Radion Stabilization by Stringy Effects in General Relativity

0.3 MB·English

by  Subodh P. Patil

Radion Stabilization by Stringy Effects in General Relativity
Subodh P. Patil1) and Robert Brandenberger2,1
1) Department of Physics, Brown University,
Providence, RI 02912, USA
2) Physics Department,
McGill University, 3600 rue Universit´e,
Montreal, QC H3A 2T8, CANADA
E-mail: patil,[email protected]
Weconsidertheeffectsofagasofclosedstrings(treatedquantummechanically)onabackground
where one dimension is compactified on a circle. After we address the effects of a time dependent
background on aspects of the string spectrum that concern us, we derive the energy-momentum
tensor for a string gas and investigate the resulting space-time dynamics. We show that a variety
oftrajectories arepossiblefortheradiusofthecompactifieddimension,dependingonthenatureof
5 the string gas, including a demonstration within the context of General Relativity (i.e. without a
0 dilaton) of a solution where the radius of the extra dimension oscillates about the self-dual radius,
0 withoutinvokingmatterthatviolatesthevariousenergyconditions. Inparticular,wefindthatinthe
2 casewherethestringgasisinthermalequilibrium,theradiusofthecompactifieddimensiondynam-
n ically stabilizes at the self-dual radius, after which a period of usual Friedmann-Robertson-Walker
a cosmologyofthethreeuncompactifieddimensionscansetin. Weshowthatourradionstabilization
J mechanism requires a stringy realization of inflation as scalar field driven inflation invalidates our
1 mechanism. Wealsoshowthatourstabilizationmechanismisconsistentwithobservationalbounds.
1

I. INTRODUCTION dimensionsisstabilizedattheself-dualradius. Morepre-
4
v cisely, the expansion of the three large dimensions leads
7 todampedoscillationsinthe“radion”abouttheself-dual
In the early days of string cosmology, it was realized
3 value. Thus,inthecontextofabackgrounddescribedby
that superstrings had an effect on space-time dynamics
0 Dilaton Gravity, radion stabilization is a natural conse-
1 that was qualitatively quite different from that of parti-
quence of brane gas cosmology [28].
0 cles or fields. In particular, it was realized that string
4 winding modes could provide a confining mechanism for
Atthepresenttime,however,thedilatonismostlikely
0 certaincompactdirectionsinsuchawayastoallowonly
fixed(see,however,[11]foranalternatescenario). Thus,
/
h three spatial dimensions to grow large [1]. Key to this it is of interest to explore how the inclusion of string
t realization are the T-duality of the spectrum of string
- (and brane) winding and momentum modes influences
p states, and the fact that the background is described by the dynamical evolution of the radion in a background
e Dilaton Gravity, and not by General Relativity with a
space-time described by General Relativity (GR). There
h
fixeddilaton(thisiscrucialinorderthatthebackground
is another motivation for studying this issue. Another
:
v equationsobeytheT-dualitysymmetry). Thearguments corneroftheM-theorymodulispaceis11-dsupergravity.
i of[1]wereputonafirmerbasisbytheanalysisof[2](see
X In [12] it was found that a brane gas in this background
also [3]).
also admits a region in the phase space of initial condi-
r
a Starting point of the considerations of [1] is the as- tions in which only three spatial dimensions can become
sumption that all spatial dimensions begin at close to large, although this corner may not be consistent with
the self-dual radius (the string scale), and that matter holographic entropy bounds [13] (see also [14] where the
consists of a hot gas of string states. The considerations considerations in this corner of M-theory moduli space
of [1] were more recently applied to “brane gas cosmol- was extended to spaces with more general topologies).
ogy” [4, 5], a scenario in which the initial string gas is Motivatedbytheseconsiderations,weinthispaperstudy
generalizedto be a gas of all brane modes. It was shown a simplified problem, namely the questions of how a gas
thatgiventhehotdenseinitialconditionsassumedin[1], ofwinding andmomentummodes ofstringswinding one
the string winding modes are the last modes to fall out compactified spatial dimension (taken to be a circle) ef-
of equilibrium and thus dominate the late time dynam- fects the evolution of the radius (the radion). We start
ics. Hence [4], the inclusion of brane degrees of freedom with initial conditions in which the three spatially non-
does not change the prediction that only three dimen- compactdimensionsareexpanding. Wefindthatthegas
sions to growlarge. The dynamical equations describing of string winding and momentum modes gives a natu-
the growth of the three dimensions which can become ral radion stabilization mechanism. Our approach is to
largeweresolvedin[5](see also[6]). In[7],itwasshown consider the effect of strings on 5D space-time dynamics
that isotropy in these large dimensions is a consequence (with the extra spatial dimension compactified to a cir-
ofthedynamics. In[8],itwasfoundthatifboththemo- cle) by adding the appropriate matter term to the stan-
mentumandwindingmodesofthestringsareincludedin dard Einstein-Hilbert action. We will derive this term
the dynamical equations, the radius of the compactified shortly(seealso[15]forasimilarderivation). Theresult-
2
ing energy-momentum tensor leads to a novel behavior partly captured by the Adiabatic Theorem, which is the
when inserted into the Einstein equations. We will find statementthatgiventwosystemswithHamiltoniansthat
thatwecangenerateanon-singularbouncingsolutionfor canbecontinuouslyinterpolated,theninaprecisesense,
the radius of the compactified dimension in the context the eigenstates of the initial system will evolve into the
of GR (without a dilaton) while respecting the Domi- eigenstates of the final system if this interpolation takes
nant Energy Condition for the matter content. Specifi- place slowly enough. Slow enough in simple quantum
cally, the radionperforms damped oscillationsabout the systems usually means that the variation happens over
self-dual radius. Initially, we study a pure state of mat- much longer time scales than the characteristic time of
terwithspecificquantumnumbersobeyingtheT-duality the system (by which we mean the time associated with
symmetry. However,wewillfindthatwecanrathernat- the typical energy of the system:τ 1). Having said
∼ E
urallyextendtheanalysistoagasofthesestringsinther- this, were we to study QFT in places where the met-
mal equilibrium (with a bath of gravitons and photons), ric varies a lot more rapidly (at the edges of black holes
withthe resultthatthe radiusofthe compactdimension or in the very early Universe) we invariably have to ac-
isdynamicallystabilizedattheselfdualradiusR=√α, countforthecurvatureofspace. Thus,wecanhopethat
′
where 2πα is the inverse of the string tension (see also theeffects ofatime dependentbackgroundonthe closed
′
[16] for a study of string gases in thermal equilibrium). string spectrum only require minor modifications to the
flatspacespectrum, ifthis time dependence isslowcom-
In addition, we find that our model evolves according
pared to the characteristic time of the string dynamics.
to standard Friedmann-Robertson-Walker (FRW) cos-
We show in the Appendix that this is indeed the case,
mologyafterthe compactdimensionhasbeen stabilized,
and in what follows we will stay within this regime.
and that the resultant stabilization is incompatible with
The Outline of this paper is as follows: we first derive
anysubsequentinflationaryepochdrivenbyabulkscalar
the energy-momentumtensorof a string gas(the deriva-
field(forstring-specificideasonhowtogenerateinflation
tionhere is moregeneralthanthe onegivenin[15]). We
in brane gas cosmology see [17]). However this conclu-
then insert this tensor into the Einstein equations and
sion can be avoided if some form of stringy inflation is
study the dynamics of the radius of the compact dimen-
realised where strings are produced in re-heating.
sion,assumingthatthethreelargespatialdimensionsare
Before we can turn to any of this, we will have to ad-
in the expanding phase. First, we consider a pure state
dress a question of principle concerning the string spec-
of matter. Next, we extend the discussion to a thermal
trum in a cosmological context (this issue is also being
state. In Section IV, we discuss the late time dynam-
studiedin[18]). ThequestionofformulatingStringThe-
ics and show that the stabilization of the radion is not
ory in a time-dependent background is a current and
compatible with inflation in the three large spatial di-
active area of research. However, we are primarily in-
mensions, assuming the simplified description of matter
terested in the behavior of strings in a background that
which we are using.
evolvesonacosmologicaltimescale. Ascanbeseenfrom
A few words on our notation: Greek indices typically
the FRW equations, the cosmological time scale H 1
− stand for 5-dimensional space-time indices, Roman in-
(where H is the Hubble expansion rate) is larger than
dices i,j,... are associated with the non-compact spatial
the characteristic microscopic time σ 1 (where σ4 is the
− dimensions, and Roman indices a,b,... are string world-
matter energy density) by a factor of m /σ, where m
pl pl sheet coordinates. The 5-dimensionalPlanck mass is de-
is the Planck mass. Thus, away from singular epochs in
noted by M (or M in abbreviated form). We also
thehistoryoftheUniverse,thecosmologicaltimescaleis work in natuprl5al units5(c = ~ = k = 1) where we pick
goingtobemany,manyordersofmagnitudelongerthan B
energy to be measured in electron volts.
the characteristic time scale of the string dynamics, and
hence we should be able to inherit many of the features
of the string spectrum in a static space-time (with some
II. THE ENERGY-MOMENTUM TENSOR
obvious modifications). We justify this intuition more
rigorouslyintheAppendix,butwefeelthatitmightsuf-
Tostudyhowagasofstringsaffectsspace-timedynam-
ficeatthispointtoremindthereaderoftheapproximate
ics, we need to derive the energy-momentum tensor of
irrelevance of a time dependent background for a much
suchagas. Webeginbystudyingtheenergy-momentum
more familiar theory: Quantum Field Theory (QFT).
tensorofasingleclosedstring. StartingwiththeNambu-
Although quantum fields in curved spaces exhibit sev-
Goto action
eral qualitatively different features from quantum fields
1
in flat spaces[29], we still manage to do a lot of sensible S = − d2σ√ h, (1)
NG
(andspectacularlysuccessful)flatspace-timeQFTcalcu- 2πα′ Z −
lationsdespite the persistentHubble expansionofspace- where h denotes the world sheet metric
ab
time. Thereasonforthisiseasytosee: thecontributions
h =∂ Xµ∂ Xνg (X) (2)
to masses, to scattering amplitudes, to the structure of ab a b µν
the Hilbert space of our theory, etc., that come from (and h is its determinant), we see that any variation in
termsthatdependonderivativesofthe metricareinthe the space-time metric g induces a variation in the in-
µν
present epoch highly suppressed and irrelevant. This is duced world-sheet metric (where the unmatched indices
3
indicate that we perturb only the λβ component of the where we use the inverse metric to write the metric con-
metric): tributions in the denominator. Thus, the single string
space-time energy-momentum tensor becomes
g (X) g (X)+δλδβδD(Xτ yτ)
µν → µν µ ν − 2 δS
Tλβ = − (3)
δgµν √ gδg
λβ
hab(σ) → hab+∂aX|λ∂bXβδ{Dz(Xτ −}yτ) 1− δD−2(Xi yi)√ hhab∂aXλ∂bXβ
= . − − .
δhab 2πα′ X0 X D √ g g00gDD
| ′ | − −
| {z }
Now, varying the Nambu-Goto action with respect to Insertingtheexplicitformoftheinveprseworld-sheetmet-
the space-time metric (performing a perturbation δg
αβ ric
which acts on the metric as givenabove)will give us the
.
space-time energy-momentum tensor of a single string: 1 h h 1 X µX Xµ X
hab = 22 − 12 = ′. µ′ − . . µ′
δSNG = 1 d2σ√ hhabδhab h(cid:18)−h21 h11 (cid:19) h −Xµ Xµ′ XµXµ !(4)
δg (y) −4πα −
λβ ′ Z and using the constraints on the world-sheet fields [30]
1
= d2σ√ hhab∂ Xλ∂ XβδD(Xτ yτ).
−4πα′ Z − a b − PµX′µ =0 (5)
We must first discuss the meaning of the expression
P Pµ+X X µ =0, (6)
d2 σ δD(Xτ yτ) µ µ′ ′
−
Z we can write (3) as
= d2σδ(X0 y0)δ(X1 y1)...δ(XD yD).
Z − − − Tλβ = −1 .δD−2(Xi−yi) [X′λX′β X.λX.β] (7)
In order to change the variable of integration, we need 2πα′|X0 X′D| −detgij −
to apply dσa = dXλ and sum over all the zeroes of
Xλ[σ] yλ when∂paeXrfλorming the integration. However, Next, we solve for X.0pusing the constraint (6) which
since w−e considering modes winding one particular spa- becomes
tial direction, there are precisely two coordinates that . . . . . .
are monotonic functions of the world-sheet parameters: 0 = X0X0 +XiXi +XDXD
−
X0 being a monotonic function of σ0 and XD being a X 0X 0+X iX +X DX ,
monotonic function of σ1 (the Dth direction is taken to − ′ ′ ′ i′ ′ D′
be compact). Thus, where we have explicitly used the backgroundmetric
d2 σδD(Xτ[σ] yτ) gµν =diag( 1,a2(t),a2(t),a2(t),b2(t)). (8)
− −
= dσ0dσ1δ(X0[σ] y0)δ(X1[σ] y1)...δ(XD[σ] yD)
− − − It is consistent with the equations of motion in a (slow
dX0 dXD enough) time varying backgroundto set X 0 =0 [31], so
= X.0 |X′D|√−g00gDD that ′
| |
δ(X0 y0)δ(XD yD)δD 2(Xi yi), . 2
× − − − − X0 =PiPi+X′iXi′+PDPD+X′DXD′ . (9)
where we include the metric factors in the last line so
Inaddition(in aslowlytime dependent background)the
that we can take the delta functions in X0 and XD to
righthand side can be expressedin terms of the familiar
be properly normalized. With this result, we get:
oscillator expansion. Accounting for the zero mode op-
erators explicitly, we get the center of mass momentum
δS δD 2(Xi yi)
NG = − − fromthespatialzeromodesandthewindingenergyfrom
δgλβ −4πα′ −g00gDD the zero mode terms in the compactified direction. The
dX0 dXD other modes give us the left and right moving oscillator
×Z X.0 |X′Dp|δ(X0−y0)δ(XD−yD)√−hhab∂aXλ∂bXtβerms (see [19] for details):
| |
1 . 2 n wb
=−4πα g00gdd X0= gijpipj + α (N+N −2)+(b)2+(α )2, (10)
′ − r ′ ′
δD 2(Xi yi)
− . p− √ hhab∂aXλ∂bXβ , wherenandw arethe quantumnumbersformomentum
× X0 X D − X0=y0,XD=yD and winding in the compact direction, respectively, and
| ′ | (cid:12)
(cid:12)
(cid:12)
4
N and N are the levels of the left- and right-moving os- accountforthemetricfactorsinthisexpression,wewrite
cillatormodes ofthe string,respectively. The expression µ(t) as µ (t)/a3(t) since this is how a number density
0
above is none other than the energy of the string. Using explicitlydependsonthemetric. Now,realizingthatthis
the level matching constraints is an energy density, we can introduce this gas of strings
as matter interacting with the gravitationalfield by just
N +nw N=0, (11) addingthefollowingtermtothegravitationalpartofthe
−
action:
we finally end up with [32]
S = d5x√ gǫ (18)
. 4 n wb int − −
X0= gijp p + (N 1)+( + )2. (12) Z
i j
α − b α
r ′ ′ (see e.g. Section 10.2 of [20]).
Now, we are ready to evaluate (7) for a single wound Realizing now that the metric factors in the denom-
.
string. We have an explicit expression for X0 and we inator of the expression for the energy density can be
know that X 5 = wb in units of α for a wound string, written as a3 = det(gij) and b = √g55, we can write
′ ′
| | | | the above equation as:
thefactorof w beingcanceledbythesummationoverall
p
| |
(w intotal)zeroesofthe argumentofthe delta function.
d5x√ g µ (t)
Thus, component by component, we get: S = − 0
int
−Z detgij√g55 2π
T0 = ρ (13)
0 − p 4 n wb
pip + (N 1)+( + )2
1 δ3(Xi yi) 4 n wb × i α − b α
= − pip + (N 1)+( + )2 r ′ ′
−2π a3b i α − b α µ (t)
r ′ ′ = d5x√ g 0
00
− − 2π
Z
Ti =p (14) 4 n wb
i pip + (N 1)+( + )2
i
1 δ3(Xi yi) pipi ×r α′ − b α′
= −
2π a3b pip + 4 (N 1)+(n + wb)2 Byourmetricansatzandtheisotropyofthedistribution
q i α′ − b α′ of the momenta, we have that pipi =a−2(p32 + p32 + p32).
Using this fact, it is straightforward to show that the
T5 =r (15) energy-momentum tensor derived from this interaction
5
1 δ3(Xi yi) n2 w2b2 term is:
= − b2 − α′2
2π a3b pipi+ α4′(N −1)+(nb + wαb′)2 T00 =−ρ=−21πaµ30b pipi+ α4 (N −1)+(nb + wαb)2
q r ′ ′
(note that we label the extra spatial coordinate by “5”) (19)
where we ignore off-diagonal components since we are
about to apply these expressions to an isotropic gas of
1 µ p2/3
strings. Ti =p= 0 (20)
However, we wish to present at this point another i 2πa3b pip + 4 (N 1)+(n + wb)2
derivationofthisresultwhichisrathermoredirect. Con- i α′ − b α′
q
sider the energy of a single wound string:
E2 =pipi+ α4′(N −1)+(nb + wαb′)2. (16) T55 =r = 21πaµ30b pipi+ α4′(nbN22 −−w1α2)′b2+2 (nb + wαb′)2 (21)
A spatially uniform gas of such strings with the same q
quantum numbers would have a 5-dimensional energy which is exactly what we would get from (13), (14) and
density (15)werewetoconstructahydrodynamicalaveragewith
an isotropic momentum distribution.
µ(t) 4 n wb We now investigate some simple aspects of our result.
ǫ= pipi+ (N 1)+( + )2, (17) The first thing to note is that T5, which is the pressure
2πb α − b α 5
r ′ ′
alongthecompactdirection,getsanegativecontribution
whereµ(t)isthenumberdensityofstrings. Wedivideby from the winding of our strings and a positive contribu-
2πb since this energy will be uniformly distributed over tion from the momentum along this direction. The spa-
the lengthofthe string. The momentumthat appearsin tial pressure is always positive, and for the simple case
thisexpressionisnowthemomentumsquaredofagasof n = w = 0, N = 1, which describes a gas of gravi-
strings whose momenta have identical magnitudes, but tons moving in the non compact directions, we obtain
whose directions are distributed isotropically. To fully r =0,p=ρ/3.
5
Since we are about to study the effects of this energy- III. SPACE-TIME DYNAMICS
momentum tensor on space-time, we should make sure
that the energy-momentum tensor is covariantly con- We start with the Einstein tensor derived from the
served, or else it will not be consistent to equate it to metric (8):
the covariantlyconservedEinsteintensor. The covariant
. . .
conservation of Tµ a a b
ν G0 = 3 +
0 − a a b
0=∇µTνµ, ha.. .b.i a. 2 b. a.
Gi = δi 2 + + +2
where is the covariant derivative operator, implies j − j a b a ba
µ
∇ ha.. a. 2 (cid:16) (cid:17) i
. . G5 = 3 + .
0 = ρ. +3a(ρ+p)+ b(ρ+r) 5 − a a
a b h (cid:16) (cid:17) i
0 = ∂ip Equating this to 1 Tµ will give us the Einstein equa-
M3 ν
0 = ∂5r. tions. However,letpuls5focusontheequationthatgoverns
the evolution of the scale factor b. Starting with Gi and
Itisstraightforwardtocheckthatourenergy-momentum j
.. .2
tensorsatisfiesthisasanidentity. Inthecontinuityequa- eliminating a and a by adding the appropriate combi-
tion, this is due to the metric factors contained in the nations of G0 and G5, we get:
0 5
energydensity,whichupondifferentiationproduceterms
.. . b 2r ρ
that exactly cancel the terms proportional to the Hub- b+3H b+ p =0, (23)
M3 − 3 − 3
ble factors. The remaining equations are trivially sat- pl5(cid:16) (cid:17)
isfied by our setup, which assumed an axis of symme-
where H is the 3-dimensional Hubble factor. This is a
try along the compactified dimension (the Kaluza-Klein
second order, nonlinear (because of the b dependence in
setup) with homogeneous and isotropic spatial sections.
the matter terms) differential equation with a damping
One final point to note is that we have derived an
termandadrivingterm. Wewilldemonstratefurtheron
energy-momentumtensorthatexhibitspositivepressures
that the Einstein equations admit expanding solutions
along the non-compact directions and positive or nega-
for the non-compact dimensions (H > 0), and take it
tive pressuresalongthe compactifieddirection. We need
as a given for what follows. Thus, in spite of its non-
toensurethatthisnegativepressurehasaboundedequa-
linearity, we easily see that (23) describes an expanding
tion of state as otherwise our theory would be unstable.
oracontractingscalefactordependingonthesignofthe
The DominantEnergyCondition(DEC) of GeneralRel-
driving term.
ativity [21] ensures the stability of the vacuum, and re-
The first thing to notice from this equation is that
quires that the equation of state parameter ω = p/ρ be matter for which the quantity p 2r ρ vanishes will
greater than or equal to -1 (see e.g. [22] for a recent − 3 − 3
notcontributetothedynamicsofthecompactdimension.
discussion). Since the spatial pressures are always posi-
Thus, recalling that a gas of gravitons (n=w =0, N =
tive, we only need to check our equation of state for the 1) has an equation of state p = ρ, r = 0, as does a
pressure along the compact direction: 3
gas of ordinary 4-dimensional photons, we see that such
matter will not affect the dynamics of the scale factor
1 µ n2 w2b2
r = 0 b2 − α′2 b. Infact, sucha background4-dimensionalgasprovides
2πa3b pip + 4 (N 1)+(n + wb)2 an excellent candidate for a thermal bath which we will
i α′ − b α′ eventually couple our gas of winding modes to.
q n2 w2b2 First, however, we will study this driving term as it
= ρ× pip + 4 (Nb2 −1)α+′2 (n + wb)2 , is, for a gas consisting of strings with identical quantum
i α′ − b α′ numbers. Uponevaluatingthedrivingterm,wefindthat:
where the co-efficient of ρ in the above is our equation
b 2r ρ µ
0
ofstateparameter. Werewe to considerstates described p = (24)
M3 − 3 − 3 M3 a32π
by n= 1,w = n,N =1 (which aswe will see further pl5(cid:16) (cid:17) pl5
obnil,iztautrion±n)o,utthtios pb−aerathmeetreerlerveamntaisntsatbeosutnhdaetdgaivsebuvsasrtieas- −nb22 − 23nαw′ + w3α2b′22 − 4(N3α−′1) ,
[33]: × pipi+ α4′(N −1)+(nb + wαb′)2
q
fromwhichweinferthatmomentummodesandoscillator
1 ω 1 (22) modes lead to expansion of the scale factor, whereas the
− ≤ ≤ winding modes produce contraction. Exactly what hap-
Thus, we have verified that the spectrum of string pens, of course, depends on the values of the quantum
states satisfies the DEC, and in doing so ensured our- numbers. It should be recalled that the quantum num-
selves of sensible space-time dynamics arising from the bers are subject to the constraint nw+N 0 coming
≥
string gas, the topic we will turn our attention to next. from the level matching conditions (see Eq. 11).
6
At this point, we wish to mention that the “Quantum
4
Gravity Effects” required to stabilize the extra dimen-
sions in earlier attempts [23, 24] at Kaluza-Klein cos-
3 mology find a stringy realization here, in that all that
was requiredfor radius stabilization wasmatter that de-
2 pended on the size of the extra spatial dimensions in a
non-trivial way.
To round off the discussion, we wish to demonstrate
1
that our assumption of an expanding scale factor a(t) is
consistent with an oscillating scale factor b(t). Consider
1 2 3 4 the two Einstein equations that do not contain second
derivatives of b:
FIG. 1: Potential term for n = w = 1,N = 1. The .
− ± ρ b
horizontal axis is b (in string units), the vertical axis gives = H2+H ,
the potential in units of , where is the prefactor on the 3M53 b
U U
right hand side of (25). r .
− = H +2H2.
3M3
5
Let us pick a particular set of quantum numbers. As These equations imply that
we shall see later, the most interesting case is when .
n = w = 1, N = 1, in which case the driving term . b 1
becom−es: ± H −2Hb =−3M3(2ρ+r). (27)
5
1 + 2 + b2 The resulting equation for H has the integrating factor
b p 2r ρ = 2µ0 −b2 3 3 , 1/b2, and hence the solution:
Mp3l5(cid:16) − 3 − 3(cid:17) Mp3l5a32π√α′ α′pipi+(1b−b)2 b(t) 2 b2(t) t (2ρ+r)
q (25) H(t)=H0 b − 3M3 dt′ b2(t) . (28)
where b is the scale factor in units of √α′. Quite gener- (cid:16) 0 (cid:17) 5 Z0 ′
ically, we can explore features of the “potential” energy Now, from the discussion surrounding (22), we see that
that will yield this driving term. We see that, since the ρ r ρ. Thus, the contribution to the integral in
≥ ≥ −
denominator is strictly positive, and the driving term theaboveisstrictlypositive,andthe secondtermonthe
changes sign at b= 1, this value will be a minimum of right hand side of (28) can at most take on the value:
the potential energy, and hence a point of equilibrium.
Numerical integration of the driving term yields the po- b2(t) t ρ(t′)
dt . (29)
tential energy curve of Figure 1 [34] where the potential M3 ′b2(t)
5 Z0 ′
is plotted in units of 2µ0 as a function of b.
Mp3l5a32π√α′ Thus, we see that if we pick the initial conditions for
BecauseoftheHubbledampingtermintheequationof H appropriately, the scale factor a can be taken to be
motionfor b (whichis obtainedby dividing (23)through expanding (H > 0) regardless of the detailed motion of
byafactorof√α),thescalefactorwillperformdamped b. In fact, if we assume that H starts out positive, then
′ 0
oscillationsabouttheminimumofthepotentialtowhich H(t) will remain so if
itwillevolvewithrapiditydepending onthe value ofthe
“spring constant” multiplying the driving term: 1 t 2ρ+r
H dt , (30)
0 ≥ 3M3 ′b2(t)/b2
2µ 5 Z0 ′ 0
0
k= . (26)
M3 a32πα where the eventual stabilization of b and the 1/Vol de-
pl5 ′ pendence of ρ and r will bound the integral, which im-
Thus, we have established that a gas of string modes plicitly depends on H itself. This implicit dependence
with non zero winding and momentum numbers in the works in our favor in that the larger we make H , the
0
compact direction will provide a dynamical stabilization smallertheintegralbecomesandsowecanimaginepick-
mechanism for the radion, provided that the three non- ing an initial H such that a persistent expansion of the
0
compactdimensions are expanding (such a behaviorwas non-compact dimensions results. Note, however that if
already found in an early study [10] of the dynamics of in the spirit of brane gas cosmology, we assume that all
stringwindingmodes -wethankScottWatsonfordraw- spatial dimensions are starting out with the same size
ing our attention to this paper). We will address the and instantaneously static, then it may not be possible
phenomenology of this stabilization mechanism further to evolve to a situation in which three large spatial di-
on, simply stating for now that we can obtain a robust mensions are expanding. This is, in fact, the result that
stabilization mechanism which is consistent with obser- emergesfromtheworkof[13],atleastinacertainregion
vational bounds. of phase space.
7
IV. THERMAL STRING GASES where the Boltzmann weight in the summation depends
on the values of the quantum numbers. We remind the
In what we have done so far, we have just considered reader that the sum is restricted by the level matching
thebehaviorofthesizeoftheextradimensioninarather condition N+nw 0. For completeness we also remind
≥
artificial setting, namely imposing a gas of strings with the reader of the resulting equation of motion for b:
a fixed set of quantum numbers. One expects the early
Universe to be in a state of thermal agitation, and it is 0 = .b.+3H b. +µrefeβEref (34)
inevitablethattransitionsbetweendifferentenergylevels Mp3l5a32π
willbeinducedinthestringgas. Thus,tohaveanyhope e βE n2 2nw w2b2 4(N 1)
−
of realistically applying our setup to cosmology,we need + −
× √E −b2 − 3α 3α2 − 3α
tostudy the effects ofplacingthe stringgasinathermal n,wX,N,p2 (cid:16) ′ ′ ′ (cid:17)
bath. Referring to our expression for the energy density
The summation which has to be performed in order
ofastring(19),weseethatagasofstringswithdifferent
to obtain the driving term is quite formidable, were it
quantum numbers will have the energy density:
not for a rather special feature of string thermodynam-
µ 4 n wb ics. Consider the argument of the exponential in the
ρ= n,w,N,p2 pip + (N 1)+( + )2
a3b2π i α − b α Boltzmann factor:
r ′ ′
n,wX,N,p2
(31) 4 n wb
with densities µn,w,N,p2 for each given set of quantum βEn,w,N,p2 = β pipi+ α (N −1)+(b + α )2
r ′ ′
numbers. The expressionsforthe pressureterms pandr
β n
athreesdiemnisliatrielys maroedgiifiveedn.bIyf wtheeaBreolitnzmthaenrnmawleeigqhutilibrium, = √α′rα′pipi+4(N −1)+(b+bw)2
µn,w,N,p2 =eβErefe−βEn,w,N,p2µref, (32) We see that when the energy is expressedin terms of di-
mensionlessvariables,wepull outa factorof√α. Thus,
′
where the subscript “ref” refers to some arbitrary refer-
theargumentoftheexponentialintheBoltzmannweight
ence energy level. is β/√α times a term of order unity. To be able to ne-
′
Whatconstitutesthethermalbathtowhichthestring
glect all but the first few terms in the summation, we
gas is coupled to? We know from the discussion at the
need the Boltzmann factor to be considerably less than
end of Section II that gravitons described by unwound
unity, i.e. that
strings propagate in the non-compact directions with an
equation of state p = ρ/3. Introducing a gas of ordi- β
nary photons will also add a 4-dimensional component e−√α′ ≪1.
to the energy-momentum tensor with the same equation Thus, if this condition is satisfied, then the terms which
of state. Such particles offer us an ideal candidate for
dominate the sum will be those whose quantum num-
a thermal bath, for two reasons. Firstly, thermal equi- bers render them nearly massless, since any state with
librium demands a coupling of some kind between the
even one of its quantum numbers being different from
gas of winding modes and the gas of gravitons and pho- the nearly massless combination will produce a term of
tons. Such a coupling is readily provided by the tree- order unity times β/√α.
′
level reaction w+ w hµν via which winding modes of Let us take a closer look at the above condition. We
→
equalandoppositewinding scattertoproduce4-dgravi-
knowfromstringthermodynamicsthatthereexistsalim-
tons. This thermalization mechanism will, at non-zero iting temperature – the Hagedorn temperature T [25].
H
temperatures, create an equilibrium where there will be Thus, for us to even be able to apply thermodynamics,
an ever-present non-zero winding mode density due to
we need to be well below this temperature, which for all
gravitonsscatteringintowindingmodes(andvice-versa). the string theories is of the order of T 1/√α. Thus,
This thermal bath has the further property that it does β √α, andsoifweareatatempeHra∼turemuc′hlower
notaffectthedynamicsoftheextradimensionotherthan H ∼ ′
than the Hagedorn temperature, i.e. T T or equiva-
H
through the Hubble factor (which it influences), since ≪
lently β β, then
H
the driving term is only sensitive to the combination ≪
p ρ/3 2r/3whichvanishesforthegravitonandphoton √α β, (35)
co−mpon−ents of the energy-momentum tensor. ′ ≪
Withthe aboveinmind, the drivingtermforthescale whichisexactlywhatweneedfortheBoltzmannweights
factor b becomes: of higher mass states to be negligible. So, even if the
b 2r ρ µrefeβEref thermalbathhasatemperatureofonlyoneorderofmag-
p = (33) β
Mp3l5 (cid:16) − 3 − 3(cid:17) Mp3l5a32π n10itu5dewhbieclhowcletahrelyHlaegtseduosrnignteomrepaenraytuterrem, twhehnosee−e√nαe′rg∼y
× n,wX,N,p2e−βE −pinbp22i+−α243n′α(w′N+−w31α2)b′22+−(nb4(+N3α−′wα1b′))2 , iisna−dniymtheninsgioontlehsesruthnaitnszqeroα.′pTiphiis+tr4a(nNsl−ate1s)i+nt(onbu+sbbewin)g2
q
8
able to neglect all states other than those that are mass- Inadditiontothemasslessstategivenbyn=w =0,N =
less. Thus, the summation now becomes very tractable, 1 (the graviton), and the 8 other massless states that
andwecanalsohavefaithinourtruncationofthestring appear at the self-dual radius (which are given by N =
spectrum to the lightest states all the way up to very 1,n= w= 1; N =0,w=n= 1; N =0,w=0,n=
− ± ±
high temperatures (T T /10). Before we carry on we 2 and N = 0,w = 2,n = 0), there are additional
H
∼ ± ±
should remark that exactly massless states have a non- massless states at further special radii
zero momentum given by the thermal expectation value
of E = p =3/β.
| |
Let us then proceed to evaluate (33), so that we can
2
evolve b in time using (34), recalling that now we only b= ; w = m,n=0,N =0
m ±
sum over the massless and near massless states subject | |
tothe levelmatchingconstraint. Letus beginnearb=1, m
b= | | ; n= m,w =0,N =0
i.e. b=1+Γ. Thenforthe casethatΓ=1,weonlyhave 2 ±
6
one truly massless state: n = w = 0, N = 1. This term m ǫ Z.
will not contribute to the driving force for b since
−nb22 − 23nαw′ + w3α2b′22 − 4(N3α−′1) =0. (36) Thus, athalf-integer multiples and andhalf integer frac-
(cid:16) p (cid:17) tions of the self-dual radius, two massless modes appear
| |
and will thus yield the dominant contributions to the
Thus,thenextlighteststatewhichhasquantumnumbers
driving term. These contributions again exactly cancel
N = 1,n = w = 1 will dominate the evolution of b.
− ± attwice the self dual radius,and at half the self dualra-
The level matching constraints N +nw 0 ensure that
≥ dius. Howeveringeneral,wewillgeta drivingtermthat
there are no more nearly massless states (Note we only
yields expansion at half integer points above twice the
consider states with positive mass squared - any tachy-
self dual radius and similarly, contraction at half integer
onic states are posited to be absent from our spectrum).
fractions below half the self-dual radius. However since
Such states will contribute:
we posit that we begin at or near the self-dual radius,
e√−αβ′q(1b−b)2+α′p2 −b12 + b32 + 23 . (37) wcoendairteiognusasraatnistfeyed to stay locked near it if our initial
(cid:16) (cid:17)
(1 b)2+αp2
b− ′
q
Expanding b as 1 + Γ and ignoring terms higher than
b(0) √α
quadratic in Γ results in a contribution to (37) of: ∼ ′
√e−αβ|pp| 83Γ , b. (0) ≤ √k=sM532aµ320πα′ ,
′| |
(cid:0) (cid:1)
Since there are two such terms which yield identical
contributions, the sum total of the contributions from
where the last condition constrains the initial “veloc-
the near massless states yields the equation of motion
ity” of the scale factor to be such that it cannot roll
.. . µ 8Γ overthe “hump”inthepotentialenergysurroundingthe
Γ+3H Γ+ =0, (38)
M32πa3 pα3/2 3 metastable equilibrium at b √α.
5 | | ′ ∼ ′
(cid:0) (cid:1)
wherethe exponentialfactorgets cancelledifwe usethis
Thus, we have demonstrated in the context of GR
masslessstateasourreferencestate,asin(32). Theform
how a string gas in thermal equilibrium with a bath
of this equation clearly shows that Γ will tend to zero if
of gravitons and photons will dynamically stabilize the
it starts out on either side of this value.
scale factor of the compact direction if we begin close
However, to confirm that Γ = 0 is a genuine point of
to that radius. Thermal equilibrium with the graviton
equilibrium, we need to confirm that the extra massless
bath ensures a persistent non-zero density of such wind-
states that appear at this radius (8 in all) contribute
ingmodes. Onecannowimaginethat,atsomepoint,the
in such a way so that their sum vanishes. This can be
winding mode gas becomes decoupled from the graviton
verified by a straightforwardcalculation [35].
gas, i.e. falls out of thermal equilibrium. In this situa-
However,we wish to point out that as long as we stay
tion, we are left with an unchanging driving term of the
in thermal equilibrium with the graviton gas, this equi-
form (24), which yields the potential in Fig. 1, which
libriumisactuallymetastable. Thereasonforthisiseasy
willguaranteeradionstabilizationatthe selfdualradius
to see from the formula for the mass of a winding mode:
for the remainder of the Universe’s dynamics. We now
turn our attention to the possible connection between
n 2 this mechanism and inflationary and standard Big Bang
α′m2 = b +w b +4(N −1). (39) cosmology.
(cid:16) (cid:17)
9
V. LATE TIME EVOLUTION B. Matter Dominated Evolution
Recall the Einstein tensor for our metric setup: Reconsidering (23):
. . .
a a b
G0 = 3 +
0 − a a b .. . b 2r ρ
Gi = δi h2a.. + .b.i+ a. 2+2b. a. b+3H b+Mp3l5(cid:16)p− 3 − 3(cid:17)=0,
j − j a b a ba
ha.. a. 2 (cid:16) (cid:17) i We see that any matter with the equation of state of
G5 = 3 + .
5 − a a non-relativistic dust (p = 0), can only drive the expan-
h (cid:16) (cid:17) i sion of the radion if it is of a 3-dimensional nature (i.e.
We know that the dynamics of the scale factor b in the r 0). This is surely to be a cause for concern when
≡
situations we studied above cause it to undergo damped considering that at late times, one (naively) expects 3-
oscillationsaroundtheselfdualradius. Wedemonstrated dimensionalnon-relativisticdusttobethemajordriving
in a previous section how the “spring constant” of this componentoftheexpansionoftheuniverse,whichwould
evolution will lock in to this equilibrium fairly rapidly. normally invalidate our stabilization mechanism in the
We can then study the evolution of the Universe after present epoch.
. ..
radius stabilization, which implies that b=b= 0 and p However we wish to remind the reader that present
−
2r/3 ρ/3=0. The resulting Einstein equations are: day observations demand that a significant fraction of
−
the energy density of the universe, which also drives the
G0 = 1 T0 H2 = ρ presentdaymatter dominatedexpansion,be in the form
0 M3 0 → 3 ofcolddarkmatter–whosenatureisasofyetcompletely
5
1 . 1 unknown. Thereisasignificantamountofinterestis the
Gii = M3Tii → H=−2(ρ+p) prospect that extra-dimensional matter or extra dimen-
5 sional effects might account for the ‘missing’ matter in
1 2r ρ
G5 = T5 p =0 the universe. In what follows, we find that the only way
5 M3 5 → − 3 − 2
5 to make our stabilization mechanism consistent with a
.
Tµ =0 ρ+3H(ρ+p)=0, ‘matter dominated’ epoch is to introduce extra dimen-
∇µ ν →
sional cold dark matter. We propose a candidate for
where the 55 equationis preciselythe equilibriumcondi- thiscolddarkmatterwhichisnaturallycontainedinour
tion. Thus,werecognizeintheabovethebasicequations framework, and discuss other possibilities which might
of FRW cosmology. We now consider how one achieves have a natural realisation within the general brane gas
the two important epochs of late time FRW cosmology, framework (note that a similar proposal was made in
namelytheradiationdominatederaandthematterdom- [26]).
inated era. We see that in order for the driving term in (23) to
correspond to a stable minimum at the self dual radius
for matter which obeys the equation of state for non-
A. Radiation Dominated Evolution relativistic dust (p = 0), we need to consider matter for
which:
If we assume that the density of 4-d matter gas is far
greater than the density of the winding mode gas, then r =−ρ/2|b=√α′. (42)
thiswillbethedominantcomponentthatdrivestheevo-
lution of the macroscopic dimensions. If the 4-d matter That is, we require the dominant component of the en-
has anequation ofstate parameterw, then the solutions ergy density which is driving the expansion of the uni-
to the Einstein equations become: verse be such that it preserves the stability of the ra-
dion at the self dual radius. Matter which exhibits such
ρ(t) a−3(1+w) (40) an equation of state will surely have to be massive (else
∝
there will be a non-zero pressure along the non-compact
or directions for any non-zero energy density). In addition,
such matter will have to be something beyond presently
2
a(t) t3(1+w) . (41) supposed dark matter candidates (WIMPS, supersym-
∝
metricrelicsetc.) asitwillneccesarilyhavetobe‘extra-
Thus, for a 4-d graviton and photon background, we dimensional’ in nature. We now show that our model
get that a(t) t1/2, and so we reproduce a late time naturally contains such a candidate. Recalling the dis-
∝
FRWevolutionthatisconsistentwithstandardBig-Bang cussion surrounding (22), we see that the equation of
cosmology immediately after the end of inflation, whilst state parameter for a gas of strings with a particular set
maintaining radius stabilization. ofquantumnumbersisobtainedfromthefollowingequa-
10
tion: contextofbranegascosmology,this is anappealing idea
as one might need higher dimensional branes to stabi-
n2 w2b2
lizecompactsub-manifoldsthatdonotadmittopological
r=ρ b2 − α′2 , (43)
× pip + 4 (N 1)+(n + wb)2 one-cycles(andhencedonotadmitwindingmodes). We
i α′ − b α′ willinvestigatethis possibilityfurtherinafuture report.
wherethemomentumwillbesettozero(orisvanishingly Finally,wewishtoaddressthe effectsofanintermediary
small) in order to satisfy p = 0 (c.f. (20)). In particu- epoch of scalar field driven inflation on our stabilization
lar,since we arelooking for states whichcan satisfy (42) mechanism.
at the self dual radius, we need to find the appropriate
quantum numbers which have an equation of state pa-
rameter w = 1/2 at b = √α, which reduces to the C. Intermediate (Non-Stringy) Inflation
− ′
following condition:
We find that our mechanism for radius stabilization
3n2 w2+4(N 1)+2nw =0, (44) might not be compatible with an intermediate epoch of
− −
bulk scalar field driven inflation. To investigate this, we
and we have to be mindful of the level matching con- first consider the energy-momentum tensor of an almost
straint: N +nw 0. As expected, it turns out that the homogeneous inflaton field:
≥
massless states that satisfy these conditions have an en-
ergy density proportional to p, whereas the pressure is φ.2 φ.2 φ.2
proportionalto p2/3,andhe|nc|eonecannothaveanon- Tµ =diag +V(φ) , V(φ) ,..., V(φ)
| | ν − 2 2 − 2 −
zero pressure without having a vanishing energydensity. (cid:16) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (4(cid:3)7(cid:17))
The first massive states which satisfy (44) are repre- Adding this energy-momentum tensor to our string gas
sented by the quantum numbers: yields a non-trivial contribution to the driving term in
the equation of motion for b. This driving term then
p =0 , N =2 ; n=0 , w = 2. (45)
takes the form
| | ±
These states contribute to the energy momentum tensor 2b µrefeβEref
V(φ)+ (48)
as follows (c.f. (19) - (21)): − 3M3 M3 a32π
5 pl5
2µ√2 e−βE n2 2nw w2b2 4(N 1)
+ −
p=0 , ρ= 2πa3α′ , r =−ρ/2, (46) n,wX,N,p2 √E (cid:16)−b2 − 3α′ 3α′2 − 3α′ (cid:17)
where the factor of α′ in the denominator comes from fromwhichitiseasytoseethattheinflatoncontribution
two factors of √α′– one from the metric factor b which will drive expansion in the extra dimension. In general,
has stabilized at the self dual radius, and the other as this term will compete with the string gas contribution
the pre-factor of the non zero rest mass of this string which,aswehaveseen,drivescontraction,ifweareabove
state. As we willsee in the next section,were we to look the self-dual radius. However, this competition is short
at fluctuations of the radion around the self dual radius, lived,asthe factorofa3 inthe denominatorofthe string
these states also provide a stable equilibrium at b=√α′ gas driving term will quickly render it irrelevantand the
in a phenomenologically acceptable manner. Thus, tak- scale factor will then expand according to
ingquestionsofconsistencywithobservationforgranted
for the moment, we see if these particular string states .. . 2b
b +3H b V(φ)=0. (49)
are taken to dominate the present energy density of the −3M3
5
universe, then by the Einstein equations derived at the
Recallingthat duringthis (slowroll)inflationH, andby
startofthissection,wecanadmitanepochofdustdriven
FRWexpansion(p=0,a t2/3)atlatetimes,consistent the time-time Einstein equation, also V(φ) are almost
∝ constant, we can solve the above equation, with the re-
with radius stabilization.
sulting two solutions:
However, there are many further issues that will have
to be resolved if we are to take this idea of stringy dark
3H 1+ 1+ 8V(φ) t
matter seriously, which we postpone to a future report. b(t) exp− 2 r 9H23M53
At present, however, we wish to state that there are ∝ (cid:0) (cid:1)
indications that such stringy dark matter might have b(t) exp−32H 1−r1+9H8V23(φM)53 t.
the right clustering properties at the level of first order ∝
(cid:0) (cid:1)
perturbation theory, in that local overdensities of this Substituting in the Einstein equation H2 = V(φ)/3M3
5
stringy dark matter induces gravitational clustering in gives us
the remaining 3-dimensions.
We also wish to point out that in certain situations, b(t) exp−32H 1+√1+98 t e−3.56Ht
non-relativistic p-branes might also be able to provide ∝ ∝
us with a matter content with satisfies (42) [15]. In the b(t) exp−32H(cid:0)1−√1+98(cid:1)t e0.56Ht.
∝ ∝
(cid:0) (cid:1)