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\line{\hfil DFTT-03/94}
\line{\hfil January 1994}
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\centerline {\grande Relaxed Bounds on the Dilaton Mass}
\vskip 1 true cm
\centerline{\grande In a String Cosmology Scenario}
\vskip 2true cm
\centerline{M.Gasperini}
\centerline{\it Dipartimento di Fisica Teorica dell'Universit\`a,}
\centerline{\it Via P.Giuria 1, 10125 Torino, Italy,}
\centerline{\it and}
\centerline{\it Istituto Nazionale di Fisica Nucleare, Sezione di Torino}
\vskip 2 cm
\centerline{\medio Abstract}
\noindent
We discuss bounds on the dilaton mass, following from the cosmological
amplification of the quantum fluctuations of the dilaton background, under the
assumption that such fluctuations are dominant with respect to the classical
background oscillations. We show that if the fluctuation spectrum grows with
the frequency the bounds are relaxed with respect to the more conventional case
of a flat or decreasing spectrum. If the growth of the spectrum is fast
enough, the allowed range of masses
may become compatible with models of supersymmetry breaking, and with a
universe presently dominated by a relic background of dilaton dark matter.
\vskip 1 true cm
\noi
--------------------------------
\vskip 1 true cm
To appear in {\bf Phys. Lett. B}
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\footline={\hss\rm\folio\hss}
\pageno=1
\centerline{\bf Relaxed
bounds on the dilaton mass in a string cosmology scenario.}
It is known that the coherent oscillations around the minimum of the potential
of a cosmic scalar background [1-4], such as the dilaton [5-7], put severe
cosmological bounds on the allowed value of the mass $m$ of that scalar field.
Such oscillations, however, become coherent, and then constrain the mass, only
when the possible spatial inhomogeneities of the background are negligible.
Spatial inhomogeneities of the dilaton background can arise either because of
thermal fluctuations, or because of the gravitational amplification of quantum
fluctuations. If the spatial inhomogeneities of thermal origin are diluted by
a subsequent inflationary expansion (or if they are absent simply because the
maximum temperature scale is lower than the Planckian temperature required for
thermal equilibrium), then their contribution to the energy
density (in a mode expansion of the background oscillations) becomes negligible
with respect to the mass contribution. The coherent oscillations of the
background,
with frequency $m$, can then dominate the dilaton energy density beginning at a
scale $H\simeq m$.
The same happens for the energy density stored in the quantum fluctuations, if
they are amplified with a spectrum which is flat or decreasing with frequency.
Also in that case the mass coherent contribution begins to dominate (with
respect to spatial inhomogeneities) the fluctuation
energy at $H \simeq m$, thus
avoiding a significant relaxation of the cosmological bounds, even for
negligible amplitude of the classical background oscillations [2].
If, on the contrary, the quantum fluctuations are amplified with a spectrum
which grows with frequency, then their energy density may
stay dominated by the small
scale inhomogeneities until values of $H$ much lower than $m$. In that case the
bounds on $m$ can be alleviated, provided the classical background solution
approaches enough the minimum of the potential before the scale drops to
$H=m$.
A growing spectral distribution of the dilaton fluctuations is a typical
outcome of the
"Pre-big-bang" inflationary scenario [8,9], suggested by the duality properties
of the string cosmology equations [10]. In such context, as we shall see in
this paper, the relaxation of the bound has two important consequences: the
allowed range of dilaton masses may become compatible with a) model of
supersymmetry breaking able to provide a natural resolution of the gauge
hierarchy problem [11], and with b) the possibility that our universe is
presently dominated by a relic background of dilaton dark matter. (It should be
mentioned that a growing scalar perturbation spectrum is also predicted by the
"hybrid inflation" model proposed by Linde [12] and recently generalized to the
class of "false vacuum inflation" [13] (see also [14]). Growing spectra,
moreover, have been shown to be necessary for a simultaneous fit of the COBE
anisotropies and of the observed bulk motion and large voids structures on a
$50 Mpc$ scale [15]).
In view of its importance, we start by recalling the standard arguments leading
to the bounds on the dilaton mass based on the coherent oscillations of the
classical background [1-7]. For scales $H>m$ one thus finds that $\phi$ approaches asymptotically a constant value
$\phi_1$,
$$
\phi =\phi_1 +\phi_2({H\over H_1})^{1/2} \eqno(2)
$$
so that, for $H<>m$, lies at a distance of order unity (in Planck units) from
the minimum $\phi_0$ of the potential.
One might thus be led to think that the bounds could be evaded if, owing to some
mechanism, the initial amplitude of the classical background oscillations would
be lowered to $|\phi-\phi_0|<am$ begins to oscillate at a scale $H_k=k/a_k$, with an amplitude
$\chi_k$ which is initially of order $H_1$, and which decreases in time as
$\chi_k \simeq H_1(H/H_k)^{1/2}$. When $H\simeq m$, the
non-relativistic modes ($k/a\me m$) begin to
oscillate, with initial amplitude $H_1$ and frequency $\simeq m$,
and they immediately become dominant
with respect to the other modes, as their contribution to $\rho_\chi$ decreases
like $a^{-3}$ instead of $a^{-4}$. For $H\leq m$ we are thus in a situation
where, beside the energy density stored in the possible coherent oscillations of
the classical background, $\rho_\phi = \rho_m(a_m/a)^3$, we must have,
necessarily, also some energy stored in the coherent oscillations of the
quantum fluctuations, with
$$
\rho_\chi \simeq m^2 H_1^2({a_m\over a})^3 \eqno(15)
$$
In this paper we want to discuss how the cosmological
bounds on the dilaton mass are relaxed when we move from a scenario in which
the fluctuation spectrum is flat or decreasing ($n\leq 1$), to another scenario
characterized by a growing ($n>1$) spectrum. We shall thus assume, henceforth,
that the classical oscillations (whose initial amplitude is model-dependent)
are always negligible, $\rho_\phi < \rho_\chi$, and that all
bounds on $m$ follow
from the cosmological amplification of the quantum fluctuations only. We will
obtain, in this way, the {\it maximum} (approximately
model-independent) allowed region in
parameter space.
The energy density (15) is smaller than the radiation energy $\rho_\ga$ when
$H=m$, but it grows with respect to $\rho_\ga$ as the curvature scale decreases
in time, until it equals $\rho_\ga$ at an initial scale $H_i$. If $H_i < H_2$,
which means
$$
m< H_2({ M_p\over H_1})^4 \eqno(16)
$$
(recall that $H_2$ denotes the usual matter-radiation transition scale of
eq.(3)), then $\rho_\chi$ stays always smaller than the critical density. If,
however, $H_i>H_2$ then
$$
H_i= m({H_1 \over M_p})^4 \eqno(17)
$$
and the dilaton must have already decayed, $H_d>H_0$, to avoid contradictions
with the presently observed matter density. This implies
$$
m \Me (M_p^2 H_0)^{1/3} , \eqno(18)
$$
where $H_0\sim 10^{-61}M_p$ is the present curvature scale. The dilaton decay
generates an entropy $\Da S= (T_r/T_d)^3$, where $T_r$ is the reheating
temperature (7) and $T_d$ the radiation temperature at the dilaton decay epoch,
namely
$$
T_d=T_i({a_i\over a_d})=(M_p H_i)^{1/2}({H_d\over H_i})^{2/3}
\simeq ({m^{11}\over M_pH_1^4})^{1/6} \eqno(19)
$$
This gives
$$
\Da S = {H_1^2\over m M_p} \eqno(20)
$$
If $m<10^4 GeV$ the reheating temperature is too low ($<1 MeV$) to allow
nucleosynthesis: we must impose that nucleosynthesis already occurred,
$H_i10^4 GeV$, the nucleosynthesis scale is
subsequent to dilaton decay, and the only
possible constraint is [5-7] $\Da S \me 10^5$ in order to preserve primordial
baryogenesis.
These are the conditions to be imposed if $mH_1=k_1/a_1$, then all
modes are always non-relativistic, and the spectral energy distribution becomes
[19]
$$
k{d \rho_\chi \over dk} \simeq H_1^4 ({ a_1 \over a})^3
({m\over H_1})^2 ({k\over k_1})^{n-1} \eqno(21)
$$
This gives, for $n=1$,
$$
\rho_\chi(t) \simeq m^2 H_1^2 ({a_1\over a})^3 \eqno(22)
$$
The scale $H=H_1$ marks the beginning of coherent oscillations with frequency
$m$ and initial energy $\rho_i=m^2 H_1^2$ (hence $m< M_p$ to avoid over-critical
density). There are no further bounds on $m$, in this case, as the fluctuation
energy is dissipated before a possible dilaton dominance. Indeed, the scale
$H_i=H_1(m/M_p)^4$ corresponding to the equality $\rho_\chi =\rho_\ga$ is
always smaller than the decay scale, $H_d=m^3/M_p^2>H_i$.
The values of $(m,H_1)$ allowed by the previous constraints are shown in
{\bf Fig.1}. One can see that the bounds on $m$ are relaxed but, as stressed
in [2], too low values of $H_1$ are in general required to be compatible with
an interesting range of masses. The preferred supersymmetry breaking scale
$m\sim 1 TeV$, for instance, is forbidden unless $H_1\me 10^{-8} M_p$. Similar
values of $H_1$ ($\sim 10^{-6} - 10^{-9} M_p$) are required to be compatible
with the possibility of a not yet decayed, and presently dominating, dilaton.
The situation becomes even worse for quantum fluctuation with a decreasing
spectrum ($n<1$). In that case the initial amplitude $\chi_i$
of the coherent oscillations become larger, $\chi_i \simeq
H_1(H_1/m)^{(1-n)/4} >H_1$ and, as a consequence, a lower inflation scale $H_1$
is required to be compatible with the same given value of $m$.
String cosmology, however, suggests a scenario in which the standard
radiation dominated era is preceeded by a so-called pre-big-bang phase,
describing the evolution from a flat, weakly coupled initial state [8]. The
universe super-inflates, bends up and heats up to a maximum scale $H_1$, after
which curvature and temperature begin to decrease. The dilaton grows up to the
strong coupling regime, and its settlement to a constant value marks the
beginning of the standard cosmological evolution. The particle production
associated with the transition between pre and post-big-bang regime is
characterized by a growing spectrum [8,9], and imposes the constraint
$H_11$ in eq.(13), the scale $H=m$ marks the beginning of coherent
oscillations (with frequency $m$) of the mode $k_m=ma_m$, which provides
a non-relativistic contribution to the fluctuation energy density,
$$
\rho_\chi(t)\simeq m^2H_1^2 ({m\over H_1})^{(n-1)/2}({a_m\over a})^3
\eqno(23)
$$
This contribution grows in time with respect to $\rho_\ga$, so that we
can repeat exactly the same phenomenological discussion previously
applied to the case of a flat spectrum, with the
only difference that the
energy density (15) is now replaced by eq.(23). As a consequence, we
obtain bounds which depend on the spectral index $n$ and which are, in
particular, less and less constraining as $n$ is growing (because
$\rho_\chi$ decreases with $n$).
There is a limit, however, on such a relaxation, because for $n\geq 2$
the contribution (23) to the total fluctuation energy becomes
subdominant with respect to the contribution of the maximum frequency
mode $k_1$. Indeed, this mode becomes non-relativistic at a scale
$H_{nr}$ such that $k_1/a_{nr}=m$,
namely
$$
H_{nr}= {m^2\over H_1} H_{nr}$, namely for $(m/H_1)^{n-2}>1$, which
implies $n<2$.
In order to discuss to which {\it maximum} extent the cosmological
bounds on $m$ are possibly relaxed in the case of growing dilaton
spectrum, we shall thus concentrate, in what follows, on the case $n\geq
2$ (see [19] for a discussion of the ``interpolating" case $1H_2$, where
$$
H_i={m^2\over M_p}({H_1 \over M_p})^3 \eqno(28)
$$
but the dilaton decays before becoming dominant, $H_d>H_i$. If on the contrary,
$H_i>H_2$ and $H_dH_i$ (with the scale $H_i$ of eq.(28)), and that $\Da S =(T_r/T_d)^3
\me 10$. Since
$$
T_d=T_i({a_i\over a_d}) \simeq (M_pH_i)^{1/2}({H_d\over H_i})^{2/3}=
({m^{10}\over M_pH_1^3})^{1/6} \eqno(29)
$$
the produced entropy to be bounded is, in this case,
$$
\Da S = ({H_1^3 \over m M_p^2})^{1/2} \eqno(30)
$$
The other possibility, $T_r >1 MeV$, allows a nucleosynthesis phase subsequent
to dilaton decay, so that the only bound is imposed by primordial baryogenesis,
$\Da S \me 10^5$. The case $m>H_1$, finally, provides the only bound $mH_1$ from the closure density; e) $m=H_2(M_p/H_1)^4$, upper bound
on $m$ from the present matter-to-radiation energy density ratio; f)
$H_1^2/m M_p = 10^5$, upper limit on entropy production from primordial
baryogenesis; g) $H_1^2/m M_p = 10$, upper limit on entropy from
nucleosynthesis.
\vskip 1 cm
\noi
{\bf Fig.2} Maximum allowed region (inside the full lines) in the case of
quantum fluctuations amplified with a growing spectrum (with
$n\geq 2$), and dominant with
respect to the classical background oscillations. The lines marked by $a,b,c,d$
are the same as in Fig.1. The other lines correspond to: e) $m=(H_1H_2)^{1/2}
(M_p/H_1)^2$, upper bound on $m$ from the present matter-to-radiation energy
density ratio; f) $(H_1^3/m M_p^2)^{1/2}=10^5$, upper limit on entropy
production from primordial baryogenesis; g) $(H_1^3/m M_p^2)^{1/2}=10$, upper
limit on entropy production from nucleosynthesis.
\end
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