PHYSICAL REVIEW D 100, 024010 (2019)

Effective field theory for black holes with induced scalar charges

Leong Khim Wong,1 Anne-Christine Davis,1 and Ruth Gregory2,3 1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2Centre for Particle Theory, Durham University, South Road, Durham DH1 3LE, United Kingdom 3Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

(Received 19 March 2019; published 8 July 2019)

While no-hair theorems forbid isolated black holes from possessing permanent moments beyond their mass, electric charge, and angular momentum, research over the past two decades has demonstrated that a black hole interacting with a time-dependent background scalar field will gain an induced scalar charge. In this paper, we study this phenomenon from an effective field theory (EFT) perspective. We employ a novel approach to constructing the effective point-particle action for the black hole by integrating out a set of composite operators localized on its worldline. This procedure, carried out using the in-in formalism, enables a systematic accounting of both conservative and dissipative effects associated with the black hole’s horizon at the level of the action. We show that the induced scalar charge is inextricably linked to accretion of the background environment, as both effects stem from the same parent term in the effective action. The charge, in turn, implies that a black hole can radiate scalar waves and will also experience a “fifth force.” Our EFT correctly reproduces known results in the literature for massless scalars, but now also generalizes to massive real scalar fields, allowing us to consider a wider range of scenarios of astrophysical interest. As an example, we use our EFT to study the early inspiral of a black hole binary embedded in a fuzzy dark matter halo.

DOI: 10.1103/PhysRevD.100.024010

I. INTRODUCTION resemblance to astrophysical environments, even a minimally coupled, real scalar field can exhibit interesting phenomenol- The uniqueness theorems pioneered by Israel [1] (see ogy around a black hole. A classic example is the inflaton. also Ref. [2] for a recent review) tell us that black holes are Neglecting backreaction, Jacobson [19] showed that the remarkably simple objects characterized only by their solution near the event horizon1 is given by the Kerr metric mass, electric charge, and angular momentum. Even if surrounded by an effectively massless scalar, one considers more general field theories interacting with gravity, the general rule, summarized by the “no-hair” 2GMr r − r theorems [2–12], is that there are no additional charges that ϕ ϕ ϕ_ þ þ ðt; rÞ¼ 0 þ 0 t þ − log − ; ð1:1Þ a black hole can carry. These precise theorems are rþ r− r r− predicated on several crucial assumptions, however, which ϕ ϕ_ “ ” if violated can lead to a variety of new solutions. Many such where 0 þ 0t is the background coasting solution and r mark the locations of the inner and outer horizons in Boyer- examples are known today, including colored black holes _ [13], black holes with a cosmic string [14], and black holes Lindquist coordinates. Although valid only while ϕ0t is with complex massive scalar or Proca hair supported by sufficiently small, this solution nonetheless remains a good rotation [15,16], or real scalar hair supported by exotic effective description of the inflationary epoch from the gravitational couplings [17,18]. perspective of the black hole, whose light-crossing time is In this paper, we revisit a different kind of circumvention much shorter than cosmological timescales. of the no-hair theorems. By relaxing the assumptions of Let us now zoom out on this solution by expanding in stationarity and asymptotic flatness, which bear little powers of GM=r. We can write Q ϕ ¼ Φ þ þ Oð1=r2Þ;Q¼ −A∂ Φ; ð1:2Þ 4πr t Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1Solutions that extend all the way to the cosmological horizon the author(s) and the published article’s title, journal citation, have also been found [20,21], albeit only for spherical black and DOI. Funded by SCOAP3. holes.

2470-0010=2019=100(2)=024010(28) 024010-1 Published by the American Physical Society WONG, DAVIS, and GREGORY PHYS. REV. D 100, 024010 (2019)

8π where A ¼ GMrþ is the area of the event horizon. The charge impacts the inspiral in other ways. Moreover, _ first term, Φ ¼ ϕ0 þ ϕ0t, describes the background scalar previous analytic studies have all been limited to massless field that persists independently of the black hole. The scalar-field backgrounds varying at most linearly with effect of the black hole is to “drag” the scalar, leading to the space and time. In this case, results can be obtained by Coulomb-like potential in the second term, whose dimen- appropriating Damour and Esposito-Far`ese’s calculations sionless numerator Q is called the induced scalar charge.2 [29] for the inspiral of binary neutron stars in scalar-tensor More recently, Horbatsch and Burgess [22] applied this theories, since the derivatives of Φ are constant. New result to models of the Universe in which late-time “technology” will have to be developed, however, for acceleration is driven by a rolling scalar. In such cases, backgrounds that are more complicated functions of space _ all black holes should be dressed with a charge Q ¼ −Aϕ0, and time. This generalization is worth exploring, since which they argue enables a black hole to radiate energy and many scenarios beyond the Standard Model predict the momentum into scalar waves. Furthermore, as scalar existence of massive (pseudo)scalar fields that can form radiation would lead to a faster decay in the orbital period localized, gravitationally bound objects, which resist col- of a binary, they arrive at the constraint lapse by oscillating in time [30–34]. The prime example is a ffiffiffiffiffiffiffiffiffi galactic fuzzy dark matter halo formed by an ultralight p _ −1 −7 −1 −22 −21 4πGjϕ0j ≲ ð16 daysÞ ¼ 7 × 10 s ð1:3Þ scalar of mass μ ∼ 10 –10 eV [35–38]. Whether black holes can be used to probe such configurations is an on any rolling scalar in the vicinity of the quasar OJ287. interesting question. (Of course, black hole superradiance This bound stems from the supermassive black hole binary already provides a way of probing new fundamental fields at the center of the quasar having an inspiral consistent with [39–49]. Our work explores a complementary avenue, as it the predictions of general relativity in vacuum to within an does not rely on rotation and only pertains to fields with a uncertainty of 6% [23–25]. While by no means a spec- Compton wavelength much larger than the black hole.) tacularffiffiffiffiffiffiffiffiffi bound (a slow-rolling scalar should satisfy We push forward by constructing an effective field p _ −18 −1 4πGjϕ0j ≪ H0 ¼ 2 × 10 s ), that black holes are theory (EFT) ala` Goldberger and Rothstein [50,51], which sensitive to this value at all is interesting. Black holes describes black holes in terms of worldlines furnished with observed by LIGO have also been used to constrain this composite operators that capture finite-size effects. The key effect [26], but the bound obtained is much looser. benefit of this description is the ability to disentangle It is worth emphasizing that this behavior is not unique to questions about the long-distance, infrared (IR) physics we rolling scalars: Black holes will develop scalar charges are interested in—such as the trajectory of the black hole— when embedded in any arbitrary scalar-field environment, from the short-distance, ultraviolet (UV) physics transpir- as long as the background scalar evolves in time relative to ing near its horizon. Information about the latter is the black hole’s rest frame. This intuition is supported by accessible to distant observers, like ourselves, through numerical relativity simulations [27,28], which show that the way it impacts the black hole’s multipolar structure. scalar radiation is also emitted by black holes moving Mathematically, this is characterized in the EFT by through background scalar fields (even static ones) that are Wilsonian coefficients, whose values can be determined spatially inhomogeneous. In general, an analytic descrip- by matching calculations with the “full theory.” As we are tion of such systems is not possible, except when there doing purely classical physics, we have the advantage of exists a large hierarchy between the length and timescales knowing what this UV completion is—it is just general of the black hole and its environment. In this limit, which relativity. will be our focus, the black hole can be approximated as a This paper is organized as follows: We begin in Sec. II by point particle traveling along the worldline of some solving perturbatively the Einstein-Klein-Gordon field effective center-of-energy coordinate. The general defini- equations for a black hole interacting with a massive scalar tion for the scalar charge should then be field. This generalizes Jacobson’s result and will be later used to fix Wilsonian coefficients. We then construct the _ QðτÞ ≔−AΦ(zðτÞ); ð1:4Þ EFT in Sec. III. The main novelty of our approach is the way we obtain the black hole’s effective action: By μ where τ is the proper time along the worldline z ðτÞ. integrating out composite operators localized on its world- This brings us to the motivation for this work: Can we line using the in-in formalism, we obtain an action understand the full extent to which generic scalar-field expressed in terms of correlation functions that can environments affect the motion of black holes embedded systematically account for both conservative and dissipa- within them? To date, only the flux of scalar radiation has tive effects. Contained in these correlation functions are the been studied, but it is possible that a black hole’s scalar aforementioned Wilsonian coefficients. We find that the coefficient responsible for the induced scalar charge also 2Other definitions in the literature differ on minus signs and sets the accretion rate of the background scalar onto the factors of 4πG. We find this definition the most natural. black hole. This inextricable connection is the EFT’sway

024010-2 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019) of saying that the charge arises as a natural consequence of separable in Boyer-Lindquist coordinates ðt; r; θ; φÞ; thus, ingoing boundary conditions at the horizon. one can make the ansatz [52,53] The remainder of the paper is concerned with explo- −iωtþimφ ring our EFT’s broader phenomenological implications. ϕ ∝ e RlmðrÞSlmðθÞ; Section IV presents the derivation of the universal part of the equation of motion for the black hole’s worldline, where the integers ðl;mÞ label different angular-momen- demonstrating that the black hole experiences a drag force tum states. To obtain an analytic solution, we further restrict 3 −1 −1 due to accretion and a fifth force due to its scalar charge. attention to near-horizon distances r ≪ maxðω ; μ Þ We then specialize to the case of a black hole binary and truncate the solution to first order in Mω and Mμ. embedded in a fuzzy dark matter halo in Sec. V. In addition With these simplifications, the angular part of the solution imφ m to the effects already discussed in earlier sections, our EFT SlmðθÞe reduces to the spherical harmonics Yl ðθ; φÞ, also provides a natural language for calculating two other while the radial part is [53] effects not unique to black holes but common to any − iPm − massive body: dynamical friction and the gravitational ∝ r rþ −l l 1;1 − 2 ; r r− RlmðrÞ − 2F1 ; þ iPm − ; force exerted by the halo. Finally, our calculations are r r− rþ r− combined with observations of OJ287 to constrain the ð2:1Þ allowed local density of fuzzy dark matter. The result is a very weak upper bound, which is unsurprising, since having imposed ingoing boundary conditions at the future typical halos are too dilute to leave any observable imprints event horizon. The parameter Pm is defined to be in the binary’s inspiral. The paper concludes in Sec. VI, − 2 ω where we discuss some potential future applications of our ≔ am Mrþ Pm − ; ð2:2Þ EFT, which may lead to better observational prospects. rþ r− Note that while we use the usual ℏ ¼ c ¼ 1 units (except in Sec. II where we also set G ¼ 1), in this paper the reduced where a is the specific angular momentum of the Planck mass is defined by m2 ¼ 1=ð32πGÞ to be consistent black hole. Pl ’ with the EFT literature. As we did with Jacobson s result, let us zoom out on Eq. (2.1) to obtain a coarse-grained description valid at ≪ ≪ ω−1 μ−1 II. SCALAR MULTIPOLE MOMENTS distances M r maxð ; Þ. The two dominant IN THE FULL THEORY terms are ∝ l −l−1 We start by considering what happens when a black hole RlmðrÞ r þ Clmr ; ð2:3Þ of mass M is embedded within a background environment ϕ μ with relative coefficients, accurate to first order in Mω and comprised solely of a Klein-Gordon field of mass . The μ problem is analytically tractable under four conditions: M ,givenby[53,54] (1) As perceived by an observer in the rest frame of the 2 Yl −1 ðl!Þ black hole, the timescale ω on which the back- − − 2lþ1 2 4 2 Clm ¼ iPmðrþ r−Þ 2l ! 2l 1 ! ðj þ PmÞ: ground varies is much longer than the black hole ð Þ ð þ Þ j¼1 light-crossing time, Mω ≪ 1. ð2:4Þ (2) Similarly, the background is assumed to vary on a R length scale that is much greater than the black These expressions can now be used to read off a black ’ R ≪ 1 hole s radius, M= . hole’s scalar multipole moments. (3) The Compton wavelength μ−1 of the scalar is also assumed to be much greater than the size of the black hole, Mμ ≪ 1. A. Scalar charge (4) The energy density in the scalar field is dilute Consider first the l ¼ 0 mode. At distances enough that, in the immediate vicinity of the black M ≪ r ≪ maxðω−1; μ−1Þ, the solution reads hole, its backreaction onto the geometry is subdomi- −iωt −1 nant to the black hole’s own spacetime curvature. ϕ ¼ Φ0e ð1 þ C00r Þ; ð2:5Þ Rather than being seen as just simplifying assumptions, these should be considered defining characteristics for what having included an overall amplitude Φ0 for the field. From 2 ω it means to be a background environment. Eq. (2.4), C00 ¼ Mrþi , and for real scalar fields, taking The last condition implies that the scalar behaves like a the real part of Eq. (2.5) yields test field near the horizon of the black hole. By neglecting its backreaction, the problem of studying the effect of the 3This suffices for our purposes, since larger distances are well black hole on ϕ reduces to one of solving the Klein-Gordon within the purview of our EFT. We only need this full-theory equation on a fixed Kerr background. This equation is calculation to resolve the UV physics near the horizon.

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8π Φ ω 16π 4 ϕ Φ ω Mrþ 0 sin t M d ¼ 0 cos t þ : ð2:6Þ pj 0 ¼ − ð∇ΦÞ: ð2:12Þ 4πr a¼ 3 dt It should be readily apparent that this reproduces Eq. (1.2): The same procedure can be repeated for l ≥ 2; hence, For a background environment of the form Φ ¼ Φ0 cos ωt we learn that a black hole gains not just a scalar charge in the vicinity of the black hole, the full scalar field behaves when immersed in an arbitrary scalar-field environment as ϕ ¼ Φ þ Q=ð4πrÞ, with the scalar charge Q defined by Φðt; xÞ, but an infinite set of multipole moments. In Eq. (1.4) as before. practice, however, it often suffices to keep only the scalar charge and, in the case of rotating black holes, the spin- dependent dipole moment. Higher multipole moments are B. Higher multipole moments suppressed by ever greater powers of M=R, making their Now suppose our scalar field is not quite homogeneous phenomenology increasingly irrelevant. but has a linear gradient: Φ ¼ðb · xÞ cos ωt. This induces a l 1 dipole moment in the scalar, via the ¼ mode, whose III. THE EFFECTIVE ACTION solution is The systems of interest in this paper are all governed by X1 the action4 ϕ −iωt −2 m θ φ ¼ bme ðr þ C1mr ÞY1 ð ; Þ: ð2:7Þ Z m¼−1 pffiffiffiffiffiffi 1 1 S g; ϕ −g 2m2 R − ∂ϕ 2 − μ2ϕ2 : : f½ ¼ Pl 2 ð Þ 2 ð3 1Þ The constants bm ∼ Oð1=RÞ are related to the Cartesian x components of the vector b ¼ðb ;b ;b Þ via x y z When the length and timescales of its environment are rffiffiffiffiffiffi rffiffiffiffiffiffi 2π 4π much greater than those of the black hole, the latter can be approximated as an effective point particle traveling along a b1 ¼ 3 ðbx þ ibyÞ;b0 ¼ 3 bz: ð2:8Þ worldline zμðτÞ with 4-velocity uμ, normalized to satisfy μ u uμ ¼ −1. This description emerges after integrating out Unlike the l ¼ 0 case, C1m has a term that is independent of ω: short-wavelength modes from the full theory to generate the effective action [50] i 2 C1m ¼ − amM þ OðMωÞ: ð2:9Þ 3 S ¼ Sf½g; ϕþSp½z; g; ϕ: ð3:2Þ Substituting this back into Eq. (2.7) reveals that in the The first term Sf now governs only the remaining long- presence of a nontrivial background scalar gradient wavelength modes of the fields g; ϕ , while the dynamics ∇Φ b ω ð Þ ¼ cos t, black holes also acquire a spin-dependent of the worldline and its interaction with the fields living in dipole moment, the bulk are given by the point-particle action Sp. Performing this integration generally leads to an infinite p xˆ 4π 2 ϕ Φ · p aM Sˆ ∇Φ O ω number of terms in S , which can be organized according ¼ þ 2 ; ¼ ð × Þþ ðM Þ; p 4πr 3 to relevancy as an expansion in three small “separation-of- ð2:10Þ scale parameters,” R ≪ 1 ω ≪ 1 μ ≪ 1 where Sˆ is the unit vector along the black hole’s spin axis. GM= ;GM ; and GM : ð3:3Þ Notice that the dipole moment p survives in the static limit ω → 0. The no-hair theorems are still circumvented here In this section, we discuss how to systematically construct because a linear spatial gradient Φ ∼ b · x violates the Sp and determine the most relevant terms needed to assumption of asymptotic flatness. describe the interaction of a black hole with its scalar-field Spherical black holes can also attain higher-order environment. moments, although the effect is suppressed by one power of Mω relative to the spinning case. Setting a ¼ 0 in A. Worldline degrees of freedom Eq. (2.4) yields Finite-size effects are modeled in the EFT by introducing a set of composite operators fqLðτÞ; …g localized on the l! 4 ð Þ 2 2lþ2 ω worldline, which represent short-wavelength degrees of Clmja¼0 ¼ 2l ! 2l 1 ! ð MÞ i : ð2:11Þ ð Þ ð þ Þ R R 4 d Note thatR we writeR x ¼ d x as shorthand. Later, we will d 2π d Upon substitution into Eq. (2.7), we find that the spin- also write p ¼ d p=ð Þ for integrals over momentum independent part of the dipole moment is variables.

024010-4 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019) freedom (d.o.f.) living near the horizon [51,55,56]. Using dynamics, but we have also neglected to write these standard EFT reasoning, we then construct the effective down explicitly in Eq. (3.4) since their exact forms are action by writing down all possible terms that couple these unknown to us. Without detailed knowledge of their operators to the long-wavelength fields ðg; ϕÞ in a way that kinetic terms, integrating out qL leaves us with an effective is consistent with the symmetries of the theory. In this case, action expressed in terms of their correlation functions they are general covariance, worldline reparametrization hqLðτÞ…qL0 ðτ0Þi,5 which can be reconstructed through a invariance, and worldline SO(3) invariance. (We restrict series of matching calculations with the full theory. The attention to spherical black holes for simplicity; the gener- situation simplifies tremendously, however, if we assume alization to rotating ones is left for the future.) These steps that the dynamics of these operators is fully characterized lead us to the “intermediary” point-particle action by their two-point correlation functions. Far from being just Z X∞ Z convenient, this assumption is linked to the test-field L approximation in Sec. II and is thus valid under the Ip ¼ − M þ q ðτÞ∇Lϕ þ: ð3:4Þ τ τ l¼0 conditions outlined therein.

The first term is the familiar action for a point mass M.The B. Integrating out second term accounts for all possible interactions between the black hole and the real scalar field ϕ. Analogous terms Because we are interested in studying the real, causal that couple other worldline operators to the curvature tensors evolution of a system, rather than calculating in-out scatter- are also present, but these have been omitted from Eq. (3.4) ing amplitudes, the appropriate language required for inte- and will be neglected in this paper, since they become grating out the worldline operators is the in-in, or closed time – important only at much higher orders in perturbation theory path (CTP), formalism. (See Refs. [60 64] for classic texts – [50,57]. Note that conventional multi-index notation is being on the subject and Refs. [57,65 67] for applications similar used [58]: Theworldline operators arewritten as qL ≡ qˆ{1…ˆ{l , to the present context.) At its heart, this formalism converts the standard version of Hamilton’s variational principle, whereas ∇ ≡∇ˆ …∇ˆ denotes the action of multiple L {1 {l which is inherently a boundary value problem, into an initial covariant derivatives. The indices ˆ{ ∈ f1; 2; 3g label the value problem. It accomplishes this by doubling all dynami- three directions in the black hole’s rest frame that are μ cal d.o.f. Ψ → Ψ1; Ψ2 and allowing the two copies to mutually orthonormal to one another and to the tangent u ð Þ evolve independently subject to appropriate boundary con- of the worldline. ditions. Physical observables are obtained by making the Traces of ∇Lϕ are redundant operators; hence, they can −2 identification Ψ1 ¼ Ψ2 ¼ Ψ at the end. Following Galley be absorbed into redefinitions of qL n, where n counts the [68], we will refer to this identification as “taking the number of traces [55,56]. As a result, the worldline physical limit.” operators qLðτÞ can be taken to be symmetric and trace free (STF). The set of all STF tensors of rank l generates an irreducible representation of SO(3) of weight l [59]; thus, 1. Fixed worldlines μ L the worldline operators admit an interpretation as dynami- The d.o.f. of our EFT are Ψ ¼fz ;gμν; ϕ;q g, and we l 0 cal multipole moments of the black hole [51]. The ¼ wish to integrate out qL. It will be instructive to begin by τ operator qð Þ must therefore be responsible for the induced considering a simplified problem in which we fix the metric l 1 ˆ{ τ scalar charge, while the ¼ operator q ð Þ will lead to and worldline to be nondynamical. Under this restriction, l the induced dipole moment. The th operator, in turn, the intermediary point-particle action (3.4) reads corresponds to the lth multipole moment. As its name suggests, the intermediary point-particle Z action (3.4) is not yet the end of the story. At the moment, it Ip ¼ ðq1ϕ1 − q2ϕ2Þþ ð3:6Þ is comprised of both UV d.o.f., which a distant observer τ cannot directly probe, and the IR d.o.f. ðz; g; ϕÞ that we ultimately care about. While it is possible to perform when recast in the in-in formalism. We focus on the l 0 calculations directly with this action (see, e.g., ¼ operator to streamline the discussion, although the Refs. [51,56]), for our purposes it will be instructive— generalization to higher multipole moments is straight- ∈ 1 2 and more convenient—to integrate out qL and obtain a truly forward. Introducing CTP indices a; b f ; g allows us effective point-particle action: to write

Z ab a q1ϕ1 − q2ϕ2 ¼ c qaϕb ¼ qaϕ : L L Sp½z; g; ϕ¼−i log Dq expðiIp½z; g; ϕ;q Þ: ð3:5Þ

5Expectation values are taken with respect to the ground state Being dynamical variables in their own right, the worldline of the worldline theory, which corresponds to a classical, operators qL come with kinetic terms that govern their unperturbed black hole. Hawking radiation can be neglected.

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X∞ Z Note that all our d.o.f. Ψa ¼ðΨ1; Ψ2Þ innately come with a 1 χLL0 τ τ0 ∇ ϕA τ ∇ ϕB τ0 downstairs index; indices are raised with the CTP met- Sp ¼ AB ð ; Þ L ð Þ L0 ð Þ: ð3:14Þ 2 τ τ0 ab l¼0 ; ric c ¼ cab ¼ diagð1; −1Þ. The assumption that the dynamics of qðτÞ is fully characterized by its two-point functions implies that Eq. (3.5) is a Gaussian integral that can be evaluated 2. Dynamical worldlines exactly to yield Having gained a sense for how this calculation proceeds, Z Z let us now integrate out qL in the general case when all our 1 μ L a 0 a b 0 d.o.f. Ψ ¼fz ;gμν; ϕ;q g are dynamical. Complications Sp ¼ hqaiϕ þ χabðτ; τ Þϕ ðτÞϕ ðτ Þ: ð3:7Þ 2 0 τ τ;τ arise when there are two copies ðz1;z2Þ of the worldline for one If nonvanishing, the vacuum expectation value hq i in the black hole, each with their own proper times, since the a L τ τ first term describes a permanent scalar charge of the black operators q1 ð 1Þ appear to be living on the first copy z1ð 1Þ, L τ hole. From what we know of the no-hair theorems, this whereas q2 ð 2Þ live on the second. How, then, should we must be zero, leaving us with only the linear response in the integrate out these worldline operators, given that they second term. The matrix of two-point functions is [62,64] appear to be living on different spaces? The resolution comes by recalling that zμ are merely parametrizations in a given coordinate chart. The worldline χF χ− χ ¼ ð3:8Þ γ∶ I → M I ⊂ R ab χ χ itself is a map from the interval to the þ D bulk, four-dimensional manifold M. When there are two copies z , there are also two maps γ , but there is still only (see Appendix A for details on the individual two-point a a one underlying manifold I. Let λ or σ be the coordinate on functions) and satisfies the symmetry property I used to parametrize both copies of the worldline 0 0 simultaneously. The tangent to each worldline is written χabðτ; τ Þ¼χbaðτ ; τÞ: ð3:9Þ μ μ μ as z_a ¼ dza=dλ. (We reserve u for when the worldline is L ≡ In most circumstances, it is more convenient to work in a parametrized by its proper time.) The operators qa L τ I different basis called the Keldysh representation. Define the qa ð aÞ are pulled back onto via the map average and difference of our two copies as, respectively, Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ μ ν dza dza 1 τaðλÞ¼τaðλiÞþ dσ −ga;μν(zaðσÞ) ; ð3:15Þ Ψ ≔ Ψ Ψ Ψ ≔ Ψ − Ψ λ dσ dσ þ 2 ð 1 þ 2Þ; − 1 2: ð3:10Þ i Ψ Ψ Ψ 0 where it should be understood that the CTP index a above In the physical limit (PL), þjPL ¼ and −jPL ¼ . This transformation can also be written in index notation as is acting as a placeholder and is not to be summed over. We are always free to choose the lower integration limit λ and i 1 1 the initial value τaðλiÞ. The intermediary point-particle Ψ ¼ Λ aΨ ; Λ a ¼ 2 2 ; ð3:11Þ action thus reads A A a A 1 −1 Z ∈ − χ with A; B fþ; g. Similarly, CTP tensors like ab trans- Ip ¼ ½−Mτ_1ðλÞþτ_1ðλÞq1(τ1ðλÞ)ϕ1(z1ðλÞ) − ð1 ↔ 2Þ: a b λ form as χAB ¼ ΛA ΛB χab. Using the identities in Eq. (A2), ð3:16Þ 1 χ χ χ 2 H R : 3:12 AB ¼ ð Þ As before, we focus only on the l ¼ 0 operator, since it is χA 0 straightforward to generalize the following steps for l ≥ 1. Because the transformation is linear, the identity in Clearly, Eq. (3.16) suggests we need better notation. To Eq. (3.9) holds also in this basis. Indices can still be raised that end, we begin by generalizing the CTP metric to a set and lowered with the CTP metric, which in this represen- of tensors defined by tation reads 8 > 1 1 <> þ a1 ¼ a2 ¼¼an ¼ ; 01 … AB a1 an −1 2 c ¼ cAB ¼ : ð3:13Þ c ¼ > a1 ¼ a2 ¼¼an ¼ ; ð3:17Þ 10 :> 0 otherwise: Repeating similar steps for the higher multipole moments and using the no-hair theorems to infer that With these at our disposal, one can verify by direct hqLi¼0, in general we have evaluation that Eq. (3.16) is equivalent to

024010-6 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019) Z Z Λ A Λ a Λ AΛ b δb − aτ_ λ λ J a λ where a is the inverse of A , satisfying a A ¼ a. Ip ¼ M c að Þþ qað Þ ð Þ; ð3:18Þ AA0 λ λ Having explicit expressions for X will be useful. The same argument that allowed us to neglect χH earlier also given sources J a defined by allows us to neglect X −−. Taken together with the sym- Z metry property in Eq. (3.23), we conclude that it suffices to a abcd ð4Þ know only the following two components: Δ ðλ; xÞ ≔ c δ(λ − τbðσÞ)δ (x − zcðσÞ)τ_dðσÞ; Zσ Z 1 J a λ ≔ abcΔ λ; ϕ þþ 0 ð Þ c bð xÞ cðxÞ: ð3:19Þ X ðx; x Þ¼ ½2χRðΔ1Δ10 − Δ2Δ20 Þ x 2 λ;λ0 −χ Δ Δ Δ − Δ In this form, Eq. (3.18) is reminiscent of the simplified Cð 1 þ 2Þð 10 20 Þ; ð3:25aÞ problem in Sec. III B 1, apart from two minor differences: Z The manifold I is parametrized by λ rather than τ, and the 1 X −þ 0 2χ Δ Δ Δ Δ scalar field ϕa is here replaced by J a. These prove to be no ðx; x Þ¼ ½ Rð 1 10 þ 2 20 Þ 4 λ;λ0 obstacle to evaluating the functional integral, which yields −χ Δ1 − Δ2 Δ10 − Δ20 : 3:25b Z Z Cð Þð Þ ð Þ 1 a 0 a a0 0 Sp ¼ −M c τ_aðλÞþ χaa0 ðλ; λ ÞJ ðλÞJ ðλ Þ: λ 2 λ;λ0 In both cases, judicious use of Eqs. (A1) and (A2) has been made to express X AA0 only in terms of the retarded ð3:20Þ propagator χR and the commutator χC (and χH, which is then discarded). The definition in Eq. (A1g) can then be Before proceeding any further, let us remark that the χ ≡ χ used to infer that Hadamard propagator H þþ appears in this action þ flanked by two powers of J ≡ J −, which vanishes in χ τ τ0 − χ τ0 τ χ τ τ0 the physical limit. This implies that when we extremize the Rð ; Þ Rð ; Þ¼ Cð ; Þ; ð3:26Þ action S to obtain the equations of motion for the system, 6 χH will never contribute; thus, we set χH ¼ 0 from now on. which in Fourier space reads The hard work is over at this point, but the result in ˜ ˜ ˜ ˜ Eq. (3.20) is not yet written in a form convenient for χCðωÞ¼χRðωÞ − χRð−ωÞ¼2iImχRðωÞ: ð3:27Þ calculations. Specifically, we want to make manifest its ϕ dependence on a. Using the definitions in Eq. (3.19),we Thus, once we know χ , we also know χ . write R C Z Z 1 C. Matching calculations a aa0 0 0 Sp ¼ −M c τ_aðλÞþ X ðx; x ÞϕaðxÞϕa0 ðx Þ; λ 2 x;x0 So far, the effective action we have constructed is fully ð3:21Þ generic and can account for finite-size effects of any spherical compact object interacting with a real scalar expressed in terms of the correlation functions field. We will now specialize to black holes exclusively Z by fixing the form of the retarded propagator. On general grounds, we expect χ to depend on both the X aa0 0 ≔ abc a0b0c0 Δ Δ χ R ðx; x Þ c c b b0 cc0 : ð3:22Þ black hole mass M and the scalar field mass μ. Dimensi- λ;λ0 onal analysis and the assumption of spherical symmetry are This is the desired end result. In the definition above, the sufficient to deduce that two-point functions all depend on the same argument, χ ≡ χ λ λ0 χ˜LL0 ω δLL0 2lþ1 ω μ cc0 cc0 ð ; Þ, and a primed index denotes dependence R ð Þ¼ ðGMÞ FlðGM ;GM Þ; ð3:28Þ on primed variables, i.e., Δb ≡ Δbðλ; xÞ whereas Δ ≡ Δ λ0; 0 0 b0 b0 ð x Þ. As a generalization of Eq. (3.9),itis where δLL is the identity on the space of STF tensors of easy to show that rank l. The dimensionless functions Fl almost certainly depend in a complicated way on their arguments. However, X aa0 0 X a0a 0 ðx; x Þ¼ ðx ;xÞ: ð3:23Þ for low-frequency sources, we can expand in powers of the first argument to obtain These correlation functions can be written in the Keldysh representation by utilizing the transformation rule 6Since the worldline is reparametrization invariant, the two- τ τ0 τ − τ0 X AA0 0 X aa0 0 Λ AΛ A0 point functions depend on and only via their difference , ðx; x Þ¼ ðx; x Þ a a0 ; ð3:24Þ making them amenable to a Fourier transform.

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X∞ part of the field action not included in Z0. Further details ð2nÞ ω 2n ð2nþ1Þ ω 2nþ1 Fl ¼ ½Fl ðGM Þ þ iFl ðGM Þ can be found in Appendix B. n¼0 At leading order, hφðxÞi is sourced only by terms in Sp ð0Þ ð1Þ ð2Þ 2 ¼ Fl þ iFl GMω þ Fl ðGMωÞ þ; ð3:29Þ that are linear in φ. Moreover, the worldline can be held fixed when computing field expectation values; hence, it ðnÞ suffices to work with the simplified action in Sec. III B 1. where the dimensionless coefficients Fl themselves admit an expansion in the remaining argument GMμ. Naturally, Substituting the field decomposition (3.30) into the action the finite size of the black hole sets the UV cutoff for this (3.14), we obtain EFT, and only the first few terms in this expansion are X∞ Z needed in practice when GMω ≪ 1. It is also worth χLL0 τ τ0 ∇ φ τ ∇ Φ τ0 O φ2 Sp ¼ R ð ; Þ L −ð Þ L0 ð Þþ ð Þ; τ τ0 remarking that this series cannot capture nonperturbative l¼0 ; effects like quasinormal-mode resonances, but we do not ð3:33Þ expect such effects to be important in the low-frequency limit. The terms in Eq. (3.29) even in ω are time-reversal having used Eq. (3.9) to simplify terms. Using the Fourier symmetric and constitute what is called the “reactive” part χ ð1Þ of the black hole’s response. On the other hand, the odd representation of R and concentrating on the F0 term for terms break time-reversal symmetry and are responsible for now, we find dissipative processes. Z Z We can now determine the values of each of these ð1Þ 2 −iωðτ−τ0Þ 0 Sp ⊃ ½F0 ðGMÞ iω þe φ−ðτÞΦðτ Þ Wilsonian coefficients by a matching calculation. To make τ;Zτ0 ω contact with our results in Sec. II, we ought to compute ð1Þ 2 _ ¼ − F0 ðGMÞ φ−ðτÞΦðτÞþ; ð3:34Þ field expectation values. While working with the full fields τ ðg; ϕÞ earlier was advantageous to manifestly preserve general covariance, to compute observables, we split where the second line follows from integrating by parts. ð1Þ The Wilsonian coefficient F0 characterizes the leading- hA;μν order, low-frequency dissipative response and is respon- ϕA ¼ ΦA þ φA;gA;μν ¼ g¯A;μν þ : ð3:30Þ mPl sible for the induced scalar charge of the black hole. To see this, we compute The background fields ðg¯A; ΦAÞ describe a scalar-field environment that persists independently of the black hole. φ φ h ðxÞi¼h þðxÞijPL As these fields are nondynamical, we can immediately fix Z δ2Z 3 2 ð1Þ _ 0½j; J ¼ð−iÞ ðGMÞ F0 ΦðτÞ − ¯ Φ ¯ Φ ¯ Φ 0 τ δjþ x δj z τ ðgþ; þÞ¼ðg; Þ; ðg−; −Þ¼ : ð Þ ð ð ÞÞ ðj;JÞ¼0 1 Z Fð Þ Moreover, we will no longer have need to refer to the full 0 τ τ ¼ 16π GRðx; zð ÞÞQð Þ; ð3:35Þ metric explicitly, so let us drop the overbars and denote the τ background metric by gμν. Being much smaller than its where G is the retarded propagator for the scalar field. The environment, a black hole sources fluctuations ðh; φÞ in the R way this is written suggests that fields that can be treated perturbatively. Expectation values of these fields can be computed by ð1Þ 16π taking appropriate derivatives of the generating functional F0 ¼ ; ð3:36Þ Z and indeed this is true. We verify this by considering (so as Z φ Φ φ ½j; J¼ DhD expðiS½z; g þ h=mPl; þ Þ to reproduce the scenario in Sec. II) a black hole at rest at Z the origin, zμðτÞ¼ðτ; 0Þ, around which the background pffiffiffiffiffiffi μν − φA A field behaves as × exp i gð jA þ hμνJA Þ ; ð3:31Þ x 2 2 μν ΦðxÞ¼Φ0 cos ωt þðb · xÞ cos ωt þ Oðr =R Þ: where ðjA;JA Þ are arbitrary sources. This is approximated in perturbation theory by working with Recall that R denotes a typical length scale of the back- b ∼ O 1 R int ground, j j ð = Þ, and we will further assume a Z ð Þ Z 2 2 ½j; J¼expðiSf þ iSpÞ 0½j; J; ð3:32Þ gravitationally bound state such that ω < μ . Moreover, let us suppose that Φ ∼ OðεÞ is sufficiently weak not just in Z where 0½j; J is the (gauge-fixed) generating functional the vicinity of the black hole but everywhere in spacetime, ðintÞ for the propagators of the free fields and Sf denotes the such that the background admits the weak-field expansion

024010-8 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019) g ¼ η þ H, where H ∼ Oðε2Þ is the backreaction of Φ onto We can now deduce the following by induction: Power ðnÞ the geometry. To leading order in ε, it suffices to evaluate counting indicates that the coefficient Fl is responsible the integral in Eq. (3.35) on flat space. Integrals of this form for effects appearing at order ðGM=RÞlðGMωÞn at the will need to be evaluated many times in this paper, and the earliest. Being accurate only to first order in GMω, the general technique is reviewed in Appendix C. The result is limitations of our results in Sec. II preclude determining the values for any coefficient with n ≥ 2. The n ¼ 1 pffiffiffiffiffiffiffiffiffiffi QðtÞ − μ2−ω2 coefficients have a one-to-one mapping with the objects hφðxÞi ¼ e r; ð3:37Þ 4πr Clmja¼0 in Eq. (2.11), so can all be determined, up to corrections in GMμ, by following the same procedure that in total agreement with the full theory. Note that the led to Eqs. (3.36) and (3.38). Yukawa suppression is to be expected here, despite it 0 For the n ¼ coefficients, the vanishing of Clmja¼0 in not featuring in our results in Sec. II, since the latter the static limit ω → 0 implies concentrated only on distances r ≪ maxðω−1; μ−1Þ. The same procedure can be repeated for the higher multi- ð0Þ Fl ≃ 0 ∀ l; ð3:40Þ pole moments; the spin-independent dipole moment in Eq. (2.12), for instance, is reproduced by our EFT provided up to possible corrections quadratic in GMμ. These coefficients are the scalar analog of a black hole’s tidal ð1Þ F1 ¼ 16π=3: ð3:38Þ Love numbers, and Eq. (3.40) implies that they vanish identically when μ ¼ 0. (The same result is obtained in What about the other Wilsonian coefficients? To start Ref. [70] by different means.) It is well known that the with, consider the following three terms also present in the (gravitational) tidal Love numbers also vanish [70–74], effective action: which in the EFT translates to the vanishing of analogous Z Wilsonian coefficients that couple the black hole to the ð0Þ ð2Þ 3 ̈ curvature tensors. This presents a fine-tuning problem, as Sp ⊃ ðF0 GMΦ − F0 ðGMÞ Φ τ there is no apparent symmetry in the EFT that would make ð0Þ 3 ˆ{ this vanishing technically natural [75,76]. A potential þ F1 ðGMÞ ∂ Φ∂ˆ þÞφ−: ð3:39Þ { resolution has recently been put forward [77], but for These constitute the most relevant terms characterizing the now we will just accept Eq. (3.40) at face value. (Note that reactive part of the response. Two comments are worth for scalars, this problem is unrelated to the no-hair making at this stage: First, this part of the action could just theorems, which only tell us that there are no permanent L 0 as easily have been constructed by writing down all allowed scalar multipole moments; hq i¼ .) contractions between uμ, the fields ðg; ϕÞ, and their derivatives (see, e.g., Ref. [69]). This bottom-up approach D. Worldline vertices cannot account for dissipative processes, however, hence When working to leading, nontrivial order in the our more comprehensive and systematic route of integrat- separation-of-scale parameters, it suffices to keep only ð1Þ ing out worldline operators. Our second comment is that the F0 coefficient. At this order, the retarded propagator Eq. (3.39) is exactly the action Horbatsch and Burgess [22] for qðτÞ is simply took to be responsible for the induced scalar charge, but Z from what we have learned this cannot be true. The 0 −iωðτ−τ0Þ χRðτ; τ Þ¼A iωe ; ð3:41Þ conclusions of their paper are nonetheless still valid, since ω their arguments do not rely on a specific form for the action. φ ð0Þ Computing h ðxÞi as before, we find that the F0 term while its commutator χC is just twice that. In fact, when generates a scalar-field profile due to a charge proportional written in this way, Eq. (3.41) is valid not only for spherical ð0Þ to Φ, whereas the F1 coefficient is responsible for a dipole black holes, but for rotating ones as well. We conclude this section by substituting Eq. (3.41) back moment proportional to ∂iΦ. Neither of these features are present in the full theory; thus, demanding consistency with into the point-particle action Sp to obtain simplified the predictions of general relativity forces us to conclude expressions for the worldline vertices. This process will ð0Þ ð0Þ also help elucidate the rich physical content currently that F ¼ F ¼ 0. More precisely, these coefficients are 0 0 1 hidden in the correlation functions X AA x; x0 . We begin zero up to possible quadratic-order corrections in GMμ, ð Þ by decomposing the fields according to Eq. (3.30) to obtain since our calculations in Sec. II are accurate only to linear the expansion order in GMμ and GMω. Accordingly, the value of the 2 ð Þ ∞ ∞ coefficient F0 , which predicts a contribution to the scalar X X n ;nφ 2 ̈ 2 ð h Þ charge proportional to ðGMÞ Φ ∼ O(ðGMωÞ ), cannot be Sp ¼ Sp ; ð3:42Þ 0 0 determined at present. nh¼ nφ¼

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FIG. 1. Examples of worldline vertices. The graviton h is drawn as a helical line, the scalar φ is drawn as a dashed line, and each insertion of the background scalar Φ is denoted by a dotted line terminating in a circle. The black hole worldline, which is held nondynamical while the fields are being integrated out, is depicted as a solid line. The physical interpretation for each vertex is as follows: (a) kinetic term for the black hole leading to the geodesic equation, (b) correction to the kinetic term due to accretion of the background scalar, (c) a graviton sourced by a black hole of constant mass, (d) correction to the graviton vertex due to mass growth by accretion, and (e) induced scalar charge of the black hole.

where the integers ðnh;nφÞ count the number of field 2. Graviton terms perturbations appearing in each term. Diagrammatic rep- Two terms appear in the point-particle action that are resentations for the first few in this series are drawn linear in the graviton h. In both cases, they emerge from in Fig. 1. having expanded the metric appearing in the definition of τ_ τ_ δτ_ O 2 the proper time, aðg þ h=mPlÞ¼ aðgÞþ a þ ðh Þ. 1. Scalar terms The first-order piece is The scalar field enters the point-particle action only Z 1 pffiffiffiffiffiffi μν through the second term in Eq. (3.21). Decomposing ϕ δτ_1ðλÞ¼− −gh1 μνðxÞt ðx; λÞ; ð3:47Þ 2m ; 1 according to Eq. (3.30), it becomes Pl x

Z with a similar expression holding for δτ_2 after relabeling 1 μ þþ 0 Aþ 0 AA0 1 ↔ 2 _ _ _ _ν ðX ΦΦ þ 2X φAΦ þ X φAφA0 Þ; ð3:43Þ . Writing gðz1; z1Þ¼gμνðz1Þz1z1 as shorthand, 2 x;x0 μ ν ð4Þ μν z_1z_1 δ (x − z1ðλÞ) having used Eq. (3.23) to simplify the second term and t1 ðx; λÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ð3:48Þ − _ _ −g writing Φ0 ≡ Φðx0Þ for brevity. Note that the full metric is gðz1; z1Þ still hiding in X AA0 , so this can be further expanded to is the contribution to the energy-momentum tensor of a unit generate an infinite series of terms with n ≥ 0 and h point mass when it is at the position λ along the worldline 0 ≤ nφ ≤ 2. Here, we concentrate on terms that depend φ z1. The total energy-momentum tensor of a point mass M is only on . then obtained by simply integrating over the worldline: A discussion of the first term in Eq. (3.43) is postponed φ Z until Sec. IV. The second term, linear in and drawn in μν μν Fig. 1(e), sources the induced scalar charge and can be Ta ðxÞ¼M ta ðx; λÞ: ð3:49Þ rewritten as λ Z Substituting this expansion into the point-mass term ffiffiffiffiffiffi R ð0;1Þ p A −M caτ_ ⊂ S , we get the familiar contribution Sp ¼ −gQ ðxÞφAðxÞð3:44Þ a p x Z 1 ffiffiffiffiffiffi ð1;0Þ ⊃ p− a μν upon defining the induced charge density of the black hole, Sp 2 ghμνðxÞTa ðxÞ: ð3:50Þ mPl x Z 1 A A This vertex is drawn in Fig. 1(c). Even without explicit Q ðxÞ ≔ pffiffiffiffiffiffi X þðx; x0ÞΦðx0Þ: ð3:45Þ −g x0 calculation, we know that this term sources the gravita- tional potential ∼GM=r of the black hole. ð1;0Þ The reader will not be surprised to learn that, in the physical The second contribution to Sp comes from the term limit, Z X þþ 0 Φ Φ 0 Z 4 ðx; x Þ ðxÞ ðx Þ: ð3:51Þ δð Þ( − τ ) 0 Q x zð Þ τ Q 0 x;x þðxÞjPL ¼ pffiffiffiffiffiffi Qð Þ; −ðxÞjPL ¼ : τ −g To unpack this, substitute in Eq. (3.25a) and integrate over ð3:46Þ the delta functions contained in Δa. Most of the terms will vanish, since χR is purely dissipative at leading order, so is This result is derived in Appendix A. therefore odd under time reversal. By definition, χC is also

024010-10 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019) odd under time reversal. One therefore finds that the only between a black hole’s scalar charge and its accretion rate nontrivial part of Eq. (3.51) is can be viewed as a special case of the fluctuation-dis- Z sipation theorem. 1 What about the second term in Eq. (3.53)? It is a constant τ_1τ_20 χCðτ1; τ20 ÞΦðz1ÞΦðz20 Þ: ð3:52Þ 2 λ;λ0 contribution to the black hole mass, but one that generically diverges in the limit λf → ∞. Physically, this IR divergence Recall, for brevity, that (un)primed indices denote is signaling the breakdown of our EFT at late times. This functions of (un)primed variables; e.g., τ1 ≡ τ1ðλÞ whereas makes intuitive sense, since an increase in the black hole’s 0 z20 ≡ z2ðλ Þ. At this stage, we can expand the metric mass must be compensated for by a depletion of the entering via the proper times to first order in h. surrounding scalar-field environment. Eventually, the black Technical details of this derivation are relegated to hole will grow to be nearly as massive as its dwindling Appendix A. The end result is environment, at which point there is no longer a good Z separation of scales. Accordingly, we should only trust this 1 0 Sð ; Þ ⊃− δτ_a λ δM λ − δM λ ; 3:53 EFT for a limited duration of time. Within its period of p ð Þ½ að Þ að fÞ ð Þ δ λ λ validity, it is safe to just absorb Mþð fÞ into a renorm- alization of the constant M appearing in the Lagrangian, δ where the function M1 is defined by such that M represents the mass of the black hole at the Z Z λ λ point when initial conditions are specified. _ f 0 _ 0 δM1ðλÞ ≔ A dσΦ(z1ðσÞ) dσ Φ(z2ðσ Þ) Another way to see that our EFT cannot be valid for all λ λ i i times is to differentiate Eq. (3.56) to obtain the accretion 0 × δ(τ1ðσÞ − τ2ðσ Þ): ð3:54Þ rate _ _ 2 One obtains the definition for δM2 by interchanging 1 ↔ 2. δMðτÞ¼AΦ (zðτÞ): ð3:57Þ The integration limits ðλi; λfÞ appearing in these formulas are the initial and final times at which appropriate boundary Notice that the horizon area A appearing on the rhs is that conditions are specified according to the in-in formalism. defined at some fixed time. This is only a good approxi- mation provided δM ≪ M. A more precise formula would Using the expression for δτ_a in Eq. (3.47), the first term in Eq. (3.53) yields see the constant A replaced by the instantaneous area AðτÞ, Z Z but doing it properly would require a resummation involv- 1 0 ffiffiffiffiffiffi h μνðxÞ μν ing higher-order terms. It will be interesting to explore how ð ; Þ ⊃ p− a; abcδ λ ; λ Sp g c Mbð Þtc ðx Þ: ð3:55Þ to do so in the future, but in practice we expect typical x 2m λ Pl scalar-field environments to be dilute enough that When compared with Eq. (3.50), we recognize that this Eq. (3.57) is a valid approximation for long enough periods vertex, drawn in Fig. 1(d), describes a graviton sourced by a of time. black hole whose mass is slowly growing due to accretion of the background scalar. Indeed, in the physical limit, the IV. WORLDLINE DYNAMICS increase in mass as a function of the proper time is Having successfully constructed our effective action, we Z τ now wish to understand its phenomenological implications. δ τ δ τ0Φ_ 2( τ0 ) Mð Þ¼ MþjPL ¼ A d zð Þ ; ð3:56Þ Two classes of observables are worth calculating in this τ λ ð iÞ theory: field expectation values, which tell us about gravitational and scalar radiation, and the equation of which is exactly what we would predict from the full theory motion for the worldline. The general method for comput- by calculating the flux of the scalar across the horizon ing the former has already been discussed in Sec. III C.For [19,21,78,79]. What is remarkable here is that we did not instance, Eq. (3.35) can be used to determine the profile of put this result in by hand. After performing matching scalar waves (at leading order) radiated by a black hole calculations to reproduce the correct behavior of the scalar μ traveling along some worldline z ðτÞ. charge, our EFT immediately gives us the correct accretion To determine the trajectory of this worldline, we inte- rate for free. This is proof that our formalism is working grate out the bulk fields to obtain a new effective action correctly and, more importantly, that the physics governing [57,65,66] these two effects are one and the same. Indeed, their Z magnitudes are both set by the same Wilsonian coefficient ð1Þ Γ½z ¼−i log Dh Dφ expðiSÞ F0 ¼ 16π. Interestingly, this coefficient manifests as a scalar charge when it appears in the retarded propagator χR ð0;0Þ sum of connected but is responsible for setting the accretion rate when ¼ Sp þ : ð4:1Þ Feynman diagrams appearing in the commutator χC. In this light, the relation

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Its equation of motion is then obtained from the extrem- We now have to expand this to first order in z−. There are ization condition two routes from which z− emerges: from expanding the proper times τ_ → τ_ þ δτ_ and from expanding the arguments

δΓ of the background scalar Φ. The method for performing the μ ¼ 0: ð4:2Þ δz− first of these expansions has already been established, with PL the final result given in Eq. (3.53). After renormalizing the The sum of Feynman diagrams in Eq. (4.1) stems from IR-divergent part, we find the backreaction of the black hole onto the background Z Z ð0;0Þ a _ μ fields, leading to a number of self-force effects including Sp ⊃− δτ_ δMa ¼ − ðδMaμ − δMuμÞz−: ð4:7Þ radiation reaction from the emission of gravitational and λ τ scalar waves. If present, interactions with other compact Second, we expand the arguments of Φ and use the objects would also appear in this sum. We believe there is antisymmetry property of χ to obtain little to be gained from discussing these terms in generality C Z here. Rather, they are better understood through examples 1 ð0;0Þ ⊃− χ τ τ0 Φ( τ )∂ Φ( τ0 ) μ and so are left to be explored further in Sec. V. Sp 2 Cð ; Þ zð Þ μ zð Þ z− In this section, we concentrate on the part of the equation Z τ;τ0 ð0;0Þ of motion for the worldline arising from S ⊂ Γ, which μ p ¼ QðτÞ∂μΦ(zðτÞ)z−; ð4:8Þ applies universally to black holes embedded in any scalar- τ field environment. This part of the action reads where the second line follows after writing χ in Fourier Z Z C 1 space and then integrating by parts. ð0;0Þ − aτ_ X þþ 0 Φ Φ 0 Sp ¼ M c a þ ðx; x Þ ðxÞ ðx Þ: ð4:3Þ Combining the results in Eqs. (4.5), (4.7), and (4.8),we λ 2 0 x;x learn that the equation of motion for the worldline (neglecting backreaction effects) is The two terms are drawn in Figs. 1(a) and 1(b), respec- ≔ tively. Note that this action is a functional of zþ μ _ μ μν ½M þ δMðτÞa ¼ −δMðτÞu þ QðτÞg ∂νΦ: ð4:9Þ ðz1 þ z2Þ=2 and z− ≔ z1 − z2, which give the average and difference of the coordinates of the two worldline The terms involving δM administer a drag force on the black copies ðz1;z2Þ, but do not themselves correspond to world- hole due to accretion, whereas the remaining term involving a lines. Of course, the average coordinate tends to a descrip- derivative on Φ must be interpreted as a scalar fifth force. tion of the physical worldline, z j ¼ z, whereas þ PL The reader familiar with scalar-tensor theories will find this z− 0. The latter suggests that we can easily solve jPL ¼ last term a little odd, seeing as the fifth force usually appears Eq. (4.2) by Taylor expanding the action in powers of μν μ ν in the equation of motion as Qðg þ u u Þ∂νΦ [80]. In fact, z− and reading off the linear coefficient. τ_ 2 we can easily put Eq. (4.9) into such a form since, by Performing this expansion for 1 (note z1 ¼ zþ þ z−= ), 2 definition, we obtain τ_1ðz1Þ¼τ_1ðzÞþδτ_1 þ Oðz−Þ, where _ _ 2 _ ν δM ¼ AΦ ¼ −QΦ ¼ −Qu ∂νΦ: ð4:10Þ 1 μ d μ δτ_1 ¼ aμz− − ðuμz−Þ ; ð4:4Þ 2 dτ Thus, an equivalent way of writing Eq. (4.9) is

μ α μ μ μν μ ν with a ≔ u ∇αu denoting the acceleration of the world- ½M þ δMðτÞa ¼ QðτÞðg þ u u Þ∂νΦ: ð4:11Þ line. Being interested only in the physical limit, we have → already taken the liberty of sending zþ z and para- In Sec. III D 2, we saw that the physics of the scalar charge metrizing it by the proper time τ. The result for δτ_2 is and of accretion were one and the same, having emerged similar up to the change of sign z− → −z−. Using this from the same term in the point-particle action. Here, this expansion, the point-mass term in the action gives connection is made manifest at the level of the equations of Z Z motion: The scalar fifth force due to this charge includes the a μ 2 drag force due to accretion. It is impossible for one to exist −M c τ_a ¼ −M aμz− þ Oðz−Þ: ð4:5Þ λ τ without the other.

As for the second term in Eq. (4.3), we demonstrated in V. BINARY BLACK HOLES Sec. III D 2 that it simplifies to IN FUZZY DARK MATTER HALOS Z 1 We have so far been limited in our discussion to the τ_ τ_ χ τ τ Φ Φ general features of our EFT, which apply universally to 2 1 20 Cð 1; 20 Þ ðz1Þ ðz20 Þ: ð4:6Þ λ;λ0 black holes embedded in any scalar-field environment.

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There is further insight to be gleaned from specializing to circumstances, the virial theorem relates the orbital sepa- concrete systems. To complete this paper, we explore one ration of the binary to the typical size GM and the such example involving a black hole binary embedded in a characteristic velocity v of its constituents; v2 ∼ GM=a. galactic fuzzy dark matter (FDM) halo. While the calcu- For most of its inspiral, v ≪ 1, allowing us to study the lations in this section apply to astrophysical black holes of evolution of this system in the nonrelativistic, post- any size, our focus will center on supermassive black holes, Newtonian (PN) limit. for which effects stemming from the scalar charge Q are the Furthermore, when v is small, the system neatly sepa- largest, since Q ∝ A. rates into a “near zone” and a “far zone.” Following Galactic halos in FDM models consist of a central Refs. [50,87,88], these two zones are dealt with one at a (pseudo)solitonic core that is surrounded by an envelope time by constructing a tower of EFTs. To that end, we split of fluctuating density granules arising from wave interfer- ence [81–84]. The core resists further gravitational collapse ðh; φÞ → ðh;¯ φ¯ Þþðh; φÞ: by coherently oscillating in time at a frequency ω that is ’ ω ≈ μ 7 essentially set by the scalar s mass, , and has a The fields ðh; φÞ on the rhs represent potential modes that typical length scale R determined by the scalar’s de Broglie mediate forces in the near zone (at distances r ∼ a), wavelength, whereas ðh;¯ φ¯ Þ denote radiation modes in the far zone ≳ μ −1 −1 (r a=v). The former are always off shell, whereas the R ∼ 400 vvir pc −22 −1 ; latter can go on shell and propagate to infinity. The 10 eV 300 km s potential modes are integrated out first to obtain a new effective action governing the dynamics of the binary where vvir denotes the virial velocity of the halo. coupled to the remaining radiative d.o.f., As galaxies merge, the black holes at their centers form a Z binary that inspirals for eons before ultimately coalescing ¯ φ¯ − φ [85]. In this section, we use our EFT to determine how the Seff½zκ; h; ¼ i log DhD expðiSÞ; ð5:2Þ binary’s early inspiral is affected when situated inside 8 P an FDM halo’s core. For simplicity, we will focus 9 where S ¼ S þ κS κ is the original gauge-fixed effec- exclusively on systems for which the orbital separation a f p; tive action [cf. Eq. (3.2)] and the index κ ∈ f1; 2g labels the is much smaller than the typical length scale R of the 1010 individual members of the binary. The flux of radiative background. Even a gargantuan M⊙ black hole has a modes off to infinity can be calculated at this stage using radius that extends only to a few milliparsecs; thus, it is S . The effective action for the self-consistent motion of easy to envision comfortably fitting not just one black hole, eff the worldlines is obtained after also integrating out the but a binary of supermassive black holes within such a radiation modes [cf. Eq. (4.1)]: distance. Calculations are straightforward in this regime because the constituents of the binary perceive a local Z Γ − ¯ φ¯ environment that is effectively spatially homogeneous: ½zκ¼ i log DhD expðiSeffÞ: ð5:3Þ Φ Φ μ Υ O R ¼ 0 cosð t þ Þþ ða= Þ; ð5:1Þ A convenient way to perform these integrations in perturbation theory is with the use of Feynman diagrams, where Υ is some arbitrary phase. Let ε ¼ Φ0=m be a Pl which can be organized to scale in a definite way with the dimensionless parameter that characterizes the local density expansion parameters of our EFT. Schematically, each term of this halo. Typical FDM halos satisfy the condition ε ≪ 1 in the effective action Γ scales as [see also Eq. (5.15) later]; hence, the scalar field backreacts onto the geometry only weakly. As a result, we can expand Γ ∼ L1−lv2nεp1 ðGMμÞp2 ; the background metric as g ¼ η þ H about Minkowski space, where H ∼ Oðε2Þ is the gravitational potential of the where L ∼ Mav is the characteristic angular momentum of halo. the binary. The integer l counts the number of loops in a Provided that background gradients ∂H are not given Feynman diagram, and since L ≫ 1 for astrophysical too strong (a more precise statement will be made in black holes, only the tree-level contributions are needed Sec. VA4), the dominant force acting on the black holes n is still their mutual gravitational attraction. In such [50]. The integer or half-integer counts the order in the usual PN expansion, which is supplemented by two addi- tional parameters, ε and GMμ, that characterize the impact 7This relation holds up to small, negative corrections from a nonrelativistic binding energy [30–34], which we neglect. 8Binaries outside the core may find their orbital inspiral stalled 9We gauge fix the potential mode h with respect to the ¯ at kiloparsec scales due to interactions with FDM fluctuations, background g ¼ η þ H þ h=mPl to preserve gauge invariance which pump energy into the orbit [38,86]. of Seff [50].

024010-13 WONG, DAVIS, and GREGORY PHYS. REV. D 100, 024010 (2019) Z 10 X of the scalar-field environment on the binary. The terms QA xi∂ φ¯ 0 κ ðxÞ i AðxÞ: ð5:8Þ with p1 ¼ p2 ¼ constitute the standard PN equations for κ x a binary in vacuum [58] and need not be revisited here. Effects involving the scalar field first appear when In the physical limit, this leads to the expectation value 2 p1 ¼ p2 ¼ . As in earlier parts of this paper, we work Z only at leading nontrivial order; hence, our EFT is, in fact, ∂ φ¯ ⊃− x; 0 0 Pi 0 organized as an expansion in just two small parameters: v h ðxÞi i GRðt; t ; Þ ðt Þ; ð5:9Þ ∂x t0 and εGMμ. sourced by the binary’s scalar dipole moment A. Phenomenology X i i In what follows, we discuss five distinct physical effects P ðtÞ¼ QκðtÞzκðtÞ: ð5:10Þ that arise when a black hole binary is embedded in an FDM κ halo. Concomitantly with some explicit calculations, we also establish power counting rules to determine the order The master integral in Appendix C can be used to at which they appear in the PN expansion. As the effects we evaluate Eq. (5.9). Keeping only the radiative part that discuss span a range of 4.5PN orders, a comprehensive and reaches an observer at infinity, we find systematic expansion of Γ in powers of v is far beyond the Z 1 ∞ dω scope of this paper. We will limit ourselves to deriving only φ¯ ⊃− 2 xˆ P˜ ω −iðωt−krÞ h ðxÞi 4π Re 2π · ð Þike ; ð5:11Þ the leading-order expression for each effect. r μþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 − μ2 1. Scalar dipole radiation where the wave number k ¼ . Finally, we integrate the ðt; rÞ component of the scalar’s energy- It is only fitting that we begin our discussion with the momentum tensor over a spherical shell of radius r and phenomenon that started it all. In the PN limit, the discard terms that vanish in the limit r → ∞ to obtain the t coordinate time can be used to parametrize the worldlines; radiated power hence, the charge densities in Sec. III D 1 reduce to Z A A 2 2 ¯ ¯ Qκ ðxÞ¼QκðtÞδκ ðxÞð5:4Þ F ϕ ¼ −r d Ω∂rhφi∂thφi: ð5:12Þ at leading order in v, where the delta function For a circular binary with orbital frequency Ω, the flux at a

ð3Þ distance r is δ κðxÞ ≔ δ (x − z κðtÞ) ð5:5Þ a; a; 16π 2 − 2 mPl 2 8=3 2 M1 M2 localizes the integral to be along the ath copy of the κth F ϕ ¼ ðεGMμÞ ðGMΩÞ ν 3 M worldline. For the background in Eq. (5.1), the scalar μΦ μ Υ Ω4 Ω4 charge is QκðtÞ¼Aκ 0 sinð t þ Þ. The radiation mode 3 þ 3 − v θ Ω − 2μ v− φ¯ couples to the binary via the term × þ Ω4 þ ð Þ Ω4 Z Ω2 Ω2 X þ − ⊃ QA φ¯ −θðΩ − 2μÞv v−ðv þ v−Þ cos ϖ ; ð5:13Þ Seff κ ðxÞ AðxÞ: ð5:6Þ þ þ Ω4 κ x

where M ¼ M1 þ M2 is the total mass of the binary and Definite scaling in v is achieved by multipole expanding ν 2 the radiation mode as [50] ¼ M1M2=M is its symmetric mass ratio. Four worthy observations can be made here: First, the φ¯ t; x φ¯ t; 0 xi∂ φ¯ t; 0 5:7 terms in square brackets signify that scalar waves emanate Að Þ¼ Að Þþ i Að Þþ ð Þ Ω Ω μ at two frequencies, ¼ . This is to be expected ˜ i about the binary’s barycenter, which we place at the origin. since the dipole moment P ðωÞ is the convolution of Qκ and Substituting this back into Eq. (5.6), we find that the zκ. The two waves travel with different group velocities 1 − μ2 Ω2 1=2 monopole term ∝ φ¯ Aðt; 0Þ does not radiate at this PN order v ¼ð = Þ , and the third line in Eq. (5.13) but merely describes the total scalar charge of the binary. accounts for their interference after they accumulate a ϖ 2μ 2Υ − Ω − Ω The dominant channel for scalar radiation is the dipole phase difference ¼ t þ ð þvþ −v−Þr. moment, whose term in the action reads Second, the presence of step functions indicates that the Ω larger-frequency mode þ is radiated throughout the entire 10Of the three separation-of-scale parameters we started with in history of the inspiral, whereas the lower-frequency mode Eq. (3.3), only GMμ survives because we neglect spatial Ω− is radiated only when Ω− > μ. This stems from the variations of Φ and have set ω ≈ μ. simple fact that only sources with frequencies greater than

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TABLE I. Post-Newtonian power counting rules for black hole binaries embedded in fuzzy dark matter halos. All derivatives ∂μ scale in the same way, except spatial derivatives acting on the potential modes, which are denoted by the 3-momentum p, and spatial derivatives on Φ, which vanish. The rules involving the radiation modes assume Ω ≫ μ for simplicity. ¯ h; φ h; φ¯ Φ=mPl H ∂μ p M=mPl δM=mPl Q ffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi p 2 p p p v=a v=a εμa=v ðεμa=vÞ v=a 1=a Lv LvðεGMμÞ2v−3 LvðεGMμÞ

R the scalar’s mass can deposit energy into on-shell modes. where dt ∼ a=v, since the orbital period is the key Third, observe that the flux vanishes entirely in the equal- timescale in this system. Integrating out the radiation mass limit. We can understand this by noticing in Eq. (5.10) modes, two copies of this vertex linked by a propagator that the dipole moment becomes proportional to the generate a term in Γ that scales as Lv3ðεGMμÞ2. Hence, position of the barycenter when M1 ¼ M2. Finally, as a scalar radiation reaction first appears at 1.5PN order, albeit sanity check, we note that Eq. (5.13) reduces to the correct suppressed by two powers of εGMμ. For typical FDM expression [Eq. (2.37) of Ref. [22] ] in the massless limit halos [81,82,89,90], μ → 0 with Qκ → const. Let us clarify when our result for F ϕ is valid. It relies on ρ 2 ε μ 2 ∼ 2 10−16 M the multipole expansion in Eq. (5.7), which holds if the ð GM Þ × 100 −3 1010 ; p Ω2 −μ2 1=2 M⊙pc M⊙ larger-frequency mode, with momentum j j¼ð þ Þ , satisfies ajpj ≪ 1. Writing a2p2 ¼ a2Ω2 þ 2a2μΩ, we can ð5:15Þ rephrase this as two conditions: We require a2Ω2 ≪ 1 and 2 2 2 2 a μΩ ≪ 1. The first of these equivalently reads v ≪ 1,so where ρ ¼ μ Φ0=2 is the local energy density. is always satisfied during the early inspiral. The second can It is instructive to compare this with gravitational 5 be rewritten as μav ≪ 1 or a ≪ 1=ðGMμ2Þ and signifies radiation reaction, which scales as Lv (2.5PN order). that the binary cannot be too widely separated; Our power counting rules then tell us that the energy radiated in scalar waves is suppressed by ðεGMμÞ2=v2 −1 μ −2 relative to gravitational waves. Indeed, this simple estimate ≪ 10 M a pc 10 −22 : ð5:14Þ is consistent with a more detailed calculation. In the Ω ≫ μ 10 M⊙ 10 eV limit, the ratio of F ϕ to the leading quadrupolar flux of gravitational waves F g [58] is We may regard this condition as an IR cutoff for the validity of our EFT when applied to this system. F 5 ε μ 2 − 2 This kind of scaling analysis can also be used to establish ϕ ∼ ð GM Þ M1 M2 2 μ Υ F 48 2 sin ð t þ Þ: ð5:16Þ power counting rules, which enable a quick estimate of the g v M relative sizes of different effects. (The rules developed here and later in this section are summarized in Table I.) For Taking v ≈ 0.1 and ðεGMμÞ2 ≈ 2 × 10−16, this ratio is at simplicity, we will concentrate on the later stages of the most 2 × 10−15. Clearly, the impact of scalar radiation on inspiral (Ω ≫ μ) when discussing radiative effects, since the inspiral of the binary is unlikely to be observable. That this is when they are most pronounced. In this regime, the said, effects appearing at lower PN orders may have better 4-momentum of φ¯ satisfiesR p ∼ Ω ∼ v=a; thus, the propa- observational prospects; hence, the remainder of this gator scales as hφ¯ φ¯ i ∼ d4p=p2 ∼ ðv=aÞ2, and so φ¯ ∼ v=a section concentrates on terms in Γ that arise from integrat- when appearing as an internal line in a Feynman diagram. ing out the potential modes. Similar reasoning implies h¯ ∼ v=a [50]. In position space, ∂ the 4-momentum pμ translates into a derivative μ; thus, 2. Scalar fifth force ∂μ ∼ v=a when acting on the radiation modes. Time The potential mode φ couples to the scalar charge of the derivatives acting on the background scalar can be arranged φ¯ to scale in the same way by taking Φ=m ∼ εμa=v, such black hole in the same way as , namely through the term pffiffiffiffiffiffiPl that ∂ Φ ∼ μεm . Consequently, Q ∼ Lv εGMμ after Z t Pl pffiffiffiffiffiffi ð Þ ∼ ⊃ QA φ using the relation M=mPl Lv [50]. We use these rules to Sp;κ κ ðxÞ AðxÞ: ð5:17Þ deduce that Eq. (5.8) scales as x Z The diagrams in Fig. 2 arise from connecting two copies of 2 pffiffiffiffi i a v 3=2 dtQx ∂iφ¯ ∼ Qa ∼ Lv ðεGMμÞ; this vertex by a propagator. Using standard Feynman rules v a (outlined in Appendix B), they yield

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Q1Q2 F1 ⊃− n; ð5:22Þ 4πr2

where r ¼ z1 − z2, r ¼jrj, and n ¼ r=r. Naturally, inter- changing the labels 1 ↔ 2 yields an identical force acting on the second black hole.11 We obtained this result by keeping the black holes at rest, but nothing changes at this PN order had they been allowed FIG. 2. (a) The exchange of a potential-mode scalar between to move freely, since any departure from the static case the worldlines mediates an attractive scalar fifth force. (b) Self- must depend on v. When working to higher orders, we energy diagram that is pure counterterm. Its mirror inverse, in achieve definite scaling in powers of v by expanding the which the scalar propagates to and from the top worldline, is φ included implicitly since we do not distinguish between the two propagator for the potential mode about its instantaneous solid lines. limit: X Z 1 1 0 2 − μ2 ˜ 1 ðp Þ 2 δþ 0 δ− 0 0 GRðpÞ¼ 2 2 ¼ 2 þ 2 þ : ð5:23Þ Fig: ¼ QκðtÞ κ ðxÞGRðx; x Þ κ0 ðx ÞQκ0 ðt Þ: μ p p 0 p þ κ;κ0 x;x ð5:18Þ The instantaneous part 1=p2 is responsible for the inverse- square law force; hence, the potential mode has 3-momenta The sum over terms with κ ¼ κ0 leads to the self-energy satisfying p ∼ 1=a, or, in other words, spatial derivatives diagram in Fig. 2(b), which is pure counterterm and acting on φ scale as 1=a. In contrast, the oscillating vanishes identically in dimensional regularization [50] background forces the energy of the scalar to have two (at leading order in GMμ). Only the cross terms κ ≠ κ0 pieces that scale differently: jp0j ∼ μ þ v=a, such that in Fig. 2(a) have interesting physical consequences. As we ½ðp0Þ2 − μ2=p2 ∼ v2 þ μav. Thus, we see that assuming did in Sec. IV, the equations of motion for the worldlines φ propagates instantaneously is valid only under the can be read off after expanding each term in Γ to first order conditions v2 ≪ 1 and μav ≪ 1, which are the same in z−. We use the fact that conditions we derived earlier for the radiation modes; cf. Eq. (5.14). ∂ þ i ð3Þ 2 For the power counting rules, it suffices to neglect the δκ ðxÞ¼−z− δ (x − zκðtÞ) þ Oðz−Þ; ð5:19aÞ ∂xi subleading μav dependence when working to leading order in GMμ,12 such that ðp0Þ2 − μ2 ∼ v2=a2, while time deriv- − ð3Þ atives of φ scale with v=a. Taken together, these consid- δκ ðxÞ¼δ (x − zκðtÞ) þ Oðz−Þð5:19bÞ pffiffiffi erations imply φ ∼ v=a. Similar relations apply to the potential-mode graviton h (see Table I), whose propagator to write admits the analogous quasi-instantaneous expansion [50] Z X ∂ 2 z ; 0 z zi 1 1 0 2 Fig: ðaÞ¼ i ½QKQκ0 GRðt; κ t ; κ0 Þ −;κ: ˜ ðp Þ 0 ∂zκ D ðpÞ¼ ¼ 1 þ þ : ð5:24Þ κ≠κ0 t;t R p2 p2 p2 ð5:20Þ These power counting rules tell us that Fig: 2 ∼ Lv0ðεGMμÞ2; thus, the scalar fifth force is a Newtonian- It is instructive to first evaluate this integral while order effect. holding the black holes fixed at their respective positions. This permits use of the master integral in Appendix C, 3. Accretion which returns We already encountered the drag force from accretion in X Z Sec. IV in a fully relativistic setting. When expanded in ∂ QκQκ0 2 zi Fig: ðaÞ¼ i −;κ: ð5:21Þ ∂zκ 4πjzκ − zκ0 j κ≠κ0 t 11Note that these labels now distinguish between the members of the binary. Equations of motion are always given in the In general, the κth black hole obeys an equation of motion physical limit, so there are no longer any CTP indices floating a F around. of the form Mκ κ ¼ κ. Taking the functional derivative of 12At higher orders, it becomes necessary to factor it out Fig. 2(a) with respect to z−, we learn that the first black hole explicitly; for instance, by working with the complex field ψ experiences the scalar fifth force instead, defined from φðxÞ ∝ ½e−iμtψðxÞþH:c:.

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calculated using our EFT framework. We discuss the external force due to the halo’s gravitational potential here, before turning to dynamical friction in Sec. VA5. As we did for the radiation modes, we preserve definite scaling in v by multipole expanding

x 0 xi∂ 0 FIG. 3. Leading-order diagrams accounting for accretion. As Hμνðt; Þ¼Hμνðt; Þþ iHμνðt; Þ with earlier diagrams, their mirror inverses (in which the back- 1 xixj∂ ∂ 0 ground scalar interacts with the top worldline) are included þ 2 i jHμνðt; Þþ ð5:26Þ implicitly. about the binary’s barycenter. Note Hμν must depend on the powers of v, the leading term in Eq. (4.6) is proportional to spatial coordinates—despite Φ being (approximately) just a 2 v2 and is depicted in Fig. 3(a).13 Schematically, function of time—if it is to be a consistent solution at Oðε Þ Z to the background field equation 1 2 −3 2 Fig: 3ðaÞ ∼ dt δMðtÞv ∼ Lv ðεGMμÞ : 1 2 8π ∂ Φ∂ Φ η μ2Φ2 Rμν ¼ G μ ν þ 2 μν : ð5:27Þ

In the presence of a second black hole, an additional This equation enforces the relation R ∼ ∂∂H ∼ ε2μ2, which diagram contributes at this order: Fig. 3(b) accounts for the is satisfied provided all derivatives acting on H scale as 2 change in the gravitational force between the black holes ∂μ ∼ v=a, while taking H ∼ ðεμa=vÞ . due to their increasing masses. Notice that only one of the Although it is possible to stick with the general multipole black holes is accreting in this diagram; the diagram in expansion in Eq. (5.26), it is far more convenient if we O ε4 which both are accreting first appears at ð Þ and, thus, is work in Fermi normal coordinates [91,92]. We then have neglected. 14 that both Hμνðt; 0Þ and ∂iHμνðt; 0Þ¼0 in this gauge, Even without detailed calculation, it is easy to correctly whereas intuit that Fig. 3 leads to the force 1 ∂ ∂ 0 − 0 _ GðM1δM2 þ M2δM1Þ i jH00ðt; Þ¼ R0i0jðt; Þ; ð5:28aÞ F1 ⊃−δM1v1 − δM1a1 − n: ð5:25Þ 2 r2 1 2 −1 5 ∂ ∂ 0 − 0 Formally, this is a . PN effect but is still subleading to 2 i jH0kðt; Þ¼ 3 R0ikjðt; Þ; ð5:28bÞ the Newtonian-order interactions ∼Lv0 due to suppression by two powers of εGMμ. The negative-power scaling in 1 1 ∂ ∂ 0 − 0 v indicates that the effects of accretion—in contrast to 2 i jHklðt; Þ¼ 3 Rkiljðt; Þ: ð5:28cÞ radiation reaction—are most pronounced at the very early stages of the inspiral when the binary is widely separated. At leading order in the PN expansion, the only contribution Consequently, future space-based gravitational-wave detec- involving Hμν comes from expanding the point-mass term tors like LISA are unlikely to be sensitive to this effect. in the action: Rather, pulsar timing arrays or other astronomical obser- Z X 1 vations may prove more suitable when attempting to Γ ⊃ a ( z ) c 2 MκH00 t; a;κðtÞ observe, or at least constrain, the impact of an FDM halo κ t on a supermassive black hole binary. We will return to the X Z subject of constraining FDM models in Sec. VB. − 0 zi zj ¼ MκR0i0jðt; Þ −;κ þ;κ; ð5:29Þ κ t 4. Background gravitational potential which gives rise to the force The three effects discussed so far—scalar radiation, the fifth force, and accretion—all stem from the interaction Fi ⊃− i 0 zj between a black hole’s horizon and the scalar field. Two κ R 0j0ðt; ÞMκ κ: ð5:30Þ other effects, which are not unique to black holes but which influence the motion of any massive body, can also be The effect of this force on binary pulsars has previously been studied in Ref. [93]. Its effect on black hole binaries is analogous and will be discussed briefly in Sec. VB. 13There would also have been an Oðv0Þ term if Φ were spatially inhomogeneous, which would yield the scalar fifth force 14 ∝ ∂iΦ exerted by the background; cf. Eq. (4.9). The linear terms ∂iH would be nontrivial if ∂iΦ ≠ 0.

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Power counting tells us that this external force first appears at −3PN order; Eq: ð5.29Þ ∼ Lv−6ðεGMμÞ2. This inverse scaling with v, which we first met in Sec. VA3,is signaling a second type of IR breakdown of our EFT.15 To see this, recall that our perturbative expansion is predicated on the virial relation v2 ∼ GM=a, which holds only if the 0 FIG. 4. Feynman diagrams constituting (a) dynamical friction Newtonian-order interactions ∼Lv are the dominant terms and (b),(c) the backscattering of gravitons. Details about the in the action. This demands that the binary satisfy the interaction vertices in the bulk can be found in Appendix B. condition v6 ≫ ðεGMμÞ2, which can equivalently be written as a3 ≪ GM=ðεμÞ2 or most transparently as need not agree. We have, however, verified that our EFT ðεμa=vÞ2 ∼ H ≪ 1. For small enough velocities or large approach correctly reproduces the results in Appendix A of enough orbital separations, our scaling rules naively Ref. [98] when working under similar assumptions. suggest that H can attain values of order one, at which Let us return to our own result in Eq. (5.32):Power point it stops being a weak perturbation to the Minkowski counting tells us that dynamical friction first appears at metric. Before this can happen, spatial variations of −1PN order. In contrast, the diagrams in Figs. 4(b) Φ become relevant and must be taken into account. and 4(c)—which depict the backscattering of gravitons Thus, a multipole expansion of the background fields is off the gravitational potential and energy density of the halo valid only if —scale with ε and GMμ in the same way but appear earlier −2 at PN order. Evaluating these diagrams proves to be ρ −1=3 1=3 more challenging, however, and is for the time being left as ≪ 80 M a pc −3 10 : ð5:31Þ 100 M⊙pc 10 M⊙ an open problem.

This is a second, independent IR cutoff for our EFT, which B. Observational constraints must be satisfied in addition to Eq. (5.14). We conclude this section by exploring how well obser- vations of OJ287 can be used to constrain FDM models. 5. Dynamical friction The supermassive black hole binary at the center of the quasar has an orbital period that decays at a rate P_ ∼ 10−3, The final effect we wish to discuss is the drag force due which is consistent with the predictions of vacuum PN to dynamical friction. It arises because the gravitational theory to within an uncertainty of 6% [25]. Hence, the field of a black hole, or any massive body, perturbs the effects discussed in Sec. VA should not hasten or stall the medium through which it moves, forming a wake in the inspiral by more than δP_ 6 × 10−5. This condition can latter that then exerts a gravitational pull back on the object. j j¼ be translated into an upper bound on the local FDM density Although usually considered in the context of collisionless ρ in the vicinity of the quasar. Although PN corrections are or gaseous media [94–97], recent studies have begun needed to accurately predict the evolution of the inspiral exploring what modifications are needed to account for due to gravitational-wave emission [25], it suffices to treat the wavelike nature of FDM [38,86,98]. Our EFT formal- effects involving the scalar field as first-order perturbations to ism provides a natural language for calculating the force _ that dynamical friction exerts on a massive body. The the Kepler problem when determining their contribution to P. interaction of a black hole with its gravitationally induced The general method for performing such calculations is wake is depicted in Fig. 4(a) and yields described at length in Chap. 3 of Poisson and Will [99];in what follows, we will simply quote the required formulas. 2 _ 2 2 Consider the effective-one-body Kepler problem Fκ ⊃−16πðGMκÞ Φ vκ vˆκ: ð5:32Þ GM ̈r þ n ¼ f; ð5:33Þ The derivation is presented in Appendix D. r2 This formula relies on the assumption that the binary is where r ¼ z1 − z2 is the separation of the binary of total tight enough to satisfy the condition μav ≪ 1 [cf. Eq. (5.14)] mass M, n ¼ r=r, and f is an additional force (per unit such that the scalar can be approximated as propagating mass) acting on the system, which we will treat as a small instantaneously at leading PN order. In Refs. [38,86,98],the perturbation. After time-averaging over one orbit, the force impact of dynamical friction within an FDM halo is studied f μ ≳ 1 results in a secular decay of the orbital period given to first in the opposite regime av (objects orbiting the center order by of a galaxy, for instance, satisfy this condition). Con- Z sequently, our results cannot be directly compared and they 3 2π pffiffiffiffiffiffiffiffiffiffiffiffiffi _ GM f λ 1 − 2 f n hPi¼ 4 3 du½ð · Þ e þð · Þe sin u; ðGMΩÞ = 0 15The first type has to do with a breakdown at arbitrarily late times; see the last few paragraphs of Sec. III D 2. ð5:34Þ

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TABLE II. Parameters of the supermassive black hole binary in This procedure automatically excludes any contribution f quasar OJ287, reproduced from Ref. [25]. Errors have been from bkg, since the Riemann tensor is proportional to omitted for any quantity accurate to at least three significant cosð2μt þ 2ϒÞ. It is still possible to extract a meaningful figures. The intrinsic period P is determined by rescaling the constraint by choosing ϒ such that we calculate the value measured on Earth by the scale factor ð1 þ zÞ−1 [23]. The maximum possible value of jhP_ ij (as Ref. [93] does for uncertainty on P_ is at the 1σ level. binary pulsars), but we will not elect to do so and will f Parameter Value instead simply concentrate on acc. It turns out that the constraint we derive from f is several orders of magni- Redshift z 0.306 acc 10 tude better than what we would get from f . This is Primary black hole mass M1 1.83 × 10 M⊙ bkg 8 f ∝−r_ Secondary black hole mass M2 1.50 × 10 M⊙ because acc has a component ( ) that is always ’ Primary dimensionless spin parameter χ1 0.381 opposing the binary s motion. f Eccentricity e 0.657 The contribution from acc to the orbital period decay is Intrinsic orbital period P 9.24 yr 2 Orbital period decay P_ ð99 6Þ × 10−5 48πG MPð2ν − eÞρ E½hP_ i ¼− : ð5:38Þ acc 1 − e _ where Ω is the orbital frequency of the unperturbed binary Requiring that this have a magnitude less than jδPj¼ and e is its eccentricity. The unit vector λ points along the 6 × 10−5 imposes the upper bound direction orthogonal to both n and the binary’s angular 9 −3 momentum vector. The trajectory along the orbit is ρ ≲ 2 × 10 M⊙ pc ð5:39Þ parametrized by the eccentric anomaly u, which can be related to the coordinate time t via Kepler’s equation, at the 1σ level for the local density of FDM. Note that Ωðt − t0Þ¼u − e sin u. The orbital parameter t0 is called Eq. (5.38) assumes that the black holes are spherical for the time of pericenter passage and can be set to zero in this simplicity (even though the spin of the primary black hole calculation without loss of generality. has been measured), since that is good enough for deriving The power counting rules established in Sec. VA can be an order-of-magnitude constraint. As a final step, it is used to infer that, of the five effects we calculated, the necessary to check that this bound is consistent with the IR forces due to the halo’s gravitational potential and accretion cutoffs in Eqs. (5.14) and (5.31). While the second is easily will provide the largest contributions to P_ , since they scale satisfied for the case of OJ287, which has an orbital with the most negative powers of v. For this reason, we separation a ≈ 56 mpc, the first of these tells us that our concentrate only on these two effects. Respectively, they conclusions are valid only for scalars with a mass exert the forces μ ≪ 8 × 10−22 eV. The constraint in Eq. (5.39) is very weak, as FDM halos fi − i 0 rj 100 −3 bkg ¼ R 0j0ðt; Þ ; ð5:35Þ are expected to have core densities of around M⊙ pc [81,82,89,90]. Accordingly, we conclude that typical dark _ _ matter halos are too dilute to leave any observable imprints δM1 δM2 GðδM1 þ δM2Þ f ¼ − þ Mνr_ − n: ð5:36Þ in the inspiral of a black hole binary. This is entirely in line acc M2 M2 r2 1 2 with our expectations going in. Nonetheless, the work in this section is still useful for illustrating how our EFT Substituting these forces into Eq. (5.34), we obtain an framework can be used to make quantitative predictions. In expression for hP_ i that is a function of the local density ρ the next section, we will briefly comment on other scalar- we wish to constrain, the known orbital parameters as field environments with greater observational potential that summarized in Table II, and one unknown: the phase factor are worth exploring in future work. Υ of the background relative to our zero of our time. Not knowing what value this parameter ought to have, we can obtain a conservative estimate for hP_ i by marginalizing VI. CONCLUSIONS Υ 16 over assuming a uniform prior. The resulting expect- We have developed a worldline EFT that accurately ation value is describes how black holes in general relativity interact with Z minimally coupled, real scalar fields. Stringent no-hair 1 2π E½hP_ i ¼ dϒhP_ i: ð5:37Þ theorems limit the kinds of terms that are allowed in the 2π 0 effective action—in particular, black holes are not permit- ted any permanent scalar multipole moments of their 16 — By randomly sampling values of ϒ ∈ ½0; 2πÞ and observing own but we still uncover a rich phenomenology how they affect the value of hP_ i, we have verified that our when accounting for finite-size effects. Being an extension assumption of a uniform prior does not bias our conclusions. of Goldberger and Rothstein’s construction [50,51], the

024010-19 WONG, DAVIS, and GREGORY PHYS. REV. D 100, 024010 (2019) novelty of our approach is in the integrating out of and the fifth force—can be traced back to a single parent composite operators localized on the worldline, which term in the effective action. encode information about UV physics transpiring near We illustrated how this EFT can be used to make the horizon. This procedure proved to be a powerful quantitative predictions by studying the early inspiral of method for generating new terms in the effective action a black hole binary located in the core of a fuzzy dark never before considered in the literature. Central to this matter halo. This example was useful as a case study, since achievement was our use of the in-in formalism of quantum a series of approximations made performing calculations field theory, which enabled the accounting of dissipative straightforward, but ultimately, typical halos in these effects at the level of the action. models are too dilute to leave any observable imprints in Our EFT reveals that the motion of a black hole the binary’s inspiral. This is no cause for discouragement, embedded in a scalar-field environment exhibits three however, as there are still other examples of scalar-field features that distinguish it from other compact objects: environments worth studying, which may have greater First, the black hole experiences a drag force due to observational potential. At least two come to mind: Even accretion of the background scalar field, which proceeds if an ultralight scalar field is not produced in large at a rate that is uniquely determined (at leading order) by abundances during the early Universe, rapidly rotating the area of its horizon. Second, a scalar-field environment black holes with radii coincident with the scalar’s induces a scalar charge onto the black hole, granting it the Compton wavelength can quickly generate a corotating ability to radiate energy and momentum into scalar waves. condensate of the field through a superradiant instability Third, the onset of this scalar charge also stipulates that a [39–44]. Such a system is outside the regime of validity of black hole must move under the influence of a fifth force. our point-particle EFT, since there is no separation of scales Of these three effects, accretion is the most natural and between the scalar condensate and its host black hole, but unsurprising. Accordingly, many studies [89,100,101] have our EFT is perfectly poised to study what would happen appreciated its importance, which in optimal scenarios may to a much smaller black hole orbiting this system. In even dominate over radiation reaction in driving the more exotic scenarios, it is also possible to envision a evolution of a black hole’s inspiral [101]. However, typical stellar-mass black hole orbiting a supermassive, compact estimates for the accretion rate often rely on the absorption horizonless object like a boson star [33]. Both of these cross section for free, collisionless, nonrelativistic particles extreme-mass-ratio inspiral scenarios have been studied in [102], which is strictly valid only for a black hole moving the past [100,101], albeit using a Newtonian approach with slowly through a gas of such particles. In contrast, we are finite-size effects included in an ad hoc fashion. Our EFT often more interested in the motion of a black hole through provides a systematic framework for extending these results a background field that is localized and bound by its own into the fully relativistic regime while also accounting for self-gravity. To qualify as a background, the total mass of effects associated with the black hole’s induced scalar this configuration must also be much greater than that of the charge, hitherto unexplored. This points to one exciting black hole. In such cases, the correct accretion rate is direction for future work. determined from computing the flux of this scalar field Also in the future, it will be interesting to extend our EFT ’ across the horizon [19,21,78,79]. As we pointed out earlier, to include a black hole s spin and to push its capabilities what is remarkable is that this accretion rate emerges beyond leading, nontrivial order in the separation-of-scale naturally from first principles in our EFT. Importantly, parameters. The novel techniques we have employed when our equation for the resulting drag force works not only in constructing the effective action are also likely to be the Newtonian regime, but holds in a fully relativistic invaluable when modeling the interactions of black holes setting. or other compact objects with external scalar, vector, or Less obvious is the fact that black holes gain scalar tensor fields. charges when embedded in a scalar-field environment. The prediction of scalar radiation originates with Horbatsch and ACKNOWLEDGMENTS Burgess [22], but to the best of our knowledge, we are the It is a pleasure to thank Cliff Burgess, Vitor Cardoso, first to point out that a black hole can experience a fifth Bogdan Ganchev, Joe Keir, Jorge Santos, Ulrich Sperhake, force mediated by a minimally coupled scalar field. While and Ira Rothstein for helpful comments and discussions. This scalar radiation and fifth forces are par for the course in work has been partially supported by STFC Consolidated alternative theories of gravity, owing to a nonminimal Grants No. ST/P000371/1, No. ST/P000673/1, and No. ST/ coupling between the scalar and one or more curvature P000681/1. L. K. W. is supported by the Cambridge tensors [80,103,104], the effects discussed in this paper Commonwealth, European and International Trust, and emerge as necessary and inescapable consequences of Trinity College, Cambridge. R. G. is also supported in accretion of the background scalar onto the black hole. part by Perimeter Institute. Research at Perimeter Institute Our EFT exposes this connection in no uncertain terms, is supported by the Government of Canada through the showing that all three effects—accretion, scalar radiation, Department of Innovation, Science and Economic

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Development and by the Province of Ontario through the In these last three equations, all two-point functions have Ministry of Research and Innovation. the same indices LL0 and arguments ðτ; τ0Þ, which have been suppressed for readability. APPENDIX A: DERIVING 2. Charge density THE POINT-PARTICLE ACTION In the main text, we defined the induced charge density This Appendix collates several technical details used in of the black hole as deriving the point-particle action Sp in Sec. III. Z 1 QAðxÞ ≔ pffiffiffiffiffiffi X Aþðx; x0ÞΦðx0Þ: ðA3Þ 1. Two-point correlation functions −g x0 Let us review several key features of two-point corre- lation functions. The basic ingredients are the Wightman To obtain the end result in Eq. (3.46), we substitute in X Aþ functions explicit expressions for and simplify. The two cases A ∈ fþ; −g must be treated separately, but since the steps − χLL0 τ τ0 ≔ L τ L0 τ0 are almost identical, it suffices to work through just one i þ ð ; Þ hq ð Þq ð Þi; ðA1aÞ example. Let us do Qþ. Using Eq. (3.25), we obtain − χLL0 τ τ0 ≔ L0 τ0 L τ Z i − ð ; Þ hq ð Þq ð Þi; ðA1bÞ ffiffiffiffiffiffi p− Qþ χ λ τ Δ −χ λ τ Δ τ_ Φ g ¼ f½ Rð ; 10 Þ 1 Cð ; 10 Þ þ 10 ðz10 Þ from which all other two-point functions can be built. λ;λ0 − χ λ τ Δ −χ λ τ Δ τ_ Φ Respectively, we define the Feynman, Dyson, Hadamard, ½ Rð ; 20 Þ 2 Cð ; 20 Þ þ 20 ðz20 Þg ðA4Þ and Pauli-Jordan propagators as after integrating over the delta functions in Δa0 . We write − χLL0 τ τ0 ≔ L τ L0 τ0 Δ Δ Δ 2 τ ≡ τ λ0 ≡ λ i F ð ; Þ hTq ð Þq ð Þi; ðA1cÞ þ ¼ð 1 þ 2Þ= , a0 að Þ, and za0 zað Þ for brev- ity. Now substitute in explicit forms for χR and χC, given by − χLL0 τ τ0 ≔ L τ L0 τ0 i D ð ; Þ hT q ð Þq ð Þi; ðA1dÞ Eq. (3.41), to obtain Z Z LL0 0 L L0 0 pffiffiffiffiffiffi −iχ ðτ; τ Þ ≔ hfq ðτÞ;q ðτ Þgi; ðA1eÞ þ −iωðλ−τ20 Þ H −gQ ¼ A ½Δ1τ_20 iωe Φðz20 Þ − ð1 ↔ 2Þ: λ;λ0 ω − χLL0 τ τ0 ≔ L τ L0 τ0 i C ð ; Þ h½q ð Þ;q ð Þi; ðA1fÞ ðA5Þ

where T and T denote the time-ordering and anti-time- Recognizing that τ_20 iω can be rewritten as a derivative ordering operators, respectively. Whether minus signs or d=dλ0 acting on the exponential, and likewise for 1 ↔ 2,we factors of i appear on the lhs is simply a matter of find − χ convention. Note also that i C is nothing but the com- Z mutator. Last but not least, we define the retarded and ffiffiffiffiffiffi p− Qþ − Δ Φ_ δ λ − τ − 1 ↔ 2 advanced propagators by g ¼ A ½ 1 ðz20 Þ ð 20 Þ ð Þ ðA6Þ λ;λ0 χLL0 τ τ0 ≔ θ τ − τ0 χLL0 τ τ0 R ð ; Þ ð Þ C ð ; Þ; ðA1gÞ after integrating by parts. Finally, integrating over the remaining delta functions in Δ1;2 gives us χLL0 τ τ0 ≔−θ τ0 − τ χLL0 τ τ0 A ð ; Þ ð Þ C ð ; Þ; ðA1hÞ Z Z ð4Þ δ x − z1 Qþ − ð Þ τ_ Φ_ δ τ − τ − 1 ↔ 2 where θðxÞ is the Heaviside step function. ¼ A pffiffiffiffiffiffi 1 ðz20 Þ ð 1 20 Þ ð Þ: λ −g λ0 Not all of these two-point functions are independent. Notice from their definitions that ðA7Þ

− χLL0 τ τ0 χL0L τ0 τ Repeating similar steps to obtain an expression for Q , þ ð ; Þ¼ − ð ; ÞðA2aÞ we recognize the following pattern: If we define and, furthermore, the identity θðxÞþθð−xÞ¼1 implies Z Z ð4Þ δ (x − z1ðλÞ) Q ≔− ffiffiffiffiffiffi τ_ λ Φ_ ( λ0 ) χ χ − χ χ − χ 1ðxÞ A p− 1ð Þ z2ð Þ R ¼ F − ¼ þ D; ðA2bÞ λ g λ0 δ(τ λ − τ λ0 ) χ χ − χ χ − χ × 1ð Þ 2ð Þ ðA8Þ A ¼ F þ ¼ − D; ðA2cÞ and define Q2ðxÞ by interchanging 1 ↔ 2 in the above χ χ χ χ χ ∓ H ¼ F þ D ¼ þ þ −: ðA2dÞ Q ≡ Q equation, then the charge densities in the

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Keldysh representation are obtained through the usual obtained after swapping the integration limits on λ and σ transformation rule in Eq. (3.10). by using the identity Z Z Z Z λf λ λf λf 3. Accretion rate dλ dσ ¼ dσ dλ λ λ λ σ We now turn to deriving the accretion rate. Our starting i i Z i Z Z λf λf σ point is ¼ dσ dλ − dλ : ðA14Þ λ λ λ Z i i i 1 Sp ⊃ τ_1τ_20 χCðτ1; τ20 ÞΦðz1ÞΦðz20 Þ: ðA9Þ 2 λ;λ0 APPENDIX B: PROPAGATORS AND BULK As we did in the main text, we perturb the proper time such VERTICES IN WEAKLY CURVED SPACETIMES that τa → τa þ δτa. The terms linear in δτa are Z As we did for the point-particle action in Sec. III D,we 1 substitute the decomposition (3.30) into the field action Sf τ_1τ_20 χCðτ1; τ20 ÞΦðz1ÞΦðz20 Þ 2 λ λ0 to obtain the series ; Z ∂ χ λ δτ_1 ð1Þ C X∞ X∞ × þ dσδτ_1ðσÞ ðn þnφÞ τ_ χ S S h ; B1 1 C λi f ¼ f ð Þ Z 0 0 ∂ χ λ0 nh¼ nφ¼ δτ_20 ð2Þ C þ þ dσδτ_2ðσÞ ; ðA10Þ τ_20 χ λ C i where recall the integers ðnh;nφÞ count the powers of the field perturbations appearing in each term. Since the ∂ where we write ðnÞ to mean the derivative with respect to background ðg; ΦÞ is assumed to be a valid solution of the nth argument. Using the explicit expression for χ in C the field equations, there are no terms with nh þ nφ < 2. Fourier space, this becomes With general relativity being a gauge theory, it is necessary Z Z that we supplement Sf with a gauge-fixing term ala` −iωðτ1−τ 0 Þ A Φðz1ÞΦðz20 Þ e 2 Faddeev and Popov, λ λ0 ω ; Z Z λ pffiffiffiffiffiffi 2 − − μν A B × δτ_1τ_20 iω − τ_1τ_20 ðiωÞ dσδτ_1ðσÞ Sgf ¼ gcABg Gμ Gν ; ðB2Þ λ x Z i 0 λ A 2 which imposes the gauge condition Gμ ≈ 0. If we impose þ δτ_20 τ_1iω þ τ_1τ_20 ðiωÞ dσδτ_2ðσÞ : ðA11Þ λ i the generalized Lorenz gauge Just as we did when deriving the charge density, recognize 1 ζ A ν A A A Gμ ¼ ∇ hμν − h gμν − φ ∇μΦ ðB3Þ that each appearance of τ_1iω can be replaced by a derivative 2 2 mPl −d=dλ acting on the exponential, and likewise each factor τ_ ω λ0 of 20 i can be replaced by d=d . Having done so, defined in terms of an arbitrary constant ζ, the part of the Eq. (A11) simplifies to field action quadratic in the perturbations is Z Z  λ Z d d 2 1 pffiffiffiffiffiffi Φ Φ 0 σδτ_ σ ð Þ A αβμν αβμν B A ðz1Þ ðz2 Þ d 1ð Þ 0 S ¼ cAB −g hαβðP □ − M Þhμν λ λ0 dλ λ dλ f 2 ; Z i x λ0 1 − ζ d d A μ ν μν α 2 B − dσδτ_2ðσÞ δðτ1 − τ20 Þ: ðA12Þ þ hμν½2∇ Φ∇ − g ð∇αΦ∇ þ μ ΦÞφ dλ0 λ dλ 2m i Pl  ζ A B μ ν A 2 B Integrating by parts then yields − hμνφ ∇ ∇ Φ þ φ ð□ − μ Þφ : ðB4Þ m eff Z Z Z Pl λ λ f _ _ A dλ dσδτ_1ðσÞΦðz1Þ Φðz20 Þδðτ1 − τ20 Þ This is expressed in terms of three background quantities: 0 λi λi λ 1 − ð1 ↔ 2Þ: ðA13Þ αβμν αμ βν αν βμ − αβ μν P ¼ 2 ðg g þ g g g g Þ; ðB5aÞ Note that λ ; λ correspond to the initial and final times at ð i fÞ μ2Φ2 which boundary conditions are to be specified according to Mαβμν 2 gαμRβν − Rαμβν − Pαβμν; ¼ ð Þ 4 2 ðB5bÞ the in-in formalism. The final result in Eq. (3.53) is mPl

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2ζ2 μ2 μ2 ∇ Φ∇αΦ eff ¼ þ 2 α : ðB5cÞ mPl

The convenient gauge choice ζ ¼ 1 exchanges derivative interactions between the different field perturbations in FIG. 5. Examples of bulk vertices. The graviton h is drawn as a favor of simpler algebraic ones, but nonetheless an arbitrary φ Φ ≠ 0 helical line while the scalar is drawn as a dashed line. Insertions background with will lead to a quadratic action of the background fields are denoted as dotted lines terminating in that mixes the graviton with the scalar. In general, these a given shape. The circle, filled square, and empty square mixing terms must be treated nonperturbatively, meaning correspond to the background scalar Φ, the mass tensor M, we cannot speak of a propagator for h and a separate and the background gravitational potential H, respectively. propagator for φ [105]. An exception to this rule is when Φ ∼ O ε is itself a ð Þ ζ 0 17 weak perturbation living on top of a vacuum geometry. In in the Lorenz gauge ¼ . Note that the tensor P is here such cases, the background solution admits its own defined in terms of the Minkowski metric. expansion in the small bookkeeping parameter ε, namely One finds that both propagator matrices are symmetric under the simultaneous interchange of the arguments x ↔ x0 and the CTP indices A ↔ B. As a result, appro- 1 2 0 2 3 Φ ¼ Φð Þ þ Oðε Þ;g¼ gð Þ þ gð Þ þ Oðε Þ: priate relabeling of dummy indices and integration varia- bles can always be used to replace the advanced The example of a fuzzy dark matter halo we consider in propagators ðGA;DAÞ in a Feynman diagram with the Sec. V admits this expansion; the vacuum spacetime is retarded propagators ðGR;DRÞ. Moreover, much like with 0 the Hadamard propagator χ for the worldline operators in described by the Minkowski metric, gð Þ ¼ η, which is only H 2 Sec. III B 2, one also finds that ðG ;D Þ are always weakly perturbed by the gravitational potential gð Þ ≡ H of H H flanked by at least two quantities that vanish in the physical the halo. This Appendix establishes the Feynman rules for limit. Consequently, they never contribute to the (classical) backgrounds of this form. equations of motion [57,65] and can therefore be neglected. Taken together, these observations tell us that only the 1. Free-field propagators retarded propagators are needed for calculating physical Since Φ ∼ OðεÞ is assumed to be small, the mixing terms observables. For our purposes, it is most convenient to in the second and third lines of Eq. (B4) can be treated write the scalar’s retarded propagator as Z perturbatively as interactions. Hence, the graviton and eip·ðx−x0Þ scalar now have their own propagators, which are defined G x; x0 : Rð Þ¼ − 0 ϵ 2 p2 μ2 ðB9Þ on flat space. The gauge-fixed generating functional for the p ðp þ i Þ þ þ free fields, which we introduced in Sec. III C, is then The graviton’s retarded propagator DR has an identical expression, except with μ 0. Z ¼ i αβμν Z A 0 B 0 0½j; J¼exp JαβðxÞDAB ðx; x ÞJμνðx Þ 2 0 2. Bulk vertices x;xZ We treat every term in the field action Sf not included in i A 0 B 0 × exp j ðxÞGABðx; x Þj ðx Þ : ðB6Þ the generating functional Z0 perturbatively as an interac- 2 0 x;x tion vertex. Three are relevant for the purposes of this paper. At linear order in ε, the aforementioned mixing terms give Directly analogous to Eq. (3.12), the scalar field has a us the vertex matrix of propagators given by μν hA α 2 LhφΦ ¼ ½2∂μΦ∂ν − ημνð∂ Φ∂α þ μ ΦÞφA; ðB10Þ 1 2m G G Pl G ¼ 2 H R ðB7Þ AB drawn in Fig. 5(a). The second vertex, depicted in Fig. 5(b), GA 0 is an effective mass for the graviton, in the Keldysh representation, whereas the matrix of 17 graviton propagators reads We must set ζ ¼ 0 when Φ ∼ OðεÞ ≪ 1, as the last term in Eq. (B3) is now smaller than the others. Were we to choose a ζ nonzero value for , the gauge-fixing term would still attempt to 1 O ε0 αβμν D D enforce the Lorenz gauge at ð Þ and then subsequently attempt αβμν 2 H R A 1 D ¼ P ðB8Þ to enforce the condition φ ∇μΦ ≈ 0 at Oðε Þ. This would be AB 0 DA undesirable.

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1 A αβμν We begin by expressing both G and f in Fourier space L 2M ¼ − hαβM h μν; ðB11Þ R h 2 A; to find 2 M ∼ O ε Z Z Z 0 0 where the mass tensor ð Þ at leading order. e−ip ðt−t Þeip·x G ˜ ω −iωt0 The final vertex in Fig. 5(c) comes from expanding the ½f¼ fð Þe − 0 ϵ 2 p2 μ2 : ðC2Þ background metric in the graviton’s kinetic term to first t0 ω p ðp þ i Þ þ þ order in H: Integrating over t0 generates a delta function which imposes 0 0 the condition p ¼ ω. Also integrating over p then gives 1 1 ∂ pffiffiffiffiffiffi αβρσ L 2 ffiffiffiffiffiffi − A □ h H ¼ Hμν p ð ghαβðP ÞhA;ρσÞ : Z Z 2 −g ∂gμν p x g¼η ei · G ˜ ω −iωt ½f¼ fð Þe 2 2 2 : ðC3Þ ðB12Þ ω p p þ μ − ðω þ iϵÞ

This generates a large number of terms that derivatively The integral over momentum space must be evaluated couple Hμν to the gravitons. We will omit writing down an separately depending on the sign of the real part of explicit expression, since it will not be needed for any of k2 ¼ðω þ iϵÞ2 − μ2. When k2 ≤ 0, the scalar gives rise our calculations in this paper. to the Yukawa potential Z pffiffiffiffiffiffiffiffiffiffi 3. Position-space Feynman rules ip·x − μ2−ω2r ω ≔ e ⊃ θ − 2 e Ið Þ 2 2 ð k Þ : ðC4Þ Let us schematically denote each bulk vertex as p p − k 4πr Z hnh φnφ If instead k2 > 0, we expect this equation to describe Sf ⊃ Vf ; ðB13Þ x nh! nφ! spherical waves emanating from the origin. Indeed, per- forming the integral yields where all indices have been suppressed. In general, Vf is a 2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi derivative operator acting on the fields. The worldline θðk Þ 2 2 2 2 IðωÞ ⊃ ðθðωÞei ω −μ r þ θð−ωÞe−i ω −μ rÞ: ðC5Þ vertices in Sec. III D are denoted in a similar way by 4πr replacing subscript f’s with subscript p’s. The position- space Feynman rules for our EFT are then as follows: The complete result for IðωÞ is formed by taking the sum of these two equations. Substituting this back into Eq. (C3), (1) Each bulk vertex gives an appropriate factor of iVf, we find that we can write while each worldline vertex gives a factor of iVp. (2) Each graviton or scalar line corresponds to an Z αβμν 1 − ˜ −iωtþikðωÞr appropriate propagator matrix, either iDAB or G½f¼ fðωÞe ; ðC6Þ 4πr ω −iGAB, respectively. (3) All CTP and spacetime indices are to be summed 2 over, and all spacetime points are to be integrated where the root of k is defined as ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi over, except those corresponding to external legs. 2 2 2 2 (4) Divide each diagram by the appropriate symmetry i μ − ω ω ≤ μ kðωÞ ≔ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC7Þ factor. sgnðωÞ ω2 − μ2 ω2 > μ2: If the diagram being computed has no external legs, we choose to additionally multiply by a factor of −i such that it Occasionally, it will be convenient to simplify this constitutes a term in the effective action Γ rather than one further by exploiting the fact that the Fourier transform in iΓ. of a real source fðtÞ satisfies f˜ð−ωÞ¼f˜ ðωÞ, while our definition for kðωÞ satisfies kð−ωÞ¼−kðωÞ. Thus, an APPENDIX C: MASTER INTEGRAL equivalent expression is Z Many occasions in the main text call for the evaluation of 1 ∞ ω d ˜ − ω ω an integral of the form G½f¼ 2Re fðωÞe i tþikð Þr: ðC8Þ 4πr 0 2π Z 0 0 G½fðt; xÞ ≔ GRðt; x; t ; 0Þfðt Þ; ðC1Þ t0 APPENDIX D: AN EFT APPROACH TO DYNAMICAL FRICTION which describes the leading-order expectation value for φ due to a time-dependent source fðtÞ at rest at the origin. In Here we derive the drag force in Eq. (5.32) due to the interest of efficiency, let us discuss how to evaluate this dynamical friction. Taking its nonrelativistic limit, the integral (on flat space) once for an arbitrary source fðtÞ. graviton vertex in Eq. (3.50) reduces to

024010-24 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019) Z 0 Mκ −iq ðt−t0Þ ⊃ δA on e and then integrating by parts. These steps Sp;κ 2 κ ðxÞhA;00ðxÞðD1Þ mPl x give us at leading order in v. This can be used in conjunction with Z M2 d the Feynman rules in Appendix B to obtain 4 δ − 0 0 0 Fig: ðaÞ¼ 4 ðt t Þ W0ðt; t ÞþW1ðt; t Þ 0 16m 0 dt Z Pl t;t Z M 2 d2 Fig:4 a δþ x δ− x0 0 δþ δ− 0 x − x0 3 ð Þ¼ ð Þ ð Þ þW2ðt; t Þ 02 ðxÞ ðx Þj j ; 2m x;x0;y;y0 dt x x0 Z Pl ; eip·ðx−yÞ eiq·ðy−y0Þ eip0·ðy0−x0Þ ðD6Þ × 2 2 02 p;p0;q p q p αβ 0 0 μν 00 − 0 0 −4Φ_ Φ_ 0 × P00αβVhφΦðy ;q ÞP00μνVhφΦðy ; q Þ; ðD2Þ with W2ðt; t Þ¼ ðtÞ ðt Þ. Determining expressions for W0 and W1 will not be necessary. Now expand δ x in powers of z− according to having kept only the instantaneous part of the propagators; ð Þ Eq. (5.19). To linear order in z−, the term involving W2 cf. Eqs. (5.23) and (5.24). We have also suppressed the index κ since this force acts independently on each member yields of the binary. The vertex functions in the third line read Z 8π 2 0 _ _ 0 i : 4 ⊃− GM δ t − t Φ t Φ t z− t 1 Fig ðaÞ 3 ð Þ ð Þ ð Þ ð Þ ð Þ αβ 0 0 0 _ 0 2 0 t;t0 P00αβV ðy ;q Þ¼ ½−2iq Φðy Þ − μ Φðy Þ: ðD3Þ hφΦ 2 2 mPl d ∂ × jz ðtÞ − zðt0Þj3 : ðD7Þ 02 ∂zi þ dt þðtÞ PL The two terms in square brackets scale with different powers of GMμ and v, but it will be instructive to keep both of them around in this derivation. It turns out that the Evaluating the derivatives, the second line becomes second term μ2Φ provides no contribution whatsoever to the force. 6ðr · vÞv 3v2r 3ðr · vÞ2r 3ðr · aÞr i þ i − i − 3jrja − i ; ðD8Þ We first simplify Eq. (D2) by performing a number of jrj jrj jrj3 i jrj trivial integrations. Integrating over p0 and p00 produces delta functions that enforce the conditions y0 ¼ x0 ≡ t and 0 0 where we write r ≡ z t − z t0 for brevity. Defining y0 ¼ x0 ≡ t0, respectively. Moreover, integrating over y ð Þ ð Þ − 0 r v 2a 2 O 3 and y0 enforces the conservation of 3-momentum along the s¼t t , we Taylor expand ¼s ðtÞþs ðtÞ= þ ðs Þ entire diagram, p ¼ p0 ¼ q. The result is and substitute it back into Eq. (D7) to obtain Z Z Z M2 eiq·ðx−x0Þ : 4 δþ x δ− x0 W q0; t; t0 ; 4 ⊃−16π 2 Φ_ 2v2vˆ z Fig ðaÞ¼16 4 ð Þ ð Þ q6 ð Þ Fig: ðaÞ ðGMÞ · −; ðD9Þ mPl x;x0 q t ðD4Þ after integrating over s. Notice that only the Oðs0Þ terms with Wðq0;t;t0Þ¼½−2iq0Φ_ ðtÞ−μ2ΦðtÞ½2iq0Φ_ ðt0Þ−μ2Φðt0Þ. contribute to the force because of the delta function δðsÞ. We perform the integral over q by utilizing the standard The desired result can already be read off from Eq. (D9), identity meaning that the terms involving W0 and W1 do not contribute. This is easy to see, since Z ddq eiq·r 1 Γðd=2 − αÞ r2 α−d=2 ¼ ; ðD5Þ ∂ d q2 α d=2 Γ α 4 0 0 3 2 ð2πÞ ð Þ ð4πÞ ð Þ W0ðt; t Þ jz ðtÞ − zðt Þj ∼ Oðs Þ; ∂zi ðtÞ þ þ PL 0 ∂ while the integral over q is performed by replacing each 0 d 0 3 W1ðt; t Þ jz ðtÞ − zðt Þj ∼ OðsÞ: 0 0 0 0 0 ∂zi þ factor of iq in Wðq ; t; t Þ with a derivative d=dt acting dt þðtÞ PL

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[1] W. Israel, Event horizons in static vacuum space-times, [24] M. J. Valtonen et al., A massive binary black-hole system Phys. Rev. 164, 1776 (1967). in OJ 287 and a test of general relativity, Nature (London) [2] P. T. Chruściel, J. L. Costa, and M. Heusler, Stationary 452, 851 (2008). black holes: Uniqueness and beyond, Living Rev. Rela- [25] L. Dey et al., Authenticating the presence of a relativistic tivity 15, 7 (2012). massive black hole binary in OJ 287 using its general [3] R. Ruffini and J. A. Wheeler, Introducing the black hole, relativity centenary flare: Improved orbital parameters, Phys. Today 24, No. 1, 30 (1971). Astrophys. J. 866, 11 (2018). [4] J. D. Bekenstein, Nonexistence of Baryon number for [26] N. Yunes, K. Yagi, and F. Pretorius, Theoretical physics static black holes, Phys. Rev. D 5, 1239 (1972). implications of the binary black-hole mergers GW150914 [5] S. W. Hawking, Black holes in the Brans-Dicke theory of and GW151226, Phys. Rev. D 94, 084002 (2016). gravitation, Commun. Math. Phys. 25, 167 (1972). [27] J. Healy, T. Bode, R. Haas, E. Pazos, P. Laguna, D. M. [6] K. S. Thorne and J. J. Dykla, Black holes in the Dicke- Shoemaker, and N. Yunes, Late inspiral and merger of Brans-Jordan theory of gravity, Astrophys. J. 166, L35 binary black holes in scalar-tensor theories of gravity, (1971). Classical Quantum Gravity 29, 232002 (2012). [7] S. L. Adler and R. B. Pearson, “No-hair” theorems for the [28] E. Berti, V. Cardoso, L. Gualtieri, M. Horbatsch, and U. Abelian Higgs and Goldstone models, Phys. Rev. D 18, Sperhake, Numerical simulations of single and binary 2798 (1978). black holes in scalar-tensor theories: Circumventing the [8] T. Zannias, Black holes cannot support conformal scalar no-hair theorem, Phys. Rev. D 87, 124020 (2013). hair, J. Math. Phys. (N.Y.) 36, 6970 (1995). [29] T. Damour and G. Esposito-Far`ese, Tensor-multi-scalar [9] J. D. Bekenstein, Novel “no-scalar-hair” theorem for black theories of gravitation, Classical Quantum Gravity 9, 2093 holes, Phys. Rev. D 51, R6608 (1995). (1992). [10] A. Saa, New no-scalar-hair theorem for black holes, [30] D. A. Feinblum and W. A. McKinley, Stable states of a J. Math. Phys. (N.Y.) 37, 2346 (1996). scalar particle in its own gravational field, Phys. Rev. 168, [11] T. P. Sotiriou and V. Faraoni, Black Holes in Scalar-Tensor 1445 (1968). Gravity, Phys. Rev. Lett. 108, 081103 (2012). [31] P. Jetzer, Boson stars, Phys. Rep. 220, 163 (1992). [12] A. A. H. Graham and R. Jha, Nonexistence of black holes [32] E. Seidel and W.-M. Suen, Oscillating Soliton Stars, Phys. with noncanonical scalar fields, Phys. Rev. D 89, 084056 Rev. Lett. 66, 1659 (1991). (2014); Erratum, Phys. Rev. D 92, 069901(E) (2015). [33] S. L. Liebling and C. Palenzuela, Dynamical boson stars, [13] P. Bizon, Colored black holes, Phys. Rev. Lett. 64, 2844 Living Rev. Relativity 20, 5 (2017). (1990). [34] L. Visinelli, S. Baum, J. Redondo, K. Freese, and F. [14] A. Achúcarro, R. Gregory, and K. Kuijken, Abelian Higgs Wilczek, Dilute and dense axion stars, Phys. Lett. B 777, hair for black holes, Phys. Rev. D 52, 5729 (1995). 64 (2018). [15] C. A. R. Herdeiro and E. Radu, Kerr Black Holes with [35] W. Hu, R. Barkana, and A. Gruzinov, Fuzzy Cold Dark Scalar Hair, Phys. Rev. Lett. 112, 221101 (2014). Matter: The Wave Properties of Ultralight Particles, Phys. [16] C. Herdeiro, E. Radu, and H. Rúnarsson, Kerr black holes Rev. Lett. 85, 1158 (2000). with Proca hair, Classical Quantum Gravity 33, 154001 [36] L. Berezhiani and J. Khoury, Dark matter superfluidity and (2016). galactic dynamics, Phys. Lett. B 753, 639 (2016). [17] K. Yagi, N. Yunes, and T. Tanaka, Slowly rotating black [37] D. J. E. Marsh, Axion cosmology, Phys. Rep. 643,1 holes in dynamical Chern-Simons gravity: Deformation (2016). quadratic in the spin, Phys. Rev. D 86, 044037 (2012); [38] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ultra- Erratum, Phys. Rev. D 89, 049902(E) (2014). light scalars as cosmological dark matter, Phys. Rev. D 95, [18] T. P. Sotiriou and S.-Y. Zhou, Black hole hair in general- 043541 (2017). ized scalar-tensor gravity: An explicit example, Phys. Rev. [39] R. Brito, V. Cardoso, and P. Pani, Superradiance, Lect. D 90, 124063 (2014). Notes Phys. 906, 1 (2015). [19] T. Jacobson, Primordial Black Hole Evolution in Tensor- [40] A. Arvanitaki and S. Dubovsky, Exploring the string Scalar Cosmology, Phys. Rev. Lett. 83, 2699 (1999). axiverse with precision black hole physics, Phys. Rev. D [20] S. Chadburn and R. Gregory, Time dependent black holes 83, 044026 (2011). and scalar hair, Classical Quantum Gravity 31, 195006 [41] A. Arvanitaki, M. Baryakhtar, and X. Huang, Discovering (2014). the QCD axion with black holes and gravitational waves, [21] R. Gregory, D. Kastor, and J. Traschen, Evolving black Phys. Rev. D 91, 084011 (2015). holes in inflation, Classical Quantum Gravity 35, 155008 [42] A. Arvanitaki, M. Baryakhtar, S. Dimopoulos, S. Dubovsky, (2018). and R. Lasenby, Black hole mergers and the QCD axion at [22] M. W. Horbatsch and C. P. Burgess, Cosmic black-hole advanced LIGO, Phys. Rev. D 95, 043001 (2017). hair growth and quasar OJ287, J. Cosmol. Astropart. Phys. [43] H. Yoshino and H. Kodama, Gravitational radiation from 05 (2012) 010. an axion cloud around a black hole: Superradiant phase, [23] A. Sillanpää, S. Haarala, M. J. Valtonen, B. Sundelius, and Prog. Theor. Exp. Phys. 2014, 043E02 (2014); Probing the G. G. Byrd, OJ 287: Binary pair of supermassive black string axiverse by gravitational waves from Cygnus X-1, holes, Astrophys. J. 325, 628 (1988). Prog. Theor. Exp. Phys. 2015, 061E01 (2015).

024010-26 EFFECTIVE FIELD THEORY FOR BLACK HOLES WITH … PHYS. REV. D 100, 024010 (2019)

[44] R. Brito, V. Cardoso, and P. Pani, Black holes as particle [63] R. D. Jordan, Effective field equations for expectation detectors: Evolution of superradiant instabilities, Classical values, Phys. Rev. D 33, 444 (1986). Quantum Gravity 32, 134001 (2015). [64] E. Calzetta and B. L. Hu, Closed-time-path functional [45] R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. formalism in curved spacetime: Application to cosmologi- Dvorkin, A. Klein, and P. Pani, Stochastic and Resolvable cal back-reaction problems, Phys. Rev. D 35, 495 (1987). Gravitational Waves from Ultralight Bosons, Phys. Rev. [65] C. R. Galley and M. Tiglio, Radiation reaction and Lett. 119, 131101 (2017); Gravitational wave searches for gravitational waves in the effective field theory approach, ultralight bosons with LIGO and LISA, Phys. Rev. D 96, Phys. Rev. D 79, 124027 (2009). 064050 (2017). [66] C. R. Galley, A nonlinear scalar model of extreme mass [46] O. A. Hannuksela, K. W. K. Wong, R. Brito, E. Berti, and ratio inspirals in effective field theory: I. Self-force through T. G. F. Li, Probing the existence of ultralight bosons with third order, Classical Quantum Gravity 29, 015010 (2012). a single gravitational-wave measurement, Nat. Astron. 3, [67] O. Birnholtz, S. Hadar, and B. Kol, Theory of post- 447 (2019). Newtonian radiation and reaction, Phys. Rev. D 88, [47] V. Cardoso, Ó. J. C. Dias, G. S. Hartnett, M. Middleton, P. 104037 (2013). Pani, and J. E. Santos, Constraining the mass of dark [68] C. R. Galley, Classical Mechanics of Nonconservative photons and axion-like particles through black-hole super- Systems, Phys. Rev. Lett. 110, 174301 (2013). radiance, J. Cosmol. Astropart. Phys. 03 (2018) 043. [69] T. Damour and G. Esposito-Far`ese, Gravitational-wave [48] M. J. Stott and D. J. E. Marsh, Black hole spin constraints versus binary-pulsar tests of strong-field gravity, Phys. on the mass spectrum and number of axionlike fields, Rev. D 58, 042001 (1998). Phys. Rev. D 98, 083006 (2018). [70] B. Kol and M. Smolkin, Black hole stereotyping: Induced [49] D. Baumann, H. S. Chia, and R. A. Porto, Probing ultra- gravito-static polarization, J. High Energy Phys. 02 (2012) light bosons with binary black holes, Phys. Rev. D 99, 010. 044001 (2019). [71] T. Damour and A. Nagar, Relativistic tidal properties of [50] W. D. Goldberger and I. Z. Rothstein, Effective field theory neutron stars, Phys. Rev. D 80, 084035 (2009). of gravity for extended objects, Phys. Rev. D 73, 104029 [72] T. Binnington and E. Poisson, Relativistic theory of tidal (2006). Love numbers, Phys. Rev. D 80, 084018 (2009). [51] W. D. Goldberger and I. Z. Rothstein, Dissipative effects in [73] S. Chakrabarti, T. Delsate, and J. Steinhoff, Effective the worldline approach to black hole dynamics, Phys. Rev. action and linear response of compact objects in New- D 73, 104030 (2006). tonian gravity, Phys. Rev. D 88, 084038 (2013). [52] D. J. Rowan and G. Stephenson, The Klein-Gordon [74] S. Chakrabarti, T. Delsate, and J. Steinhoff, New perspec- equation in a Kerr-Newman background space, J. Phys. tives on neutron star and black hole spectroscopy and A 10, 15 (1977). dynamic tides, arXiv:1304.2228. [53] S. Detweiler, Klein-Gordon equation and rotating black [75] I. Z. Rothstein, Progress in effective field theory approach holes, Phys. Rev. D 22, 2323 (1980). to the binary inspiral problem, Gen. Relativ. Gravit. 46, [54] P. Pani, V. Cardoso, L. Gualtieri, E. Berti, and A. Ishibashi, 1726 (2014). Perturbations of slowly rotating black holes: Massive [76] R. A. Porto, The tune of love and the nature(ness) of vector fields in the Kerr metric, Phys. Rev. D 86, 104017 spacetime, Fortschr. Phys. 64, 723 (2016). (2012). [77] R. F. Penna, Near-horizon Carroll symmetry and black [55] B. Kol, The delocalized effective degrees of freedom of a hole Love numbers, arXiv:1812.05643. black hole at low frequencies, Gen. Relativ. Gravit. 40, [78] L. A. Ureña-López and A. R. Liddle, Supermassive black 2061 (2008); Int. J. Mod. Phys. D 17, 2617 (2008). holes in scalar field galaxy halos, Phys. Rev. D 66, 083005 [56] S. Endlich and R. Penco, A modern approach to super- (2002). radiance, J. High Energy Phys. 05 (2017) 052. [79] R. Gregory, D. Kastor, and J. Traschen, Black hole [57] C. R. Galley and B. L. Hu, Self-force on extreme mass thermodynamics with dynamical lambda, J. High Energy ratio inspirals via curved spacetime effective field theory, Phys. 10 (2017) 118. Phys. Rev. D 79, 064002 (2009). [80] Y. Fujii and K.-I. Maeda, The Scalar-Tensor Theory of [58] L. Blanchet, Gravitational radiation from post-Newtonian Gravitation (Cambridge University Press, Cambridge, sources and inspiralling compact binaries, Living Rev. England, 2003). Relativity 17, 2 (2014). [81] H.-Y. Schive, T. Chiueh, and T. Broadhurst, Cosmic [59] K. S. Thorne, Multipole expansions of gravitational radi- structure as the quantum interference of a coherent dark ation, Rev. Mod. Phys. 52, 299 (1980). wave, Nat. Phys. 10, 496 (2014). [60] J. Schwinger, Brownian motion of a quantum oscillator, [82] H.-Y. Schive, M.-H. Liao, T.-P. Woo, S.-K. Wong, T. J. Math. Phys. (N.Y.) 2, 407 (1961). Chiueh, T. Broadhurst, and W.-Y. Pauchy Hwang, Under- [61] L. V. Keldysh, Diagram technique for nonequilibrium standing the Core-Halo Relation of Quantum Wave Dark processes, Zh. Eksp. Teor. Fiz. 47, 1515 (1965) [Sov. Matter from 3D Simulations, Phys. Rev. Lett. 113, 261302 Phys. JETP 20, 1018 (1965)]. (2014). [62] K.-C. Chou, Z.-B. Su, B.-L. Hao, and L. Yu, Equilibrium [83] B. Schwabe, J. C. Niemeyer, and J. F. Engels, Simulations and nonequilibrium formalisms made unified, Phys. Rep. of solitonic core mergers in ultralight axion dark matter 118, 1 (1985). cosmologies, Phys. Rev. D 94, 043513 (2016).

024010-27 WONG, DAVIS, and GREGORY PHYS. REV. D 100, 024010 (2019)

[84] J. Veltmaat and J. C. Niemeyer, Cosmological particle-in- [95] J. Binney and S. Tremaine, Galactic Dynamics, 2nd ed. cell simulations with ultralight axion dark matter, Phys. (Princeton University Press, Princeton, NJ, 2008). Rev. D 94, 123523 (2016). [96] E. C. Ostriker, Dynamical friction in a gaseous medium, [85] Q. Yu, Evolution of massive binary black holes, Mon. Not. Astrophys. J. 513, 252 (1999). R. Astron. Soc 331, 935 (2002). [97] H. Kim and W.-T. Kim, Dynamical friction of a circular- [86] B. Bar-Or, J.-B. Fouvry, and S. Tremaine, Relaxation in a orbit perturber in a gaseous medium, Astrophys. J. 665, fuzzy dark matter halo, Astrophys. J. 871, 28 (2019). 432 (2007). [87] J. Huang, M. C. Johnson, L. Sagunski, M. Sakellariadou, and [98] V. Lora, J. Magaña, A. Bernal, F. J. Sánchez-Salcedo, and J. Zhang, Prospects for axion searches with advanced LIGO E. K. Grebel, On the mass of ultra-light bosonic dark through binary mergers, Phys. Rev. D 99, 063013 (2019). matter from galactic dynamics, J. Cosmol. Astropart. Phys. [88] A. Kuntz, F. Piazza, and F. Vernizzi, Effective field theory 02 (2012) 011. for gravitational radiation in scalar-tensor gravity, J. Cos- [99] E. Poisson and C. M. Will, Gravity: Newtonian, Post- mol. Astropart. Phys. 1905 (2019) 052. Newtonian, Relativistic (Cambridge University Press, [89] N. Bar, D. Blas, K. Blum, and S. Sibiryakov, Galactic Cambridge, England, 2014). rotation curves versus ultralight dark matter: Implications [100] M. C. Ferreira, C. F. B. Macedo, and V. Cardoso, Orbital of the soliton-host halo relation, Phys. Rev. D 98, 083027 fingerprints of ultralight scalar fields around black holes, (2018). Phys. Rev. D 96, 083017 (2017). [90] M. Bošković, F. Duque, M. C. Ferreira, F. S. Miguel, and [101] C. F. B. Macedo, P. Pani, V. Cardoso, and L. C. B. Crispino, V. Cardoso, Motion in time-periodic backgrounds with Into the lair: Gravitational-wave signatures of dark matter, applications to ultralight dark matter haloes at galactic Astrophys. J. 774, 48 (2013). centers, Phys. Rev. D 98, 024037 (2018). [102] W. G. Unruh, Absorption cross section of small black [91] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation holes, Phys. Rev. D 14, 3251 (1976). (Freeman, San Francisco, 1973). [103] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, [92] E. Poisson, A. Pound, and I. Vega, The motion of point Modified gravity and cosmology, Phys. Rep. 513,1 particles in curved spacetime, Living Rev. Relativity 14,7 (2012). (2011). [104] C. M. Will, The confrontation between general [93] D. Blas, D. L. Nacir, and S. Sibiryakov, Ultralight Dark relativity and experiment, Living Rev. Relativity 17,4 Matter Resonates with Binary Pulsars, Phys. Rev. Lett. (2014). 118, 261102 (2017). [105] P. Zimmerman, Gravitational self-force in nonvacuum [94] S. Chandrasekhar, Dynamical friction I. General consid- spacetimes: An effective field theory derivation, Phys. erations: The coefficient of dynamical friction., Astrophys. Rev. D 92, 064040 (2015). J. 97, 255 (1943).

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