Ringing in de Sitter spacetime

Hydrodynamics is a universal effective theory describing relaxation of quantum field theories towards equilibrium. Massive QFTs in de Sitter spacetime are never at equilibrium. We use holographic gauge theory/gravity correspondence to describe relaxation of a QFT to its Bunch-Davies vacuum - an attractor of its late-time dynamics. Specifically, we compute the analogue of the quasinormal modes describing the relaxation of a holographic toy model QFT in de Sitter.


Introduction
Isolated strongly interacting systems typically 1 reach a thermal equilibrium state at late times of its dynamical evolution. An approach towards equilibrium is governed by hydrodynamics -a universal effective theory organized as derivative expansion of the local velocity gradients to the temperature of the final equilibrium state. One example is the relativistic hydrodynamics of conformal gauge theories developed in [3,4].
As an effective description, gradient expansion of the gauge theory hydrodynamics has zero radius of convergence due to the existence of the non-hydrodynamic modes in equilibrium plasma [5,6]. Whenever gauge theory allows for a dual holographic description [7,8] in terms of classical supergravity, its thermal equilibrium state is represented by a black hole/black brane in the gravitational dual [9]. Furthermore, linearized hydrodynamic and non-hydrodynamic excitations about the equilibrium state are mapped to the quasinormal modes (QNMs) of the corresponding dual black hole [10]. QNMs encode the information about the relaxation of the near-equilibrium state of a gauge theory plasma [11][12][13][14]. Implicit in the above overview was an assumption that QFT dynamics occurs in Minkowski spacetime. Using holographic correspondence 2 , it was argued in [21,22] that massive gauge theories in de Sitter spacetime are not in equilibrium at late times: while Bunch-Davies (BD) vacuum is the late-time attractor of a dynamical evolution of a QFT state, the co-moving entropy production rate is nonzero. In this paper we make the first step addressing the question: 1 There are some exceptions to this lore: condensed matter systems with many-body localization [1]; holographic models with phase-space restricted dynamics [2]. 2 For early work on gauge theories in de Sitter within holographic framework see [15][16][17][18][19].
What is the effective theory of the relaxation towards Bunch-Davies vacuum of a massive QFT?
We restrict our attention to a simple holographic toy model of a 2 + 1-dimensional massive QF T 3 with the effective dual gravitational action 3 : The four dimensional gravitational constant κ is related to the ultraviolet (UV) con- φ is a gravitational bulk scalar with (1.5) Following [21], in the next section we describe gravitational dynamical setup encoding de Sitter evolution of spatially homogeneous and isotropic states of the boundary field theory. We study the late-time attractor of the evolution in section 2.1. In section 2.2 we compute the spectrum of linearized fluctuations of the boundary theory around its BD vacuum. In section 2.3 we use fully nonlinear characteristic formulation of asymptotically AdS dynamics [23] and establish that generic homogeneous and isotropic states of the boundary theory indeed "ring-down" to BD vacuum with frequencies computed in section 2.2. We conclude in section 3. 3 We set the radius L of an asymptotic AdS 4 geometry to unity.

Holographic gravitational dynamics
A generic state of the boundary field theory with a gravitational dual (1.1), homogeneous and isotropic in the spatial boundary coordinates x = {x 1 , x 2 }, 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 (1.1) we obtain the following equations of motion: as well as the Hamiltonian constraint equation: and the momentum constraint equation: In (2.2)-(2.4) we denoted ′ = ∂ ∂r ,˙= ∂ ∂t , and d + = ∂ ∂t + A ∂ ∂r . The near-boundary r → ∞ asymptotic behaviour of the metric functions and the scalar encode the mass parameter Λ and the boundary metric scale factor a(t) ≡ e Ht : is the residual radial coordinate diffeomorphism parameter [23]. An initial state of the boundary field theory is specified providing the scalar profile φ(0, r) and solving the constraint (2.3), subject to the boundary conditions (2.5). Equations (2.2) 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 φ (t) of the prescribed boundary QFT initial state.
Specifically, extending the asymptotic expansion (2.5) for {φ, A}, the observables of interest can be computed following the holographic renormalization of the model: where the terms in brackets, depending on arbitrary constants {δ 1 , δ 2 }, encode the renormalization scheme ambiguities. Independent of the renormalization scheme, these expectation values satisfy the expected conformal Ward identity Furthermore, the conservation of the stress-energy tensor is a consequence of the momentum constraint (2.4): From now on we choose a scheme with δ i = 0.
One of the advantages of the holographic formulation of a QFT dynamics is the natural definition of its far-from-equilibrium entropy density. A gravitational geometry (2.1) has an apparent horizon located at r = r AH , where [23] d + Σ Following [24,25] we associate the non-equilibrium entropy density s of the boundary QFT with the Bekenstein-Hawking entropy density of the apparent horizon (2.14) Using the holographic background equations of motion (2.2)-(2.4) we find Following [21] it is easy to prove that the entropy production rate as defined by (2.15) is The holographic evolution as explained above is implemented in section 2.3, adopting numerical codes developed in [26,27].

Bunch-Davies vacua of holographic toy QF T 3
Following [21], the equations for the late-time attractor of the evolution (a Bunch- Introducing a new radial coordinate x ≡ H r , (2.18) and denoting we find along with an algebraic expression for g: Vacuum solution has to satisfy the boundary conditions (2.5), and remain nonsingular for x ∈ (0, x AH ], where the location of the apparent horizon x AH is determined from [21] Without loss of generality we fix the diffeomorphism parameter λ so that We will always have x AH > 1 3 . It is straightforward to construct an analytic solution to (2.20) as a series expansion in conformal symmetry breaking parameter which determines following (2.21) (2.27) From (2.22), the apparent horizon is located at  For generic p 1 we have to resort numerics. Details of the numerical implementation are explained in [21]. Fig. 1  Note that in vacuum P v = −E v , thus following (2.10), In [22] it was argued that the vacuum of a massive QFT in de Sitter has a constant "entanglement" entropy density s ent , related to the comoving entropy production rate R at late times. Specifically, parameterizing the comoving entropy production from (2.15) as  Following [22], the surface gravity of the apparent horizon equals (−H). the horizon [28]. In case of infalling Eddington-Finkelstein coordinates (as in (2.1)), the horizon boundary condition is replaced with the regularity at the trapped surface (the apparent horizon). We stress again that it is the boundary conditions at the horizon and the asymptotic boundary that determine the spectrum of fluctuations.

Spectrum of vacuum linearized fluctuations
In analogy to QNMs, we consider linearized fluctuations of the system (2.2)- (2.4) about the late-time attractor solution (2.17). To this end, we define the fluctuations with the harmonic time dependence of frequency ω as follows: where the connection coefficients  that the vacuum equations of motion have a coordinate singularity 5 when A v (x = x singularity ) = 0. In our case x singularity = 1 3 , see (2.23). This coordinate singularity occurs always before the apparent horizon: x AH > x singularity . Turns out that the connection coefficients C i,j are singular at x singualarity , and requiring that this is just a coordinate singularity and the fluctuating fields H i are smooth across this point and extend all the way to the apparent horizon x AH , provides the second boundary condition on the spectrum of fluctuations.
To recap: the spectrum of linearized fluctuations about Bunch-Davies vacuum is determined from: • Dirichlet conditions at the AdS boundary on the non-normalizable modes of the dual gravitational bulk fluctuating fields; • regularity condition for bulk fluctuating fields at the location A v = 0.
It is instructive to solve (2.35) perturbatively in the conformal deformation parameter p 1 , using perturbative expansion for the BD vacuum (2.25)-(2.26). Introducing to leading order k = 0 we find:  The leading order solution (2.42) can be extended to higher orders in O(p 1 ). For example, for n = 2 mode we find: where we fixed the diffeomorphism parameter λ(t) to all orders in p 1 requiring that A(t, x = 1 3 ) = 0 . Note that n = 2 mode is purely dissipative. In fact, we find that all modes except for n = 3 are purely dissipative. For example,

44)
6 This is also confirmed comparing with the relaxation to BD vacuum in the full nonlinear dynamics as explained in section 2.3. whilê Although the spectrum is determined solving (2.35) on the radial interval x ∈ (0, x singularity ), we verified that the solution can indeed be smoothly extended to the full interval x ∈ (0, x AH ). For example, the radial profile H 1 (x) at p 1 = 1 is presented in fig. 4.

Fully nonlinear dynamics and relaxation to BD vacuum
In this section we report results of the fully nonlinear evolution of the toy holographic QFT defined by a dual gravitational action (1.1). Numerical implementation parallel the codes developed in [26,27], and will not be discussed here. In what follows we focus on the model 7 with p 1 = 1. We use the radial coordinate as in (2.18) and evolve in dimensionless time τ ≡ Ht. As in [23] we adjust the diffeomorphism parameter λ(t) so that the apparent horizon is always at x AH = 1. We set the initial condition for the evolution as where A is the amplitude. We also need to supply the initial energy density (see (2.12)) We verified that BD vacuum is indeed the attractor of long-time dynamics by choosing different initial states for the evolution, i.e., different profiles φ initial and/or µ initial . Fig. 5 represent a typical dynamical evolution of the boundary QFT state from the initial condition (2.46). As times τ = Ht 2 the state relaxes to BD vacuum. The relaxation process is studied in further details as follows.
We use the last 1000 data points, corresponding to time interval τ ∈ [5.6, 6] and fit the observed O φ (τ ) with a single QNM ansatz: where α i are constant free parameters. α 1 is expected to agree with the BD expectation value and α 3 should approximate the frequency of the lowest BD QNM mode, i.e., 7 The discussion is generic for the parameter set with stable and convergent evolution of the code. n = 2. We find that the BD vacuum expectation value is correct with a relative error of ∼ 3 × 10 −6 and the relative error in the frequency,  (2.51) The residual δ is presented in fig. 6. The quality of approximation suggests that the QNMs computed in section 2.2 are all the modes defining the relaxation of the theory to its BD vacuum.

Conclusion
A surprising fact discovered in [21,22] is that a vacuum of a massive QFT in de Sitter space-time has a constant entropy density s ent . We stress that it is important that both the Hubble constant is nonzero, and that the theory is non-conformal. For example, in a simple 2+1 dimensional holographic toy model discussed here where c is a UV central charge of the model and Λ is a mass scale of the theory.
Thermal equilibrium states have entropy. (Non)-hydrodynamic modes in equilibrium plasma owe their existence to this entropy -no entropy, nothing to excite. By analogy, the nonvanishing vacuum entropy of a massive QFT in de Sitter suggests that there should be analogous QNM-like excitations about its Bunch-Davies vacuum. In this paper we showed that this is indeed the case.