p_1\sim (H_1/M_{\rm P})^{1/2}10^{-4}$ eV, then all modes today are non-relativistic, and the $\Om_1$ branch of the spectrum disappears. Note also that the relativistic branch of the spectrum is typically growing \cite{1} in minimal string cosmology models ($0<\da \leq 3$), while the non-relativistic spectrum may have a flat or decreasing branch if $\da \leq 1$. The present amplitude of the non-relativistic spectrum is controlled by two basic parameters, $H_1$ and $m$. In minimal string cosmology models $H_1$ is typically fixed at the string scale, $H_1 \simeq M_{\rm s}$, and the question {\em ``How strong is today the relic dilaton background?"} thus becomes {\em ``How large is the dilaton mass?"} We would like to have a mass small enough to avoid the decay and to resonate with the present gravitational antennas, but large enough to correspond to a detectable amplitude. It is possible? The answer depends on the value of the dilaton mass. From the theoretical side we have no compelling prediction, at present. From the phenomenological side, however, we know that the allowed values of mass (and thus of the range of the dilatonic forces) are strictly correlated to the strength of the dilaton coupling to macroscopic matter. We are thus lead to the related question: {\em ``How large is the dilaton coupling?"} For a better formulation of such a question we should recall that the motion of a macroscopic body, in a gravi-dilaton background, is in general non-geodesic, because of the direct coupling to the gradients of the dilaton field \cite{2}, \beq {du^\mu\over d\tau}+ \Ga_{\a\b}^\mu u^\a u^\b+q \nabla^\mu \phi=0, \label{5} \eeq induced by the effective ``dilatonic charge" $q$ of the body, which controls the relative strength of scalar-to-tensor forces. So, how large is $q$? The experimental value of $q$, as is well known, is directly constrained by tests of the equivalence principle and of macroscopic gravity, which provide exclusion plots in the $\{m, q^2\}$ plane (see for instance \cite{11}). From a theoretical point of view, on the other hand, there are two possible (alternative) scenarios. A first possibility is that, by including all required loop corrections into the effective action, the dilatonic charge $q$ becomes large and composition-dependent \cite{12}. In that case one has to impose $m \gaq 10^{-4}$ eV, to avoid contradiction with the present gravitational experiments, which exclude testable deviations from the standard Newton's law down to the millimeter scale \cite{13}. The alternative possibility is that, when including loop corrections to all orders, the resulting effective charge of ordinary macroscopic matter turns out to be very small ($q \ll1$)and universal \cite{14}. In that case the dilaton mass can be very small, or even zero. In the following discussion we will accept this second (phenomenologically more interesting) possibility, relaxing however the assumption of universal dilaton interactions (which is not very natural, and which seems to require fine-tuning). We shall thus assume that the relic dilatons are weakly coupled to matter, and light enough to have not yet decayed ($m \laq 100$ MeV), so as to be available to present observations. The amplitude of the background, as already noticed, depends however on $m$, and a larger mass corresponds in principle to a stronger signal. In this range of parameters, on the other hand, there is an upper limit on the mass following from the critical density bound, \beq \int^{p_1} d(\ln p) \Om(p) \laq 1, \label{6} \eeq which has to be imposed to avoid a Universe over-dominated by dilatons. So, which values of mass may correspond to the stronger signal, i.e. to an (almost) critical dilaton background? The answer depends on the shape of the spectrum. If $m>p_1$ and $\da \geq 1$, so that all modes are non-relativistic and the spectrum is peaked at $p=p_1$, the critical bound reduces to $\Om_2(p_1)\laq 1$, and implies \beq m<\left(H_{\rm eq}M_{\rm P}^4/H_1^3\right)^{1/2}, \label{7} \eeq which, for $H_1 \sim M_{\rm s}\sim 10^{-1} M_{\rm P}$, is saturated by $m \sim 10^2$ eV. This means that the density of the relic dilaton background may approach the critical upper limit for values of mass which are too large (as we shall see later) to fall within the resonant band of present gravitational detectors. If $\da<1$ the above bound is relaxed, but the mass is still constrained to be in the range $m>p_1$. Let us thus consider the case $m