Black hole thermodynamics with dynamical lambda

Published for SISSA by Springer Received: August 11, 2017 Accepted: October 3, 2017 Published: October 18, 2017 Black hole thermodynamics with dynamical lambda a Centre for Particle Theory, Durham University, South Road, Durham, DH1 3LE, U.K. b Perimeter Institute, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada c Amherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts, 710 N Pleasant St., Amherst, MA 01003, U.S.A. E-mail: r.a.w.gregory@durham.ac.uk, kastor@umass.edu, traschen@umass.edu Abstract: We study evolution and thermodynamics of a slow-roll transition between early and late time de Sitter phases, both in the homogeneous case and in the presence of a black hole, in a scalar field model with a generic potential having both a maximum and a positive minimum. Asymptotically future de Sitter spacetimes are characterized by ADM charges known as cosmological tensions. We show that the late time de Sitter phase has finite cosmological tension when the scalar field oscillation around its minimum is underdamped, while the cosmological tension in the overdamped case diverges. We compute the variation in the cosmological and black hole horizon areas between the early and late time phases, finding that the fractional change in horizon area is proportional to the corresponding fractional change in the effective cosmological constant. We show that the extended first law of thermodynamics, including variation in the effective cosmological constant, is satisfied between the initial and final states, and discuss the dynamical evolution of the black hole temperature. Keywords: Black Holes, Black Holes in String Theory ArXiv ePrint: 1707.06586 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP10(2017)118 JHEP10(2017)118 Ruth Gregory,a,b David Kastorc and Jennie Traschenc Contents 1 Introduction 1 2 Pure de Sitter to de Sitter flows 2.1 Engineered flow 2.2 Slow-roll analysis 2.3 Cosmic hair 3 3 6 7 11 14 16 17 17 4 Dynamical thermodynamics 4.1 Analysis of horizon growth 4.2 Two first laws 4.3 Temperature and mass for evolving black holes 19 19 20 20 5 Illustrative example 22 6 Concluding remarks 25 A Cosmological tension 25 1 Introduction The physics of black holes in the early universe is an important subject, about which relatively little is known. The system is both interactive and dynamic, combining effects of cosmic expansion with accretion of matter onto the black hole. The gravitational thermodynamics is non-equilibrium, involving time dependent areas and surface gravities for the black hole and cosmological horizons, such that formulating an appropriate definition of temperature proves to be complicated. Still, physical implications of primordial black holes have been heavily researched, including a recent revival of interest that they may provide the dark matter and the progenitors of the massive black holes detected by LIGO [1–10]. In this paper we explore cosmological black holes in a relatively well-controlled setting that is also of physical interest, namely the evolution of black holes in slow-roll inflation. We consider transitions between distinct early and late time de Sitter phases, with evolution driven by a scalar field rolling slowly between a maximum and minimum of its potential, both of which are assumed to be positive. The initial and final states are described –1– JHEP10(2017)118 3 Schwarzschild-quasi de Sitter spacetimes 3.1 The scalar field behaviour 3.2 Growth of the event horizons 3.2.1 Cosmological event horizon 3.2.2 Black hole event horizon by Schwarzchild-de Sitter (SdS) metrics with different values of the black hole mass and cosmological constant. We begin in section 2 by analyzing pure vacuum to vacuum transitions, without a black hole present. Exact results are obtained for an ‘engineered’ example and compared with the results of a perturbative slow-roll calculation. The growth of the cosmological horizon between the two de Sitter vacua is found to obey the extended first law [11, 12], which in the absence of a black hole is given by (1.1) where the pressure P is provided by the effective cosmological constant, V is the thermodynamic volume, and T is the initial temperature of the de Sitter horizon. Hence δP is dynamically generated by the scalar field. Depending on the parameters of the scalar field potential there are two qualitatively distinct ways that the metric can approach the late time de Sitter metric. The motion of the scalar field is either overdamped or underdamped as it settles into the true vacuum, corresponding to a slow-roll or an oscillatory relaxation respectively. Although in both cases the metric decays exponentially fast to de Sitter, the ADM cosmological tension [13] is infinite for the overdamped case, but finite for the oscillatory evolution. This behavior is analogous to behavior found for AdS black holes with scalar fields, as well as in studies of AdS domain-wall/cosmology dualities [14–29] in which a scalar field gets a negative mass-squared from resting at the maximum of a potential, and back-reaction generates an infinite ADM mass. In the second part of the paper, we add a black hole to the cosmology and solve for the evolution of the scalar field and metric in a perturbative “slow-roll” approximation. This generalizes the calculations of [30] to potentials in which both the initial and final states are approximately de Sitter, and we focus on thermodynamic aspects of the evolution. The black hole grows due to accretion of the scalar field, but the cosmological horizon is subject to competing influences. While the growing black hole tends to pull the cosmological horizon further in, the decaying cosmological constant makes it expand. We find that the expansion dominates. For a potential interpolating between initial and final values of the cosmological constant, Λi and Λf , we find that the change in both the black hole and cosmological horizon areas evolve proportionally to |δΛ| = |Λf − Λi | times a factor that depends on properties of the Schwarzchild-de Sitter (SdS) spacetime. This is summarized in equation (4.1) below. We then show that the extended first law of thermodynamics [11, 12], that relates the sum of T δS contributions from each horizon to VδP , is satisfied between the initial and final SdS phases. The area growth calculations only depend on the temperatures of the background static spacetime, but one would like to go further and work with a dynamical temperature. As a first step, we utilize a definition of the dynamical black hole temperature Tdyn based on the Kodama vector [31]. This is very nearly that of a Schwarzschild-de Sitter black hole with the instantaneous values of the effective cosmological constant and black hole radius. We show how the black hole temperature relaxes from its initial value to its final asymptotic value, at a rate dependent on the rolling of the scalar and the local horizon –2– JHEP10(2017)118 T δS = VδP surface gravity. In section 5, we demonstrate this explicitly for a simple potential, finding analytic expressions for the dynamical area and temperature. 2 Pure de Sitter to de Sitter flows and use a mostly minus signature. Assuming an FRW form for the metric ds2 = dτ 2 − a2 (τ )dx2 , the equations of motion for the system are given by  2   ȧ 1 2 1 φ̇ + W (φ) = a 3Mp2 2 ∂W =0 φ̈ + 3H φ̇ + ∂φ (2.2) where H = ȧ/a. For the pure dS-dS flow, we can simply numerically integrate these FRW equations for any desired potential, however for the analysis of the black hole set-up, it is useful to have analytic solutions, or approximate solutions, to use to explore the dynamical evolution of the horizons. 2.1 Engineered flow As a first method, we start with an interpolating Ansatz for the scalar field, and use the Hamilton-Jacobi formulation [35] to engineer a potential W (φ) that corresponds to this flow. This approach has been used, for example, to generate smooth analytic domain wall solutions in the presence of gravity [36], and is very similar to the “fake supersymmetry” approach described in [37]. Briefly, the Hamilton-Jacobi approach uses φ as a time coordinate, writing H = H(φ). The equations of motion (2.2) then imply that H′ = − φ̇ 2Mp2 –3– (2.3) JHEP10(2017)118 Our goal in this paper is to investigate the effect of a dynamically generated cosmological constant on the growth of black hole and cosmological horizons, and to explore the thermodynamic relations for such evolving black hole systems. We begin by examining a pseudo-de Sitter spacetime, where the cosmological constant varies in time, with no blackhole present. As noted in [25], this is a double-analytically continued version of an AdS flow, thus an analogue of the C-theorem [32–34] tells us that the cosmological constant must always flow to lower values in time. In accordance with this, we consider a real scalar field φ with potential W (φ) and assume that W (φ) has a maximum Wi at φ = φi , and a minimum, Wf at φ = φf . If the scalar field starts off at φi at early times, and rolls to φf at late times, the cosmological constant Λ will make a transition between the values Wi /Mp2 to Wf /Mp2 , where Mp2 = 1/8πG, at early and late times. We take the action for the coupled Einstein plus scalar field system to be Z
√ 1 S= (2.1) d4 x −g −Mp2 R + (∇φ)2 − 2W (φ) 2 where the superscript prime denotes a derivative with respect to φ. With this identification, the Friedmann equation takes the form of a first order NLDE for H(φ) H ′2 − W (φ) 3H 2 + =0 2 2Mp 2Mp4 (2.4) A judicious choice of evolution for φ(τ ) that allows φ̇ to be re-expressed as a function of φ then gives an H ′ that can be integrated up to give H and hence W (φ) using (2.4). For example, if we suppose that the flow of the scalar is (2.5) then we find √ λ 2 (η − φ2 ) H′ = − 2Mp2 ⇒ H(φ) = H0 − √ λ φ(3η 2 − φ2 ) 6Mp2 (2.6) where H0 is an integration constant. The Hamilton-Jacobi equation (2.4) then finally determines the scalar field potential to be W (φ) = 3Mp2 √ λ φ(3η 2 − φ2 ) H0 − 6Mp2 !2 − λ 2 (φ − η 2 )2 2 (2.7) Taking H0 > 0, the potential W (φ) has a local maximum at φ = −η and a minimum at φ = η. Correspondingly, the Hubble parameter makes a transition from a larger value Hi associated with the higher vacuum energy at early times, to a smaller value Hf for the lower vacuum energy at late times, given by Hi = H0 + √ λη 3 /3Mp2 , H f = H0 − √ λη 3 /3Mp2 , (2.8) Integrating the expression for H(τ ), obtained by plugging (2.5) into (2.6), yields the scale factor √ √ η2 η2 2 λητ ) + λητ ) (2.9) log a(τ ) = H0 τ − log cosh( sech ( 3Mp2 12Mp2 √ We see that for |τ | & 1/ λη, ln a is approximately linear in τ with the appropriate Hubble constants given by (2.8), thus ln a is a smoothed out step function, as we would expect. In the late time limit, the cosmological scale factor and scalar field are approximately given by     η 2 −Hf Γτ Hf τ −Hf Γτ /2 a(τ ) ≃ Ke 1− e , φ ≃ η 1 − 2e (2.10) 2Mp2 √ 2 2 where K = 2(η /3Mp ) and we have defined ΓHf = 4 λη in order to facilitate comparison to the late time behavior in subsequent examples. Note that with this notation, the expression for the scalar field (2.5) becomes φ(τ ) = η tanh(Hf Γτ /4). For fixed Hf the parameter Γ then allows one to interpolate between slow-roll behavior for Γ ≪ 1 and a sudden change for Γ ≫ 1. –4– JHEP10(2017)118 √ φ(τ ) = η tanh( λη τ ) We are interested in the evolution of the future cosmological horizon. If a light signal is emitted at time τ , then as the reception time goes to infinity the signal is received at a co-moving coordinate separation rc (τ ) away Z ∞ dτ ′ (2.11) rc (τ ) = a(τ ′ ) τ The cosmological horizon radius dc (τ ) is the corresponding proper distance dc (τ ) = a(τ )rc (τ ) . (2.12) a(τ ) ≈ eHi τ Θ(−τ ) + eHf τ Θ(τ ) (2.13) Performing the integral (2.11) in this case gives an approximate expression for the evolution of the cosmological horizon radius     1 1 1 1 Hi τ dc (τ ) ≈ − Θ(τ ) (2.14) + Θ(−τ ) + e Hi Hf Hi Hf figure 1 shows the approximate expression compared to the exact one. We see in both the exact and approximate results that the cosmological horizon interpolates between dc = 1/Hi at early times and dc = 1/Hf at late times. These correspond respectively to the Killing horizons of the early and late time de Sitter phases, for the Killing vectors ξ∗ = P (∂/∂τ ) − H∗ j xj (∂/∂xj ), where H∗ = Hi or Hf respectively. It is satisfying to see that even the simple step-function approximation for a(τ ) captures the teleological behavior of the horizon as τ increases to the transition time τ = 0. It is interesting to look at the change in the cosmological horizon area Ac = 4πd2c over the evolution from early to late times, which is given by ! 1 1 δAc = 4π (2.15) − Hf2 Hi2 An extended first law for de Sitter black holes, including variation in the cosmological constant was derived in [11, 12]. For the case of no black hole (see also [38]), this reduces to Tc δSc = VδP (2.16) where Tc is the temperature of cosmological horizon, Sc = Ac /4 is its entropy, P = −Λ/8π is the cosmological pressure, and V is called the thermodynamic volume. The extended first law, in this simple case, relates the change in cosmological horizon entropy to the change in cosmological constant. For a de Sitter spacetime the thermodynamic volume is equal to V= 4πd3c 3 –5– (2.17) JHEP10(2017)118 Evaluating the integral (2.11) for the cosmological horizon radius dc (τ ) numerically with the scale factor (2.9) gives a smooth evolution between the initial Hubble horizon, Hi−1 , and the final horizon, Hf−1 , depicted in figure 1 by a blue line. It is interesting to compare this exact result to the approximate solution for rc which is gotten by using a step-function approximation for a(τ ), i.e. the volume of a Euclidean sphere of radius dc , while the horizon temperature is T = 1/(2πdc ). If we consider a limiting case of our evolution such that Hf = Hi + δH, where δH/Hi ≪ 1, then the change in horizon area (2.15) is given to leading order by δAc = − 8πδH Hi3 (2.18) and it is easily verified that the first law (2.16) is satisfied. In section 4 we will see that the first law for a slow-roll inflationary spacetime with a black hole also obeys the appropriate extension of (2.16). 2.2 Slow-roll analysis In the previous subsection we assumed a form for φ that interpolated between two de Sitter phases, and found the exact potential and scale factor, such that the cosmological evolution was a prescribed flow from the maximum of the potential to the minimum. The approach to the true vacuum was overdamped rather than oscillatory, corresponding to a slow-roll evolution. In this section we show that if one starts with a classic double well potential, with positive minima, then the behavior of the scalar field is qualitatively the same as that previously assumed. For an analytic treatment we take the standard slow roll assumption [39] that the scalar field evolution is friction dominated, so that 3H φ̇ ≃ −W ′ (φ) . –6– (2.19) JHEP10(2017)118 Figure 1. The actual cosmological horizon evolution for the Hamilton-Jacobi engineered potential (blue) vs. the approximation (red-dashed). Now define slow-roll parameters ε(φ) = Mp2 W ′2 ≪ 1, 2 W2 Γ(φ) = 2Mp2 W ′′ ≪1 W (2.20) where Γ = Γ(η) is the slow-roll parameter evaluated at the minimum. The Einstein constraint equation evaluated at late times, when φ = η, gives the standard relation between the effective cosmological constant Λf = Wf /Mp2 and the Hubble parameter in the final de Sitter phase, Wf (2.22) Hf2 = 3Mp2 The parameter ε ∼ η 2 Γ2 /Mp2 , and we assume that Γ, ε ≪ 1. Since the time dependent part of φ is already first order, in its equation of motion (2.19) we can approximate H ≈ Hf . The solution for φ is then      Hf Γ Hf Γ η2 η2 Hf Γτ /4 2 τ e = 1 + tanh τ (2.23) φ = sech 2 4 2 4 Here an integration constant determining the transition point has been set to zero, but one has the freedom to replace τ in (2.23) with (τ − τ0 ). Hence we see that for the double well potential (2.21) the scalar field has a tanh(Hf Γτ ) type behavior, similar to the engineered case. This makes sense since the potentials (2.7) and (2.21) have the same qualitative features for evolving from the maximum to a minimum. At late times the field goes like   1 −Hf Γτ /2 φ≃η 1− e (2.24) 2 which has the same decay rate to the true vacuum as in the previous example (2.10). The scale factor approaches its final de Sitter form with corrections that fall off like e−Hf Γτ . 2.3 Cosmic hair As shown in [13], asymptotically future de Sitter (AFdS) spacetimes carry cosmic hair, encoded in the exponential fall-off terms in the metric, analogous to the ADM charges of a black hole. For black holes, the ADM integrals are computed at spatial infinity, while for AFdS spacetimes, these integrals are computed at future infinity. With the AFdS boundary conditions established in [13], the exponentially small size of the corrections to de Sitter in this regime are compensated by the exponentially growing spatial volume, giving –7– JHEP10(2017)118 The second parameter Γ (related to the standard slow roll “eta” parameter by a factor of two) is relevant for evolution near a minimum of W , where ε = 0. While φ is rolling slowly, it will be approximately linear in cosmological time, although the behaviour near each critical point will be modified. Now let W be a double well potential with a max at φ = 0 and a min at φ = η,  
Γ 2 2 2 W (φ) = Wf 1 + (2.21) φ −η 16η 2 Mp2 U (φ) = W (φ) − Wf ≥ 0 (2.25) For the scalar field evolution the energy density and pressure are given by 1 ρ = φ̇2 + U (φ) ≥ 0, 2 1 p = φ̇2 − U (φ) ≥ 0 2 (2.26) Since ρ ≥ 0 the weak energy condition is satisfied. However, ρ + 3p = 2(φ̇2 − U (φ)), which is negative in the early time de Sitter phase, so that the strong energy condition is not satisfied. Nonetheless, we find that the metric and scalar field still approach the late time de Sitter vacuum exponentially fast. This decay is generic, determined by the fact that near a minimum of the potential the wave equation for φ becomes that of a damped simple harmonic oscillator, with damping provided by the Hubble expansion. However, we show that in the overdamped case the detailed decay rate is too slow to result in a finite cosmological tension charge. Assume that the scalar field potential W (φ) has a maximum and a minimum so that there can be two de Sitter phases as the field evolves from an initial value φi at the 1 Cosmic hair in the form of cosmological tension charges is fully consistent with the results of reference [40], even though this work is often mischaracterized as a “cosmic no-hair theorem”. –8– JHEP10(2017)118 finite results for what were referred to as cosmological tension charges.1 While e.g. the ADM mass and angular momentum of a black hole are associated with asymptotic time translation and rotation symmetries, cosmological tension is associated with asymptotic spatial translation symmetries. It is analogous to the ADM spatial tension charges defined for black brane spacetimes in [41] (see also [42] for the asymptotically AdS case). It was found [13] that cosmological tension in a given spatial direction captures the leading order correction of the scale factor in that direction to its limiting late time de Sitter behavior. In this paper, we consider transitions between early and late time de Sitter phases. Although it diverts us briefly from our main line of development, we explore in this section how cosmological tension behaves in the Einstein-scalar field cosmologies considered here. These represent an interesting and qualitatively different set of examples from those explored in [13], where we can compute the cosmological tensions associated with the approach to the late time de Sitter limit. Assuming that the minimum of the potential is at a finite value of φ, we find that the cosmological tensions will be finite, if the approach to the minimum is underdamped. However, in the overdamped case, the corrections to de Sitter behavior fall off too slowly, and the cosmological tension charges diverge. In both the engineered example in section 2.1 and the approximate analytic solution for the slow-roll example in section 2.2, the scalar field and metric indeed decay to the late time de Sitter vacuum exponentially fast. This agrees with expectations based on [40], which showed that an initially expanding, homogeneous spacetime with cosmological constant Λ > 0, matter fields satisfying the weak (ρ ≥ 0) and strong (ρ + 3p ≥ 0) energy conditions, and non-positive spatial curvature, falls off to de Sitter exponentially quickly at late times. To align our scalar field system with the assumptions of [40], we take the cosmological constant to be given in terms of the final value of the scalar potential Λ = Wf /Mp2 . The scalar field then moves in the shifted potential maximum Wi to a final value φf at the minimum Wf , as in the two examples above. We want to solve equations (2.2) for φ(τ ) and a(τ ) at late times as φ approaches the minimum at φf . Assume that the second derivative of the potential is non-zero at φf , so that near the minimum the scalar field potential can be approximated by 1 W (φ) ≃ Wf + Wf′′ (φ − φf )2 2 (2.27) To leading order the wave equation (2.2) for φ then reduces to the ODE for a damped simple harmonic oscillator (2.28) which has the solutions φ(τ ) = φf + φ1 e−β± Hf τ , β± =  p 3 1 ± 1 − 2Γ/3 2 (2.29) where φ1 is a constant and Γ = 2Wf′′ /3Hf2 . Overdamped oscillation results from real values of β, corresponding to Γ ≤ 3/2, and in this case β− is the dominant mode. From (2.20) we see that slow-roll evolution corresponds to Γ ≪ 1, giving φ = φf + φ1 e−ΓHf τ /2 (slow-roll) (2.30) in agreement with the late time slow-roll evolution in (2.24). The late time behavior of φ determines the late time behavior of the scale factor. As φ approaches its value at the minimum of the potential W (φ) the Einstein constraint equation becomes  2   ȧ 1 ′′ 1 2 1 2 Wf + Wf (φ − φf ) + φ̇ (2.31) ≃ a 3Mp2 2 2 the evolution of the scale factor in this regime can be written as a(τ ) ≃ eHf τ + δa(τ ) (2.32) where the deviation δa(τ ) from the late time de Sitter evolution is small. The leading terms relate the late time value of the Hubble parameter to the value of the potential at its minimum, as in (2.22), while the next order terms give δa(τ ) = −
φ21 2 Hf2 + Wf′′ eHf (1−2β− )τ β− 2 24Mp β− (2.33) In the slow-roll approximation, with Γ ≪ 1, the late time limit of the scale factor is then ! 2H 2 φ 1 f (slow-roll) (2.34) e−ΓHf τ a(τ ) = eHf τ 1 − 8Mp2 in agreement with (2.10). Critically damped motion occurs when Γ = 3/2. In this case the two roots coincide, and the second linearly independent mode goes like e−βHf τ ln τ . –9– JHEP10(2017)118 φ̈ + 3Hf φ̇ + Wf′′ (φ − φf ) = 0 Underdamped oscillations occur when Γ > 3/2 [35], and at this point there is a qualitative change in the aymptotic behavior of φ and a(τ ) as the fields oscillate around the minimum with decaying amplitude, φ(τ ) = φf + φ1 e−3Hf τ /2 sin ωτ    1 2 ′′ −3Hf τ Hf τ 1−e a=e φ W + C1 cos 2ωτ + C2 sin 2ωτ 36Mp2 1 f (2.35) δ ȧ dv ≃ eHf (3−2β− )τ ads (2.36) where dv = e3Hf τ is the area element on a constant time slice of the background de Sitter metric. As τ → ∞ this is finite and non-zero only for the critically damped case of β− = 3/2, whereas the integral diverges for all the overdamped cases which have β− < 3/2. However, in the critically damped case the two modes corresponding to β± coincide, and the dominant late time mode is instead a second linearly independent solution that goes like (ln τ )e−3Hf τ /2 and also leads to a divergent tension. Hence it is only the oscillatory modes that yield a finite cosmological tension T . Since the tension involves an integration over an infinite spatial area, we make the spatial coordinates periodic with period L, and the finite quantity is the tension per unit area. For the underdamped case, where the tension is finite, one then finds 9 2T (2.37) = φ21 Hf Mp2 L2 4 where φ21 sets the magnitude of φ − φf and δa. To summarize, there are two qualitatively distinct ways for φ to decay to the minimum of the potential, either with or without oscillations. These behaviors are distinguished by finite or infinite cosmological tension. Note that if φ were coupled to other fields to reheat the universe, this would also distinguish different physical mechanisms for the reheating. Lastly, it is interesting to see how these different asymptotics look in AdS. In four dimensions the metric of a planar AdS black hole approaches AdS at spatial infinity exponentially fast, like e−3y/ℓ , where y is a proper length radial coordinate and ℓ is the AdS length scale. However, the relevant area element grows like e3y/ℓ , so the resulting boundary term for the ADM mass is finite. A significant amount of research has been done on the behavior 2 Note that the Hamilton-Jacobi method in section 2.1 will only produce monotonic scalar flows and hence will not generate oscillatory cases such as those considered here. 3 Since the metric is isotropic, all three tensions, associated with invariance in the three spatial directions, are the same. – 10 – JHEP10(2017)118 where ω 2 = Wf′′ − Hf2 , and C1 and C2 are constants whose precise expressions will not be needed. The underdamped,2 oscillatory, cases all decay at the same rate, which is precisely that needed for a finite, non-zero, cosmological tension3 T . The ADM charges are defined in terms of boundary integrals, which require a number of further definitions in order to specify. We present this material in a short appendix to the paper. In computing T we average over a period, and the oscillatory terms average to zero. For the non-oscillatory solutions (2.33), the key ingredient in the boundary integral is the behavior of a term like of scalar fields in AdS/ CFT, and particularly analogous to the issue of future asymptotically de Sitter spacetime are studies of AdS black holes with scalar hair, AdS domain-wall scenarios, and holographic domain walls [14–29]. In these situations the potential for the scalar field is negative, and the field approaches a maximum of the potential, which provides an effective negative mass-squared. The two modes of the field decay like e−∆± y/ℓ ,  p 3 2 2 ∆± = 1 ± 1 + 4m ℓ /9 , 2 ′′ m2 = W∞ <0 (2.38) 3 Schwarzschild-quasi de Sitter spacetimes In section 2 we analyzed de Sitter to de Sitter evolutions, with no black hole present. We now investigate the effects of a black hole on this transition, and in particular how the black hole and cosmological horizons evolve. Exact solutions of dynamical black holes include the McVittie metric [44], which is simply SdS in cosmological coordinates in the most physical case, and examples of maximally charged multi-cosmological black holes both without [45] and with scalar fields [46]. Accretion of fields and growth of cosmological black holes has been studied in different approximations and numerically, addressing scalar field cosmologies, generalizing the properties of Killing black holes, and the dynamics of accretion [31, 38, 47–62]. In this paper we follow the approach of [30] and systematically apply perturbation theory and the slow roll approximation to the Einstein-scalar field equations, identify the horizons, and use well-behaved coordinates on the horizons to compute the behavior of the scalar field and thermodynamic properties. Our results yield analytic expressions for the evolution of horizon areas, temperatures, thermodynamic volume, and local pressure. We find that the generalised first law of thermodynamics is satisfied between the final and initial SdS states. Though the results are given in terms of a general potential W (φ), it is most straightforward to make a thermodynamic interpretation in the case that W has a maximum and a minimum, since then the initial and final states are equilibria – 11 – JHEP10(2017)118 with the constraint m2BF ≤ m2 ≤ m2BF + 1 < 0, where m2BF = −9/(4ℓ2 ) is the Breitenlohner-Freedman bound [14, 16–21, 43]. This formula is analogous to the scalar field decay in the cosmological case (2.29), with the substitution of a negative potential in which φ approaches a maximum, for the positive potential with φ approaching a minimum. The dominant far field mode with ∆− decays too slowly for the spacetime to have a finite ADM mass, as we have found for the late time de Sitter behavior with real β− . In the AdS case a finite ADM charge can be constructed by combining contributions from the gravitational and scalar fields [15, 16, 22, 23]. A similar situation arises in AdS domain wall spacetimes in which a scalar field potential interpolates in the radial direction between two AdS vacua. The interpolation between AdS vacua corresponds to an RG flow between two CFTs, and has previously been compared to de Sitter to de Sitter transitions [24]. A topic for future analysis is to work out the combination of cosmological tension with a scalar field contribution to form a finite generalized ADM tension, appropriate for a slow roll approach to de Sitter. but are nonsingular at the horizons, and indeed facilitate the analysis of horizon behaviour 1/2 that are now located at “U = 0” or “V = 0”. Here, B0 is a fiducial length scale that maintains dimensional consistency, and dΩ2II is the metric on the 2-sphere. This form of the metric also clearly identifies the main physical degree of freedom of the gravitational field as the B−function, since ν communicates with the remaining gauge freedom of conformal transformations in the U, V plane. In the absence of any scalar evolution, the physical degree of freedom in B corresponds to the mass of the black hole as we now briefly review. (This discussion follows [30, 63]). In this null gauge, the Einstein equations become r 1 B0 2ν e − (B,U φ,V + B,V φ,U ) (3.2) φ,U V = −W,φ (φ) B 2B  W (φ) 1/2 1/2 B,U V = 2 (3.3) B − B −1/2 e2ν B0 2 Mp   φ,U φ,V 1 W (φ) −1/2 1/2 −3/2 ν,U V = e2ν B0 − B +B (3.4) 2 2 Mp 2Mp2 B,V V = 2ν,V B,V − Bφ2,V /Mp2 B,U U = 2ν,U B,U − (3.5) Bφ2,U /Mp2 (3.6) If φ is constant, equations (3.5) and (3.6) give B,U B,V + G′ (U ) = log √ + F ′ (V ) 2ν = log √ B0 B0 (3.7) where F and G are in principle arbitrary functions, expressed here as derivatives for convenience. From this, we deduce that B = B [F (V ) + G(U )] and – 12 – 1/2 e2ν B0 = F ′ G′ B ′ (3.8) JHEP10(2017)118 described by a static SdS metric. The main results of this section are summarized in equations (3.37), (3.40), and (3.42), (3.45), which give the total change in the cosmological and black hole horizon areas respectively. The reader uninterested in the derivation can skip to those results without loss of continuity. For a slowly rolling scalar field, the spacetime will be adiabatically de Sitter (or SdS), with small, “time-dependent” corrections. The idea therefore is to first find a solution with a constant φ, then to correct this perturbatively for a rolling φ. Given that we have a cosmological evolution in ‘time’, together with the black hole giving us a ‘radial’ dependence of our geometry, our analysis should capture the dependence on these two coordinates. Since the black hole and cosmological event horizons represent coordinate singularities in the standard SdS metric, we follow the methodology of reference [30] that treated black hole evolution for a scalar with an exponential potential, using null coordinates for the metric that encode the dependence on two parameters: r B0 2 2ν (3.1) dU dV − BdΩ2II , ds = 4e B Then (3.3) can be integrated up to give B′ = 4W0 3/2 B − 4B 1/2 + µ 3Mp2 (3.9) where the integration constant µ is nonzero if a black hole is present. Substituting these expressions back into the metric gives   W0 µ √ ds2 = −16 1 − dF dG − BdΩ2II . (3.10) B − 3Mp2 4 B F ↔− (t + r⋆ ) , 4 G↔ (t − r⋆ ) 4 (3.11) (where r⋆ is the standard tortoise coordinate), then transform to t, r coordinates we recover the SdS metric: dr2 − r2 dΩ2II . (3.12) ds2 = N (r)dt2 − N (r) We have written the SdS potential as N (r) = 1 − 2GM H2 − H02 r2 = − 0 (r − rc )(r − rh )(r − rN ), r r (3.13) where H02 = W0 /3Mp2 , and we identify the black hole horizon (if present) as rh , the cosmological horizon as rc , and the remaining zero of N as rN = −(rc + rh ). Note, the roots of N are related to the physical parameters via Λ = 3H 2 = 3/(rc2 + rh2 + rh rc ) 2GM = H 2 rc rh (rc + rh ) and the tortoise coordinate r⋆ is given explicitly by Z 1 dr 1 r − rN r − rc r − rh 1 ⋆ log = log log r (r) = + + . N (r) 2κc rc 2κh rh 2κN rN (3.14) (3.15) Here, κi are the usual surface gravities at the individual horizons, 2κi = N ′ (ri ). Finally, although F and G are null coordinates, the metric still has coordinate singularities at the black hole and cosmological event horizons. These can of course be removed locally by using the standard Kruskal coordinates: • r → rc • r → rh 1 1 exp [κc (t + r⋆ )] , U = exp [κc (t − r⋆ )] 2κc 2κc 1 1 v= exp [κh (t + r⋆ )] , u = − exp [−κh (t − r⋆ )] 2κh 2κh V = (3.16) however, no global maximal extension of the SdS coordinates is possible. Given that we are interested in the future evolution of the spacetime, we could choose to use {u, V } as nonsingular coordinates, however, in practice it is easier to analyse physics near the event horizons in the local Kruskals. – 13 – JHEP10(2017)118 Comparison with the SdS metric suggests that we identify µ = 8GM , and choose a √ radial coordinate r = B. It is then fairly clear that if we identify F and G with the advanced and retarded null coordinates 3.1 The scalar field behaviour Prior to starting the analysis of an evolving black hole, it is useful to relate this notation to our previous discussion with cosmological time. With the SdS black hole, the natural solution is expressed in static gauge, as we have discussed above, but it is useful to see how the null and static coordinates relate to the cosmological coordinates. Setting µ = 0, and integrating (3.9) gives B[X] = 1 tanh2 [−2H0 X] H02 (3.17) X=− 1 1 − H0 r r⋆ = ln 2 4H0 1 + H0 r (3.18) Taking V ± U and using (3.16) then relates these expressions to our canonical cosmoR logical coordinates (via conformal time η̂ = dt/a) as 1 1 ln [−H0 (U + V )] = t + log(1 − H02 r2 ) H0 2H0 re−H0 t ρ = (V − U ) = p 1 − H02 r2 τ =− (3.19) In the cosmological slow-roll approximation with no black hole, φ only depends on cosmological time τ . Even though this is a more involved expression in the static gauge, we still have the notion that φ depends on a single function of t and r. In [30], it was found that in the slow-roll approximation for the case of an exponential scalar potential, in the presence of a black hole φ depended linearly on a variable x(t, r), which was a generalization of the cosmological time coordinate in (3.19). Here we maintain this notion of slow-roll, and look for a similar x(t, r) that reduces to τ as the black hole area goes to zero. In this slow-roll approximation, derivatives of φ and W ′ (φ) are assumed to be small quantities relative to the overall magnitude of the potential, thus (3.3)–(3.6) reduce to the pure cosmological constant equations we have just discussed, and (3.2) to leading order requires only these background forms of the metric functions to find the evolution of φ due to its potential in the presence of the black hole. Substituting in these forms, (3.2) is φ,F G 2 ∂W − (φ,F + φ,G ) = 4 N (r) r ∂φ (3.20) (recalling that r is a function of F + G). We now look for a variable x(t, r), such that φ is predominantly a function of x. That is, x must be suitably chosen to render (3.20) an ODE for φ when the φ′′ term is neglected. Now   φ,F G x,F x,G ′′ x,F + x,G ′ x,F G 2 − (φ,F + φ,G ) = φ + −2 φ (3.21) N (r) r N N r so let x = t + ξ[r] – 14 – (3.22) JHEP10(2017)118 where X = F + G < 0. Using (3.11), (3.15), and κc = −H0 = −rc−1 , then, B′ x,F = 2 + ξ ′ √ = 2(1 − N ξ ′ ) 2 B B′ x,G = −2 + ξ ′ √ = −2(1 + N ξ ′ ) 2 B (3.23) N 2 ξ ′2 − 1 ′′ (r2 N ξ ′ )′ ′ ∂W φ + φ = N r2 ∂φ (3.24) and (3.20) becomes We can now read off our requirement for slow-roll as C + r3 r2 N (3.25) The final determination of the constant of proportionality and the integration constant C is determined by the boundary conditions that the field is purely ingoing at the black hole horizon and purely outgoing at the cosmological horizon, leading to x = t − r⋆ + rc r 1 rh rc r − rh r − rN ln + ln ln − κh rh 2κh rh rN rc − rh r0 (3.26) where r⋆ is given in (3.15). The new x coordinate reduces to the expression for the cosmological coordinate τ in (3.19) when rh = 0. This implies that   1 rc2 rh2 (rc + rh ) − r3 (rc2 + rh2 ) ′ (3.27) ξ = N r2 (rc3 − rh3 ) and dropping the φ′′ term, the slow-roll equation becomes 3γφ′ (x) = − where γ= ∂W ∂φ rc2 + rh2 Atot = 3V rc3 − rh3 (3.28) (3.29) Here Atot = 4π(rc2 +rh2 ) is the total horizon area, and V = 4π(rc3 −rh3 )/3 the thermodynamic volume, of the de Sitter black hole system. The volume V, with rh = 0, arose in the thermodynamics of the cosmological horizon in the absence of a black hole, equations (2.16) and (2.17). It will again enter the thermodynamics of the black hole−cosmological system in subsequent sections. It is interesting to compare (3.28) to the cosmological slow-roll equation, where γ = H0 . Clearly as rh → 0, the two slow-roll equations coincide, but we can now explicitly see the effect of the black hole on the friction for φ−motion. As rh is switched on, the denominator in (3.29) decreases and the numerator increases, hence γ is larger for larger black holes at fixed Λ.4 Thus the effect of a black hole is to further slow down the slow roll inflation, 4 It is a little more subtle, since for fixed H 2 , i.e. W , rc decreases as rh increases, however, it is easy to check that the overall effect of increasing the black hole size is to increase γ. – 15 – JHEP10(2017)118 ξ′ ∝ and for black holes very close to the Nariai limit, the evolution becomes arbitrarily slow. Indeed, expanding γ for small and large black holes, demonstrates this effect clearly: ( H0 (1 + GM H0 ) M → 0 γ∼ (3.30) 2 r → r c h 3(rc −rh ) 3.2 Growth of the event horizons What does the resulting evolution look like? Two fundamental geometrical properties of the spacetime are the areas of the black hole and cosmological horizons. Since the black hole is accreting the scalar field, we expect the black hole to grow. A bigger black hole will tend to pull in the cosmological horizon. However, the effective cosmological constant is decreasing, which leads the cosmological horizon to grow. We will see that the second effect dominates, and that both horizons grow. Ideally, we would calculate the full gravitational back reaction throughout the spacetime, as described in [30], however, this process is rather involved, and somewhat specious to the main theme of our discussion, namely the evolution of the event horizons. As it turns out, this is fairly straightforward to extract. Note that in the background solution, the horizon (a null surface) is at fixed r, i.e. fixed B. Taking the local Kruskals at each horizon, given in equation (3.16), the cosmological event horizon is defined as V = 0, and parametrised by U whereas the black hole event horizon is defined as u = 0, and parametrised by v. In the vicinity of the horizons of the background SdS we have:  V = U exp [2κc r⋆ ] ≈ U (r − rc ) as r → rc (3.31) 1 1 ⋆ u = − 2 exp [2κh r ] ≈ − 2 (r − rh ) as r → rh 4κh v 4κh v To get the horizon growth, the idea is to expand the Einstein equations (3.5) for the black hole horizon, and (3.6) for the cosmological horizon, by using the fact that B,u (B,V ) is zero on the black hole (cosmological) event horizon in the background spacetime. δB,U U = 2ν,U δB,U − Bφ2,U /Mp2 • r → rc • r → rh δB,vv = 2ν,v δB,v − Bφ2,v /Mp2 (3.32) (3.33) Here the derivative of φ terms enter as a perturbative source. Lastly, we will need the expression for ν at each horizon: • r → rc rc N rc rN (r) ≈− 2 2 ≈− 2 4κc U V 4κc U (r − rc ) 2κc U 2 rN (r) rh N ≈ 2κh rh =− 2 ≈ (r − rh ) 4κh uv e2ν = • r → rh e2ν – 16 – (3.34) JHEP10(2017)118 Finally, given that the coefficient of the φ′′ term contains a 1/N factor, we must check that this term remains small. From (3.27), we see that N ξ ′ → ±1, at the black hole and cosmological horizons, thus the term multiplying φ′′ is in fact regular at the horizons, and thus overall stays small. It is therefore consistent to drop the second derivative term as long as |φ′′ | ≪ γφ′ . 3.2.1 Cosmological event horizon Starting with the cosmological event horizon, as r → rc , x ∼ κ−1 c ln(2κc U )+const., and from (3.34) we have ν,U ≃ −1/U along the cosmological horizon. Hence (3.32) gives dx rc2 rc2 φ′2 φ′2 = − 2 2 2 κ c Mp U κc Mp dU Z Z 2 2 rc rc ∂W ′2 =− φ dx = dφ 2 2 κ c Mp 3γκc Mp ∂φ (U δB),U U = − ⇒ (U δB),U rc2 (Wi − W [φ(U )]) 3γκc Mp2 (3.36) Here we have set δB = 0 when φ takes its initial value φi , which is as U → −∞ and Wi = W (φi ). Integrating again gives Z 0
rc2 Wi − W [φ(U ′ )] dU ′ (3.37) δB = − 2 3γ|κc |Mp U U where we have written −κc = |κc | to clarify that the change in the horizon area is positive. As U → −∞, φ → φi , and we get δB → 0, but as we go to future infinity, or U → 0 , then φ → φf and 1 δB = (3.38) r2 (Wi − Wf ) , 3γ|κc |Mp2 c This means that the total change in cosmological horizon radius is positive, and given by δrc = δ p B(rc ) = δB 1 rc (Wi − Wf ) = 2rc 6γ|κc |Mp2 (3.39) When there is no black hole, then rh = 0, rc = 1/H, |κc | = H, and the change in horizon radius is Hf−1 − Hi−1 , in agreement with equation (2.14) and figure 1. Finally, this gives for the change in the cosmological horizon area δAc = 8πrc δrc δAc = Ac (Wi − Wf ) 3γ|κc |Mp2 (3.40) This can be written in terms of the change in the early and late time effective cosmological constants by using ΛI = WI /Mp2 . Since Wi > Wf , corresponding to evolution of φ by rolling down the potential, the cosmological horizon grows. It is of interest to find how much the presence of a black hole affects the growth of the cosmological horizon, for a fixed potential. In equation (3.40) the quantities Ac , κc and γ all depend on rh . Looking at the limits, the effect is of course negligible for very small black holes, but as the size of the black hole horizon becomes comparable to that of the cosmological horizon, the total change in Ac is diminished by a factor of approximately two-thirds. 3.2.2 Black hole event horizon The analysis for the black hole horizon area is similar to that for cosmological horizon, but now we use the Kruskal coordinates u, v. The black hole horizon is at u = 0, and – 17 – JHEP10(2017)118 =− (3.35) v is the coordinate along the horizon. As r → rh , we have x ∼ κ−1 h ln(2κh v)+const. From equation (3.34) it follows that ν is constant on the event horizon, and the perturbed equation for B becomes r2 δB,vv = − 2 h2 2 φ′2 (3.41) κh M p v which is solved by rh2 v δB = − 3γκh Mp2 ∞ v W [φ(v ′ )] − W [φ(0)]
dv ′ v ′2 (3.42) We have set a constant of integration equal to zero that would have lead to an increase in the area that grows linearly in v, rather than a constant value as is consistent with asymptotically dS boundary conditions. As v → 0, we get δB → 0, but as v → ∞ (or as we go to future infinity), rh2 (Wi − Wf ) (3.43) δB = 3γκh Mp2 This means that the change in black hole horizon radius is δrh = δ p δB rh B(rh ) = (Wi − Wf ) = 2rh 6γκh Mp2 (3.44) One sees that δB,v 6= 0 at v = 0, which is equivalent to the observation that the event horizon begins to move out before matter has crossed it, which follows from the teleological nature of its definition. The corresponding change in the area of the black hole horizon is δAh = Ah (Wi − Wf ) 3γκh Mp2 (3.45) The area of the black hole horizon increases, as is expected due to accretion of the scalar field. The fractional rate of area increase is less for the black hole than for the cosmological horizon, since (3.40) and (3.45) imply |κc | (δAh /Ah ) = <1 (δAc /Ac ) κh (3.46) Now that we have the changes in horizon radii, as a check we can compute the change in Λ that would be required by the SdS relation equation (3.14), using the changes in horizon radius computed by integrating the horizon radii, equations (3.39) and (3.44) δΛ = −3H 4 ((2rc + rh )δrc + (2rh + rc )δrh )   H 4 (Wf − Wi ) rc rh (2rc + rh ) − (2rh + rc ) = γMp2 2κc 2κh   2 4 2 r H δW δW rc = − 2 h = 2 2 2 γMp H (rc − rh ) H (rc − rh ) Mp as required. – 18 – (3.47) JHEP10(2017)118 Z 4 Dynamical thermodynamics In this section we explore a number of aspects of the thermodynamics of the slow roll evolution of black holes between initial and final Schwarzschild-de Sitter states that we have established in the last section. 4.1 Analysis of horizon growth where 2πTα = |κα | are the horizon temperatures and Λi,f = Wi,f /Mp2 are the initial and final values of the cosmological constant. For a fixed potential W , and hence fixed values for Λi,f , how much does a black hole with initial radius rh grow? Using the formulae for the temperatures and areas in Schwarzschild-de Sitter spacetime (3.14), we can express the change in the areas in terms of rh , Λi , and δΛ = Λi − Λf . For the black hole horizon, one finds that |δΛ| 2rh (rc2 + rh2 + rc rh ) (4.2) δAh = Ah Λi (2rh + rc )(rh2 + rc2 ) while the corresponding expression for δAc is obtained by interchanging rh and rc . However, this is not quite what we want, since rc is still dependent on rh and Λi . This dependence can be dealt with exactly using (3.14), but it is most useful to focus on the limits where the black hole horizon is either small or comparable in size to the cosmological horizon. One finds that in these limits, the change in the black hole horizon area is given by p p |δΛ| × (rh Λi ) , rh Λi ≪ 1 δAh ≃ 2Ah √ 3Λi (4.3) p |δΛ| , rh Λi ∼ 1 ≃ Ah Λi We see that the fractional growth in area, δAh /Ah , is parametrically suppressed for small black holes, while it is of order |δΛ|/Λi for large ones. Likewise, one can ask how much the cosmological horizon is “pulled back” by the black hole, compared to the case with no black hole. In the limiting cases of small and large black holes, one finds that the change in the cosmological horizon area is given by p |δΛ| δAc ≃ 12π 2 , rh Λi ≪ 1 Λi (4.4) p |δΛ| ≃ 4π 2 , rh Λi ∼ 1 Λi For small black holes, the effect of the black hole on the cosmological horizon growth is negligible. While for large black holes, it can be reduced by as much as 2/3. For a black hole with initial area 1/100 of the cosmological horizon area, one finds that the diminution effect is a factor of 1/10. – 19 – JHEP10(2017)118 Let us further analyze the results for evolution of the black hole and cosmological horizons found in the last section. Using the definition (3.29) of γ as well as the thermodynamic volume and total horizon area, both equations (3.40) and (3.45) can be written in terms of thermodynamic quantities as V Aα δAα = (Λi − Λf ) , α = h, c (4.1) 2πTα Atot 4.2 Two first laws Two independent first laws for asymptotically de Sitter black hole spacetimes can be derived [12]. One relates the change in area of the black hole horizon to the change in mass, while the other relates the change in area of the cosmological horizon to the change in mass. Including the possibility of a change in the cosmological constant, each of these laws has an additional term proportional to a thermodynamic volume times δΛ, i.e. a term of the form Vα δΛ, where α = h, c. One can take the difference of the two first laws, such that the mass term drops out giving (4.5) where V is the thermodynamic volume between the black hole and cosmological horizons, which was introduced above, and we have set |κI |δAI = 8πTI δSI and Λ = −8πP . Here we are studying the evolution from one Schwarzschild-de Sitter spacetime to another, where the change is effected by the rolling scalar field. Combining the changes in the horizon areas computed in the previous sections, and given in equations (3.40), and (3.45), one can check that (4.5) is indeed satisfied for these evolutions. This requires use of (3.29), which implies that (Ah + Ac )/(3γ) = V. This result is interesting because a dynamical scalar field is beyond the scope of applicability of the derivation in [12], and yet our results for a black hole in slow-roll inflation are still found to satisfy the first law, applied to the differences between the initial and final de Sitter phases. This agreement suggests that the first law might be satisfied continuously along the evolution, an idea that we return to in the discussion. 4.3 Temperature and mass for evolving black holes A definition of temperature for dynamical black holes was discussed in [31], which proposes that a generalized surface gravity is √ 2κdyn = − ⋆ d ⋆ d B (4.6) where the Hodge dual ⋆ refers to the 2D spacetime perpendicular to θ and φ, or the U − V part, and the right hand side is evaluated on the horizon.5 On the u − v subspace, one has ⋆ du = du , ⋆dv = −dv , −1 ) ⋆du ∧ dv = −(guv Evaluating (4.6) on the black hole horizon at u = 0, one finds that     √ B,uv B,u B,v e−2ν B,u du − B,v dv √ √ = − + 2κdyn = − ⋆ d ⋆ d B = − ⋆ d 2 4B B0 2 B √ B,u B,v −2ν −1/2 W (φ) B+ e B0 = B −1/2 − Mp2 4B (4.7) (4.8) where the Einstein equation (3.6) has been used in obtaining the second line. One might guess that for slow-roll evolution the black hole temperature would instantaneously be 5 This formula differs by a minus sign (4.6) from that in [31] due to using different signatures. – 20 – JHEP10(2017)118 Th δSh + Tc δSc = VδP that of a Schwarzschild-de Sitter spacetime with the value of Λ and rh at that time, and this turns out to be almost the case. Using the function N (r) in (3.13), the black hole temperature in Schwarzschild-de Sitter is 4πTsds = r1h − Λrh . For the evolving spacetime, this gives the temperatures in the initial and final states, where rh and Λ taking their initial and final values. Defining a quasi-static temperature that interpolates between the initial and final values along a sequence of Schwarzschild-de Sitter spacetimes as 4πTqs (v) = W (v) 1 rh (v) − rh (v) Mp2 (4.9) B,u B,v −2ν −1/2 κh v e B0 ≃ − 2 δB,v 4B rh (4.10) So the dynamical temperature is given by 2πTdyn (v) = 2πTqs (v) − κh v δB,v rh2 (4.11) where the derivative of δB, which follows from (3.41) (or from (3.42)), is given by Z ∞  ′ rh2 δW [φ(v)] ′ dv δB,v = − δW [φ(v )] ′2 − (4.12) 3γκh Mp2 v v v Now consider the late time behaviour of this temperature, when the integral in δB R is approximately given by6 δW/v 2 ≃ δW/v. Substituting this in to δB and (4.12), at late times the dynamical temperature is given by the quasi-static approximation, which expanded to first order is   Wi rh2 (Wi − W [φ(v)]) 2 Tdyn ≃ Tqs ≃ Tsds,i − (4.13) 1+ − 6γκh rh 24πγκh Mp2 Mp2 where Tsds,i is the temperature of the initial SdS spacetime. A definition of the dynamical mass can be found by considering the first law relating the change in mass to the changes in black hole area and Λ. Perturbations about a static black hole with positive Λ satisfy [12] δM = Th δSh − Vh δP (4.14) where Vh = 4πrh3 /3 is the black hole thermodynamic volume, here given for SdS spacetime. As noted in the context of the first law formulated between the two horizons (4.5), a dynamical cosmological constant due to a scalar field potential is beyond the scope of the derivation of (4.14). However, since we found that (4.5) is true for our solutions, let us assume that (4.14) also holds as well and see what it implies for the mass. This is equivalent 6 For more detailed discussion of the asymptotic behaviour of B, see section 5, where the dynamics of the black hole system is computed in detail for a sample potential. – 21 – JHEP10(2017)118 then this matches the first two terms in (4.8). We can evaluate the last term in (4.8) pertur−1/2 batively. In the background solution, one has B,v = 0 and B,u e−2ν B0 = −4κh v, so that to assuming that at late times, when the stress-energy is again dominated by the smaller effective Λf , that the metric can be put into static SdS form with the evolved values of rh and rc . The final mass Mf is then given by the SdS relations (3.14). Explicitly, substituting δAh from (3.45) and δΛ = (Wf − Wi )/Mp2 into (4.14) gives δM = r c rh |δΛ| M 2 Λ + rh (4.15) rc2 Meanwhile, Vh δP = so 4πB 3/2 δ(−W ) 3 (4.17) √ δB 4π 3/2 √ − 2πW BδB − B δW 3 4G B   √ 1 W 3/2 = B− B δ 2G 3Mp2 Tdyn δSh + Vh δP = (4.18) u=0 This quantity is defined on the black hole horizon, and if our assumptions are correct, is equal to δM in (4.15). It also suggests that the mass is given by   W 1 √ 3/2 B (4.19) B− M= 2G 3Mp2 u=0 In the static case, where B = r2 , this is precisely the definition of M . Supporting this interpretation is that integrating (3.5), gives that to leading order the additional piece in the dynamical temperature is Z Z 1 1 2 δB,v = − 2 Bφ,v = − 2 Tvv (4.20) Mp Mp that is, the ‘mass’ contribution due to scalar accretion onto the black hole. In general, to complete the argument, we would want to show that the quantity evaluated at u = 0 is equal to a quantity defined on future spacelike infinity. 5 Illustrative example In this paper, we have derived general results for the accreting black hole. It is helpful to illustrate these with a test-case example, using our standard double well potential (2.21), – 22 – JHEP10(2017)118 Analogous to the check we did on the change in Λ (3.47), one can vary M directly from (3.14), substitute in our results for the changes in rh , rc and Λ, and find the same answer as in (4.15). Lastly, it is interesting to assemble the terms on the right hand side of (4.14) as follows. We have that   vκh δB,v 1 W (φ) √ −1/2 B− Tdyn δS = B − δB 4G Mp2 B   (4.16) W (φ) √ 1 −1/2 2 B − = B δB + O(δB) 4G Mp2 which will allow us to explore the effects of varying the black hole mass and slow roll parameters. Recall that the solution to the slow-roll equation, 3γφ′ = −W ′ (φ), for (2.21) is 2 2 φ =η eHi Γx/2γ 2 (5.1) 2 eHi Γx/2γ + 1 We start by focussing on the black hole event horizon where κh x ≃ log(2κh v), and compute the dynamical horizon area and temperature as a function of v̂ = 2κh v. Writing a = ΓHi2 /2γκh , we have 3Hi2 Γη 2 v̂ a (2 + v̂ a ) 16 (v̂ a + 1)2 (5.2) and hence (3.42) gives A = 4πB = A0  a∆v̂ 1− I[v̂, a] 8  (5.3) Here ∆ = η 2 /Mp2 represents the strength of the gravitational interaction of the scalar field, and Z ∞ a y (2 + y a )dy I[v̂, a] = − y 2 (1 + y a )2 v̂    (5.4) 1 1+a (1 + a(1 + v̂ a )) 1 + a −a 1, + ; ; −v̂ 1 − F =− 2 1 av̂(1 + v̂ a ) av̂ a a is a dimensionless integral. Meanwhile, the dynamical temperature is found to be   v̂ v̂ a (2 + v̂ a ) a∆ + I[v̂, a] κh (3γrh + 1) Tdyn = Tinit + 32π (1 + v̂ a )2 rh (5.5) Having extracted the dimensionful dependence, we can now see how the various parameters impact the evolution of the scalar and hence the black hole horizon. First, it is clear that the overall gravitational strength of the scalar, ∆, simply scales the overall magnitude of the variation of the area and temperature. The parameter a on the other hand not only scales the magnitude of these variations, but also their rapidity, as would be expected since a is directly dependent on the rate of slow-roll of the scalar field. Given the expressions for γ and κh , we see that a∼ Γ(r2 /r2 + rc /rh + 1) Γ ΓHi2 = 2 c2 h < 2γκh 2 (rc /rh + 1)(rc /rh + 2) (5.6) It is perhaps most useful to display the variations of the black hole variables as a function of the local (normalised) Eddington-Finkelstein advanced time co-ordinate on the event horizon, V̂EF = H(t+r⋆ ) = κHh log[2κh v] ∼ Hx. Figure 2 shows a plot of the variation of horizon area with advanced time, while figure 3 shows the variation of temperature. In both cases, rather large values of ∆ and Γ have been chosen to emphasize the effects. The differing gradations of “slow-roll” on display arise because of the differing black hole: cosmological horizon ratios — recall that the true slow-roll parameter γ, given in (3.29), for the scalar in the presence of the black hole has a strong dependence on the geometry. – 23 – JHEP10(2017)118 δW = W [φ] − W [0] = − /i Δ=1 rh =7 1.0030 Γ=0.05 1.0025 rh =5 2 1.0020 H =1/175 rh =3 1.0010 rh =1 1.0005 -400 -200 0 200 400  V EF Figure 2. Illustration of evolution of the horizon area (Ah /Ai ) as a function of the EddingtonFinkelstein advanced time coordinate, scaled by the Hubble parameter, on the black hole horizon. We see that for a larger initial black hole, the increase in black hole horizon area is both larger and more gradual, than for small black holes. T/Ti 1.0000 Δ=1 Γ=0.05 0.9999 rh =1 2 0.9998 H =1/175 0.9997 rh =3 rh =5 0.9996 rh =7 -400 -200 0 200 400  V EF Figure 3. Illustration of evolution of temperature (T /Ti ) as a function of the Eddington-Finkelstein advanced time coordinate on the black hole horizon. We see that for larger black holes the decrease in black hole horizon temperature is both larger and more gradual than for small black holes. – 24 – JHEP10(2017)118 1.0015 6 Concluding remarks Acknowledgments The authors would like to thank Kostas Skenderis for helpful conversations. RG is supported in part by STFC (Consolidated Grant ST/J000407/1). RG also acknowledges support from the Wolfson Foundation and Royal Society, and Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. A Cosmological tension The ADM cosmological tension charges for an asymptotically future de Sitter spacetime were constructed in [13]. Generally, an ADM charge corresponding to an asymptotic sym- – 25 – JHEP10(2017)118 Central concepts in gravitational thermodynamics are horizon areas and temperatures, and the relations between them. The analysis presented here allows the study of these quantities in a “mildly” dynamical setting, namely the evolution from one SdS spacetime to a second SdS with a smaller cosmological constant, in a perturbative and slow-roll approximation. An advantage of these boundary conditions is that the initial and final states are equilibria with approximate Killing horizons and associated temperatures. Within these approximations we have solved the Einstein plus scalar field system to extract the growth of the black hole and cosmological horizons for a general scalar potential that has a maximum and a minimum. The results are expressed in terms of the change in the effective cosmological constant, as dictated by the potential, and geometrical properties of the initial spacetime. Using a proposed definition of dynamical temperature, the temperatures of each horizon are found to decrease between the initial and final values as the horizons grow. One of the interesting features of the solutions is that the first law of thermodynamics (4.5), formulated between the two horizons, holds between the initial and final SdS states. This brings up two questions for further study. One is to derive the first law including the stress-energy of a scalar field, which generates additional contributions at the black hole and cosmological horizons, and verify that these contributions vanish for our solutions. Second, this result suggests that the first law might be satisfied not only between the early and late time SdS metrics, but continuously along the evolution. More strongly, is there a solution for the metric functions which illustrates that the metric is quasi-SdS at each time? In general, it would be advantageous to have explicit expressions for the full metric across the range of r and x. This would facilitate analyzing the flow of energy-momentum throughout the volume, to test the definition of dynamical temperature by studying the near-horizon metric, and to follow the evolution of the mass-like quantity that interpolates between the initial and final mass parameters. Another interesting direction for further work is to transform the analysis to cosmological coordinates, which are likely more convenient for ascertaining how a black hole affects the important predictions of inflation, such as the spectrum of quantum perturbations and reheating. metry of a spacetime is defined via Hamiltonian perturbation theory [64]. We follow the general prescription presented in [41], which we briefly outline here. The construction starts with foliating the spacetime by (D − 1)-dimensional hypersurfaces Σ with unit normal na such that the metric can be decomposed as gab = (n · n)na nb + sab where B a = F (D̄a h − D̄b hab ) − hD̄a F + hab D̄b F 1 + √ β b (π̄ cd hcd s̄a b − 2π̄ ac hbc − 2pa b ) s (A.2) and D̄a is the covariant derivative operator compatible with the background metric s̄ab . Note that to simply define the charge Q one only needs the symmetry (and the foliation) asymptotically. However, if the symmetry holds througout the spacetime, this set-up can then be used to prove a first law [41]. The ADM mass results when a spacetime has an asymptotic static Killing field at spatial infinity and choosing Σ to be a timelike slice with unit timelike normal, na na = −1. On the other hand, the construction can be used for a cosmological spacetime with an asymptotic spatial translation Killing field ξ and taking Σ to have a unit spacelike normal, na na = 1. Then the boundary ∂Σ is in the asymptotic future. The resulting ADM charge is a cosmological tension. If the spacetime is anisotropic but homogeneous, as was considered in [13], then the tensions are distinct. The inflationary spacetimes studied in this paper are isotropic and homogeneous, so there are three equal tensions. Using the formulae derived in [13] to process the boundary term (A.2) for the underdamped asymptotically de Sitter case, and averaging over a period of oscillation, gives the cosmological tension (2.37). In the overdamped and critically damped cases, the metric functions decay too slowly to balance the growth of the volume element, and the tension diverges. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. – 26 – JHEP10(2017)118 where na na = ±1 and sab denotes the induced metric on the slice(s), satisfying the orthogonality relation sab nb = 0. Let (sab , π ab ) denote the Hamiltonian initial data on a slice Σ and (hab , pab ) be perturbations linearized about the background denoted by (s̄ab , π̄ ab ). Furthermore, let ξ a be a Killing vector of the background, which we project along the slice Σ and its normal according to ξ a = F na + β a such that na β a = 0. The ADM charge corresponding to the Killing vector ξ a is then defined by an integral over a (D − 2)-dimensional boundary at infinity on Σ given by Z 1 dac B c (A.1) Q(ξ) = − 16πG ∂Σ∞ References [1] B.J. Carr, K. Kohri, Y. Sendouda and J. 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