Evolving black holes in inflation

ACFI-T18-05 DCPT-18/13 Evolving Black Holes in Inflation Ruth Gregorya,b ,∗ David Kastorc ,† and Jennie Traschenc‡ arXiv:1804.03462v2 [hep-th] 3 Jul 2018 a Centre for Particle Theory, Durham University, South Road, Durham, DH1 3LE, UK b Perimeter Institute, 31 Caroline St, Waterloo, Ontario N2L 2Y5, Canada c Amherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts, 710 N Pleasant St, Amherst, MA 01003, USA (Dated: July 5, 2018) Abstract We present an analytic, perturbative solution to the Einstein equations with a scalar field that describes dynamical black holes in a slow-roll inflationary cosmology. We show that the metric evolves quasi-statically through a sequence of Schwarzschild-de Sitter like metrics with time dependent cosmological constant and mass parameters, such that the cosmological constant is instantaneously equal to the value of the scalar potential. The areas of the black hole and cosmological horizons each increase in time as the effective cosmological constant decreases, and the fractional area increase is proportional to the fractional change of the cosmological constant, times a geometrical factor. For black holes ranging in size from much smaller than to comparable to the cosmological horizon, the pre-factor varies from very small to order one. The “mass first law” and the “Schwarzchild-de Sitter patch first law” of thermodynamics are satisfied throughout the evolution. ∗ r.a.w.gregory@durham.ac.uk † kastor@physics.umass.edu ‡ traschen@physics.umass.edu 1 I. INTRODUCTION The dynamics of black holes in cosmology is important for understanding several questions about the very early universe. During very hot phases a population of black holes will affect the behavior of the plasma and impact upon phase transitions, and could even have a significant effect on the the geometry near the big-bang. On the late time, or ‘cooler’ side, there has been interesting recent research [1–4] about the possibility of an early population of black holes that could seed galaxies and provide progenitors for the larger black holes detected by LIGO [5, 6]. Such situations are highly interactive involving classical accretion, the expansion of the universe, as well as classical and quantum mechanical radiation exchange. Known analytic solutions include the stationary Kerr-Reissner-Nordstrom de Sitter metric [7], the cosmological McVittie black hole spacetime [8], which has an unphysical pressure field except when it reduces to Schwarzschild-de Sitter [9], and cases of multi, maximally charged, black holes [10–13] that exploit fake supersymmetries [14, 15]. A range of approximations and numerical analyses have been used to study accretion and estimate the growth of black holes in these interactive systems, including [16–39]. Many of these studies incorporate test stress-energy and infer time rate of change of the black hole mass by computing the flux of stress-energy across the horizon. In this paper we continue our studies of the evolution of black holes in inflationary cosmologies by solving the full coupled scalar field plus Einstein equations in the slow-roll and perturbative approximations. This system has the advantage that it is only “mildly dynamical”, and quantities can be computed in controlled approximations. While mild, we anticipate that the solution presented here will be useful for computing observational signals of black holes that are present during the inflationary epoch, and helpful in developing techniques applicable to hotter early universe situations. Slow-roll inflationary cosmologies are described by quasi-de Sitter metrics, in which the cosmological constant, Λ, is provided by the potential of a scalar field φ, and Λ slowly evolves in time as φ rolls down a potential. If a black hole is included, one thinks about the spacetime as being approximately described by quasi-Schwarzschild-de Sitter metrics, with both Λ and the mass parameter M changing slowly. But is this picture correct? Here we answer this question in the affirmative. We show that the black hole plus scalar field system evolves through a sequence of very nearly Schwarzschild-de Sitter (SdS) metrics as 2 the inflaton rolls down the potential. In earlier work, Chadburn and Gregory [28] found perturbative solutions for the black hole and scalar field cosmology, as the field evolves slowly in an exponential potential. Their null-coordinate techniques were particularly useful for analyzing the behavior of the horizon, and computing the growth of the horizon areas. Recently we extended the work of [28] to general potentials [40], with a focus on evolutions that interpolate between an initial SdS and final SdS with a smaller cosmological constant Λ. The analysis yielded geometrical expressions for the total change in the horizon areas. Further, it was found that the “SdSpatch” first law was obeyed between the initial and final SdS states. The SdS-patch first law [41] only involves quantities defined in the portion of the spacetime between the black hole and cosmological horizons, see equation (9) below. That result supports the picture of a quasi-static evolution through SdS metrics, which we address in detail in this paper. Consider Einstein gravity coupled to a scalar field with potential W (φ) governed by the action S= Z 4 √ d x −g  Mp2 R − (∇φ)2 − 2W (φ) 2  (1) where Mp2 = 1/8πG is the reduced Planck Mass in units with ~ = c = 1. Varying the action yields Einstein’s equation Gab = Tab /Mp2 with the stress-energy tensor given by  1  Tab = ∇a φ∇b φ − gab (∇φ)2 + 2W (φ) 2 (2) and the scalar field equation of motion is given by g ab ∇a ∇b φ = ∂W . ∂φ (3) The coordinate system used in [40] was chosen to facilitate analysis of the horizons and clarify the degrees of freedom of the metric, but did not prove amenable for finding the full time-dependent corrections to the metric throughout the SdS patch. In this paper we aim to find the metric in a transparent and physically useful form. In particular, the metric ansatz ultimately promotes the SdS parameters Λ and M to time dependent functions Λ(T ) and M (T ) in a natural way. The solution determines dΛ/dT and dM/dT , which are found to be proportional to (dφ/dT )2 . Additional analysis gives the rate of change of the area of the black hole horizon Ab to be Ab dAb = dT κb Mp2 3  dφ dT 2 (4) where κb is the surface gravity of the background SdS black hole, and T a suitable time coordinate to be identified. A similar relation holds for the cosmological horizon, which also grows in time. In the slow-roll approximation, dφ/dT is proportional to ∂W/∂φ, so the complete solution is known once the potential and the initial black hole area are specified. It is then shown that both the SdS-patch first law, and the more familiar mass first law, are obeyed throughout the evolution. The analysis starts by asking: “Who sees the simple evolution?” In Section II we review basics of the static SdS spacetime, and then take some care in choosing a natural time coordinate T such that for a slowly rolling scalar, φ only depends on T . Transforming SdS to the new coordinate system then facilitates finding a useful anzatz for the time dependent metric. The necessary conditions on the potential for this approximation to be valid are found in §II C. In Section III the linearized Einstein equations are solved. Section IV analyzes the geometry near the horizons. In Section V the rate of change of the horizon areas is found, both the SdS-patch first law and the mass first law are shown to hold throughout the evolution, and a candidate definition of the dynamical surface gravity is computed. Conclusions and open questions are presented in Section VI. II. SCHWARZSCHILD-DE SITTER WITH A SLOWLY EVOLVING INFLATON Finding the metric and scalar field of a black hole in general in an inflationary cosmology is a complicated dynamical problem, even in the spherically symmetric case. On the other hand, in the case that the inflaton and the effective cosmological constant change slowly, there is an expectation that the metric evolves quasi-statically through a sequence of SdSlike metrics, with Λ approximately equal to the value of the potential at that time. In this paper we address this expectation, and show that the picture is remarkably accurate within the slow-roll and perturbative approximations, with some small adjustments. To be concrete, we assume that the scalar potential W (φ) has a maximum where the field starts, and a minimum to which it evolves, so that the metric is initially SdS with an initial value of the black hole horizon area, then evolves to an SdS with different values of Λ and the black hole area. The first step is to come up with a workable ansatz for the metric that is compatible with a quasi-SdS evolution. The key question is then to whom does the dynamical spacetime 4 look simple? Analytically, the equations are likely to be simpler using a time coordinate T in which the scalar field only depends on T . This would imply that the potential W (φ) also only depends on T , and hence is consistent with the idea that the potential acts as a slowly changing cosmological constant with W (φ) ≃ Λ(T ). In the initial unstable SdS phase φ is a constant, so a time-dependent φ would then be first order in a perturbative expansion. Hence when solving the wave equation for φ(T ), the metric takes its background or zeroth order values, which allows us to find the preferred coordinate T without knowing the back-reacted metric. In this section we review some required properties of SdS metrics, then analyze the slowroll wave equation in SdS to find T . We then transform SdS to this new coordinate system, and base our ansatz for the back-reacted metric on this new slicing of SdS, as we know it is compatible with an evolution of φ that only depends on T . Finally, we identify the conditions for an analog of the slow roll approximation to be valid. A. Schwarzschild-de Sitter spacetime The starting point for our construction is Schwarzschild-de Sitter (SdS) spacetime dr2 + r2 dΩ2 , ds = −f0 (r)dt + f0 (r) 2 2 f0 (r) = 1 − 2GM Λ − r2 r 3 (5) where M is interpreted as the mass, and Λ is a positive cosmological constant. The coupled Einstein-scalar field system (1) will have SdS solutions if the potential W (φ) has a stationary point φ0 , such that W (φ0 ) > 0. SdS solutions then exist having φ = φ0 everywhere and Λ = W (φ0 )/Mp2 . The SdS metric is asymptotically de Sitter at large spatial distances. For GM ≤ √ Λ/3, the SdS metric function f0 (r) has three real roots, which we will label as rc , rb and rn , and the SdS metric function can be rewritten as f0 (r) = − Λ (r − rc )(r − rb )(r − rn ) 3r (6) The absence of a term linear in r in f0 (r) implies that rn = −(rb + rc ), and we will assume that rc ≥ rb ≥ 0. The parameters M and Λ of the SdS metric are then related to these roots according to GM = rc rb (rb + rc ) , 2(rb2 + rc2 + rb rc ) 5 Λ= 3 rb2 + rc2 + rb rc (7) Provided also that M > 0, the SdS metric describes a black hole in de Sitter spacetime, with black hole and cosmological Killing horizons at radii rb and rc respectively. Letting the subscript h denote either horizon, the surface gravities κh at the two horizons are found from the formula κh = f0′ (rh )/2 to be κb = Λ (rc − rb )(2rb + rc ) , 6rb κc = − Λ (rc − rb )(2rc + rb ) 6rc (8) where we note that κc is negative. The horizon temperatures are given by Th = |κh |/2π and the horizon entropies are given by GSh = Ah /4 = πrh2 . The thermodynamic volume V is another relevant thermodynamic quantity, arising as the coefficient of the δΛ term when the first law of black hole thermodynamics is extended to include variations in the cosmological constant [42]. See [43] for an excellent review on this topic. As shown in [41], two different first laws may be proved for de Sitter black holes. The derivation of the first, and less familiar, of these focuses on the region between the black hole and de Sitter horizons, and relates the variations in areas of the two horizons to the variation of the cosmological constant: Tb δSb + Tc δSc + Mp2 VdS δΛ = 0 . (9) We refer to this statement as the SdS-patch first law. It holds for arbitrary perturbations around SdS that satisfy the vacuum Einstein equations with cosmological constant Λ. The thermodynamic volume VdS for an SdS black hole works out to be simply the Euclidean volume between the horizons, which can be written equivalently in the two forms VdS =
4π 4π 3 rc − rb3 = (rc − rb ) 3 Λ (10) In particular, even though the de Sitter patch first law (9) does not include an explicit δM term, it holds for perturbations within the SdS family in which both the parameters M and Λ, or equivalently rb and rc , are varied, as can be verified straightforwardly using the formulae above. An alternative approach to a de Sitter patch first law is contained in [44], in which the volume is varied. An SdS patch Smarr formula [41, 45], can be obtained by integrating the first law (9) with respect to scale transformations, giving (in a general dimension D) (D − 2)Tb Sb + (D − 2)Tc Sc − 2Mp2 VdS Λ = 0 (11) where the factors of (D − 2) and −2 arise respectively from the scaling dimensions of the horizon entropies and cosmological constant. In this paper we will be working in D = 4. 6 (See also [46] and references therein for earlier “phenomenological” arguments for an SdS Smarr relation.) A second, independent first law, which we call the mass first law, can be derived [41] by considering a spatial region that stretches from the black hole horizon out to spatial infinity, and is given by δM − Tb δSb + Mp2 Vb δΛ = 0 (12) The thermodynamic volume Vb that arises in the mass first law is given in SdS by the Euclidean volume of the black hole horizon1 , Vb = 4π 3 r . 3 b Integrating the mass first law (12) also gives a second independent Smarr formula (D − 3)M − (D − 2)Tb Sb − 2Mp2 Vb Λ = 0 B. (13) Convenient time coordinate for slow roll dynamics As in [40], we now suppose that Λ is an effective cosmological constant that varies in time as a scalar field, described by the action (1), evolves in its potential W (φ). Our first step is to identify a preferred time coordinate T , such that the value of the scalar field throughout the spacetime depends only on T , and hence makes viable a scenario in which the effective cosmological constant Λ also only depends on this time. Such a time coordinate was identified for perturbative slow-roll evolution with an exponential potential in [28], and for a general slow transition in [40],       1 r − rn rc 1 rb rc r rc − r r − rb 1 + + ln ln ln − + ln (14) x = t− 2κc rc 2κb rb 2κb rb 2κn rn rc − rb r0 where rn = −(rb + rc ), as above, is the negative root of the SdS metric functionf0 (r), κn = f0′ (rn )/2 is the surface gravity evaluated at this root, and r0 is an arbitrary constant of integration. Starting from first principles we review the argument that leads to our choice of a new time coordinate, which then gives SdS in an interesting new set of coordinates. Let us begin 1 A third first law, which we call the cosmological first law, can also be obtained by considering the spatial region stretching from the cosmological horizon out to spatial infinity. The cosmological first law has the same form as (12) with the subscript b replaced by c, and with the corresponding thermodynamic volume given in SdS by Vc = 4π 3 3 rc . However, only two of the first laws are independent. For example, subtracting the black hole and cosmological horizon versions, the mass term drops out giving the relation (9) that only involves the geometry of the horizons. 7 by making a coordinate transformation of the SdS metric (5) T = t + h(r) , (15) where, for now, h(r) is an arbitrary function of radius. The SdS metric then has the form2  1  ′ 2 2 2 ′ 1 − (f0 h ) dr2 + r2 dΩ2 (16) ds = −f0 dT + 2f0 h dT dr + f0 We look for solutions to the scalar wave equation (3) in these coordinates with φ = φ(T ), so that the wave equation then reduces to − dW 1 1 2 (1 − (f0 h′ ) )φ̈ + 2 ∂r (r2 f0 h′ )φ̇ = f0 r dφ (17) Assuming that the scalar field evolution may be approximated as a slow roll, we neglect the φ̈ term on the left hand side. Under the assumption that φ = φ(T ), the right hand side of the wave equation depends only on T and hence it is necessary that 1 ∂r (r2 f h′ ) = −3γ 2 r (18) where γ is a constant to be determined, and the factor of −3 is included for convenience. The wave equation then becomes dW , (19) dφ giving 3γ the interpretation of a damping coefficient. Equation (18) can be integrated to − 3γ φ̇ = obtain the required function h(r) in the coordinate transformation (15), with the result   β 1 ′ −γr + 2 (20) h = f0 r Here, β is a further integration constant that is determined, along with γ, by regularity at the two horizons3 . Specifically, we require that a solution φ(T ) be an ingoing wave at the black hole horizon and an outgoing wave at the cosmological horizon, so that the scalar field is regular and physical in the SdS bulk. These conditions can be understood in terms of Kruskal-type coordinates which are smooth at the horizons Uh = − 2 3 1 −|κh |(t−r⋆ ) e , |κh | Note that for the Schwarzschild metric, the choice dh dr f Vh = 1 κh (t+r⋆ ) e κh (21) = ±1 corresponds to outgoing/ingoing Eddington- Finkelstein coordinates. The constants β and γ will be key features of the slowly evolving black hole solutions found in Section III, determining the relation between the rates of change of the mass and cosmological constant. 8 where the subscript h = (b, c) designates either the black hole (b) or cosmological (c) horizons, and r⋆ is the tortoise coordinate defined by dr⋆ = dr f0 (r) (22) The black hole horizon is located at r⋆ = −∞, while the cosmological horizon is at r⋆ = +∞. It follows that Ub = 0 on the black hole horizon, with Vb a coordinate along the horizon. For the cosmological horizon, the situation is reversed with Vc = 0 on the horizon and Uc a coordinate along the horizon. Hence a regular solution for the scalar field must behave like φ(T ) ≃ φ(t + r⋆ ) near the black hole horizon, and φ(T ) ≃ φ(t − r⋆ ) near the cosmological horizon. This implies that the function h(r) in the coordinate transformation (15) has the boundary conditions h(r) ≃ +r⋆ and h(r) ≃ −r⋆ as r approaches rb and rc respectively. It follows from the definition of the tortoise coordinate (22) that the derivative of h must behave like ′ h (r) ≃      +1 , 2κb (r−rb ) r → rb −1 , 2κc (r−rc ) r → rc (23) On the other hand, equation (20) implies that as r approaches rb or rc , we must have     β  +1 −γrb + r2 , r → rb 2κb (r−rb ) b (24) h′ (r) ≃    β  +1 −γrc + 2 , r → rc 2κc (r−rc ) rc Comparing these two conditions we see that the constants β and γ must satisfy − γrb + β = 1, rb2 −γrc + β = −1 rc2 (25) which we can solve to obtain γ= rc2 + rb2 , rc3 − rb3 β= rc2 rb2 (rc + rb ) rc3 − rb3 (26) Having found these expressions for γ and β, we can now integrate equation (20) and determine the function h(r) that specifies the time coordinate T . After some algebra, one finds that T agrees with the coordinate x in (14), discovered in the specific case of an exponential scalar field potential [28]. Combining the formulae (16) and (20) now gives the SdS metric in the stationary patch coordinates (T, r), ds2 = −f0 dT 2 + 2η0 drdT + 9 dr2 (1 − η02 ) + r2 dΩ2 f0 (27) T v=const. u=const. Black Hole Horizon Cosmological Horizon r FIG. 1. A plot showing how the null lines u = t − r⋆ =const, v = t + r⋆ =const appear in the (T, r) coordinate system. Note how the v-lines meet the black hole horizon and the u-lines the cosmological horizon at finite T . where f0 h′ ≡ η0 (r) = −γr + β r2 (28) with γ and β as given in (26). Noting that η0 (rb ) = 1 and η0 (rc ) = −1, one sees that the boundary conditions on the scalar field have yielded the result that near the black hole horizon, the time coordinate T coincides with the ingoing Eddington-Finkelstein coordinate, while near the cosmological horizon it becomes the outgoing coordinate. Hence the (T, r) coordinates are well behaved in the region between and on the horizons. Figure 1 shows schematically how the null (t ± r⋆ ) coordinates (u, v) appear in this alternate system. These properties will be useful when analyzing the near-horizon behavior of our dynamical solutions in Section IV B. The constants γ and β are both positive, and the damping coefficient 3γ in the evolution equation (19) for the scalar field has a simple geometrical interpretation as the ratio 10 of the sum of the black hole and cosmological horizon areas Atotal = 4π(rc2 + rb2 ) to the thermodynamic volume VdS = 4π(rc3 − rb3 )/3 from the SdS patch first law (9), 3γ = Atotal . VdS (29) as was noted in [40]. The damping of the scalar field evolution is greatest for small VdS , which corresponds to a large black hole with radius approaching that of the cosmological horizon. C. Interlude on slow-roll approximation Before proceeding with finding the solution, we outline more precisely the approximations within which we will be working. The resulting conditions on the potential are essentially the same as those in slow-roll inflation without a black hole [47, 48], summarized in equations (32) and (34) below. The picture is that we have a scalar potential with a shallow gradient so that the kinetic contribution of the scalar energy momentum tensor and the time-dependent corrections to W are sub-dominant to the zeroth order contribution from the potential. In inflation, we summarise the slow-roll conditions in terms of the acceleration of the universe: “Ḣ/H < 1”. Here however we are in stationary coordinates for SdS spacetime, therefore we must identify our slow-roll in a slightly different way. However, the useful parameters turn out to be the same as the conditions familiar for typical inflationary models. Similar to slow roll inflation, we will demand that the stress-energy is dominated by the contribution of the scalar potential, and that the φ̈ term be subdominant to the φ̇ term in the scalar equation of motion. That is, 1 − η2 2 φ̇ ≪ W , f 1 − η2 1 φ̈ ≪ 2 f r
′ r2 η φ̇ (30) Since φ̇ is assumed perturbatively small, the metric coefficients in the above relations can be taken to be their zeroth order values. Note that (1 − η02 )/f0 is monotonically decreasing between the two horizons, and (r2 η0 )′ /r2 = −3γ is constant. Hence we can manipulate the first relation to give ′2 1 − η02 φ̇2 η ′ (rb ) φ̇2 2 (2rc3 + 3rc2 rb + rb3 )(rb2 + rc2 + rc rb ) W ′2 2W ≪ 1 (31) <− = ≃ M p f0 W κb W 3Λ (2rb + rc )(rb2 + rc2 )2 W W2 11 suggesting that we use the slow-roll parameter ε= ′2 2W Mp 2 W ≪1 (32) to quantify how “small” the kinetic term of the scalar is: φ̇2 . εW . For the second relation, we take the leading order approximation to the φ−equation, (19), and differentiate to find φ̈, giving ′′ 2 (2rc3 + 3rc2 rb + rb3 )(rb2 + rc2 + rc rb ) ′′ 1 − η02 φ̈ 2W < W ≃ M ≪1 p 3γf0 φ̇ 3Λ (2rb + rc )(rb2 + rc2 )2 W (33) suggesting again that we use the parameter δ = Mp2 W ′′ ≪1 W (34) to quantify how small the variation of the kinetic term is: φ̈ . φ̇δ/rc . Note that this is again similar to the ‘eta’ parameter of the traditional slow-roll approach, here renamed as δ. We see therefore that the slow-roll relations are more or less the same as for the FRW cosmology without a black hole, however, the black hole introduces a radial dependence in the stationary patch coordinates that is easily bounded. The main difference between slow-roll with and without a black hole is the γ−parameter responsible for the friction of the √ scalar field. Since Λ = O(rc−2 ), γ = O( Λ)/(1 − rb /rc ) can be very much greater than the Hubble parameter that usually serves as the friction term in inflationary slow-roll, although it should be noted that the time with respect to which the scalar is rolling is the stationary patch time coordinate T . III. SOLUTION FOR THE DYNAMICAL METRIC In the last section we analyzed the slow-roll evolution of a test scalar field φ with potential W (φ) in a background SdS spacetime. We found a suitable time coordinate T , such that there exist solutions φ = φ(T ) that are purely ingoing at the black hole horizon and outgoing at the cosmological horizon. Our next goal is to solve the back-reaction problem in perturbation theory to obtain the leading corrections to the SdS black hole metric as the scalar field rolls down its potential. We will present a preview of the results here and then provide their derivation in the next subsection. We start by making an ansatz for the form 12 of the metric, taking the form of the SdS metric in the (T, r) coordinates (27) but allowing the metric functions to depend on both the T and r coordinates ds2 = −f dT 2 + 2ηdrdT + dr2 (1 − η 2 ) + r2 dΩ2 f (35) where f = f (r, T ) and η = η(r, t). To be specific, we assume that the scalar field starts at time T = T0 with a value φ0 corresponding to a maximum of the potential. The initial cosmological constant is given by Λ0 = W (φ0 )/Mp2 , and we assume that we start with an SdS black hole, so that the unperturbed metric has the form (27) with f (r, T ) = f0 (r; M0 , Λ0 ) = 1 − 2GM0 Λ0 2 − r r 3 (36) where M0 is the initial black hole mass. Corresponding to the values of M0 , Λ0 are initial values rb0 , rc0 of the black hole and cosmological horizon radii, which in turn give the parameters β, γ for the unperturbed SdS spacetime via (26). Moving forward in time, the scalar field φ evolves via the scalar equation. In the background SdS spacetime, φ is constant, and therefore the time-dependent part of φ is perturbative and to leading order evolves according to the wave equation (19) in the background SdS spacetime. We note for future use that the time coordinate is related to the evolving scalar field by 1 T =− 3γ Z dφ (dW/dφ) (37) With this set-up, we can now state our results for the evolution of the spacetime geometry. The physical picture is that for slow-roll evolutions, the metric should transition through a progression of SdS metrics, with the parameters M and Λ varying slowly in time. We will show that the solutions come very close to realizing this expectation. Explicitly, we find that to leading order in perturbation theory, a metric of the form (35) satisfies the Einstein equations with stress-energy coming from the rolling scalar field, and metric functions given by f = fQS (r, T ) + δf (r, T ) , η = η0 (r) + δη(T, r) (38) where fQS (r, T ) = f0 (r; M (T ), Λ(T )) = 1 − Z r 1 − η02 ′2 ′ 1 φ̇2 δf (r, T ) = − r dr , 2r Mp2 rb0 f0 13 2GM (T ) Λ(T ) 2 − r r 3 Z r(1 − η02 ) T φ̇2 dT δη(r, T ) = 2 2f0 T 0 Mp (39) with Ṁ = dM = 4πβ φ̇2 , dT Λ̇ = −3γ φ̇2 Mp2 (40) We will also demonstrate that the function δf above is transient, and sub-dominant to the time dependent part of fQS (r, T ). Thus the time dependent black hole and cosmological horizon radii rh (T ) are well approximated by the zeros rhQS (T ) of the quasi-statically evolving portion fQS (r, T ). By construction these zeros are related to the time dependent mass and cosmological constant M (T ) and Λ(T ) according to the SdS formulae in (7). Hence the time derivatives of M (T ) and Λ(T ) in (40) can be converted into time derivatives of the horizon radii rh (T ), giving ṙh = rh φ̇2 , 2|κh | Mp2 h = (b, c) (41) Expressions for the time dependent mass and cosmological constant can be obtained by using the integral Z T T0 φ̇2 dT ′ = − 1 [W (φ(T )) − W (φ(T0 ))] 3γ (42) Integrating Ṁ , Λ̇ and ṙh using (42) we arrive at the results Λ(T ) ≈ W (φ(T )) , M (T ) ≈ M0 + β [W0 − W (φ(T ))] 6γ (43) and rh ≈ rh0 + rh0 [W0 − W (φ(T ))] 6γ|κh | (44) and similarly for the function δη(r, T ) in (39). Note, we use the approximate equality above, as in the slow roll approximation it is possible for M and Λ to have additional slowly varying contributions (such as φ̇2 ) whose derivatives are order φ̈ or higher, and hence would not contribute to (40) at this order. We have denoted here T0 as an initial time, which could be at T0 → −∞, when the scalar field is at the top of the potential with value φ0 , the metric is SdS with mass parameter M0 , Λ0 = W (φ0 )/Mp2 , and the horizon radii are the corresponding rh0 determined by (7). Note that the value of the scalar field potential, and hence the effective cosmological constant, is decreasing in time, so that the mass and both horizon radii are increasing functions of time. 14 A. Solving the Einstein equations We now present the derivation of the quasi-de Sitter black hole solutions given above. With the assumption that φ = φ(T ), there are two independent components of the stress tensor, TT T Tαβ  1 + η2 2 φ̇ |gT T | , = W (φ) + 2f   1 − η2 2 = −W (φ) + φ̇ gαβ 2f  and (45) otherwise. and the Einstein tensor is: " # ˙ η f 1 GT T = 2 (1 − f − rf ′ ) − |gT T | r rf # " ˙ η f 2 η̇ 1 + grr Grr = − 2 (1 − f − rf ′ ) + r rf r(1 − η 2 ) " # 2 ˙ 1 (1 − η ) f GrT = − 2 (1 − f − rf ′ ) − grT r rηf "  2 ·· # Gφφ 1 (η − 1) f ′′ f ′ η ′ f˙ η̇f ′ η̇ gθθ = + − + + + η̇ ′ + Gθθ = 2 r 2f 2f r 2 f sin2 θ First consider the linear combination relation and then GT T |gT T | + Grr grr = Mp−2  TT T |gT T | GT T |gT T | + GT r gT r = φ̇2 f˙ = −rη 2 Mp  rr , which yields + Tgrr η̇ = r(1 − η 2 ) φ̇2 2f Mp2 Mp−2  TT T |gT T | + TT r grT  (46) , which gives the (47) (48) thus both of our metric functions have a time-dependence determined by a radial profile times the kinetic energy of the scalar field. Since the quantity φ̇2 is already perturbative, we need only consider the metric functions on the right hand sides of equations (47) and (48) to leading order, i.e. f → f0 and η → η0 , the zeroth order SdS functions given in (5) and (28). So in perturbation theory, these equations give explicit expressions for the time evolution of f and η. Equations (47) and (48) can now be substituted back into the Einstein tensor in (46) to eliminate f˙ and η̇. One then finds that the T T , T r, and rr components of the normalized 15 Einstein equation all reduce to W (φ) (1 − η02 ) φ̇2 1 ′ + (1 − f − rf ) = r2 Mp2 2f0 Mp2 (49) There remains the θθ equation, which using (47) and (48) becomes  φ̇2 1 f′ W 1  − f ′′ − = 2+ (1 − η02 ) − rη0 η0′ − r(1 − η02 )f0′ 2 r Mp 2f0 Mp2 (50) Taking the derivative with respect to r of the Einstein equation (49) gives (50). So the equations of motion for the coupled Einstein scalar field system have been reduced to four equations, namely (47), (48) and (49), together with the slow-roll equation (19) for the scalar field. Physical considerations now give insight as to the perturbative form of the metric functions f (r, T ) and η(r, T ). During slow roll evolution in an inflationary universe without a black hole the scalar field potential W (φ) provides a slowly varying effective cosmological constant. The stress-energy is dominated by the potential, with small corrections coming from the kinetic contribution of the scalar field. As the potential changes, the metric is approximately given by a progression of de Sitter metrics. With a black hole present, for sufficiently slow evolution, one expects that the metric will be close to an SdS metric with slowly evolving parameters M (T ) and Λ(T ). Looking at the expressions for the first three normalized components of the Einstein tensor (46), one sees that if f = f0 , then the terms in the first round parentheses are all proportional to Λ0 . Since these terms only depend on radial derivatives, if we use the ‘quasi-static’ metric function fQS (r, T ) defined in (39) then these terms will be proportional to Λ(T ). Further, looking also at the normalized stressenergy components (45), one sees that the Einstein equations suggest a strategy of balancing Λ(T ) with W [φ(T )] and then equating the remaining terms. These are corrections to a pure cosmological equation of state from the kinetic energy of the scalar field. Based on this reasoning, we adopt an ansatz of the form f (r, T ) = fQS (r, T ) + δf (r, T ) (51) η(r, T ) = η0 (r) + δη(r, T ) It may appear redundant to have the time dependence for f explicit in both the function fQS (r, T ), as well as δf (r, T ). However, it turns out this is a convenient way of packaging the time dependence implied by the Einstein equations in a physically relevant manner. 16 One might also question the asymmetry in (51) between f (r, T ) and η(r, T ). However, this is simply what turns out to work. The next step is to substitute (51) into the Einstein equations, show that the ansatz allows a solution, and solve for the four functions M (T ), Λ(T ), δf (r, T ), and δη(r, T ). Starting with f (r, T ), we substitute our ansatz into equations (49) and (47), to get W [φ(T )] (1 − η02 ) φ̇2 1 ∂ + (rδf ) = r2 ∂r Mp2 2f0 Mp2   Λ̇r2 2GṀ β φ̇2 2 ˙ + − δf = −γr + 3 r r Mp2 Λ(T ) − (52) (53) respectively. According to our physical picture, we try for a solution such that the cosmological constant tracks the value of the scalar potential, Λ(T ) = W [φ(T )]/Mp2 . Taking the time derivative of this, and using the slow-roll equation (19) for the scalar field, we get φ̇2 Λ̇(T ) = −3γ 2 , Mp (54) consistent with the O(r2 ) terms in (53). Next, we integrate (52) (after cancelling the Λ and W terms) to obtain 1 δf (r, T ) = − 2r Z r ′ ′ 21 dr (r ) rb0 − η02 f0  φ̇2 Mp2 (55) Here, we have chosen for convenience to fix the lower limit of the integral to be the initial black hole horizon radius rb0 , so that δf (r0b , T ) = 0; an alternate lower limit would correspond to a shift in δf ∝ φ̇2 /r that without loss of generality can be absorbed in M (T ) – recall that within the slow roll approximation, the rate of change of φ̇2 is ignorable. Since second derivatives of φ are being neglected in the slow roll approximation, equation (55) implies that to this order d δf ≃ 0 dT (56) Ṁ (T ) = 4πβ φ̇2 (57) and hence we read off from (53) This rate of change of M corresponds to the accretion of scalar matter into the black hole. We have therefore demonstrated the quasi-static form of the solution for f , subject to the slowly varying δf . 17 It is interesting to compare this expression for Ṁ to that obtained for fluid accretion onto an asymptotically flat black hole by Babichev et al. [31, 33], Ṁ (V ) = −4πr2 TVr , (58) where V is the Eddington-Finkelstein advanced time coordinate for the black hole, and we have used the more general expression from [33] to facilitate comparison. Taking the cosmological constant tending to zero, that is, rb ≪ rc , we find that our expression for T reduces to the Eddington-Finkelstein coordinate V , and β ≈ rb2 . Finally, using the expression for the energy momentum tensor, φ̇2 = −TVr giving precisely the expression (58) of Babichev et al. Finally, we integrate (48) to obtain r(1 − η02 ) δη(r, T ) = 2f0 Z T T0 φ̇2 dT ′ + η1 (r, T ) Mp2 (59) where η1 is a slowly varying function (η̇1 ≈ 0) that will turn out to be proportional to φ̇2 , and we require that δη is zero at the start of the slow roll process T0 . This completes the derivation of the slow-roll solution. The evolution of all the functions, M (T ), Λ(T ), δf (r, T ), and δη(r, T ) in the quasi-SdS metric (51) are determined in terms of the unperturbed SdS metric and φ̇(T ), which is in turn determined through the slow roll equation (19) by the slope dW/dφ of the scalar potential. While the quasi-SdS solution is not quite as simple as the static SdS spacetime, we will see that the quasi-SdS form allows us to study the thermodynamics of both the black hole and cosmological horizons in an intuitive way. IV. HORIZONS In this section we locate the horizons rh (T ) of the dynamical metric. In the background static SdS metric, M0 and Λ0 are related to the horizon radii rb0 and rc0 by the formulae (7). Since our solutions almost track a sequence of SdS metrics we expect that the black hole and cosmological horizons are given by almost the same SdS formulae, with M (T ) and Λ(T ) replacing the constant parameters. However, “almost” is the operative word, since both f and η receive modifications. Nonetheless, our findings are that: • After an initial time period, the δf corrections become negligible; 18 • η(T, r) still satisfies the needed boundary conditions, and as a result, • the simple quasi-static formulae for the horizon radii apply after that initial period. The derivation proceeds in two steps. We first find the time dependent zeros of f (T, r) and identify the “start up” time interval. Second, we look at the form of the metric near the zeros of f and show that these are horizons. We close this section by comparing the results to those in our previous paper [40] which used a different coordinate system for the evolving system. A. Finding zeros of f (r, T ) Here we show that after an initial time period estimated below, the horizons rh (T ) are given by the zeros of fQS , that is, the rh (T ) with h = (b, c), have the same algebraic relation to M (T ) and Λ(T ) as do the quantities in static SdS. In the unperturbed SdS spacetime, the black hole and cosmological Killing horizons are located at the zeros of f0 (r), so we start by finding the zeros of f (r, T ), which we designate by rz,h (T ). That is, f (rz,h (T ), T ) = 0 (60) where f (r, T ) is given in equation (38) by f (r, T ) = fQS (r, T ) − with I(r) = Z r rb0 1 φ̇2 I(r) 2r Mp2 (1 − η02 ) ′2 ′ r dr f0 (61) (62) Denote the zeroes of the quasi-static metric function fQS (r, T ), defined in (39), as rhQS (T ), so that   fQS rhQS (T ), T = 0 (63) It follows that the quasi-static radii rhQS (T ) are given in terms of the time dependent mass and cosmological constant parameters M (T ) and Λ(T ) by the SdS relations (7). Let us now compare the rz,h (T ) to the rhQS (T ). Since by construction I(rb0 ) = 0, these two types of zeroes coincide at the black hole horizon4 , rz,b (T ) = rbQS (T ) 4 (64) Recall that the factor φ̇2 preceding I(r) in (61) is perturbative. It is therefore sufficient to evaluate the integral at the unperturbed horizon radius. 19 On the other hand, when r = rc0 in I, equation (60) becomes − Λ(T ) 3 1 φ̇2 I(rc0 ) = 0 rz,c + rz,c − 2GM (T ) − 3 2 Mp2 (65) which is not the same as the condition (63). Hence, the radii rcQS (T ) and rz,c (T ) differ, with the difference arising from I(rc0 ), which enters the polynomial equation (65) as a contribution to the time dependence of the mass term M (T ). Let us compare the magnitudes of the last two terms in (65). One might be concerned that I(rc0 ) would diverge due to the factor of 1/f0 in the integrand in (62). However, the zeros of 1 − η02 cancel the zeros of f0 at rb0 and rc0 . Explicitly, 1 − η0 = (r − rb0 ) P (r, rb0 , rc0 ) , 3 3 (rc0 − rb0 )r2 1 + η0 = (rc0 − r) P (r, rc0 , rb0 ) 3 3 (rc0 − rb0 )r2 (66) where P (r, d, e) = r2 (d2 + e2 ) + re2 (d + e) + de2 (d + e). Therefore the integral may be rewritten as 3 I(r) = 3 3 2 Λ(rc0 − rb0 ) Z r dr′ rb0 P (r′ , rb0 , rc0 )P (r′ , rc0 , rb0 ) r′ (r′ − rN ) (67) which is manifestly finite at r = rc0 . So the final term in (65) is then proportional to the instantaneous value of φ̇2 times this finite, time-independent, quantity, while from (40) we know that the change in M (T ) from its initial value M0 depends on the accumulated change of φ̇2 integrated over time (plus a possible slowly varying term). Therefore, after a start up time the contribution to rz,c coming from φ̇2 I(rc0 ) will be small compared to the contribution from the evolution of [M (T ) − M0 ]. Explicit comparison of these terms shows that the shift due to the I(rc0 ) term is negligible5 after a time interval (T − T0 ) ≃ rb0 ln(rc0 /rb0 ) for small black holes with rb0 ≪ rc0 , and after (T − T0 ) ≃ rb for large black hole with rb0 of order rc0 . Note that if the initial time T0 is taken to be −∞ then the extra term is irrelevant6 . After this initial time period the zeros of f (r, T ) coincide with the zeros of fQS (r, T ) rz,b (T ) = rbQS (T ) , rz,c (T ) = rcQS (T ) (68) and hence evolve in a quasi-static way determined by the evolution of the quasi-SdS parameters M (T ) and Λ(T ). It may seem odd that the additional function δf is needed to solve the Einstein equation at early times, rather than at late times. This likely illustrates the 5 6 For clarity, we have dropped numerical constants in making the following estimates. Naturally, a different gauge choice could have been made to set I = 0 at rc , and the correction would show up at rb . 20 teleological nature of the horizon, as it starts to grow in anticipation of the influx of stressenergy from the evolving scalar field. This effect was also noticed in a simpler inflationary example with no black hole in [40]. B. Near horizon behavior In this section we find the form of the metric near the zeros of f and show that it takes the form of a black hole or cosmological horizon. This then implies that the functions rz,h (T ) do locate the horizons, and combining with the results of the previous section, we will have demonstrated that the horizon radii evolve quasi-statically, rh (T ) = rz,h (T ) ≃ rhQS (T ) (69) after an initial period as discussed above. Let us recall what the horizons look like in the static SdS metric, starting with the static time coordinate t, and r. Defining the radial tortoise coordinate by dr⋆ = dr/f0 and the ingoing Eddington-Finklestein coordinate at the black hole horizon by v = t + r⋆ , the SdS metric near the black hole horizon is approximately given by ds2 ≃ −2κb (r − rb )dv 2 + 2drdv + r2 dΩ2 (70) Although the metric component gvv ≃ −2κb (r − rb )dv 2 vanishes at the horizon, the metric is non-degenerate there. The vector ∂/∂v is null and ingoing at the horizon. To study the cosmological horizon, one uses outgoing coordinates with u = t − r⋆ . Then near the cosmological horizon the metric becomes ds2 ≃ 2|κc |(r − rc )dv 2 − 2drdu + r2 dΩ2 (71) The vector field ∂/∂u is outgoing and null at the horizon and the cross term in the metric changes sign. One can also look at the metric near the horizons of SdS using the the stationary coordinate T given in (14). As noted earlier, T interpolates between the ingoing null EddingtonFinklestein coordinate v on the black hole horizon, and the outgoing null coordinate u on the cosmological horizon. So there is no need to transform the coordinates, we just look at the metric functions near the horizons. As r → rb the SdS metric becomes ds2 ≃ −2κb (r − rb )dT 2 + 2dT dr + 21 Cb 2 dr + r2 dΩ2 κb (72) where Cb = −η0′ (rb ) = (2rc3 + 3rc2 rb + rb3 )[rb (rc3 − rb3 )]−1 > 0. The essential difference between (70) and (72) is that grr = 0 in the (v, r) coordinates, while grr is non-zero but finite in the (T, r) coordinates. This is due to the fact that T is not identical to v near rb . Similarly, near the cosmological horizon of SdS the metric in stationary cooordinates has the form ds2 ≃ 2|κc |(r − rc )dT 2 − 2dT dr + Cc 2 dr + r2 dΩ2 |κc | (73) and one sees explicitly that T has become an outgoing null coordinate. Here Cc = −η0′ (rc ) > 0. The analysis of the time-dependent metric in (38) near the zeros of f proceeds in a similar way. Take the time T to be sufficiently large that the zeros of f are well approximated by the zeros of fQS . Further, to avoid repetition of bulky notation, we will simply abbreviate rhQS (T ) = rh (T ), and the end result justifies this replacement. As r approaches rb (T ), the metric function f behaves like f ≃ −2κb (T ) (r − rb (T )) (74) where κb (T ) is defined as the derivative 1 κb (T ) = f ′ |r=rb (T ) 2 (75) At present we will not ascribe any significance to κb (T ) as a possible time-dependent surface gravity or temperature. The behavior of the metric function grr = (1 − η 2 ) f (76) in the time dependent solution (38) appears to be more subtle, since although the zeros of f in the denominator are cancelled by zeros in the numerator for exact SdS, it is not immediately obvious whether this is still true for the perturbed functions f and η. However, (47) and (48) together imply that grr is in fact independent of T , provided only that φ = φ(T ), hence grr remains regular even if the locations of the zeros of f shift. Let us now confirm this, by deriving explicitly the locations of these zeros. Denote rh0 as an horizon radius in the unperturbed SdS spacetime (a zero of f0 ) and expand the time-dependent horizon radius around this as rh (T ) = rh0 + xh (T ) 22 (77) where the shift in the position of the zero is assumed small, |xh |/rh0 ≪ 1. rh (T ) is defined as a zero of f (r, T ), therefore xh can be deduced by Taylor expanding the relation f (T, rh (T )) = 0 around rh0 and using the expression we have derived for f (r, T ), yielding Z T 2 I(rh ) φ̇2 φ̇ rh ′ dT − η0 (rh ) xh (T ) = 2 2κh 2rh Mp2 T 0 Mp thus verifying (44). Expanding η near each horizon similarly shows Z rh (1 − η02 ) T φ̇2 ′ + η1 (rh , T ) η(rh (T )) = η0 (rh ) + xh η0 (rh ) + 2 2f0 T 0 Mp I(rh ) φ̇2 = η0 (rh ) + η1 (rh , T ) − 2rh Mp2 (78) (79) i.e. fixing η1 = I(r)φ̇2 /2rMp2 , we have that η(rh (T )) = η0 (rh0 ) = ±1, explicitly confirming that grr remains regular at the new horizon radii. Therefore although the metric function gT T vanishes at rh (T ), the metric is nondegenerate there, as was the case in the unperturbed SdS spacetime. The cross term +2dT dr in the metric at the black hole horizon illustrates the ingoing nature of the T coordinate near that horizon, where it becomes a null coordinate that is well-behaved7 , a property that was built into the choice of T to ensure ingoing boundary conditions for φ(T ). A similar analysis applies to the cosmological horizon as r approaches rc (T ). The cross term in the metric in this case will have the opposite sign illustrating the outgoing nature of the coordinate T near the cosmological horizon. The preceeding subsections are summarized by the formula for rh (T ) given in (44), valid after an initial start-up time. C. Calculation in null coordinates In this subsection we show that these results for the growth of the horizon radii agree with the calculations in our previous paper [40]. This is a useful exercise since the first paper worked in null coordinates, in which it was simple to locate the horizons but difficult 7 The vector (∂/∂T ) is null at r = rb and is normal to the surface dr = 0, and the second null radial direction is defined by (−η0′ (rb (T ))/κb )dr + 2dT = 0. So the two null directions on the black hole horizon are Ta = ∂ , ∂T Ka = − normalized so that T a Ka = −1. 23 η0′ (rb (T )) ∂ ∂ − 2κb ∂T ∂r (80) to find the metric throughout the region. In contrast, the coordinates used in the current paper allow us to find the metric straightforwardly, but the horizon structure requires more work. Comparison requires further processing of the results of [40] using the slow-roll approximation requirements derived earlier. In [40], we showed that the linearized Einstein equations on the horizon reduced to simple second order ODE’s in terms of the advanced time coordinate Vb on the black hole horizon, and Uc on the cosmological horizon: Ac φ̇2 d2 (U δA ) = − c c dUc2 κ2c Mp2 Uc Ab φ̇2 d2 δA = − , b dVb2 κ2b Mp2 Vb2 (81) where Ah is the area of the respective horizon. In the background SdS metric, these coordinates are related to T by κc Uc = exp[κc (T − T0′ )] κb Vb = exp[κb (T − T0 )] , (82) Integrating (81) gave the general expressions for the horizon areas, from which we can deduce the values of horizon radii as rb Vb δrb = − 6γκb Mp2 rc δrc = − 6γκc Mp2 Uc Z ∞ Vb 0 Z Uc δW [φ(Vb′ )] dVb′ Vb′2 (83) δW [φ(Uc′ )]dUc′ At first sight, these look somewhat different from the values of δrh obtained from (44), however, armed with the slow roll analysis for integrating the Einstein equations, let us re-examine these expressions. Integrating by parts in δrb , and using κb Vb = exp[κb (T − T0 )] we find   Z ∞ rb κb T 2 −κb T ′ ′ δW [φ(Vb )] − 3γe δrb = − φ̇ e dT 6γκb Mp2 T Similarly, the rc equation can be integrated by parts to give   Z ∞ rc −κc T 2 κc T ′ ′ −δW [φ(Uc )] + 3γe δrc = − φ̇ e dT 6γκc Mp2 T (84) (85) To approximate these integrals, we use (34) to compare how rapidly φ̇2 is varying compared to the exponential, in other words, the magnitude of r2 + rc rb + rb2 4rh δ W ′′ 1 d 2 = c 2 < 3δ ≪ 1 φ̇ = γκh rc + rb2 2rh + rh̄ φ̇2 κh dT 24 (86) (where the ‘h̄’ subscript stands for ‘the other’ horizon). Thus φ̇2 varies much more slowly than the exponential, and we can approximate each of these integrals by φ̇2 e−|κh |T /|κh | yielding   rh 3γ 2 δrh ≃ − δW − φ̇ 6γ|κh |Mp2 |κh | (87) Let us now compare these two terms. At the start of slow-roll, i.e. at the local maximum of W, 1 W ≃ W0 + W ′′ δφ2 ⇒ 2 δφ ∝ e−W ′′ T /3γ as T → −∞ (88) hence 3γ φ̇2 2W ′′ = < 2δ ≪ 1 κh δW 3κh γ (89) thus δW is initially the dominant term in (87). Moreover, throughout slow-roll, examining the rate of change of each term, ˙ = −3γ φ̇ , δW 2 3γ φ̇2 κh !· = −3γ φ̇ 2 2W ′′ κh γ ⇒ 3γ φ̇2 κh !· ˙ ≪ δW (90) shows that if δW is dominant initially, it will remain so throughout the transition between vacua, meaning that the evolution of the horizon area is dominated by the shift in the cosmological constant which is the first term in (87), in agreement with (44). V. HORIZON AREAS, BLACK HOLE MASS, AND FIRST LAWS In this section we compute the rates of change for the areas of the black hole and cosmological horizons, the black hole mass, and the thermodynamic volume between the horizons. We then show that the de Sitter patch first law (9) and the mass first law (12) are satisfied throughout the evolution. In our previous paper [40] we showed that (9) held between the initial and final SdS states, relating the total changes in these quantities over the evolution. In this earlier work we lacked the detailed form of the time dependent metric and were unable to verify the mass first law equation, though we inferred the value of the late time mass based on assuming that the mass first law was true. Here, equipped with the solution for the metric (38), we are able to do more. 25 A. Horizon area growth and first law A priori one expects that the black hole horizon area should be increasing in time due to accretion of scalar field stress-energy. However, for the cosmological horizon there are competing influences on its area that tend in opposite directions. For fixed Λ the cosmological horizon gets pulled in as the black hole mass grows, while for fixed black hole mass the cosmological horizon grows as Λ shrinks. The solutions showed that both horizon radii are increasing, so the latter effect dominates the behavior of the cosmological horizon. Throughout this section we will work in the late time limit defined above in §IV A, so the black hole and cosmological horizons are located at the zeros of the quasi-static metric function fQS (r, T ). This implies that rb (T ) and rc (T ) are given in terms of the time dependent mass M (T ) and vacuum energy Λ(T ) by the same relations (7) that apply in the unperturbed SdS spacetime. The expressions for Ṁ and Λ̇ in the time dependent solution (38) then yield the expressions given in (41) for r˙b and r˙c . It follows that the rate of change of the black hole and cosmological horizon areas is Ȧh = Ah φ̇2 , |κh | Mp2 h = (b, c) (91) Now it can be checked that the SdS-patch first law (9) holds in a dynamical sense. Summing the expressions for the rate of growth of the horizon areas (91) gives |κb |Ȧb + |κc |Ȧc = Atot φ̇2 Mp2 (92) Using the formulae for Λ̇ in (40) and the thermodynamic volume of the de Sitter patch VdS , equation (29), gives V Λ̇ = −Atot φ̇2 Mp2 (93) and so |κb |Ȧb + |κc |Ȧc = −V Λ̇ (94) Interpreting the variations in (9) as time derivatives, and translating |κh |Mp2 δAh = Th δSh , we see that the SdS-patch first law (9) holds between successive times. Hence the decrease of the effective cosmological constant due to the scalar field rolling down its potential goes into increasing the black hole and cosmological horizon entropies. It is also straightforward to check that the mass first law (12) is satisfied throughout the evolution for the time dependent solutions. Plugging in for Ṁ and Λ̇ using (38), and Ȧb 26 using (91), one finds that the mass first law reduces to the first relation in (25), that is, the evolution satisfies Ṁ − |κb |Ȧb + Vb Λ̇ = 0 (95) Mp2 The accumulated growth in time for each horizon is obtained by integrating the expressions (91) for A˙h , which at late times gives δAh (T ) = − Ah VdS δΛ 3|κh |Atot (96) where δAh (T ) ≡ Ah (T ) − Ah0 and δΛ = Λ0 − Λ[φ(T )]. The prefactors all refer to values in the initial SdS spacetime, and keeping in mind that δΛ(T ) is negative, δAh is positive. Hence the change in area of each horizon between T0 and T is proportional to the initial horizon area times the change in the effective cosmological constant. It is also interesting to note that during the evolution that the fractional increase in area, times the magnitude of the surface gravity, is the same for both horizons |κc | δAb δAc = |κb | Ac Ab (97) The geometrical quantities appearing in equation (96) are not all independent. The initial SdS spacetime is specified by two parameters, which we have been taking to be the initial black hole horizon radius and the initial value of the potential, and so one wants formulae that only depend on rb and Λ. Substituting in the surface gravity and thermodynamic volume gives, for the black hole horizon, δAb = Ab |δΛ| 2rb (rc2 + rb2 + rc rb ) Λ (2rb + rc )(rb2 + rc2 ) (98) The corresponding expression for δAc is obtained by interchanging rb and rc . In (98) rc is still an implicit function of rb and Λ. We will display the results in the limits of small and large black holes. For the black hole horizon area one finds  p p  2 Ab 2|δΛ| √ Λr , Λrb2 ≪ 1 b 3Λ δAb ≃ p  Ab |δΛ| , Λrb2 ∼ 1 Λ (99) One sees that the fractional growth is parametrically suppressed for small black holes, and of order |δΛ|/Λ for large ones. Likewise, one can examine how much the growth of the cosmological horizon is suppressed by the presence of the black hole. One finds  p  12π |δΛ| Λrb2 ≪ 1 , Λ2 δAc ≃ p  4π |δΛ| Λrb2 ∼ 1 , Λ2 27 (100) The small black hole result is the same as if there were no black hole. On the other hand, the growth of the area of the cosmological horizon can be diminished by as much as a factor of two-thirds for large black holes. This suggests that the effect of a large black hole on the spectrum of CMBR perturbations created during slow roll inflation is worthy of further study. While both horizon areas increase, the black hole gets smaller compared to the cosmological horizon in certain ways. The volume VdS between the horizons increases according to V̇dS = 2π  r3 rc3 − b |κc | |κb |  φ̇2 Mp2 (101) Since |κc | < κb [49] the right hand side is positive. Another measure is how the difference between the black hole and cosmological horizon areas changes in time: Ac (T ) − Ab (T ) − (Ac0 − Ab0 ) = δAc − δAb = 24π |δΛ(T )| (rc − rb )(rc + rb )3 (102) Λ2 (rc2 + rb2 )(2rc + rb )(2rb + rc ) Since |δΛ(T )| increases with time, the black hole is getting smaller in comparison to the cosmological horizon. To unravel the parameter dependence of (102), we again look at different limiting cases  p |δΛ|   12π Λrb2 , 2  Λ   δAc − δAb ≃ 6π |δΛ| , Λ2      32π |δΛ| (1 − pΛr2 ) , b 3 Λ2 p Λrb2 ≪ 1 rb = 21 rc p Λrb2 ∼ 1 (103) We see in this case that the effect is parametrically suppressed both for very small and very large black holes and most prominent in the intermediate regime. B. Dynamical temperature We have not yet discussed horizon temperature for the time dependent quasi-SdS black holes (38). These represent non-equilibrium systems, for which it is not clear that a welldefined notion of dynamical temperature should exist. Nevertheless, given that our system is only slowly varying, we might expect that an adiabatic notion of temperature makes sense [50, 51] and indeed candidate definitions have been suggested in the literature. We will focus, in particular, on the proposal [52] which defines a dynamical surface gravity κdyn for outer trapping horizons in nonstationary, spherically symmetric spacetimes. In this 28 construction, the Kodama vector [53] substitutes for the time translation Killing vector of a stationary spacetime in providing a preferred flow of time. Further, the authors use a variant of the tunneling method of [54], adapted to the non-stationary setting, to argue that particle production has a thermal form with temperature Tdyn = κdyn /2π. The dynamical surface gravity is defined in [52] as κdyn = 1 ⋆ d ⋆ dr 2 (104) where the Hodge ⋆ refers to the two dimensional T, r subspace orthogonal to the 2-sphere, and the quantity is to be evaluated at the horizon. The Hodge duals of the relevant forms are given by ⋆ dr = f dT − hdr , ⋆dr ∧ dT = 1 (105) Using these expressions, and evaluating (104) on the black hole horizon, one finds that κdyn (T ) = (f ′ + η̇) = κb (T ) + O(εδ) (106) where κb (T ) was defined in equation (75) as the derivative of the perturbed metric function f (r, T ) evaluated at radius rb (T ). Note that although the contribution δf (r, t) to the metric function vanishes at rb0 , its derivative is non-zero and therefore κb (T ) gets contributions both from fQS and from δf , with the result that κb (T ) = κQS (T ) − Cb 2 φ̇ 2κb (107) Here the quasi-static surface gravity κQS (T ) is given by the SdS relation (8) using rb (T ) and rc (T ), and expanding to linear order in the perturbative quantities. These time-dependent corrections to the radii rh (T ) are proportional to the integral over time of φ̇2 , but the second term in (107) depends on the instantaneous value of φ̇2 . So after an initial time period the latter term is small compared to the first, and the dynamical surface gravity is well approximated by κb (T ) ≃ κQS (T ) (108) We see that the proposal of [52] yields a simple, intuitive result. This is in agreement with our results in [40] where the calculation was done using null coordinates. 29 VI. CONCLUSION In this paper we have found a tractable form for the metric of a black hole in a slowroll inflationary cosmology, to first order in perturbation theory, which one can readily understand in terms of expectations for the slowly evolving system. The solution directly gives the time dependence of Λ and M and it is straightforward to then find the time dependent horizon areas, the thermodynamic volume, and the dynamical surface gravity. A topic for future study is to compute the flux of the energy-momentum of the scalar field across surfaces of constant r. The flux is ingoing at the black hole horizon, and outgoing at the cosmological horizon, and it would be good to understand our results for the growth of the horizons and the mass in terms of the fluxes in greater detail, as was done in [28]. Further, there must be a transition surface between the horizons where the flux vanishes, so mapping out the flux throughout the domain would be of interest. A related issue is to look at the energy density and pressure variation across spatial slices that interpolate to a standard inflationary cosmology in the far field. The range of validity of the approximate solution we have derived is another issue for further study. Conservatively, one assumes that |δΛ/Λ| must be small. However, since the slow-roll conditions require that the derivatives of the potential must be small, one might ask if larger accumulated change in W is allowed, as long as the evolution is slow enough. In any case, our approximation is equally valid as the slow roll approximation in the inflationary evolution of the early universe. It is of interest to calculate the perturbations from inflation with the black hole present, as such signatures in the CMBR may be a method for detecting, or inferring, primordial black holes [55]. Although the black hole itself is small-scale, the wavelength of its signature on modes that re-enter the horizon at late times is stretched with the modes themselves. A related problem is to compute the Hawking radiation in this metric by extending the methods of [56] to the quasi-static case. Such a calculation is needed to support the interpretation of the dynamical surface gravity as a physical temperature. While this quasi-static set-up would only represent a step in understanding horizon temperature in a dynamical setting, this metric is one of the few known dynamical examples where the cosmological and black hole temperatures are not equal. There are examples of elegant analytic descriptions in which an evolving physical system tracks a family of static solutions, such as charge-equal-to mass black holes [57–59], or 30 magnetic monopoles [60, 61], with small relative velocities. In these cases there is a BPS symmetry of the zeroth order time-independent solution, which apparently protects small perturbations from being too disruptive. There is not obviously any such symmetry in the black hole plus scalar field system, and yet the slow-roll dynamics is analogous. One avenue for future study is to see if there is an underlying reason for this behavior, which in turn could lead to a more fundamental understanding. ACKNOWLEDGMENTS RG is supported in part by the Leverhulme Trust, by STFC (Consolidated Grant ST/P000371/1), and by the Perimeter Institute for Theoretical Physics. 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