Constraining Black-Hole Horizon Effects by LIGO-Virgo Detections of Inspiralling Binary Black Holes

Constraining black-hole horizon effects by LIGO-Virgo detections of inspiralling binary black holes

Kwun-Hang Lai and Tjonnie Guang Feng Li Department of Physics, The Chinese University of Hong Kong, Sha Tin, Hong Kong. (Dated: September 24, 2018) General relativity predicts mass and spin growth of an inspiralling black hole due to an energy-momentum flux flowing through the black-hole horizon. The leading-order terms of this horizon flux introduce 2.5 and 3.5 post-Newtonian corrections to inspiral motions of binary black holes. The corrections may be measurable by gravitational waves detectors. Since the proper improvements to general relativity is still a mystery, it is possible that the true modified gravity theory introduces negligible direct corrections to the geodesics of test masses, while near horizon corrections are observable. We introduce a parameterization to describe arbitrary mass and spin growth of inspiralling black holes. Comparing signals of gravitational waves and a waveform model with parameterized horizon flux corrections, deviations from general relativity can be constrained. We simulate a set of gravitational waves signals following an astrophysical distribution with horizon flux modifications. Then, we perform a Bayesian analysis to obtain the expected constraints from the simulated response of the Advanced LIGO-Virgo detector network to the simulated signals. We show that the constraint can be improved by stacking multiple detections. The constraints of modified horizon flux can be used to test a specific class of modified gravity theories which predict dominant corrections near black-hole horizons over other types of corrections to general relativity. To support Hawking’s area theorem at 90% confidence level, over 10000 LIGO-Virgo detections are required. Within the lifetime of LIGO and Einstein Telescope, a future ground-based gravitational wave detector, near horizon corrections of modified gravity theories are potentially detectable if one of the modified gravity theory is true and the theory predicts a strong correction.

I. INTRODUCTION A gravitational wave emitted by a binary black-hole merger consists of three phases: inspiral, merger, ringdown. The properties of black hole horizons during the merger, includ- In the observable universe, regions near black-hole horizons ing the area theorem, are discussed in [16]. To extract the contain the most extreme spacetimes. The strong curvature information of black hole horizons from a gravitational wave nature of black-hole horizons provides a stage for interesting detection, Cabero et al. [17] proposed an observational test by physical phenomena, both classically and quantum mechani- checking the consistency between the initial black-hole area cally. A set of thermodynamics laws governing the evolution in the inspiral phase and the final black-hole area in the ring- of black hole horizons is predicted by general relativity and down phase. In this paper, we propose another way to extract quantum physics [1,2]. It is natural to ask if we can observe the evolution of black-hole horizons from the inspiral phase deviations from general relativity in such a strong curvature only. regime. In fact, modified gravity theories can predict significantly different features of black holes or black-hole-like ob- In general relativity, the inspiral motion of a binary black jects. For example, a specific type of scalar-tensor gravity holes can be described by Post-Newtonian (PN) formal- predicts black holes with zero temperature [3]. It is possible ism [18]. The inspiral motion and the corresponding grav- to construct theories which predict observable deviations from itation waves in modified gravity theories can be described general relativity in the near horizon regime, while effects on under parameterized frameworks including the Parameter- the orbital motion of test particles are negligible. Examples in- ized Post-Newtonian (PPN) [19] and the Parameterized Post- clude scalar-tensor-vector gravity thermodynamics and quan- Einsteinian [20] formalism, where deviations from general arXiv:1807.01840v3 [gr-qc] 21 Sep 2018 tum corrections to black-hole entropy. Those corrections may relativity are parameterized by a set of independent variables. lead to violation of the area theorem. Constraining near hori- Through comparing parameterized waveforms and gravita- zon modifications with observational data can rule out some tional waves detected by Laser Interferometer Gravitational- of the theories in this category and hint at the correct form of Wave Observatory (LIGO), parameters of the parameterized gravity. waveforms can be constrained by Bayesian analysis [21, 22]. There are no significant deviations from general relativity Currently, X-ray binary systems [4,5], measurements of found so far [13, 20]. stellar orbital motions around supermassive black holes [6– 8] and gravitational waves [9–13] are the main ways to ob- Waveforms in the PN formalism can be derived without serve black holes, and provide the possibility of observing the considering the internal structure of the objects, which is physics of black-hole horizons. X-ray binary systems require known as the effacement principle [23]. For the purpose of complicated models to describe atomic processes [14], which detecting gravitational waves, the effacement principle is suf- makes it hard to extract the information of horizons. The ficient [24]. However, modified gravity theories may pre- Event Horizon Telescope is a telescope network that observes dict observable corrections near black-hole horizons, while astronomical objects near supermassive black hole horizons direct effects on geodesics of black holes are negligible. Dif- directly [15]. In contrast, we propose an alternative method ferent phenomena have been investigated to understand the to extract the physics of horizons through observing binary effects of near-horizon corrections [25], including the tidal black-hole merger by gravitational waves. Love number [26, 27], echoes [28, 29], and tidal heating with 2

LISA binaries [30]. In this paper, we investigate the con- infinity [36], while AH is defined as the outermost marginally straint on a modified horizon flux1, predicted by some modi- trapped surface [37]. During the inspiral phase of a binary fied gravity theories, by simulating responses of an Advanced black hole coalescence, we show that EH and AH are indistin- LIGO-Virgo detector network to waveforms in a modified PN guishable by the LIGO detectors in AppendixA. As a result, formalism. The constraint can be used to test those modi- we do not discriminate between AH and EH cross-section ar- fied gravity theories with dominating corrections near black- eas in a time slice, the horizon areas of the two black holes are hole horizons. The methodology has a similar philosophy denoted as Ai. to [30], but we focus on the stellar-mass black-hole binaries which are detectable by the Advanced LIGO-Virgo detector network other than the supermassive black-hole considered B. Growth of black hole area, mass and spin in [30]. Since the surface gravity on a spherically-symmetric stellar-mass black hole is stronger than that on a spherically- By adopting the ingoing Kerr coordinates (v, r, θ, φ), the symmetric supermassive black hole, some modified gravity growth of black hole horizon can be written as [32] theories may predict stronger near-horizon corrections for a stellar-mass black hole, and thus the corrections may be better dA I i = ΘdS, (1) constrained by the Advanced LIGO-Virgo detector network. dv This paper is structured as follow: Sec.II describes the origin of the horizon flux and the parameterized modification of where Θ denotes the expansion scalar which quantifies how the flux; Sec. III shows the simulated constraints of the param- the black-hole area changes, and dS denotes the differential eterized modification; and Sec.IV discusses the limitations of surface area of the horizon. the methodology and points to future research. The first law of black hole thermodynamics and the mode decomposition of matter fields can be written as (see [32] for details) II. PARAMETRIZED HORIZON EFFECTS IN GRAVITATIONAL WAVEFORMS κ hA˙ i = hm˙ i − Ω hJ˙ i, (2) 8π i i H i In this section, we review the horizon flux into a black hole, derived by Alvi [31], Poisson [32], and Chatziioannou et al. [33, 34]. Assuming that the spacetime around a black- ω hm˙ i = hJ˙ i , (3) hole horizon deviates from general relativity, we introduce a i µ,ω µ i µ,ω parameterization to model the effect of the horizon flux to a gravitational waveform. The applications of this parameteri- where ΩH is the angular velocity of the unperturbed black zation are also discussed. hole, the brackets denote accumulative growths while elimi- nating the highly oscillating modes, and note that µ here refer to an index of the Fourier modes. After solving the Teukol- A. Configuration and notations sky’s equation, we have [33]

We focus on the inspiral phase of a binary black hole coales- dJi cence, which allows us to separate and study the black-hole = (Ω − Ω)C0 , (4) dx H x horizon effects. Black holes i (i = 1, 2) are described by mass mi and angular momentum Ji, where m1 < m2, and where we require Ji to stay parallel/anti-parallel to the orbital angular momentum for simplicity, which is sufficient to cover dmi 0 = Ω(ΩH − Ω)Cx, (5) most of the sources to which LIGO is sensitive [35]. The di- dx 2 mensionless spin parameters are defined as χi = Ji/mi . For convenience, we show some of the equations in total mass 2 M = m + m and symmetric mass ratio ν = m m /M . dAi 8π 1 2 1 2 = − (Ω − Ω)2C0 , (6) The PN velocity is denoted by x = (πMf)1/3, where f is the dx κ H x frequency of the orbital motion. Throughout the paper, we use the geometrized units c = G = 1. where Ω is angular velocity of the tidal field created by its 0 Each black hole is bounded by a few horizons. The event companion black hole respectively. Cx < 0 [33] is a function horizon (EH) and the apparent horizon (AH) are two of the of m1, m2, χ1, χ2 and x. Together with dx/dt > 0 [38], most important horizons of black holes. EH is defined as the Eq. (6) implies Hawking’s area theorem hdAi/dti > 0. boundary of a region where no null curve can reach future null Eqs. (4) and (5) can be interpreted as the angular momentum flux and mass flux into a Kerr black-hole horizon. Mass flux and angular momentum flux flow into a general horizon are well defined, though it is derived in a different ap- 1 Horizon flux is another name for the tidal heating effect. proach [39]. 3

C. 2.5PN and 3.5PN horizon terms in frequency domain dJ1 dJ1 Eqs. (4) and (5) can be integrated in the PN barycentric frame → (1 + β ) , (12) dx 1 dx to give a 2.5PN and 3.5PN phase term (AppendixB). We follow the logic of [40] to derive the effect of the horizon flux in the TaylorF2 formalism2. dJ2 dJ2 In the frequency domain, the TaylorF2 gravitational wave- → (1 + β2) , (13) form h(f) with the horizon effect can be written as dx dx where the unmodified flux follows Eqs. (4) and (5). In gen- F 2 −i[ΨF 2(f)+ΨF 2(f)] eral, α1, α2, β1, β2 depend on x. In the PN framework, h(f) = A (f)e H , (7) α1, α2, β1, β2 can be expressed as polynomial series of x. To where AF 2(f) and ΨF 2(f) are the amplitude and the phase investigate the simplest case of the horizon flux modification, of the TaylorF2 waveform respectively without the horizon we restrict α1, α2, β1, β2 to be constant throughout the paper. effect (we refer to [38, 41] for the 3.5PN aligned spins Tay- With the parameterized mass and spin flux modifications, F 2 the modified TaylorF2 2.5PN and 3.5PN horizon phase terms lorF2 waveform without the horizon terms), and ΨH (f) corresponds to the horizon effect phase term. are given by F 2 The horizon flux introduces extra 2.5PN (ΨH,5) and 3.5PN F 2 (ΨH,7) phase terms. The phase terms are given by F 2 x F 2 ΨH = 1 + 3 ln ΨH,5(α1, α2) xreg (14) 2 F 2 F 2 x F 2 2 F 2 + x ΨH,7(α1, α2, β1, β2), ΨH = 1 + 3 ln ΨH,5 + x ΨH,7, (8) xreg

F 2 where ΨH,5 = (1 + α1)C5α1 + (1 + α2)C5α2, F 2 F 2 ΨH,7 = (1 + α1)C7α1 + (1 + α2)C7α2 (15) ΨH,5 = C5α1 + C5α2, (9) F 2 + (1 + β1)C7β1 + (1 + β2)C7β2, ΨH,7 = C7α1 + C7α2 + C7β1 + C7β2, where C ,C ,C ,C ,C ,C are given and where we ignore the effect of spins on the 5α1 5α2 7α1 7α2 7β1 7β2 in (B29). innermost stable orbit and choose the PN veloc- ΨF 2 is a function of α and α , while ΨF 2 is a function of ity of the Schwarzschild innermost stable orbit H,5 1 2 H,7 √ α , α , β and β . When α = α = β = β = 0, Eq. (14) x = x = 1/ 6 as the stopping condition of the 1 2 1 2 1 2 1 2 ISCO reg reduces to the horizon terms in general relativity, given by waveform. C ,C ,C ,C ,C ,C are given in 5α1 5α2 7α1 7α2 7β1 7β2 Eqs. (8) and (9). AppendixB. The TaylorF2 waveform with the parameterized phase terms is utilized to analyze the constraint on the horizon effect D. Modified mass and spin flux in Sec. III. Even though the TaylorF2 model can accurately describe the early inspiral phase [38], it fails to capture the higher frequency late-inspiral-merger-ringdown phase [24]. The variation of mass and spin produces observable effects For investigative purposes, we use TaylorF2 for both signal in the inspiral phase. To test any theory which deviates from simulations and data analysis instead of analyzing real sig- general relativity in the near horizon regime, we insert mass- nals (such as GW150914 [9]) to avoid misinterpretation of the growth parameters α1, α2 and spin-growth parameters β1, β2 errors caused by the inaccurate late-inspiral-merger-ringdown to parameterize the mass and spin flux deviations on black band in the TaylorF2 model as a violation of general relativity. hole 1,2 respectively: However, to test general relativity with real LIGO signals, a more detailed study on the potential systematic error induced dm1 dm1 → (1 + α1) , (10) by the inaccuracy of the model is required. With the simplify- dx dx ing assumption in this paper, the results can be interpreted as an optimistic estimation compared to an actual study on real signals. dm dm 2 → (1 + α ) 2 , (11) dx 2 dx E. Application of the horizon effect parameterization

If a theory predicts observable deviations from general rela- 2 While we agree on the 2.5PN phase term, we compute a slightly different 3.5PN phase term comparing to Eq. (4.40) in [40]. However, the difference tivity around black-hole horizons, these deviations can be pa- is negligible for the cases considered in this paper, see the discussion in rameterized by non-zero α1, α2, β1, β2. Furthermore, if the AppendixB. theory predicts negligible direct effects on geodesics of test 4

Sec.IIE, we focus on the minimal parameterization α = TABLE I: Expressions of the area theorem (AppendixC1), scalar- 1 α = β = β = α. The following binary black hole param- tensor-vector gravity thermodynamics (AppendixC2) and quantum 2 1 2 gravity correction to black-hole entropy (AppendixC3) in terms of eters are free parameters in the inference process: α, masses, α. Symbols α0, Ξ are the independent parameters in the correspond- aligned/anti-aligned (with orbital angular momentum) spins, ing models, while A is the area of a black hole. inclination, angular sky location and distance. Noise is simulated assuming that all three Advanced LIGO-Virgo detectors Theory α are operational at their design sensitivity [48, 49]. To estimate the constraints on horizon effects from multiple LIGO-Virgo Area theorem [43] ≥ −1 detections, multiple α = 0 (without modiﬁcation of general

γ√ relativity) events are simulated and analyzed. Scalar-Tensor-Vector Gravity thermodynamics [44] = − 1+γ+ 1+γ

2 Quantum corrections to black-hole entropy [45] = 256π Ξ A A. Dependence between constraints of α and binary black holes parameters

The area theorem discussed in Sec.IIE motivate us to search for α at order 1. Therefore, it is sufficient to set the prior of masses, α1, α2, β1, β2 are the leading order measurable deviations. α to be uniformly distributed within −50 < α < 50. We To quantify the requirement where corrections from hori- investigate the constraints on α of different kinds of binary zon effects dominate, we consider a point-mass binary system black holes by Bayesian analysis. Eq. (15) indicates that constraints on α are related to total mass M, symmetric mass in a modified gravity theory. The orbital energy Eorb can be expressed as a deviation δ from the PN energy E , ratio ν and spin parameters χ1 and χ2. To investigate the PN dependencies between the above parameters and constraints on α, we fix all inclinations (0°), sky locations (longitude = 120°, latitude = -70°) and distances (400 Mpc) of the sim- Eorb = EPN(1 + δ). (16) ulated gravitational waves events in this section. We simu-

The dominating order of EPN is proportional to the orbital sep- late 3 sets of 50 gravitational waves events: (1) chirp mass 3/5 1/5 aration 1/r, where r is proportional to 1/x2. If δ can be ex- Mc = (m1m2) /M uniformly distributed from 5M pressed in a power law, i.e., δ ∝ 1/rn + O(1/rn+1), where to 35M , χeff = 0.9 and ν = 0.16, (2) ν distributed from n > 3.5, then δ has no effect on the 3.5PN waveform. There- 0.08 to 0.25, Mc = 30M and χeff = 0.9, (3) χ1 and χ2 fore, the lowest order correction from the horizon effect can uniformly distributed from −0.9 to 0.9, Mc = 30M and become the leading order correction to general relativity. ν = 0.16. For each event, a posterior distribution of α is ob- To relate the parameterization to a specific modified grav- tained, and the width (∆α) of the 90% confidence interval of ity theory, the horizon effect should be derived from the the- α is calculated from the posterior distribution. The results are ory. Following the philosophy of deriving the horizon effect summarized in Fig.1, where the variables χ1 and χ2 are col- in general relativity [32], the metric perturbation equation, the lected into a single variable χeff. Fig.1 shows that constraints gravitational flux flowing into a black hole horizon, and the on α improve with decreasing Mc and η, while the constraints black hole thermodynamics laws are potentially essential to are only weakly dependent on χeff. As demonstrated in Fig.1, derive the horizon effect in a modified gravity theory. How- the dependencies are quantified by the normalized covariances ever, horizons may not exist in some modified gravity theo- between the variables and ∆α. Among the three variables, ries [42], which means the derivation process of the horizon the constraint of α depends most on Mc. Therefore, keeping effect may not be valid in those theories. η fixed, a cleaner constraint on the horizon effect can be ob- If we adopt the parameterization α1 = α2 = β1 = β2 = α, tained by analyzing a lower mass binary black-hole merger, then deviations from general relativity including violation of despite a lower signal-to-noise ratio (SNR). The constraint on black-hole area theorem, scalar-tensor-vector gravity thermo- the horizon effect from multiple LIGO-Virgo detections (see dynamics and quantum corrections to black-hole entropy can Sec. III B) depends on the astronomical source mass distri- be expressed in terms of α (see TableI and AppendixC). bution. For example, an astrophysical distribution favoring Since the complete modification due to a modified gravity the- lower mass black-hole mergers implies a cleaner constraint ory are not included, TableI suggests some signatures of the on the horizon effect by LIGO-Virgo detections. modification instead of providing a precise description of the waveform in a modified gravity theory. B. Stacking posteriors of multiple events

III. CONSTRAINTS ON HORIZON EFFECTS The constraint on α from a single event may not be strong enough to prove or rule out any theory. Stacking posteriors To study the constraints on α1, α2, β1, β2, we simulate wave- of different events can give a much better constraint if there forms and analyze the simulated data using the nested sam- is no systematic bias in the posteriors [50]. Suppose there are pling [46] algorithm in LALInference [47]. Motivated by n independent detection events {H1,H2, ..., Hn}. For each 5

FIG. 2: Median (dots) and 90% conﬁdence interval (cross hairs) for α as a function of number of events. Injections were performed using α = 0 (red line), distributed uniform in χ1 and χ2, isotropic in sky location and uniform in volume, with (top) source masses (5,5) M distributed from 100 Mpc to 200 Mpc, (bottom) source masses (30,30) M distributed from 100 Mpc to 600 Mpc. As the number of events increases, constraints on α improve and approach α = 0.

gate constraints on α from multiple events with different total masses while keeping the mass ratio fixed. We simulate multiple α = 0 waveforms which are distributed uniform in FIG. 1: 90% confidence interval (∆α) for α versus (top) Mc, (mid- χ1 and χ2, isotropic in sky location and uniform in volume. dle) η, (bottom) χeff. The dependencies are quantified by their nor- Binaries with source masses (5,5) M distributed from 100 malized covariances (top) (0.52), (middle) (0.27), and (bottom) Mpc (redshift ∼ 0.02) to 200 Mpc (redshift ∼ 0.05) are sim- (−0.03). Constraints on α improve with decreasing M and η, c ulated to investigate an optimistic case since Fig.1 shows that while the constraints are only weakly dependent on χeff. The result suggests that an astrophysical distribution favoring lower mass we can constrain α better for smaller chirp mass. Addition- black-hole mergers implies a cleaner constraint on the horizon effect ally, binaries with source masses (30,30) M from 100 Mpc by LIGO-Virgo detections. (redshift ∼ 0.02) to 600 Mpc (redshift ∼ 0.15) are simulated to investigate how much can we constrain α in a higher source mass case. The source masses of most of the future event Hi, a posterior distribution is obtained to give a prob- LIGO detections of binary black hole mergers are expected to ability density function P (α|Hi). The multiple events poste- be in between (5,5) M and (30,30) M [51]. Priors of α are rior P (α|H1, ..., Hn) can be calculated by enlarged to be uniformly distributed in −500 < α < 500 to allow larger fractional deviation from general relativity (com- n n pared to the prior in Sec. III A) during the Bayesian analysis, Y Y which helps avoid truncation effects of the posterior when the P (α|H1, ..., Hn) ∝ P (α) P (Hi|α) ∝ P (α|Hi). horizon effect is weak. The waveforms are analyzed using i=1 i=1 (17) LALInference. Fig.2 shows the constraints on α of the Note that we make use of the uniform prior of α, P (α) = two sets of simulated events, where the constraints of multi- const. ple events are calculated by Eq. (17). In Sec. III A, we show that the dependency between the In our simulations, 100 gravitational-wave events of (5,5) +3.3 chirp mass and the constraint on α is stronger than between and (30,30) M binaries constrain α within −2.8−4.9 and +5.6 the mass ratio and the constraint. To understand the quanti- −1.1−4.9 (90% confidence interval) respectively. The bet- tative behavior of the constraint on α, we decided to investi- ter constraint of the (5,5) M (lower chirp mass) case con- 6

firms the relation in Sec. III A. The width (∆α) of the 90% It is possible that extra corrections to general relativity other confidence interval of α is ∼ 10. The independence of the than those from the horizon effect correction exist. Depend- simulated events suggest that the multiple events posteriors ing on the functional form of the extra corrections, we may can be approximated√ by a gaussian distribution, which im- not be able to discriminate the extra corrections from the hori- plies ∆α ∝ 1/ N, where N is the number of events. When zon effect due to degeneracies between the parameters of the N ∼ 10000, ∆α ∼ 1, suggesting that it is possible to prove horizon effect and parameters describing these extra effects. the area theorem α > −1 (Sec.C1) at 90% confidence level. Therefore, the horizon effect parameterization is not guaran- The number of events required to support/disprove a modified teed to be a fully model-independent way to extract the infor- theory depends on the corresponding horizon effects parame- mation of black-hole horizons. Instead, we suggest that the ter in the theory (see Sec.IIE). Approximately, 100 detections horizon effect parameterization can be used to test a modified of gravitational waves can constrain γ and Ξ to γ > −0.99 gravity theory only if the theory predicts dominant horizon 2 2 and −150M < Ξ < 150M respectively. The constraint effect correction over other types of corrections to general rel- rules out a range of parameters in those theories, which pro- ativity. Theories falling into this category can be ruled out by vides a ground for further theoretical research. comparing the predicted horizon effects with the constraints obtained by gravitational waves detections. When extracting the horizon effects from real signals, cau- IV. DISCUSSION tions should be taken to avoid interpreting systematic errors as a violation of general relativity. Since the TaylorF2 model is In this paper, we have developed a practical connection be- inaccurate in the late-inspiral-merger-ringdown band, system- tween horizon effects predicted by abstract theories and ob- atic errors arise naturally when a real signal is analyzed by the servable gravitational waves. During the inspiral phase of a inaccurate model [53]. Applying a low-pass filter to the re- binary black hole coalescence, horizon effects can be sepa- move the late-inspiral-merger-ringdown signal from the data helps to ensure that the violation detected does not come from rated and parameterized by mass-growth parameters α1, α2 a misinterpretation of the late-inspiral, merger and ringdown and spin-growth parameters β1, β2 to describe the mass and spin flux deviations on black hole 1,2 respectively. With the waveform. However, the low-pass filter reduces the strength of the signal, which may weaken the constraint of the hori- minimal parameterization α1 = α2 = β1 = β2 = α, we show that the 90% confidence interval of α can be constrained zon effect, and systematic bias can be caused by the abrupt to ∆α ∼ 10 with 100 gravitational waves detections from bi- frequency cut [54]. On the other hand, improper calibrations nary black-hole mergers by the Advanced LIGO-Virgo detec- of detectors can also induce systematic errors [55]. Without tor network. If the modified gravity theory further satisfies proper treatments, the systematic errors can exceed the statis- the condition where the predicted horizon effect correction tical errors when we try to extract weak horizon effects from dominates over other types of corrections to the geodesic mo- real signals [56]. To establish a rigorous constraint on the tions, then it is suitable to test the theory by the constraints horizon effects, studies should be conducted in the future to on the horizon effect. For example, the black hole area the- understand the influence of various systematic errors to the orem, scalar-tensor-vector gravity thermodynamics and quan- constraint. tum corrections to black-hole entropy are suitable to be tested Even though the analytical expression of the horizon flux or constrained by their horizon effects. is used to analyze the detectability of the horizon effects in The parameterized test proposed in this paper can be con- this paper, there are a few theoretical subtleties in the topic. sidered as a subset of LIGO’s parameterized test [20], but aims While following similar derivation steps as [40], we obtain a at separating and measuring the horizon effect alone. Theoret- slightly different numerical expression of the TaylorF2 hori- ically, LIGO’s parameterized test is general enough to capture zon effects in Eqs. (B29) and (B30) (see AppendixB for de- a large set of modified gravity theories. However, the possibil- tails). Besides, Chatziioannou et al. [33] point out a disagree- ity where horizon effect corrections dominate in some theories ment with the extreme mass ratio result in [57]. We do not use motivates us to study the slightly more specific parameterized the unsettled PN order expression in our derivation. Moreover, horizon effect instead of considering the full generic test. Al- we do not consider the full correction from a self-consistent though only the parameterization α1 = α2 = β1 = β2 = α theory in Sec.IIE. Other corrections including metric per- has been considered in our study, a generic α1, α2, β1 and β2 turbation corrections and gravitational flux corrections could as a function of the parameters of the binary black holes is also either amplify or reduce the horizon effect predicted by ther- possible in some modified gravity theory. The different func- modynamics. Due to the uncertainties, rather than positioning tional form of α1, α2, β1 and β2 can lead to a significantly our result as a rigid reference, the paper aims at investigating different constraint. In principle, if a generic parameterized the possibility of constraining horizon effects using advanced test has been done on a detected signal jointly on the 2.5PN gravitational-wave detectors and the potential of applying a and 3.5PN phase terms, it is possible to reinterpret the pos- constraint to theories. However, even if the numerical details terior, and search for a specific functional form of degenera- are not accurate, the effects are expected to be of a similar cies predicted by the horizon effect, but extra efforts should be order of magnitude. spent to avoid bias introduced by the choice of prior probabil- Assuming that the interpretation in Sec.IIE is correct, the ity distributions [52]. Instead, it is cleaner to constrain horizon area theorem requires approximately ∼ 10000 events to give effects directly by the method proposed in the paper. a reasonable constraint at 90% confidence level depending on 7 the masses of black holes. LIGO may not be able to detect κ ∼ 1/mi implies that large amount of events [51]. However, Einstein Tele- scope is more sensitive than LIGO, especially at lower fre- δr m δm ∼ i i ∼ 10−12, (A2) quencies, which means signals can be detected starting from rAH T mi a lower frequency [58, 59]. Thus, the constraints on α can also be improved by detecting longer signals. Together with which is negligible unless we can probe the horizon at that 2 the potentially higher event rate [59], we may have enough scale. Note that the slowly varying approximation O(m ˙ i ) ∼ events in Einstein Telescope to test area theorem in the fore- 0 is valid only for the inspiral phase. seeable future. On the other hand, if the strongly-spinning, supermassive black-hole binaries are considered, the LISA detector can pose a reasonable constraint on the horizon effect Appendix B: Fully modified horizon flux term even by the detection of a single event [30]. However, the near-horizon correction predicted by a specific modified gravity theory may be mass-dependent, such as the 1/A depen- In this section, we follow the logic of [40] to compute the full dence of α of the quantum corrections to black-hole entropy. modified horizon flux term. The theory predicts a stronger correction when the area of the Define the following symbols for computation convenience, black hole is smaller, which means a lower mass of the black hole. Thus, instead of using the supermassive black-hole bi- √ naries detectable by the LISA detector, this kind of theories ∆ = − 1 − 4ν, (B1) can be better constrained by the advanced LIGO-Virgo detector network or the Einstein Telescope. Despite all the exciting future outlook, the work in this paper has demonstrated that constraints can still be posed on M 2 M 2 S = (1 + ∆)2χ + (1 − ∆)2χ , (B2) the horizon effects of modified gravity theories using the Ad- l 4 1 4 2 vanced LIGO-Virgo detector network. The results motivate us to further investigate the theoretical ground of horizon effects in different modified gravity theories and extend signatures of M 2 M 2 horizon effect modification to suitable models which can de- Σ = − (1 + ∆)χ + (1 − ∆)χ . (B3) scribe late inspiral, merger or ringdown phase accurately [60]. l 2 1 2 2 Ultimately, constraints on horizon effects can be improved by extracting the effects from full inspiral-merger-ringdown sig- Integration of Eqs. (46) and (47) of [33] gives individual nals. mass and spin variations δm1, δm2, δJ1 and δJ2 as a function of PN velocity x. With the parameters α1, α2, β1, β2 introduced in Sec.IID, the variations are translated into Acknowledgement δM, δν, δSl, δΣl at their leading PN order, where δ means the deviation from the initial value, the PN velocity x = (1/3) The work described in this paper was partially supported (πMf) is also modified due to the mass modification, by a grant from the Research Grants Council of Hong Kong CUHK 24304317 and by the Direct Grant for Research from 1 7 the Research Committee of the Chinese University of Hong δM = M[(1 + α1)Cm1σ1 + (1 + α2)Cm2σ2]x , (B4) Kong. 56

Appendix A: Difference between event horizons and apparent 1 δν = ν[(1 + α )C σ + (1 + α )C σ ]x7, (B5) horizons during the inspiral phase 56 1 ν1 1 2 ν2 2

Consider a spherically symmetric black hole which is growing at a constant rate. The difference between the EH radius and 1 the AH radius in a time slice is given by [61] δS = M 2[(1 + β )C σ + (1 + β )C σ ]x4, (B6) l 32 1 S1 1 2 S2 2 m˙ δr = r − r = 2 i + O(m ˙ 2), (A1) EH AH κ i

1 2 4 where mi, m˙ i denote the Misner-Sharp mass and its time δΣl = M [(1 + β1)CΣ1σ1 + (1 + β2)CΣ2σ2]x , (B7) derivative (with respect to ingoing Eddington-Finkelstein co- 32 ordinate) of the black hole, and κ denotes the dynamical surface gravity [62]. For a typical binary black holes detected by LIGO, δx 1 δM −6 = , (B8) δmi/mi ∼ 10 [40], timescale T ∼ 1s, mi ∼ M , x 3 M 8

Since the masses of the black holes are varying, it is natural

2 to add masses m1, m2 together with E + δE to be the total σ1 = χ1(1 + 3χ ), 1 energy of the system Etotal, 2 σ2 = χ2(1 + 3χ2), 2 Cm1 = − (1 + ∆)ν + (3 + ∆)ν , Etotal = E + δE + m1 + m2. (B18) 2 Cm2 = − (1 − ∆)ν + (3 − ∆)ν , dE /dt = −F − 2 Effectively, the energy balance equation total Cν1 = (1 + ∆)ν − 2(2 + ∆)ν , δF can be written as 2 (B9) Cν2 = (1 − ∆)ν − 2(2 − ∆)ν , 2 ∂(E + δE) CS1 = − (1 + ∆)ν + (3 + ∆)ν , = −F (x), (B19) ∂t eff 2 m1,m2,J1,J2 CS2 = − (1 − ∆)ν + (3 − ∆)ν , 2 CΣ1 = (1 + ∆)ν − 2ν , C = − (1 − ∆)ν + 2ν2. Σ2 Feff = F + δF + (1 + Γ1)FH1 + (1 + Γ2)FH2, (B20)

Note that m1 < m2, m1 = M(1+∆)/2, m2 = M(1−∆)/2. We denote the energy and the energy flux of the binary without horizon effects by E and F respectively. The first two dm1/2 FH1/2 = (1 + α1/2) , (B21) leading PN energy terms E0,E1, the leading spin-orbit cou- dt SO pling energy E0 , the first two leading PN energy flux terms SO F0,F1 and the leading spin-orbit coupling flux term F are 0 where dm1/2/dt can be calculated by Eq. (43) in [33]. The summerized below [18, 38]: factors Γ1, Γ2 are introduced (to the leading order) to describe the energy change with respect to the mass and spin absorp- Mν 2 E0 = − x , (B10) tion, 2

SO ∂E0 1 ∂E0 Mν 3 1 4 Γ1/2 = + 2 , E = − (− − ν)x , (B11) ∂m1/2 Ωm ∂χ1/2 1 2 4 12 m2/1,χ1,χ2 1/2 χ2/1,m1,m2 (B22)

SO ν 14 5 E0 = − ( Sl + 2∆Σl)x , (B12) 2M 3 3 3∆ ν Γ = − + + x2, (B23) 1 4 4 6 32 F = ν2x10, (B13) 0 5 3 3∆ ν 2 32 1247 35 Γ2 = − − + x . (B24) F = ν2(− − ν)x12, (B14) 4 4 6 1 5 336 12

The TaylorF2 waveform phase without horizon effects can 32 ν2 5 F SO = (−4S − ∆Σ )x13. (B15) be described by two master equations, 0 5 M 2 l 4 l dΨ By substituting M/ν/Sl/Σl → M/ν/Sl/Σl + − 2πt = 0, (B25) δM/δν/δSl/δΣl, the mass and spin variations induce df an energy variation δE from Eqs. (B10), (B12) and an energy ﬂux variation δF from (B13), (B15) at 3.5PN order,

1 δx 2 dt πM dE/dx δE = − (Mδν + νδM + 2Mν )x + 2 = 0. (B26) 2 x (B16) df 3x F 1 ν 14 5 − ( δSl + 2∆δΣl)x , 2 M 3 Horizon effects are integrated into the formalism by substituting E → E + δE and F → Feff. FH1/2 contributes to the 32 δx 2.5PN phase, while δE, δF, Γ1/2FH1/2,E1FH1/2,F1FH1/2 δF = (2νδν + 10ν2 )x10 contribute to the 3.5PN phase. The TaylorF2 phase of the pa- 5 x (B17) rameterized horizon effect ΨF 2 is: 32 ν2 5 H + (−4δS − ∆δΣ )x13. 5 M 2 l 4 l 9

F 2 x F 2 2 F 2 ΨH = 1 + 3 ln ΨH,5(α1, α2) + x ΨH,7(α1, α2, β1, β2), (B27) xreg √ where xreg can be substitued as the innermost stable orbit 1/ 6.

F 2 ΨH,5 = (1 + α1)C5α1 + (1 + α2)C5α2, F 2 (B28) ΨH,7 = (1 + α1)C7α1 + (1 + α2)C7α2 + (1 + β1)C7β1 + (1 + β2)C7β2,

5 C = χ 3χ2 + 1 (∆ν − ∆ + 3ν − 1) , 5α1 128ν 1 1 5 C = − χ 3χ2 + 1 (∆ν − ∆ − 3ν + 1) , 5α2 128ν 2 2 5 C = χ 1740χ2∆ν2 + 4827χ2∆ν + 580∆ν2 + 1819∆ν − 4371χ2∆ − 1667∆ + 828χ2ν2 7α1 14336ν 1 1 1 1 1 + 13569χ2ν + 276ν2 + 5153ν − 4371χ2 − 1667, 1 1 (B29) 5 C = − χ 1740χ2∆ν2 + 4827χ2∆ν + 580∆ν2 + 1819∆ν − 4371χ2∆ − 1667∆ − 828χ2ν2 7α2 14336ν 2 2 2 2 2 2 2 2 − 13569χ2ν − 276ν − 5153ν + 4371χ2 + 1667 , 15 C = − χ 3χ2 + 1 (18∆ν − 59∆ + 136ν − 59) , 7β1 4096 1 1 15 C = χ 3χ2 + 1 (18∆ν − 59∆ − 136ν + 59) . 7β2 4096 2 2

Note that if no modiﬁcation is made towards general relativity, α1 = α2 = β1 = β2 = 0. Eq. (15) reduces to

F 2 ΨH,5 = C5α1 + C5α2, F 2 (B30) ΨH,7 = C7α1 + C7α2 + C7β1 + C7β2.

Substituting α1 = α2 = β1 = β2 = 0, we successfully Appendix C: Examples of horizon effect parameterization F 2 reproduce the result of [40] until Eq. (B26), ΨH,5 in Eq. (B30) is also the same as that in [40]. However, the numerical value In the following, we demonstrate the relation between some F 2 of ΨH,7 in Eq. (B30) is slightly different from that in [40]. speciﬁc theories and the horizon effect parameterization. F 2 Comparing to [40], ΨH,7 in Eq. (B30) contains an extra term

5 2 1. Area theorem − [(−1 + ∆(ν − 1) + 3ν)χ1(1 + 3χ ) 14366 1 (B31) 2 − (1 + ∆(ν − 1) − 3ν)χ2(1 + 3χ2)], It is well known that if the null energy condition is satisfied, where we notice that this extra term can be written as then the areas of black holes are non-decreasing [43]. Dur- ing the inspiral phase, the positivity of area growth is shown 5 δM − . (B32) by Eq. (6) where dA/dx is always positive. By adopting 256x7ν M α1 = α2 = β1 = β2 = α, and assuming that the first law We fail to resolve the origin of this difference. Fortunately, of black hole thermodynamics is unmodified, the change of a for the parameter range considered in this paper, the difference black hole area can be interpreted as is negligible. For example, if we substitute ν = 0.25, χ1 = dA dA χ2 = 0.5, the extra term is only 0.02% of the numerical value → (1 + α) . (C1) F 2 dx dx of ΨH,7. If we substitute ν = 0.1, χ1 = χ2 = 0.5, the F 2 extra term reduces to 0.006% of the numerical value of ΨH,7. If α < −1 is observed in future gravitational-wave events, the The relatively small value of the extra term assures that it has area theorem will be proven wrong. In Sec. III B, we show negligible influence on both the qualitative and quantitative that the stacking of gravitational wave constraints of α should behavior of the constraints analyzed in this paper. be able to tell whether α < −1 from observations. 10

2. Scalar-Tensor-Vector Gravity γ < 0 is not considered to be a physical case in [44] since it refers to a complex gravitational charge. However, γ > −1 Scalar-Tensor-Vector gravity introduces an extra scalar field is well defined for both the temperature and the entropy, and and a vector field with the standard Einstein-Hilbert action, only observations can tell whether it is physical or not. which predicts a corrected temperature and an corrected en- Also, note that Mureika et al. derive a zeroth PN order tropy for a static black hole [44]. correction in the theory [44], while Moffat suggests that the correction is just phenomenological and it is not necessarily 1 1 T (γ) = √ √ , (C2) true within the theory [63]. 2πGN mi (1 + 1 + γ)(1 + γ + γ)

2 p 2 SA(γ) = πGN mi (1 + 1 + γ) 3. Quantum corrections to black-hole entropy 1 1 1 (C3) − ln √ , 2 2 4πGN (1 + γ + 1 + γ) Even though many quantum gravity theories predict general relativity corrections at the Planck scale, it is possible to G γ where N is the Newtonian gravitational constant, and is a observe some corrections at the black hole thermodynamics non-negative real number representing the modification. Note level. Effective field theory of quantum gravity predicts a log- γ α γ = 0 that we use to denote the in [44]. corresponds to arithmic correction towards the Schwarzschild black-hole en- unmodified general relativity. tropy [45, 64]. Again, the correction is just an approximate This correction is calculated for a non-rotating black hole, correction to the Kerr black hole solution. so it is just an approximate correction towards the more real- istic rotating solution. It is reasonable to assume black holes still follow the first 2 A law of black hole thermodynamics with an modified entropy, Sbh = SBH + 64π Ξ ln , (C6) AQG

˙ ˙ T hSAi = hm˙ ii − ΩH hJi, (C4) where Sbh is the corrected entropy, SBH = A/4 is the or- dinary black-hole entropy, AQG is a quantum gravity area where T = κ/2π, SA = A/4 recovers general relativity (see scale, and Ξ represents a massless-particles contribution in the Eq. (2)). model (refer to [45] for the details). For simplicity, assuming that only corrections on the first law contribute to the mass and spin flux corrections, which is Similar to Sec.C2, the correction can be translated to α. not precise in a full modified gravity theory since the pertur- 2 bation does not follow Teukolsky’s equation. The flux correc- dSbh 256π Ξ α = − 1 = . (C7) tions can be translated to α = α1 = α2 = β1 = β2, dSBH A

T (γ)dSA(γ) γ 2 α = − 1 = − √ . (C5) Note that α is mass dependent, because 1/A = 1/16πmi . T (0)dSA(0) 1 + γ + 1 + γ

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