Probing Einstein-Dilaton Gauss-Bonnet Gravity with the Inspiral and Ringdown of Gravitational Waves

Probing Einstein-dilaton Gauss-Bonnet Gravity with the inspiral and ringdown of gravitational waves

Zack Carson and Kent Yagi Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA (Dated: May 19, 2020) Gravitational waves from extreme gravity events such as the coalescence of two black holes in a binary system fill our observable universe, bearing with them the underlying theory of gravity driving their process. One compelling alternative theory of gravity – known as Einstein-dilaton Gauss-Bonnet gravity motivated by string theory – describes the presence of an additional dilaton scalar field coupled directly to higher orders of the curvature, effectively describing a “fifth force” interaction and the emission of scalar dipole radiation between two scalarized black holes. Most previous studies focused on considering only the leading correction to the inspiral portion of the binary black hole waveforms. In our recent paper, we carried out inspiral-merger-ringdown consistency tests in this string-inspired gravity by including corrections to both the inspiral and ringdown portions, as well as those to the mass and spin of remnant black holes, valid to quadratic order in spin. We here extend the analysis by directly computing bounds on the theoretical coupling constant using the full inspiral-merger-ringdown waveform rather than treating the inspiral and merger-ringdown portions separately. We also consider the corrections valid to quartic order in spin to justify the validity of black hole’s slow-rotation approximation. We find the quasinormal mode corrections to the waveform to be particularly important for high-mass events such as GW170729, in which the dilaton fields’ small-coupling approximation fails without such effects included. We also show that future space-based and multiband gravitational-wave observations have the potential to go beyond existing bounds on the theory. The bounds presented here are comparable to those found in via the inspiral-merger-ringdown consistency tests.

I. INTRODUCTION ultimately finding agreement with Einstein’s famous theory. Observations on the solar-system scale where gravity is weak and approximately static [20], or the strong-field, On September 14, 2015, the Laser Interferometer static observations of binary pulsar systems [21, 22], even Gravitational-wave Observatory (LIGO) in Hanford and cosmological observations [23–27], and extreme-gravity Livingston chirped with activity as they, for the first time observations of GWs [2–4,9–12], have all ultimately ever, observed the iconic gravitational wave (GW) signal found results remarkably consistent with the predictions from the explosive coalescence of two black holes (BHs) of GR. 1.4 billion lightyears away. Aptly named GW150914 [1] by the LIGO/Virgo Collaborations (LVC), this historic Even with the substantial list of past observational suc- detection has ushered in an entirely new era of obser- cess, we must continue to test GR. While this theory still vational astrophysics, finally allowing us to probe the explains all of our gravitational observations, there still extreme gravity regime of spacetime [2–4], where the remains several open questions which could potentially fields are strong, non-linear, and highly dynamical. GWs be explained by alternative theories of gravity. To give a such as these carry multitudes of information across the few examples, the accelerated expansion of the universe universe regarding the local spacetime properties of the due to dark energy [28–31], the inconsistent galactic rota- event, including clues highlighting the underlying the- tion curves due to dark matter [31–35], the matter/anti- ory of gravity driving the show [5–7]. For the past 100 matter asymmetry in the current universe [32, 35], the inflationary period of the early universe [30–32, 35], or arXiv:2003.00286v2 [gr-qc] 16 May 2020 years, Einstein’s theory of general relativity (GR) has remained at its post as the prevailing theory of gravity, even the question of unifying GR and quantum mechan- despite GW150914 and the following 10 events [8] all be- ics [28, 30–32, 34, 35] all remain open to this day. Sev- ing found to be consistent with his theory [9–12]. Even eral modified theories of gravity have been proposed to though the marvel of modern engineering that is the cur- date, many of which have been found to explain some of rent LVC infrastructure [13] might not yet be sensitive the open questions remaining. Similar to the historical enough to expose the subtle signs of a theory beyond Newtonian description of gravity, these advanced theo- GR, the next generation of ground- and space-based GW ries could potentially reduce to GR in weak-gravity envi- detectors [14–19] promise hefty sensitivity improvements ronments, and activate in the un-probed extreme-gravity across the GW frequency spectrum. This may yet prove spacetimes, such as outside binary BH mergers. to finally be enough to study the traces of a new hidden In particular, we consider an interesting classifica- theory of gravity describing our universe. tion of gravitational theories known as (massless) scalar- For the past 100 years, GR has been put under the tensor theories, in which a massless scalar field is intro- microscope, with countless observations and tests per- duced. Specifically, we focus our attention on a particular formed in a wide variety of spacetime environments, all string-inspired scalar-tensor theory known as Einstein- 2 dilaton Gauss-Bonnet (EdGB) gravity, where a dilaton rately to look for their consistency. Moreover, BHs were scalar field is coupled to a quadratic curvature term in assumed to be slowly-rotating, and corrections were de- the action [36–39], with coupling parameter α. With rived up to quadratic order in spin in the previous anal- this new interaction in hand, BHs can become scalar- ysis. Here, we also estimate how much higher-order cor- ized [4, 40–47] (similar to conducting spheres becoming rections in spin may affect bounds on α by deriving cor- electrically charged), and a new fifth force interaction rections to quartic order in spin. can be experienced between two such objects in a binary We now briefly summarize our findings. With the orbit. Similar to analogous interactions found in nature EdGB corrections to the inspiral signal as well as the (i.e. electromagnetic dipole radiation), such binary sys- remnant BH mass, spin, and QNMs in hand, we de- tems would decay faster than proposed by GR through rive current and projected future bounds on the EdGB additional scalar dipole radiation. coupling parameter α. As a first step calculation, we adopt the Fisher analysis [64] technique, which is known The current observational constraint found√ on the EdGB coupling parameter to date has been set to α . 2 to agree well with Bayesian analyses for loud enough km [36, 48–54]. Previous work on constraining EdGB signals [65, 66], such as GW150914 [3]. We first con- gravity with GWs from BBH mergers mainly focused sider four GW events, in order of increasing mass: on looking at the correction in the inspiral due to the GW170608 [67], GW151226 [68], GW150914 [1], and scalar dipole emission [3, 52–54]. BH quasinormal modes GW170729 [69]. We find that GW events detected dur- (QNMs) can also be used to probe this theory [50], while ing the O1/O2 runs by LVC√ detectors have varying suc- Ref. [51] estimated a rough bound on the theory from cess on the constraint of α while varying the type of the dephasing due to the scalar field radiation com- EdGB corrections introduced to the template waveform puted via numerical relativity simulations. Addition- (inspiral only, axial or polar QNMs only, or both). We ally, see Ref. [55] where the authors found constraints find that for more massive events, the inclusion of cor- on dipole emission with space-based detector LISA, as rections to the merger-ringdown are necessary in order to well as multiband observations. See also a recent analy- satisfy the small coupling approximation 16πα2/M 4 1 sis by Ref. [56], where the first numerical relativity model for the total mass M of a binary. This stresses the need for the inclusion of merger-ringdown corrections to the of an EdGB merger-ringdown waveform was√ presented, finding a coupling parameter constraint of α ≤ 11 km. template waveform, especially for more massive events In this investigation, we probe EdGB gravity with where such contributions become important. Further, GWs from BBH mergers by including both inspiral and we find that future GW150914-like events detected by ringdown corrections. The former correction is computed CE [14], LISA [16], or the multiband combination of the using the commonly used parameterized post-Einstenian two improve the constraints considerably, going beyond (ppE) formalism [57], in which generic amplitude and current bounds. The results are similar to what were phase modifications are introduced into the inspiral GR found in Refs. [70–72], in which we only include correc-√ waveform, and the mapping to EdGB is known [3, 54, 58]. tions to the inspiral. The resulting “constraints” on α The latter corrections are computed with the EdGB cor- found here are comparable to those from the IMR con- rections to the individual QNM ringing frequency and sistency test [63]. We also find that√ higher-order terms damping time found in Ref. [59] (see also [50, 60, 61]). in spin only affect the constraints α by 1.6% at most, Moreover, we take into account EdGB corrections to the justifying the slow-rotation approximation. final mass and spin of the remnant BH as a function of The outline of the paper is as follows. We begin in the initial masses and spins, which can be estimated from Sec.II discussing the theory of EdGB gravity. We also corrections to the orbital energy and angular momentum explain the resulting corrections we computed to the in- found in Ref. [62]. spiral and merger-ringdown gravitational waveforms, as This analysis follows closely along with that of Ref. [63] well as the remnant BH mass and spin. SectionIII de- by the same authors, in which we considered the same scribes the Fisher analysis methodology used to derive EdGB corrections in GW signals (while template wave- bounds on EdGB gravity. In Sec.IV we present and forms were still those in GR) and studied the inspiral- comment on our results, and investigate the effects of merger-ringdown (IMR) consistency tests of GR. In par- including higher-order BH spin corrections into the grav- ticular, the EdGB coupling parameter α was allowed to itational waveform. Finally, we offer concluding remarks vary until the inspiral and merger-ringdown portions of as well as a discussion of the various caveats to our work the waveform began to disagree with each other to a sta- in Sec.V. Throughout this paper, we have adopted geo- tistically significant degree. At this point, the assump- metric units such that G = 1 = c. tion of GR in the template waveform fails, and evidence of an emergent theory of gravity can be presented. The IMR consistency test was applied to a specific non-GR II. EINSTEIN-DILATON GAUSS-BONNET theory for the first time in Ref [63]. GRAVITY In this paper, we directly estimate the measurement accuracy of α using the full IMR waveform, rather than In this section, we discuss the theoretical background treating the inspiral and merger-ringdown portions sepa- relevant to the analysis. We split this discussion into 3 two distinct sections: beginning with a brief introduction Scalar charges in EdGB gravity with a linear coupling into EdGB gravity, followed up with the various EdGB as in Eq. (3) only anchor to BHs [4, 40–44], and not to corrections to the gravitational waveform, including the other objects such as neutron stars [39, 58]. Such scalar inspiral, the quasinormal ringdown modes of the remnant charges s depend on the BH’s mass, spin, and the EdGB BH, and finally the remnant BH’s mass and spin. coupling parameter, and have been found to be [4, 44]

p 2 2 1 − χi − 1 + χi α si = 2 2 . (4) A. Theory χi mi Here χ ≡ |S~ |/m2 are the dimensionless spins of the ith In this paper, we focus on the string-theory inspired i i i BH with mass m and spin angular momentum S~ . EdGB theory of gravity. In this particular theory, the i i EdGB gravity may be treated as an effective field the- “dilaton” scalar field ϕ is coupled to a quadratic curva- ory only if the correction S to the action is much ture term in the action. Correspondingly, the Einstein- EdGB smaller than the Einstein-Hilbert action S . Such an Hilbert action is modified by the additional coupling term EH assumption allows one to neglect the higher-order curva- and the scalar field kinetic term [36–38] ture terms of order O(R3), which correspond to cubic in Z √ 1 the coupling parameter O(α3)3. This approximation is S = d4x −g f(ϕ)R2 − ∇ ϕ∇µϕ , (1) EdGB GB 2 µ known as the small coupling approximation, and enforces the requirement that [58, 73] where g is the determinant of the metric g and R is µν GB 16πα2 the curvature-dependent Gauss-Bonnet invariant given ζ ≡ 1, (5) by M 4 for binaries with total mass M ≡ m + m . If this in- 2 abcd ab 2 1 2 RGB ≡ RabcdR − 4RabR + R . (2) equality fails to be upheld, constraints on α are deemed to be invalid, as the assumption S S no longer f(ϕ) is a function of ϕ and some types of string theory EdGB EH effectively reduces to the correction in Eq. (1) with an holds. Typically, constraints on the EdGB√ coupling pa- 2 rameter are presented for the quantity α, which has exponential coupling between the scalar field and RGB. If one expands such a function about a fiducial value of units of length (commonly in km). Current constraints on this quantity have been found to be 107 km from solar the scalar field ϕ0, the leading constant term does not 2 system observations [74], and O(1km) from theoretical contribute to the field equations since RGB is a topologi- cal invariant. Thus, the leading effect arises from a linear considerations, and observations of BH low-mass X-ray coupling, and in this paper, we consider1 binaries, neutron stars, and GWs [36, 48, 49, 51–54]. f(ϕ) = αϕ , (3) B. Corrections to the gravitational waveform where α is the coupling parameter of the theory. In scalar-tensor theories of gravity including EdGB In this section, we describe the various corrections to gravity, compact objects can accumulate scalar monopole the gravitational waveform taken into account in this charges, which in turn source a scalar field. This analysis. This includes corrections to the inspiral portion effect is naturally analogous to the classical ef- of the waveform, to the remnant BH’s QNMs, and finally fects of electric/mass/color charges sourcing the elec- to the remnant BH’s mass and spin predictions. We refer tric/gravitational/strong fields. Pairs of such scalarized the readers to [63] for more details and actual expressions. objects will then give rise to a new “fifth force” inter- Finally, we point out a recent analysis in Ref. [56], in action between them, altering their ensuing trajectories. which a numerical relativity binary BH merger-ringdown This effect is dependent on the internal structures of the waveform in EdGB gravity was presented for the first compact objects, therefore violating the strong equiva- time. See also Ref. [51] for the scalar radiation during lence principle, one of the fundamental pillars of GR. BH binary mergers in EdGB gravity, and Ref. [75] for a While two such compact objects orbiting each other in similar analaysis but in dynamical Chern-Simons gravity. a binary system will decay under the emission of gravitational radiation (as predicted by GR)2, the new scalar interaction will additionally induce scalar dipole radia- 1. Inspiral tion. This effect will of course accelerate the coalescence process more than the predictions of GR estimate. In our analysis, we consider the commonly-used ppE formalism [57]4 to enact EdGB corrections to the inspi-

1 Additionally, refer to [45–47] for more general couplings. 2 Gravitational radiation is also modified from GR in EdGB grav- 3 ϕ is of O(α), and thus Eq. (1) is of O(α2). ity, though such an effect enters at higher order than the scalar 4 ppE phase corrections have a one-to-one correspondence to the dipole radiation in the binary evolution. inspiral corrections in the generalized IMRPhenom formalism [2] 4 ral gravitational waveform. The ppE formalism allows they merge together. The remnant BH then relaxes down one to modify the phase and amplitude of the GR wave- to its final state via the radiation of QNMs [78]. QNMs form with generic parameterized corrections, in a theory- can be described by just two parameters: the ringdown agnostic way. The generalized ppE waveform can there- frequency fRD and the damping frequency fdamp [78, 79]. fore be written as fRD and fdamp are described by the remnant BH’s mass

b and spin Mf and χf (from the BH no-hair theorem), ˜ a i(ΨGR(f)+βppEu ) hppE = AGR(f)(1 + αppEu )e , (6) which in turn only depend on m1, m2, χ1, and χ2 of the original BH binary system obtained through numerical where ΨGR and AGR are the GR phase and amplitude re- relativity simulations [80]. See Ref. [81] where similar 1/3 spectively, u = (πMf) is the effective relative velocity corrections to the QNMs were made, and bounded by of the compact objects with GW frequency f, and chirp future observations of multiple GW events. Additionally 3/5 2 mass M ≡ Mη , where η ≡ m1m2/M is the symmet- see Refs. [82, 83] where they developed a new general for- ric mass ratio. Finally, the ppE parameters αppE (βppE) malism to map ringdown corrections to specific theories classify the magnitude of the amplitude (phase) modifi- of gravity. cations to the waveform which enter at a (b) power of However, within the EdGB viewpoint of gravity, the velocity. The parameters a and b are related to the post- QNMs additionally depend upon the coupling parameter Newtonian (PN) order n by a = 2n and b = 2n−5, where ζ. In this analysis, we attempt to model corrections to terms entering the waveform at nPN order are propor- the ringdown and damping frequencies f and f up 2n RD damp tional to (u/c) relative to the leading-order term. to first order in ζ, like so In this investigation, we solely consider corrections to 2 the GR waveform derived from the EdGB theory of grav- fRD = fRD,GR + ζfRD,ζ + O(ζ ), (9) ity. Such a theory affects the waveform amplitude at 2 fdamp = fdamp,GR + ζfdamp,ζ + O(ζ ), (10) −1PN order (a = −2) with magnitude [76]

where fRD,GR and fdamp,GR are the GR QNM frequency 2 2 2 (EdGB) 5 (m s˜2 − m s˜1) predictions [80, 84], and f and f are the first α = − ζ 1 2 , (7) RD,ζ damp,ζ ppE 192 M 4η18/5 order EdGB corrections. These quantities can be read off from [59]5, as can be found in [63]. withs ˜i = simi/α. Similarly, the waveform phase is mod- We consider both the l = 2 axial QNMs, as well as the ified at −1PN order (b = −7) with magnitude [77] l = 2 polar QNMs. As discussed in Ref. [59], the QNM non-spinning components have been computed for both 5 (m2s˜ − m2s˜ )2 β(EdGB) = − ζ 1 2 2 1 . (8) of these modes. The spinning components of the axial ppE 7168 M 4η18/5 modes were then obtained by adopting the null geodesic correspondence6 [88] since such modes do not couple to To have consistency with other EdGB corrections to be the scalar field perturbation. The spinning components explained later, we only keep up to quadratic order in of the polar modes in EdGB gravity is currently un- BH spins. However, in Sec.IVB we consider EdGB cor- 4 known, though based on the claim in [59], we assume rections to the waveform up to O(χ ), comparing it to the polar modes have the same spin dependence as the those found here. For the remainder of this paper, cor- axial modes to carry out a rough estimate on the lat- rections labeled “inspiral” correspond to the addition of ter. In this analysis, we include EdGB corrections into both phase and amplitude corrections to the GR inspiral the merger-ringdown waveform up to linear order in ζ waveform. We note that Tahura et al. [54] showed that and quadratic in χf using the above prescription, with corrections to the GR amplitude is not as important as corrections like so labeled as “axial/polar QNMs”. See those in the phase, but we keep the former in this paper Sec.IVB for a demonstration of the inclusion of spin for completeness. effects into the remnant BH QNMs, where we include corrections up to O(χ4), and also remove all spin effects. 2. Ringdown 3. Remnant BH mass and spin While the ppE formalism described above allows us to include EdGB corrections to the inspiral description of In addition to the direct waveform modifications dis- the waveform, we can additionally model corrections to played in the preceding sections, a post-merger remnant the ringdown waveform. As the orbits of the inspiraling BHs decay under the emission of gravitational radiation, they eventually become close enough to each other to enter plunging orbits, where a common horizon is formed as 5 Reference [59] follows a slightly different EdGB notation than considered here, beginning with the coupling parameter α in the 0 action as well as their definition of ζ . The√ quantities can be mapped to our definitions by letting ζ0 → 4 ζ. 6 In Refs. [85–87], the null geodesic correspondence was used to used by LVC [3]. approximate corrections to rotating BHs as well. 5

BH in EdGB gravity will settle down into a non-GR final III. FISHER ANALYSIS mass and spin configuration, due to the increased levels of energy and angular momentum radiation. This effect In this section, we provide a brief introduction to the will also indirectly modify the gravitational waveform. In analysis methods√ utilized in this paper. We estimate the GR, the final spin angular momentum of the post-merger maximum size α can take under a GW observation of a BH can be roughly approximated to be the sum of the binary BH merger, while still remaining consistent within spin angular momentum of the initial BHs and the orbital the detector noise. angular momentum of a particle with mass µ = m1m2/M To this end, we employ the Fisher analysis techniques orbiting about the remnant BH at the radius of the in- described in Refs. [64, 92–94]. A more comprehensive nermost stable circular orbit (ISCO), rISCO [89]. More Bayesian analysis can be used instead to give more ac- specifically, the full expression for a spin-aligned system curate results, yet is significantly more computationally is found to be [3, 89, 90] expensive. For loud enough events however, the two have been shown to agree well [65, 66]. For example, the dif- µLorb(M, χf , rISCO) = M(Mχf − as − δmaa), (11) ference in the bounds on the leading PN correction to where as,a ≡ (m1χ1 ± m2χ2)/2 are the symmetric/anti- the inspiral portion of the waveform between Fisher and symmetric combinations of spins, δm ≡ (m1 − m2)/M Bayesian analyses are small [3] for GW150914. Similarly, is the weighted mass difference, and Lorb is the specific regarding the inspiral-merger-ringdown consistency tests, orbital angular momentum. Similarly, the final mass of Fisher results agree well with those from Bayesian anal- the remnant BH Mf can be expressed in relation to the yses for GW150914 [63]. specific orbital energy Eorb of a particle with mass µ In the following investigation, we utilize the non- orbiting at rISCO as precessing, sky-averaged IMRPhenomD waveform [80,

M − Mf = µ [1 − Eorb(Mf , χf , rISCO)] . (12) 84] together with the EdGB corrections described in Sec.II. Namely, the inspiral corrections to the ampli- Here E ≡ 1 − E is equivalent to the binding energy b orb tude and phase in Eqs. (6)–(8) are added to the inspiral of the particle. portion of the IMRPhenomD waveform. The ringdown We here make an assumption that the above GR pic- and damping frequencies in the merger-ringdown portion ture also holds in EdGB gravity and derive corrections of the waveform is modified as Eqs. (9)–(10). The mass to M and χ . To do so, we take into account the EdGB f f and spin of the remnant BH entering in these frequencies corrections to E , L and r . Unfortunately, these orb orb ISCO needs to be related to the initial masses and spins, which expressions are not known to all orders in the BH spin. are modified according to Eqs. (15)–(16). Thus, the grav- Thus, we use the expressions valid to quadratic order in itational waveform non-GR effects are all parameterized spin presented in [62]. In addition, there is a scalar in- into the single EdGB parameter ζ (or α). The resulting teraction between two scalarized BHs, and thus Eq. (12) template waveform consists of needs to be modified to a θ = (ln A, φc, tc, M, η, χs, χa, ζ) , (17) M − Mf =µ [1 − Eorb(Mf , χf , rISCO) M5/6 −Escalar(µ, M, χf , rISCO, ζ)] . (13) √ z where A ≡ 2/3 is a frequency-independent part 30π DL Here of the amplitude in Fourier space with redshifted chirp 2 ! mass Mz ≡ M(1+z) and redshift z, DL is the luminosity ζ χf M E (µ, M, χ , r , ζ) = 1 − (14) distance, χa,s ≡ (χa ± χ2)/2 are the symmetric and anti- scalar f ISCO η2 4 r symmetric combinations of dimensionless spins, and φc is the scalar interaction energy at r between two par- and tc are the coalescence phase and time. ISCO Let us now explain how one can obtain posterior prob- ticles with scalar charges [91] taken up to quadratic order a in spin. See Sec.IVB for an investigation into higher- ability distributions on template parameters θ using a order spin effects up to O(χ4) for each EdGB correction Fisher analysis. With the assumption that both the considered here. prior and template parameter distributions are Gaus- sian [64, 92]7, the root-mean-square errors on parameters Similar to the merger-ringdown corrections to the i QNM ringing and damping frequencies, we consider cor- θ can be found to be q rections to the remnant BH mass and spin to linear order i ˜−1 ∆θ = Γii , (18) in ζ and quadratic (and also quartic) in χf . The complete expressions for Mf and χf can then be written as where Γ˜ is the effective Fisher matrix given by 2 Mf = Mf,GR + ζMf,ζ + O(ζ ), (15) 1 Γ˜ = Γ + δ , (19) 2 ij ij (0) ij χf = χf,GR + ζχf,ζ + O(ζ ), (16) 2 (σθi ) where Mf,GR and χf,GR are the GR predictions of the final mass and spin from the numerical relativity fits of 7 Ref. [80], and Mf,ζ and χf,ζ are the resulting EdGB cor- A Bayesian analysis can utilize more accurate probability distri- rections at first order in ζ. butions, such as uniform. 6

−4 while the Fisher information matrix can be written as regime 10 −1 Hz. For the above detectors, flow is found to be 23 Hz, 1 Hz, and f respectively, where [79] ! 4yrs ∂h ∂h Γij ≡ . (20) −5/8 ∂θi ∂θj −2 M f4yrs = 1.09 × 10 Hz (23) 28M σ(0) is the root-mean-square error on the Gaussian prior θi is the frequency 4 years prior to merger. Similarly, fhigh for the ith parameter. In the above expression, the no- is found to be 1 Hz for LISA, and 4,000 Hz for O2 and tation (a|b) represents the inner product weighted by the CE, such that the GW spectrum is sufficiently small com- detector noise spectral density S (f) n pared to the detector sensitivity at fhigh. We consider four binary BH GW coalescence events Z fhigh a˜∗˜b + ˜b∗a˜ (a|b) ≡ 2 df, (21) in our analysis, ranging from small to large total mass M. In order of increasing total mass, we have cho- flow Sn(f) sen GW170608 [67], GW151226 [68], GW150914 [1], and where fhigh,low represent the detector-dependent high GW170729 [69] with total masses M, chirp masses M, and low cutoff frequencies, as are tabulated and described symmetric mass ratios η and luminosity distances (DL) in Ref. [72]. Finally, if one wishes to combine the obser- tabulated in TableI. Specifically, we perform Fisher anal- vations from multiple detectors with Fisher matrices ΓA yses to estimate the extraction uncertainty on the EdGB and ΓB, the resulting effective Fisher matrix can be found parameter ζ using each detector and GW event consid- to be ered above. We choose the fiducial values of χs = χa = φc = tc = 0. We also choose ζ = 0 (GR), effectively ˜tot A B 1 making the resulting root-mean-square error on ζ to in- Γij = Γij + Γij + δij. (22) (0) 2 dicate the amount of non-GR “fuzziness” one can expect (σθi ) the parameter to observe while still remaining consistent We adopt Gaussian priors corresponding to |φc| ≤ π, and with GR, within the detector noise. |χs,a| ≤ 1. Additionally, we investigate the effect of each type of EdGB correction present in the template waveform: inspiral and ringdown effects. In particular, we consider -18 GW150914 aLIGO O2 the following five cases in which we perform a Fisher 10 CE -19 4yr LISA analysis:

] 10 GWs

-1/2 -20 1. Inspiral: EdGB corrections only in the inspiral 10 waveform.

| [Hz -21 GW170729 ~ |h 10 GW170608

1/2 2. Axial QNMs: EdGB corrections only in the ring- -22 10 down waveform for the case of purely axial QNMs. , 2f

1/2 -23 n 10 S GW151226 3. Polar QNMs: Same as 2 but with polar QNMs. -24 10 4. Inspiral+Axial QNMs: Combination of 1 and 2, -25 10 with corrections to both the inspiral and ringdown -4 -3 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 portions. f [Hz] 5. Inspiral+Polar QNMs: Same as 4 but for polar p FIG. 1. Sensitivities Sn(f) of the gravitational-wave in- QNMs. terferometers aLIGO O2, CE, and LISA considered in this We include remnant BH mass and spin corrections within analysis.√ We additionally display the characteristic am- plitudes 2 f|h˜(f)| for GW events GW170608, GW151226, only the latter four cases listed above. Within each of GW150914, and GW170729 with 4 years prior to merger dis- the above listed cases, we compare the results from each played as orange stars. detector and event considered.

For the remainder of this analysis, we consider the fol- IV. RESULTS lowing GW detectors, with sensitivities shown in Fig.1. We consider the current-generation LIGO/Virgo 2nd ob- Now let us discuss the fundamental results obtained in serving run (O2) [13] detector, as well as the future- this investigation. In particular, we present our results planned third generation Cosmic Explorer (CE) [14] de- from the GW tests of GR discussed in Sec.III, comment- tector with a ∼ 100 times improvement in the frequency ing on the estimated constraints given on the EdGB cou- range 1 − 104 Hz, and ﬁnally the future space-based de- pling parameter α observed by various GW events, GW tector LISA [16], with advanced sensitivity in the mHz detectors, and types of EdGB corrections introduced to 7

Event M (M ) M (M ) η χ1 χ2 DL (Mpc) GW170608 [67] 19.0 7.92 0.233 0.5 −0.66 340 GW151226 [68] 21.7 8.89 0.226 0.5 −0.36 440 GW150914 [1] 65.0 28.1 0.247 0.32 −0.44 410 GW170729 [69] 83.5 36.2 0.249 0.60 −0.57 2,900

TABLE I. List of binary BH GW events investigated in this analysis, including the most and least massive events yet detected (GW170729 and GW170608 respectively), along with their total mass M, chirp mass M, symmetric mass ratio η, BH spins χ1,2, and the luminosity distance DL used in this paper.

2 60 10 GW170729

1 [km] [km] 10 1/2 1/2 α α 40 GW170608 GW151226 GW150914 GW170729 0 30 10

Polar Axial Polar Axial Inspiral Inspiral

Inspiral+AxialInspiral+Polar Inspiral+AxialInspiral+Polar FIG. 3. Same as Fig.2 but for various GW events √FIG. 2. 90%-credible upper bounds on the EdGB parameter α for the most massive binary BH event detected to date: GW170608, GW151226, GW150914, and GW170729 (in or- GW170729. Such bounds are organized into six categories der of increasing mass). The dashed horizontal lines represent (represented by the columns in each panel) of EdGB correc- the small coupling approximation ζ 1 for events of the same tions introduced to the GR waveform as discussed in Sec.III: color, representing invalid constraints when placed above the inspiral, axial QNMs, polar QNMs, inspiral+axial QNMs, and corresponding line. The dotted black√ horizontal line corre- inspiral+polar QNMs. Observe the importance of including sponds to the current constraint of α ≤ 2 km. non-GR effects in the merger-ringdown waveform for massive events, as the small-coupling approximation (valid only in the shaded region) becomes invalid otherwise. constraining EdGB gravity with large mass binaries for which the contribution of the ringdown is larger. √ Figure3 displays the upper bounds on α observed on the template waveform. We first perform our main anal- O1/O2 runs for each GW event considered in this analy- ysis with EdGB corrections to the waveform to O(χ2) in sis. We observe several points in regards to this. Firstly, BH spin, followed up by a discussion and demonstration the smaller total mass events correspond to stronger con- of corrections to O(χ4). straints, as was noted in similar analyses [71, 72].√ This is because the expressions in Eqs. (7)–(8) minimize α for both minimal mass ratio and, more notably, the individ- 2 A. O(χ ) corrections to BH spin ual mass. Second, we observe that the type of EdGB corrections to the waveform does not strongly affect the two We start by performing the analysis with EdGB cor- more massive events (GW150914, and GW170729), while 2 rections to O(χ √) in BH spin. Figure2 presents the up- the two lighter events (GW170608 and GW151226) ob- per bounds on α from the most massive binary BH serve a deterioration in constraining EdGB gravity when event detected to date, GW170729, with various EdGB only including the axial/polar QNMs. This is because the corrections considered. Observe first that when only fraction of the ringdown portion in the observed wave- the inspiral correction is considered, the bound is be- form becomes larger for larger-mass binaries, and hence yond the validity of the small coupling approximation, QNM corrections become more important for these bi- while those become valid once we consider corrections to naries. Third, we observe that for more massive events QNMs. This shows the importance of the latter when such as GW170729, the inclusion of only inspiral EdGB 8

√ √ α [km] α [km] GW150914-like events Detector (this paper) (IMR consist. [63]) 1 10 aLIGO 17 15 CE 5 8 [km] LISA 0.6 – 1/2

α Multiband 0.3 0.2 0 10 CE LISA Multiband TABLE II. Comparison√ between the current and future upper bound on α obtained in this paper and the IMR consistency tests of GR [63]. Such constraints were formed from Axial Polar GW150914-like events, with the both inspiral and axial QNM Inspiral EdGB effects included in the waveform template. Observe how constraints obtained from both tests produce compara- Inspiral+AxialInspiral+Polar ble results on the detectability of EdGB effects in the GW signal. FIG. 4. Same as Fig.3 but for future GW150914-like events detected by the ground-based detector CE, space-based detector LISA, and the multi-band observation between the two. √ √ α (χ2) α (χ4) frac. diff. We note that no space-based or multiband bounds appear in GW Event the axial/polar QNMs columns, because space-based detec- [km] [km] [%] tors such as LISA can not observe the merger-ringdown effects GW170608 [67] 2.29 2.28 0.4 for GW150914-like events occurring at high frequencies. GW151226 [68] 2.76 2.75 1.1 GW150914 [1] 17.16 17.15 0.1 GW170729 [69] 28.71 28.29 1.5 effects results in an invalid constraint due to violation of √ TABLE III. Constraints on α obtained with EdGB correc- the small coupling approximation, as already shown in tions to the waveform up to quadratic order in BH spin (2nd Fig.2. Similar conclusions were made in Ref. [63] for column), and quartic order in BH spin (3rd column). The last massive events, in which the merger-ringdown portion column shows the fractional difference between the two. We of the gravitational waveform began to make significant observe that such results agree to within 1.5% in all cases, contributions to the IMR consistency test, compared to with the largest difference appearing for the most massive the inspiral portion. event GW170729. Next we consider the future detectability of EdGB effects in the waveform.√ Figure4 displays the possible B. O(χ4) corrections to BH spin upper bound on α observed by CE, LISA, and the multiband observation between the two for GW150914- In this section we compute EdGB corrections to the like events, which indeed lie in the multiband detectabil- gravitational waveform up to quartic order in BH spin, ity region displayed in Ref. [72]. We note that EdGB to check the validity of the slow-rotation approximation effects with only axial/polar QNM corrections can not to quadratic order in spin used in the previous subsec- be probed by LISA (thus multiband observations give tion. We begin by expanding the expressions for the the same result as CE detections alone) due to its cutoff inspiral dipole radiation and QNM corrections already frequency of 1 Hz. We observe that LISA observations computed in Sec.IIB to quartic order in BH spin. Next alone can improve the ability to probe EdGB gravity by we compute corrections to r , E and L via the roughly one order of magnitude from CE observations ISCO orb orb EdGB spacetime metric gEdGB found in Ref. [95], where alone, with little difference made by the addition of ax- αβ they computed each element up to 5th order in BH spin ial/polar QNM corrections. Multiband observations fur- χ. The orbital energy and angular momentum can be ther improve the bound by about a factor of two. Notice obtained from gEdGB by simultaneously solving the equa- also that the LISA and multiband bounds are stronger αβ tions V (r) = 0 and d V (r) = 0 for E and L with than current bounds [36, 48, 49,√ 51–54]. See also TableII eff dr eff orb orb for a comparison between the α constraints found in effective potential given by this paper, and the one in Ref. [63] through the IMR con- 2 EdGB EdGB 2 EdGB E g + 2EorbLorbg + L g sistency tests. In particular, the latter analysis utilized V (r) = orb φφ tφ orb tt − 1. eff EdGB 2 EdGB EdGB the same EdGB corrections to the gravitational waveform (gtφ ) − gtt gφφ used here, and then tested the consistency between the (24) inspiral and merger-ringdown signals for varying values Finally, the location of the ISCO is given by d of α. We find that the bounds from the two analyses are dr Eorb(rISCO) = 0. comparable to each other. With the above corrections to the entire gravitational 9 waveform to quadratic order√ in spin, we estimate con- EdGB merger-ringdown contributions begin to hold high straints on EdGB parameter√ α. In particular, we com- significance. When only the inspiral corrections to the pute constraints on α for each GW event considered in waveform (as is typically considered) were applied, the this analysis: GW150914, GW151226, GW170608, and small-coupling approximation ζ 1 failed to be upheld. GW170729 as detected on the O2 detector, with non-zero Only upon the inclusion of the merger-ringdown correc- fiducial BH spins. We compare these results with those tions does this quantity√ become satisfied, allowing for of the main analysis, with corrections to only quadratic valid constraints on α. We found that future space- order in spin.√ TableIII presents a comparison between based and multiband observations can place bounds that constraints on α obtained from (i) waveforms with cor- are stronger than current bounds√ on EdGB gravity. We rections to quadratic order in BH spin, and (ii) to quartic also found that the constraints on α agree well with the order in BH spin. We find that such results agree with recent similar analysis in Ref. [63] by the same authors. each other to between 0.1% and 1.5%, with the latter In this analysis, the same EdGB corrections to the wave- resulting from the massive BHs in GW170729, in which form were made and the inspiral and merger-ringdown spin effects become more important as it seems to have signals were tested for consistency with increasing values the largest final spin out of the 4 GW events considered of α injected into the signal, until they failed to remain here. Therefore we conclude that the effect of higher- consistent with each other. We improved upon Ref. [63] order spin corrections to the gravitational waveform has by considering effects at higher order in spin to justify the up to a ∼ 1.5% effect on our predictions, which validates use of slow-rotation approximation. We found that such our order-of-magnitude estimation presented in this pa- higher-order corrections only change our results up to a per including up to quadratic order. maximum of 1.5%. We additionally investigate the effect Finally, we consider the effect of including spin effects of spin corrections to the remnant BH QNMs, finding into the remnant BH QNMs. For example, in dynam- that their inclusion has a negligible impact on parameter ical Chern-Simons (dCS) gravity, all of the ingredients estimation. required to correct the full waveform considered here are available, with the exception of the QNM spin correc- As noted by Ref. [63], we take note here that the preceding analysis falls short by several assumptions made tions. Here, we remove all EdGB spin effects to the rem-√ nant QNM corrections and compute constraints on α. throughout, however we present it as a first step to- We find the constraint to be 27.58 km for GW170729 wards going beyond including the leading correction to with which the contribution of the ringdown is most sig- the inspiral only or considering QNM corrections only nificant out of the 4 GW events considered. Such a con- under the BH spectroscopy. In particular, we offer sev- straint agrees very well with those tabulated in TableIII eral caveats to the EdGB waveform corrections displayed for spin corrections to both O(χ2) and O(χ4). Therefore, in this paper, to be considered as future improvements on similar analyses. In this investigation, we have made we conclude that spin effects in the remnant BH QNMs√ make only a negligible impact on constraints on α in use of the IMRPhenomD GR waveform, which makes EdGB gravity. several assumptions that hold strong and true in GR. However, in alternative theories of gravity (like EdGB gravity), such assumptions may fail. Firstly, polar and axial QNMs are found to be exactly identical in GR, while V. CONCLUSION AND DISCUSSION those in EdGB gravity are different in general. Secondly, we would additionally need to consider corrections to the EdGB gravity is a proposed scalar-tensor theory of intermediate merger signal, rather than just the inspi- gravity with curvature coupling to the dilaton scalar field. ral and post-merger ringdown signals. Finally, in our This string-inspired theory predicts the scalarization of analysis we consider only leading order PN corrections BHs [40–43], calling forth “fifth force” interactions be- to the inspiral waveform using the ppE formalism, while tween orbiting BHs in a binary system and giving rise to higher orders could yet be taken into account. Having scalar dipole radiation that predicts an increased rate of said this, the bounds presented in this paper should serve inspiral between them. In this analysis, we have mod- as valid order-of-magnitude estimates. This is because eled the resulting EdGB effects throughout various parts the correction√ to the waveform are linearly proportional of the gravitational waveform, including the inspiral, the to ζ ∝ ( α)4, which means systematic errors due to mis- characteristic ringdown QNMs, and finally, to the final modeling the waveform is suppressed by a fourth-root mass and spin properties of the remnant BH. With these power. new tools in hand, we offer predictions on the future detectability of such EdGB effects present in the gravita- Faults such as the ones listed above can be remedied by tional waveform. the full construction of an EdGB (or any non-GR theory) We studied the detectability of EdGB effects in an ob- waveform. Work in this direction is already in progress served GW signal by introducing various combinations of such as Ref. [51], where the scalar field dynamics during EdGB modifications to the inspiral and merger-ringdown binary BH mergers have been expressed in EdGB grav- portions of the waveform. In particular, we discovered ity. Very recently, the EdGB correction to the merger- that for more massive events such as GW170729, the ringdown waveform from a binary black hole has been 10 computed [56]8. We plan to compare such numerical- the QNMs (for dCS gravity, these are only known for non- relativity waveforms with the simple analytic model pre- spinning BHs [99–101]), the simple “patchwork” analysis sented here to quantify the validity of the latter. presented here could be revisited, without the need for a full non-GR waveform. Other future avenues include In the preceding investigation, we considered inspiral- repeating the calculations using a Bayesian analyses. ringdown waveform modifications from the EdGB theory of gravity as one given example. Future analyses could, given all the necessary ingredients described above, re- ACKNOWLEDGMENTS peat the entire investigation using any given modified theory of gravity. By simply knowing the leading PN Z.C. and K.Y. acknowledge support from NSF Award corrections to the inspiral portion (known for most mod- PHY-1806776 and the Ed Owens Fund. K.Y. would like ified theories of gravity [76]), corrections to the specific to also acknowledge support by a Sloan Foundation Re- orbital energy Eorb and angular momentum Lorb, (known search Fellowship, the COST Action GWverse CA16104 for theories such as dCS gravity [98]), and corrections to and JSPS KAKENHI Grants No. JP17H06358.

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