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CERN--TH-6077/91
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SCALE FACTOR DUALITY FOR CLASSICAL AND QUANTUM STRINGS \\
\vspace{10mm}
G. Veneziano \\
{\em Theory Division, CERN, CH-1211 Geneva 23, Switzerland} \\
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\vspace{10mm}
\centerline{Abstract}

\noindent
Duality under inversion of the  cosmological scale factor  is discussed
  both for the classical motion of strings in
cosmological backgrounds and
for genus zero, low energy effective actions. The
string-modified,  Einstein-Friedmann
equations are then shown to possess physically-inequivalent,
 duality-related solutions which generically describe
  the "decay" of an initial, perturbative, flat $D=10$
 superstring vacuum towards a more interesting strong-coupling state
 through an appealing pre-big-bang cosmological scenario.


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CERN--TH-6077/91 \\
April 1991
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\section{INTRODUCTION}

Undoubtedly, one of the deepest quantum symmetries of string theory
is (target space) duality.
In its simplest form \cite{1}, duality says that a (closed)
string moving on a
circle of radius $R$ is equivalent to one which moves on a
 circle of radius
$\lambda_{s} ^{2}/R$ where $\lambda_{s} ^2 = 2 \alpha ^{\prime}\hbar$
is the fundamental
length parameter (Planck constant) of string theory.
Duality has been extended
to more complicated situations \cite{2,3,4} and
is more generally termed
modular invariance (in target space).
It is believed to be an exact symmetry
\cite{5}, at least order by order in the string-loop expansion.

Duality appears to have far-reaching consequences, such as
introducing a minimal
 compactification scale \cite{6}, restricting the
 possible form of scalar (or super)
potentials \cite{7} and determining some characteristics
 of non-perturbative supersymmetry
breaking \cite{8}. It could also have lots to do with
the notion of a minimal
observable scale $O(\lambda_{s})$ in string collisions
\cite{9} and with extended
forms \cite{10} of the Uncertainty   and Equivalence
 principles, although
these concepts appear to retain their validity
irrespectively of compactification.

Under duality the roles of $X'$ and $P\sim\dot{X} $
(winding number and momentum
for zero modes) are interchanged. Indeed, a somewhat
simplified (see below)
derivation of duality \cite{3} consists of performing
a canonical transformation
on the string's position and momentum variables which
are integrated over
in the (Hamiltonian) path integral defining the partition
function $Z$. If no anomaly
gets in the way, this immediately leads to a symmetry
 of $Z$ under certain discrete
changes of the metric and torsion background
fields, which is one of the possible
definitions of duality \cite{3}.

So far duality has been fully discussed
and implemented for a variety
of \underline {constant} background
fields $G_{\mu\nu}, B_{\mu\nu}, ..$..and
there has been some controversy as to the possibility
of extending it beyond such static situations\cite{11}.

In this paper I wish to present some evidence, both
at the classical and at the quantum level, that duality
is  a very useful concept even  for time-dependent backgrounds.
Being a symmetry of the effective action (including classical
stringy sources), at least to lowest order
in derivatives and in the string loop expansion, duality
will relate, in general, physically inequivalent cosmological
solutions of the string-modified Einstein-Friedmann
  equations (which include a non-trivial  dilaton).

Unlike the usual $R$-duality, this symmetry does not
rest on compactification and connects (physically) expanding to
(physically) contracting Universes. To distinguish it $R$-duality,
 I shall refer to it as scale-factor-duality (SFD).

In Sect.2 I shall present some heuristic arguments
for SFD starting from
the classical motion of strings in cosmological backgrounds.
In Sect.3 SDF
will be substantiated by the analysis of the
low-energy, tree-level
string effective action. The final result will
 be a system of modified,
SFD-invariant Einstein equations coupled to
the SFD-invariant classical
sources of Sect.2. In Sect.4 I shall present
some explicit solutions and
physical considerations, while Sect.5 will contain some
conclusions and a rather  speculative outlook.

In this paper I shall only present the general
ideas and some results.
A more detailed account, as well as extensions
 to more general backgrounds, will
be given elsewhere \cite{12,13}.

\section{SCALE FACTOR DUALITY AT THE CLASSICAL LEVEL}
\vspace{1 cm}
The possibility of extending $R$-duality to
time-dependent scale factors
 comes naturally from the study \cite {14,15,16}
of classical string propagation
in homogeneous, isotropic, cosmological backgrounds:

$$g_{\mu\nu}= diag(-1, {a }^2 \left(t\right)) \; ;
\; \mu ,\nu = 0,1,...\left(D-1
\right) $$
In the orthonormal
gauge, the corresponding string
 equations of motion and constraints read
 (i = 1,2, ..., D-1; X$^{0}\rm \equiv $ t)
%]|Expr|[(($^<c$$A^:4;,= X>_^;)=b""0;,= ,M$^X_(";),B0;,,]R<2("|
%|dR("dt> <c%#D^<c!=Q($$^<c!$1^$^X_(#;),G i>_^2;,,M $^<c!$1^$^<c$$1|
%|^X>_^;)i>_^2>^:5;2.W^:4;)i_>]|[
$$\rm {\ddot{X}}^{0}-{X}^{\prime\prime 0}=a{da \over dt}
\sum\nolimits\limits_
{i}\left[{{\left({{X}^{\prime i}}\right)}^{2}-{\left({{\dot{X}}^{i}}
\right)}^{2}}\right]$$
$$\rm {\ddot{X}}^{i}-{X}^{\prime \prime i}={2 \over a}\;{da \over
dt}({X}^{\prime 0}{X}^{\prime i}-{\dot{X}}^{0}{\dot{X}}^{i}), $$
 %]|Expr|[(*$^<c!$1^$^<c$$1^:4;,= X>_^;)=b""0>_^= 2;,,K <c!$1^$|
%|^X_(";),G0>;, ,] $^R_^;)2;, <c%#D^<c!=Q($$^<c!$1^$^X_(#;),G i>|
%|_^2;,,K $^<c!$1^$^<c$$1^X>_^;)i>_^2>^:5;2.W^:4;)i_>]|[
$$\rm {\left({{\dot{\rm X}}^{0}}\right)}^{2}+
\left({{X}^{\prime 0}}\right)=
{a}^{2}\sum\nolimits\limits_{i}
\left[{{\left({{X}^{\prime i}}\right)}^{2}+{\left({{\dot{X}}^{i}}
\right)}^{2}}\right]$$
$$\rm {X}^{\prime 0}{\dot{X}}^{0}=a^{2}\sum\nolimits\limits_{i}
{\dot{X}}^{i}
{X}^{\prime i}.\eqno {(2.1)}  $$
Asymptotic solutions to the system (2.1) were
discussed in \cite{16} (see also \cite {14,15} ) for the case
 $ a(t)\rightarrow
 \infty $
  in the form of a systematic large-$a(t)$ expansion.
Of particular interest
was the superinflationary case ($\dot {H}\equiv
{d\over dt} \left({\dot a\over a}\right) > 0 $)
 of which a typical representative is
$$\rm a(t) = (-t)^{-\gamma} \;\;  (\gamma > 0) \eqno{(2.2)}$$
For such backgrounds, a regime of "high instability" was found
to develop inevitably at late times.  It is
 characterized asymptotically
by \cite {16}:

1. Proportionality of world-sheet time $\tau$ and of
 conformal time $\eta$
where, as usual,
$$\rm ad \eta  \equiv  dt \eqno{(2.3)}$$

2. The stretching ("freezing") of spatial string coordinates:
$$ \rm  {X}^{\prime i} >> {\dot {X}}^{i} \;\; (i=1,2...D-1)
\eqno{(2.4)}$$

3. A negative pressure which, in the ideal gas approximation,
takes the asymptotic
value:
$$ \rm p = -{ \rho \over (D-1)}   \;  ;\; \rho = energy\;
density \eqno{(2.5)}$$
In ref. \cite{17} the case of a rapidly contracting Universe, e.g.
$$\rm   \tilde{a} (t) = (-t)^{\gamma} = {a(t)}^{-1} \;\;
 (\gamma > 0) \eqno{(2.6)}$$
was also considered and solved at "late" times
 ($t \rightarrow 0, \tilde{a} \rightarrow
0$). In this case, the asymptotic solution was instead characterized  
by:

$\tilde{1}$. Proportionality between $\tau$ and $\tilde{\zeta}$ where:

$$\rm  d \tilde{ \zeta}  \equiv  \tilde{a} d\tilde{t} \eqno{(2.7)}$$

$\tilde{2}$. Very fast shrinkage of the string:

$$ \rm  {\tilde{X}}^{\prime i} << \tilde{{\dot{X}}^{i}} \;
\eqno{(2.8)}$$

$\tilde{3}$. An equation of state typical of an ultrarelativistic gas:

$$ \rm \tilde {p} = {\tilde{ \rho} \over (D-1)}  \;  \eqno{(2.9)}$$

Comparison of properties 1,2,3 and $\tilde{1},\tilde{2},\tilde{3} $
 suggests \cite{17} some relation
between these two regimes and the corresponding solutions.
 Since $\tilde{a} ={a}^{-1}$,
the result $\eta = \tilde{\zeta} = \tau$ implies, up to a constant,

$$\rm t = \tilde{t}  \eqno{(2.10)}$$

A detailed study of the equations of motion and constraints
shows that, indeed, one can transform any
solution of the system of equations for the
background metric (2.1) into a solution for the "dual" metric
 by the replacements
$$\rm t \rightarrow  \tilde{t}= t $$
$$ \rm {X}^{\prime i}\rightarrow  \tilde{X}^{\prime i} =
 a^{2} \dot{X}^{i} \equiv P^{i}$$
$$ \rm {P}^{i}\rightarrow  \tilde{P}^{i} \equiv
 a^{-2} \tilde{\dot{X}}^{i} =  X^{\prime i}
\eqno{(2.11)}$$
where the latter two equations are consistent with each other thanks
to the equations of motion.

>From (2.11) and from the definition of energy and pressure density for
an ideal, isotropic gas of strings:
$$\rm {T}_{0}^{0} =\rho \; ; \; {T}_{i}^{j}= - \delta_{i}^{j} \; p$$

$$\stackrel{\rm \mu\nu}{\rm T(x)} =  {1\over \sqrt {- g}
\pi\alpha^{\prime}} \sum_{n}
  \int d\sigma d\tau
({{\dot{X}}^{\mu}_{n}{\dot{X}}^{\nu}_{n}-{X}^{\prime \mu}_{n}
{X}^{\prime \nu}_{n}})\delta
\left({X-x}\right). \eqno{(2.12)}$$
one finds, after use of (2.11),

$$   \sqrt {- g}\rho = \sqrt {-\tilde{ g}}\tilde{\rho} \;\; ; \;
\; \sqrt {- g} p = -\sqrt {-\tilde{ g}} \tilde{p} \eqno{(2.13)}$$
in agreement with (2.5) and (2.9).

A "surprise" comes when one writes \cite{17} the \underline{usual}
 Einstein-Friedmann equations
in the presence of these classical stringy sources: SFD gets badly
broken! Indeed, the left- and right-hand sides of the
equations transform differently
under $ a\rightarrow
\tilde{a} = {a}^{-1}$.
For instance,
one combination of Einstein's equations reads:

$$ \dot{H} = {\ddot{a} \over a} - \left({{\dot{a}\over a}}\right ) ^2
=  -{ 8 \pi G_{D} \over \left({D-2}\right)} \left({\rho +p}\right).
\eqno{(2.14)}$$
The l.h.s. of (2.14) is clearly odd under $ a\rightarrow
\tilde{a} = {a}^{-1}$, while the r.h.s., owing to eq.
(2.13), has no definite symmetry.
As we shall discuss in the next section, the solution of this
puzzle lies in the string's modification of Einstein's equations.
Since these equations reflect the absence of conformal anomalies
in the quantum theory, we can also
say that it is the consistent quantization of strings that restores  
SFD.
\vspace{1 cm}

\section{SCALE FACTOR DUALITY FOR THE EFFECTIVE ACTION}
\vspace{1 cm}

The way duality is implemented at the quantum level is
a little subtle
even for usual time-independent $R$-duality. It involves
a non-trivial
transformation \cite {18} of both the metric $G_{\mu\nu}$
and of the Fradkin-Tseytlin dilaton
 \cite {19} $\phi$ which appear in the Euclidean 2D action as:
$$\rm S =  1/2 \int d^{2} z\sqrt{-\gamma}(\gamma^{\alpha\beta}
\partial_{\alpha}
  {X} ^{\mu}\partial_{\beta} {X} ^{\nu}  G_{\mu\nu}
+ (R/ 4\pi )  \phi ) \eqno(3.1)$$
With this definition of the background fields (and up to
 a numerical factor),
the tree-level string effective action  takes,
 the form  \cite {20}:
$$\rm \Gamma_{eff} = \int d^{D}x \sqrt{-G} e^{-\phi} [ R +
 \partial_{\mu}\phi\partial^{\mu}\phi - V + higher\; deriv.].
\eqno(3.2)$$
where $V = {\left(D-10\right)\over 3}$ and $R$ is the
scalar curvature constructed out of $G_{\mu\nu}$.

It is clear from (3.2) that even the cosmological term (proportional
to $(D-10)$) is \underline {not} invariant under
 $ G \rightarrow G^{-1}$ unless,
at the same time, $\phi$ is also changed:
$$  G \rightarrow G^{-1} ;\; \phi \rightarrow \phi - trln G. \eqno  
(3.3)
$$

A possible way to understand why $\phi$ has to transform non-trivially
under duality, in spite of the fact it does not couple to
$\dot{X}$ or $ X^{\prime}$ in (3.1), is to recall that the
 fields appearing
in $\Gamma_{eff}$ are "renormalized" fields
while those appearing in  S  are "unrenormalized".
The path-integral "proof" of duality \cite {3} implies that
the \underline {bare} dilaton is left invariant. However,
   the relation between bare and renormalized $\phi$ involves,
at one loop order,
precisely a term proportional to $trlnG$ (see e.g.  \cite {21})
 This is a generalization
of an  observation by de Alvis $\cite{22}$ that, at the linearized
level (around a flat $G$), the dilaton mixes with the trace of $G$.

Thus, if the bare dilaton is invariant under duality, the renormalized
dilaton will transform as in (3.3). We have actually checked
that the coefficient in front of the $trlnG$ term is the correct one
if dimensional regularization is used.
More generally, since we expect the renormalized and bare
dilaton to be related
by a local transformation, some generalization of (3.3)
 should also work at
higher orders with just a more complicated, but local,
 transformation law for $\phi$,
as recently found to be the case at the next order \cite {23}.

 Let us now proceed to the case of time-dependent
scale factors. Take
for instance
$$\lambda_{s} ^2 G_{\mu\nu}= diag(-1, {a_i}^2 \left(t\right)) \; ;
 \; B_{\mu\nu}=0 \; ;
\; \phi = \phi \left(t\right) \eqno{(3.4)}$$.

We may ask if the symmetry under (3.3) survives in this case.
Surprisingly
perhaps, the answer is in the affirmative at least up
 to second order in the
space-time derivatives (slowly varying fields).  Furthermore, for
backgrounds of the type (3.4), the symmetry
group appears to be extendible from a single $Z_2$ to
 ${Z_2}^{d}$ defined
 for each $ i= 1,2,.. d $ by:
$$\rm a_{i}(t) \rightarrow a^{-1}_{i}(t) \; ;\; \phi \rightarrow
\phi - 2 ln a_{i}  \eqno{(3.5)}$$
This symmetry can be further extended, as shown in \cite{12}.

If one looks at the effective, low energy string action as
being a generic Brans-Dicke-type \cite{24} modification
of Einstein's gravity, one concludes that the $\omega$ parameter
of Brans-Dicke is fixed in string theory, by duality,
to take the value $-1$.
   Such a value is unacceptably small, phenomenologically,
 unless the dilaton
picks up a mass \cite{25}, presumably through the same mechanism that
breaks supersymmetry.

We may ask at this point whether SFD is  really
distinguished from $R$-duality even in the case of compact
dimensions. In order to see that this is so, consider
the case of
 a single circle
of fixed radius $R$ and a time-dependent scale factor $a(t)$.
 A naive
extension \cite{26} of $R$-duality would connect this situation
to the one in which
$$ aR/\lambda_{s} \rightarrow (aR/\lambda_{s})^{-1} \eqno{(3.6)}$$
However, these  $R$-duality-related situations  both
describe   a contracting or expanding circle
according to whether $aR/\lambda_{s}$ approaches the fixed point
value $1$ or moves away from it.
  Instead, by fixing
   $a(0)=\tilde{a}(0)=1$ and by letting the two evolve
according to (3.5),
 we are connecting, through SFD, a physically expanding
($ aR/\lambda_{s} \rightarrow \infty $) to a physically
contracting ($ aR/\lambda_{s} \rightarrow 1$) Universe.

It follows from the previous discussion that the correct
interpretation of SFD is not that of a true symmetry, but
rather of a group acting on the vacuum manifold and
  transforming solutions of the field equations into other
 (generally inequivalent) solutions.   SFD thus appears
  to be a generalization of Narain's construction \cite{27} to
the case of possibly non-compact, time-dependent
backgrounds. This analogy will be made
much more complete in ref. \cite{12}.

The fact that SFD is a symmetry of the action
 can be verified immediately at the level of the equations
of motion that
 follow from (3.2). In the general case these read:

$$\rm    R - \partial_{\mu}\phi \partial^{\mu}\phi +
 2 D_{\mu} D^{\mu} \phi
 - V = 0$$
$$\rm R_{\mu\nu} +D_{\mu} D_{\nu} \phi =0.
 \eqno{(3.7)}$$
where $D_{\mu}$ denotes the usual covariant
 derivative of General Relativity.

For our ansatz (3.4) eqs. (3.7) can be reduced
to the following set of independent equations:
$$\rm \sum H_{i}^{2} - (\dot{\phi}-\sum H_{j})^{2}+ T^{-2}  = 0$$
$$ \dot {H_{i}} - H_{i}(\dot{\phi}-\sum H_{j}) =0$$
$$ H_{i}\equiv \dot{a}_{i}/a_{i}\;,\; T\equiv\lambda_{s} V^{-1/2}
= \lambda_{s} {\left(D-10\over 3 \right)}^{-1/2} \eqno{(3.8)}$$
which are clearly invariant (respectively even and odd) under SFD.

We note, incidentally, that eqs.(3.8) admit, for $D>10$,
the particular
solution \cite{28}:
$$\rm \phi = Qt \;\;  ;  \; Q^{2} = T^{-2} = (D-10)/3\lambda_{s} ^{-2}
 \eqno{(3.9)}$$
which is well known to provide an exactly conformal invariant theory
to all orders in $\lambda_{s}$. However, as discussed in \cite{15} (see
also \cite{29}), these solutions do not seem to describe a physical
expansion of the Universe.
In this paper we shall stick to the claim  \cite{15} that the role of
Einstein's metric is played in string theory by $G_{\mu\nu}
\lambda_{s}^{2}$.

Before giving solutions to the above equations, let us combine
the results
obtained so far by coupling the classical string sources
described in sect.2
to the string-modified Einstein equations of this section.
This straightforward excercise yields the following modification
of eqs.(3.8)

$$\rm \sum H_{i}^{2} - (\dot{\phi}-\sum H_{j})^{2}+ T^{-2}
 = - \kappa
e^{\phi} \rho $$
$$ \dot {H_{i}} - H_{i}(\dot{\phi}-\sum H_{j})  = 1/2 \kappa
e^{\phi} p_{i}$$
$$ \kappa e^{\phi} = \alpha' e^{\phi} \lambda_{s}^{D-4} \sim 8\pi
G_{N}$$
$$ \dot{\rho} + \sum H_{i} (\rho + p_{i}) = 0. \eqno{(3.10)}$$
where we notice that, in this case, one
of the field equations, instead of being  redundant, yields the energy
conservation equation.

It is now clear, after use of (2.13), that, unlike for the case of
 Einstein's equations, the left- and
right-hand sides  of (3.10) transform in the same way under SFD.


\section{SOLUTIONS AND PHYSICAL CONSIDERATIONS}
\vspace{1 cm}


 In this paper I shall consider in some detail the case in which
the effect
of classical string matter can be neglected, commenting briefly
on more general situations. A detailed study of the general case
is being made and will appear elsewhere \cite{13}.  The general
solution of eqs. (3.7) can be written explicitly and reads $(D>10)$
$$\rm a_i(t) = a_i(-\infty)(\tanh(\pm \tau /2))^{\alpha_{i}}$$
$$\rm \bar{\phi}(t) \equiv \phi (t) -\sum ln a_{i}(t)
= -ln \sinh(\pm \tau) + \bar{\phi}_{0}$$
$$ \tau = t/T  \; ,\; \sum \alpha_{i}^{2} = 1 \; \eqno{(4.1)}$$
Two remarks are in order:
\begin{itemize}
\item[i)] in general the solutions are defined on a half-line in $t$.
 Equivalently, there is in general a singularity at some finite value  
of
$t$ here taken conventionally to be $t=0$.
If, for the moment, we consider  solutions defined for $t<0$
 the minus sign
has to be chosen in eq. (4.1);
\item[ii)] the known, all-order solution (3.8) can be recovered  
formally
by taking the limit $t\rightarrow - \infty $.
\end{itemize}

The solution (4.1) is general since it depends
on $2D-1 $ integration
 constants (this includes the time at which the
singularity occurs) and, indeed,
the solution is completely determined once the
initial values of $a_{i} , H_{i} $ and
 $\phi$ are given.
 It describes a cosmological evolution
from $t=-\infty$ to $t=0$ whereby the Universe is
 initially very flat
(and isotropic):
$$\rm H_i = - \alpha_{i} /T \sinh^{-1}(-\tau)
\rightarrow 0 \eqno{(4.2)}$$
and the D-dimensional coupling
$$\rm \lambda_{D} = exp(\phi) \rightarrow const.
exp( \tau ) \rightarrow 0  \eqno{(4.3)}$$
is very weak. Hence, the initial state is perturbative from the
point of view of both the $\sigma$-model and the
string-loop expansion.

As long as $\tau\ll -1$ ($t \ll-T$), things do not
 change much.
However, from $\tau = O(-1)$ onward, scale factors
 begin to vary
faster and faster. Depending upon the signs of the $\alpha_{i}$'s, some
dimensions undergo (super)inflation or very fast
 contraction, i.e. precisely
the kind of behaviours found in \cite{17} from solving Einstein's
equations in the presence of highly unstable
strings in a self-consistent
way. This  certainly gives a hint that
the addition of classical string sources will not
modify qualitatively the solutions of the pure
 gravity-plus-dilaton
system. Notice here that it is SFD that allows,
for any solution with
an expanding dimension, one with a reciprocally
 contracting dimension:
anisotropic cosmologies are natural alternatives to
isotropic ones in SFD-invariant theories.

As $t \rightarrow 0^{-}$,  $H_i$  blows up
like ${\alpha_{i}\over t } $ while
the behaviour of $\lambda_{D}$ depends on the
actual value of $\sum\alpha_{i}$ via
$$\lambda_D \sim (a)^{\sum \alpha_{i}-1} (1-a^{2})\; ,
 \; a\equiv \tanh(-\tau/2)
\rightarrow 0 \eqno{(4.4)}$$

Thus $\lambda_{D}$ goes to zero, to a finite constant or
 to infinity for
 $\sum\alpha_{i}> 1$, $\sum\alpha_{i}=1$ and
 $\sum\alpha_{i}< 1$, respectively.
 However, the relevant, effective coupling in the
 expanding dimensions after
they have become much larger than all the contracting ones
 is related to $\lambda_{D}$ by
$$\lambda_{eff}^{-1} = \lambda_{D}^{-1}\prod_{contr.\;dim's} a_i
 \eqno{(4.5)}$$
and is thus easily seen to be always growing as one approaches
 the singularity.
In conclusion, our solutions represent an evolution
 from flat D-dimensional
space-time and weak coupling to a regime of high
 curvatures  and
large coupling through a period of super-inflation
 and dimensional
reduction. The duration of inflation and the
corresponding "e-folding" factor
$$\rm N\equiv ln({a_{final}\over a_{initial}}) \;\eqno{(4.6)}$$
are determined by requiring that, at the end of inflation, our
approximations are still valid, implying, at least:
$$\rm -t_{final} \ge\lambda_{s}\; ; \;  \vert H_{final}
\vert < O(\lambda_{s}^{-1})
\; ; \; \lambda_{D}<O(1). \eqno{(4.7)}$$
Consequently one finds:
$$\rm N\le O( \lambda_{s}^{-1}T) = O((D-10)^{-1/2})
 \;\eqno{(4.8)}$$

We thus see that there is no hope of getting
a large amount of inflation in the
case of a tree-level cosmological constant.
 Paradoxically perhaps, the
amount of inflation increases for decreasing $\Lambda_0$,
 the reason being
that the smaller $\Lambda_0$, the longer inflation will last.

The above considerations suggest considering the case of
critical dimensions,
$D=10$. In this case,
   because of non-renormalization theorems, no potential
is generated at any finite order in the string
loop expansion. Neglecting non-perturbative effects,
 eqs.(4.1) still give
the general solution by letting $\tau \rightarrow 0$. One finds:
$$\rm a_i(t) = a_i(-t_{0})(-t /t_{0} )^{\alpha_{i}}$$
$$\rm \bar{\phi}(t) =  -ln (t/ t_{0}) $$
$$  \sum \alpha_{i}^{2} = 1 \; \; (D=10) \; \eqno{(4.9)}$$
The amount of inflation is only limited now
 by the initial values
of the Hubble constants $H_{i}$ and of the dilaton.
A simple calculation yields:
$$ N < min( -\phi_{initial}, -1/2 ln (\lambda_{s}^{2}
\sum H_{i}^{2})) \;
\eqno{(4.10)}$$
 If we imagine   to start, at some initial time,
with a slight perturbation
of the $D=10$ flat, weakly coupled, supersymmetric vacuum, we can
easily achieve large enough values of $N$ for solving the standard
cosmological problems.

On the other hand, non-perturbative, supersymmetry
breaking effects are expected to produce a
non-vanishing $V$ of order \cite{8}
$$\rm V \sim exp(-c\;exp(-\phi)) \ll 1 \;
, \; (c>0\; ,\; \phi \ll -1 ) \eqno{(4.11)}$$
Furthermore, the potential will depend,
 in general, on the $a_{i}$ through the
combination $ r_{i}(t) \equiv a_{i}R/\lambda_{s}$ if the ith.
direction is a circle of radius $R$ with a
 dependence of $V$ on the $r_{i}, $   itself
restricted by modular invariance \cite {7,8}.

In this case   eqs. (3.10) can be shown to become:
$$\rm \sum H_{i}^{2} - (\dot{\phi}-\sum H_{j})^{2}+
 V \lambda_{s}^{-2}
 = - \kappa e^{\phi} \rho $$
$$ \dot {H_{i}} - H_{i}(\dot{\phi}-\sum H_{j})
 +1/2 (\partial   V
/\partial \phi) + 1/2 (\partial   V
/ \partial ln {a_{i}}) = 1/2 \kappa  e^{\phi} p_{i}$$
$$ \dot{\rho} + \sum H_{i} (\rho + p_{i}) = 0. \eqno{(4.12)}$$

 The above equations appear to break SFD. Only ordinary $R$ duality
is strictly preserved in the compactified dimensions
thanks to the symmetry properties of V.
Eqs. (4.12) can no longer be solved in closed form for a generic
$V$ and their analysis will be investigated elsewhere \cite {13}.
Nonetheless, we can already anticipate that the possibility of
a long inflation and of a large e-folding factor
appears to be preserved even in this case.
\section{CONCLUSIONS AND SOME "PHILOSOPHY"}
\vspace{1 cm}
Obviously, any realistic situation will have to be
much more complicated
than the one described by the system of eqs. (3.10). Nonetheless,
we may hope  some general features of the real world to be shared
by the solutions of the simpler system.

Consider, for instance, eqs. (4.12) with some generic $V$ satisfying
(4.11) and depending  on the "radii" $r_i$ in a modular-invariant way.
Let us start evolving from a "very classical" situation, i.e.
$$ H_i\ll \lambda _{s}^{-1} \; , \; \phi \ll -1 \; \eqno{(5.1)}$$
Under these conditions, eqs. (4.12) imply,
at early times,
$$\rm  (\dot{\phi}-\sum H_{j}) \sim \pm \sqrt {\sum H_{i}^{2}} \; ,
\; {d\over dt} ( \sum H_{i}^{2})^{-1/2} \sim \mp 1 \; \eqno{(5.2)}$$
and thus a two-fold ambiguity.

We thus see that the choice of sign depends, very generally,
 on whether one
wants to describe a late- or an early-time solution.
 Since, evidently, we
do not want to describe today's Universe in terms of a Kaluza-Klein
cosmology with a time-dependent gravitational
and gauge coupling, we are
forced,  by physics, to choose the early-time solution
(upper signs in eq. (5.2)). We thus
obtain an interesting cosmological scenario whereby
  a classical, weak coupling, small curvature regime
 evolves naturally into
 a quantum era, with large curvatures and/or coupling.
This is naturally identified with the "big-bang", i.e. with the
beginning of our epoch. Of course the (semi)classical
 description exhibits
a singularity and, as such, cannot
  be continued across  $t=0$. Hopefully, this
just reflects the  inadequacy of the semiclassical picture, while
 quantum string theory has a way to go through the singularity on to
$t>0$.

What will happen during the fully quantum era is clearly
 mere speculation
 since we do not know,
at present, how to tackle such a complicated non-perturbative regime.
As already pointed out, SFD is expected to be broken as soon as higher
loops and/or compactification effects will be felt.
Hopefully, this will
select, out of all SFD-related solutions,
those evolutions where six dimensions contract
while the other three expand. During the quantum era,
 at least two nice
"miracles" should take place: the freezing of
the internal dimensions
to scales $O(\lambda_{s})$ and the freezing of
the dilaton at a value $O(1)$
with generation of a dilaton mass.
 Only under these circumstances the way could
be paved for starting a more
conventional kind of cosmology at $t>O(\lambda_{s})$.

\vspace{1 cm}
After completion of this work I became aware of ref. \cite {30}
 where the solution (4.1)
to the $\beta $ -function equations already appears.

 \vspace{1 cm}


{\bf ACKNOWLEDGEMENTS}
\vspace{1 cm}

I am grateful to M. Gasperini and N. Sanchez for stimulating
discussions which  motivated  this work as well as  for
many subsequent suggestions. I am also grateful to K. A. Meissner
for discussions and for having rechecked
 most of my own calculations. Finally, I wish to thank
 E. Alvarez, D. Amati, S. Ferrara,
  E. Rabinovici and A. Tseytlin for  useful comments.

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 \end{document}