1$ this gives the stringent constraint
\beq
m\laq \left(H_{\rm eq}M_{\rm P}^4\over H_1^3\right)^{1/2} ,
\label{632}
\eeq
which represents, in this context, the most restrictive upper bound on
the mass of (not yet decayed) dilatons. For $100$ keV $\laq m
\laq 100$ MeV a more restrictive constraint on $\Om_\chi$ is provided
in principle by the observations of the astrophysical background of
diffuse $\ga$-rays \cite{DaVi96a}; such a range of masses, however,
tends to be excluded, as will be shown below, in the context of pre-big
bang models where the inflation scale is controlled by the string mass,
$H_1\simeq M_{\rm s}$.
The critical density bound (\ref{632}) can be evaded if the dilatons are
heavy enough to decay, so that the energy stored in their coherent
oscillations was dissipated into radiation before the present epoch. The
decay scale $H_d$ is fixed by the decay rate (for instance into
two photons) as
\beq
H_d\simeq \Ga_d \simeq {m^3/M_{\rm P}^2}
\label{633}
\eeq
(which implies, in particular, that the produced dilatons have not yet
decayed, today, only if $m \laq 100$ MeV). The
decay, however, generates radiation, reheats the Universe, and is
associated in general to a possible entropy increase
\beq
\Da S \simeq
\left(T_r/T_d\right)^3,
\label{634}
\eeq
where $T_r$ and $T_d$ are the final reheating
temperature and the radiation temperature immediately
before dilaton decay, respectively. The decay process is thus the source
of additional phenomenological constraints on the dilaton spectrum,
which are complementary to the critical bound (\ref{632}), as they imply
in general a lower bound on the dilaton mass.
In order to discuss the consequences of dilaton decay let us note, first
of all, that the induced reheating is significant ($T_r>T_d$, $\Da s > 1$)
provided dilatons decay when they are dominant with respect to
radiation, i.e. for $H_d k_m .
\label{776}
\eeq
The associated energy density, $\rho_\sg^{\rm non-rel}(k) \sim
(\sg'/a)^2\sim (ma/k) \rho_\sg^{\rm rel}(k)$, differs from the previous
one by the non-relativistic rescaling $m/p$, thus leading to the
spectrum
\beq
\Om_\sg(p,t)= g_1^2 {m\over H_1} \left (H_1\over H\right)^2
\left (a_1\over a \right)^3
\left(p \over p_1\right)^{2-2\nu}, ~~~~~~~
p_m p_T$, the role of the transition
frequency, which separates modes that become non-relativistic inside
and outside the horizon, is played by $p_T$,
and the lowest-frequency
band of the non-relativistic spectrum (\ref{783}) has to be replaced by
\cite{GasVe99}:
\beq
\Om_\sg (p,t)= g_1^2 {m\over \sqrt{ H_1H_{\rm eq}}} \left({\rm
eV}\over T_m\right) \left (H_1\over H\right)^2 \left (a_1\over a
\right)^3
\left(p \over p_1\right)^{3-2\mu}, ~
p