Thermal duality and non-singular superstring cosmology

Hervé PartoucheEcole Polytechnique, Paris

Based on works in collaboration with :C. Angelantonj, I. Florakis, C. Kounnas and N. Toumbas

arXiv: 0808.1357, 1008.5129, 1106.0946

Balkan Summer Institute, « Particle Physics from TeV to Planck Scale », August 29, 2011.

Introduction

In General Relativity : Cosmological evolution leads to an initial Big-Bang singularity.

There is a belief : Consistency of String Theory is expected to resolve it.

In this talk, we would like to discuss this question.

Consistency of a string vacuum means what ?

There is no tachyon in the spectrum i.e. particles with negative masses squared.

A tachyon would mean we sit at a maximum of a potential. An Higgs mechanism should occur to bring us to a true vacuum.

It is an IR statement but string theory has the property to relate IR and UV properties into each other.

From a UV point of view, consistency means the 1-loop vacuum-to-vacuum amplitude Z is convergent in the « dangerous region » l → 0 :

Z =

� +∞

0

dl

2lVd

�ddk

(2π)d

� +∞

0dM

�ρB(M)− ρF (M)

�e−(k2+M2)l/2

In fact, there must be a cancellation between the densities of bosons and fermions.

Highly non-trivial statement, because they are .

Cancellation such that the effective density of states is that of a 2-dimensional Quantum Field Theory [Kutasov, Seiberg].

Effectively, there is a finite number of particles in the UV.

Z =

� +∞

0

dl

2lVd

�ddk

(2π)d

� +∞

0dM

�ρB(M)− ρF (M)

�e−(k2+M2)l/2

∼ eβHM

For cosmological pusposes, we should reconsider this question of consistency for string models at finite T.

The 1-loop vacuum amplitude Z is computed in a space where time is Euclidean and compactified on S

1(R0),

Z = lnTr e−βH =

Vd−1

�dd−1k

(2π)d−1

� +∞

0dM

�ρB(M) ln

� 1

1− e−β√k2+M2

�+ ρF (M) ln

�1 + e−β

√k2+M2

��

∼ eβHM ∼ e−βM

(β = 2πR0).

When , Z diverges.β < βH i.e. T > TH = 1/βH

This break down of the canonical ensemble formalism at high temperature is not believed to be an inconsistency of string theory in the UV.

Instead, it is interpreted as the signal of the occurence of a phase transition := Hagedorn phase transition.

This UV behavior can be rephrased in an IR point of view by observing the appearance of tachyons when T = TH.

Usually, one gives a vacuum expectation value to these tachyons in order to find another string background which is stable.

However, the dynamical picture of this process, in a cosmological set up, is unknown.

So, the logics in string theory at finite temperature is the following :

When we go backward in time and the temperature is growing up, we incounter the Hagedorn transition before we reach an eventual initial Big Bang singularity.

The hope is that if we understand how to deal with it, we may evade the initial curvature and infinite temperature divergences of the Big Bang present in General Relativity.

In this talk : We are going to see that string models yield deformed canonical partition functions

Z(R0) = lnTr�e−βH(−)a

�

Before annoucing what we are going to do in this talk, let us define some notations :

To describe a canonical ensemble, the fields are imposed (−)F boundary conditions along S

1(R0).

In string theory, a state corresponds to the choice of two waves propagating along the string : A Left-moving one (from right to left) and a Right-moving one (from left to right) :

F = a+ a0 or 1 mod 2

Total Right-moving fermion number of the multiparticle states

It is not a canonical partition function, but has good properties :

Converges for any R0, due to the alternative signs (no tachyon).

T-duality symmetry

R0 → 1

2R0with fixed point Rc =

1√2

has a maximal value .T = 1/β Tc =1

2πRc

This allows to interpret T as a temperature.

T < Tc =⇒ Tr�e−βH(−)a

�� Tr e−βH

In fact, the alternative signs imply the effective number of string modes which are thermalized is finite.

Thanks to the consistency of these thermal backgrounds, we obtain cosmological evolutions with :

No Hagedorn divergence.

No singular Big Bang.

Compactification on

Supersymmetries generated by Right-moving waves are broken by imposing boundary conditions along S

1(R9).

Supersymmetries generated by Left-moving waves are broken by imposing boundary conditions along S

1(R0).

Both models are non-supersymmetric ⇒ 1-loop vacuum amplitude Z is non-vanishing and may diverge :

Lightest scalar masses satisfy :

Type II superstring modelsS1(R0)× T d−1(V )× T 9−d × S1(R9)

(−)a

(−)a+a or (−)a

M2 = R20 − 2 =⇒ RH =

√2

M2 =

�1

2R0−R0

�2

≥ 0

Moreover, the tachyon free model has two kinds of fermions : Spinors and Conjugate Spinors : S8, C8.

Admits a T-duality symmetry :

At the self-dual point R0=Rc: Enhanced SU(2)L + adj matter.

R0 → 1

2R0with S8 ↔ C8

Physical interpretationIn QFT at finite T, the particles have momentum m0/R0 along S

1(R0).

In String Theory, the strings have momentum m0/R0 and wrap n0 times along S

1(R0).

However, it is possible to use symetries on the worldsheet (modular invariance) to reformulate the theory in terms of an infinite number of point-like particles i.e. with n0 = 0.

In the thermal model, for R0 > RH, the string 1-loop amplitude can be rewritten as a canonical partition function :

Z = = lnTr e−βH β = 2πR0where

where and the fermions are Spinors S8,

the excitations along S 1(R0) momenta .

In the tachyon-free model, for R0 > Rc :

Since the same procedure yields a deformed canonical partition function :

(−)a = (−)a+a (−)a

eZ = Tr�e−βH(−)a

�

β = 2πR0

for R0 < Rc : T-duality leads the same expression but

and the fermions are Conjugate Spinors C8,

the excitations along S 1(R0) have mass contributions .

β = 2π1

2R0

|m0|R0

|n0| 2R0

This contribution is proportional to the lenght of S

1(R0) : They are winding modes. (solitons in QFT language)

Thus, we have two distinct phases :

a momentum phase R0 > Rc and a winding phase R0 < Rc .

In both :

Therefore :

β > 2πRc

T = 1/β > Tc =

√2

2π

eZ = Tr�e−βH(−)a

�� Tr e−βH

Moreover,

may differ from a canonical partition function for multiparticle states containing modes with .

However, in these models, these modes have

Since T < Tc , the multiparticle states which contain them are suppressed by Boltzmann’s factor :

eZ = Tr�e−βH(−)a

�

The tachyon free model is essentially a thermal model.

a = 1

M ≥ 1

2R9= R9 =

1√2> Tc (R9 is stabilized there)

In fact, the convergence of when T ≈ Tc is due to the alternative signs.

eZ = Tr�e−βH(−)a

�

have opposite contributions when they are degenerate. (Boson, a) and (Fermion, 1− a)

This reduces the effective number of degrees of freedom which are thermalized.

For a gas of a single Bosonic (or Fermionic) degree of freedom, with Right-moving fermion number ,a

ln Tr�e−βH(−)a

�= ∓

�

k

ln�1∓(−)ae−β

√k+M2

�

Then, why is there no Hagedorn divergence ?

The pairing of degenerate can even be an exact symmetry of models.

This symmetry can exist only in d = 2 and concerns massive modes only [Kounnas].

In this case, exact cancellation in Z of all contributions from the massive modes : They remain at zero temperature.

We are left with thermal radiation for a finite number of massless particles (in 2 dimensions) :

(Boson, a) and (Fermion, 1− a)

F

V≡ − Z

βV= −nσ2

β2

In general, in any model and dimension, the cancellation is only approximate :

F

V� −nσd

βd

What happens at R0 = Rc ?

There are additional massless states in the Euclidean model : gauge bosons U(1) → SU(2) + matter in the adjoint.

They have non-trivial momentum and winding numbers m0 = n0 = ±1 along S

1(R0).

This is exactly what is needed to transform the states of the « winding phase » R0 < Rc into the states of the « momentum phase » R0 > Rc .

They trigger the Phase Transition between the two worlds.

(0, −N)(N, 0)

(N, N)

In each phase, the Euclidean low energy effective action in S

1(R0) ×Td−1 is, up to 1-loop and 2-derivatives,

2πR0 or 2π1

2R0

�dx0dd−1x βad−1

�e−2φ

�R

2+ 2(∂φ)2

�+

Z

βad−1

�

At a fixed such that R0 = Rc, there are additional massless scalars with m0 = n0 = ±1. They don’t even exist at other x0, since in each phase there are only pure momentum or pure winding modes. Their action depends on space only :

ϕαx0c

�dd−1x a(x0

c)d−1 e−2φ

�− 1

a(x0c)

2∂iϕ

α∂iϕα

�=⇒ ∂iϕ

α = a(x0c)γ

αi

Cosmology

Dilaton := String coupling

In total, the action is

The analytic continuation x0 → i x0 is formally identical :

The gradients yield a negative contribution to the pressure and vanishing energy density,

This is an unusual state equation. It arises from the phase transition and not from exotic matter.

If we approximate to solve explictly the

equations of motion, we find in conformal gauge :

Z

βad−1� nσd

βd

�dx0dd−1x βad−1

�e−2φ

�R

2+ 2(∂φ)2

�+

Z

βad−1

�−�

dx0dd−1x ad−1 e−2φ δ(x0 − x0c)

�

i,α

|γαi |

PB = −�

i,α

|γαi |2 δ(x0 − x0

c) , ρB = 0

ad−2 T

τ

eφ

ττ

lna

ac= ln

Tc

T=

1

d− 2

�η+ ln

�1 +

ω|τ |η+

�− η− ln

�1 +

ω|τ |η−

��

φ = φc +

√d− 1

2

�ln

�1 +

ω|τ |η+

�− ln

�1 +

ω|τ |η−

��

where ω =d− 2√

2

√nσd ac T

d/2c eφc , η± =

√d− 1± 1

Bouncing cosmology, radiation dominated at late/early times.Perturbative if is chosen small. Ricci curvature is small and higher derivative terms are negligeable.

NB: A similar solution with spatial curvature k=−1 also exists. It is curvature dominated.

eφc

∂τ ≤ O(eφc) =⇒

SummaryWe have seen string models describe systems at finite T, where the effective number of states which are thermalized is finite.

These models have two thermal phases, where the thermal excitations are either momentum or winding modes along S

1(R0).

Additional massless states with both windings and momenta trigger the phase transition at the maximal temperature Tc.

They induce a negative contribution to the pressure,

which implies a bounce in the string coupling and Ricci curvature during the cosmological evolution.