Thermal stability of stealth and de Sitter spacetimes in scalar-tensor gravity

Stealth solutions of scalar-tensor gravity and less-known de Sitter spaces that generalize them are analyzed regarding their possible role as thermal equilibria at non-zero temperature in the new first-order thermodynamics of scalar-tensor gravity. No stable equilibria are found, further validating the special role of general relativity as an equilibrium state in the landscape of gravity theories, seen through the lens of first-order thermodynamics.


I. INTRODUCTION
A surprising and intriguing relationship appears to exist between thermodynamics and gravitation.Two seminal works showed that both the Einstein equations of General Relativity (GR) and the field equations of metric f (R) gravity can be recovered from purely thermodynamical considerations, starting with a few assumptions [1,2].However, dealing with a modified theory of gravity requires a generalization to a non-equilibrium thermodynamical setting.These works put forward the idea that, in the landscape of gravity theories, GR could be an equilibrium state and modified gravity a non-equilibrium one.This idea was made more concrete by the recent proposal [3,4] of a first-order thermodynamics of scalartensor theories, with minimal assumptions and in a context completely different from that of spacetime thermodynamics [1,2].Scalar-tensor theories represent prototypical candidates of modified gravity and were first introduced by Brans and Dicke in [5] and then extended in [6][7][8].The first-order thermodynamical proposal relies on interpreting the scalar contributions as an imperfect fluid [9][10][11] and applying a non-equilibrium thermodynamical description [12] to it.This idea unexpectedly allows one to introduce a concept of "temperature of gravity" (which is clearly no physical temperature, but simply a temperature relative to the GR equilibrium state) and an understanding of the dissipative process leading gravity towards (or away from) the GR state of equilibrium.This proposition has been applied and tested on both different classes of theories (such as Horndeski gravity) and specific solutions of scalar-tensor theories, such as those in Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime [13,14].
A characteristic feature of scalar-tensor gravity is the existence of stealth solutions, namely solutions with the same geometry of GR solutions but with a nontrivial scalar field profile that does not contribute to the effec-tive stress-energy tensor.Current motivation to study stealth solutions comes from the possibility of detecting black hole hair in stealth black holes through gravitational wave observations [15].Indeed, "first-generation" scalar-tensor and Horndeski theories allow for stealth solutions that violate some assumptions of the no-hair theorems and for which the scalar field does not gravitate.This would in principle make it possible to observationally distinguish GR from scalar-tensor theories.Such solutions include stealth Schwarzschild (-de Sitter) black holes with a scalar field linearly dependent on time in the context of Horndeski and beyond-Horndeski gravity [16][17][18][19][20][21].
Here we are interested in stealth solutions in the framework of scalar-tensor thermodynamics, where they would correspond to different "states of gravity" away from the GR equilibrium, as explained in the following.Studying these solutions would therefore help to clarify the existence of equilibrium states different from GR and establish which gravity theories or specific solutions could approach them, extending the study of scalar-tensor thermodynamics to uncharted territory.Assessing the stability of such states is crucial: it is reasonable to expect that, due to the special status of the GR equilibrium state in the landscape of gravity theories, these other equilibria would be unstable, thus less relevant than GR.
The stability of certain stealth geometries has been previously studied with the Bardeen-Ellis-Bruni-Hwang [22][23][24][25][26] approach for cosmological perturbations in modified gravity [27][28][29][30][31][32].Here, we propose a complementary criterion based solely on our thermodynamical formalism.Insights coming from thermodynamics provide essential guidance to both approaches, as for the stealth spacetime studied in [33] with the gauge-invariant formalism.In that case, stability was assessed with the gauge-invariant criterion, while in the present work we mostly use the thermal criterion.
Stealth solutions include those of Refs.[15,[34][35][36][37][38][39][40][41][42][43][44][45][46].Often these are degenerate cases of de Sitter spaces with non-constant scalar fields, which are not as well-known as stealth solutions of the field equations.de Sitter spaces with constant scalar fields are fixed points of the dynamical system of scalar-tensor cosmology [47] and are also common in GR cosmology sourced by scalar fields.On the contrary, de Sitter spaces with a non-constant scalar field are a signature of modified gravity.Both stealth solutions and de Sitter universes with non-constant scalar fields seem peculiar and deserve investigation in the thermodynamics of scalar-tensor gravity.
We follow the notation of Ref. [48].The (Jordan frame) scalar-tensor action reads where R is the Ricci scalar, the Brans-Dicke scalar φ > 0 is approximately the inverse of the effective gravitational coupling G eff , ω(φ) is the "Brans-Dicke coupling", V (φ) is the scalar field potential, and is the matter action.The field equations are [5][6][7][8] where R ab is the Ricci tensor, ab is the trace of the matter stress-energy tensor T (m) ab , ω ,φ ≡ dω/dφ and V ,φ ≡ dV /dφ.

II. THERMAL STABILITY CRITERION
Assuming ∇ a φ to be timelike and future-oriented, it is used to define the four-velocity of an effective irrotational fluid The effective stress-energy tensor of φ in the effective Einstein equations (1.2) has the form of a dissipative fluid which, surprisingly, obeys Eckart's constitutive relations [12] and leads to identifying a "temperature of gravity" T by where K is an effective thermal conductivity.The equation illustrating the approach to (or the departure from) the GR equilibrium in scalar-tensor gravity, for theories described by the action (1.1) is [3,4,13] where 3) has two fixed points, KT = 0 and KT = const.> 0. We explore both because, if stable, they could correspond to equilibrium states other than GR.Gravitational theories with non-dynamical scalar fields have been shown to recover KT = 0 in [49], while a state with KT = const.that never approaches the GR equilibrium state was found in [33] to be metastable.Here, we complement this analysis by studying more stealth solutions and assessing their stability with a new, purely thermodynamical, criterion found as follows.
Rewriting Eq. (2.3) as where we have instability if the square of the effective mass (that we call "thermal mass") is m 2 eff < 0 and stability if m 2 eff ≥ 0. Since KT is a scalar, this stability criterion is covariant and gauge-invariant.This effective mass of scalar-tensor gravity differs from those explored in [50,51].
The thermal stability criterion m 2 eff ≥ 0 is not particularly useful in the general thermodynamics of scalartensor gravity because one does not a priori know the quantities appearing in (2.5).However, if one wants to assess the stability of specific solutions (or classes of solutions) of the field equations, (2.5) is indeed suitable.This is the goal of the rest of this work.The criterion was used in [49] to study Nordström gravity, finding it unstable.

III. STEALTH SOLUTIONS
The stealth solutions we are interested in here are special cases where Minkowski space results not from the absence of matter, but from a tuned balance between matter and the Brans-Dicke scalar or, in vacuo, between different terms in the scalar contribution to the stressenergy tensor.Stealth solutions like those studied in [34][35][36][37] are interesting since they show that Minkowski space is not necessarily devoid of matter, and the effect of gravitational coupling persists in the energy-momentum tensor even when this coupling is switched off.
Stealth solutions commonly encountered in the literature in the context of the scalar-tensor theory (1.1) are usually of two kinds: 1. g ab = η ab and φ = φ 0 e α t ; 2. g ab = η ab and φ = φ 0 |t| β , where η ab is the Minkowski metric in Cartesian coordinates, φ 0 , α, β are constants, and φ 0 > 0 so that gravity is always attractive.
thus the requirement of future-directed scalar gradient translates into the conditions φ > 0 and or, for the specific scenarios above, Thus, enforcing the future orientation of the scalar field gradient, we shall restrict to cases that satisfy the conditions In the first case which means that this solution never approaches the GR equilibrium state.If we now consider its stability from the point of view of first-order thermodynamics, we see that the effective mass is constant and given by which makes this stealth solution unstable.A stealth solution of this type was assessed in [33] with the gauge-invariant criterion for cosmological perturbations and shown to be a metastable state.
In the second case, β = 1 and β = 2 are the most relevant situations encountered in the literature.Therefore, according to our conventions, in order to have Then, if β > 0 the effective gravitational coupling behaves as the effective temperature of gravity (2.2) is and the effective mass reads If β = 1, we get m 2 eff = 0. Therefore this constant "mass" solution is marginally stable.As t → 0 − , we approach a singularity of the theory where G eff → +∞, KT → +∞, gravity becomes infinitely strong and deviates from GR drastically.Indeed, nothing could be further from a GR situation than infinitely strong gravity with Minkowski spacetime!This solution matches the idea that singularities are "hot" in the sense of the thermodynamics of scalar-tensor gravity [3,4].This situation is stable according to the thermal stability criterion (2.5).Hence, barring instabilities of a different nature, one expects this behaviour to occur in nature if singularities are present.The implication is that the GR equilibrium state is not always approached and gravity indeed departs from GR near singularities.Of course, the final theory of gravity should remove singularities, but it is clear that scalar-tensor gravity is not this final theory since it does contain spacetime singularities and singularities of G eff .
The situation where β = 2, exemplified in Sec.III B, entails m 2 eff = −2/t 2 < 0, meaning instability from the thermal point of view, while KT = 1/4π|t| and G eff = 1/φ 0 t 2 both diverge as t → 0 − , thus departing from GR at the singularity of G eff .In our formalism the t > 0 branch of the solution is not meaningful.
Most exact solutions of Brans-Dicke theories in cosmology exhibit the power-law behaviour φ = φ 0 t β [52], such as those found by O'Hanlon and Tupper [53] and Nariai [54,55].These were studied from the point of view of first-order thermodynamics in [14], and in III A and III B we consider two degenerate cases of such solutions that reduce to a Minkowski background with a non-trivial scalar field profile.
Other types of stealth solutions with Minkowski metric and non-trivial scalar include those found for a nonminimally coupled φ [35], where the field is inhomogeneous, wave-like, and does not gravitate.Their stability was studied in [56] using the Bardeen-Ellis-Bruni-Hwang gauge-invariant formalism for cosmological perturbations [22][23][24][25][26], showing mixed stability results depending on the specific choice of parameters.These solutions either do not correspond to future-oriented four-velocity u c , or are very cumbersome to discuss because ∇ a φ is timelike only in very restricted spacetime regions and for special combinations of their parameters.Therefore, they will not be examined here. with ) and 3q ± +s ± = 1.This solution has a "hot" singularity at t → 0 + , where Brans-Dicke theory departs from the GR behaviour.Although the value ω = 0 was not contemplated in [53], it is straightforward to check that it corresponds to a Minkowski space solution of the equations of vacuum Brans-Dicke cosmology with V = 0, q = 0, a(t) = 1, and linear scalar field φ(t) = φ 0 t (choosing t 0 = 1 for convenience).This is a bona fide stealth solution, which could have been introduced in Ref. [53] long before solutions with this name were noticed and appreciated [15,[34][35][36][37][38][39][40][41][42][43][44][45][46].In order for the four-velocity to be future-oriented and for G eff to be positive, it must be φ 0 < 0 and t < 0. This situation is akin to case 2. with β = 1 considered above, hence the ω = 0 O'Hanlon & Tupper solution turns out to be marginally stable according to the thermal stability criterion. 1This universe has as t → 0 − , deviating from GR.
B. Nariai solution with ω = −1/2 The Nariai solution [54,55] is a particular power-law solution for a K = 0 FLRW universe with perfect fluid matter that has P = (γ − 1) ρ (with γ = const.), Here we are interested in a cosmological constant fluid with γ = 0, P (m) = −ρ (m) , and This solution is an attractor in phase space and was used in the extended inflationary scenario [57,58].For ω = −1/2, δ = 8πρ 0 /φ 0 , the scale factor is constant and H = 0, making this a Minkowski stealth solution with non-trivial (polynomial) scalar field profile.It is a straightforward generalisation of the type 2. stealth solutions described above. 2It must be φ 0 > 0, (1 + δt) < 0 and as (1 + δt) → 0 − .In the far past t → −∞, KT → 0 and GR is approached, but the instability prevents this state from being an equilibrium alternative to GR.In fact, the thermal stability criterion yields and this solution is thermally unstable.

IV. DE SITTER SPACE SOLUTIONS
Other common solutions of scalar-tensor gravity are de Sitter ones with line element in comoving coordinates, with scale factor a(t) = a 0 e H0t , where a 0 , H 0 are constants.
In GR with a minimally coupled scalar field as the only matter source, the only possible de Sitter spaces are obtained for a constant scalar field, (H, φ) = (H 0 , φ 0 ), with both H 0 and φ 0 constant.In spatially flat FLRW cosmology, the independent dynamical variables are3 (H, φ) and the phase space is a 2-dimensional subset of the 3dimensional space H, φ, φ identified by the Hamiltonian constraint.This 2-dimensional subset is analogous to an energy surface in point particle mechanics [59,60].The points (H 0 , φ 0 ) are then all the equilibrium points of the dynamical system.
For spatially flat FLRW universes in scalar-tensor cosmology, the independent variables are still H and φ and there can be fixed points (H 0 , φ 0 ) of this dynamical system.The structure of the phase space and the fixed points for specific scalar-tensor theories are discussed extensively in [59] and [47,61], respectively.Gaugeinvariant criteria for the stability of these de Sitter fixed points (and of their degenerate Minkowski cases) are given in [62][63][64][65][66].In addition to de Sitter fixed points, in scalar-tensor cosmology there can be de Sitter spaces with non-constant scalar field, usually exponential or power-law in time.Since these are only admissible in modified gravity and not in GR, they are interesting for first-order thermodynamics.Degenerate cases of such de Sitter solutions can reproduce Minkowski space with a non-trivial scalar field and are therefore another kind of stealth solutions similar to those of the previous section.

A. de Sitter solutions of scalar-tensor gravity
This type of solution, known in many scalar-tensor theories, is found starting from the action (1.1) and reads with φ 0 a positive constant.The constants H 0 and α are related to the parameters of the specific scalar-tensor theory.Although these solutions have been known for a long time, here we consider them from the novel point of view of scalar-tensor thermodynamics.
In order to get a future-directed four-velocity of the effective φ-fluid and an attractive gravitational interaction we need to require, again, that φ > 0 and g ab ∇ a φ (∂ t ) b < 0 , (4.4) which implies φ 0 > 0 and α < 0.
We have (as in (3.4)) and this solution remains away from the zerotemperature GR state of equilibrium at all times.Is it thermally stable?We find therefore, we have stability for 3H 0 ≥ |α| and instability for |α| > 3H 0 .
In particular, it is clear that exponentially contracting FLRW universes (H 0 < 0) are always unstable.This conclusion, obtained with simple considerations in scalartensor thermodynamics, matches the result found in the literature on scalar-tensor cosmology [62] with a dynamical systems analysis which requires the complete specification of the theory.
1. Kolitch solutions of vacuum Brans-Dicke cosmology with cosmological constant Kolitch [67] found solutions of vacuum Brans-Dicke cosmology with positive cosmological constant Λ, equivalent to the linear potential V (φ) = 2Λφ.These solutions were previously noted in [68,69] and read ) For ω = −1, they reduce to the stealth solution with where, again, we must choose the lower sign to have a future-oriented four-velocity.This solution deviates from GR at all times since KT = const.> 0, but it corresponds to m 2 eff = −α 2 < 0 and is unstable.Its stability has also been studied with respect to both homogeneous and inhomogenous metric perturbations in [56], where the solution with the upper sign is found to be stable and the one with the lower sign unstable.However, the solution with the upper sign cannot be analysed in the framework of scalar-tensor thermodynamics since it entails a past-oriented ∇ a φ.
Let us consider now the de Sitter spaces (4.7), (4.8) for ω = −1: taking the lower sign we have where C is a positive real constant if ω < −3/2 and ω > −4/3.Therefore, the effective mass reads Then, if ω < −3/2 we have an expanding de Sitter universe which is thermodynamically stable, although the scalar field for such values of the coupling is phantom and therefore suffers from different types of instabilities [70].Other configurations are otherwise unstable.It is often mentioned in the literature that the O'Hanlon & Tupper solution (3.9)-(3.12)approaches de Sitter space in the limit ω → −4/3, recovering with H 0 a positive constant.Technically, this statement is not accurate since the above result is recovered by simultaneously choosing the values q + and s − of the exponents, which correspond to two distinct solutions.However, the solution above is the only de Sitter one for flat FLRW and vacuum [70].Given that α < 0, the velocity of the scalar field fluid is future-oriented and 3H 0 − |α| = 0, so this solution is marginally stable according to the thermal criterion.This solution describes expanding universes for which the effective fluid four-velocity is only future-oriented.
These expanding universes are unstable with respect to tensor modes, as can be concluded using the Bardeen-Ellis-Bruni gauge-invariant formalism for cosmological perturbations [22][23][24][25][26] in Hwang's version adapted to modified gravity [27][28][29][30][31][32].The relevant equations are summarized in Appendix A. We only need Eq. (A.14) for the gauge-invariant variable H T associated with the tensor modes which, in the background (4.13) and (4.14), becomes where k is the mode's momentum and the coefficients are given by the unperturbed a(t) and φ(t), which yields 3H+ φ/φ = 0 to zero order.With H 0 > 0, the asymptotic equation at late times t → +∞ reduces to with linear solution H(t) = α t + const.The tensor perturbation diverges and this universe is unstable.Metric f (R) gravity is a subclass of scalar-tensor theories described by the action and is equivalent [71][72][73] to a Brans-Dicke theory with φ = f (R) (a prime denotes differentiation with respect to R), ω = 0, and the potential Assuming that ∇ c R is timelike and future-oriented, the effective dissipative fluid associated with f (R) gravity has [3] where it is required that f (R) > 0 in order for the effective gravitational coupling G eff = 1/φ to be positive and for the graviton to carry positive kinetic energy, while f (R) ≥ 0 is required for local stability [74] (here ∇ c φ is timelike and future-oriented if ∇ c R is).The fact that the effective Brans-Dicke scalar field φ in f (R) gravity is tied so intimately with the Ricci scalar makes all constant curvature spaces in these theories zero-temperature states indistinguishable from GR, because this means that φ = f (R) = const.and ∇ c φ vanishes identically, together with KT .Furthermore, these states are (marginally) stable in our thermal sense because φ = 0 and the effective mass m 2 eff = φ/φ also vanishes identically.
The condition m 2 eff ≥ 0 for the thermal stability of f (R) gravity does not coincide with the stability condition of de Sitter space with respect to first order local perturbations, obtained in a gauge-invariant way ( [66] and references therein), where a zero subscript denotes a quantity evaluated on the de Sitter background.Therefore, the thermal stability condition m 2 eff ≥ 0 does not necessarily coincide with other stability notions, as could be expected.Indeed, also in Newtonian systems and in GR one has different notions of stability (thermal, dynamical, etc.) and the thermodynamics of modified gravity evidently cannot account for all possible notions of stability.

V. CONCLUSIONS
In this work, we studied the states of gravity corresponding to KT = const., which are fixed points of the effective heat equation describing the approach to (or departure from) equilibrium (2.3), in the context of first-order thermodynamics [4].These states, away from the GR equilibrium, correspond to different types of stealth solutions, which are not admitted by the Einstein equations and are thus a signature of alternative gravity [15,[34][35][36][37][38][39][40][41][42][43][44][45][46].Specifically, we studied the scalar field profiles 1. φ = φ 0 e α t (with α < 0) and 2. φ = φ 0 |t| β (with t > 0, β < 0 or with t < 0, β > 0), common in the literature.The first case has KT = const.> 0, which would correspond to a state of equilibrium at positive temperature.However, this state is unstable according to a new, purely thermal, criterion that we find (2.5).This criterion does not necessarily go hand-in-hand with other stability criteria, which should not come as a surprise, since a physical system can be subject to instabilities of different nature, with different time scales.Sometimes instability in the thermal sense (2.5) is accompanied by instability with respect to gravitational perturbations; however, this coincidence should not always be expected.
In any case, stable equilibrium states of gravity with KT = const.either do not exist or are fragile and easily destroyed by perturbations (i.e., metastable).Stealth solutions with a linear scalar field profile, as in the second case, require caution because, combining the requirements that G eff > 0 and that the effective φfluid four-velocity u a be future-oriented (essential when discussing dissipation associated with an arrow of time), one finds a singularity of the effective gravitational coupling at t = 0, which can justly be regarded as a "thermodynamical" singularity of scalar-tensor gravity.These spaces are stable according to the thermal criterion and are not destroyed by perturbations (as far as scalar-tensor gravity applies), but KT diverges at this singularity, as it does in ordinary spacetime singularities, signalling a drastic deviation from GR predicted in [3,4].This result reinforces the idea that gravity strongly deviates from GR at singularities, but now the concept of "thermodynamical singularity" is extended to include also singularities of the effective gravitational coupling G eff .These considerations, of course, do not solve the spacetime singularity problem of relativistic gravity; the temperature T introduced by scalar-tensor thermodynamics is relative to the GR state and measures the distance of the actual state of gravity from the GR state of equilibrium at KT = 0, which is still affected by the spacetime singularity problem.
The realization that stealth solutions of scalar-tensor gravity are often degenerate cases of de Sitter universes with non-constant Brans-Dicke-like scalar field prompts the consideration of these spaces (Sec.IV).It is intriguing that the cosmic no-hair theorem (when valid) can be seen in a new light from the point of view of scalar-tensor thermodynamics.(The validity, or lack thereof, of cosmic no-hair in various scalar-tensor gravities will be examined from the thermal point of view in future work).On the one hand, de Sitter spaces with constant scalar field can be attractors of the cosmological dynamics (even starting with anisotropic Bianchi models) but, when φ is constant, KT vanishes and gravity reduces to its zero-temperature GR state of equilibrium. 4On the other hand, de Sitter spaces with non-constant scalar field are known to occur in various scalar-tensor gravities (where they are not attractors of the cosmological dynamics) but are impossible in GR and are a signature of alternative gravity.In this sense, they can be regarded as generalizations of stealth solutions [77] and as such they were studied here from the point of view of first-order thermodynamics.
The results obtained for the solutions of scalar-tensor gravity analyzed here are summarized in Table I.Overall, the two general principles of first-order thermodynamics of scalar-tensor gravity are confirmed: i) gravity deviates wildly from GR near spacetime singularities and near singularities of the gravitational coupling; ii) the convergence of gravity to GR at late times is marked by KT → 0. No states of equilibrium KT = const.other than GR (corresponding to KT = 0) have been found here, except for solutions that are unstable according to various criteria and are, therefore, physically irrelevant.This result reinforces the special role of general relativity as an equilibrium state in the landscape of gravity theories, seen through the lens of first-order thermodynamics.The results above will be useful in the following developments of the first-order thermodynamical formalism.

TABLE I .
Summary of the analytical solutions studied and their thermal stability.