Exact theory for the Rezzolla-Zhidenko metric and self-consistent calculation of quasinormal modes
EExact theory for the Rezzolla-Zhidenko metric and self-consistent calculation ofquasinormal modes
Arthur G. Suvorov ∗ Theoretical Astrophysics, IAAT, University of T¨ubingen, T¨ubingen 72076, Germany
Sebastian H. V¨olkel
SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste andIFPU-Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy (Dated: February 10, 2021)A covariant, scalar-tensor gravity is constructed such that the static, spherically symmetric Rezzolla-Zhidenko metric is an exact solution to the theory. The equations describing gravitational pertur-bations of this spacetime, which represents a generic black hole possessing an arbitrary numberof hairs, can then be derived. This allows for a self-consistent study of the associated quasinormalmodes. It is shown that mode spectra are tied to not only the non-Einstein parameters in the metricbut also to those that appear at the level of the action, and that different branches of the exacttheory can, in some cases, predict significantly different oscillation frequencies and damping times.For choices which make the theory appear more like general relativity in some precise sense, wefind that a nontrivial Rezzolla-Zhidenko parameter space is permissible under current constraintson fundamental ringdown modes observed by Advanced LIGO.
I. INTRODUCTION
The recent release of data by Advanced LIGO andVirgo concerning the first half of their third observing runsaw the observed binary black-hole (BH) merger countincrease by 39 [1]. The character of gravitational waves(GWs) from these and previous coalescence events, whichare broadly categorised into inspiral, merger, and ring-down phases, can be used to place tight constraints ontheoretical departures from the theory of general relativ-ity (GR) [2–4], amongst other physics [5]. The ringdownphase is especially suited to the experimental validationof the classical no-hair theorems, which state that thepost-merger object must be a (astrophysically disturbed)Kerr BH [6–8]. In particular, a breakdown of GR inthe strong-field regime, as anticipated from various theo-retical considerations (such as non-renormalizability [9]),would be signalled by the appearance of non-Kerr fea-tures in the quasi-normal mode (QNM) spectrum of thenewborn, ringing object [10–12].A theoretical classification of possible non-Kerr fea-tures is, however, quite challenging. If one is interestedin studying tensorial GWs associated with ringdown insome particular, non-GR theory, it suffices to first de-termine the metric structure of permitted black holes within that theory, and then derive equations describ-ing their response to gravitational perturbations [14, 15].The eigenvalues of the relevant perturbation operator canbe tied to the QNM spectrum, with real parts denotingthe oscillation frequencies and imaginary parts denotingthe inverse of the damping times due to radiation reaction ∗ [email protected] Some theories even predict the existence of disjoint families ofblack holes, which complicates this procedure; see, e.g., Ref. [13]. [12]. This process is far from trivial to carry out in manycases, and often even the first step provides a computa-tional hurdle since finding exact solutions, especially forrealistic and rotating objects, is notoriously difficult. Oneway to circumvent these issues is to instead consider aparameterised spacetime metric, which arises from sometheory-agnostic considerations about what might be ex-pected of astrophysical BHs regardless of the particularsof the gravitational action [16–18]. While parameterisedapproaches have been largely successful in placing tightconstraints on departures from GR from GW data [19–21], they are inherently limited because radiation reac-tion (especially relevant for the imaginary componentsof the QNMs) cannot be accounted for self-consistently:without a governing set of field equations, backreactioncan only be studied approximately.A particular solution to this inverse problem was re-cently proposed in Ref. [22], where it was found that acovariant, scalar-tensor theory of gravity can be designedaround a particular spacetime (see also Refs. [23–25]).More specifically, there are a class of mixed scalar- f ( R )theories that possess the special property that for anygiven metric g , there exists a function f such that g be-comes an exact solution in that theory if the scalar fieldsatisfies a particular constraint equation. In this wayan explicit theory can be reverse-engineered around anygiven parameterised BH metric, such as those consideredin Refs. [16–18], and gravitational disturbances can bestudied self-consistently. It is the purpose of this work toexemplify the inverse mechanism developed in Ref. [22]by deriving relevant perturbation equations and study-ing the QNMs, via Wentzel-Kramers-Brillouin (WKB)methods, of a representative class of parameterised BHmetrics.While realistic BHs are expected to rotate, we considerstatic metrics in this work to provide a first step towardsthe building of exact and self-consistent ringdown wave- a r X i v : . [ g r- q c ] F e b forms for parameterised, non-Kerr objects (see also Ref.[26]). We additionally focus on axial perturbations, asthese are numerically easier to handle. For demonstra-tion purposes, we consider the Rezzolla-Zhidenko (RZ)[18] class of BH metrics, which represent parameteriseddepartures from a Schwarzschild, and thus vacuum GRdue to Birkhoff’s theorem, description. The metric coef-ficients in the RZ family of metrics are built from contin-ued fraction expansions, so as to be able to efficiently rep-resent a large class of non-Schwarzschild BHs with only afew terms [27, 28]. Moreover, while the RZ family is verygeneral, free parameters appearing within the metric arechosen to abide by certain recurrence relations so that thespacetime possesses a number of algebraically-desirableproperties and can withstand existing observational con-straints. We show precisely how the RZ parameters en-ter into the self-consistent perturbation equations, asso-ciated with some particular branch of an exact theory,and influence the QNM spectrum of a hypothetical, non-Schwarzschild BH.This paper is organised as follows. Section II reviewsthe particulars of a mixed scalar- f ( R ) theory, and showshow it can be used to find solutions to the gravitationalinverse problem. Section III presents a derivation for thepotential functions associated with a Regge-Wheeler-likeperturbation scheme. Some specific QNM calculationsare then given in Section IV, with emphasis placed ontheir relationship to the associated Schwarzschild andRZ values obtained from well-motivated approximationschemes. Some discussion is given in Section V. II. MIXED SCALAR-F(R) GRAVITY
The vacuum action for the mixed scalar- f ( R ) theoryconsidered in this work reads A = κ (cid:90) d x √− gf (cid:0) F ( φ ) R + V ( φ ) − χ ( φ ) ∇ α φ ∇ α φ (cid:1) , (1)where κ = (16 πG ) − , G is the (bare) Newton constant(which, together with the speed of light, is set to unitythroughout for ease of presentation), R ≡ R µν g µν de-notes the scalar curvature for the metric tensor g , and F , V , and χ are functions of the scalar field φ . The the-ory described by (1) is a particular case of a very generalclass first introduced by Hwang and Noh [29, 30], whichcan be ghost-free for various choices of f [25]; necessaryconditions for the health of Friedman-Robertson-Walkeruniverses in the theory described by (1), for example,can be deduced from equations (75) and (76) in Ref. [31](see also Sec. IV). When the function f is linear in itsargument X , where we define X ≡ F ( φ ) R + V ( φ ) − χ ( φ ) ∇ α φ ∇ α φ, (2)the action (1) reduces to the standard scalar-tensor onein the Jordan frame [32, 33]. Similarly, the f ( R ) theory ofgravity is recovered for constant scalar field and vanishing potential V [34]. For linear f and constant scalar field,the action (1) therefore reduces to the Einstein-Hilbertone.The equations of motion (EOM) for the metric andscalar fields read [22, 29, 30]0 = F ( φ ) f (cid:48) ( X ) R µν − f ( X )2 g µν + g µν (cid:3) [ F ( φ ) f (cid:48) ( X )] − ∇ µ ∇ ν [ F ( φ ) f (cid:48) ( X )] − χ ( φ ) f (cid:48) ( X ) ∇ µ φ ∇ ν φ (3)and0 = f (cid:48) ( X ) (cid:104) χ ( φ ) (cid:3) φ + dχ ( φ ) dφ ∇ α φ ∇ α φ + R dF ( φ ) dφ + d V ( φ ) dφ (cid:105) + 2 χ ( φ ) ∇ α φ ∇ α f (cid:48) ( X ) , (4)respectively. Equations (3) and (4) define the vacuumfield equations for the theory, though matter can be in-troduced in the usual way via the stress-energy tensor. A. Solving the inverse problem
As demonstrated in Ref. [22], the configuration spaceof the theory defined by (1) is so large that practically anygiven metric (spherically symmetric or otherwise) can beadmitted as an exact solution for some choice of the func-tions f, F, V , and χ . In particular, given a metric, thedynamics of the scalar field can be constrained in such away that that particular g and φ pair form an exact so-lution to equations (3) and (4) provided that f belongsto some particular class of functions.Specifically, given some metric g , if there exists ascalar-field solution φ to the kinematic constraint equa-tion X = F ( φ ) R + V ( φ ) − χ ( φ ) ∇ α φ ∇ α φ, (5)for some constant X , then, for any f such that f ( X ) = f (cid:48) ( X ) = 0, that particular g is an exact solution to(3) and (4). In general, equation (5) must be solvednumerically.As a concrete example, if φ is chosen such that X = 0,then g is a solution to the field equations for f ( X ) = X σ , (6)for any σ >
0. Theories of the form (6) represent apotentially infinitesimal deviation from standard scalar-tensor gravity at the level of the action, and generalise the f ( R ) = R σ theories considered by Buchdahl [35] andothers. These latter theories are of potential astrophysi-cal relevance since they have been successful in explain-ing the flatness of galactic rotation curves for σ ∼ − without the invocation of dark matter [36, 37]. Suchvalues for σ are also consistent with cosmological con-straints coming from primordial nucleosynthesis [38] (seealso Ref. [39]). For σ (cid:46) − , stellar structure is largelyunchanged relative to GR [40], and thus the theory iscapable of accommodating massive neutron stars, suchas J0740+6620 (with mass M ∼ . M (cid:12) [41]), for stiffequations of state. For weakly-varying scalar fields, thesefeatures are likely to persist since the theory (6) reducesto the Buchdahl one exactly for ∇ µ φ = 0 and V = 0. Acurious feature of theories with σ (cid:28) f is not analytic, which means that one cannot employTaylor expansions about small scalar curvatures to in-vestigate the Newtonian limit of the theory. Strong-fieldtests (coming from QNMs, for instance) are therefore es-pecially germane to theories such as (6). Note, however,that analytic models can also be constructed; for exam-ple, the theory with f ( X ) = X + αX + α X / α (cid:54) = 0 admits exact solutions with X = − α − .In any case, we emphasise here that we are not neces-sarily advocating for this particular theory as a realisticdescription for gravitational phenomena, especially sincehigher-order theories such as (1) are generally suscepti-ble to the Ostrogradsky instability (though see Ref. [42]).The techniques presented in this work are only meant toillustrate that self-consistent studies of QNMs for param-eterised spacetimes are possible, and that their proper-ties are sensitive to the particulars of the gravitationalaction, be it (1) or otherwise. In particular, the scalarfield solution to equation (5) depends on the functionalforms of the potential ( F, V ) and kinetic ( χ ) terms, andtherefore behaves differently in different branches of thegeneral theory (1). The QNMs are thus sensitive to theparticulars of the dynamics in the scalar sector (see Sec.III. B). III. GRAVITATIONAL PERTURBATIONS
In general, the geometric response of a background spacetime metric g to gravitational disturbances is stud-ied, at the linear level, by introducing a perturbationterm h through g µν = g µν + δh µν , where δ (cid:28) h is ‘sourced’ only by thebackground metric. In general, one may decompose h asa sum of harmonics, viz. h µν = (cid:80) (cid:96)m h µν,(cid:96)m , where eachof the terms h µν,(cid:96)m are chosen such that they respectthe angular symmetries of the spacetime (see below). Afurther decomposition of h µν,(cid:96)m into axial and polar sec-tors is possible, h µν,(cid:96)m = h axial µν,(cid:96)m + h polar µν,(cid:96)m , where the twopieces are defined by how they transform under parity[43, 44]. A separation of this sort is particularly usefulfor static spacetimes, because the axial and polar sectorsdecouple [45] (see also Ref. [46]). While a derivation ofthe (generalised) Teukolsky equations relevant for rotat-ing solutions in the theory (1) is achievable in principle Throughout this work, an overhead zero denotes a backgroundterm, i.e., an O ( δ ) quantity. using the Newman-Penrose formalism or its extensions(see Refs. [47, 48] for the f ( R ) gravity case), solving theresulting equations is computationally challenging [45].We therefore restrict our attention to non-rotating solu-tions. A. Axial modes
To give a concrete example of how the QNMs of ablack hole can be self-consistently studied and comparedbetween different theories, we focus on axial perturba-tions. Axial perturbations have the benefit that scalardegrees of freedom do not couple to the tensor sector [49],which further simplifies the analysis. The backgroundscalar field still plays a role in the behavior of the per-turbations however [49, 50] and modifies the form of theRegge-Wheeler potential [43]. Additionally, we note thateven if the perturbed scalar field δφ does not couple tothe metric, equation (4) predicts the existence of scalarGWs, the amplitudes of which may be non-negligible insome cases [51].Given some astrophysical measurements of axialQNMs, it has been shown that it is possible to recon-struct the spacetime metric using statistical techniques(e.g., the Bayesian methods detailed in Ref. [20]) if oneassumes some fixed set of EOM. However, metric and ac-tion parameters may be intertwined in the sense that thefunctional structure of the EOM may itself be dependenton the non-Schwarzschild parameters, which complicatesthis procedure. The full track of the inverse problem istherefore a non-linear one: one attempts to astrophysi-cally reconstruct the metric by matching QNM measure-ments to eigenvalues that arise from a metric-dependentoperator. As a first step towards a general solution ofthis difficult problem, we show how one can build a the-ory around a given spacetime and study its axial pertur-bations self consistently. This could be then be combinedwith statistical approaches, such as those detailed in [20],in a future study.As mentioned earlier, we work with static and spher-ically symmetric spacetimes in this work for simplicity.The background line element, in Boyer-Lindquist coordi-nates ( t, r, θ, ϕ ), is taken to be ds = − A ( r ) dt + B ( r ) dr + r dθ + r sin θdϕ . (7)Making use of the Regge-Wheeler gauge [43], a generic Theories of the form (1) generally allow for BHs with topolog-ical horizon structures also [52], though such objects are likelyruled out by electromagnetic observations of reflection spectra inaccreting black holes [53]. axial perturbation can be written as h axial µν,(cid:96)m = h ( t, r ) S (cid:96)mθ ( θ, ϕ ) h ( t, r ) S (cid:96)mϕ ( θ, ϕ )0 0 h ( t, r ) S (cid:96)mθ ( θ, ϕ ) h ( t, r ) S (cid:96)mϕ ( θ, ϕ ) ∗ ∗ ∗ ∗ , (8)where S (cid:96)mθ ( θ, ϕ ) = − csc θ∂ ϕ Y (cid:96)m ( θ, ϕ ) and S (cid:96)mϕ ( θ, ϕ ) =sin θ∂ θ Y (cid:96)m ( θ, ϕ ) for spherical harmonic Y (cid:96)m , and the as-terisk denotes a symmetric entry. The (in general com-plex) QNM frequency ω is introduced by taking a Fouriertransform of the components h and h through h , ( t, r ) = 12 π (cid:90) ∞−∞ dωe − iωt ˜ h , ( ω, r ) . (9)The explicit dependence on ω and the tildes will hence-forth be dropped for ease of presentation.The methodology used to derive the EOM for h and h in the mixed scalar- f ( R ) theory is remarkably similarto that of GR. In particular, the θϕ -component of the O ( δ )-field equations (3) allows for a direct expression of h in terms of h , viz. [ f (cid:48) ( X ) (cid:54) = 0] h ( r ) = i ωB (cid:34) h A (cid:48) + 2 Ah (cid:48) − Ah B (cid:48) B + 2 Ah F (cid:48) ( φ ) φ (cid:48) F ( φ ) + 2 Ah X (cid:48) f (cid:48)(cid:48) ( X ) f (cid:48) ( X ) (cid:35) , (10)where we note that X is a function of radius only forthe spacetime described by (7). For the special case of f (cid:48) ( X ) = 0, a similar expression to that of (10) can beimposed, though without the final term on the secondline. We note, however, that both expressions for h ( r )agree for X (cid:48) ( r ) = 0 (relevant for the inverse problemdiscussed in Sec. II. A), i.e., when the constraint equation(5) is satisfied.Moving forward, we introduce the tortoise coordinate r (cid:63) ( r ) ≡ (cid:82) dr (cid:113) B ( r ) A ( r ) , and a new variable Z through h ( r ) = rZ ( r ) (cid:118)(cid:117)(cid:117)(cid:116) B ( r ) A ( r ) f (cid:48) ( X ) F ( φ ( r )) (11)for f (cid:48) ( X ) (cid:54) = 0 and X (cid:48) ( r ) (cid:54) = 0, and h ( r ) = rZ ( r ) (cid:118)(cid:117)(cid:117)(cid:116) B ( r ) A ( r ) F ( φ ( r )) (12)otherwise, which allows us to rewrite the rϕ -componentof the field equations (3) as a Schr¨odinger-like equation.Algebraic manipulations eventually lead to the gener-alised Regge-Wheeler [43] equation in mixed scalar- f ( R )theory, which takes the familiar form0 = d dr (cid:63) Z + (cid:2) ω − V (cid:96) ( r ) (cid:3) Z. (13)In the general case where f (cid:48) ( X ) (cid:54) = 0, the potential V (cid:96) hasthe lengthy but simple form V (cid:96) ( r ) = V ,(cid:96) ( r ) − f AF f (cid:48) + 3 f (cid:48)(cid:48) X (cid:48) Bf (cid:48) A + rA (cid:48) r − AB (cid:48) B + 3 AF (cid:48) φ (cid:48) F + A X (cid:48)(cid:48) X (cid:48) + A X (cid:48) f (cid:48)(cid:48) f (cid:48) + 3 f (cid:48)(cid:48)(cid:48) A ( X (cid:48) ) Bf (cid:48) . (14)The term V ,(cid:96) ( r ), which reads V ,(cid:96) ( r ) = (cid:96) ( (cid:96) + 1) Ar + 3 ( AB (cid:48) − BA (cid:48) )2 rB + 3 F (cid:48) φ (cid:48) BF A + rA (cid:48) r − AB (cid:48) B + AF (cid:48) φ (cid:48) F + 3 A ( φ (cid:48) ) F (cid:48)(cid:48) BF + 3 AF (cid:48) φ (cid:48)(cid:48) BF , (15)is the piece that can be considered to ‘survive’ for f ( X ) = f (cid:48) ( X ) = 0 and X (cid:48) ( r ) = 0 [cf. equations (11) and (12)].The latter term (15) in particular is the one most rele-vant for solutions of the inverse problem. In this case,given some background metric, the scalar field is chosensuch that the constraint equation (5) is satisfied, and (15)differs from the GR form explicitly through the presenceof φ and F and implicitly through A and B . It is worthpointing out that the only (cid:96) -dependent part within V (cid:96) ( r ) comes from the (cid:96) ( (cid:96) + 1) A/r term, as expected of the-ories which predict that (helicity-2) gravitational wavestravel at the speed of light; in the geometric-optics limit (cid:96) → ∞ , the perturbation equation (13) reduces to thenull geodesic equation (see Ref. [20] for a discussion).Note that we have considered background spacetimesthat are vacuum in this work. This is important to men-tion since, strictly speaking, any given spacetime can alsoarise as a solution to the Einstein equations exactly ifone allows for arbitrary stress-energy tensors. Pertur-bations can be studied self-consistently in this approachalso. However, in this latter instance, it is unlikely thatthe matter sector will be associated with a physical La-grangian (e.g., arising from a fluid) or abide by well-motivated energy restrictions (e.g., dominant energy con-dition) for a generic background [54]. Such an approachis therefore not entirely physical in the study of disturbedBHs, since it would be difficult to explain the presenceof exotic matter following a vacuum merger event, forexample. Nevertheless, a self-consistent approach to thestudy of perturbations of arbitrary spacetimes within GRis formally possible and the EOM are simply given by δR µν = 0 if one assumes that the matter sector is unper-turbed by the ringing.This latter scheme, which we refer to as a ‘GR-like’approximation throughout, could also be used to studythe perturbations of a given spacetime (7). In the case ofconstant scalar field, the ‘GR-like’ Regge-Wheeler equa-tion (see, e.g., equations (14)–(16) in Ref. [20]) is recov-ered exactly from (14) for either linear f or for those f with the property that f (cid:48) ( X ) = 0 [expression (15)]. Inmany cases of interest therefore, one might expect thatthis and similar schemes provide a fair representation forthe QNMs, even if they are inexact (see the discussionin Ref. [20] and below). One of the aims of this presentwork is to provide a quantification for its accuracy, atleast in the simple case of a truncated RZ metric (seeSec. IV). B. Dependence of QNMs on the underlying theory
Expressions (14) and (15) imply that QNMs are, ingeneral, sensitive to the particular choices of f , F , V ,and χ . As a demonstration, consider (6) for the Brans-Dicke choices with vanishing potential, F ( φ ) = φ and χ ( φ ) = χ /φ , where the quantity χ is akin to the Brans-Dicke coupling constant. The χ → ∞ limit thereforecorresponds to a more GR-like theory; when f is linear,the theory reduces to Brans-Dicke and therefore to GRin this limit. In any case, the constraint equation (5)depends on the value of χ , and therefore the scalar fielddoes too when considering a solution to the inverse prob-lem. Since the axial potentials V (cid:96) depend on φ they alsoimplicitly depend on χ , and the QNMs shift dependingon the branch of the general theory (1) under considera-tion.This result is familiar from the study of Kerr orSchwarzschild BHs, which are known to ringdown dif-ferently in different theories of gravity [47, 48, 55], some-times so dramatically that the same BH may be unsta-ble in one theory and not another [56, 57] (see below).For the present case in the study of the inverse problem,this implies that ringdown analyses place constraints onthe metric behaviour and on the theory simultaneously;the QNMs for any given quantum numbers depend on χ and the free parameters within the metric in a non-trivial way. This adds a layer of complication from a data analy-sis perspective, since there may be a degeneracy betweenthe physical BH parameters and those parameters thatappear within the action functional (see, e.g., Ref. [58]).The dependence of the QNMs on χ implies that sta-bility can be theory-dependent. Specifically, the appear-ance of a deep enough turning point for V (cid:96) < ω < V (cid:96) is positive-definite, then it is well known that theSchr¨odinger equation (13) admits no bound states. Thiscondition is not necessary however: using the so-calledS-deformation technique [62], Kimura [63] has presentedevidence that if there exists a continuous and boundedfunction S such that0 = V (cid:96) ( r ) + S (cid:48) ( r ) − S ( r ) , (16)then stability is also assured. These techniques arenot necessary for our analysis, since we consider caseswith sufficiently small deformation parameters and largeenough χ (i.e., more GR-like values) so that the nega-tive gap is never too deep to allow for bound states. Ingeneral though, a given BH may be stable for χ val-ues above some threshold value χ c that depends on thedeformation parameters, and unstable for theories with χ < χ c . A thorough investigation of the (in)stability ofBHs in different branches of the general theory (1) will beconducted elsewhere, since the WKB method used hereis not well-suited for identifying bound states. IV. QUASI-NORMAL MODES FOR THEREZZOLLA-ZHIDENKO METRIC
The key difference between this work and others whichhave studied the QNMs of parameterised, bottom-upmetrics, is that we build an exact theory around thespacetime using the method detailed in Sec. II. A so that vacuum perturbations can be modeled self-consistently.In addition to being of theoretical interest in its ownright, this allows, in principle, for a quantification of theaccuracy of various widely-used approximations involv-ing test fields [64–66] or mild deviations at the level ofthe EOM [14, 15, 20]. Where appropriate, we keep trackof the relative difference between QNMs computed usingthe approach presented here and that of the ‘GR-like’approximation discussed in Sec. III. A, where only thefirst two terms within (15) are kept. We focus on thecase of the RZ metric described in Ref. [18] due its al-gebraic simplicity, though the methods detailed here canbe readily applied to more complicated examples.For our application we consider a subset of themost general RZ metric with non-vanishing parame-ters
M, (cid:15), a , and b , because it provides a good bal-ance of flexibility to approximate non-GR black holes[27, 28], but at the same time carries a numerically-manageable number of terms. This choice implies thatthe BHs we study match Schwarzschild exactly at firstpost-Newtonian order, because we set a = b = 0, whichcan be motivated from Solar system tests [67]. We notehowever that such constraints can be bypassed in somealternative theories of gravity, and these extra parame-ters could be straightforwardly included as well. For thistruncated RZ metric, the coefficients A and B definingthe line element (7) take the form A ( r ) = 1 − r H (1 + (cid:15) ) r + r H ( (cid:15) + a ) r − r H a r (17)and B ( r ) = (cid:16) r H b r (cid:17) A ( r ) , (18)respectively. In expression (17), r H ≡ M/ (1 + (cid:15) ) de-fines the location of the event horizon for black hole mass M , where (cid:15) , a , and b are generic (dimensionless) de-formation parameters that are, in principle, to be con-strained by observations [18]. The metric reduces to theSchwarzschild metric in the limit (cid:15) = a = b = 0. Weemphasise again that the absence of r − terms in thecoefficient A ensures that the spacetime automaticallyrespects many post-Newtonian constraints [67].For the RZ metric defined by expressions (17) and (18),it is straightforward to solve the constraint equation (5)(even when rotation is included; see Fig. 1 in Ref. [22])for the Brans-Dicke choices discussed in Sec. III. B; i.e., F = φ , V = 0, and χ = χ /φ for some positive χ .Solving for the scalar field in the constraint equation (5)with the property that X = 0, i.e.,0 = φ ( r ) R − χ φ (cid:48) ( r ) φ ( r ) B ( r ) , (19)leads to an exact solution in the mixed scalar-tensor the-ory (1) with f ( X ) = X σ for any σ >
0, as discussed inSec. II. A. In general, one should impose the boundarycondition lim r →∞ φ ( r ) = 1 , (20)so that the scalar field approaches the Newtonian value,viz. F ( φ ) →
1. Equation (19) informs us that notall arbitrary combinations of the deformation parame-ters ( (cid:15), a , b ) are permitted within the aforementionedtheory. Some combinations can lead to ghost-like insta-bilities where φ ≤ φ (cid:48) ( r ) < χ = χ = χ = χ = χ = � � � � � �� ϕ = �� � �� �� ������� ������ ( � ) � �� � � � � � � � ϕ FIG. 1. Radial scalar field profiles as solutions to equation(19) for a variety of χ values (see plot legends), where wetake M = 1, (cid:15) = 0 . a = 0 .
15, and b = − .
4. The horizon,occurring at r = r H = 20 /
11, is shown by a vertical dashedline. as solutions to equation (19) for a variety of χ valuesand a representative set of RZ parameters (henceforth,we set M = 1 throughout the rest of the work): (cid:15) = 0 . a = 0 .
15, and b = − .
4. We see that the amplitudeof the scalar field is directly proportional to the recip-rocal of χ and furthermore monotonically decreases asa function of radius, with maxima and minima attainedat the horizon and infinity, respectively. For instance,the χ = 0 . (cid:38) χ = 6 (green curve) case nearthe horizon, though at r ∼
50 the two agree to within afew percent. The scalar hair is short-ranged in all casestherefore, and would be virtually invisible at large dis-tances from the horizon even for relatively small valuesof χ . Given a solution to equation (19), one can nowproceed to evaluate the axial potentials V (cid:96) , as necessaryto compute QNMs. A. Numerical methods
In general, the Schr¨odinger-like equation (13) is subjectto some set of boundary conditions. These are chosensuch that we have pure outgoing radiation at infinity andpurely ingoing radiation at the horizon, i.e., that as r (cid:63) →±∞ , we have [68] Z ( ω, r ) ∼ e ± iωr (cid:63) . (21)Equation (13), subject to the boundary conditions (21),constitutes an eigenvalue problem for the QNMs. How-ever, since we are mostly interested in the frequenciesof the QNMs and not so much the functional form of Z itself, it is not necessary to solve equation (13) formally(see Refs. [10–12] for some classical reviews on the topic).Although there are a number of semi-analytic and nu-merical methods available, we operate with the WKBmethod [69–71] because it is especially suited for prob-lems in which other standard approaches may require sig-nificant adjustments or extensions [72]. Other techniquesinvolving continued fractions (e.g., Leaver’s method [73]),phase-integrals (e.g., [74]), or time-dependent integration(e.g., [75]) typically require an analytic understanding ofthe asymptotic properties of the potential, which may beabsent when one only has numerical potentials definedup to some finite radius at their disposal, as in our case.In the order- j WKB method, the QNM frequency is de-termined through iQ (cid:112) Q (cid:48)(cid:48) − (cid:88) j =2 Λ j ( n ) = n + 12 , (22)for overtone number n , where Q ( r (cid:63) ) ≡ ω n − V (cid:96) ( r (cid:63) ) isevaluated at the maximum of the potential and primesdenote derivatives with respect to the tortoise coordinate r (cid:63) . The Λ j in equation (22) depend on 2 j th -order deriva-tives of V (cid:96) and have lengthy forms that we will not writeout here; see, e.g., Ref. [71].While the WKB method has been extended up to 13thorder using Pad´e approximants [thereby requiring 26 th -order derivatives of V (cid:96) ( r )] [77], numerical stability andimplementation considerations limit us to the 4 th -orderscheme. The relative errors of our results have beenchecked using both 3 rd and 4 th order approximationsand are, at worst, at the percent level for a few cases(imaginary parts). For most cases however the errors aresignificantly smaller. An additional numerical check hasbeen performed using the P¨oschl-Teller approximation[78, 79] to verify the robustness of the WKB method.Note that for a few parameter combinations the (cid:96) = 2potentials develop a small negative gap left of the max-imum (see Figure 2 below), which is likely responsiblefor the percent-level errors mentioned previously. In anycase, the various numerical checks performed in this workmake us confident that the numerical routines for findingthe potential [checking that the right-hand side of (19)vanishes to machine precision] and computing the QNMs(checking agreement between 3 rd and 4 th orders and theP¨oschl-Teller approximation) give reliable results.We graph the (cid:96) = 2 potential profiles V , ( r ) (solidcurves) for two illustrative cases with a = b = 0 and M = χ = 1 but (cid:15) = − . (cid:15) = − . φ (cid:48) ( r ) = 0] are rep-resented by dashed curves. The respective domains ofthe potential functions vary with (cid:15) , because the hori-zon location r H (shown by vertical dashed lines) is sen- Throughout this work we only work with the least-damped (andtherefore most relevant for astrophysical observation) modes forbrevity, which have n = 0 by definition. However, overtones aregenerally expected to be active for a newborn BH and may be im-portant when reconstructing BH parameters from astrophysicaldata; see Ref. [76] for a discussion. FIG. 2. Comparison between the (cid:96) = 2 exact [ V , from equa-tion (15); solid curves] and GR-like (dashed curve) potentialprofiles for a = b = 0 and M = χ = 1 but (cid:15) = − . (cid:15) = − . (cid:15) (cid:46) − . sitive to this quantity. In the case of low | (cid:15) | (i.e., moreSchwarzschild-like), we see that the GR-RZ and exactpotential functions overlap almost exactly: the greatestdifference between the two curves is ∼ .
5% near therespective peaks (shown by solid diamonds). However,since the properties of the QNMs are directly tied to thelocation and value of the peak [cf. equation (22)], evena small difference can lead to a non-trivial disparity inthe real and imaginary components (see Sec. IV. B). Forthe (cid:15) = − . ∼
7% near the respective peaks.Note that a negative gap develops near the horizon in theexact potential for the large | (cid:15) | case, though is not deepenough to produce bound states for the chosen kineticvalue χ = 1. For χ (cid:28) (cid:15) (cid:28) − .
3, bound statesmay exist in principle. More generally, the disagreementbetween the GR and exact schemes scales inversely with χ and directly with | (cid:15) | (see Figures 3 and 4 below, re-spectively). B. Results
In this section we present computations of QNMs usingthe numerical methods detailed in the previous sectionfor a variety of representative cases.Fig. 3 shows the (cid:96) = 2 (blue circles) and (cid:96) = 3 (or-ange squares) QNM frequencies as functions of χ forthe RZ parameters relevant to the scalar field solutionsshown in Fig. 1 (i.e., (cid:15) = 0 . , a = 0 .
15, and b = − . χ in the range 0 . ≤ χ (cid:46) .
5, wesee a substantial and monotonic variation in both thereal ( ≈ ≈ R e ( ) l = 2 l = 3asymptotic GR-RZ I m ( ) l = 2 l = 3asymptotic GR-RZ FIG. 3. Fundamental ( n = 0) QNM frequencies ω as func-tions of χ for (cid:96) = 2 (blue circles) and (cid:96) = 3 (orange squares),where we fix (cid:15) = 0 . , a = 0 .
15, and b = − .
4. The upperpanel displays the real part of ω , and the lower panel theimaginary part. Overplotted (black dots) are the frequen-cies obtained when using the GR-like approximation schemedetailed in the text. (cid:96) = 2 QNMs. Variations on a similar scale are likewiseobserved in the (cid:96) = 3 case. Turning points occur in thegraphs once a critical value of χ ≈ . χ limit, both sets of frequencies asymptote towards par-ticular GR-like values (see below). In any case, we findthat a continuous (with respect to χ ) spectrum of fre-quencies within some bounded range can be achieved fora fixed set of RZ parameters. This result has the inter-esting implication that, given only a single measurementof the fundamental frequency of a newborn object (as inthe case of GW150914 [2]), many RZ parameter sets canbe accommodated within some branch of the generaltheory (1). This stresses the necessity of having multiple Note, however, it may be the case that that particular branchis Ostrogradsky-unstable [42] or does not respect independentconstraints coming from cosmology [36, 37], neutron-star astro-physics [40], or Solar-system dynamics [67].
QNM measurements in placing constraints on BH be-haviour and strong gravity simultaneously (see also Refs.[14, 15, 64]).Overplotted in Fig. 3 are the GR-like QNMs (blackdots), which of course do not vary as a function of χ . Forsmall of values of χ , the two schemes predict distinct be-haviour, as noted previously. In the limit χ → ∞ how-ever, the scalar dynamics are heavily suppressed (cf. Fig.1) and the curves match as the theory approaches theBuchdahl one [35], much in the same way that the Brans-Dicke theory (i.e., linear f ) approaches GR in this limit.In practice, values χ (cid:38) lead to numerical indistin-guishability between the real and imaginary componentsfor the GR-like and exact schemes for this particular setof RZ parameters. Note though that this does not meanthat the frequencies approach the Schwarzschild values:the (cid:96) = 2 fundamental mode in the Schwarzschild case(not shown) has real value Re( ω GR ) = 0 .
374 [75], for in-stance, and is still ∼
8% different from the RZ value inthis particular case even when χ → ∞ . Such a disparitydoes not, however, exceed the limits imposed by obser-vations of GW150914, where constraints on the funda-mental frequency were placed at roughly the ∼
10% levelrelative to the GR (Kerr) values at 90% confidence [2].As another example, we consider the case of a = b =0 with fixed χ = 1 but varying (cid:15) . This case is illustratedin Fig. 4 in a similar style to Fig. 3. In this instance, theGR-like scheme is approached as (cid:15) →
0, where in fact theSchwarzschild QNMs are recovered exactly [75] in bothschemes, as expected. Overall, we see that the GR-likeapproximation is a robust one: disagreements, relativeto the exact case, in the real and imaginary parts of theQNM frequencies are at most ∼
3% for | (cid:15) | (cid:46) . (cid:96) = 2. For larger values of χ and (cid:96) , the disagreementsfall even further. This adds strength to the claims madein Ref. [20], who made use of the GR-like approximationwithin a Bayesian scheme to show how QNM data can beused to reconstruct a spacetime metric to a high degree ofaccuracy. In either scheme, however, we see that betweenthe (cid:15) = 0 (Schwarzschild value) and (cid:15) = − . ≈ (cid:46)
10% leads to the constraint | (cid:15) | (cid:46) .
16 for χ = 1, though we note that such a direct comparisonwith GW data is imprecise at this stage because we donot model rotation.Although the parameter space of RZ-QNMs is verylarge (different quantum numbers, RZ parameters, andtheory variables) and an exhaustive study is impracti-cal, it is instructive to consider a few additional cases.The (cid:96) ≤ (cid:15) , a , and b are shown in Table I (II)for χ = 1 ( χ = 0 . b . The real parts ofthe frequencies are largely similar to the Schwarzschildvalues in these cases for b = − .
38, where we find R e ( ) non-GR-RZ l = 2non-GR-RZ l = 3GR-RZ l = 2GR-RZ l = 3GR Schwarzschild I m ( ) non-GR-RZ l = 2non-GR-RZ l = 3GR-RZ l = 2GR-RZ l = 3GR Schwarzschild FIG. 4. Similar to Fig. 3 though displaying ω as a functionof (cid:15) for fixed values a = b = 0 and χ = 1. Re( ω ) = 0 .
38 for χ = 1 while Re( ω ) = 0 .
368 for χ = 0 .
5, which are marginally larger and smaller thanthe Schwarzschild value Re( ω GR ) = 0 . (cid:38)
40% difference) though, implying thatthe modes would be damped out faster [since the damp-ing time ∝ −
Im( ω ) − ] than the corresponding GR case.For more negative values of b , however, even the realvalues diverge significantly from the Schwarzschild val-ues, and are thus likely ruled-out from GW observations[3]. The damping times for the cases shown in the fi-nal two columns, which have (cid:15) = 0 . b = − . a = 0 .
15 and a = 0 .
3, respectively, are very sim-ilar to each other, and only weakly depend on (cid:96) . Thisimplies that even high- (cid:96) modes may be important inthe early characterisation of BH ringdown for some non-Schwarzschild metrics, since they may not necessarily bedamped out faster than their low- (cid:96) counterparts.
V. DISCUSSION
Recently, a solution to the gravitational inverse prob-lem was presented in Ref. [22]: given some metric g , acovariant, scalar-tensor theory of gravity [with action (1)] can be designed such that that particular g is an exact tothe vacuum field equations (3) and (4) (see Sec. II. A).A practical application of this result is that bottom-upBH metrics can be assigned to an exact theory of gravity,which allows for a self-consistent study of their perturba-tions. In this work, we derive the EOM describing axialperturbations of static, spherically symmetric spacetimesin this theory (Sec. III). As a demonstration of the math-ematical machinery, we compute the QNMs for the RZmetric for a variety of non-GR parameters using WKBmethods (Sec. IV). The approach presented here is notunique to the RZ metric, and the method described herecan be readily adapted to practically any class of static,parameterised BH metrics.While realistic BHs rotate in reality, understanding theQNM spectrum of static objects is still useful in charac-terising hypothetical signatures of modified gravity in thestrong-field regime. In particular, V¨olkel and Barausse[20] have shown how ringdown data can be used to re-construct the local spacetime metric given a theory ofgravity (see also Refs. [14, 15, 64]). In that work, how-ever, a solution to the inverse problem was not available,and so various approximations for the EOM describingQNMs had to be used. This work may therefore help tomaximise the information gleaned from future GW mea-surements when combined with statistical analyses alongthe lines presented in Ref. [20]. For large values of theBrans-Dicke parameter χ , however, this extra step maynot be necessary since we found that the exact resultsare well-approximated by the GR-like scheme.At present, constraints on departures from the GR fun-damental frequency are at roughly the ∼
10% level at90% confidence [2]. For the RZ metric specifically, wefind that this corresponds to a constraint | (cid:15) | (cid:46) .
16 for χ = 1 when a = b = 0, as can be seen from Fig. 4.There is of course additional uncertainty since the QNMfrequencies scale with χ and the other RZ parameterstoo; see Fig. 3 and Tabs. I and II. Overall, we validatethe results of Ref. [20] where GR-like perturbation equa-tions were used, and find that a non-trivial RZ parameterspace is consistent with current ringdown bounds.There are several directions in which extensions of thiswork would be worthwhile. One of these is to includerotation, as mentioned previously: the difficulty in thisis largely computational in nature, since the perturba-tion equations are in general coupled and solving the as-sociated eigenvalue problem requires more involved tech-niques, such as generalizations of Leaver’s method [73] ordirect time domain computations. In particular, the in-verse problem as presented here is still relatively straight-forward to handle for stationary spacetimes (see the ex-ample given in Ref. [22]), and so this aspect of the workis not difficult to extend. In a similar way, we haveonly looked at axial QNMs here, though polar pertur-bations are expected to carry ∼
50% of the GW energyaway from a newborn BH due to Regge-Wheeler-Zerilliisospectrality (though cf. Refs. [80, 81]), and are there-fore astrophysically important. Further investigation of0
TABLE I. Selected (cid:96) ≤ χ = 1 /φ (i.e., χ = 1). The columnsheaded by vectors ( q , q , q ) refer to the values of the RZ parameters: (cid:15) = q , a = q , and b = q , respectively. Thecomplex frequencies are given in geometrical units, though can be converted into physical units through a multiplication by(2 π × × M (cid:12) /M ) Hz (see, e.g., Ref. [10]). For reference, the Schwarzschild values are given by ω GR = 0 . − . i for (cid:96) = 2 and ω GR = 0 . − . i for (cid:96) = 3. Quantum number (0,0,-0.38) (0,0,-0.57) (0.1,0.15,-0.19) (0.1,0.3,-0.19) (cid:96) = 2 0 . − . i . − . i . − . i . − . i(cid:96) = 3 0 . − . i . − . i . − . i . − . i(cid:96) = 4 0 . − . i . − . i . − . i . − . i(cid:96) = 5 1 . − . i . − . i . − . i . − . i TABLE II. Similar to Tab. I but for χ ( φ ) = 1 / (2 φ ), i.e., χ = 0 . Quantum number (0,0,-0.38) (0,0,-0.57) (0.1,0.5,-0.19) (0.1,0.3,-0.19) (cid:96) = 2 0 . − . i . − . i . − . i . − . i(cid:96) = 3 0 . − . i . − . i . − . i . − . i(cid:96) = 4 0 . − . i . − . i . − . i . − . i(cid:96) = 5 1 . − . i . − . i . − . i . − . i the existence of bound states for non-RZ BHs using theS-deformation technique associated with equation (16) isalso interesting, since it is known that even Kerr blackholes can be unstable in some theories [56, 57]. Using themethods presented here, one could attempt to map outthe space of stable BH solutions in a theory-dependentmanner (e.g., by considering stability as a function of χ ). Finally, the scalar-tensor class of theories (1) is notthe only type of theory that can be designed to solvethe inverse problem. Mixed vector- f ( R ) theories (e.g.,generalised Proca theories [82]) also provide examplesof solutions to the inverse problem; see Ref. [22] for adiscussion. It would be worthwhile to study the QNM spectrum of RZ or other black holes in these theories infuture. ACKNOWLEDGMENTS
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