JHEP10(2017)118 Springer October 3, 2017 August 11, 2017 October 18, 2017 : : : Received Accepted Published , c Published for SISSA by https://doi.org/10.1007/JHEP10(2017)118 [email protected] , and Jennie Traschen c . 3 David Kastor 1707.06586 The Authors. a,b c Black Holes, Black Holes in String Theory

We study evolution and thermodynamics of a slow-roll transition between early , [email protected] Amherst Center for FundamentalDepartment Interactions, of Physics, University710 of N Massachusetts, Pleasant St., Amherst,E-mail: MA 01003, U.S.A. [email protected] Centre for Particle Theory,South Durham Road, University, Durham, DH1 3LE,Perimeter U.K. Institute, 31 CarolineWaterloo, Street ON, North, N2L 2Y5, Canada b c a Open Access Article funded by SCOAP ArXiv ePrint: of thermodynamics, including variationbetween in the the initial effective and cosmological finaltemperature. constant, states, is and satisfied discuss the dynamical evolution of theKeywords: black hole cosmological tension when the scalarwhile field the oscillation cosmological around tension its in minimumin the is the overdamped underdamped, case cosmological diverges. and Wefinding black compute hole the that variation horizon the areasfractional fractional between change in change the the effective early in cosmological and constant. horizon late We show area time that the is phases, extended first proportional law to the corresponding Abstract: and late time de Sitter phases,hole, both in a in scalar the field homogeneous modelminimum. case with and Asymptotically a in generic future the potential presence de havingknown of Sitter both a as spacetimes a black maximum cosmological are and characterized tensions. a by positive ADM We charges show that the late time de Sitter phase has finite Ruth Gregory, Black hole thermodynamics with dynamical lambda JHEP10(2017)118 ]. 10 – 1 20 11 17 – 1 – 3 17 16 19 19 14 25 25 22 6 3 20 7 1 3.2.1 Cosmological3.2.2 event horizon Black hole event horizon 4.1 Analysis of4.2 horizon growth Two first4.3 laws Temperature and mass for evolving black holes 3.2 Growth of the event horizons 2.1 Engineered flow 2.2 Slow-roll analysis 2.3 Cosmic hair 3.1 The scalar field behaviour In this paper we exploreis cosmological black also holes of in a physicalconsider relatively interest, well-controlled transitions setting namely between that the distinct evolutiondriven early of by and black late a holes time scalar intial, de slow-roll both field Sitter inflation. of rolling phases, which We slowly with are evolution between assumed a to maximum be positive. and minimum The of initial and its final poten- states are described cosmic expansion with accretion ofdynamics matter is onto non-equilibrium, the involving black timeblack hole. dependent hole The areas and gravitational and cosmological thermo- surfacetemperature horizons, proves gravities to such for be the that complicated.have formulating Still, been physical an implications heavily appropriate of researched, primordial definition includingthe black of a holes dark recent matter revival and of the interest progenitors that of they the may massive provide black holes detected by LIGO [ 1 Introduction The physics of black holesatively in little the is early known. universe The is system an is important both subject, interactive about and which dynamic, rel- combining effects of 6 Concluding remarks A Cosmological tension 5 Illustrative example 4 Dynamical thermodynamics 3 Schwarzschild-quasi de Sitter spacetimes Contents 1 Introduction 2 Pure de Sitter to de Sitter flows JHEP10(2017)118 ], ], is 12 12 (1.1) , , δP based 11 11 dyn ] in which T 29 – is the thermo- 14 V times a factor that , is satisfied between | i Λ δP V − f Λ | = | Λ δ | δP V = – 2 – T δS , we find that the change in both the black hole and ] to potentials in which both the initial and final states f 30 ] is infinite for the overdamped case, but finite for the oscil- contributions from each horizon to 13 and Λ ]. This is very nearly that of a Schwarzschild-de Sitter black by analyzing pure vacuum to vacuum transitions, without a i is the initial temperature of the de Sitter horizon. Hence 31 2 T δS T is provided by the effective cosmological constant, P ) below. We then show that the extended first law of thermodynamics [ 4.1 The area growth calculations only depend on the temperatures of the background In the second part of the paper, we add a black hole to the cosmology and solve for the We begin in section As a first step, weon utilize the a Kodama definition of vectorhole the [ with dynamical the black hole instantaneousradius. temperature values We of show the howasymptotic the effective value, black cosmological hole at constant temperature a and relaxes rate black from dependent hole its on initial the value rolling to its of final the scalar and the local horizon depends on properties of the Schwarzchild-deequation Sitter (SdS) ( spacetime. This isthat summarized in relates the sumthe of initial and final SdS phases. static spacetime, but one would like to go further and work with a dynamical temperature. to competing influences.horizon further While in, the theexpansion decaying growing cosmological dominates. black constant For makes hole a itcosmological potential tends expand. constant, interpolating to We between Λ find pull initial thatcosmological and the the horizon final cosmological areas values of evolve the proportionally to evolution of the scalargeneralizes field and the metric calculations in ofare a [ approximately perturbative de “slow-roll” Sitter, and approximation.black we This hole focus grows on due thermodynamic to aspects accretion of of the the evolution. scalar The field, but the cosmological horizon is subject ADM cosmological tension [ latory evolution. This behaviorscalar fields, is as analogous well to asa in behavior scalar studies found of field for AdS gets AdS domain-wall/cosmologyand dualities a black back-reaction [ generates holes negative an with mass-squared infinite from ADM resting mass. at the maximum of a potential, dynamically generated by the scalarpotential field. there Depending are on two thetime qualitatively parameters de distinct of Sitter ways the metric. scalar that Theas field the motion it of metric settles the can into scalar the approachrespectively. field true is the Although vacuum, either in late corresponding overdamped both or to cases underdamped a the slow-roll metric or decays an exponentially oscillatory fast relaxation to de Sitter, the which in the absence of a black hole is given by where the pressure dynamic volume, and cosmological constant. black hole present. Exactwith results the are results obtained for ofhorizon an a between ‘engineered’ the perturbative example two slow-roll and de calculation. compared Sitter vacua The is growth found of to the obey cosmological the extended first law [ by Schwarzchild-de Sitter (SdS) metrics with different values of the black hole mass and JHEP10(2017)118 2 p ). − at φ 2 ( f /M (2.3) (2.2) (2.1) φ i dτ H , and a W i = = φ 2 H = ds φ at i W
) φ ( ) that corresponds to this  ) φ ( φ W ( 2 W at early times, and rolls to W − i 2 + φ ) 2 φ ˙ φ ∇ 2 1 ) has a maximum 2 p  as a time coordinate, writing φ ] tells us that the cosmological constant + ( ( ˙ 2 φ p φ M 34 R 2 1 W M 2 – p 3 − 32 M – 3 – = = = 0 − 0 2 ], and is very similar to the “fake supersymmetry” g H  ∂φ ˙ ∂W a a − 36  √ + x ˙ ] to engineer a potential φ 4 , at early and late times. d H 35 ], this is a double-analytically continued version of an AdS ) then imply that Z πG , we demonstrate this explicitly for a simple potential, finding 25 8 + 3 1 2 ) and assume that / 2.2 5 ¨ φ ]. φ ( = . If the scalar field starts off at 37 f = 1 W S φ 2 p = M φ . For the pure dS-dS flow, we can simply numerically integrate these FRW at f a/a W , where 2 p = ˙ , the equations of motion for the system are given by 2 with potential H x /M d φ f ) Briefly, the Hamilton-Jacobi approach uses We take the action for the coupled Einstein plus scalar field system to be τ W ( 2 approach described in [ The equations of motion ( 2.1 Engineered flow As a first method,Hamilton-Jacobi we formulation start [ with anflow. interpolating This Ansatz approach for has thesolutions been scalar used, in field, for the example, and presence to use of generate the gravity smooth [ analytic domain wall where equations for any desired potential,useful to however have for analytic the solutions, analysis orevolution of approximate of solutions, the the to black horizons. use hole to set-up, explore it the dynamical is and use a mostlya minus signature. Assuming an FRW form for the metric field minimum, late times, the cosmologicalto constant Λ will make a transition between the values thermodynamic relations for suchpseudo-de evolving Sitter black spacetime, hole where systems. thehole cosmological We present. constant begin varies by As in examining time,flow, noted with a in thus no [ an black- analoguemust always of flow the to lower C-theorem values in [ time. In accordance with this, we consider a real scalar analytic expressions for the dynamical area and temperature. 2 Pure de SitterOur to goal de in Sitter this flows ical paper constant is on to the investigate growth the of effect black hole of and a cosmological dynamically horizons, generated and cosmolog- to explore the surface gravity. In section JHEP10(2017)118 i φ H (2.8) (2.7) (2.10) for the ). f H 2.4 ) then finally ) (2.6) 2  ) (2.9) 2 the parameter Γ 2.4 φ ), yields the scale 2 τ/ f and a minimum at − ) Γ λητ ) 2 , ) using ( 2.6 f H 2 φ η 2 η p √ φ η H ( ( ( − 1 and a sudden change − 2 M − (3 H e 3 W 2 φ . With this identification, = 2 /  2 φ φ p 3 ( λ sech ) into ( φ − with the appropriate Hubble 2 2 M p λ √ 1 λη 6 τ 2.5 2  M √ − η η − 2 12 − 0 4). For fixed in order to facilitate comparison to ' ! 0 ) (2.5) ) and hence H = 0 (2.4) to be re-expressed as a function of τ/ 2 H ) + λη ˙ ) Γ φ φ H 4 p f λη τ φ = √ ) = , φ ( − H M f λητ √ φ 2 ( 2 W η = 4 √  H τ f (3 + Γ φ tanh( tanh( H f 2 ,H 2 p – 4 – 2 p η η H λ 2 p H M − M 3 √ 2 e 6 M log cosh( ) that allows ⇒ 2 ) = p ) = 3 τ − 2 2 τ p τ / − ( ( η M 2 ( 3 2 ) has a local maximum at 0 is a smoothed out step function, as we would expect. φ 0 ), obtained by plugging ( ) 2 η φ M φ φ is approximately linear in 2 τ 3 ( a λη H H ( φ − a

√ W − H 1 − 2 p τ +  2 , ln 0 M 0 η τ ( f H λη H 2 p H λ √ ), thus ln ) becomes = / M ) = 3 √ ) = i Ke 1 and we have defined Γ 2 τ φ 2.8 2.5 ) ( ( H − 2 ' & p a | that can be integrated up to give ) W M = τ 3 τ 0 | 0 / ( log 2 a 0, the potential H H η ( > is an integration constant. The Hamilton-Jacobi equation ( 0 = 2 0 1. H H K . Correspondingly, the Hubble parameter makes a transition from a larger value  η In the late time limit, the cosmological scale factor and scalar field are approximately For example, if we suppose that the flow of the scalar is = where the late time behavior in subsequentfor examples. the Note scalar that with field thisthen ( notation, allows the expression one tofor interpolate Γ between slow-roll behavior for Γ constants given by ( given by Integrating the expression for factor We see that for φ associated with the higherlower vacuum vacuum energy energy at at late early times, given times, by to a smaller value Taking where determines the scalar field potential to be then gives an then we find the Friedmann equation takes the form of a first order NLDE for A judicious choice of evolution for where the superscript prime denotes a derivative with respect to JHEP10(2017)118 π = = 8 c ∗ / over d ξ Λ 2 c (2.17) (2.15) (2.12) (2.16) (2.11) − πd = P = 4 ) numerically c τ A ( c ]), this reduces to d 38 ) (2.14) τ Θ( f 1 H which is gotten by using a 4 is its entropy, / c c r ) (2.13) ) + A τ τ = − Θ( ! by a blue line. It is interesting to c τ 2 . i f Θ( S ) 1 1 ) 0 ) captures the teleological behavior of 0 H  H τ τ τ respectively. It is satisfying to see that = 0. e τ ( ( ( i dτ c f − τ a a 3 c δP H r e ) 2 H ) + V f 3 ∞ πd τ 1 τ  τ ( H 4 = i or Z − a 1 c

i – 5 – H = Θ( π H δS τ ) = V ) = − c i τ τ T = H f ( ( = 4 at late times. These correspond respectively to the ) away ) is the corresponding proper distance 1 e c c ∗ c H τ r ), τ d f ( ( H is called the thermodynamic volume. The extended first τ ≈ c  c ( ) for the cosmological horizon radius δA , depicted in figure r ) d V a /H 1 + τ ( − f i ]. For the case of no black hole (see also [ a 2.11 1 H ) in this case gives an approximate expression for the evolution H = 1 12  c , ), where ) gives a smooth evolution between the initial Hubble horizon, d j ≈ 2.11 11 ) 2.9 τ ( c , then as the reception time goes to infinity the signal is received at a ∂/∂x ( d τ j increases to the transition time x j τ P ∗ H is the temperature of cosmological horizon, c shows the approximate expression compared to the exact one. We see in both − T ) at early times and 1 , and the final horizon, It is interesting to look at the change in the cosmological horizon area Evaluating the integral ( We are interested in the evolution of the future cosmological horizon. If a light signal i 1 − i /H ∂/∂τ is the cosmological pressure, and law, in this simplein case, cosmological relates constant. the For change a in de cosmological Sitter horizon spacetime the entropy to thermodynamic volume the is change equal to An extended firstconstant law was for derived in de [ Sitter black holes, including variation in thewhere cosmological the evolution from early to late times, which is given by the exact and approximate results1 that the cosmological horizon interpolatesKilling between horizons of the( early and lateeven time the de simple Sitter step-function phases, approximationthe for for horizon as the Killing vectors of the cosmological horizon radius figure compare this exact resultstep-function to approximation for the approximate solution for Performing the integral ( with the scale factorH ( is emitted at time co-moving coordinate separation The cosmological horizon radius JHEP10(2017)118 , δH (2.18) (2.19) + i H ] that the = 39 f H we will see that the 4 ) is given to leading order by , while the horizon temperature is 2.15 . c d ) that interpolated between two de Sitter φ 3 i ( φ 0 H πδH 8 W ) is satisfied. In section − – 6 – = ' − 2.16 ˙ c φ H δA 3 ). 1, then the change in horizon area (  2.16 ). If we consider a limiting case of our evolution such that i c πd . The actual cosmological horizon evolution for the Hamilton-Jacobi engineered potential (2 δH/H / For an analytic treatment we take the standard slow roll assumption [ = 1 previously assumed. scalar field evolution is friction dominated, so that In the previous subsection we assumedphases, a and form found for the exact potentialwas and a scale prescribed factor, flow such that fromto the the the cosmological maximum evolution true of vacuum theevolution. was potential overdamped to In rather the this than minimum. sectionwith oscillatory, The positive we corresponding approach minima, show to then that the a if behavior slow-roll of one the starts scalar with field a is qualitatively classic the double same well as that potential, first law for a slow-rollextension inflationary of spacetime ( with a black hole also obeys the2.2 appropriate Slow-roll analysis where and it is easily verified that the first law ( Figure 1 (blue) vs. the approximation (red-dashed). i.e. the volumeT of a Euclidean sphere of radius JHEP10(2017)118 . f . H τ Γ (2.23) (2.21) (2.22) (2.24) f ≈ ). The , H η is rolling H − e = 2.10 φ φ  τ 1 (2.20) Γ f 4 = 0. While  H ε 00   1. Since the time dependent W 2 ) have the same qualitative W
2 p 2 = 0 and a min at ) we can approximate  η φ 2.21 M , ε − , where 2.19  2 2 1 + tanh W φ , gives the standard relation between  ) = 2 τ/ η Γ φ 2 2 p 2 f ). Hence we see that for the double well ) and ( η 2 and the Hubble parameter in the final de 0 = p Γ( H ) type behavior, similar to the engineered f M τ 2 − τ p 2 = 2.7 φ Γ M W e η − Γ 3 4 1 2 f τ /M 16 τ/ = f H Γ – 7 – − f 2 f , 1 W H 1 + 1 H  e = ], the exponentially small size of the corrections to de  η  ) with ( f  f τ 13 ' 2 2 W 0 Γ φ , and we assume that Γ f 4 2.23 2 W p W H ) = 2 p in (  φ 2 /M ( M 2 τ Γ W 2 sech η ) = 2 2 φ η ∼ ( ε ε = is then 2 φ ], asymptotically future de Sitter (AFdS) spacetimes carry cosmic hair, φ be a double well potential with a max at ) is the slow-roll parameter evaluated at the minimum. The Einstein con- ) the scalar field has a tanh( 13 η W 2.21 is already first order, in its equation of motion ( φ The parameter Now let encoded in the exponentialof fall-off a terms black inwhile hole. for the AFdS spacetimes, metric, For these analogous blackboundary integrals conditions are to holes, established computed in the at the [ future ADMSitter infinity. ADM in charges With integrals this the AFdS regime are are computed compensated by at the spatial exponentially growing infinity, spatial volume, giving which has the same decayscale rate factor to approaches the its true final vacuum de as Sitter in form the2.3 with previous corrections example that ( Cosmic fall off hair like As shown in [ case. This makesfeatures sense for evolving since from the the maximum potentials to ( a minimum. At late times the field goes like Here an integration constant determining thehas transition the point freedom has to been replace potential set ( to zero, but one part of The solution for where Γ = Γ( straint equation evaluated at latethe times, effective cosmological when constant Λ Sitter phase, slowly, it will beeach approximately critical linear point in will cosmological be time, modified. although the behaviour near The second parameter Γof (related two) is to relevant the for standard evolution slow near roll a “eta” minimum parameter of by a factor Now define slow-roll parameters JHEP10(2017)118 )), φ ( . The at the U 2 p ], which i − φ 40 /M 2 f ˙ φ W While e.g. the = 2( 1 p 0) energy conditions, + 3 0 (2.26) ≥ ρ p ≥ becomes that of a damped ) ], we take the cosmological + 3 φ φ ( ρ 40 U − 0 (2.25) 2 ˙ φ ≥ 1 2 f = W and the approximate analytic solution for ) has a maximum and a minimum so that − ] for the asymptotically AdS case). It was φ ) ( 0) and strong ( 2.1 42 φ ( W ≥ – 8 – , p 0 W ρ ≥ ) ) = φ φ ( ( , the scalar field and metric indeed decay to the late time ] (see also [ U U 41 2.2 + 2 ˙ φ 1 2 = ρ , we find that the cosmological tensions will be finite, if the approach φ ], where we can compute the cosmological tensions associated with the ap- 13 0 the weak energy condition is satisfied. However, ] that cosmological tension in a given spatial direction captures the leading order ≥ 13 ], even though this work is often mischaracterized as a “cosmic no-hair theorem”. ρ 40 0, matter fields satisfying the weak ( Assume that the scalar field potential In both the engineered example in section In this paper, we consider transitions between early and late time de Sitter phases. Cosmic hair in the form of cosmological tension charges is fully consistent with the results of refer- 1 > cosmological tension charge. there can be two de Sitterence phases [ as the field evolves from an initial value not satisfied. Nonetheless, wetime find de that Sitter the vacuum metricthat exponentially and near fast. a scalar minimum field This ofsimple still decay harmonic the approach oscillator, is potential the with generic, the late dampingshow wave determined provided equation that by by for the in the Hubble the fact expansion. overdamped However, case we the detailed decay rate is too slow to result in a finite For the scalar field evolution the energy density and pressure are given by Since which is negative in the early time de Sitter phase, so that the strong energy condition is To align our scalarconstant field to be system given withscalar in the field terms then assumptions of moves of the in final [ the value shifted of potential the scalar potential Λ = the slow-roll example in section de Sitter vacuum exponentiallyshowed fast. that This an agrees initiallyΛ with expanding, expectations homogeneous based spacetime onand non-positive with [ spatial cosmological curvature, falls constant off to de Sitter exponentially quickly at late times. proach to the lateat time a de finite value Sitter of to limit. the minimum Assuming is that underdamped.Sitter the behavior However, minimum in fall the of off overdamped too the case, slowly, potential and the is the corrections cosmological to tension de charges diverge. correction of the scale factor in that direction toAlthough its it limiting diverts late us timetion briefly de how from Sitter cosmological behavior. our tension mainhere. behaves line These in of represent the an development, Einstein-scalarexplored interesting we field and in explore cosmologies qualitatively [ in different considered this set of sec- examples from those ADM mass and angulartranslation momentum and of rotation a symmetries, blackspatial translation cosmological hole symmetries. are tension It associated is isfor with associated analogous black to asymptotic with brane the time spacetimes asymptotic ADMfound spatial in [ tension [ charges defined finite results for what were referred to as cosmological tension charges. JHEP10(2017)118 ) we (2.31) (2.33) (2.27) (2.29) . 2.20 τ ln τ f , so that near f βH φ − 2. In this case the e /  3 / τ  ) 2 2Γ ) the Einstein constraint approaches the minimum − ˙ φ φ β − ( φ 2 1 2 − 1 W + (1 ). p f 2 2 (slow-roll) (2.34) ) ) H f  f e φ φ 1 2.24
) = 0 (2.28) ) (2.32) , as in the two examples above. We 00  f f − − τ f ! 3 2 ( φ (slow-roll) (2.30) τ is the dominant mode. From ( W φ φ f W ( ( δa − − = H then reduces to the ODE for a damped + 00 00 f f Γ β 1, giving 2 φ 2 f +  ( φ − W W τ/ τ e H ) at late times as 00  1 2 f f 1 2 f 2 f τ 2 2 − p H ( H W β , β + – 9 – Γ H + e a 1, the late time limit of the scale factor is then M . Overdamped oscillation results from real values 2 1 ) for − f τ f + 8 2 f φ e ' f − ˙  1 φ W W H ) 2.2 H β f φ − 2 3 p τ 2 1   ' ) and ( / β 1 H φ + 2 00 p τ a M ) f − determines the late time behavior of the scale factor. (

f at the minimum e φ 1 M φ τ ( 1 24 W φ + 3 φ f 3 f φ ¨ − φ φ H W = 2, and in this case e ' + / φ ) for 2 f 3 ) = φ ) from the late time de Sitter evolution is small. The leading  τ ) = 2.2 ≤ ), while the next order terms give τ ˙ ( a a τ ( ). Critically damped motion occurs when Γ = 3 (  δa ) = a δa τ 2.22 ( 2.10 φ to a final value i W is a constant and Γ = 2 1 φ approaches its value at the minimum of the potential . Assume that the second derivative of the potential is non-zero at The late time behavior of f , corresponding to Γ φ φ β in agreement with ( two roots coincide, and the second linearly independent mode goes like its minimum, as in ( In the slow-roll approximation, with Γ the evolution of the scale factor in this regime can be written as where the deviation terms relate the late time value of the Hubble parameter to the value of the potential at As equation becomes see that slow-roll evolution corresponds to Γ in agreement with the late time slow-roll evolution in ( where of To leading order thesimple wave harmonic equation oscillator ( which has the solutions want to solve equations ( at the minimum the scalar field potential can be approximated by maximum JHEP10(2017)118 2. we / 3 , and T (2.37) (2.36) (2.35) < L − β coincide, and  β  is the AdS length ` ωτ sin 2 to decay to the minimum 2 φ C + . The ADM charges are defined , so the resulting boundary term ωτ T ) as the fields oscillate around the ], and at this point there is a qual- y/` 3 τ τ 3 ) ( were coupled to other fields to reheat e cos 2 35 f − a 1 β φ H 2 2 [ will only produce monotonic scalar flows and C 2 1 / − . Since the tension involves an integration φ 3 + (3 and 2.1 9 4 . f T 00 f > φ H δa = e W 2 1 2 – 10 – ' φ L 2 T and p 2 p are constants whose precise expressions will not be 2 dv f 1 M 2 M ˙ a φ ds ωτ C δ 36 a −  sin φ τ f and 2 is a proper length radial coordinate and H 1 τ/ oscillatory, cases all decay at the same rate, which is precisely y 3 f C 2 − H e 3 − − e 1 1 this is finite and non-zero only for the critically damped case of , and φ , where 2  f and also leads to a divergent tension. Hence it is only the oscillatory τ + H is the area element on a constant time slice of the background de Sitter y/` f 3 2 f H τ − − φ e f τ/ → ∞ e 00 ), the key ingredient in the boundary integral is the behavior of a term like f f H = 3 H τ W 3 e ) = a − τ 2.33 = sets the magnitude of e = ( ) φ 2 2, whereas the integral diverges for all the overdamped cases which have 2 1 τ / ω φ dv Lastly, it is interesting to see how these different asymptotics look in AdS. In four di- To summarize, there are two qualitatively distinct ways for Underdamped oscillations occur when Γ Note that the Hamilton-Jacobi method inSince section the metric is isotropic, all three tensions, associated with invariance in the three spatial directions, = 3 2 3 − scale. However, the relevant areafor element the grows ADM mass like is finite. A significant amount of researchhence has will been not done generate on oscillatory the cases behavior such asare those the considered same. here. finite or infinite cosmological tension.the universe, Note this that if would also distinguish different physical mechanismsmensions for the the metric of reheating. a planartially AdS fast, black like hole approaches AdS at spatial infinity exponen- where of the potential, either with or without oscillations. These behaviors are distinguished by like (ln modes that yield a finiteover cosmological an tension infinite spatial area,the we finite make quantity the is spatialtension coordinates is the finite, periodic tension one with per then period finds unit area. For the underdamped case, where the where metric. As β However, in the criticallythe damped dominant case late the time two modes mode corresponding is to instead a second linearly independent solution that goes specify. We present thisaverage material over in a a period,solutions short and ( appendix the to oscillatory the terms paper. average to In zero. computing For the non-oscillatory where needed. The underdamped, that needed for a finite,in non-zero, terms cosmological of tension boundary integrals, which require a number of further definitions in order to itative change in theminimum aymptotic with behavior decaying of amplitude, JHEP10(2017)118 ]. ), W φ 24 ( W , ) is the . In the 2 − ` y/` β  (4 ∆ / 9 − e − = 0 (2.38) 2 BF < approaching a minimum. m 00 φ ∞ W = 2 0, where , m <  ] and systematically apply perturbation 9 ]. In these situations the potential for the / 30 ]. This formula is analogous to the scalar 2 ]. A similar situation arises in AdS domain 29 ` + 1 – 2 43 , ]. Accretion of fields and growth of cosmological 23 m – 11 – 14 , 2 BF ), with the substitution of a negative potential in 21 46 – m decays too slowly for the spacetime to have a finite 22 , 1 + 4 2.29 − 16 ≤ , p 16 , 2 14  ], which is simply SdS in cosmological coordinates in the 15 m 1 44  ≤ 3 2 = 2 BF ].  m 62 ∆ – 47 ] and with scalar fields [ , 45 38 , we analyzed de Sitter to de Sitter evolutions, with no black hole present. 31 2 approaches a maximum, for the positive potential with φ In this paper we follow the approach of [ the evolution of horizonWe areas, find temperatures, that thermodynamic the generalised volume,initial first and law SdS local of pressure. states. thermodynamicsit is satisfied Though is between the most the final results straightforwardhas and to are a make given maximum a in and thermodynamic terms a interpretation of minimum, in a the since general case then potential that the initial and final states are equilibria field cosmologies, generalizing theaccretion properties [ of Killing black holes, and thetheory dynamics and of the slow rollhorizons, approximation and to use the well-behaved Einstein-scalarthe coordinates field scalar equations, field on identify and the the thermodynamic horizons properties. to Our compute results the yield behavior analytic of expressions for the black hole and cosmologicalinclude horizons the evolve. McVittie Exact metric solutionsmost [ of physical dynamical case, black holes both and without examples [ ofblack maximally holes has charged been multi-cosmological studied in black different holes approximations and numerically, addressing scalar 3 Schwarzschild-quasi de Sitter spacetimes In section We now investigate the effects of a black hole on this transition, and in particular how two AdS vacua. The interpolationtwo between CFTs, AdS vacua and corresponds has toA an previously topic RG for been flow future between comparedscalar analysis to field is contribution de to to Sitter workroll form out to approach a the to de finite combination de Sitter generalized of Sitter. ADM transitions cosmological tension, tension [ appropriate with for a a slow The dominant far field modeADM with mass, ∆ as weAdS have case found a for finitegravitational the ADM and late charge scalar time can fields dewall be [ spacetimes Sitter in constructed behavior which by a with scalar combining field real contributions potential interpolates from in the the radial direction between with the constraint Breitenlohner-Freedman bound [ field decay in the cosmologicalwhich case ( cally de Sitter spacetime arescenarios, studies and of holographic AdS domain blackscalar walls holes field with [ is negative, scalar and hair, thean AdS field approaches domain-wall effective a negative maximum mass-squared. of the The potential, which two provides modes of the field decay like of scalar fields in AdS/ CFT, and particularly analogous to the issue of future asymptoti- JHEP10(2017)118 (3.8) (3.3) (3.4) (3.5) (3.6) (3.1) ] that treated black (3.2) ,V 2 ) p 0 φ 30 . Given that we have a ,U B M 0 φ ,U φ ) (3.7) 2 φ G 0 V ,V ( 0 F − B F 2 is a fiducial length scale that , = / + 2 II 1 2 0 2 + 2 / / / Ω ,V B 1 0 1 0 0 1 0 ν φ ,U B 2 B B B Bd ν e ,U ν B 2 √ 2 B e  − e ( 2 /  3 2 B 1 / − 2 1 2 2 p p B − ) = log dUdV − is the metric on the 2-sphere. This form of the 0 B U + ν /M /M B ( 2 2 II B 0 2 – 12 – − = 0”. Here, e 2 ,V )] and 2 ,U / ), which give the total change in the cosmological Ω ) give G 1 0 2 r d U / B V − ν ( B 1 Bφ + Bφ 2 3.6 B 3.45 G e B 0 r ) − − ) ) ,V 2 φ B p 2 φ p ), ( φ ( ,V ,U = 4 ( ) + B ( √ M M 2 B B V W ,φ ]). ) and ( W ( 3.42 ,V ,U plane. In the absence of any scalar evolution, the physical ds  W = 0” or “  F ν ν 63 [ 3.5 communicates with the remaining gauge freedom of conformal 1 2 , − = log U B ν 30 ν = = 2 = = 2 = 2 U, V 2 corresponds to the mass of the black hole as we now briefly review. = ), and ( B B ,UU ,UV ,UV ,UV ,V V ν φ B B B 3.40 , then to correct this perturbatively for a rolling are in principle arbitrary functions, expressed here as derivatives for con- φ ), ( G 3.37 function, since and − B F is constant, equations ( In this null gauge, the Einstein equations become For a slowly rolling scalar field, the spacetime will be adiabatically de Sitter (or SdS), φ where venience. From this, we deduce that If degree of freedom in (This discussion follows [ that are now locatedmaintains at dimensional consistency, “ and metric also clearly identifies theas main the physical degree oftransformations freedom in of the the gravitational field metric that encode the dependence on two parameters: but are nonsingular at the horizons, and indeed facilitate the analysis of horizon behaviour with a constant cosmological evolution in ‘time’, together withof the black our hole giving geometry, us a ourSince ‘radial’ dependence analysis the black should hole capturethe and the cosmological standard dependence event SdS horizons metric, onhole represent we these coordinate evolution follow two singularities for the coordinates. in a methodology scalar of with reference an [ exponential potential, using null coordinates for the equations ( and black hole horizonskip areas to respectively. those results The without reader loss uninterested of in continuity. thewith derivation small, can “time-dependent” corrections. The idea therefore is to first find a solution described by a static SdS metric. The main results of this section are summarized in JHEP10(2017)118 as } (3.9) , the (3.15) (3.11) (3.13) (3.14) (3.16) (3.10) (3.12) h r u, V with the ). Note, { h r . G ). i

r + ( N , and choose a 0 c r and )] , r N N ? ) ( r − r F N GM − = r r )] −

i . ? t = κ − r coordinates we recover 2 II ( = 8 r h log Ω − N µ κ )( t, r t r N ) h ( − Bd 1 ? c κ r r 2 κ ) . as − c − 4 r − 2 II + exp [ r h t µ N

Ω r ( h )( exp [ h d c 1 κ + c r 2 r + 2 dF dG h 2 r κ 1 ↔ 2 r h / − 2 − − 1  r ) − r r h B

( B = = ) + r 4 2 r 2 µ 0 2 c √ ( r r + 4 − log dr H ( N c 2 / h , u r − / − ,U ( 1 κ 3 − h 2 )] = – 13 – B )] 2 B r = 3 ? ,G ? c 2 2 p 2 r 0 p 0 + 2 r r dt ) r ) 2

? 2 0 M W + M W H r c + r 3 4 ( 3 t H r H t 4 ( c ( + N h r − c − = = is nonzero if a black hole is present. Substituting these − t κ κ 0 1 ( r = Λ = 3 µ are null coordinates, the metric still has coordinate singu-

 B is given explicitly by 2 r GM G ? GM exp [ exp [ 16 ds log 2 r ↔ − 2 , and the remaining zero of c h . It is then fairly clear that if we identify c − c 1 κ κ 1 F κ 1 r − B and 2 2 = 2 , and we identify the black hole horizon (if present) as √ 2 2 F p = = = ds = v M ) ) = 1 V 3 r r r ( ( / dr 0 c h N N are related to the physical parameters via r r W Z N → → = r r is the standard tortoise coordinate), then transform to ) can be integrated up to give 2 0 ) = are the usual surface gravities at the individual horizons, 2 • • ? r i r H ( 3.3 κ ? r Finally, although Comparison with the SdS metric suggests that we identify however, no global maximalare extension interested of in the the SdSnonsingular future coordinates coordinates, evolution however, is of in possible. the practicehorizons it spacetime, in Given is the we that easier local could we to Kruskals. choose analyse to physics use near the event larities at the black holelocally and by cosmological using event the horizons. standard These Kruskal can coordinates: of course be removed and the tortoise coordinate Here, cosmological horizon as the roots of We have written the SdS potential as where (where the SdS metric: radial coordinate advanced and retarded null coordinates where the integration constant expressions back into the metric gives Then ( JHEP10(2017)118 due φ (3.20) (3.18) (3.19) (3.21) . That ) is x 3.2 ) reduce to the 0 φ term is neglected. 3.6 ) to leading order  00 ) ,G φ )–( 2 3.2 ], it was found that in x r , 2 0 1 3.3 r 30 only depends on cosmo- + − c H r φ ) are assumed to be small ,F − − φ x ( . In [ 2 0 r = when the

− 0 W ] (3.17) r r φ log(1 0 0 ∂φ H ∂W X and 0 H H 0 − ), which was a generalization of the ,F G N t 1 and H x H − 2 = 2  t, r φ is predominantly a function of 1 1 + c ( ) = 4 ) as ] (3.22) −

+ κ x [ + φ r ,G t [ 2 00 ln φ ξ φ 0 dt/a 2 + + 1 r ,G H R ) an ODE for )] = tanh t 2 t 0 4 x ), and ,F 0 2 V – 14 – 0 N = = H φ 1 H ,F = ( H 3.20 η + x − x ) gives ). − 3.15 r 2 ? 2 as the black hole area goes to zero. U re r G 1 ). Here we maintain this notion of slow-roll, and look ), such that ( 3.9 ] = − τ 0 ), ( ) = − + p ) then relates these expressions to our canonical cosmo- ) X t, r H [ ,G r ( 3.19 = F ( φ − x ,F G B 3.11 ) = φ 3.16 N X + ln [ U 0 ,F − 1 φ H ( V − r 2 depends on a single function of 0. Using ( = = ( φ − ρ τ and using ( ) ) that reduces to r depended linearly on a variable ( is a function of U ,F G G < φ t, r φ r N (  = 0, and integrating ( + . Even though this is a more involved expression in the static gauge, we still x τ µ V F = X must be suitably chosen to render ( We now look for a variable In this slow-roll approximation, derivatives of Taking Setting x is, Now so let (recalling that quantities relative to the overallpure magnitude of cosmological the constant potential, equations thusrequires ( we only these have background just formsto of discussed, its the and potential metric ( functions in to the find presence the of evolution the of black hole. Substituting in these forms, ( have the notion that the slow-roll approximation for theof case a of black an hole exponentialcosmological scalar time potential, coordinate in in the ( presence for a similar In the cosmological slow-roll approximationlogical with time no black hole, logical coordinates (via conformal time ˆ where Prior to starting theto analysis our of previous an evolving discussionsolution black with is hole, cosmological expressed it in time. is staticthe gauge, useful With null as to the and we relate SdS have static discussed this black coordinates above, notation hole, relate but it to the is the natural useful cosmological to coordinates. see how 3.1 The scalar field behaviour JHEP10(2017)118 . ) 0 C H 2.16 (3.27) (3.28) (3.29) (3.24) (3.25) (3.26) (3.23) = γ 0 r r = 0, arose in the ln cosmological system h h 3 the thermodynamic r r c − / r ) h − 3 h r c r . r increases, however, it is easy to γ −  h 3 c ) , with − ) r 0 r 2 h

V ( r is switched on, the denominator π ∂φ N ) Nξ ∂W is larger for larger black holes at 0 r + h N r γ 2 c r = − = 4 r Nξ 0 ) ( r V φ 3 3 h V tot

− 2(1 + 0 r r 3 ) A 0 decreases as − ln ∂φ 3 ∂W − − r c h 2 = r ) = 3 c N − r r Nξ , h c 2 r + h 2 2 3 h h = 2(1 ( r r r r 0 B r r κ 2 W C ( motion. As 2 r ) = + 0 B B √ + − − x – 15 – c 2 ∝ + B ( √ 2 3 + c c 0 φ r 0 0 , i.e. 2 r r ( 00 ξ

2 0 ξ coordinate reduces to the expression for the cosmo- 2 h = 0. φ h ξ γφ r H r = 1 x h 3 2 c h ) to the cosmological slow-roll equation, where 2 + r r r γ − − −  r 2 0

N 3.28 ξ = 2 + = 1 N 2 ln ,F ,G N h = ) when x x 1 0 κ ξ ). The new + 3.19 ? ) is the total horizon area, and r term, the slow-roll equation becomes 2 3.15 h in ( r − 00 τ t φ + 2 c 0, the two slow-roll equations coincide, but we can now explicitly see the r = ( x → π h r Thus the effect of a black hole is to further slow down the slow roll inflation, = 4 ) becomes ). It will again enter the thermodynamics of the black hole is given in ( 4 ? ) decreases and the numerator increases, hence tot r 3.20 2.17 A It is interesting to compare ( This implies that We can now read off our requirement for slow-roll as 3.29 It is a little more subtle, since for fixed 4 effect of the black hole onin the ( friction for fixed Λ. check that the overall effect of increasing the black hole size is to increase thermodynamics of the cosmological horizon inand the ( absence of a blackin hole, equations subsequent ( sections. Clearly as where Here volume, of the de Sitter black hole system. The volume and dropping the where logical coordinate The final determination of theis determined constant of by the proportionality boundary andhorizon conditions the and that integration purely the constant outgoing field at is the purely cosmological ingoing horizon, at leading the to black hole and ( then, JHEP10(2017)118 ) ,V B = 0, ( (3.31) (3.33) (3.30) (3.34) (3.32) V ,u ) for the = 0, and B 3.5 u 2 p 2 2 p c h U c r r /M c factor, we must check r κ /M 2 ,U → → 2 1, at the black hole and 2 ,v /N r r Bφ Bφ h ≈ − − →  r − c ) 0 h 0 r c ,U ) as κ ,v r h 2 r → Nξ → δB − δB ≈ h − r ,U ,v N . Taking the local Kruskals at each M r ( c ) ν ν is in fact regular at the horizons, and r 2 r B h ( 00 r v U ) ) as N 2 h φ 2 c 0 c = 2 = 2 1 h κ − r κ r 4 term contains a 1 4 r ,vv ( − 00 ,UU r φ ( ≈ δB ≈ − GMH ≈ − δB – 16 – ] ), we see that ) U ) ? , i.e. fixed h r ) uv r r r ( ≈ r 2 h h − 2 ( (1 + UV ] 3.27 c κ κ 2 c ? 0 r rN 4 r κ 3( rN H c 4 − κ ( ), the cosmological event horizon is defined as = = exp [2 ∼ v ν ν 2 2 2 h 3.16 γ 1 e e κ exp [2 ) for the cosmological horizon, by using the fact that 4 U terms enter as a perturbative source. Lastly, we will need the − whereas the black hole event horizon is defined as c h 3.6 h c ], however, this process is rather involved, and somewhat specious r r φ r r = = U 30 for small and large black holes, demonstrates this effect clearly: → → V u → → r r γ    r r . . In the vicinity of the horizons of the background SdS we have: 0 • • v • • at each horizon: γφ ν |  00 φ | To get the horizon growth, the idea is to expand the Einstein equations ( Ideally, we would calculate the full gravitational back reaction throughout the space- Finally, given that the coefficient of the expression for is zero on the black hole (cosmological) event horizon in the background spacetime. Here the derivative of black hole horizon, and ( turns out, this isthe fairly horizon straightforward to (a extract. nullhorizon, surface) Note given is that in at in the fixed and equation background parametrised ( solution, by parametrised by is decreasing, which leadseffect the dominates, cosmological and horizon that to both grow. horizons grow. We willtime, see as that described the in second [ to the main theme of our discussion, namely the evolution of the event horizons. As it 3.2 Growth ofWhat the does event horizons the resultingthe evolution spacetime look are like? thehole Two areas is fundamental of accreting geometrical the properties the blackwill of scalar hole tend to and field, pull cosmological we in horizons. expect the the cosmological Since horizon. black the hole However, black the to effective grow. cosmological constant A bigger black hole that this term remainscosmological small. horizons, From thus ( the termthus multiplying overall stays small.long as It is therefore consistent to drop the second derivative term as Indeed, expanding and for black holes very close to the Nariai limit, the evolution becomes arbitrarily slow. JHEP10(2017)118 γ by and φ and (3.35) (3.38) (3.37) c c 0 , then δr c = 0, and , κ → c πr u → −∞ ) gives U A )+const., and U = 8 U 3.32 c c κ δA dφ . ln(2 ) (3.39) 1 1 0 f ∂φ ∂W − c W κ dU Z , and the change in horizon
− (3.36) 2 ∼ p i , which is as )] ) the quantities dx H i 0 dU )]) , x M 2 φ W 0 2 U c c ( = ) (3.40) ) U , ( r φ c ( f 3.40 f c | φ r 2 c r ) and figure p φ [ γκ [ 2 W W p κ 3 2 c | M W r , corresponding to evolution of c → , M − − W f = | 2.14 κ i i c − 1 r − i κ − /H W W | dx i ( ( 2 γ W 0 2 c > W 2 = p 6 W . The black hole horizon is at r φ = 1 ( i 0 2 2 p 0 M U c 2 = | p U c W Z φ r c Z u, v M – 17 – c A | 2 κ 2 M p p r 2 | c c 0, but as we go to future infinity, or c 1 U δB 2 r 2 2 c c γ κ 2 M p M | r r 3 2 c c γκ = 0, → γ M κ κ 3 2 c takes its initial value 3 | is diminished by a factor of approximately two-thirds. h = ) = r c r . Since − − − c c to clarify that the change in the horizon area is positive. c δB φ κ r = along the cosmological horizon. Hence ( | | 2 ( A p = = = c γ δA B 3 κ | δB /U /M ,U − 1 p ) I ,UU = δ ) = W c = κ ' − UδB c = − δB = 0 when ( UδB ,U I δr ( , and we get , in agreement with equation ( ν i 1 φ δB − i ⇒ H → . Looking at the limits, the effect is of course negligible for very small black h − φ r , 1 ). Integrating again gives − f i ) we have φ H ( and 3.34 f W → −∞ φ It is of interest to find how much the presence of a black hole affects the growth of the Finally, this gives for the change in the cosmological horizon area = U i → ical horizon, the total change in 3.2.2 Black holeThe event horizon analysis for thebut black now hole we horizon use area the is Kruskal similar coordinates to that for cosmological horizon, constants by using Λ rolling down the potential, the cosmological horizon grows. cosmological horizon, for a fixedall depend potential. on In equationholes, ( but as the size of the black hole horizon becomes comparable to that of the cosmolog- This can be written in terms of the change in the early and late time effective cosmological When there is noradius black is hole, then As φ This means that the total change in cosmological horizon radius is positive, and given by W where we have written Here we have set Starting with the cosmologicalfrom event ( horizon, as 3.2.1 Cosmological event horizon JHEP10(2017)118 ) (or as (3.47) (3.42) (3.41) )+const. 3.44 v h → ∞ κ v ln(2 ) and ( 1 − h  κ 3.39 h h 0 2 κ 2 p r 0 ∼ ) (3.44) 2 0, but as ), using the changes in v dv f ) M δW c x
r W → = 3.14 + − (0)]  i ) δB h φ ) [ r h h W ) (3.45) ) (3.43) r ( (2 δr f W f 2 ) p 2 c − 1 (3.46) 0 − W 2 h − W r , we have c M r φ c h r < h h c )] − 2 − 0 ( κ + r r r i v 2 2 i v | 0, we get h 2 γκ p ( ) c h 2 h r W H 6 W h φ κ r ( → κ [ ( M | r → 2 p 2 h 2 − p = r v W κ + = + (2 ) M h h M c c h h 2 h − ) r ) h r – 18 – r A r δB h c ∞ 2 δr v = γκ (2 ) γκ − Z 2 c 3 h /A /A is constant on the event horizon, and the perturbed 3  c r c r h v r ) = ,vv ) , rather than a constant value as is consistent with ν 2 = ( i p h = v + 2 δA r δA h δB ( ( c W ( M H 2 h r h 2 B δB r p − δA  ) imply = 0, which is equivalent to the observation that the event ((2 γκ f p 3 δ 4 2 γM p v W ( δW − H = 3.45 4 4 3 γM h = H H − δr = = δB 6= 0 at ) it follows that ) and ( Λ = δ ,v 3.34 becomes 3.40 δB B Now that we have the changes in horizon radii, as a check we can compute the change is the coordinate along the horizon. As as required. in Λ that wouldhorizon radius be computed required by integrating by the the horizon SdS radii, equations relation ( equation ( field. The fractional rate ofhorizon, area since increase ( is less for the black hole than for the cosmological nature of its definition. The corresponding change in the area of the black hole horizon is The area of the black hole horizon increases, as is expected due to accretion of the scalar One sees that horizon begins to move out before matter has crossed it, which follows from the teleological asymptotically dS boundary conditions.we As go to future infinity), This means that the change in black hole horizon radius is We have set ain constant the of integration area equal that to grows zero linearly that in would have lead to an increase From equation ( equation for which is solved by v JHEP10(2017)118 (4.3) (4.4) (4.1) (4.2) . However, c r 3. For a black / and are the initial and h . This dependence i r 2 p ), we can express the /M , and hence fixed values 1 3.14 1 and Λ ) can be written in terms i,f W h  ∼ W r i i ) ) h, c 3.45 grow? Using the formulae for h 2 c Λ Λ as well as the thermodynamic = r r 1 = h c 1 p p γ . For the black hole horizon, one r r + h h i,f f  ∼ 2 h + Λ i i r 2 h , α ) and ( ) of Λ Λ , r − )( r ) ) c i , is parametrically suppressed for small f i p p r + h h h Λ Λ 3.40 3.29 2 c + r /A − p ( h h h i , r r h Λ = Λ r , r | r ( δ (2 2 i | (Λ 2 δA Λ is obtained by interchanging 2 i – 19 – Λ δ Λ × | α for large ones. Likewise, one can ask how much | c is still dependent on Λ tot δ i | Λ i A π i c A Λ | δ π δA , r r | Λ , and ), but it is most useful to focus on the limits where 12 4 α i Λ | / h 3Λ | δ i | Λ V A ' ' √ Λ πT Λ δ ,Λ 3.14 | δ 2 h c h | = h r A h = δA 2 A α ' ' δA h δA 100 of the cosmological horizon area, one finds that the diminution δA / 10. / are the horizon temperatures and Λ | α κ | = α πT , how much does a black hole with initial radius i,f For small black holes,negligible. the While effect for of large thehole black with black holes, initial hole area it on 1 caneffect the be is cosmological a reduced factor horizon by of as growth 1 much is as 2 black hole. In thein limiting the cases cosmological of horizon small area and is large given black by holes, one finds that the change We see that the fractionalblack growth holes, in while area, itthe is cosmological of horizon is order “pulled back” by the black hole, compared to the case with no can be dealt withthe exactly black using hole ( horizonOne is finds either that in small these or limits, comparable the in change size in the to black the hole cosmological horizon horizon. area is given by change in the areas infinds terms that of while the corresponding expression for this is not quite what we want, since where 2 final values of the cosmologicalfor constant. Λ For athe fixed temperatures potential and areas in Schwarzschild-de Sitter spacetime ( Let us further analyze thefound results in for evolution the of last thevolume section. black and hole total Using and horizon cosmological theof horizons area, definition thermodynamic both ( quantities equations as ( In this section weevolution of explore black a holes numberhave established between of in initial aspects the and last of final section. the Schwarzschild-de4.1 thermodynamics Sitter of states the that Analysis slow we of roll horizon growth 4 Dynamical thermodynamics JHEP10(2017)118 ), V − (4.6) (4.8) (4.5) 3.40 ], and U 12 . 0 ν πP 2 8 B − , or the − e √ ) (4.7) φ Λ, i.e. a term of 1 ], which proposes  δ subspace, one has − uv ,v g 31 v and ( B B − θ − 4 ,u 2 / u B = and Λ = 1 − 0 I + dv B δS ν ∧ I 2 ,uv 2 − On the B e πT 5 − ,v  . This result is interesting because B = 8 B B δP V 4 ,u = ] due to using different signatures. I √ V B  31 δA | = ) = + dv I = 0, one finds that c γ κ ,v B | u ? d ? d (3 B δS √ c / B dv , ?du ) ) − T − c √ 2 − – 20 – φ p 2 ( = A + du M = h W + ,u ) from that in [ dyn h δS B − κ 4.6 ) is indeed satisfied for these evolutions. This requires h  A 2 2 T / ) has been used in obtaining the second line. One might 1 4.5 ? d − . One can take the difference of the two first laws, such that 3.6 − B = = h, c = B refers to the 2D spacetime perpendicular to √ ? α du , ?dv = ? d ? d ) on the black hole horizon at ? du − Λ, where 4.6 ), which implies that ( δ = α ), one can check that ( V is the thermodynamic volume between the black hole and cosmological horizons, ]. One relates the change in area of the black hole horizon to the change in mass, dyn 3.29 κ V 12 2 3.45 Here we are studying the evolution from one Schwarzschild-de Sitter spacetime to This formula differs by a minus sign ( 5 where the Einstein equationguess ( that for slow-roll evolution the black hole temperature would instantaneously be Evaluating ( where the Hodge dual part, and the right hand side is evaluated on the horizon. we return to in the discussion. 4.3 Temperature and massA definition for of evolving temperature black forthat dynamical holes black a holes generalized was surface discussed gravity in is [ use of ( a dynamical scalar fieldyet is our beyond results the for scope aapplied of black to applicability hole the of in differences the slow-rollsuggests derivation between inflation that in the the are [ first initial still law found and might to be final satisfy satisfied de continuously the along Sitter first the phases. law, evolution, an This idea agreement that which was introduced above, and we have set another, where the changein is the effected horizon byand the areas ( rolling computed scalar in field. the Combining previous the sections, changes and given in equations ( the form the mass term drops out giving where Two independent first lawsrived for [ asymptotically de Sitterwhile black the hole other spacetimes can relatesmass. be the Including de- the change possibility in ofhas a area an change of additional in the term the cosmological proportional cosmological constant, to each horizon of a to these thermodynamic laws the volume change times in 4.2 Two first laws JHEP10(2017)118 δB ), a ), at (4.12) (4.11) (4.13) (4.14) (4.10) 4.5 , so that ) pertur- 4.12 v h 4.8 κ 4 and ( −  ), the black hole = 2 h at that time, and δB r 2 h h / r  , where the dynamics of 1 3.13 )), is given by − 0 γκ 5 )] 6 v B ( and Λ taking their initial ν − φ 3.42 2 v h [ ) in ( − r 2 h r 2 e p ( r . For the evolving spacetime, i ,u δW ,v ) (4.9) h N M ,v r v W B ( − , see section δB Λ ) is true for our solutions, let us h δB 0 B v 2 r 2 0 h − v h ) 1 + 2 h r 4.5 v dv that interpolates between the initial h 2 κ v p r h  δP 1 ) (or from ( ( κ r )] h 0 M − v W = 0 and )]) = ) ( 2 3.41 p v v φ − V ' − ,v ( − [ ( M h 2 φ . Substituting this in to sds B ) [ qs / h v 1 δW δS ( 1 πT W − ). We can evaluate the last term in ( 0 πT h h – 21 – ∞ r T B πγκ − δW/v v ν 4.8 i 2 Z 24 =  − ' ) = 2 W ) = ] e ( v 2 p v 2 ( ( ,v 12 δM − M qs B 2 h B h dyn r 4 ,u πT δW/v γκ B πT sds,i 4 3 quasi-static temperature 2 T R − , which follows from ( 6 ' = δB qs ,v T ' δB ). However, since we found that ( 3 is the black hole thermodynamic volume, here given for SdS spacetime. ) also holds as well and see what it implies for the mass. This is equivalent / dyn 3 h 4.14 T 4.14 πr is the temperature of the initial SdS spacetime. = 4 sds,i h T V A definition of the dynamical mass can be found by considering the first law relating Now consider the late time behaviour of this temperature, when the integral in For more detailed discussion of the asymptotic behaviour of 6 dynamical cosmological constant due toderivation a of scalar ( field potentialassume that is ( beyond the scope of the the black hole system is computed in detail for a sample potential. black hole with positive Λ satisfy [ where As noted in the context of the first law formulated between the two horizons ( where the change in mass to the changes in black hole area and Λ. Perturbations about a static is approximately given by late times the dynamicalexpanded temperature to first is order given is by the quasi-static approximation, which where the derivative of So the dynamical temperature is given by then this matches the first twobatively. terms in In ( the background solution, one has this turns out totemperature be in Schwarzschild-de almost Sitter the isthis case. gives 4 the temperatures Using in the theand initial final function and final values. states, Defining where and a final values along a sequence of Schwarzschild-de Sitter spacetimes as that of a Schwarzschild-de Sitter spacetime with the value of Λ and JHEP10(2017)118 h ), r 2.21 (4.18) (4.19) (4.15) (4.20) (4.16) = 0 is directly u M ) as follows. 4.14 . Supporting this M δW 2 2 ) / δB 3  ). Explicitly, substituting B δB and Λ, and find the same ,v ( π =0 c 3 u 4 O vv ), one can vary

3.14 δB B T h , r  + − =0 h 2 u / r

Z vκ 3.47 3 ) (4.17) δB  2 p | 2 B − 1  / BδB 2 W Λ p M 3 ) gives Λ B B δ √ − | − B M W ( √ √ 2 3 δ p ) ) M = 4.14 2 2 2 πW φ φ p p / 2 h M − W ( ( 2 3 2 ,v r 3 h M M 3 B r − W W + c − πB Bφ into ( √ r 2 c 4 – 22 – B − −  r 2 B p Z 2 2 δ √ / / = √ 2 δB = p 1 1 G G  1 /M ), gives that to leading order the additional piece in 1 − − 4 2 ) M i δP G 1 B B h 3.5 2 δM − = =   W , this is precisely the definition of V 2 = = r − G G 1 1 δP 4 4 f ,v h = M V W = = δB is then given by the SdS relations ( B + f δS h M Λ = ( δS dyn δ ). It also suggests that the mass is given by T dyn ). T 4.15 ) and 4.15 in ( , that the metric can be put into static SdS form with the evolved values of 3.45 ), substitute in our results for the changes in f δM 3.14 . The final mass c from ( r Lastly, it is interesting to assemble the terms on the right hand side of ( h equal to a quantity defined on future spacelike infinity. 5 Illustrative example In this paper, weillustrate have derived these general with results a for test-case the example, accreting using black our hole. standard It double is well helpful potential to ( that is, the ‘mass’complete contribution due the to argument, scalar we accretion would onto want the to black show hole. that In the general, quantity to evaluated at In the static case,interpretation where is that integratingthe ( dynamical temperature is This quantity is definedequal on to the black hole horizon, and if our assumptions are correct, is so Meanwhile, Analogous to thefrom check ( we didanswer as on in the ( change in ΛWe have ( that effective Λ and δA to assuming that at late times, when the stress-energy is again dominated by the smaller JHEP10(2017)118 ), ) is (5.5) (5.6) (5.1) (5.2) (5.3) (5.4) 3.29 ), and 2.21 v h . Writing κ v h on the other κ ), for (  a φ log(2 a ( 0 = 2 , given in ( −  ˆ γ v ' ] v W − − ; x v, a h a [ˆ = κ I 0 ) a 2 h a ˆ v 2 r 1 + Γ v γφ ; shows a plot of the variation + 1 a < + 1)  2 ] , ) a 2 (2 + ˆ 1 a ) v a  v (ˆ a ˆ v v, a 1 + 2) v [ˆ + 1) 2 F I h η 2 h v (2 + ˆ Γ . Figure γ /r + 1 a 8 2 2 /r i c ∆ˆ (1 + ˆ 16 shows the variation of temperature. In − ˆ c v γ r a x/ r 2 H 1 Hx 3 Γ 3 x/ 2 i −  + ∼ Γ − H + 1) 1 2 2 h i a ] + 1)( e h ˆ v v  H 2 /r h h e a 0 2 – 23 – c γr 2 κ 1 + r /r A [0] = η 2 c (3 2 r + Γ( h dy = ) W ( = ) κ a log[2 a )) 2  y − = a y h φ ) ] πB H κ v π a h 2 φ ∆ i v [ a 32 H (1 + = 4 γκ (2 + W 2 ) = Γ 2 a (1 + ˆ + ? y , we see that A y a (1 + ˆ r = h ∼ ˆ v κ + init ∞ a a t ˆ v T ( δW Z (1 + H = and − − is directly dependent on the rate of slow-roll of the scalar field. Given represents the strength of the gravitational interaction of the scalar γ = = a dyn 2 ] = p T EF ) gives ˆ , we have V /M v, a h [ˆ 2 I η 3.42 γκ 2 / 2 i H It is perhaps most useful to display the variations of the black hole variables as a Having extracted the dimensionful dependence, we can now see how the various pa- = Γ of horizon area with advancedboth time, cases, while figure ratherThe large differing values gradations of of “slow-roll”cosmological ∆ on horizon and display ratios arise Γ — becausefor have recall of the been that the scalar the differing chosen in black true the to hole: slow-roll presence emphasize of parameter the the black effects. hole has a strong dependence on the geometry. function of the local (normalised)event Eddington-Finkelstein horizon, advanced time co-ordinate on the is clear that themagnitude overall of gravitational the strength variation ofhand of the not the only scalar, area scales ∆,be and the simply expected temperature. magnitude scales since of the The thesethe overall parameter expressions variations, for but also their rapidity, as would rameters impact the evolution of the scalar and hence the black hole horizon. First, it is a dimensionless integral. Meanwhile, the dynamical temperature is found to be Here ∆ = field, and a and hence ( parameters. Recall that the solution to the slow-roll equation, 3 We start bycompute focussing the dynamical on horizon the area and black temperature hole as a event function horizon of where ˆ which will allow us to explore the effects of varying the black hole mass and slow roll JHEP10(2017)118 EF  EF V  V =5 =3 =1 =7 h h h h =1 =3 =5 =7 r r r r h h h h r r r r 400 400 ) as a function of the Eddington- i A / h 200 200 A ) as a function of the Eddington-Finkelstein i T/T i i 0 0 – 24 – T/T / 1.0010 1.0005 1.0030 1.0025 1.0020 1.0015 0.9998 0.9997 0.9996 1.0000 0.9999 -200 -200 =1/175 2 =1/175 -400 2 Δ=1 Γ=0.05 H -400 Δ=1 Γ=0.05 H . Illustration of evolution of the horizon area ( . Illustration of evolution of temperature ( Figure 3 advanced time coordinate on thein black black hole hole horizon. horizon We see temperature that is for both larger larger black and holes the more decrease gradual than for small black holes. Finkelstein advanced time coordinate,We scaled see by that the for Hubble amore parameter, larger gradual, on initial than the black for hole, black the small hole increase black horizon. in holes. black hole horizon area is both larger and Figure 2 JHEP10(2017)118 – 25 – . This would facilitate analyzing the flow of energy-momentum x ]. Generally, an ADM charge corresponding to an asymptotic sym- and 13 r ), formulated between the two horizons, holds between the initial and final SdS 4.5 One of the interesting features of the solutions is that the first law of thermodynam- by the Province of Ontario through the Ministry ofA Research, Innovation and Science. Cosmological tension The ADM cosmological tensionwere charges constructed for in [ an asymptotically future de Sitter spacetime The authors would likeported to in thank part Kostas by STFCport Skenderis (Consolidated from for Grant the helpful ST/J000407/1). Wolfson Foundation conversations. RG andical also Royal Physics. RG Society, acknowledges and is sup- Research Perimeter sup- at Institutethrough Perimeter for the Theoret- Institute Department is of supported Innovation, by Science the and Government Economic of Canada Development Canada and nient for ascertaining how aas black the hole spectrum affects of the quantum important perturbations predictions and of reheating. inflation,Acknowledgments such time? In general, itacross would the be range advantageous of to havethroughout explicit the expressions volume, for to thenear-horizon full test metric, metric and the to definition followbetween of the the evolution dynamical of initial temperature the and by mass-likework quantity studying final is that to the mass interpolates transform parameters. the analysis to Another cosmological interesting coordinates, direction which are for likely further more conve- ing the stress-energy of ahole scalar and field, cosmological which horizons, generates and additional verifySecond, contributions that this at these result the contributions suggests vanish black for thatand our the solutions. late first law time might SdS bea metrics, satisfied solution not but for only continuously between the along the metric the early functions evolution. which More illustrates strongly, that is the there metric is quasi-SdS at each Using a proposed definitionare of found dynamical to temperature, decrease between the the temperatures initial of and each finalics horizon values ( as the horizonsstates. grow. This brings up two questions for further study. One is to derive the first law includ- with approximate Killing horizons andtions associated we have temperatures. solved the Within Einstein thesehole plus approxima- and scalar field cosmological system horizons tominimum. for extract the a The growth results general of the are scalarconstant, black as expressed potential dictated in that by terms has the of a potential, the maximum and change and geometrical properties in a of the the effective initial cosmological spacetime. Central concepts in gravitational thermodynamics arethe horizon relations areas between them. and temperatures, The and analysisin presented a here “mildly” allows the dynamical study setting, ofSdS namely these with quantities the a evolution from smaller oneAn cosmological SdS advantage constant, of spacetime these in to boundary a a conditions second perturbative is and that slow-roll the initial approximation. and final states are equilibria 6 Concluding remarks JHEP10(2017)118 . a ). 1. n ab ab s − ] to ¯ π (A.1) (A.2) , = 13 ab a s n a ], then the n 13 ). In the over- ]. We follow the 64 2.37 ) b F b a = 0. p ¯ D is then defined by an integral 2 a ab a β − h ξ c a n bc B + c h ab F s ac da a π ¯ + 2¯ D ∞ b Σ h − ∂ n a Z b − such that a n ) ¯ ) s and taking Σ to have a unit spacelike normal, a ab n ) denote the Hamiltonian initial data on a slice πG β cd 1 ξ h · – 26 – h b ab one only needs the symmetry (and the foliation) 16 + n ¯ cd D a , π − π Q (¯ 1)-dimensional hypersurfaces Σ with unit normal − = ( ab b ]. s F n h ], which we briefly outline here. The construction starts β − ) = ab a s 41 ξ g = 41 ¯ 1 ( D D Σ is in the asymptotic future. The resulting ADM charge √ ) for the underdamped asymptotically de Sitter case, and a ), which permits any use, distribution and reproduction in ( Q ξ ∂ F + A.2 denotes the induced metric on the slice(s), satisfying the or- = = 0. Let ( a ab b B s n ab CC-BY 4.0 s be a Killing vector of the background, which we project along the slice a This article is distributed under the terms of the Creative Commons 1 and ξ  ) be perturbations linearized about the background denoted by (¯ ab = 2)-dimensional boundary at infinity on Σ given by , p a − ab n is the covariant derivative operator compatible with the background metric ¯ a h D n a = 1. Then the boundary ¯ D If the spacetime is anisotropic but homogeneous, as was considered in [ The ADM mass results when a spacetime has an asymptotic static Killing field at The ADM charge corresponding to the Killing vector a n a damped and critically damped cases,growth of the the metric functions volume element, decay and too the slowly to tensionOpen balance diverges. Access. the Attribution License ( any medium, provided the original author(s) and source are credited. tensions are distinct. Thehomogeneous, inflationary so spacetimes there studiedprocess are in the this three boundary paper equal term areaveraging tensions. ( isotropic over and a Using period the of formulae oscillation, derived gives the in cosmological [ tension ( spatial infinity and choosing ΣOn to be the a other timelike slice hand,asymptotic with spatial unit the translation timelike construction Killing normal, field cann be used foris a a cosmological cosmological tension. spacetime with an and Note that to simplyasymptotically. define the However, if charge thethen symmetry be holds used througout to the prove spacetime, a this first set-up law can [ where over a ( where thogonality relation Σ and ( Furthermore, let Σ and its normal according to general prescription presented in [ with foliating the spacetime bysuch ( that the metric can be decomposed as metry of a spacetime is defined via Hamiltonian perturbation theory [ JHEP10(2017)118 , ] ]. D 2 + 1 Class. SPIRE , ]. Class. 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