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\line{\hfil  DFTT-29/94}
\line{\hfil July 1994}
\vskip 2 cm
\centerline {\grande  The Inflationary Role of the Dilaton}
\vskip 0.5 true cm
\centerline{\grande in String Cosmology}

\vskip 1.5true cm 
\centerline{M.Gasperini}
\centerline{\it Dipartimento di Fisica Teorica, Universit\'a di Torino,}
\centerline{\it Via P.Giuria 1, 10125 Turin, Italy,}
\centerline{\it and}
\centerline{\it INFN, Sezione di Torino, Turin, Italy}

\vskip 1.5 true cm
\centerline{\medio Abstract}

\noindent
It is stressed that the kinematical problems of the standard 
cosmological model can be solved by a phase of accelerated 
contraction of the cosmic scale factor. Such a behaviour is only a 
particular case of a more general ``pre-big-bang" inflationary scenario, 
arising naturally in a string cosmology context, and driven 
asymptotically by the free dilaton field without any self-interaction 
potential and/or vacuum contribution.

\vskip 2 true cm
\centerline{----------------------------------}
\vskip 1 true cm
\noi
To appear in the {\bf Proceedings of the First Int. 
Workshop on Birth of the Universe and Fundamental Physics},  
{\it Roma, May 18-21, 1994} (Springer-Verlag, Berlin)

\vfill\eject 

\footline={\hss\rm\folio\hss}
\pageno=1


\centerline{\bf THE INFLATIONARY ROLE OF THE DILATON IN STRING COSMOLOGY}
\bigskip
\centerline{M. GASPERINI}
\centerline{\small{\it Dipartimento di Fisica Teorica, Universit\'a di
Torino, }}
\centerline{\small{\it Via P.Giuria 1, 10125 Turin, Italy,}}
\centerline{\small{\it and INFN, Sezione di Torino, Turin, Italy}}
\bigskip
\centerline{ABSTRACT}
\midinsert
\narrower
\noi
It is stressed that the kinematical problems of the standard 
cosmological model can be solved by a phase of accelerated 
contraction of the cosmic scale factor. Such a behaviour is only a 
particular case of a more general ``pre-big-bang" inflationary scenario, 
arising naturally in a string cosmology context, and driven 
asymptotically by the free dilaton field without any self-interaction 
potential and/or vacuum contribution.
\endinsert
\bigskip
\noi
{\bf 1. Introduction: Inflation Without Inflation}

\noi
It is well known that various problems of the standard cosmological 
scenario can find a natural explanation [1], if we assume that the phase 
of radiation dominance is preceeded in time by a phase of accelerated 
expansion (the so-called inflation), characterized by $H>0$, 
$\ddot a>0$, where $a$ is the scale factor of an isotropic metric, $H=
\dot a /a$, and the dot denotes differentiation with respect to cosmic 
time. What seems to be presently less known, however, is the existence 
of another possible solution of the standard kinematical problems 
(suggested by recent studies of string cosmology [2,3,4]), based on the 
simple assumption that the evolution of our universe is time-symmetric 
with respect to some given past curvature scale $H_1\sim 1/t_1>0$.

Suppose, in particular, that our universe is radiation-dominated up to 
$H_1$, i.e. $a_+(t)\sim (t+t_1)^{1/2}$ for $t>0$. By time symmetry, the 
scale factor is to be continued to negative times as
$$
a_-(t)\sim (t_1-t)^{1/2}, ~~~~~~~~~~~~~~~t<0 \eqno(1.1)
$$
This metric describes accelerated contraction, $\dot a<0$, $\ddot a<0$, 
and growing curvature scale, $|\dot H|>0$, up to $H_1$, and it is 
still an exact 
solution of the radiation-dominated cosmological equations (with 
negligible spatial curvature), as the Einstein equations are invariant 
under time-reversal transformations. Of course, near $t=0$, a suitable 
modification of both $a_{\pm}$ and of the source stress tensor is to be 
expected, in order to guarantee the presence of a bounce at $|H|\sim H_1
$, and a continuous matching of the two branches of the radiation-
dominated evolution. As a consequence, $a_{\pm}$ are to be regarded only 
as the asymptotic form of the solution, for $|H|<<H_1$. This 
asymptotic form is already enough, however, to show that the horizon and 
flatness problems [1] are absent in a radiation-dominated universe, if 
its evolution is time-symmetric.

In fact, in the primordial contracting phase (1.1) the proper size of a 
causally connected region shrinks in time like the scale factor, while 
the proper size $d_e(t)$ of the event horizon,
$$
d_e(t)=a(t)\int_t^{t_1}{dt^\pr \over a(t^\pr)}={1\over 2}(t_1-t)\eqno
(1.2)
$$
shrinks linearly in cosmic time, i.e. faster than $a(t)$. This means 
that causally connected scales are ``pushed out" of the horizon (just 
like in conventional inflationary expansion). The horizon problem may 
thus be solved, provided the ratio
$$
{a(t)\over d_e(t)}\sim \eta^{-1}\eqno(1.3)
$$
($\eta$ is the conformal time coordinate, defined by $dt=ad\eta$) grows 
enough during the contraction ($t<0$), so as to compensate the 
subsequent decrease of $a(t)/t$ during the phase of decelerated 
expansion ($t>0$). To obtain, in particular, a causally 
connected universe at the end ($\eta=\eta_2$) of the radiation-dominated 
epoch, we must require according to (1.3) that the contracting phase is 
at least as long as the subsequent expanding one, namely
$$
{|\eta_i|\over |\eta_1|}
\Me {|\eta_2|\over |\eta_1|}
\eqno(1.4)
$$
where $\eta_i$ marks the beginning of the contraction, and $\eta_1$ is 
the fixed point of the time-reversal symmetry, corresponding to the 
maximum scale $H_1$. This condition is certainly satisfied if the 
evolution is time-symmetric, irrespective of the value of $\eta_1$.

With similar arguments one can show that also the flatness problem is 
solved by a time-symmetric cosmological evolution. Indeed, the solution 
of this problem requires that the ratio between the spatial curvature 
term, $k/a^2$, and the other terms of the cosmological equations,
$$
{k\over a^2H^2}={k\over \dot a^2}\sim \eta^2 \eqno(1.5)
$$
at the end of the contracting phase ($\eta=\eta_1$) is tuned to a value 
small enough, so that the subsequent decelerated evolution eventually 
leads to a value of the ratio $\me 1$ at $\eta=\eta_2$. This implies
$$
\left(\eta_1\over \eta_i\right)^2
\me \left(\eta_1\over \eta_2\right)^2
\eqno(1.6)
$$
which is equivalent to the condition (1.4), and which is again satisfied 
by a time-symmetric cosmological evolution.

The presence of a long enough period of accelerated contraction, which 
may follow in particular from the hypothesis of time-symmetric cosmology 
(like in the previous example), has thus all the kinematical effects of 
inflation, with no need of introducing some ``ad hoc" inflaton 
potential, slow-rolling conditions, fine-tuning, and so on, like in 
conventional inflationary models [5]. A possible objection to the 
complete equivalence of inflation and time-reversal symmetry is that the 
effective potential, $V(\eta)=(d^2a/d\eta^2)/a$, appearing in the scalar 
and tensor perturbation equations [6], is vanishing both for the 
expanding and contracting radiation-dominated evolution, in which $a\sim 
|\eta|$. This implies that no parametric amplification of the 
perturbations is possible, in such context. However, in a truly time-
symmetric evolution the radiation-dominated contraction should be 
preceeded by the time-reversed of the dust-dominated evolution, which also 
describes accelerated contraction with $a\sim |t|^{2/3}\sim |\eta|^2$. 
The transition between the two contracting phases amplifies 
perturbations with just a flat Harrison-Zeldovich spectrum, even if only 
for those comoving scales $k$ which are smaller than the ``height" 
$|\eta_2|^{-1}$ of the effective potential barrier, namely for scales 
entering inside the horizon during our present matter-dominated epoch.
\vskip 1 cm
\noi
{\bf 2. The Pre-Big-Bang Scenario}

\noi
Another objection to the ``inflationary" properties of a phase of 
accelerated contraction is that such a phase seems unable to dilute the 
possible production of topological defects, such as monopoles or 
strings. A possible answer to this objection is that strings shrinks in 
contracting backgrounds and, as a consequence, they are eventually 
diluted by the same amount as in the conformally related 
superinflationary (expanding) backgrounds [3]. Moreover, if it is true 
that monopoles 
inflate together with the metric background in which they are embedded, 
as recently argued in [7], then they should shrink, just like strings, 
if the background is contracting. So accelerated contraction is not 
incompatible, in principle, with the dilution of unwanted topological 
relics.

However, what it should be stressed at this point is that the previous 
example of phase with accelerated contraction and growing curvature, 
obtained through a time-reversal transformation from the standard 
radiation-dominated cosmology, is only a particular case of 
``pre-big-bang" scenario [8,2-4]. Such a scenario, describing a universe 
which starts from a flat and vacuum initial state, and approaches a 
maximum curvature scale (identified with the big-bang) through a period 
of accelerated evolution, finds a natural motivation in the context of 
the equations obtained from the low energy string effective action 
(rather than in the context of Einstein's cosmological equations). In 
their minimal version such equations contain, besides the metric and the 
matter sources, also a non-minimally coupled scalar field (the dilaton). 
For a homogeneous and isotropic, spatially flat, $(d+1)$-dimensional 
background, with perfect fluid sources ($p/\rho=\ga=const$), such 
equations can be written explicitly as [2,8]
$$
(\dot \phi -dH)^2-dH^2=8\pi G \r e^\phi
$$
$$
2\dot H-2H(\dot \phi-dH)=8\pi G \ga \r e^\phi
$$
$$
(\dot \phi -dH)^2-2\ddot \phi +2d \dot H + dH^2=0 \eqno(2.1)
$$
(where $G$ is the $d$-dimensional analog of the Newton constant). They 
are invariant not only under the usual time-reversal transformation 
$t\ra -t$, but also under the ``duality" transformation [9,10]
$$
a \ra a^{-1},~~~~~~ \phi \ra \phi -2d\ln a,~~~~~ \ga \ra -\ga \eqno(2.2)
$$

By combining duality and time reversal transformations it is thus 
possible, in a string cosmology context, to associate to any given 
standard ``post-big-bang" solution, with decreasing curvature scale, 
a related ``pre-big-bang" solution whose curvature has a specular 
behaviour with respect to the chosen scale of time-symmetry, but which 
is expanding, in an accelerated (inflationary) way. Consider, for 
instance, the general solution of eqs.(2.1), which becomes, at low 
enough curvature scales [2,3],
$$
a(t)\sim |t|^{2\ga /(1+d\ga^2)},~~~~ \phi 
\sim {d\ga -1\over \ga}\ln a, ~~~~ \r\sim a^{-d(\ga+1)}\eqno(2.3)
$$
For $\ga>0$ and $t>0$ the metric describes a decelerated expansion with 
decreasing curvature, characterized by
$$
\dot a >0,~~~~ \ddot a <0,~~~~ \dot H<0 \eqno(2.4)
$$
By performing the duality transformation (2.2), plus a time inversion, 
we are led to a metric describing accelerated expansion 
(of the so-called superinflationary type) and increasing curvature scale, 
characterized by
$$
\dot a >0,~~~~ \ddot a >0,~~~~ \dot H>0 \eqno(2.5)
$$
The particular case of the radiation-dominated solution ($\ga=1/d$) 
corresponds to a constant dilaton,
$$
a\sim t^{2/(d+1)},~~~~ \phi = const, ~~~~ \r \sim a^{-(d+1)},
~~~~ t>0 \eqno(2.6)
$$
and it is associated to a pre-big-bang phase dominated by a gas of 
stretched strings [11] (with $\ga=-1/d$), characterized by 
superinflationary expansion, growing curvature and dilaton coupling,
$$
a\sim (-t)^{-2/(d+1)},~~~~ \phi = 2 d \ln a, ~~~~ \r \sim a^{-(d-1)},
~~~~ t<0
\eqno(2.7)
$$

The particular case of pre-big-bang discussed in the previous section 
($a\sim (-t)^{1/2}$, $\phi =const$) is thus only a trivial example 
obtained from the solution (2.6) through time-reversal, which is the 
only possible transformation in the context of the Einstein cosmological 
equations, where there is no dilaton and the duality symmetry cannot be 
implemented. In the string cosmology context, on the contrary, the dilaton 
is present, and the associated duality symmetry has various important 
consequences.

First of all, with a superinflating (instead of contracting) 
pre-big-bang scenario the scale factor evolves monotonically 
in time, and 
this leads to the interesting possibility of self-dual cosmological 
solutions [8], characterized by $a(-t)=a^{-1}(t)$, and connecting in a 
smooth way the two duality-related pre and post-big-bang regimes (see 
however [12] for non-monotonic but smooth solutions). Of course, a 
superinflating pre-big-bang metric may become contracting when 
transformed in the Einstein frame (where the dilaton is minimally 
coupled to the metric). In that frame, however, the contraction is 
accelerated and the curvature is still growing, so that the solution of 
the standard kinematical problems is preserved [2-4], as discussed also 
in Section 1.

The presence of a non-trivial dilaton background, moreover, contributes 
to the effective potential appearing in the linearized perturbation 
equation [13,2,3], and then to the parametric amplification of the 
perturbations. The potential corresponding to the solution (2.7), in 
particular, is no longer flat, unlike the case of the solution (2.6) 
(for $d=3$). But the main consequence of a dilaton background is that, 
in the high curvature regime, the general solution of the 
equations (2.1) 
becomes dilaton-dominated (quite independently of the matter equation of 
state), and the universe enters a phase of dilaton-driven 
superinflation, with [2,3]
$$
a\sim (-t)^{-1/\sqrt d},~~~~~~~~~~~~~~~ \phi \sim (d+\sqrt d)\ln a 
\eqno(2.8)
$$
When the sources are, in particular, a diluted gas of non-interacting 
fundamental strings, and we impose as initial condition the flat and 
cold string perturbative vacuum ($a=const$, $\phi= -\infty$), then the 
general solution of the system of background string equations and string 
equations of motion shows that the beginning of the superinflationary 
evolution of the metric, and of the phase of dilaton dominance, are 
nearly coincident at the same time scale [3]. In that case the dilaton 
field is completely responsible for the phase of accelerated evolution.
\vskip 1 cm
\noi
{\bf 3. Discussion}

\noi
According to the string cosmology equations, a phase of dilaton-driven 
isotropic superinflation can thus be arranged with no need of a 
simultaneous shrinking of the internal dimensions (like in the 
conventional realizations of superinflation [14]) or of some fine-tuned 
dilaton potential. Indeed, at low energy scales (with respect to the 
string tension $(\ap)^{-1/2}\simeq 10^{-1}~M_p$) the dilaton potential 
can appear at a non-perturbative level only, and it is expected to be 
negligible, as $V(\phi)\sim \exp [-\exp(-\phi)]$.

A string-motivated dilaton potential must play a fundamental role, 
however, in the transition 
from the accelerated evolution with growing curvature 
of the pre-big-bang regime, to the 
subsequent standard cosmological evolution. As recently discussed 
in [15], a smooth transition seems to be impossible without including 
in the low energy 
string effective action  higher curvature corrections 
and the contribution of a non-perturbative dilaton potential. The 
``graceful exit" from the pre-big-bang to the post-big-bang regime is 
indeed the main unsolved problem, at present, of this scenario, even if 
recent results on exact string solutions (to all orders in $\ap$) are 
encouraging [16]. The non-perturbative potential, in particular, is 
expected to force the dilaton to a minimum which gives it a mass, and 
freezes the value of the Newton constant to its present value.

In addition, the transition from the growing to decreasing curvature 
regime should give rise to a copious non-adiabatic production of 
radiation, in order to explain the large amount of entropy presently 
observed on a cosmological scale. An interesting possibility is that 
such entropy be produced in the decay of the dilatons (or of other 
massive particles) created from the vacuum by the expanding metric 
background [3,17]. In such case, the dilaton freezing is likely to occur 
not simultaneously to, but sensibly after the bounce of the curvature at 
a maximum scale (of Planckian order); moreover, the universe is expected 
to be dilaton-dominated even in the first period of the post-big-bang 
era, before the radiation production. But for what concerns these 
aspects of string cosmology work is still in progress [18], and I hope to 
be able to report more details in some future publication.

\vskip 1 cm
\noi
{\bf Acknowledgements.}

\noi
I am very grateful to M. Giovannini and G. Veneziano for a 
fruitful collaboration which led to the results reported 
in this paper. I wish to thank also F. Occhionero for his kind 
invitation, and for the perfect organization of this excitant 
Workshop on the various aspects of primordial cosmology.

\vskip 1 cm
\centerline{\bf References.}

\item{1.}A. Guth, Phys. Rev. D23 (1981) 347

\item{2.}M. Gasperini and G. Veneziano, Mod. Phys. Lett. A8 (1993) 3701

\item{3.}M. Gasperini and G. Veneziano, Phys. Rev. D50 (1994) No.4

\item{4.}See also M. Gasperini, ``Phenomenological aspects of the 
pre-big-bang scenario in string cosmology", to appear in Proc. of the 
2nd Journ\'ee Cosmologie, Paris, June 1994 (World Scientific, Singapore) 
(astro-ph/9406056)

\item{5.}See for instance E. W. Kolb and M. S. Turner, ``The early 
universe" (Addison-Wesley, Redwood City, Ca, 1990)

\item{6.}See for instance V. Mukhanov, H. A. Feldman and R. Brandenberger, Phys. Rep.
215 (1992) 203

\item{7.}A. Linde, ``Monopoles as big as the universe", Preprint 
SU-ITP-94-2 (astro-ph/9402031);

A. Vilenkin, ``Topological inflation", CTP-MIT Preprint 
(hep-th/9402085)

\item{8.}M. Gasperini and G. Veneziano, Astropart. Phys. 1 (1993) 317

\item{9.}G. Veneziano, Phys. Lett. B265 (1991) 287

\item{10.}M. Gasperini and G. Veneziano, Phys. Lett. B277 (1992) 256

\item{11.}M. Gasperini, N. S\'anchez and G. Veneziano, Int. J. Mod. Phys.
A6 (1991) 3853; Nucl. Phys. B364 (1991) 365

\item{12.}C. Angelantonj, these Proceedings. See also Diploma Thesis, 
Universita' dell'Aquila (1994) (unpublished)

\item{13.}M. Gasperini and M. Giovannini, Phys. Rev. D47 (1993) 1529

\item{14.}D. Shadev, Phys. Lett. B317 (1994) 155;

R. B. Abbott, S. M. Barr and S. D. Ellis, Phys. Rev. D30 (1994) 720;

E. W. Kolb, D. Lindley and D. Seckel, Phys. Rev. D30 (1994) 1205;

\item{15.}R. Brustein and G. Veneziano, ``The graceful exit problem in 
string cosmology", CERN-TH.7179/94

\item{16.}E. Kiritsis and C. Kounnas, ``Dynamical topology change in 
string theory", CERN-TH.7219/94 
(hep-th/9404092)

\item{17.}M. Gasperini, Phys. Lett. B327 (1994) 214

\item{18.}M. Gasperini, M. Giovannini and G. Veneziano, in preparation

\end




