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\begin{document}
%%% start CERN preprint title page %%%%%%%%%%%%%
\begin{titlepage}
\begin{flushright}
DFTT-32/98\\
gr-qc/9806073
\end{flushright}
\vspace{2 cm}
\begin{center}
\Large\bf Weighing the String Mass \\
with the COBE Data
\end{center}
\vspace{1.5cm}
\begin{center}
M. Gasperini\\
{\sl Dipartimento di Fisica Teorica, Universit\`a di Torino,}\\
{\sl Via P. Giuria 1, 10125 Turin, Italy}\\
and\\
{\sl Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, Italy}\\
\end{center}
\vspace{1.5cm}
\begin{abstract}
\noi
In the context of the pre-big bang scenario the large-scale CMB
anisotropy can be seeded by a primordial background of very light (or
massless) axion fluctuations. In that case the slope of the temperature
anisotropy spectrum, allowed by present observations, defines an allowed
range of values for the string mass scale. Conversely, from the
theoretical expected value of the string scale we can predict the slope
of the anisotropy spectrum. In both cases there is a remarkable
agreement between observations and theoretical expectations.
\end{abstract}
\vspace{1.5cm}
\begin{center}
------------------------------
\vspace{1.5cm}
Published in \\
{\sl Proc. of the Euroconference ``Fifth Paris Cosmology
Colloquium"}\\
Observatoire de Paris, 3-5 June 1998 --
Eds. H. J. De Vega and N. Sanchez\\
(Observatoire de Paris Publications, 1999) p. 317-330
\end{center}
\vspace{1.5cm}
\vfill
\end{titlepage}
%\thispagestyle{empty}
%\vbox{}
%\newpage
%%% end CERN preprint title page %%%%%%%%%%%%%
\normalsize\textlineskip
\thispagestyle{empty}
\setcounter{page}{1}
%\copyrightheading{} %{Vol. 0, No. 0 (1993) 000--000}
\vspace*{0.11truein}
\fpage{1}
\centerline{\bf WEIGHING THE STRING MASS WITH THE COBE DATA}
\vspace*{0.27truein}
\centerline{\footnotesize MAURIZIO GASPERINI}
\vspace*{0.015truein}
\centerline{\footnotesize\it Dipartimento di Fisica Teorica,
Universit\`a di Torino,}
\baselineskip=10pt
\centerline{\footnotesize {\it Via P. Giuria 1, 10125, Turin, Italy}}
\baselineskip=10pt
\centerline{\footnotesize and {\it Istituto Nazionale di Fisica Nucleare,
Sezione di Torino, Turin, Italy}}
\vspace*{0.3truein}
\abstracts
{In the context of the pre-big bang scenario the large-scale CMB
anisotropy can be seeded by a primordial background of very light (or
massless) axion fluctuations. In that case the slope of the temperature
anisotropy spectrum, allowed by present observations, defines an allowed
range of values for the string mass scale. Conversely, from the
theoretical expected value of the string scale we can predict the slope
of the anisotropy spectrum. In both cases there is a remarkable
agreement between observations and theoretical expectations. }
{}{}
\vspace*{0.225truein}
\pub{DFTT-32/98;~~~~~~~~~~ E-print Archives: gr-qc/9806073}
\vspace*{0.8pt}\textlineskip
\textheight=7.8truein
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\alph{footnote}}
\vspace*{0.125truein}
%\runninghead{Introduction} {Introduction}
\renewcommand{\theequation}{1.\arabic{equation}}
\setcounter{equation}{0}
\section{Introduction}
\label{sec:1}
\noindent
The aim of this paper is to review, and briefly discuss,
a possible mechanism for generating the large-scale CMB anisotropy,
based on a primordial background of axion fluctuations acting as
seeds for scalar metric perturbations\cite{1,2,3}.
Such a mechanism is particularly appropriate
to pre-big bang models\cite{4}
formulated in a string cosmology context, since in that case it seems
difficult\cite{5} to generate the observed anisotropy through the
standard inflationary mechanism. Let me explain why.
At very large angular scales, the temperature anisotropy spectrum is
determined by the metric fluctuation spectrum $\Phi_k$ through the
well-know Sachs-Wolfe (SW) effect\cite{6}:
\beq
\left(\Da T\over T\right)_k \sim \Phi_k .
\label{11}
\eeq
Metric
fluctuations, directly amplified by the accelerated evolution of the
background, have a spectrum that depends on the value of the
Hubble scale at the time of horizon crossing,
\beq
\Phi_k \sim \left(H\over M_p\right)_k
\label{12}
\eeq
($M_p$ is the Planck mass). In the standard de Sitter (or quasi-De
Sitter) inflationary scenario $H$ is constant in time, so that the
spectrum is scale invariant. A typical normalization of the
spectrum, corresponding to inflation occurring roughly at the GUT
scale,
\beq
{H\over M_p} \sim {{\rm GUT~curvature~scale \over PLANCK~scale}}
\sim 10^{-5},
\label{13}
\eeq
is thus perfectly consistent with the anisotropy observed
at the present horizon scale, $\Da T/T \sim 10^{-5}$, and with the fact
that the spectrum is scale-invariant.
Why this simple mechanism does not work in a string cosmology
context? In string cosmology models the curvature scale grows with
time, so that the spectrum of metric fluctuations (\ref{12}) grows
with frequency. In addition, the natural inflation scale corresponds to
the string scale, so that the normalization of the spectrum, at the
end-point frequency $k_1$, is controlled by the ratio
\beq
\left(H\over M_p\right)_{k_1}
\sim {{\rm STRING~curvature~scale \over PLANCK~scale}} \sim
10^{-2}.
\label{14}
\eeq
We are thus led to the situation qualitatively illustrated in Fig. 1.
For pre-big bang models the slope of the spectrum is too steep, and
the normalization too high, to be compatible with COBE
observations\cite{7}. The slope is so steep, however, that the
contribution of metric fluctuations to $\Da T/T$ is certainly negligible
at the COBE scale. So, on one hand there is no contradiction with
observations, namely the COBE data cannot be used to rule out
pre-big bang models. On the other hand, the problem remains: how to
explain the observed anisotropy if the contribution of metric
fluctuations is so small?
\begin{figure}[htb]
\vspace{10cm}
\special{psfile=f1.ps angle=0 hscale=75 vscale=75 voffset=-50
hoffset=-40}
{\caption{\label{fig:f1}
{\sl The contribution of primordial metric fluctuations to $\Da T/
T$, in the pre-big bang scenario, is expected to be negligible at
the COBE scale. }}}
\end{figure}
A possible answer to this question comes from the observation that
the previous argument applies to the primordial spectrum of metric
fluctuations, {\em directly} amplified by the accelerated
evolution of the background. There is an additional indirect
contribution to the final metric perturbation spectrum, however,
arising from the quantum fluctuations of other fields (let me call
them, generically, $\sg$), amplified during inflation. Even if such
fluctuations are eventually negligible as sources of the metric
background, $\r_\sg \ll \r_c$, their inhomogeneous stress tensor
generates metric fluctuations according to the standard gravitational
equations, and they can act as ``seeds" for temperature anisotropies
through the SW effect, as before:
\beq
{\r_\sg \over \r_c} \sim \Phi \sim {\Da T\over T}
\label{15}
\eeq
Why the seed mechanism can work? First of all because, unlike metric
perturbations, there are fields whose fluctuations can be amplified
with a flat spectrum even in the context of the pre-big bang
scenario.
Second because the contribution to $\Da T/T$ is quadratic in the seed
fields, and not linear like in case of metric perturbations. So, even if
the amplitude of seed fluctuations is still normalized at the string
curvature scale, the square of the amplitude is
not very far from the expected value $10^{-5}$:
\beq
{\Da T \over T} \sim \Phi \sim \sg^2
\sim \left({\rm STRING~curvature~scale \over PLANCK~scale}\right)^2
\sim 10^{-4}.
\label{16}
\eeq
In addition, we must recall that the string normalization is
imposed at the end-point of the spectrum\cite{8} (roughly, at the
GHz scale), while COBE observations constrain the spectrum at the
present horizon scale ($\sim 10^{-18}$Hz). A very small (blue) tilt of
the seed field spectrum is thus enough to make compatible the COBE
normalization and the string normalization, as illustrated in Fig. 2.
\begin{figure}[htb]
\vspace{10cm}
\special{psfile=f2.ps angle=0 hscale=75 vscale=75 voffset=-50
hoffset=-40}
{\caption{\label{fig:f2}
{\sl The amplitude of metric fluctuations induced by seeds may be
consistent both with the COBE and the string normalization of the
spectrum. }}}
\end{figure}
The basic question now becomes: are there fields, in the context of
the pre-big bang scenario, whose fluctuations can be amplified with a
flat enough spectrum, so as to seed metric fluctuations and to fit
consistently the observed anisotropy?
In the following Sections I will present two possible examples that
seem to be promising: the case of massless and massive axion
fluctuations.
\vskip 1 cm
\renewcommand{\theequation}{2.\arabic{equation}}
\setcounter{equation}{0}
\section{Massless axions as seeds of large-scale anisotropy}
\label{sec:2}
\noindent
A first possible candidate for seeding the large-scale anisotropy is a
stochastic background of massless pseudoscalar fluctuations\cite{9}.
I will
take, as a particular example, the so-called ``universal" axion of
string theory, namely the four-dimensional dual $\sg$ of the
Kalb-Ramond antisymmetric tensor $H_{\mu\nu\a}$, appearing in the
low-energy string effective action:
\bea
&&
S=-\int d^4x \sqrt{-g} e^{-\phi}\left[R+(\pa_\mu \phi)^2 -{1\over 12}
H_{\mu\nu\a}^2 \right], \nonumber\\
&&
H^{\mu\nu\a} =e^{\phi} \ep^{\mu\nu\a\b} \pa_\b \sg .
\label{21}
\eea
The whole discussion can be applied, however, to any type of
pseudoscalar fluctuation amplified with a flat enough primordial
energy spectrum.
I will concentrate my discussion on three points. First, I have to show
that such axion fluctuations can be amplified with a final
scale-invariant distribution of their spectral energy density
$\Om_\sg$:
\beq
\Om_\sg(k,\eta) = {d\r_\sg(k,\eta)\over \r_c d \ln k} \sim
{\rm scale~invariant}.
\label{22}
\eeq
Second, I will show that the scalar metric fluctuation on a given scale
$k$, at the time the scale re-enters the horizon, is precisely
determined by the axion energy distribution evaluated at the
conformal time of re-entry, $\eta_{re} \simeq k^{-1}$:
\beq
\Phi_k(\eta_{re}) \sim \Om_\sg (k, \eta_{re}).
\label{23}
\eeq
Third, I will show that the dominant contribution to the SW effect
comes from a scale at the time it re-enters the horizon, so that the
final temperature spectrum exactly reproduces the primordial seed
spectrum:
\beq
\left(\Da T\over T\right)_k \sim \Phi_k(\eta_{re}) \sim \Om_\sg (k,
\eta_{re}).
\label{24}
\eeq
These last two results are far from being trivial, being a consequence
of the particular time-dependence of the Bardeen spectrum induced
by axion fluctuations (these results
do not apply, for instance, to electromagnetic
fluctuations\cite{1}). I will give in this paper only a sketch of the
arguments leading to the above results. A detailed derivation can be
found in Refs. [1,2].
1) The possibility of a
flat axion spectrum\cite{9} can be easily checked
by considering the axion perturbation equation written in terms of the
canonical variable, $\psi=\sg \xi$, and of the ``pump" field
$\xi=ae^{\phi/2}$ (here $\phi$ is the dilaton, and $a$ is the
four-dimensional scale factor of the string frame metric). In the
conformal time gauge one obtains from eq. (\ref{21}) the effective
action
\beq
S= \int d^3x d\eta~ a^2 e^\phi \left(\sg^{\prime 2} +\sg
\nabla^2\sg \right),
\label{25}
\eeq
and the perturbation equation (for the Fourier modes $\psi_k$)
\beq
\psi_k'' +\left[k^2-(\xi''/\xi)\right]\psi_k =0
\label{26}
\eeq
(the prime denotes differentiation with respect to the conformal time
$\eta$). Assuming, for instance, a power-law evolution of the pre-big
bang background, $\xi \sim |\eta|^\a$, the perturbation equation
reduces to a Bessel equation, and the normalized solution can be
written in terms of the Hankel functions $H_\nu$ as
\beq
\psi_k = \eta^{1/2} H_\nu^{(2)}(k\eta), ~~~~~~~~~~~~~~~~~
\nu= |\a-1/2|,
\label{26a}
\eeq
where the Bessel index $\nu$ depends on the kinematic of the
background. The spectral energy density, for modes re-entering in the
radiation era, depends finally on the background as
\beq
\Om_\sg (k) \sim k^{3-2\nu}.
\label{27}
\eeq
If we take now a very simple, higher-dimensional but isotropic
vacuum solution of the string cosmology equations, in $d=3+n$ spatial
dimensions\cite{10},
\beq
a \sim |\eta |^{-1/(1+\sqrt{d})}, ~~~~~~~~~~~~~~~~~
e^{\phi/2} \sim |\eta |^{-{3+\sqrt d \over
2(1+\sqrt d)}},
\label{28}
\eeq
we find that the spectral index depends on $d$,
\beq
3-2\nu= {\sqrt d -3\over 1+\sqrt d},
\label{29}
\eeq
and that the spectrum may be flat ($3-2\nu=0$), in particular\cite{11},
for $d=9$.
It is important to stress that a flat spectrum is possible, in the
previous background, for axion
fluctuations, but impossible for metric fluctuations which are
characterized by a different pump field\cite{12}, $\xi=a e^{-\phi/2}$.
With this pump field, in the same background (\ref{28}), both the
power $\a$ and the Bessel index $\nu$ are independent of $d$, and
the spectrum is always growing with a cubic slope, in any number of
dimensions: \beq
\xi = ae^{-\phi/2} \sim |\eta|^{1/2}, ~~~~~~~~~~~~~~~~~~
3-2\nu=3.
\label{210}
\eeq
2) Let us now compute the spectrum of metric perturbations seeded
by a flat, primordial distribution of axion fluctuations. Define, as
usual, the power spectrum of the Bardeen potential, $P_\Phi(k)$, in
terms of the Fourier transform of the two-point correlation function:
\beq
\int {d^3 k\over (2 \pi k)^3} e^{i {\bf k}\cdot({\bf x}-{\bf x'})}
P_\Phi(k) =\langle \Phi(x) \Phi(x')\rangle
\label{211}
\eeq
(the brackets denote spatial average, or expectation value if
perturbations are quantized). The square root of the two-point
function, evaluated at a comoving distance $k^{-1}$, represents the
typical amplitude of fluctuations on a scale $k$:
\beq
\left(\langle \Phi(x) \Phi(x')\rangle\right)^{1/2}_{|x-x'|=k^{-1}}
\sim k^{3/2} |\Phi_k| .
\label{212}
\eeq
Define also the power spectrum of the seed stress tensor, in the same
way (no sum over $\mu$, $\nu$):
\beq
\int {d^3 k\over (2 \pi k)^3} e^{i {\bf k}\cdot({\bf x}-{\bf x'})}
P_\mu^\nu(k) =\langle T_\mu^\nu(x) T_\mu^\nu(x')\rangle
-\langle T_\mu^\nu (x)\rangle ^2 .
\label{213}
\eeq
Metric fluctuations and seed fluctuations are related by the
cosmological perturbation equations. By taking into account the
important contribution of the off-diagonal components of the axion
stress tensor one finds, typically, that Bardeen spectrum and axion
energy density spectrum are related by\cite{1}:
\beq
P_\Phi^{1/2}(k) \sim G \left(a \over k\right)^2 P_\r^{1/2}(k),
\label{214}
\eeq
where
\bea
\int {d^3 k\over (2 \pi k)^3} e^{i {\bf k}\cdot({\bf x}-{\bf x'})}
P_\r(k) &=&\langle \r_\sg(x) \r_\sg(x')\rangle
-\langle \r_\sg(x) \rangle^2\nonumber \\
&\sim & \langle \sg^{\prime 2}(x) \sg^{\prime 2}(x')\rangle
-\langle \sg^{\prime 2}(x)\rangle^2 + ...
\label{215}
\eea
It may be interesting to note that the two-point correlation function
of the energy density becomes a four-point function of the seed
field, since the energy is quadratic in the axion field.
Using the condition of stochastic average for the axion field\cite{1},
\beq
=\langle \sg'({\bf k}, \eta)\sg^{\prime \ast}
({\bf k'}, \eta)\rangle= (2 \pi)^3 \da^3 (k-k') \Sigma(
{\bf k}, \eta),
\label{216}
\eeq
we find that the energy density spectrum reduces to a convolution of
Fourier transforms,
\beq
P_\r(k) \sim {k^3\over a^4} \int d^3p~ \Sigma(p)
\Sigma(|k-p|)+ ... ~~,
\label{217}
\eeq
which is dominated by the region $p\eta \sim 1$ for a flat enough
axion spectrum\cite{1,2}. By expressing the convolution through the
spectral energy density $\Om_\sg$, and evaluating the Bardeen
potential at the time of re-entry $\eta_{re} \sim k^{-1}$, we are led
finally to relate the Bardeen spectrum and the axion spectrum as
\beq
P_\Phi^{1/2} (k, \eta_{re})
\sim k^{3/2} \left|\Phi_k(\eta_{re}\right|
\sim \Om_\sg(k, \eta_{re}).
\label{218}
\eeq
In the next (and last) step of my discussion I will explain why
we are interested in metric fluctuations evaluated at the time of
re-entry.
3) Let us come back, finally, to the seed contribution to $\Da T
/T$. In the multipole expansion of the temperature anisotropies,
\beq
\left\langle{\delta T\over T}({\bf
n}){\delta T\over T}({\bf n}') \right\rangle_{{~}_{\!\!({\bf n\cdot
n}'=\cos\vartheta)}} ~~=~~~~
{1\over 4\pi}\sum_\ell(2\ell+1)C_\ell P_\ell(\cos\vartheta)~,
\label{219}
\eeq
the coefficients $C_\ell$, at very large angular scales
($\ell \ll 100$), are determined by the SW effect as
follows\cite{1,13}:
\beq
C_\ell^{SW} ={2\over \pi}\int
d~(\ln k) \left \langle \left[\int^{k\eta_0}_{k\eta_{dec}} d(k\eta)~
k^{3/2} (\Psi -\Phi)({\bf k},
\eta)j_{\ell}^{\prime}\left(k\eta_0-k\eta \right)\right]^2
\right\rangle .
\label{220}
\eeq
Here $\Phi$ and $\Psi$ are the two-independent components of the
gauge-invariant Bardeen potential, and
$j_\ell$ are the spherical Bessel functions. Eq. (\ref{220}) takes into
account both the
``ordinary" and the ``integrated" SW contribution, namely the
complete distortion of the geodesics of the CMB photons (due to
shifts in the gravitational potential), from the time of decoupling
$\eta_{dec}$ down to the present time $\eta_0$. By inserting the
Bardeen potential determined by the axion field, one now finds that
the time integral is dominated by the region $k\eta \sim 1$. Using eq.
(\ref{218}) we obtain
\bea
C_\ell^{SW} &\sim & \int d (\ln k)~ k^{3/2} |\Phi_k(\eta_{re})|^2
j_\ell (k\eta_0)|^2
\nonumber\\
&\sim & \int d (\ln k)~ \Om_\sg^2 (k, \eta_{re})
\label{221}
\eea
($j_\ell$ are the spherical Bessel functions).
Here is why it was important to evaluate the Bardeen spectrum
at the time of re-entry, $\eta_{re} \sim k^{-1}$.
From the final expression that gives the multipole coefficients
in terms of the axion spectral distribution\cite{1,2}
we can extract, in
particular, the value of the quadrupole coefficient $C_2$:
\beq
C_2 \simeq \Om_\sg ^2 (k_0, \eta_0) \simeq
\left(M_s\over M_p\right)^4 \left(k_0\over k_1\right)^{n-1},
\label{222}
\eeq
where $n$ is the spectral index (sufficiently near to $1$)
characterizing the primordial axion distribution, $k_0$ is the comoving
scale of the present horizon, and $k_1$ the end-point of the spectrum,
namely the maximal amplified comoving frequency.
The peak amplitude of the axion spectrum, at the end-point
frequency, is controlled by the fundamental ratio between string and
Planck mass, $M_s/M_p$. The quadrupole coefficient, on the other
hand, is presently determined by COBE as\cite{14}
\beq
C_2 = (1.9 \pm 0.23) \times 10^{-10}.
\label{223}
\eeq
This experimental value, inserted into eq. (\ref{222}), implies a
relation between the string mass and the spectral index of the
temperature anisotropy, which is illustrated in Fig. 3.
\begin{figure}[htb]
\epsfxsize=11cm
\centerline{\epsfbox{f3.ps}}
\centerline{\parbox{11.5cm}{\caption{\label{fig:f3}
{\sl Relation between string mass and spectral index of CMB
anisotropy, obtained by combining the COBE normalization of the
spectrum with the prediction of an axionic seed model of the
anisotropy. The dashed lines correspond to the experimentally
allowed range of the spectral index. The shaded area corresponds to
the theoretically expected value of the string scale. }}}}
\end{figure}
The experimentally allowed range of the spectral index\cite{15} is, at
present,
\beq
1\leq n \leq 1.4
\label{224}
\eeq
(I have excluded the allowed values $0.8 \leq n\leq 1$, which
would imply in our case an
over-critical axion production). It is remarkable that
the corresponding allowed range of the string mass is perfectly
compatible with theoretical expectations\cite{16},
\beq
0.01~\laq~ M_s/M_p~ \laq~ 0.1
\label{225}
\eeq
(see Fig. 3). Conversely, the above expected range for $M_s$ implies a
spectral index around $1.1$ or $1.2$ (see again Fig. 3), which is also
in very good agreement with observations.
It must be stressed, however, that eq. (\ref{222}) is valid in the
assumption that the inflation scale of pre-big bang models exactly
coincides with the string mass scale. If the two scales
were slightly different, an additional source of
uncertainty would be introduced into the relation (\ref{222}).
\vskip 1 cm
\renewcommand{\theequation}{3.\arabic{equation}}
\setcounter{equation}{0}
\section{Massive axions as seeds of large-scale anisotropy}
\label{sec:3}
\noindent
Up to now the discussion was devoted to massless pseudoscalar
perturbations. It is likely, however, that axions become massive in
the post-inflationary era: it is thus important to consider this
possibility also.
Let me say immediately that also in the massive case the seed
mechanism can work\cite{3}, and let me introduce the main
differences between the massless and the massive case.
A first difference is the relation between the Bardeen potential
$\Phi$ and the
axion energy density $\r_\sg$. In the massless case the perturbation
equations, taking into account the important contribution of all the
off-diagonal terms of the axion stress tensor, lead to\cite{1}
\beq
\Phi_k \sim G \left(a\over k\right)^2 \r_\sg (k).
\label{31}
\eeq
In the massive case, on the contrary, the axion stress tensor can be
approximated as a diagonal, perfect fluid stress tensor, and we
obtain\cite{3}
\beq
\Phi_k \sim G a^2 \eta^2 \r_\sg (k).
\label{32}
\eeq
Also, in the massless case the convolution (\ref{217})
for the axion energy density
is dominated by the region\cite{1,2} $p \sim \eta^{-1}$, while in the
massive case by\cite{1,3} $p\sim k$. In the massless case the
integrated SW effect is the dominant one\cite{1}, while in the
massive case the ordinary SW effect is dominant\cite{3}.
In spite of all these differences, the final result is similar, and in
both cases the quadrupole coefficient is determined by the axion
spectral energy density as
\beq
C_2 \sim \Om_\sg^2 (k_0, \eta_0).
\label{33}
\eeq
In the massive case, however, the axion spectrum is affected by
non-relativistic corrections. In order to include such corrections, it
is convenient\cite{1} to distinguish between modes that become
non-relativistic ($k/a H$), and modes that become non-relativistic when they are still
outside the horizon ($k/a {T_m\over {\rm eV}}.
\label{35}
\eea
Here $H_{eq} \sim 10^{-27}$ eV is the Hubble scale at the time of
matter-radiation equilibrium, and $T_m$ is the temperature scale of
mass generation (for instance, $T_m \sim 100$ MeV if axions become
massive at the epoch of chiral symmetry breaking). In both cases the
slope is the same as that of the massless spectrum, $3-2\nu$.
The amplitude of the spectrum now depends on the axion mass, and
the constraint imposed by the COBE normalization (\ref{223})
necessarily bounds the allowed range of masses.
This might represent a problem, in general: since the slope cannot be
too steep at low frequency (according to eq. (\ref{224})), the allowed
mass could be too low to be compatible with realistic axion models.
This conclusion, based on the effect illustrated in Fig. 4,
refers however to a relativistic spectrum characterized by a constant
slope. On the other hand, it is quite easy to imagine, and to implement
in practice, a model of background in which the relativistic axion
spectrum is flat enough at low frequency (as required by a fit of the
large-scale anisotropy), and much steeper at high frequency. A simple
example is illustrated in Fig. 5, where I have compared two
spectra. The first one is flat everywhere, except for non-relativistic
corrections. The second one is flat at low frequency, and steeper at
high frequency. It is evident that the steeper and the longer the
high-frequency branch of the spectrum, the larger is the suppression
of the amplitude at low frequency, and the larger is the axion mass
allowed by the COBE normalization at $\om=\om_0$.
\begin{figure}[htb]
\vspace{10cm}
\special{psfile=f5.ps angle=0 hscale=75 vscale=75 voffset=-50
hoffset=-40}
{\caption{\label{fig:f5}
{\sl Two examples of axion spectra with non-relativistic corrections.
Note the common normalization at the end-point frequency $\om_1$,
in spite of the different slopes in the different frequency
regimes. }}}
\end{figure}
We have analysed this possibility\cite{3}
in an explicit two-parameter model of background, including exact
solutions of the low-energy string cosmology equations with
classical string sources. The allowed region in parameter space turns
out to be consistent with a very wide range of axion masses, from the
equilibrium scale $m\sim 10^{-27}$ eV up to $m \sim 100$ MeV (higher
masses are not acceptable, because of the axion decay into photons).
We can say, therefore, that there is no fundamental incompatibility
between a fit of the large-scale anisotropy, and an axion mass in the
expected range of conventional axion models.
\vskip 1 cm
\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
\section{Conclusion}
\label{sec:4}
\noindent
A stochastic cosmic background of pseudoscalar fluctuations,
produced with a flat enough primordial spectrum, can seed the
observed CMB anisotropy {\em at very large angular scales}. The
end-point normalization of the spectrum imposed by the string
cosmology scenario, and the observational normalization at the COBE
scale, are consistent both for massless and massive fluctuations.
In spite of these promising results, it should be clearly stressed that
this approach to CMB anisotropy is only the first step of a much longer
research program, still to be implemented. Many important questions
are still waiting for an answer, among which the crucial one, in my
opinion, concerns the possible differences {\em at smaller angular
scales} between this mechanism and the standard inflationary
mechanism of anisotropy production. In particular: is the statistic
non-Gaussian? are there shifts in the position of the Doppler peak?
etc ...
The answer to these questions is at present unclear, but we hope to
provide answers in future papers.
\vspace{1cm}
{\it Acknowledgments:\/} I am very grateful to Ruth Durrer, Mairi
Sakellariadou and Gabriele
Veneziano for a fruitful collaboration which led to the
results reported in this paper.
I wish to thank also the Hector De Vega and Norma Sanchez for
their kind invitation, and for the perfect organization of this
interesting Euroconference.
\vskip 1 cm
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