Generating Scale-Invariant Power Spectrum in String Gas Cosmology ·  · 2006-02-21Generating Scale-Invariant Power Spectrum in String Gas Cosmology ... expansion of space is sufficiently

Generating Scale-Invariant Power Spectrum in String Gas Cosmology

Ali NayeriAli NayeriJefferson Physical LaboratoryJefferson Physical Laboratory

Harvard UniversityHarvard University

Canadian Institute for Theoretical Astrophysics

Thursday, January 19, 2006In collaboration with:

Robert BrandenbergerCumrun Vafahep-th/0511140

String Gas Cosmology : A Scenario with Scale-Invariant Scalar Power

Spectrum

• Our goal and our vision.• Review of Brandenberger-Vafa, a.k.a, string

gas cosmology.• String cosmology and dilaton.• Statistical mechanics of closed strings.• Calculating the power spectrum.• Open strings[?]• Conclusion.

Our GoalHere we study the generation of cosmological fluctuations during the early Hagedorn phase of string gas cosmology using the tools of string statistical mechanics. Since this early phase is quasi-static, the Hubble radius (1/H(t)) is very large (infinite in the limit of the exactly static case). The approximation of thermodynamic equilibrium is justified on scales smaller than the Hubble radius. In this phase, a gas of closed strings induces a scale-invariant spectrum of scalar metric fluctuations on all scales smaller than the Hubble radius. Provided that the expansion of space is sufficiently slow, these scales will include all scales which are currently being probed by cosmological observations:

h(δM)2i= hE2i− hEi2 = T2CV .

Our Vision

• Provided that the spectrum in these fluctuations is not distorted at the time of the transition from the Hagedorn phase to the usual phase of radiation-domination of standard cosmology (which is unlikely), it follows that string gas cosmology will lead -without invoking a period of inflation - to a scale-invariant spectrum of adiabatic curvature fluctuations.

Brandenberger-Vafa String Cosmology

• First attempt to understand cosmology in its stringy phase.• The behavior of temperature, as a function of radius, in a

compact topology, i.e., toroidal geometry is obtained.• While temperature is T-dual under transformation a →

1/a, the Friedmann equation is not. • The universe is nonsingular .• A stringy attempt to explain spacetime dimensionality:

“decompactification”• More on thermodynamic side rather than dynamical side.

Behavior of Temperature

Dilatonic String Cosmology

A =Z q−(10)g d10xe−2φ

h(10)R+4(∇φ)2 + ...

i,

ds2 = −dt2 +(d)Xıa2dx2ı +

(9−d)Xı

a2sdx02ı ,

Dilatonic String Cosmology: Field Equations

−(d)μ2 − (9− d)ν2+ ϕ2 = eϕE ,

μ− ϕμ= 12eϕPd ,

ν − ϕν = 12eϕP9−d ,

ϕ − (d)μ2− (9− d)ν2 = 12eϕE .

V = (2πqα0)9a(d)a(9−d)s ≡ (2π

qα0)9e(d)μe(9−d)ν .

ϕ ≡ 2φ− (d)μ − (9− d)ν ,

a(t) = eμ(t) , as(t) = eν(t) ,

Equations in Terms of Original Dilaton

−(d)μ2−(9−d)ν2+[φ−(d)μ−(9−d)ν]2 = eφρ ,

μ− [φ − (d)μ− (9− d)ν]μ = 12eφpd ,

ν − [φ − (d)μ− (9− d)ν]ν = 12eφp9−d ,

[φ− (d)μ− (9− d)ν]− (d)μ2− (9− d)ν2 = 12eφρ .

Hagedorn Era

Late time behavior in Hagedorn era:

φ(t) = ln

"16π2/E

t(t+ c)

#

μ(t) = A+B lnµ

t

t+ c

¶

eφ ∼ const.t2

, a(t) ∼ const.

Classical String Cosmology

Hagedorn Phase

Pre-Big Bang Cosmology

String Gas Cosmology

Post-Bang Radiation EraPre-Bang Radiation Era

Quantum String Cosmology

Behavior of Temperature

Microcanonical Description

Ω(E) ≡Xıδ(E −Eı) ,

S(E) ≡ lnΩ(E) .

1

T (E)= β(E) ≡

µ∂S

∂E

¶V=

µ∂ lnΩ

∂E

¶V,

P ≡ Tµ∂S

∂V

¶E= T

µ∂ lnΩ

∂V

¶E,

CV ≡µ∂E

∂T

¶V= −β2

̶E

∂β

!V

= −"T2

̶2S

∂E2

!V

#−1.

Canonical DescriptionZ(β) =

Xıe−βEı ,

Z(β) =Z ∞0dEe−βEΩ(E) ,

Ω(E) =Z L+i∞L−i∞

dβ

2πieβEZ(β) ,

ΔE

hEi ≡vuuthE2ihEi2 − 1 ∝

1qhEi

.

hEi ≡ −Z−1(∂Z/∂β) = −(Z 0/Z)

hE2i ≡ Z−1(∂2Z/∂β2) = (Z 00/Z)

CV = β2³hE2i− hEi2

´= β2

ÃZ 00

Z− Z

02

Z2

!= β2

∂

∂β

ÃZ 0

Z

!= −β2∂hEi

∂β.

Statistical Mechanics of the Ideal String Gas

Density of States for the Ideal String Gas

When the string coupling is sufficiently small, gs << 1, and the local spacetime geometry is close to flat Rd + 1 over the length scale of the finite size box of volume V = Rd, there are two distinct regimes that characterizes the statisticalmechanics of string thermodynamics: the massless modes with field theoretic entropy :

and the highly excited strings with

One simple realization of this setup for a string background is a spatial toroidalcompactification with d dimensions of size L and 9 - d dimensions of string scale size. Small string coupling ensures us that we can measure energies with respect to the flat time coordinate.

S ∝ Ed/(d+1)

S ∝ E

Universe in Hagedorn Phase

Stringy Block Universe

R

R

Density of Single State

• For a highly excited closed string represented as a random walk in atarget space, the energy ε of the string is proportional to the length of the random walk. Thus the number of the strings with a fixed starting pointgrows as e(βH ε), with TH = 1/βH is the Hagedorn temperature. This explains the bulk of the entropy of highly energetic strings.

• Closed strings correspond to random walks that must close on themselves. This overcounts by a factor of roughly the volume of the walk, denoted Vwalk(ε). The global translation of the random walk in volume V = R d and (1/ε) due the fact that any point in the string can be considered as a new starting point are other factors that contribute to the number of closed string. Therefore, the final result is

ωclosed(ε) ∼ V.1

ε.eβHε

Vwalk(ε).

Density of Single State: Two limiting cases

1. Volume of the random walk is well-contained in d spatial dimensions (i.e., R >> (ε 1/2)) which corresponds to a string in d non-compact dimensions. In this case the V(ε) ~ ε d/2, the density of states per unit volume is

2. Volume of the random walk is space-filling (R << (ε1/2)) and saturates at order V which corresponds to d compact dimensions that contains the highly excited string states. Hence,

ωclosed(ε)/V ∼eβHε

εd/2+1, for RÀ √ε .

ωclosed(ε) =eβHε

ε, for R¿√ε .

Total DensityThe total density of states, Ω(E), can be ob-

tained. The partition function Z(β) can be

evaluated explicitly in the one loop approxi-

mation. Near the Hagedorn temperature, we

can assume Maxwell-Boltzman statistics and

thus treat the system quasiclassically. Then

we can write Z(β) = exp [z(β)], where z(β) is

the single-string partition function (free ther-

mal energy)

z(β) =

Z ∞0dεω(ε)e−βε .

The singular part of the partition function at

finite volume for closed strings is given by a

set of poles of even multiplicity gı = 2kı

Zsinguları,closed ∼Ã

βı

β − βı

!kı,

with kı = k0 = 1 for the leading Hagedorn

singularity βı = βH.

The regular part of the free energy to leading

energy is

zreguları ∼ aHVD−1−ρHVD−1(β−βı)+O(V (β−βı)2) ,where aH and ρH have dimensions of number

and energy density on the world-volume, re-

spectively.

Whenever the specific heat is positive (and

large), there is a correspondence between the

canonical and microcanonical ensembles and

thus the saddle point approximation is appli-

cable. A necessary condition for this is that

γ < 1, ensuring the canonical internal energy

E(β) ∼ ∂βz(β) diverges at the Hagedorn sin-

gularity that is these systems are unable to

reach the Hagedorn temperature since their re-

quire an infinite amount of energy to do so.

For these systems the Hagedorn temperature

is limiting, and this is true for the closed strings

the Hagedorn temperature is non-limiting for

any model in which Dc > 4. In other words,

stable canonical (i.e., no phase transition) can

be achieved for the closed strings in low di-

mensional thermodynamical limits d < 2.

The Partition Function

The case of closed strings has been studied in

great details by Jain et. al [1992]. The leading

singularity at very high and finite volume is

always a simple pole of the partition function

at the Hagedorn singularity,

Zclosed(β) = (β − βH)−1.Zreg(β) .

The Total Density of StatesThe next to leading singularity of the partition

function β1 is a pole of order k1 = 2D − 2,located at

βH − β1 ∼

⎧⎪⎨⎪⎩α03/2R2

, for RÀ `sR2√α0, for R¿ `s

with α0 ∼ O(1) in string units. The total den-sity of the states is

Ωclosed ≈ βHeβHE+aHVD−1

⎡⎢⎢⎢⎢⎢⎣1−(βHE)

2D−3

(2D− 3)! e−(E−ρHVD−1)/R2| z

δΩ(1)

⎤⎥⎥⎥⎥⎥⎦ ,

Entropy and Temperature

• The entropy of the universe in the Hagedorn phase is then

and therefore the temperature will be

S(E,R) ' βHE+ aHV + ln³1+ δΩ(1)

´,

T (E,R) '⎛⎝βH + ∂δΩ(1)/∂E

1+ δΩ(1)

⎞⎠−1 ' THÃ1+

βH − β1βH

δΩ(1)

!.

T ≡ [(∂S/∂hEi)V ]−1

Correlation FunctionCμνσλ = hδT μνδTσλi= hT μνTσλi− hTμνihTσλi

= 2Gμβ√−G

∂

∂Gβν

ÃGσδ√−G

∂ lnZ

∂Gδλ

!

+2Gσβ√−G

∂

∂Gβλ

ÃGαδ√−G

∂ lnZ

∂Gδν

!, (1)

with δTμν = Tμν − hTμνi and

hT μνi= 2Gμα√−G

∂ lnZ

∂Gαν,

Energy FluctuationNow if we divide the universe inside the Hub-

ble radius, H−1, to small blocks of size `s ¿R ¿ H−1, where R is almost independent oftime during the Hagedorn phase. The par-

tition function Z = exp (−βF), where F =

F (β√−G00,R) is the string free energy with

β√−G00 = T−1

√−G00. Therefore C0000, be-comes

C0000 = hδρ2i= hρ2i− hρi2

= − 1R6

∂

∂β

ÃF + β

∂F

∂β

!= − 1

R6∂hEi∂β

,

=T2

R6CV (1)

Specific HeatThe specific heat CV can be obtained from the

entropy of the system,

CV ≡ −"T2

̶S(E,R)

∂E

!V

#−1

=R2

α03/2T1

1− T/TH, (1)

and thus

C0000 = hδρ2i= hρ2i− hρi2

=T

α03/2R41

1− T/TH(2)

Comparison

• Specific heat for compact and large space is positive:

• Specific heat for non-compact dimensions is negative

• The specific heat of the ideal gas of point particles is positive

CV =R2

α03/2T1

1− T/TH,

CV = −2

D+1

µβHE −

D+1

2

¶2

CV =V

1− d/(d+1)ρd/(d+1)

Metric Fluctuation• On scales larger than the Hubble radius, gravity dominates the dynamics and metric

fluctuations play the leading role. We will calculate here the spectrum of scalar metric fluctuations, fluctuation modes which couple to the matter sources. In the absence of anisotropic stress, there is only one physical degree of freedom, namely the relativistic generalization of the Newtonian gravitational potential. In longitudinal gauge, the metric then takes the form

• On scales smaller than the Hubble radius, the gravitational potential Φ is determined by the matter fluctuations via the Einstein constraint equation (the relativistic generalization of the Poisson equation of Newtonian gravitational perturbation theory)

ds2 = −(1+ 2Φ)dt2+ a(t)2(1− 2Φ)dx2 ,

∇2Φ = 4πGδρ .

Spacetime diagram of the fluctuation modes

Power SpectrumPΦ(k) ≡ k3|Φ(k)|2 ∼ kn−1

= 16π2G2k−4h(δρ)2i= 16π2G2α0−3/2

T

1− T/TH(1)

Note that the amplitude is suppressed by the

ratio (`Pl/`s)4, In order to obtain the observed

amplitudeof fluctuations, a hierarchy of lengths

of the order of 103 is required. This is consis-

tent with our initial assumption that the string

coupling constant should be really small since

(`Pl/`s) = gs ∼ 10−3¿ 1.

Tensor Modes

Ciiii = hδTii

2i= hT ii2i− hTiii2

≈ 10

3

(1− T/TH)`3sR4

(ln

"`3sT

R2(1− T/TH)

#Ã1− 2

5ln

"`3sT

R2(1− T/TH)

#!)

≈ 4

3

T(1− T/TH)`3sR

4ln2

"R2

`3sT(1− T/TH)

#, (1)

Conclusion I• Here, we have studied the generation and evolution of

cosmological fluctuations in a model of string gas cosmology in which an early quasi-static Hagedorn phase is followed by the radiation-dominated phase of standard cosmology, without an intervening period of inflation.

• In order to compute the spectrum of metric fluctuations at late times, we have applied the usual general relativistic theory of cosmological perturbations. Whereas this is clearly justified for times t > tR, its use at earlier times is doubtful.

Conclusion II• We have also assumed that the metric perturbation variable Φ does

not change on super-Hubble scales during the transition between the Hagedorn phase and the radiation-dominated phase of standard cosmology. These assumptions are well justified in the context of the usual relativistic perturbation theory. However, the fact that the Hagedorn phase is described by a dilaton gravity background and not by a purely general relativistic background may lead to somemodifications.

• Although our cosmological scenario provides a new mechanism for generating a scale-invariant spectrum of cosmological perturbations, it does not solve all of the problems which inflation solves. Inparticular, it does not solve the flatness problem. Without assuming that the three large spatial dimensions are much larger than thestring scale, we do not obtain a universe which is sufficiently large today.

Future HopesOur scenario may well be testable observationally. Taking into accountthe fact that the temperature T evaluated at the time ti(k) whenthe scale k exits the Hubble radius depends slightly on k, the formula leads to a calculable deviation of the spectrum from exact scaleinvariance. Since T(ti(k)) is decreased in gas k increases, a slightly redspectrum is predicted. Since the equation of state does not change by orders of magnitude during the transition between the initial phase and the radiation-dominated phase as it does in inflationary cosmology, the spectrum of tensor modes is not expected to be suppressed compared to that of scalar modes. Hence, a large ratio of tensor to scalar fluctuations might be a specific prediction of our model. This issue deserves further attention.

Acknowledgement

Hereby, I would like thank the man in the far “right” for many useful discussions and helps!

Open Strings• Let's consider a highly excited string between Dp- and Dq-branes. In addition to the leading exponential degeneracy for random walks with a fix starting point on the Dp-brane, there would be a degeneracy factor due to the fixing of the endpoints on each brane

(VNNVwalkND).(Vwalk

NNVwalkDN)

where N and D refer to Neumann and Dirichlet boundary conditions. Finally there would be an overall factor due to the translation of the walk in the excluded NN volume which is VNN/Vwalk

NN. Thus for the open string, thedensity of the states looks like

where Vwalkopen = Vwalk

NN.VwalkND.Vwalk

DN.VwalkDD is the total volume of the

random walk.

Two Limiting Cases:

Summary:Summary:

• where $V| = VNN$ and $V⊥ = VDD$ are the volumes• transverse and perpendicular to the $D$-brane. $0 · do ·• d⊥ = dDD$ is the number of dimensions transverse to the• brane with no windings and $Vo$ is the volume of this space (which• is $\calO(1)$ in string units when there are windings in all• directions) . Similarly, $d_c$ is the number of dimensions in which• closed strings have no windings and again $V_c$ is the volume of• this space. Both $\gamma_o$ and $\gamma_c$ are• $\varepsilon$-dependent critical exponent. The `effective' number• of large spacetime dimensions (i.e, the total number of $NN + DD$• dimensions) as a function of $\varepsilon$,• \be• D_o(\varepsilon) = d_NN + d_o(\varepsilon) \,,• \ee

• with $d_NN = p + 1$ being the $p$ spatial non-compact Neumann• directions of the $Dp$-brane.

Total Density of StatesTotal Density of States