Thermal duality and non-singular superstring cosmology Hervé Partouche Ecole Polytechnique, Paris Based on works in collaboration with : C. Angelantonj, I. Florakis, C. Kounnas and N. Toumbas arXiv: 0808.1357, 1008.5129, 1106.0946Balkan Summer Institute, « Particle Physics from TeV to Planck Scale », August 29, 2011.
IntroductionIn General Relativity : Cosmological evolution leads to aninitial Big-Bang singularity.There is a belief : Consistency of String Theory is expected toresolve 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 : +∞ +∞ dl d d k −(k2 +M 2 )l/2Z= Vd d dM ρB (M ) − ρF (M ) e 0 2l (2π) 0
+∞ +∞ dl d d k −(k2 +M 2 )l/2Z= Vd dM ρB (M ) − ρF (M ) e 0 2l (2π)d 0 In fact, there must be a cancellation between the densities of bosons and fermions. Highly non-trivial statement, because they are βH M . ∼ e 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.
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 1 time is Euclidean and compactiﬁed on S (R0), (β = 2πR0 ). −βH Z = ln Tr e = d−1 +∞ √ d k 1 −β k2 +M 2Vd−1 dM ρB (M ) ln √ + ρF (M ) ln 1 + e (2π)d−1 0 1− e−β k2 +M 2 βH M −βM ∼e ∼e When β < βH i.e. T > TH = 1/βH , Z diverges.
This break down of the canonical ensemble formalism at hightemperature is not believed to be an inconsistency of stringtheory in the UV.Instead, it is interpreted as the signal of the occurence of aphase transition := Hagedorn phase transition.This UV behavior can be rephrased in an IR point of view byobserving the appearance of tachyons when T = TH.Usually, one gives a vacuum expectation value to thesetachyons in order to ﬁnd another string background which isstable.However, the dynamical picture of this process, in acosmological set up, is unknown.
So, the logics in string theory at ﬁnite temperature isthe 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.
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) : 0 or 1 mod 2 F =a+a ¯ In this talk : We are going to see that string models yield deformed canonical partition functions −βH a ¯ Z(R0 ) = ln Tr e (−)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 1 1 R0 → with ﬁxed point Rc = √ 2R0 2 1 T = 1/β has a maximal value Tc = . 2πRc −βH a ¯ −βH T < Tc =⇒ Tr e (−) Tr e This allows to interpret T as a temperature.
In fact, the alternative signs imply the eﬀective number ofstring modes which are thermalized is ﬁnite.Thanks to the consistency of these thermal backgrounds, weobtain cosmological evolutions with : No Hagedorn divergence. No singular Big Bang.
Type II superstring modelsCompactiﬁcation on S 1 (R0 ) × T d−1 (V ) × T 9−d × S 1 (R9 )Supersymmetries generated by Right-moving waves are brokenby imposing (−)a boundary conditions along S 1(R9). ¯Supersymmetries generated by Left-moving waves are broken byimposing (−)a+¯ or (−)a boundary conditions along S 1(R0). aBoth models are non-supersymmetric 1-loop vacuumamplitude Z is non-vanishing and may diverge : Lightest scalar masses satisfy : √ M 2 = R0 − 2 =⇒ RH = 2 2 2 2 1 M = − R0 ≥0 2R0
Moreover, the tachyon free model has two kinds of fermions :Spinors and Conjugate Spinors : S8, C8. Admits a T-duality symmetry : 1 R0 → with S8 ↔ C 8 2R0 At the self-dual point R0=Rc: Enhanced SU(2)L + adj matter.
Physical interpretationIn QFT at ﬁnite T, the particles have momentum m0/R0along S 1(R0).In String Theory, the strings have momentum m0/R0 and wrapn0 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 : −βHZ= = ln Tr e where β = 2πR0
In the tachyon-free model, for R0 > Rc :Since (−)a = (−)a+¯ (−)a the same procedure yields a a ¯deformed canonical partition function : Z −βH a ¯ e = Tr e (−) where β = 2πR0 and the fermions are Spinors S8, |m0 | 1 the excitations along S (R0) momenta . R0 for R0 < Rc : T-duality leads the same expression but 1 β = 2π and the fermions are Conjugate Spinors C8, 2R0 the excitations along S 1(R0) have mass contributions |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 : β > 2πRc √ 2 Therefore : T = 1/β > Tc = 2π
Z −βH a ¯ Moreover, e = Tr e (−)may diﬀer from a canonical partition function for multiparticlestates containing modes with a = 1 . ¯ However, in these models, these modes have 1 1 M≥ = R 9 = √ > Tc (R9 is stabilized there) 2R9 2 Since T < Tc , the multiparticle states which contain them are suppressed by Boltzmann’s factor : Z −βH a ¯ −βH e = Tr e (−) Tr e The tachyon free model is essentially a thermal model.
Then, why is there no Hagedorn divergence ? In fact, the convergence of eZ = Tr e−βH (−) when T ≈ Tc is a ¯due to the alternative signs. For a gas of a single Bosonic (or Fermionic) degree of freedom, with Right-moving fermion number a , ¯ √ 2 ln Tr e−βH (−)a = ∓ ¯ ln 1∓(−)a e−β ¯ k+M k (Boson, a) and (Fermion, 1 − a) have opposite contributions ¯ ¯ when they are degenerate. This reduces the eﬀective number of degrees of freedom which are thermalized.
The pairing of degenerate (Boson, a) and (Fermion, 1 − a) ¯ ¯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) : F Z nσ2 ≡− =− 2 V βV βIn general, in any model and dimension, the cancellation is onlyapproximate : F nσd − d V β
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 . (0, −N) (N, 0) (N, N) They trigger the Phase Transition between the two worlds.
Cosmology In each phase, the Euclidean low energy eﬀective action in 1 S (R0) ×T d−1 is, up to 1-loop and 2-derivatives, 0 d−1 d−1 −2φ R 2 Z dx d x βa e + 2(∂φ) + d−1 2 βa 1 2πR0 or 2π Dilaton := String coupling 2R0 At a ﬁxed such that R0 = Rc, there are additional massless 0 xc scalars ϕ with m0 = n0 = ±1. They don’t even exist at other α x0, since in each phase there are only pure momentum or pure winding modes. Their action depends on space only : d−1 0 d−1 −2φ 1 α α α 0 α d x a(xc ) e − ∂ ϕ ∂i ϕ 0 )2 i ¯ =⇒ ∂i ϕ = a(xc )γi a(xc
In total, the action is 0 d−1 d−1 −2φ R Z d−1 −2φ dx d x βa e + 2(∂φ)2 + d−1 − 0 d−1 dx d xa e 0 δ(x − x0 ) c α |γi | 2 βa i,α The analytic continuation x0 → i x0 is formally identical : The gradients yield a negative contribution to the pressure and vanishing energy density, PB = − |γi |2 δ(x0 − x0 ) , α c ρB = 0 i,α This is an unusual state equation. It arises from the phase transition and not from exotic matter. Z nσd If we approximate d to solve explictly the βad−1 β equations of motion, we ﬁnd in conformal gauge :
a Tc 1 ω|τ | ω|τ | ln = ln = η+ ln 1 + − η− ln 1 + ac T d−2 η+ η− √ d−1 ω|τ | ω|τ | φ = φc + ln 1 + − ln 1 + 2 η+ η− d−2√ √ where ω= √ nσd ac Tc eφc , d/2 η± = d−1±1 2 d−2 T φ a e τ τ τ Bouncing cosmology, radiation dominated at late/early times. Perturbative if eφc is chosen small. ∂τ ≤ O(eφc ) =⇒ Ricci curvature is small and higher derivative terms are negligeable.NB: A similar solution with spatial curvature k=−1 also exists. It iscurvature dominated.
SummaryWe have seen string models describe systems at ﬁnite T, wherethe eﬀective number of states which are thermalized is ﬁnite.These models have two thermal phases, where the thermalexcitations are either momentum or winding modes along S 1(R0).Additional massless states with both windings and momentatrigger 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 curvatureduring the cosmological evolution.