Gravitational reheating at strong coupling

We use gauge/gravity correspondence to study the gravitational reheating of strongly coupled gauge theories in the rapid exit from the long inflationary (de Sitter) phase. We estimate the maximal reheating temperature of holographic models in $d$-spatial dimensions, which scale invariance is explicitly broken by a source term for an operator ${\cal O}_\Delta$ of dimension $\frac d2+\frac 12<\Delta<d+1$, as well as the top-down ${\cal N}=2^*$ and the cascading gauge theories. The reheating is most efficient when the inflationary phase Hubble constant is much larger than the mass scale of the conformal symmetry breaking of the theory, and for theories with the conformal symmetry breaking operators that are close to marginal.


Introduction and summary
Gauge theory/string theory correspondence [1,2] is a valuable tool to study dynamics of strongly coupled gauge theories, where alternatives are scarce to non-existing.It served as an inspiration of novel insight into quantum field theory non-equilibrium dynamics, as, e.g., in [3][4][5].
Holography was instrumental in discovery of de Sitter dynamical fixed points (DFPs) of a Quantum Field Theory [6,7].Specifically, consider d + 1-spacetime dimensional quantum field theory, QFT d+1 , in Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe with a scale factor a(t), ds 2 d+1 = −dt 2 + a(t) 2 dx 2 . (1.1) Assuming that the FLRW Universe is asymptotically de Sitter, an arbitrary initial state of the QFT d+1 would typically evolve to a de Sitter DFPan internal state of the theory with spatially homogeneous and time-independent onepoint correlation functions of its stress-energy tensor T µν , and a set of gauge-invariant local operators {O i }, along with strictly positive divergence of the entropy current S µ at late times, lim t→∞ ∇ • S > 0 . (1. 3) The everlasting entropy production implied by (1.3) indicates that a DFP is a genuinely non-equilibrium state of the QFT.A given theory might have multiple distinct DFPs, characterized by a local order parameter of a spontaneously broken global symmetry; there can be phase transitions between DFPs -these and other aspects of a de Sitter DFP of a quantum field theory with a holographic dual were explored in [6][7][8][9][10][11][12][13][14][15][16][17][18].
To understand the astrophysical consequence of a DFP it is useful to unpack (1.3).
Assuming spatial homogeneity and isotropy, we associate the entropy current S µ to a comoving observer u µ ≡ (1, 0) as S µ = s(t)u µ , where s(t) is the physical entropy density.Then, where the late-time, t → ∞, limit of the physical entropy density s(t), provided this limit exists, was called in [9] the vacuum entanglement entropy (VEE) density lim t→∞ s(t) = s ent . (1.5) In other words, asymptotically finite entropy production rate of a QFT de Sitter DFP is due to asymptotically finite physical entropy density.The latter point dramatically differentiates conformal and non-conformal theories: for a CFT it is not the physical, but rather the comoving entropy density s comoving (t) ≡ a(t) d s(t) that is asymptotically constant 1 in de Sitter space-time; the physical entropy density of a CFT rather vanishes in asymptotically de Sitter space-time.As a result, the entropy production rate for a CFT in asymptotically de Sitter space-time vanishes as well -a CFT can not have de Sitter DFPs 2 .Consider now a realistic cosmology where an extended period of the accelerated expansion (the inflationary de Sitter phase) is followed by the inflationary exit, and subsequently the Hot Big Bang.In [19] it was argued that the Hot Big Bang (HBB) is a generic outcome of the fast exit from the de Sitter DFP, irrespectively of the inflaton physics 3 .The argument is very simple, and we review it here for completeness: • While technical details of the inflationary exit will not be important, it is useful to keep in mind a specific model.To this end we assume that the Hubble parameter H evolves as . (1.6) Here the constant energy scale γ specifies the exit-rate from the inflation, and the scale parameter a(t) is normalized as a(t = 0) = 1.For a rough estimate, in this model the exit from inflation occurs during the time frame t ∈ ∝ (−γ −1 , γ −1 ), so that the scale factor changes as ln a(t) • During the inflationary exit, the comoving entropy density can only increase, so 1 See section 2 of [11] for a pedagogical holographic review.
2 One can view this as a reflection of the fact that a CFT dynamics in de Sitter is Weyl equivalent to a dynamics in Minkowski space-time; given that Minkowski dynamics leads to equilibration, the same must be true for a CFT late-time de Sitter evolution. 3In particular, whether or not the inflaton couples directly to the theory to be reheated.
where we denoted by s e the physical entropy density of the post-inflationary QFT state.
• The post-inflationary state e is nonequilibrium; its subsequent evolution leads to its thermalization with the Hot Big Bang thermal entropy density s HBB ≥ s e ≥ s ent . (1.10) • To put a lower bound on the maximal HBB reheating temperature T HBB , one can interpret the VEE density s ent of a theory de Sitter DFP as an equilibrium thermal entropy density of the theory s eq (T ) at the DFP effective temperature, T HBB ≥ T DF P , where s eq (T ) In this paper we strengthen the claim of [19], in particular we argue that the reheating is very efficient, provided the inflationary stage Hubble scale H is much higher than any mass scale Λ of the theory to be reheated, i.e., when We now summarize our results: Consider a massive QFT d+1 that is a deformation of a CFT d+1 by a single operator To the leading order in O Λ H we find where Following (1.11), the reheating temperature T HBB is at least as high as that of T DF P in (1.14), i.e., the reheating is very efficient in the phenomenologically relevant regime H ≫ Λ.Note that QFTs with operators of higher dimensions achieve larger reheating temperature relative to the mass scale of the theory; however O ∆ can not be marginal -from (1.15), (1.16) In fact, we expect that for an exactly marginal operator O d+1 the reheating temperature exactly vanishes, since the theory remains conformal in this case.
Typically a massive QFT has nonzero sources for a set of relevant operators {O ∆ i }.
For example, N = 2 * gauge theory [20][21][22] is a mass-deformation of N = 4 supersymmetric Yang-Mills theory by a pair of operators {O 2 , O 3 }.We show that in such cases the maximal reheating temperature is still bounded from below by (1.14), where ∆ = max{∆ i }, and Λ is the mass scale of the source of the maximal dimension operator in the range d 2 + 1 2 < ∆ < d + 1.The Klebanov-Strassler cascading gauge theory [23,24] does not have relevant operators; it does have a single marginal, but not exactly marginal, operator [25].In this case we find that the maximal reheating temperature is bounded from below by T DF P with where Λ is the strong coupling scale of the confining cascading gauge theory 4 .
The rest of the paper is organized as follows.In section 2 we derive (1.14).In section 3 we combine the earlier results on N = 2 * de Sitter DFPs [6] and its Minkowski spacetime thermodynamics [27] to derive the corresponding T DF P .In section 4 we combine the earlier results on the cascading gauge theory de Sitter DFPs [12] and its hightemperature Minkowski space-time thermodynamics [28,29] to derive the corresponding The reheating mechanism introduced in [19] and further developed here is a strong coupling (holographic) version of the preheating in "non-oscillatory" (NO) models [30,31].To make a closer connection it would be interesting to study finite-N and finite 't Hooft coupling corrections to the holographic gravitational reheating discussed here.

Efficient near-conformal reheating in QFT d+1
Consider a holographic toy model of d + 1-dimensional massive QFT d+1 with the effective dual gravitational action5 : The d + 2-dimensional gravitational constant κ is related to the ultraviolet (UV) conformal fixed point CFT d+1 central charge, and φ is a gravitational bulk scalar with which is dual to a dimension ∆ operator O ∆ of the boundary theory.QFT d+1 is a relevant deformation of the UV CFT d+1 as in (1.13).We study QFT d+1 dynamics in FLRW Universe (1.1).

Holographic gravitational dynamics of QFT d+1
A generic state of the boundary field theory with a gravitational dual (2.1), homogeneous and isotropic in the spatial boundary coordinates x = {x 1 , • • • , x d }, leads to a bulk gravitational metric ansatz with the warp factors A, Σ as well as the bulk scalar φ depending only on {t, r}.From the effective action (2.1) we obtain the following equations of motion: as well as the Hamiltonian constraint equation: and the momentum constraint equation: In (2.4)-(2.6)and below we denoted ′ = ∂ ∂r , ˙= ∂ ∂t , and The nearboundary r → ∞ asymptotic behavior of the metric functions and the scalar encode the mass parameter Λ and the boundary metric scale factor a(t): ) is the residual radial coordinate diffeomorphism parameter [32].An initial state of the boundary field theory is specified providing the scalar profile φ(0, r) and solving the constraint (2.5), subject to the boundary conditions (2.7).Equations (2.4) can then be used to evolve the state.
The subleading terms in the boundary expansion of the metric functions and the scalar encode the evolution of the energy density E(t), the pressure P (t) and the expectation values of the operator O ∆ (t) of the prescribed boundary QFT initial state.
These observables can be computed following the holographic renormalization of the model as, e.g., for models discussed in [6] and [10].Our interest here is however the dynamics of the non-equilibrium entropy density s(t).The background metric (2.3) has an apparent horizon located at r = r AH , where [32] Following [33,34] we associate the non-equilibrium entropy density s of the boundary QFT d+1 with the Bekenstein-Hawking entropy density of the apparent horizon Using the holographic background equations of motion (2.4)-(2.6)we find From the Hamiltonian constraint (2.5) and the boundary asymptotic (2.7), i.e., is manifestly positive; thus we conclude that during the holographic evolution, the comoving entropy density can only increase 12) The latter establishes the vital fact in harvesting the vacuum entanglement entropy of a de Sitter DFP, see (1.8) and (1.10).

de Sitter DFP of QFT d+1
Following [6], the equations for the late-time attractor of the evolution (a de Sitter DFP) can be obtained from (2.4)-(2.6)taking t → ∞ limit with identification Introducing a new radial coordinate and denoting we find along with an algebraic expression for g: . (2.17) A de Sitter DFP solution has to satisfy the boundary conditions (2.7), and remain nonsingular for x ∈ (0, x AH ], where the location of the apparent horizon x AH is determined from [6] Without the loss of generality we fix the diffeomorphism parameter λ so that We will always have x AH > 1 3 .For general values of Λ H , solutions representing de Sitter DFPs have to be found numerically.Luckily, the regime of the efficient reheating happens when the conformal symmetry breaking parameter is small; in the latter case the solution to (2.16) can be constructed perturbatively in p 1 .To proceed, we set where the p 1 → 0 solution represents the UV CFT d+1 de Sitter vacuum solution, and following (2.7), the bulk scalar solution is normalized in the limit x → 0 as Furthermore, the location of the AH is determined from (2.18) as

.24)
As z ≡ 1 − x → 0 + the bulk scalar solution diverges, (2.25) While there is no simple closed form solution for s (2) and a (2) , it is straightforward to extract their (singular) asymptotes as z → 0 + , Finally, we can compute the leading in the conformal symmetry breaking parameter p 1 (see (2.20)) VEE density s ent : or with c sing given by (2.25).
The computations leading to (2.29) are valid when ∆ is strictly above the Breitenlohner- It is straightforward to extend the analysis when ∆ = ∆ BL -the only modification in (2.29) is a different from (2.25) expression for (2.30) Consider now special cases of the general formula (2.29).
• (d, ∆) = (3, 3): s ent (2.33) Note that while to leading order in the conformal symmetry breaking parameter p 1 (2.32) and (2.33) are identical when expressed in p 1 , however, because of (2.20), the Λ H scaling of s ent is different for ∆ = 3 and ∆ = 2.In fig. 1 we test predictions (2.32) and (2.33) against the fully nonlinear in p 1 computations of s ent in these two models.

Effective temperature T DF P
Following the general arguments in section 1, the maximal reheating temperature of a QFT d+1 from the rapid exit of the prolonged inflationary stage is at least as high as the effective DFP temperature T DF P , computed from where s eq is the thermal equilibrium entropy density of the theory in Minkowski spacetime, and s ent is the VEE density of the inflationary phase -in the case of prolonged inflation a de Sitter DFP of the theory.Note that we compare the dimensionless quantities in (2.34).s eq can be computed from the thermal equation of state of the QFT d+1 , and since our theory is massive, s eq will depend both on T and Λ.For an efficient reheating from the phenomenological perspective we need i.e., upon the inflationary exit, the de Sitter DFP state of the theory is reheated to a much higher temperature than any mass scale of the theory -for a QCD this would a reheating way above the confinement scale of the theory.From the entropy perspective, effective reheating implies that Remarkably, the condition (2.36) implies that the de Sitter DFP must be in the regime of perturbatively small conformal symmetry breaking parameter p 1 (2.20) of the QFT d+1 .Indeed, from (2.29) we have so that (2.36) is indeed true when H ≫ Λ and ∆ > ∆ BL = 1 2 + d 2 .In the regime (2.35), the thermal equation of state of a QFT d+1 is that of the UV CFT d+1 .The thermal entropy density of the corresponding holographic CFT d+1 can be easily computed, we find Consider now some special cases of the general formula (2.39).
• (d, ∆) = (2, 2): • (d, ∆) = (3, 3): when H ≫ Λ . (2.41) We conclude this section with the conjecture that T DF P computed in the regime H ≫ Λ is actually the maximal reheating temperature T HBB itself, i.e., the bound in (1.11) is saturated.The reason being is that for an efficient reheating, the de Sitter DFP state of the theory must be very entropic from the QFT d+1 perspective, see (2.36).Thus, following the ideas of the "eigenvalue thermalization hypothesis" [35] it must be to a good approximation thermal.This conjecture passes the test of the full-fledged simulations of (d, ∆) = (2, 2) model discussed in [19].In fig. 2 we compare the temperatures (the left panel) and the entropy densities (the right panel) in the model at de Sitter DFPs (the black curves) and upon thermalization following the inflationary exit (the blue curves).The dashed red curves represent the leading approximation in the small conformal symmetry breaking parameter (2.20) given by ( Sitter DFP, however, the entropy production during the thermalization process vanishes as Λ H → 0. Correspondingly, the maximal Hot Big Bang reheating temperature T HBB is always larger than T DF P , however, the difference between the two also vanishes as The massive holographic models discussed in section 2 involved a deformation of a UV CFT d+1 by a single relevant operator O ∆ .More common are holographic models 6 T HBB is referred to as T max r in [19].s HBB is the same as the equilibrium thermal entropy density of the model in Minkowski space-time evaluated at the temperature T = T max r . where multiple relevant operators are sources -e.g., the four (d = 3) space-time dimensional N = 2 * gauge theory [20,21].Here, there are two relevant operators {O 2 , O 3 } with the sources being the masses of the bosonic p ∆=2  [6,11] and Minkowski space-time thermodynamics of the model was analyzed in [27].Using the results from these and the correspondence relation (2.34), we produce in fig. 3 the maximal reheating temperature in N = 2 * gauge theory (the blue curve).The dashed red curve represents the leading approximation to T DF P in the limit of small conformal symmetry breaking m H → 0, see 8 (2.41).This example illustrates that the regime of the efficient reheating is controlled by the relevant operator with the largest scaling dimension.

Reheating in the cascading gauge theory
The general formula (2.39) implies that the most efficient reheating occurs in theories with the largest dimension sourced operator O ∆ .Since renormalizability of a QFT d+1 requires that sources are introduced only for marginal and/or relevant operators, and exactly marginal operators do not reheat (1.16), we look for a model with a marginal, but not exactly marginal sourced operator.An example of just such a model is the d = 3, N = 1 supersymmetric SU(N + M) × SU(N) Klebanov-Strassler cascading gauge theory [23][24][25][26].de Sitter DFPs of the cascading gauge theory were studied extensively in [12]; here, 7 There is a consistent truncation of the N = 2 * holographic model to m f = 0 and m b = 0.
However, in this case the deformation operator is O 2 with ∆ = 2 = ∆ BF , and thus it can not lead to an efficient reheating. 8The 1 2 scaling exponent is correct, but the prefactor must be modified to account for the fact that the bulk scalar dual to O 3 does not have a canonical kinetic term; furthermore, the precision holography in this case requires that m ∝ Λ, but is not equal to it [21,22].
with Λ being the strong coupling scale of the theory [26].The high-temperature Minkowski space-time thermodynamics of the theory was analyzed in [28,29], From (2.34) we estimate the lower bound on the maximal reheating T DF P as in (1.17).

3 )Figure 2 :
Figure 2: (The left panel) The maximal reheating temperature T HBB (the blue curve, from [19]) and the effective temperature T DF P (the black curve) of the de Sitter dynamical fixed point of (d, ∆) = (2, 2) model as a function of the conformal symmetry breaking parameter.(The right panel) The maximal thermal entropy density of the Hot Big Bang s HBB (the blue curve, from [19]) and the VEE density of the de Sitter DFP s ent (the black curve) in this model.The red dashed curves represent the leading approximations for H ≫ Λ.

Figure 3 :
Figure 3: The lower bound (the blue curve) on the maximal reheating temperature T DF P of N = 2 * gauge theory in the rapid exit from the prolonged inflationary stage.The dashed red curve corresponds to small conformal symmetry breaking parameter, i.e., for H ≫ m, in (d, ∆) = (3, 3) model, see (2.41).

1 ∝ m 2 b and the fermionic p ∆=3 1 ∝
m f components of the N = 2 hypermultiplet.For generic values of m b = m f the supersymmetry in the model is completely broken.We consider 7 the N = 2 supersymmetry preserving masses, m b = m f ≡ m. de Sitter DFPs of the model were studied in .32) model (the left panel), and (d, ∆) = (3, 2) model (the right panel).Solid curves are numerical fully non-linear in p 1 (see (2.20)) computations of s ent ; the dashed red curves represent the leading order analytic approximations (2.32) and (2.33) valid when [19])for the temperature and (2.31) for the entropy density.Computations of the maximal reheating temperature T HBB and the Hot Big Bang thermal entropy density s HBB (the blue curves in fig.2) require holographic simulations of the rapid, i.e., in the regime γ ≫ H, inflationary exit (1.6) of the corresponding model.For (d, ∆) = (2, 2) model this was done in[19]6 .Notice that upon thermalization, the equilibrium entropy density is always larger than the corresponding VEE s ent of the de