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On T-Duality in Brane Gas Cosmology

Timon Boehm 1 '* and Robert Brandenberger 2, ^

1 Departement de Physique Theorique, University de Geneve,

24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland.

2 Institut d'Astrophysique de Paris, 98bis Blvd. Arago, F-75014 Paris, France,

Physics Departm

, Brown University, Providence, RI 02912, USA.
(Dated: February 1, 2008)

In the context of homogeneous and isotropic superstring cosmology, the T-duality symmetry of
string theory has been used to argue that for a background space-time described by dilaton gravity
with strings as matter sources, the cosmological evolution of the Universe will be nonsingular. In
this Letter we discuss how T-duality extends to brane gas cosmology, an approximation in which
the background space-time is again described by dilaton gravity with a gas of branes as a matter
source. We conclude that the arguments for non-singular cosmological evolution remain valid.

PACS numbers: 98.80.Cq, ll.25.-w

INTRODUCTION

In [1] it was suggested that due to a new string theory-
specific symmetry called T-duality, string theory has the
potential to resolve the initial singularity problem of
Standard Big Bang cosmology, a singularity which also
plagues scalar field-driven inflationary cosmology [2, 3].

The framework of [1] was based on an approximation
in which the mathematical background space-time is de-
scribed by the equations of dilaton gravity (see [4, 5]),
with the matter source consisting of a gas of strings. The
background spatial sections were assumed to be toroidal
such as to admit one-cycles. Thus, the degrees of freedom
of the string gas consist of winding modes in addition to
the momentum modes and the oscillatory modes. Then,
both momentum and winding numbers take on discrete
values, and the energy spectrum of the theory is invariant
under inversion of the radii of the torus, i.e. R — > a'/R,
where a x / 2 is the string length l s . The mass of a state
with momentum and winding numbers n and u, respec-
tively, in a compact space of radius 1?, is the same as that
of a state with momentum and winding numbers w and
n, respectively, in the space of radius a'/R. This sym-
metry was used to argue [1] that as the radii of the torus
decrease to very low values, no physical singularities will
occur. Firstly, under the assumption of thermal equilib-
rium, the temperature of a string gas at radius R will
be equal to the temperature of the string gas at radius
a'/R. Secondly, any process computed for strings on a
space with radius R, is identical to a dual process com-
puted for strings on a space with radius a'/R. Therefore
there exists a 'minimal' radius a 1 I 2 in the sense that
physics on length scales below this radius can equally
well be described by physics on length scales larger than

Since the work of [1] our knowledge of string theory has
evolved in important ways. In particular, it has been re-
alized [6] that string theory must contain degrees of free-
dom other than the perturbative string degrees of free-
dom used in [1]. These new degrees of freedom are Dp-
branes of various dimensionalities (depending on which
string theory one is considering). Since the T-duality
symmetry was used in an essential way (see e.g. [7]) to
arrive at the existence of Dp-branes (p-branes for short
in the following), it is clear that T-duality symmetry
will extend to a cosmological scenario including p-branes.
However, since a T-duality transformation changes the
dimensionality of branes, it is useful to explicitly verify
that the arguments of [1] for a non-singular cosmological
evolution carry over when the gas of perturbative string
modes is generalized to a gas of branes. A model for
superstring cosmology in which the background space-
time is described (as in [4, 5]) by dilaton gravity, and
the matter source is a gas of branes, has recently been
studied under the name of "brane gas cosmology" [8, 9]
(see also [10, 11] for extensions to backgrounds which are
not toroidal, and [12] for an extension to an anisotropic
background).

In this Letter, we establish the explicit action of T-
duality in the context of brane gas cosmology on a
toroidal background. For a solution of the background
geometry appropriate for cosmological considerations in
which the radii of the torus are decreasing from large to
small values as we go back in time, we must consider T-
dualizing in all spatial dimensions. We demonstrate that
the mass spectrum of branes remains invariant under this
action. Thus, if the background dynamics are adiabatic,
then the temperature of the brane gas will be invariant
under the change R — > a'/R, i.e.

T(R) -.

-(!)■

(1)

thus demonstrating that superstring cosmology can avoid
the temperature singularity problem of standard and in-
flationary cosmology. In the appendix we make further

remarks on why we expect brane gas cosmology to be
non-singular. Another crucial assumption is that the
string coupling constant g is small (compared to one)
such that back-reactions of the string and brane gas on
the curvature of space-time can be ignored. Similarly,
our results can be used to show that the arguments for
the existence of a minimal physical length given in [1]
extend to brane gas cosmology.

The outline of this Letter is as follows. In the following
section we give a brief review of brane gas cosmology. In
Section III we (partially re-)derive the energy, the mo-
mentum, and the pressure for p-branes. Next, we review
the action of T-duality on winding states of p-branes.
The main section of this Letter is Section V in which
we show that the mass spectrum of a p-brane gas of su-
perstring theory is invariant under T-duality. Section VI
contains a discussion of some implications of the result,
and conclusions.

II. REVIEW OF BRANE GAS COSMOLOGY

As already mentioned in the Introduction, the frame-
work of brane gas cosmology consists of a homogeneous
and isotropic background of dilaton gravity coupled to a
gas of p-branes as a matter source. We are living in the
bulk 1 .

The initial conditions in the early Universe are 'conser-
vative' and 'democratic'; conservative in the sense that
they are close to the initial conditions assumed to hold
in standard big bang cosmology (i.e. a hot dense gas of
matter) , democratic in the sense that all 9 spatial dimen-
sions of critical string theory are considered on an equal
basis 2 . Thus, matter is taken to be a gas of p-branes of
all allowed values of p in thermal equilibrium. In parti-
cular, all modes of the branes are excited, including the
winding modes.

The background space-time is taken to be M x T 9 where
T 9 denotes a nine-torus. The key feature of T 9 which is
used in the analysis is the fact that it admits one-cycles
which makes it possible for closed strings to have con-

1 In tins sense, brane gas cosmology is completely different in ideol-
ogy than brane world scenarios in which it is assumed (in general
without any dynamical explanation) that we live on a specific
brane embedded in a warped bulk space-time. From the point
of view of heterotic M-theory [13], our considerations should be
viewed as applying to the 10 dimensional orhilold space-time on
which we live.

2 For consistency, critical superstring theories need a 10 dimen-
sional target space-time which is in apparent contradiction with
the observed four. Usually it is assumed that six dimensions
are compactified from the outset due to some unknown physics.
However, following the usual approach in cosmology it seems
more natural that initially a.U nine spatial dimensions were com-
pact and small, and that three of them have grown large by a
dynamical <!■ compactification process. A corresponding s
was originally proposed in [1].

served winding numbers 3 . It is also assumed that the
initial radius in each toroidal direction is the same, and
comparable to the self-dual radius a 1 ' 2 . Initially, all di-
rections are expanding isotropically with R > a 1 / 2 (the
extension to anisotropic initial conditions has recently
been considered in [12]).

As shown in this Letter, as a consequence of T-duality
symmetry, brane gas cosmology provides a background
evolution without cosmological singularities. The sce-
nario also provides a possible dynamical explanation for
why only three spatial dimensions can become large.
Winding modes (and thus T-duality) play a crucial role
in the argument. Let us first focus on the winding modes
oi fuudamcnlal si rin!>,s j 1 1.

The winding and anti-winding modes u and ui of the
strings are initially in thermal equilibrium with the other
states in the string gas. Thermal equilibrium is main-
tained by the process

u + u> <=i loops, radiatio

(2)

When strings cross each other, they can intercommute
such that a winding and an anti-winding mode annihilate,
producing fundamental string loops or radiation without
winding number. This process is analogous to infinite
cosmic strings intersecting and producing cosmic string
loops and radiation during their interaction (see e.g. [14,
15] for reviews of cosmic string dynamics).

As the spatial sections continue to expand, matter de-
grees of freedom will gradually fall out of equilibrium. In
the context of string gas cosmology with R > a 1 > 2 , the
winding strings are the heaviest objects and will hence
fall out of equilibrium first. Since the energy of a wind-
ing mode is proportional to R, Newtonian intuition would
imply that the presence of winding modes would prevent
further expansion. This is contrary to what would be
obtained by using the Einstein equations. However, the
equations of dilaton gravity yield a similar result to what
is obtained from Newtonian intuition [4] : the presence of
winding modes (with negative pressure) acts as a confin-
ing potential for the scale factor.

As long as winding modes are in thermal equilibrium,
the total energy can be minimized by transferring it
to momentum or oscillatory modes (of the fundamen-
tal string). Thereby, the number of winding modes de-
creases, and the expansion can go on. However, if the
winding modes fall out of equilibrium, such that there
is a large number of them left, the expansion is slowed
down and eventually stopped. If we now try to make d
of the original 9 spatial dimensions much larger than the
string scale, then an obstruction is encountered if d > 3:
in this case the probability for crossing and therefore for
equilibrating according to the process (2) is zero. On the

3 Recently, the scenario of [1, 8] was generalized [10, 11] to spatial
backgrounds such as Calahi-^ ati manifolds which admit 2-cycles
but no 1-cycles.

other hand, in a three-dimensional subspace of the nine-
torus, two strings will generically meet. Therefore, the
winding modes can annihilate, thermal equilibrium can
be maintained, and, since the decay modes of the winding
strings have positive pressure, the expansion can go on 4 .
As a result, there is no topological obstruction to three
dimensions of the torn: growing huge while the other six
are staying small (of size R ~ a l ^ 2 ) 5 . Large compact
dimensions are not in contradiction with observations if
their radius is bigger than the Hubble radius today.

It is not hard to include p-branes into the above sce-
nario [8]. Now the initial state is a hot, dense gas of
all branes allowed in a particular theory. In particular,
brane winding modes are excited, in addition to modes
corresponding to fluctuations of the brane. Since the
winding modes play the most important role in the dy-
namical decompactification mechanism of [1, 8], we will
focus our attention on these modes. The analogous clas-
sical counting argument as given above for strings yields
the result that p-brane winding modes can interact in at
most 2p + 1 spatial dimensions. Since for weak string
coupling and for spatial sizes larger than the self-dual ra-
dius the mass of a p-brane (with a fixed winding number
in all of its p spatial dimensions) increases as p increases,
p-branes will fall out of equilibrium earlier the larger p is.
Thus, e.g. in a scenario with 2-branes, these will fall out
of equilibrium before the fundamental strings and allow
five spatial dimensions to start to grow [8] . Within these
five spatial dimensions, the fundamental string winding
modes will then allow only a three-dimensional subspace
to become macroscopic. Thus there is no topological ob-
struction to the dynamical generation of a hierarchy of
internal dimensions.

This section is devoted to the derivation of physical
quantities describing the brane gas which determine the
cosmological evolution of the background space-time.

Starting from the Dirac-Born-Infeld action, we obtain
expressions for the energy and the momentum of a p-
brane in D = d + 1 dimensional space-time. We show
that there is no momentum flowing along the p tangential
directions. From the energy-momentum tensor one can
also define a pressure, and hence an equation of state, for

4 Since quantum mechanically, I tie I tiickness of the strings is given
by the string length [16], it is important for the brane gas sce-
nario that the initial size of the spatial sections was string scale.
Otherwise, it would always be the total dimensionality of space
which would be relevant in the classical counting argument of
[1], and there could be no expansion in any direction.

5 When making these considerations we have neglected the possi-
bility that closed strings may break up (in analogy to Hadron
fragmentation in QCD). The amplitude of this process should be
investigated quantitatively.

the whole brane gas. Even though some of the results in
this section are already known, we find it useful to give
a self-consistent overview.

Let a = (a ,a l ),i = !,-■■ ,p, denote some intrinsic
coordinates on the worldsheet of a p-brane. Its position
(or embedding) in D-dimensional space time is described
by x M = A M (cr), where fi = 0, • • ■ ,d, and the induced
metric is ~f a b = r] ll , v X^ a X" b , where a, b = 0, • • • ,p.

The Dirac-Born-Infeld action is (in the string frame)

T p f d p+1 a.

where T p denotes the tension (charge) of a p-brane, 7 =
det(7 a b), and <f> is the dilaton of the compactified theory.
For our adiabatic considerations, we assume that it is
constant (taking its as\ mptotic value), and absorb it into
a physical tension

1

1

" (2tt)p ga'iP+ 1 )/ 2 '

(4)

where in the final step we have used the expression for T p
(see e.g. [7]). Note that for any p-brane t p goes like 1/g,
and that hence, in the weak string coupling regime which
we are considering, the branes are heavy. The action (3)
can be written as an integral over D-dimensional space-
time

S p = f d D x (-t p J ' dP +1 a5 {D \x^ - X^(a))y/^\ .

As the integration domain is a torus, both integrals are
finite.

Varying the action (5) with respect to the background
metric and comparing with the usual definition of the
space-time energy-momentum tensor, one obtains 6

T^(xn = (6)

-Tp J d p+1 <j5 {D) (x^ - X lt (<r))y/=rr ab X°X? b .

The DBI action (3) is invariant under p + 1
reparametrizations a — > &(o), and we can use this free-
dom to choose

-V=7,

= 0.

Notice that det(7ij) = — \/— 7 and "f lk ^kj = &)■ By
choosing this gauge, we do not specify a particular em-
bedding which will be convenient later when treating a
brane gas, where the branes have arbitrary orientations.
Furthermore, it is consistent to set X° = a .

6 In curved backgrounds the et
plied by ^f=g.

X»). (8)

To calculate the energy E p of a p-brane, (
that in the gauge (7)

7 a6 X a X b = 7 00 = - 7 L

=> T 0C V) = Tp f d p+1 a5 (D) {x"

Writing x M = (t,x n ),n = 1, • • • , d, and splitting the
delta- function into a product, the integral over a can be
carried out. The energy density of a p-brane in d spatial
dimensions is

p p = T 00 (t, x n ) =t p f dPa6^ d \x n - X n (t, a 1 )), (9)

and its total energy is

E p = d d xp p = t p d p a = T p Vol p . (10)

The volume of a p-brane in its rest frame, Vol p , is finite as
the brane is wrapped around a torus. Eq. (10) provides
a formula for the lowest mass state, M p = E p , which
will be used in section V. As expected intuitively, the
minimal mass is equal to the tension times the volume of
a brane.

To calculate the space-time momentum P™ of a p-
brane, one first evaluates

=> T 0n (x») = r p f dP +1 a5^ {x" - X"(<r))X%(U)

Proceeding similarly as before, the total momentum of a
p-brane is found to be

P^ = T p f d?<jX n {t,(J l ),

(12)

where the dot denotes the derivative w.r.t. time t.
The gauge conditions (7) can be written as = 7oi =
X m X m ^, where the sum over m = 1, • • • ,d is the or-
dinary Euclidean scalar product. This is equivalent to
saying that the (spatial) velocity vector X is perpendic-
ular onto each of the tangential vectors X™. Therefore,
only the transverse momentum is observable 7 . Assum-
ing that the brane is a pointlike classical object w.r.t.
the transverse directions, this momentum is not quan-
tized despite of the compactness of space. In particular,
the question whether there might exist a T-duality cor-
respondence between transverse momentum modes and
winding modes does not arise. Moreover, we neglect the

7 Note the analogy with 1 1 , lefects in field theory, where

also only the transverse momentum of the defects - here taken
to be straight - is observable.

possibility of open strings travelling on the brane which
would in fact lead to a non-zero tangential momentum. 8
Hence, in what follows, we focus on the zero modes of
p-branes.

Finally, the pressure V p of a p-brane is given by aver-
aging over the trace T™. First, notice that

7 a6 X™X m , 6
= j 00 X™X mfi + 7 n 7 ii + • • • + l PP lp P

+2 7 12 72i + • • • + 2 7 lp 7 P i

+2 7 23 732 + • • • + 2 7 2p 7 p2 + • • •

+27 P " 1 ' P 7 P , P -i
= --}=X" n l X mfi +p. (13)

In the first step we have used that the products of the
embedding functions can be expressed in terms of the
induced metric, e.g. ~ n = A"'['A,„.i. an( l hi the second
step the fact that j lk Jkj = Sj- Inserting this into Eq.
(6), eliminating the remaining ^—7 by — y/— 7 = 700 =
— 1 + J™ X m fi, and integrating out the er° dependence,

:/;:(/.,-"

(14)

p [<Pa5W(x'

-X n (t,a i ))[(p+l)X^X m -p}.

The quantity X m X m is the squared velocity of a point
on a brane parameterized by (£, a 1 ). We define the mean
squared velocity of the branes in the gas by averaging
over all a\ i.e. v 2 (t) = (X m X m ). In the averaged trace,
(T™), the velocity term can be taken out of the integral.

Comparing with Eq. (9),
state of a p-brane gas

■P P ^-AT-) =

: obtains the equatio

of

Pp ■

(15)

In the relativistic limit (v 2 — > 1) the branes behave like
ordinary relativistic particles: V p = \p v , whereas in the
non-relativistic limit (v 2 — > 0) V p = —^Pp- For domain
walls this result was obtained in [18].

The pressure V p and the energy density p p are the
source terms in the Einstein equations for the brane gas
[4, 8].

IV. WINDING STATES AND T-DUALITY

We briefly review some of the properties of T-duality
that are needed subsequently.

Consider a nine-torus T 9 with radii (i?i, • ■ • ,R g ). Un-
der a T-duality transformation in n-direction

and all other radii stay invariant. T-duality also acts on
the dilaton (which is constant in our case) , and hence on
the string coupling constant, as

/<\

(17)

Note, however, that the fundamental string length l s =
a 1 ' 2 is an invariant. The transformation law (17) follows
from the requirement that the gravitational constant in
the effective theory remains Invariant under T-duality. In
general, T-duality changes also the background geometry.
However, a Minkowski background (as we are using here)
is invariant.

For a p-brane on T 9 , a particular winding state is
described by a vector uo = (ui,--- ,u)g). There are
9!/[p!(9 — p)\] such vectors corresponding to all possible
winding configurations. For illustration take a 2-brane
on a three-torus: it can wrap around the (12), (13),
(23) directions, and hence there are 3!/2! = 3 vectors
LU = (iOi,LU 2 ,0),LU = (uji,0,uj 3 ),lu = (0,^2,^3).

Whereas T-duality preserves the nature of a funda-
mental string, it turns a p-brane into a different ob-
ject. To see this consider a brane with p single windings
uo = (1, ■ ■ ■ , 1, 0, • • • ,0) which represents a p-dimensional
hypersurface on which open strings end. Along the brane
the open string ends are subject to Neumann boundary
conditions. These become Dirichlet boundary conditions
on the T-dual coordinate R' n (if n denotes a tangential
direction), i.e. for each string endpoint R' n is fixed. Thus
a T-duality in a tangential direction turns a p-brane into
a (p-l)-brane. Similarly, a T-duality in an orthogonal di-
rection turns it into a (p+l)-brane (see e.g. [7] for more
details) .

Next, consider a T-duality transformation in a direc-
tion in which the p-brane has multiple windings to n > 1.
One obtains a number uj n of (p-l)-branes which are
equally spaced along this direction. As an example take
a 1-brane with winding lo\ = 2 on a circle with radius
R\. This configuration is equivalent to a 1-brane with
single winding on a circle with radius 2R\. T-dualizing
in 1-direction gives a single 0-brane on a circle of ra-
dius a! J2R\ which is equivalent to two 0-branes on a
circle of radius a 1 jR\ (see e.g. [19]). Since applying a T-
duality transformation twice in the same direction yields
the original state (up to a sign in the RR field), also the
inverse is true: a number co n of (p-l)-branes correspond
to a single p-brane with winding ui n .

So far we have discussed T-duality transformations in
a single direction. For applications to isotropic brane
gas cosmology we need to consider T-dualizing in all
nine spatial directions. Given a gas of branes, B, on
a nine-torus with radii (Ri, ■ ■ ■ , Rg) consisting of a large

number of branes of all types admitted by a particular
string theory, we want to find the corresponding gas B*
on the dual torus T* with radii (R[, ■ ■ ■ ,R' 9 ). To that
end one performs a T-duality transformation in each of
the nine spatial directions. From what we have discussed
so far, it is now easy to see that a p-brane in a winding
state lu = {ui\, • • • , ojp, 0, • • • ,0) is mapped into a num-
ber lo\ ■ ■ ■ LU p of (9-p)-branes. each of which is in a state
uo* = (0, • • • , 0, 1, . . . , 1). The (9-p)-brane wraps in the
(9-p) directions orthogonal to the original p-brane. It is
clear that the above considerations hold for any winding
configuration.

After these preparatory steps, we now turn to the main
part of this Letter.

MASS SPECTRA AND T-DUALITY

A. Masses of p-brs

3 with single

nding

In this section we show that each mass state in a brane
gas B has a corresponding state with equal mass in the
brane gas B* . Based on type II A superstring theory we
take B to consist of 0, 2, 4, 6 and 8 branes. Then, by
the discussion in the preceding section, the brane gas B*
contains 9, 7, 5, 3, 1 branes which are the states of type
IIB as we have carried out an odd number (nine) of T-
duality transformations. Notice that this follows from the
T-duality symmetry for fundamental strings, not from T-
duality arguments applied to the above brane gases which
we actually want to show. Our demonstration is done by
carrying out explicitly nine T-dualil v 1 ransiormations on
a mass state in B, and showing that there is a corre-
sponding and equal mass state in B* . In this sense the
two brane gases are T-dual.

Suppose that the branes in B are wrapped around
some of the cycles of a nine-torus with radii (Ri, • • • , -Rg).
Then, the volume Vol p of a p-brane in Eq. (10) is sim-
ply the product of the p circumferences, and the minimal
masses M p = E p (in the string frame) are

.Vo =

1

M 2 = (2^) 2 i? 9 ^ 8 r 2 =

M 4 = (2ir) 4 RgR 8 R 7 R 6 T 4 =

M 6 = (2^) 6 i? 9 • • • R 4 t 6 --

Ms = (2nfR 9 ■ ■ ■ R 2 t 8 = -

(18)
(19)
(20)
(21)
(22)

(see also [7]) where in the second step we have used ex-
pression (4) for the tension of a p-brane. For notational
convenience we have fixed a particular winding configura-
tion. The argument is generalized for arbitrary winding
configurations and winding numbers at the end of this

section. If, as we have assumed, R n > a 1 ' 2 , then the
heaviest object in the theory is the 8-brane.

The DBI action (3) is invariant under T-duality.
Hence, all formulae derived from it (energy, mass and
pressure) are valid in both the original brane gas B and
in the dual brane gas B* . Thus, the mass spectrum of
the B* brane gas is

M; = {27rYR' 7 ■ ■ ■ R[tZ =

M* = {2n) 5 R' 5 ■ ■ ■ R[t; =

M* = (2 7 r) 3 J R^^ir 3 * =

Ml = 27ri?iri* = — - .
g*a'

.9*

a 'w/2

^7

■■■R[

9*

a' 8 / 2

R',

■■■R[

9*

a'*' 2

R' 3 R' 2 R[

(24)
(25)
(26)
(27)

Since R { < a x ' 2 , the heaviest brane of the dual gas B*
is now the 1-brane. The coupling constant in B* is given
by

(28)

Note that if the radii (R\,--- , Rg) of the initial nine-
torus are bigger than the self-dual radius a 1 ' 2 , then g* <
g, and thus the assumption of a small string coupling
constant is safe.

Given the two mass spectra, one can easily verify that
each state in the brane gas B has a corresponding state
with equal mass in the dual brane gas B*:

ML,

= M„.

(29)

This establishes explicitly that the T-duality of the string
gas used in [1] extends to the brane gas cosmology of [8] .
As an explicit example, consider a 2-brane wrapped
around the 8 and 9 directions. Its mass is (19)

-l/ 2

(30)

If we replace the string coupling constant g by the dual
string coupling constant g* via (28), and the radii R$ and
Rg by the dual radii R 8 and Rg via (16), one obtains

/?'... r'_

7 ■ (31)

g*a°' z

In the above example, we have specified a particular
winding configuration for simplicity, but clearly the ar-
gument holds as well in the general case, where a p-brane
wraps around some directions n\ ■ ■ ■ n p :

M p = (277)PR ni ---R np Tj

cm

Via the same steps as in the above example, it follows
thai

Mg*_ p = (2nf-PR' mi ■ ■ ■ R' m9 _ p T9- P =

= M p ,
where {mi,- •• ,m 9 „ p } 7^ {ni,- ■ ■ ,n p }.

g* a '(10- P )/2

(33)

Multiple windings

Consider now a p-brane with multiple windings u
(wi,--- ,w p ,0, ••• ,0). Its mass is

M p (co

:,Mn •

(34)

In the B* brane gas this corresponds to a num-
ber uj\ ■ ■ ■ ijj p of (9-p)-branes each with winding oj* =
(0, • • • , 0, 1, • • ■ ,1) and mass Mg_ p . Since M g *_ p = M p ,
the total mass of this 'multi-brane' state is equal to the
mass of the original brane, namely

( Wl -- Wp )M 9 % = M J ,H,

(35)

which establishes the correspondence of B and B* in the
case of multiple windings.

One should also add fundamental strings to the brane
gas B. Since their mass squared is

(36)

'„R„'

is its clear that every fundamental string state i:
a corresponding state in B* when n n <-> iv n .

We have demonstrated explicitly how T-duality acts
on a brane gas in a toroidal cosmological background,
and have in particular shown that the mass spectrum of
the theory is invariant under T-duality. Thus, the argu-
ments of [1] which led to the conclusion that cosmological
singularities can be avoided in string cosmology extend
to brane gas cosmology.

Whereas T-duality does not change the nature of fun-
damental strings, but simply interchanges winding and
momentum numbers, it changes the nature of branes:
after T-dualizing in all d spatial dimensions, a /^-bi:ane
becomes a (d — p)-brane which, however, was shown to
have the same mass as the 'original' brane.

In [8] it was shown that the dynamical decompacti-
fication mechanism proposed in [1] remains valid if, in
addition to fundamental strings, the degrees of freedom
of type IIA superstring theory are enclosed. We briefly
comment on the decompactification mechanism in the
presence of a type IIB brane gas on a nine-torus. As

before, we assume a hot, dense initial state where, in
particular, the brane winding modes are excited and in
thermal equilibrium with the other degrees of freedom.
All directions of the torus are roughly of string scale size,
l s , and start to expand isotropically. For the 9-, 7, and
5-brane winding modes there is no dimensional obstruc-
tion to continuously meet and to remain in equilibrium,
thereby transferring their energy to less costly momen-
tum or oscillatory modes: these degrees of freedom do not
constrain the number of expanding dimensions. However,
the 3-branes allow only seven dimensions to grow further,
and out of these, three dimensions can become large when
1-brane and string winding modes have disappeard. As
far as the 'intercommutation' and equilibration process
is concerned, the 1-branes and the strings play the same
role, but since the winding modes of the former are heav-
ier (-^7 3> |y at weak coupling), they disappear earlier.

We have focused our attention on how T-duality acts
on brane winding modes. However, since in a hot and
dense initial state we expect all degrees of freedom of
a brane to be excited, we should also include trans-
verse fluctuations (oscillatory modes) in our considera-
tions concerning T-duality. To our knowledge, the quan-
tization of such modes is, however, not yet understood,
and we leave this point for future studies.

Another interesting issue is to investigate how the
present picture of brane gas cosmology gets modified
when gauge fields on the branes are included. These cor-
respond to U{N) Chan-Paton factors at the open string
ends. In this case, a T-duality in a transverse direction
yields a number N of parallel (p-l)-branes at different
positions [20].

Acknowledgments

We would like to thank S. Alexander, L. Alvarez-
Gaume, J. Fernando-Barbon, S. Foffa, S. Lelli, J.
Mourad, R. Myers, Y. Oz, F. Quevedo, A. Rissone, M.
Rozali and M. Vasquez-Mozo for useful discussions. R.B.
wishes to thank the CERN Theory Division and the In-
stitut d'Astrophysique de Paris for their hospitality and
support during the time the work on this project was
done. He also acknowledges partial support from the
US Department of Energy under Contract DE-FG02-

91ER40688, TASK A.

Appendix

In this appendix we would like to give some further
arguments for why we believe brane gas cosmology to be
non-singular. First, let us recall the results of [1]. In the
case of a string gas, where the strings are freely propa-
gating on a 9-torus, it was shown that the cosmological
evolution is free of singularities. The background space-
time has to be compact, otherwise the thermodynamical
description of strings is not sound, in particular the spe-
cific heat becomes negative at large energies. An impor-
tant assumption in the derivation was that the evolution
of the universe is adiabatic, i.e. the entropy of the string
gas is constant. Making use of this assumption, one can
find the temperature as a function of the scale factor,
T(R), without referring to the dynamics of gravity or
Einstein's equations. Using a microcanonical approach,
it was shown that there exists a maximum temperature,
called Hagedorn temperature Th, and hence there is no
temperature singularity in string gas cosmology. The
curve T(R) is invariant under a T-duality transformation,
T(R) = T (a'/R). Another crucial assumption is that the
string coupling constant, g, is small enough such that the
thermodynamical computations for free strings are appli-
cable, and that the back-reaction of the string condensate
on the background geometry can be neglected. In lack of
knowledge about brane thermodynamics we simply pos-
tulate that the statements above extend to brane gases.

We conclude by making a comment on our work in the
light of the well-known singularity theorems in General
Relativity. These theorems make a sumpl ions about the
geometry of space-time such as R^v^t, 11 > for all time-
like vectors £ M (for a textbook treatment see e.g. [21]).
By Einstein's equations this is equivalent to the strong
energy condition for matter. However, we do not trust
Einstein's equations in the very early universe as they
receive corrections which are higher order in a' as well as
g, and also they lack invariance under T-duality trans-
formations R — > l/R. Therefore we cannot invoke the
energy momentum tensor given in Eqs. (9), (11), (14) to
decide whether the universe described by our model is
goodosioally complete or not.

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