%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % BEGIN FILE: scalar4.tex % % version: 8 march 2001 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input epsf \documentstyle[epsfig,eqsecnum,aps,prd,preprint,tighten,floats]{revtex} %\documentstyle[epsfig,eqsecnum,aps,prd,tighten,floats]{revtex} %\draft % make PACS numbers print \begin{document} % \newcommand{\gras}{\mbox{\boldmath $#1$}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \newcommand{\Mc}{{\cal M}} \newcommand{\Ms}{M_{\odot}} \newcommand{\m}{\langle} \newcommand{\M}{\rangle} \newcommand{\hf}{\tilde h} \newcommand{\rf}{\tilde r} \newcommand{\yf}{\tilde y} \newcommand{\qf}{\tilde q} \newcommand{\Qf}{\tilde Q} \newcommand{\bml}{\begin{mathletters}} \newcommand{\eml}{\end{mathletters}} % %minore o circa uguale \def\laq{~\raise 0.4ex\hbox{$<$}\kern -0.8em\lower 0.62 ex\hbox{$\sim$}~} %maggiore o circa uguale \def\gaq{~\raise 0.4ex\hbox{$>$}\kern -0.7em\lower 0.62 ex\hbox{$\sim$}~} % \preprint{\vbox{\baselineskip=12pt \rightline{BA-TH/01-412}}} \rightline{BA-TH/01-412} \preprint{\vbox{\baselineskip=12pt \rightline{RCG 01/10}}} \rightline{RCG 01/10} \rightline{gr-qc/0103035} \vskip 2true cm \vspace{10mm} {\large\bf\centering\ignorespaces Detecting a relic background of scalar waves with LIGO\\ \vskip2.5pt} \bigskip {\dimen0=-\prevdepth \advance\dimen0 by23pt \nointerlineskip \rm\centering \vrule height\dimen0 width0pt\relax\ignorespaces M. Gasperini${}^{(1,2)}$ and C. Ungarelli${}^{(3)}$ \par} \bigskip {\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} \par \bgroup \leftskip=0.10753\textwidth \rightskip\leftskip \dimen0=-\prevdepth \advance\dimen0 by17.5pt \nointerlineskip \small\vrule width 0pt height\dimen0 \relax \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