially, the axion spectrum. As the
Universe expands, however, the proper momentum is red-shifted
with respect to the rest mass, and a given axion mode $k$ tends to
become non-relativistic when $\om=k/aH$, then the number $n(\om)$ of the produced axions is fixed
after re-entry, when the mode is still relativistic, and the
effect of the mass in the non-relativistic regime is a
simple rescaling of the energy density: $\Om_\sg \ra (m/\om)
\Om_\sg$. If, on the contrary, a mode becomes non-relativistic
outside the horizon, when $\om H_{eq}\sim 10^{-27}$eV
(see Sect. \ref{III}), so that non-relativistic corrections are already
effective for $\eta >\eta_{eq}$. In the non-relativistic regime
$\Om_\sg$ evolves in time like the energy density of dust matter,
$\Om_\sg\sim a^{-3}$, and then, in the matter-dominated era, it
remains frozen at the value $\Om_\sg(\eta_{eq})$ reached at
the time of matter--radiation equilibrium. By adding the
non-relativistic corrections to the generic spectrum (\ref{28}) we
thus obtain, for $\eta>\eta_{eq}$:
\bea
\Om_\sg(\om)&=&g_r^2\Om_\ga
\left(\om\over\om_r\right)^{4} n(\om),
~~~~~~~~~~~~~~~~~m<\om<\om_r,
\nonumber\\
&=&g_r^2{m\over H_r}\left(H_r\over H_{eq}\right)^{1/2}
\left(\om\over\om_r\right)^{3} n(\om) ,
~~\om_m<\om\om_T$, the role of the transition
frequency, that separates modes that become non-relativistic inside
and outside the horizon, is played by $\om_T$,
and the lowest-frequency
band of the spectrum (\ref{212}) has to be replaced by:
\bea
&&
\Om_\sg
=g_r^2\left(m\over
H_{eq}\right) \left({\rm eV}\over T_m\right)
\left(\om\over\om_r\right)^{4} n(\om) , \nonumber\\
&&
\left(m\over
H_{eq}\right)^{1/2} \left({\rm eV}\over T_m\right)>1,
~~~~~~~\om<\om_T
\label{temp}
\eea
(we have used $\om_0\sim 10^{-2}\om_{eq}
\sim 10^{-18}$ Hz, and $\om_r(t_0) \sim
g_r^{1/2} 10^{11}$ Hz). In this mass range the non-relativistic
spectrum is further enhanced with respect to eq. (\ref{212}) by the
factor $\left(m/
H_{eq}\right)^{1/2} \left({\rm eV}/ T_m\right)>1$, with a
consequently less efficient relaxation of the bounds on the axion
mass. This effect has to be taken into account when discussing
restrictions for a given model in parameter space.
\section{Massive-axion contributions to $\Da T/ T$}
\label{III}
The main results of this section have been
obtained also in \cite{1}, in the context of the ``seed" approach to
density fluctuations, by exploiting the so-called ``compensation
mechanism" \cite{19b} to estimate the relative contribution of
seeds and sources to the total scalar perturbation
potential. Here we will show, for the sake of completeness and for
the reader's convenience, that when the axion mass is sufficiently
large the contribution to the temperature anisotropies can be
quickly estimated also within the standard cosmological
perturbation formalism, with results that are the same as
those provided by the compensation mechanism. We also generalize
the results of \cite{1} to the case of a generic background
and spectrum.
We will work under the assumption that axions can be
treated as seeds for scalar metric perturbations:
the inhomogeneous axion stress tensor,
$\tau_\mu^{\nu}$, will thus represent the total source of
perturbations, without contributing, however, to the unperturbed
homogeneous equations determining the evolution of the
background. Also, we shall only consider modes that are relevant
for the large-scale anisotropy, i.e. modes that are still outside the
horizon at the time of decoupling of matter and radiation,
$\eta_{dec} \sim \eta_{eq}$. Assuming that such modes are
already fully non-relativistic,
\beq
\om < H_{eq} \eta_{eq},
\label{37a}
\eea
so that the integral (\ref{36}) becomes
\bea
&&
P_\r(k) \simeq \nonumber\\
&&
mH_r \left(k\over k_r\right)^{3}
\left( k_r\over a\right)^{6} \int {dp\over p} \left(p\over
k_r\right)^{4} {|p-k|\over k_r}n(p)n(|k-p|). \nonumber\\
&&
\label{37b}
\eea
We parametrize the slope of the relativistic axion spectrum by
$p^4 n(p) \sim p^{3-2\mu}$,
with $\mu<3/2$ to avoid over-critical axion production. For a flat
enough slope, i.e. $\mu>3/4$, it can be easily checked that the
integrand of eq. (\ref{37b}) grows from $0$ to $k$, and decreases
from $k$ to $k_m$. The power spectrum may thus be immediately
estimated by taking the contribution at $p\sim k$, namely
\beq
P_\r(k) \sim
mH_r \left( k_r\over a\right)^{6}\left(k\over k_r\right)^{8}
n^2(k).
\label{37c}
\eeq
Comparison with eq. ({\ref{212}) leads to the final result, valid for
$\eta > \eta_{eq}$,
\bea
&&
P_\Psi^{1/2}(k) \sim {G\over H^2}P_\r^{1/2} (k)\nonumber\\
&&
\sim g_r^2 \left(m\over H_{eq}\right)^{1/2}
\left(k \over k_r\right)^{4} n(k)
\sim \Om_\sg(k),
\label{38a}
\eea
where $\Om_\sg$ is the lowest frequency, non-relativistic band of
the axion spectrum.
As the axion contribution to the
Bardeen potential does not depend on time in the
matter-dominated era, the large-scale anisotropy
of the CMB temperature is determined by the ordinary,
non-integrated Sachs-Wolfe effect \cite{20} as
\cite{1,19}:
\beq
P_T^{1/2}(k)\sim P_\Psi^{1/2}(k) \sim \Om_\sg(k),
\label{38}
\eeq
where the temperature power spectrum $P_T(k)$ is defined by
\beq
\langle \Da T/T(x) \Da T/T(x')\rangle=
\int {d^3k\over (2\pi k)^3}e^{i {\bf k} \cdot ({\bf x}-{\bf x}')}
P_T (k).
\label{39a}
\eeq
Equation (\ref{38}) can be converted (see \cite{1}) into a relation for
the usual coefficients $C_{\ell}$ of the multipole expansion of
the CMB temperature fluctuations.
Also,
eqs. (\ref{38}), (\ref{38a}), and (\ref{26}) can be used to relate
the cosmological background at a given time $\eta$ directly to the
axion mass and to the temperature anisotropy at a related scale.
We find:
\beq
{\eta_{r}~\xi_{r}\over \eta ~\xi ({-\eta})} \simeq
g_r^{-1}~\left(m\over H_{eq}\right)^{-1/4}
\left[P_T^{1/4}(k)\right]_{k\eta=1} ,
\label{relation}
\eeq
where we have used the fact that, for an accelerated power-law
background, the exit time of the mode $k$ in the pre-big bang
epoch is at $k \simeq -\eta^{-1}$.
This equation can be seen as the analogue, in our context, of
the reconstruction of the inflaton potential from the
CMB power spectrum in ordinary slow-roll inflation \cite{Kolb}.
By parametrizing the slope of $\Om_\sg$ as $\Om_\sg \sim
\om^{(n-1)/2}$, it follows from eq. (\ref{38}) that $n$ can be
identified with the usual tilt parameter of the CMB anisotropy
\cite{2}, constrained by the data as:
\beq
0.8~\laq~n~\laq~1.4.
\label{39}
\eeq
The observed quadrupole amplitude, which normalizes the spectrum
at the present horizon scale $\om_0$ \cite{2a}, gives also:
\beq
\Om_\sg(\om_0)\simeq 10^{-5}.
\label{310}
\eeq
In addition, the validity of our perturbative computation, which
neglects the back-reaction of the axionic seeds, requires that the
axion energy density remains well under-critical not only at the
end point $\om_r$ (which is automatically assured by
$g_r^2<1$), but also at the non-relativistic peak at $\om=\om_m$.
We thus require
\beq
\Om_\sg(\om_m)< 0.1.
\label{311}
\eeq
From the COBE normalization (\ref{310}), imposed on the lowest
frequency end of the axion spectra (\ref{212}) and (\ref{temp}),
we can now fix the mass as
a function of the background, according to eq. (\ref{26}).
We assume, as before, that the post-big bang background is
radiation-dominated up to the string scale, i.e. $H_r
\simeq g_r M_p$, and that it becomes matter-dominated for
$H1. \label{313a}
\eea
In the next section it will be shown, with an
explicit example of
pre-big bang background, that a very large axion mass
window may in principle be compatible with the bounds
(\ref{31}) and (\ref{39})--(\ref{311}).
\section{Constraints on parameter space for a particular class of
backgrounds}
\label{IV}
In order to provide a quantitative estimate of the possible axion
mass window allowed by the large-scale CMB anisotropy, in a
string cosmology context, we will discuss here a class of
backgrounds that is sufficiently representative for our purpose, and
characterized by three different cosmological phases. We parametrize
the evolution of the
axionic pump field in these three phases as
\bea &&
\xi \sim \left|\eta\right|^{r+1/2}, ~~~~~~~ \eta <\eta_s,
\nonumber\\
&&
\xi \sim \left|\eta\right|^{-\b}, ~~~~~~~~~~\eta_s<\eta <\eta_r,
\nonumber\\
&&
\xi \sim \left|\eta\right|, ~~~~~~~~~~~~~~ \eta_r <\eta ,
\label{25}
\eea
where $\eta_s$ marks the beginning of an intermediate phase,
preceding the standard radiation era, which starts at
$\eta=\eta_r$. There is no need
to consider here also the last transition from
radiation- to matter-dominance,
at $\eta=\eta_{eq}$, since we are
assuming $m>H_{eq}$, so that the axion spectrum becomes
mass-dominated (and thus insensitive to the subsequent
transitions) before the time of equilibrium.
For the intermediate phase, $\eta_s<\eta<\eta_r$, we have two
possibilities: accelerated or decelerated evolution of the
background fields, corresponding respectively to a shrinking or
expanding conformal time parameter, $|\eta_r/\eta_s|<0$ or
$|\eta_r/\eta_s|>0$. In the first case, as $\eta$ ranges from
$-\infty$ to $\eta_r$, the ``effective potential" $V=|\xi''/\xi|$
of eq. (\ref{22}) grows monotonically from zero to $V(\eta_r)\sim
\eta_r^{-2}$, the maximal amplified frequency is
$k_r=V^{1/2}(\eta_r)\sim \eta_r^{-1}$, and all frequency modes
``re-enter the horizon" in the radiation era.
In the second case $V$ is instead decreasing from $\eta_s$ to
$\eta_r$, the maximal amplified frequency is
$k_s=V^{1/2}(\eta_s)\sim \eta_s^{-1}$, and the high-frequency
band of the spectrum, $k_s2$ the high-frequency band of the axion spectrum is
decreasing, the amplification of perturbations is thus enhanced at
the COBE scale, and the bounds on the axion mass become more
constraining instead of being relaxed. For $-1 <\b<2$ the
spectrum grows monotonically, but an explicit computation shows
that even in that case no significant widening of the mass window
may be obtained.
In this paper we will thus concentrate on the first type of
background, namely on an accelerated evolution of the pumping
field, parametrized as in eq. (\ref{25}). The axion spectrum grows
monotonically in the whole range $-2<\b<1$, but it seems natural
to assume that also the pumping field is growing, so that we shall
analyse a background with $0<\b<1$. This background includes, in
the limit $\b \ra 1$, a phase of constant curvature and frozen
dilaton, $a \simeq (-\eta)^{-1}$, $\phi \simeq$ const, which is a
particular realization of the high-curvature string phase introduced
in \cite{12,13} for phenomenological reasons, and shown to be a
possible late-time attractor of the cosmological equations when
the required higher-derivative corrections are added to the string
effective action \cite{16a}.
For the background (\ref{25}) the initial relativistic spectrum has
two branches, corresponding to modes that ``cross the horizon"
during the initial low-energy phase, $\om <\om_s \simeq H_s
a_s/a$, and during the subsequent intermediate phase,
$\om_s<\om<\om_r$. The non-relativistic corrections are to be
included according to eqs. (\ref{212}) and (\ref{temp}).
For the purpose of this paper, it will be sufficient to consider the
non-relativistic spectrum in two limiting cases only: the one
in which only the low-frequency branch of the spectrum
becomes non-relativistic, and the one in
which already in the high-frequency branch there are modes that
become non-relativistic outside the horizon. In the first
case the spectrum is given by
\bea
\Om_\sg&=&g_r^2\Om_\ga\left(\om\over
\om_r\right)^{2-2\b},
~~~~~~~~~~~~~~~~~~~~~\om_s<\om<\om_r, \nonumber\\
&=&g_r^2\Om_\ga\left(\om\over
\om_r\right)^{3-2|r|}\left(\om_s\over \om_r\right)^{2|r|-2\b-1},
m<\om<\om_s, \nonumber\\
&=&g_r^2{m\over H_r}\left(H_r\over H_{eq}\right)^{1/2}
\left(\om\over\om_r\right)^{2-2|r|}
\left(\om_s\over \om_r\right)^{2|r|-2\b-1}, \nonumber\\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~\om_m<\om(T_m/{\rm eV})^2$. In the second limiting case the
spectrum is
\bea
\Om_\sg&=&g_r^2\Om_\ga\left(\om\over
\om_r\right)^{2-2\b},
~~~~~~~~~~~~~~~~~~~~m<\om<\om_r, \nonumber\\
&=&g_r^2{m\over H_r}\left(H_r\over H_{eq}\right)^{1/2}
\left(\om\over\om_r\right)^{1-2\b},
~~~~~~\om_m<\om(T_m/{\rm eV})^2$.
The spectrum depends on five parameters: $g_r$,
$m$, $\om_s/\om_r$, $\b$ and $T_m$, the temperature scale at
which the axions become massive. The conditions to be imposed on
the axion energy density in order to fit present observations of the
large-scale anisotropy, without becoming over-critical, provide
strong constraints on the parameters $\om_s/\om_r$ and $\b$,
namely on the pre-big bang evolution of the background, as we
now want to discuss. We will show that, for $\om_0<\om_s$, the
COBE normalization
(\ref{310}) imposed on the lowest-frequency
band of the above spectra can be satified consistently
with all constraints both for $(m/H_{eq})^{1/2}T_m/$eV, and the bounds on the mass can be
significantly relaxed.
In order to discuss this possibility, it is convenient to use as
parameters the duration of the intermediate pre-big bang phase,
measured by the ratio $\om_r/\om_s$, and the variation of the
pump field during that phase,
$\xi_s/\xi_r=(\eta_s/\eta_r)^{-\b}=(\om_r/\om_s)^{-\b}$. Thus
we set
\beq
\b=-{y\over x}, ~~x=\log_{10}(\om_r/\om_s)>0, ~~
y=\log_{10}(\xi_s/\xi_r)<0.
\label{41}
\eeq
According to eq. (\ref{39}), the slope of
the non-relativistic, low-energy branch of the spectrum,
\beq
(n-1)/2=3-2|r|,
\label{42}
\eeq
is constrained by
\beq
1.4 \leq |r|\leq 1.5
\label{43}
\eeq
($n<1$ has been excluded, to obtain a
growing axion spectrum also in the limit $\eta_s\ra \eta_r$). The
COBE normalization (\ref{310}) fixes the mass as follows
\bea
&&
\log_{10}{m\over
H_{eq}}=4y+\left(4|r|-2\right)x+164-116|r|\nonumber\\
&&
-\left(1+2|r|\right)\log_{10}g_r,
~~~~~~~~\left(m\over H_{eq}\right)^{1/2} \left({\rm eV}\over T_m
\right) <1, \label{44}\\
&&
\log_{10}{m\over
H_{eq}}=2y+\left(2|r|-1\right)x+\log_{10}\left(T_m\over {\rm
eV}\right) +82-58|r|\nonumber\\
&&
-\left({1\over 2}+|r|\right)\log_{10}g_r,
~~~~~~\left(m\over H_{eq}\right)^{1/2} \left({\rm eV}\over T_m
\right) >1.
\label{44a}
\eea
The critical bound also depends on the mass: if
$m/H_{eq}<(T_m/{\rm eV})^2$ the condition (\ref{311}) has to be
imposed on eq. (\ref{213}) for $\om_m<\om_s$, and
on eq. (\ref{214}) for $\om_m>\om_s$; if, on the contrary,
$m/H_{eq}>(T_m/{\rm eV})^2$, then the condition (\ref{311}) has to
be imposed on eq. (\ref{213a}) for $\om_T<\om_s$, and
on eq. (\ref{214a}) for $\om_T>\om_s$.
Finally, we have the constraints $\log_{10}
\left(m/H_{eq}\right)>0$, see eq. (\ref{31}), and, by definition,
$x>0$, $y<0$, $y>-x$ (since $\b <1$).
The allowed region in the ($x,y$) plane, as determined by the above
inequalities, is not very sensitive to the variation of $g_r$ and $|r|$
in their narrow ranges, determined respectively by eqs. (\ref{29})
and (\ref{43}). For a qualitative illustration of the constraints
imposed by the COBE data we shall fix these parameters to the
typical values $g_r=10^{-2}$ and $|r|=1.45$ (corresponding to a
spectral slope $n=1.2$). Also, we will assume that axions become
massive at the scale of chiral symmetry breaking $T_m \simeq
100$ MeV. The corresponding allowed ranges of the
parameters of the intermediate pre-big bang phase (duration and
kinematics) are illustrated in Fig. 2.
\begin{figure}[t]
\begin{center}
\mbox{\epsfig{file=f2ax.ps,width=82mm}}
\vskip 5mm
\caption{\sl Possible allowed region for the parameters of an
intermediate pre-big phase, consistent with an axion spectrum
that does not become over-critical, and that reproduces the present
COBE observations. The dashed lines represent curves of constant
axion mass.}
\end{center}
\end{figure}
The allowed region is bounded by the bold solid curves. The lower
border is fixed by the condition $m>H_{eq}\sim 10^{-27}$ eV for
$x~\gaq ~18$, and by the condition $\b<1$, i.e. $y>-x$, for
$x~\laq ~18$. The upper border is fixed by the condition
$\Om_\sg(\om_m)<0.1$, imposed for $\om_T>\om_s$ and
$m/H_{eq}>10^{16}$.
The region is limited to the range
$\om_r/\om_s~\laq~10^{28}$, as we are considering the case in
which the axion contribution to the CMB anisotropy arises from the
low-frequency branch of the spectrum.
The dashed lines of Fig. 2 represent curves of constant axion mass,
determined by the conditions (\ref{44}), (\ref{44a}), with
$g_r=10^{-2}$, $|r|=1.45$ and $T_m=100$ MeV, namely
\bea
&&
y=-0.95x -0.9 +{1\over 4}\log_{10}\left(m\over 10^{-27} {\rm
eV}\right), ~m<10^{-11}~{\rm eV}, \nonumber\\
&&
y=-0.95x -4.9 +{1\over 2}\log_{10}\left(m\over 10^{-27} {\rm
eV}\right), ~m>10^{-11}~{\rm eV}.\nonumber\\
\label{48}
\eea
As shown in the picture, the allowed region is compatible with
masses much higher than $H_{eq}$, up to the limiting value $m\sim
100$ MeV above which the discussion of this paper cannot be
applied, since for $m>100$ MeV all the produced axions decayed
into photons before the present epoch, at a rate $\Ga \sim
m^3/M_p^2$. For the particular example shown in Fig. 2, the
allowed axion-mass window is then
\beq
10^{-27} ~{\rm eV} 18$ the
line $y=-x$ lies outside the allowed region of Fig. 2.
We conclude this section by giving a possible example of background,
which satisfies the low-energy string cosmology equations, and
which is simultaneously compatible with COBE and with higher
values of the axion mass, as illustrated in Fig. 2.
Consider the gravi-dilaton string effective action, with zero dilaton
potential, but with the contribution of additional matter fields
(strings, membranes, Ramond forms, ...) that can be approximated as
a perfect fluid with an appropriate equation of state. As discussed
in \cite{22}, the cosmological equations can in this case be
integrated exactly, and the general solution is characterized by two
asymptotic regimes. In the initial small-curvature limit,
approaching the perturbative vacuum, the background is dominated
by the matter sources. At late times, when approaching the
high-curvature limit, the background becomes instead
dilaton-dominated, and the effects of the matter sources
disappear. The time-scale marking the transition between the two
kinematical regimes (and thus the duration of the second,
dilaton-dominated phase) is controlled by an arbitrary integration
constant.
In order to provide an explicit example of this class of
backgrounds, we can take, for instance, a
$4+n$ manifold and we set
\bea
&&
g_{\mu\nu}= {\rm diag} \left(1, -a^2 \da_{ij}, -b^2 \da_{mn}\right),
\nonumber \\
&&
T_{\mu}^\nu= {\rm diag}~\r \left(1, -\ga \da_i^j,
-\ep \da_m^n\right),
\nonumber\\
&&
a\sim |t|^{\a_1}, ~~ b\sim |t|^{\a_2}, ~~
\phi =\Phi -n \ln b.
\label{410}
\eea
Here $\Phi$ is the unreduced, $(4+n)$-dimensional dilaton field and
we have called $a$ and $b$, respectively, the
external and internal scale factors, while $\ga$ and $\ep$ define the
external and internal equations of state. In the initial,
matter-dominated regime ($\eta<\eta_s$),
the string cosmology equations lead to
\cite{22}:
\beq
r={5 \ga-1\over 1+3\ga^2+n\ep^2-2\ga}-{1\over 2}.
\label{411}
\eeq
In the subsequent dilaton-dominated regime ($\eta>\eta_s$) the
equations give \cite{23}
\beq
\b={1-3\a_1\over 1-\a_1}-{1\over 2}, ~~~~ \a_1^2
={1\over 3}\left(1-n\a_2^2\right),
\label{412}
\eeq
where $\a_1, \a_2$ depend on $\ga$, $\ep$,
and on arbitrary integration constants.
The value $r=-3/2$, required for a flat
low-frequency branch of the spectrum, is thus obtained provided
internal and external pressures are related by:
\beq
n\ep^2 =-3\ga (1+\ga).
\label{413}
\eeq
On the other hand, an appropriate equation of state, motivated by
the self-consistency of this
background with the solutions of the string equations of motion
\cite{22,24}, suggests for $\ga$ the range $-1/3\leq \ga \leq 0$.
This range, together with the condition (\ref{413}),
also guarantees the validity of the so-called dominant energy
condition, $\rho \geq 0$, for the whole duration of the
low-energy pre-big bang phase. Near the singularity,
when the background enters the dilaton-dominated regime,
the kinematics, and then the value of $\b$, depends on the
integration constants. For a particularly simple choice of
such constants ($x_i=0$ in the notation of \cite{22}), one finds
$\a_1 = \sqrt{-\ga /3}$, and the value of $\b$ becomes
completely fixed by $\ga$ as
\beq
\b= { 1 - \sqrt{-3\ga} \over 1 - \sqrt{-\ga/3}}
-{1\over 2}.
\label{417}
\eeq
With $\ga$ ranging from $-1/3$ to $0$,
$\b$ ranges from $-1/2$ to
$1/2$. It is thus always possible, even in this simple
example, to implement the condition $\b<1$, in such a way as
to satisfy the properties required by the allowed region
of Fig. 2.
\section{Conclusion}
\label{V}
In this paper we have discussed, in the context of the pre-big bang
scenario, the possible consistency of a pseudoscalar origin of the
large-scale anisotropy, induced by the fluctuations of
non-relativistic Kalb--Ramond axions, with masses up to the $100$
MeV range (higher masses are not allowed by the requirement that
the axions do not decay into photons before the present epoch).
The enhancement of the low-energy tail of the axion
spectrum, due to their mass, has been shown to be possibly
balanced by the depletion induced by a steeper slope at high
frequency. We have provided an explicit example of background
that satisfies the low-energy string cosmology equations, and leads
to an axion spectrum compatible with the above requirements.
The discussion of the reported example is incomplete in many
respects. For instance, the class of models that we have
considered could be generalized by the inclusion of additional
cosmological phases; also, an additional reheating subsequent to
the pre-big bang $\ra$ post-big bang transition could dilute the
produced axions, and relax the critical density bound; and so on.
In this sense, the results discussed in Sect. \ref{IV} are to be taken
only as indicative of a possibility.
In this spirit, the main message of this paper is that in the context
of the pre-big bang scenario there is no fundamental physical
obstruction against an axion background that fits consistently the
anisotropy observed by COBE, with ``realistic" masses in the
expected range of conventional axion models \cite{5} --\cite{9}.
For a given axion mass, the corresponding anisotropy is only a
function of the parameters of the pre-big bang models. If the
axion mass were independently determined, the measurements
of the CMB anisotropy might be interpreted, in this context, as
indirect observations of the properties of a very early cosmological
phase, and might provide useful information about the
high-curvature, strong-coupling regime of the string cosmology
scenario.
\acknowledgements
We are grateful to Ruth Durrer and Mairi Sakellariadou for helpful
discussions.
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