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\rightline{BA-TH/01-412}}}
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\rightline{RCG 01/10}}}
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{\large\bf\centering\ignorespaces
Detecting a relic background of scalar waves with LIGO\\
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M. Gasperini${}^{(1,2)}$
and C. Ungarelli${}^{(3)}$
\par}
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{\small\it\centering\ignorespaces
${}^{(1)}$
Dipartimento di Fisica, Universit\`a di Bari, \\
Via G. Amendola 173, 70126 Bari, Italy\\
${}^{(2)}$
Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
Bari, Italy \\
${}^{(3)}$
Relativity and Cosmology Group\\
School of Computer Science and Mathematics,
University of Portsmouth,\\
Portsmouth P01 2EG, England
\par}
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\begin{abstract}
We discuss the possible detection of a stochastic background of massive,
non-relativistic scalar particles, through the cross correlation of the two
LIGO interferometers in the initial, enhanced and advanced
configuration. If the frequency corresponding to the mass of the scalar
field lies in the detector sensitivity band, and the non-relativistic branch
of the spectrum gives a significant contribution to energy density
required to close the Universe, we find that the scalar background can
induce a non-negligible signal, in competition with a possible signal
produced by a stochastic background of gravitational radiation.
\end{abstract}
\par\egroup
\thispagestyle{plain}
\pacs{}
\section{Introduction}
\label{S1}
In the next few years many gravitational antennas will be collecting
data, as the new interferometric detectors
(GEO, LIGO, TAMA, VIRGO) \cite{1} will join the resonant detectors
(ALLEGRO, AURIGA, EXPLORER, NAUTILUS, NIOBE) \cite{2} already in
operation, covering a frequency band from $\sim 10$~Hz up to $1$KHz.
Through the cross-correlation of these antennas it will be
possible to search for a stochastic background of
primordial gravitational radiation \cite{3}, \cite{3a}.
The detection of a relic background of gravitational
waves would allow us to reconstruct the very early stages of
cosmological evolution~\cite{4} and even an upper limit would be
significant, since there are predictions of very high relic
backgrounds \cite{5} in principle detectable already by second
generation interferometers \cite{6} (see however \cite{6a}).
A generic prediction of unified theories (such as supergravity and
superstring theories) is the existence of a gravitational multiplet which
includes, beside the usual spin-2 graviton, scalar components. In a
cosmological context, as well as the production of a primordial
background of gravitational waves, it is therefore worth investigating
the possible production of a relic background of scalar waves (for
instance, dilatons \cite{7}). In particular, if the mass of the scalar field is
small enough ($ m \laq 100$ MeV), such a background would be present
even today, and it could be accessible to direct experimental
observation. It seems thus appropriate to estimate the sensitivity of
gravitational antennas also to scalar waves.
A relic background of scalar particles can interact with a gravitational
antenna in two ways: either indirectly, through the geodesic coupling
of the detector to the induced background of scalar metric
fluctuations, or even directly \cite{8}, through the effective scalar
charge of the detector (such a direct coupling cannot be absorbed into
the metric interactions, in an appropriate frame, if the scalar
charge is non-universal). Up to now, the sensitivity to scalar waves has
been mainly studied using the indirect coupling of the antennas to the
scalar part of metric fluctuations, and considering massless scalar fields
(see for instance \cite{9}). Under those assumptions, the only difference
with respect to the case of standard gravitational radiation is
represented by the polarisation tensor of the scalar wave~\cite{10}.
However, if the detector response tensor is characterised by a
symmetric $3\times 3$ trace-free tensor (such as the differential mode
of an interferometer), the sensitivity to a scalar wave of momentum $p$
and energy $E$ generated by a massive scalar field is suppressed
by a factor~\cite{10} $(p/E)^4$ with respect to the sensitivity to an
ordinary gravitational wave\footnote{This factor coincides
with the suppression factor for the cross section of a
resonant bar to longitudinal massive scalar modes \cite{8}.}. This
suppression is ineffective only for modes which are
ultrarelativistic in the sensitivity band of the detector,
$m \ll E= (p^2+m^2)^{1/2} \simeq p$; this condition, however, occurs
when the frequency corresponding to the mass of the scalar field is much
smaller than the typical frequency of the detector, i.e. when $m \ll E_0
\sim 10 -10^3$ Hz. The scalar field would be then associated to
long-range interactions, and its coupling to the mass of the detector
should be highly suppressed (with respect to the standard gravitational
coupling), in order to agree with the existing tests of the equivalence
principle and of macroscopic gravitational interactions \cite{11}.
Hence, as far as interferometric antennas are concerned, the possible
detection of a background of (massive or massless) scalar waves would
seem to be strongly unfavoured with respect to the detection of a
graviton background (in the
literature, indeed, the possible detection of scalar waves is usually
demanded to future resonant detectors of spherical type \cite{12}).
Nevertheless, it is important to notice that, when the mass of the scalar
field corresponds to a frequency which is in the detector sensitivity
band, it is possible to obtain a resonant response also from the
non-relativistic part of the scalar waves spectrum \cite{13}. On the
other hand, unlike a relativistic background of massless particles (like
gravitons), the non-relativistic part of a relic background is not
constrained by the nucleosynthesis bound, and could saturate the critical
energy density required to close the Universe. As
suggested in in \cite{13}, if the non-relativistic branch of the spectrum is
peaked at $p \sim m$, the polarisation suppression factor and the
weakness of the scalar coupling could be compensated by the high relic
density, and such a background of scalar waves could be a
potential source for interferometric detectors.
The aim of this paper is to discuss the possible detection
of a relic stochastic background of massive scalar particles with
the two LIGO interferometers (including the enhanced and advanced
configurations). The paper is organised as follows. In Section \ref{S2} we
recall the general expression for the optimised signal-to-noise ratio (SNR), obtained by cross-correlating two detectors, with respect to a stochastic
background of massive scalar waves. In Section
\ref{S3} we apply this result to the case of the
differential mode of an interferometer,
taking into account both the geodesic coupling to the
scalar part of the metric fluctuations, and the direct coupling to the
scalar field. In Section \ref{S4} we discuss some examples, and
we show in particular that a relic background of non-relativistic scalar
particles, whose energy density provides a significant fraction of
the critical energy density, can induce
a non-negligible signal in the cross-correlation of the LIGO
interferometers (in the enhanced and advanced configurations), if
the frequency corresponding to the mass of the scalar field is in the
sensitivity band of the detectors. The main results of this paper are
finally summarised in Section \ref{S5}.
\section{Signal-to-noise ratio}
\label{S2}
We will consider a stochastic background of massive
scalar particles, described in terms of a scalar field
$\phi(\vec{x},t)$, and characterised by a dimensionless
spectrum $\Omega(p)$,
\be
\Omega(p)= {1\over \rho_c} {d \rho\over d \ln p},~~~~~~~~~~~~
\rho_c={3H_0M_P^2\over 8 \pi},
\label{21}
\ee
where $p$ is the momentum, $\rho$ is the scalar field energy density,
$\rho_c$ is the critical energy density, $H_0$ the present value
of the Hubble parameter, and $M_P$ the Planck mass. We shall assume
that the scalar field is coupled to the mass of the detector with
gravitational strength (or weaker), and that the spectrum extends in
momentum space from $p_0$ to $p_1$. The low frequency cut-off
$p_0$ may be zero (for growing spectra), or the present Hubble scale
(for decreasing spectra), while $p_1$ is an high-frequency cut-off which
depends on the details of the production mechanism.
Our starting point is the expansion of the scalar field in the
momentum space,
\be
\phi(t,\vec{x})=\int_{0}^{\infty}dp\,
\int\,d^2\hat{n}\left\{
e^{2\pi\,i[E(p)t-p\hat{n}\cdot\vec{x}]}{\phi}(p,\hat{n})
+\mbox{h.c.}\right\}\,,
\label{decomp}
\ee
where $\hat{n}$ is a unit vector specifying the propagation direction,
$\vec{p}=\hat{n}\,p$ is the momentum vector,
and the energy $E(p)$ of each mode is
\be
E(p)=\sqrt{p^2+(m/2\pi)^2}\,
\ee
(we are adopting ``unconventional" units $h=1$, so that the
proper frequency is simply $f=E$). The
background of scalar waves is assumed to be isotropic, stationary
and Gaussian~\cite{3} and satisfies the following stochastic conditions
\ba
&&\langle{\phi}(p,\hat{n})\rangle=0\,, \nonumber\\
&&\langle{\phi}(p,\hat{n})
{\phi}^*(p',\hat{n}')\rangle=
\delta(p-p')\delta^2(\hat{n}-\hat{n}')\Phi (p)\,,
\label{stoc_prop}
\ea
where, using the explicit definition of $\Omega(p)$,
\be
\Phi (p)=\frac{3\,H^2_0\Omega(p)}{8\pi^3\,p\,E^2(p)}\,.
\ee
The scalar background induces on the output
of the gravitational detector a strain
$h_{\phi}(t)$, proportional to the value of the scalar field
$\phi(t,\vec{x}_D)$ at the detector
position \cite{10},
\be
h_{\phi}(t)=\int_{0}^{\infty}dp\,
\int\,d^2\hat{n},F_{\phi}(\hat{n})\left\{
e^{2\pi\,i[E(p)t-p\hat{n}\cdot\vec{x}_D]}{\phi}(p,\hat{n})
+\mbox{h.c.}\right\}\,,
\label{26}
\ee
where $\vec{x}_D$ is the position of the detector centre of mass, and
$F_{\phi}$ is the antenna pattern. In particular,
\be
F_{\phi}(\hat n)=q e_{ab}\,D^{ab}\,
\label{27}
\ee
where $e_{ab}$ is the polarisation tensor of the scalar wave, $D_{ab}$ is
the detector response tensor, and $q$ is the effective coupling
strength of the scalar field to the detector. The explicit form of $F_\phi$
will be discussed in the next section.
The optimal strategy to detect a stochastic background requires the
cross-correlation between the output of (at least) two detectors
\cite{3}, with uncorrelated noises $n_i(t)\,,i=1,2$.
Given the two outputs over a total observation time
$T$,
\be
s_i(t)=h^i_{\phi}(t)+n_i(t)\,,\quad i=1,2,
\ee
one constructs a `signal' $S$,
\be
S=\int_{-T/2}^{T/2}dt\,dt's_1(t)s_2(t')Q(t-t')\,,
\ee
where $Q(t-t')$ is a suitable filter function, usually chosen to
optimise the signal-to-noise ratio:
\be
SNR= \langle S\rangle/ \Delta S,
\ee
where $\Delta S^2= \langle S^2\rangle-\langle S\rangle^2 $ is the
variance of $S$. In our case, we can compute the mean value $\langle
S\rangle$ by using the expansion (\ref{26}) in momentum space, the
statistical independence of the two noises (i.e.
$\langle n_1(t) n_2(t')\rangle=0$), and the fact that the noise
and the strain are uncorrelated (i.e. $\langle n_i(t)
h^i_\phi(t')\rangle=0$). By assuming that the observation $T$ is much
larger than the typical intervals $t-t'$ for which $Q \not=0$, we obtain
\be
\langle S\rangle=\frac{H^2_0}{5\pi^2}\,T\,\mbox{Re}\,
\left\{\int_0^{\infty}dp\frac{\tilde{Q}(E(p))
\Omega(p)\gamma(p)}{pE^2(p)} \right\}\,,
\label{SNR}
\ee
where
\be
\tilde{Q}(E(p))=\int_{-\infty}^{\infty} d\tau e^{2\pi\,i(E(p)\tau}Q(\tau)\,,
\ee
and
\be
\gamma(p)=\frac{15}{4\pi}\,\int d^2\hat{n}
e^{2\pi i p\hat{n}\cdot(\vec{x}_{D1}-\vec{x}_{D2})}
F^{1}_{\phi}(\hat{n})F^{2}_{\phi}(\hat{n})
\label{gamma}
\ee
is the so-called overlap reduction function \cite{3}, which
depends on the relative orientation and location of the two detectors.
In~(\ref{gamma}) the normalisation constant
has been chosen so that -- in the massless case -- one
obtains $\gamma(p)=1$ for two coincident and coaligned
interferometers.
Switching to the frequency domain ($E=f, dp/df =f/p$), the mean value
of $S$ can be written as
\be
\langle S\rangle=\frac{H^2_0}{5\pi^2}\,T\,\mbox{Re}
\left\{\int_0^{\infty}df\frac{\theta(f-\tilde{m})\,
\tilde{Q}(f)\Omega(\sqrt{f^2-\tilde{m}^2})\gamma(\sqrt{f^2-
\tilde{m}^2})} {(f^2-\tilde{m}^2)f}
\right\}\,,
\label{SNR1}
\ee
where $\tilde{m}=m/2\pi$, and $\theta$ is the Heaviside step function.
To compute the variance we will assume that for each detector
the noise is much larger in magnitude than the strain induced by the
scalar wave (i.e. $n_i(t)\gg h^i_{\phi}(t)$). One obtains~\cite{3}
\be
\Delta S^2\simeq ~~\simeq
\frac{T}{2}\,\int_{0}^{\infty}df P_1(|f|)P_2(|f|)|\tilde{Q}(f)|^2\, ,
\ee
where $P_i(|f|)$ is the one-sided noise power spectral density of the $i$-th
detector, defined by
\be
\langle n_i(t) n_j(t')\rangle=\frac{\delta_{ij}}{2}\,
\int_{-\infty}^{\infty} df e^{2\pi\,if(t-t')}P_i(|f|)\,.
\ee
Introducing the following positive semi-definite inner product
in the frequency domain,
\be
(a,b)\doteq\mbox{Re}\left\{\int^{\infty}_{0}a(f)b(f)P_1(f)P_2(f)
\right\}\, ,
\ee
the signal-to noise ratio can be written as
\be
(SNR)^2=2\,T\,\left(\frac{H^2_0}{5\pi^2}\right)^2\,
\left[\frac{(\tilde{Q},A)^2}{(\tilde{Q},\tilde{Q})}\right],
\ee
where
\be
A=\frac{\theta(f-\tilde{m})\Omega(\sqrt{f^2-\tilde{m}^2})
\gamma(\sqrt{f^2-\tilde{m}^2})}{(f^2-\tilde{m}^2)f\,P_1(|f|)P_2(|f|)}\,.
\ee
The above ratio is maximal if $\tilde{Q}$ and $A$ are
proportional, i.e. $\tilde{Q}=\lambda\,A$. With this optimal choice the
SNR reads
\be
SNR=
\left(\frac{H^2_0}{5\pi^2}\right)\,\sqrt{2\,T \,{\cal I}},
\ee
where
\be
{\cal I}=\int_{0}^{\infty}dp
\frac{\Omega^2(p)\,\gamma^2(p)}
{P_1(\sqrt{p^2+\tilde{m}^2})\,P_2(\sqrt{p^2+\tilde{m}^2})
(p^2+\tilde{m}^2)^{3/2}\,p^3}.
\label{221}
\ee
For $\gamma (p) = (df/dp) \tilde \gamma (f)$ our result
reduces to expression for the SNR already deduced in \cite{13} (modulo a
different normalisation of the overlap function).
The scalar nature of the background is encoded
into $\gamma(p)$, and will be discussed in the next Section.
\section{Antenna patterns and overlap reduction function}
\label{S3}
Given the spectrum, and the noise power spectral densities of
the two detectors, the computation of the signal-to-noise ratio
requires now the explicit expression of the overlap
reduction function $\gamma(p)$ for a pair of gravitational antennas.
As already pointed out in the Introduction, we shall restrict our
analysis to interferometric detectors. We will also consider
the interaction of the scalar waves with the differential
mode of the interferometer (see e.g.~\cite{10}),
which is described by the following symmetric, trace-free
tensor\cite{14}:
\be
D_{ab}=\frac{1}{2}\,(\hat{u}_a\hat{u}_b-\hat{v}_a\hat{v}_b),
~~~~ ~~~a,b= 1,2,3,
\label{tens_pa}
\ee
where $\hat{u}_a,\hat{v}_a$ are two unit vectors
pointing in the directions of the arms of the interferometer. We will
therefore neglect the interaction with the common mode, which is
expected to be much more noisy.
Note that in this paper we are not interested
in distinguishing a scalar signal from a tensor one, but only in estimating
the level of the signal eventually induced by a background of scalar waves.
The computation of $\gamma(p)$ requires knowledge of the
antenna pattern (\ref{27}), which describes the induced strain and
takes into account both the polarisation of the wave and the
geometrical configuration of the detector (parametrised by $D_{ab}$).
As far as the strain induced by a scalar wave is concerned,
there are two possible contributions~\cite{8}:
one corresponds to the direct interaction of the detector with the scalar
field, while the other is due to the indirect interaction with
the scalar component of the metric fluctuations induced by the scalar field
itself.
Indeed, in a general scalar-tensor theory in which
the matter fields are non-universally and
non-minimally coupled to the scalar field (for instance, gravi-dilaton
interactions in a superstring theory context \cite{15}),
the macroscopic bodies are characterised by a composition-dependent
scalar charge (which cannot be eliminated by an appropriate choice of
the frame, like the Jordan frame of conventional Brans-Dicke models),
and their motion in a scalar-tensor background is in general
non-geodesic \cite{8}. Taking into account also the direct coupling of a
test mass to the gradients of the scalar background field, it follows that
the standard equation of geodesic deviation (which is the main equation
for deriving the response of a gravitational detector) is
generalised as follows \cite{8}:
\be
{D^2 \xi^{\mu}\over D\tau^2} +R_{\alpha\beta\nu}\,^\mu u^\beta u^\nu
\xi^\alpha +q \xi^\alpha \nabla_\alpha \nabla^\mu \phi=0.
\label{32}
\ee
Here $\xi^\mu$ is the spacelike vector connecting two nearby
(non-geodesic) trajectories, and $q$ is the scalar charge per unit of
gravitational mass, representing the relative strength of scalar interaction
with respect to ordinary tensor-type interactions \cite{8}.
The small, non-relativistic oscillations of a test mass, mechanically
equivalent to the detector, are thus described by the equation:
\be
\ddot \xi^a=-\xi^b \left(R_{b00}\,^a +
q \partial_b\partial^a \phi\right).
\label{33}
\ee
The two contributions to the strain come from the gradients of the
scalar field $\phi$, and from the gradients of the scalar component
$\psi$ of the metric fluctuations (induced by $\phi$), covariantly
represented by the Riemann tensor. The two fields $\phi$ and $\psi$ are
in principle different, but not independent, being related by a set of
coupled differential equations (which are model-dependent).
The antenna pattern $F_\psi(\hat n)$,
associated to the (indirect) Riemannian part of the strain, has been
computed in \cite{10} for both massless and massive scalar fields.
We shall see that, for a traceless detector response tensor,
the function $F_\psi(\hat n)$ is also proportional to the
antenna pattern $F_\phi(\hat n)$ associated to the direct, non-geodesic
part of the strain.
Introducing the transverse and longitudinal
projectors with respect to the direction of propagation of the scalar
wave,
\be
T_{ab}=(\delta_{ab}-\hat{n}_a\,\hat{n}_b), ~~~~~
L_{ab}=\hat{n}_a\,\hat{n}_b \,,
\ee
the indirect, Riemannian contribution to the scalar pattern function
becomes \cite{10}:
\be
F_\psi(\hat n)= D^{ab}e_{ab}(\psi)=D^{ab}\left(T_{ab}+
\frac{\tilde{m}^2}{E^2}\,L_{ab}\right).
\label{35}
\ee
Using eq.~(\ref{33}) and the mode expansion for the scalar field, the
antenna pattern for the direct, non-geodesic coupling is:
\be
F_\phi(\hat n)= q D^{ab}e_{ab}(\phi)=q D^{ab}\frac{p^2}{E^2}
\,L_{ab}.
\label{36}
\ee
Since $T_{ab}= \delta_{ab}- L_{ab}$, and Tr $D=0$, it follows that
\be
F_\phi(\hat n)=-qF_\psi(\hat n)=-q \frac{p^2}{E^2}F^0_\psi(\hat n),
\label{37}
\ee
where $F^0_\psi$ is the antenna pattern corresponding to a massless
scalar wave\cite{10}. The overlap reduction
function of two interferometers, directly interacting
through a charge $q_i$ with a scalar field, can thus be written as
\be
\gamma(p)=q_1q_2\left(p\over E\right)^4\,\gamma_0(p)\,,
\label{gamma_rel}
\ee
where $\gamma_0(p)$ is the overlap reduction function
for the geodesic interaction with a massless scalar field~\cite{10}.
Using Eq.~(\ref{221}) and~(\ref{gamma_rel}), the signal-to-noise ratio is
finally given by:
\be
SNR=q_1q_2\,
\left(\frac{H^2_0}{5\pi^2}\right)\, \left[ 2 T
\int_{0}^{\infty}dp
\frac{p^5\,\Omega^2(p)\,\gamma_0^2(p)}
{P_1(\sqrt{p^2+\tilde{m}^2})\,P_2(\sqrt{p^2+\tilde{m}^2})
(p^2+\tilde{m}^2)^{11/2}}\right]^{1/2}.
\label{SNR_fin}
\ee
Note that this expression, with $q_i=1$, is also valid to estimate the
signal indirectly induced in the interferometers by the scalar metric
fluctuations through their
usual coupling to the Riemann tensor, provided $\Omega(p)$ refers to
the associated
spectrum of scalar metric fluctuations. In the following
we shall therefore use eq. (\ref{SNR_fin}) setting $q_1=q_2=1$ if the
dominant signal comes indirectly
from the stochastic background of scalar metric
fluctuations $\Omega_\psi(p)$ induced by the scalar field,
and setting instead $q_i<1$ (according to the experimental constraints)
if the dominant signal is directly due to the stochastic background
$\Omega_\phi(p)$ of massive scalar waves.
\section{Examples}
\label{S4}
We will now apply the results of the previous sections to estimate the
signal induced by a stochastic background due to a massive scalar field
on the two LIGO interferometers;
we will consider the two detectors operating
in the initial (I), enhanced (II) and advanced (III)
configurations \cite{16}. In particular, for the noise spectral density we
will use the following analytical fits:
\begin{itemize}
\item LIGO I ~\cite{Sathya}:
\begin{eqnarray}
&&P(f)=\frac{3}{2}\,P_0
\left[\left(\frac{f}{f_0}\right)^{-4}+2+
2\left(\frac{f}{f_0}\right)^2\right], \nonumber \\
&& P_0=10^{-46}\mbox{Hz}^{-1},
\quad\,f_{s}=40\mbox{Hz},
\quad\,f_{0}=200\mbox{Hz}.
\label{nligo1}
\end{eqnarray}
\item LIGO II ~\cite{BenSathya}:
\begin{eqnarray}
&&P(f)=\frac{P_0}{11}\,
\left[\left(\frac{f}{f_0}\right)^{-9/2}
+\frac{9}{2}\left(1+\left(\frac{f}{f_0}\right)^2\right)\right],
\nonumber \\
&& P_0=7.9\times10^{-48}\mbox{Hz}^{-1},
\quad\,f_{s}=25\mbox{Hz},
\quad\,f_{0}=110\mbox{Hz}.
\label{nligo2}
\end{eqnarray}
\item LIGO III ~\cite{BenSathya}:
\begin{eqnarray}
&&P(f)=\frac{P_0}{5}\,
\left[\left(\frac{f}{f_0}\right)^{-4}+2+
2\left(\frac{f}{f_0}\right)^2\right], \nonumber \\
\nonumber \\
&& P_0=2.3\times10^{-48}\mbox{Hz}^{-1},
\quad\,f_{s}=12\mbox{Hz},
\quad\,f_{0}=75\mbox{Hz}.
\label{nligo3}
\end{eqnarray}
\end{itemize}
In the above equations, $f_{s}$ is a seismic cut-off below which
the noise spectral density is treated as infinity.
We shall also assume that the non-relativistic modes in the
spectrum are the dominant ones, and their energy density almost
saturates the critical energy density, namely
\be
\int_0^m d\ln p~\Omega^{\rm non-rel}(p)\simeq 1.
\label{41}
\ee
We shall analyse, in particular, two examples of spectra.
\subsection{Minimal dilaton background}
The first example is a stochastic background of massive relativistic
dilatons, produced according to some models of early cosmological
evolution based upon string theory \cite{7}.
To illustrate the difficulty in detecting such a
background, we will consider here the dilaton spectrum obtained in the
context of a ``minimal" pre-big bang scenario and we shall assume that the
dilaton mass is enough small so that the produced dilatons have not yet
decayed into radiation with a rate $\Gamma \sim m^3/M_P^2$. This
implies $\Gamma \laq H_0$, i.e $ m \laq 100$ MeV.
Even if the mass is negligible at the beginning of the radiation era, the
proper momentum $p=k/a(t)$ is red-shifted with respect to the rest
mass because of the cosmological expansion, and all the modes in the
dilaton spectrum tend to become non-relativistic. As a consequence, the
present dilaton spectrum has in general at least three branches,
corresponding to: {\em i)} relativistic modes, with $p>m$; {\em ii)}
non-relativistic modes with $p_m~~