Spin-induced deformations and tests of binary black hole nature using third-generation detectors

Spin-induced deformations and tests of binary black hole nature using third-generation detectors

N. V. Krishnendu krishnendu@cmi.ac.in Chennai Mathematical Institute, Siruseri, 603103, India.    Chandra Kant Mishra ckm@iitm.ac.in Indian Institute of Technology Madras, Chennai, 600036, India.    K. G. Arun kgarun@cmi.ac.in Chennai Mathematical Institute, Siruseri, 603103, India.
July 20, 2019
Abstract

In a recent letter [N. V. Krishnendu et al., PRL 119, 091101 (2017)] we explored the possibility of probing the binary black hole nature of coalescing compact binaries, by measuring their spin-induced multipole moments, observed in advanced LIGO detectors. Coefficients characterizing the spin-induced multipole moments of Kerr black holes are predicted by the “no-hair” conjecture and appear in the gravitational waveforms through quadratic and higher order spin interactions and hence can be directly measured from gravitational wave observations. By employing a non-precessing post-Newtonian (PN) waveform model, we assess the capabilities of the third-generation gravitational wave interferometers such as Cosmic Explorer and Einstein Telescope in carrying out such measurements and use them to test the binary black hole nature of observed binaries. In this paper, we extend the investigations of [N. V. Krishnendu et al., PRL 119, 091101 (2017)], limited to measuring the binary’s spin-induced quadrupole moment using their observation in second generation detectors, by proposing to measure (a) spin-induced quadrupole effects using third generation detectors, (b) simultaneous measurements of spin-induced quadrupole and octupole effects, again in the context of the third-generation detectors. We study the accuracy of these measurements as a function of total mass, mass ratio, spin magnitudes, and spin alignments. These error bars provide us upper limits on the values of the coefficients that characterize the spin-induced multipoles. We find that, using third-generation detectors the symmetric combination of coefficients associated with spin-induced quadrupole moment of each binary component may be constrained to a value while a similar combination of coefficients for spin-induced octupole moment may be constrained to , where both combinations take the value of 1 for a binary black hole system. These estimates suggest that third-generation detectors can accurately constrain the first four multipole moments of the compact objects (mass, spin, quadrupole, and octupole) facilitating a thorough probe of their black hole nature.

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I Introduction

Recent detections of binary black hole (BBH) mergers by Laser Interferometric Gravitational wave Observatory (LIGO) and VIRGO gravitational wave observatory have confirmed the existence of BBHs in nature and that they merge under the effect of gravitational wave (GW) radiation reaction Abbott et al. (2016a, b, 2017a, 2017b, 2017c). Several tests of General Relativity Meidam et al. (2014); Yunes and Siemens (2013); Arun et al. (2006a, b); Berti et al. (2018a, b); Will (1977); Agathos et al. (2014); Yunes and Pretorius (2009); Will (1998); Samajdar and Arun (2017); Ghosh et al. (2017) were performed using these signals leading to the first ever bounds on potential deviation from the theory in the strong-field regime of gravity Abbott et al. (2016c, d, 2017a, 2017b, 2017c). These tests make use of the fact that the dynamics of the compact binary, and hence the gravitational waveform could be different in an alternative theory of gravity. Hence the observation of binary black holes can lead to constraints on possible departures from general relativity.

The binary black hole dynamics consists of three major phases inspiral, merger and ringdown. One can model the inspiral phase using post-Newtonian formalism Blanchet (2014) whereas numerical relativity simulations are needed to model the merger regime Pretorius (2007). In order to study the ringdown part of the dynamics, one may use black hole perturbation theory techniques Sasaki and Tagoshi (2003). While the observations till date are consistent with this binary black hole dynamics, there may still be room for explaining these observed mergers as due to mergers of some exotic compact objects Giudice et al. (2016). These exotic compact objects could mimic the properties of the black holes up to the accuracy with which we are currently able to extract the signal and its parameters. As the gravitational wave interferometers become more and more sensitive, our parameter estimation accuracies should improve dramatically enabling a thorough probe of the nature of these compact objects. Proposed third-generation ground-based detectors such as Einstein Telescope (ET) Sathyaprakash et al. (2011) and Cosmic Explorer (CE) Regimbau et al. (2012); Hild et al. (2011a, 2008) hence have the strong potential to probe the nature of compact binaries which motivates this work.

Leading candidates for these black hole mimickers include gravastars Mazur and Mottola (2004), boson stars Liebling and Palenzuela (2012) and firewalls Almheiri et al. (2013). Modeling mergers of these exotic objects is a hard problem and their direct deployment for data analysis is not likely to happen in the near future. So a more pragmatic approach would be to devise tests which are generic and model independent and are based on our solid understanding of the binary black hole dynamics. An efficient method to probe the presence of exotic compact objects is to understand the possible ways in which such objects can correct for the properties of black holes which can be detected or ruled out by introducing appropriate free parameters in the gravitational waveform. These tests are often referred to as “null tests” as the free parameters are zero for binary black holes. In order to develop such model independent null tests of black hole mimickers, it is important to identify those properties which are unique to black holes and trace their imprints on the gravitational waveform so that we can measure them from observations.

One of the characteristic properties of black holes in the general theory of relativity is related to the “No-hair” conjecture, which says that all the multipole moments of a Kerr black hole are completely specified by its mass and spin. This means that, it is always possible to relate the multipole of the Kerr black hole to the mass () and the dimensionless spin parameter () as, + = Hansen (1974); Carter (1971); Gürlebeck (2015); Ryan (1995); Geroch (1970a, b). Here and are the mass- and the current-type multipole moments, respectively. This property leads to several observational predictions unique to a black hole which are built-in to the gravitational waveform facilitating tests of black hole nature some of which are discussed below.

i.1 Tests of binary black hole nature using Gravitational Waves

The fact that a black hole cannot be tidally deformed, leads to a vanishing tidal Love number Binnington and Poisson (2009); Gürlebeck (2015). Using a gravitational wave phasing formula which contains the tidal Love numbers Flanagan and Hinderer (2008); Vines et al. (2011), one can directly measure these parameters from observations which in turn can be used to constrain the nature of the compact object constituting the binary system Cardoso et al. (2017); Sennett et al. (2017); Johnson-Mcdaniel et al. (2018). Measurement of tidal deformability parameter from gravitational wave observations for various neutron star models are also studied in different contexts Flanagan and Hinderer (2008). Recently, Cardoso et. al. Cardoso et al. (2017) have calculated the tidal deformability parameters of non-black hole compact objects (including boson stars, gravastars, wormholes, and other toy models for quantum corrections at the horizon scale) and have studied the detectability of such parameters using advanced gravitational wave detectors. In reference Sennett et al. (2017), authors studied the distinguishability of boson star systems from black holes and neutron stars by measuring the tidal deformability parameter. A rigorous formulation of this test using Bayesian inference Johnson-Mcdaniel et al. (2018) has brought the idea closer to be implemented on detected gravitational wave events.

Another way to test the black hole nature is by using the quasi-normal modes Vishveshwara (1970) of the perturbed black hole formed by the merger Dreyer et al. (2004); Berti et al. (2009); Meidam et al. (2014); Berti et al. (2009). For a Kerr black hole, all the quasi-normal modes are characterized by the mass and spin of the black hole according to the “No-hair” conjecture. Though the waveform models for exotic compact objects are less developed, there have been various attempts to calculate the quasi normal modes of boson stars Macedo et al. (2016); Berti and Cardoso (2006); Macedo et al. (2013) and gravastars Chirenti and Rezzolla (2007a); Pani et al. (2009); Chirenti and Rezzolla (2007b). These can be used to discern boson stars and gravastars from black holes.

Measurement of the so called tidal heating parameter can also be used as a tool to test the black hole nature. Consider a black hole event horizon surrounded by external gravitating objects. Rotational energy of the black hole dissipates gravitationally due to the tidal disruption of exterior matter Hartle (1973). The loss of energy and angular momentum of a Kerr black hole near the horizon can lead to non-zero values of the tidal heating parameter. The measured value of the tidal heating parameter will be zero for any system without an event horizon. The tidal heating effect shows up in the gravitational wave phasing Chatziioannou et al. (2013, 2016) which helps us to measure this effect from observations Maselli et al. (2017) and thereby test the black hole nature of the compact object.

It has been found that the multipole moment structure of a central compact object can be extracted from the dynamics of a less massive object orbiting it Ryan (1997a); Rodriguez et al. (2012); Brown et al. (2007). Reference Collins and Hughes (2004) introduced the “bumpy black holes” as a model of space-times which deviate from that of Kerr black holes. Bumpy black holes and their astrophysical importance is extensively studied in Glampedakis and Babak (2006).

Recently, Ghosh et. al. proposed a method Ghosh et al. (2017, 2018) to study the consistency of the inspiral-merger-ringdown dynamics of a binary black hole system to the one predicted by general relativity. The idea here is to infer the mass and spin parameters of the merger remnant from the post-merger part of the gravitational wave signal and ask if this is consistent with the same as inferred from the inspiral part of the gravitational wave signal (using the numerical fitting formula given in Healy et al. (2014)). This method allows one to quantify how close the observed high mass compact binary mergers are to the mergers of binary black holes in general relativity Abbott et al. (2016a, 2017a).

Figure 1: Figure displays variation of - errors in the measurement of parameters characterising spin-induced multipole moments as a function of the total mass of the binary for the three different analyses. Analysis I represents the case where is treated as an independent parameter (here are parameters characterising the spin-induced quadrupole moment of each binary component) while the antisymmetric combination of and as well as the symmetric and antisymmetric combination of parameters characterising the spin-induced octupole moment, (, ), are set to their BBH values of (), respectively. In Analysis II, both and are measured simultaneously while the antisymmetric combination and are set to their BBH values of . Finally in Analysis III, we obtain errors on and while keeping and to their BH values of . The binary is assumed to be at a distance of Mpc and is optimally oriented. The binary’s mass ratio is and posses spins of and respectively for heavier and lighter components, respectively.

i.2 Current Work

Figure 2: Figure displays variation of errors on (where are parameters characterizing the spin-induced quadrupole moment of each binary component) as a function of the binary’s total mass for three representative mass ratio cases with fixed component spins () of (top panel) and four representative spin configurations with fixed mass ratio (q) of (bottom panel).

Recently we proposed a new method to test the binary black hole nature of coalescing compact binary systems observable by ground-based and space-based gravitational wave interferometers Krishnendu et al. (2017). The method relies on measuring the spin-induced quadrupole moments of the binary constituents, which appear explicitly in the gravitational waveforms. For instance, the spin-induced quadrupole moment is given by =- where and are the mass and dimensionless spin parameter of the black hole and the coefficient , which is a measure of the spin-induced quadrupole moment, is unity for Kerr black holes, whereas it can take values roughly between - for neutron stars Laarakkers and Poisson (1999); Pappas and Apostolatos (2012); Pappas and Apostolatos (2012) and between - for boson stars Ryan (1997b). Hence an accurate and independent measurement of this coefficient for each of the binary constituents can tell us if they are indeed black holes Krishnendu et al. (2017). For this purpose, we employed the post-Newtonian (PN) waveforms for spinning compact binaries which are explicitly parametrized in terms of these coefficients (see Sec. II for more details).

It was argued in Ref. Krishnendu et al. (2017) that it would not be possible to accurately measure the deformability coefficients associated with each binary constituents () simultaneously due to the inherent degeneracies between them. However, the symmetric combination of the two, , can be measured accurately assuming the anti-symmetric combination is zero (which would mean that we work with the condition ). Since for a Kerr black hole (and hence for a BBH), an accurate measurement of is an excellent test of the BBH nature of the observed compact binary. The error bars associated with the measurement provides the upper limit on the value of allowed by the data for black hole mimicker models. These bounds, therefore, can be mapped on the parameter space of various black hole mimicker models. A statistically significant detection of could be an indication of the presence of exotic physics in play and may be followed up.

In the present work, we extend the idea of Krishnendu et al. (2017) in three ways by utilizing the enhanced sensitivity of third-generation detectors Dwyer et al. (2015); Sathyaprakash et al. (2011). Firstly, we estimate the errors on assuming a third-generation noise sensitivity and find that the enhanced sensitivity of third-generation detectors over second-generation detectors improves the estimates, roughly, by an order of magnitude (see Fig. 3). Secondly, we investigate the ability of third-generation detectors to simultaneously measure and (symmetric combination of coefficients associated with spin-induced octupole of each binary component (, )) while we set the anti-symmetric combinations of each pair of coefficients, (, ) and (, ) to zero. This would allow simultaneous measurement of the mass, spin, quadrupole and octupole moments of the source thereby permitting consistency tests between them as tests of BH nature. Thirdly, we obtain the projected bounds on and simultaneously using third-generation detectors (keeping the octupole moment coefficients to their BH values). These bounds can straightforwardly be mapped to the black hole nature of the compact object constituting the binary system leading to a much stronger test compared to the one proposed in Krishnendu et al. (2017).

A summary of our analysis is shown in Fig. 1 where the projected errors on the measurement of the spin-induced multipole moments for the three scenarios discussed above are shown as a function of total mass for a fixed mass ratio of and dimensionless spin parameters . The binary is assumed to be optimally oriented at a luminosity distance of Mpc. The projected bounds on the binary black hole nature range from to about for the choice of mass ratio and spin values depending on the type of test performed. We see in Fig. 1 that , whether measured alone (Analysis I) or together with (Analysis II) is measured with smallest errors. We also note that the addition of to the parameter space does not affect the errors on as they are relatively less correlated because of the different PN orders at which they appear unlike and which are strongly correlated as they occur together in the phasing.

The rest of the paper is organized in the following way. In section II, we review the idea of spin-induced multipole moments of compact binary system within the post-Newtonian (PN) formalism. We will briefly describe the aspects of the Fisher Information matrix in section III. Section  IV reports the results in detail. We conclude with section V.

Figure 3: The errors on , the symmetric combination of and , in the dimensionless spin parameter plane for binary’s total mass of and mass ratios of (left panel) and (right panel). In both panels, solid curve corresponds to the errors using Cosmic Explorer PSD and the errors using Adv. LIGO PSD is denoted by dashed contours. As can be seen from the plots, parameter space explored in the - plane is much larger for Cosmic Explorer compared to advanced LIGO.

Ii Spin-induced multipole moments in the post-Newtonian waveforms

Evolution of a compact binary system during the inspiral phase is accurately modeled by the post-Newtonian formalism (see Blanchet (2014) for a review). While sufficiently accurate post-Newtonian gravitational waveforms (for the purposes of detection and the parameter estimation) from compact binaries with non-spinning constituents in quasi-circular orbits were made available as early as early 2000s Blanchet et al. (2004, 2002, 1995), higher order spin effects were included through a number of recent investigations Marsat et al. (2013); Bohé et al. (2013, 2013); Marsat et al. (2014); Bohé et al. (2015); Marsat (2015); Arun et al. (2009); Kidder (1995); Will and Wiseman (1996); Buonanno et al. (2013); Mishra et al. (2016). For our purposes we choose to work with a frequency domain waveform where the spins are (anti-) aligned with respect to the orbital angular momentum Mishra et al. (2016). The state-of-the-art frequency domain waveform for compact binaries with (anti-) aligned spin components incorporates spin-orbit effects in phasing up to 4PN (leading effect appears at 1.5PN order in the phase), spin-spin effects up to 3PN (starting at 2PN) and the leading cubic-spin terms at 3.5PN. Moreover, the amplitude involves spin effects up to 2PN.

The waveform we use for our analyses contain only the leading (second) harmonic (quadrupolar mode) and its PN corrections in the amplitude, while the presence of higher modes in the waveform is neglected, and schematically reads as,

(1)

where , and denote the total mass, symmetric mass ratio and the luminosity distance to the binary system respectively. Coefficients, represents the amplitude corrections to the quadrupolar harmonic at (n/2) PN order Arun et al. (2009). The pre-factor related to the gravitational wave frequency and the total mass of the binary system as, . Here represents the phase of the waveform. Each of these and the phasing, with explicit dependence on spin-induced quadrupole (through and ) and octupole (through and ) moment parameters at respective PN orders are given in supplemental material of Krishnendu et al. (2017).

Effect of the leading spin-induced multipole moment (mass-type quadrupole, = ) in the phasing of gravitational waves from binary black hole systems was first computed in  Poisson (1998) and contributes to the gravitational wave phase at 2PN order. Here, the symbols and again represent the mass and dimensionless spin parameter for each binary component while the negative sign (by convention) indicates that the spin induces oblateness to the black hole. Post-Newtonian corrections to this at 3PN order have been computed in Bohé et al. (2015). The sub-leading, spin-induced multipole moment (current-type octupole, ) starts to contribute to the phase at 3.5PN order and was computed in Marsat (2015). Notice the spin dependences of the spin-induced multipole moments here: have quadratic (cubic) dependences on the spin parameter and first appear in the phasing formula at 2PN (3.5PN) order because these are the orders at which quadratic-in-spin (cubic-in-spin) terms start to appear in the gravitational wave phase.

Note that the relations for and assume that the binary constituents are black holes but can be generalized for a non-BH compact object by introducing coefficients that characterize the degree of deformation. For instance, we can choose to rewrite these relations as : = and where the coefficients and take the value unity for black holes whereas they deviate from unity for other types of compact objects including exotic alternatives to black holes. For example, the values of and for neutron stars, depending upon the neutron star equation of state and mass, range between - and -, respectively Laarakkers and Poisson (1999); Pappas and Apostolatos (2012); Pappas and Apostolatos (2012). The spin-induced multipole moments of a few exotic compact objects are also computed in the literature: for a particular class of spinning boson star system () can take values between (-) Ryan (1997b). Variation of quadrupole and octupole moment parameters in the boson star mass-spin parameter plane is shown respectively in Figs. 4 and 5 of Ryan (1997b). Similar computations have been done for gravastars, see for instance  Mazur and Mottola (2004); Uchikata et al. (2016) which discuss spin-induced multipole moments for thin shell gravastar models. If the observed values of spin-induced quadrupole moments are offset from black hole value, it may be interpreted as an evidence of an exotic compact object. On the other hand, if the posterior distribution for the observed value is found to be peaking at 1 with a width, the corresponding error bars can be translated into an upper bound on the allowed value of the parameter for the particular system. In this work, we compute the projected accuracies on the measurement of the spin-induced multipole moments using the semi-analytical parameter estimation technique of Fisher information matrix. The necessary details of the scheme and the analysis are presented in the next section.

Figure 4: Errors on the as a function of the total mass of the binary system for two representative 3rd generation detectors, Cosmic Explorer (CE psd) and Einstein Telescope (ET-D psd). The binary is assumed to be at a distance of 400Mpc and is optimally oriented. The binary’s mass ratio is 1.2 and spins magnitudes of 0.9 and 0.8 for heavier and lighter component, respectively. Filled- (empty-) markers represent spin orientations of each component aligned (anti-aligned) to the binary’s orbital angular momentum while squares (diamonds) represent error estimates for Cosmic Explorer (Einstein Telescope).
Figure 5: Figure displays variation of errors on (filled markers) and (unfilled markers) as a function of the binary’s total mass for three representative mass ratio cases and four representative spin-orientations with fixed component spin magnitudes () of . The four panels (left to right) represent binaries where spins of the two BH are aligned, heavier one aligned and the other anti-aligned, heavier one anti-aligned and the other aligned and both the spins are anti-aligned to the orbital angular momentum axis.

Iii Parameter estimation using the Fisher Information Matrix analysis

When we have an accurate model for the signal of interest and the expected sensitivity of the detector, Fisher information matrix approach can be used to compute the - error bars on the parameters of the signal Cutler and Flanagan (1994) assuming the noise in the detector is Gaussian-stationary, and the signal to noise ratio is high. Here we employ this approach to estimate the possible error bars on parameters associated with spin-induced multipole moment of the compact binary system. A quick review of Fisher information matrix formalism is given here. More details can be found in Cutler and Flanagan (1994).

A detector output consisting of the gravitational wave signal and the background noise can be written as,

(2)

where is the true signal which is buried in the noise and represents the set of parameters that characterizes the signal. Due to the presence of noise, the measured parameters can fluctuate about the true value leading to errors associated with their measurements. Hence measured value of , where is the true value of the parameter and is the error associated with the measurement due to noise, give us information about the parameter . From the measurement, we are interested in the probability distribution function for given the signal , . It can be shown that, for Gaussian noise in the limit of high signal to noise ratios, the posterior probability takes the form,

(3)

where is called the Fisher information matrix Rao (1945); Cramer (1946) defined as follows,

(4)

where represents the noise power spectral density of the detector and is evaluated at the true value of the parameter . Inverse of the Fisher information matrix is called the covariance matrix () and the error on each parameter is given by the square root of the diagonal entries of the covariance matrix relation,

(5)

We choose to terminate the integral of Eq. (4) at twice the orbital frequency of the inner most stable circular orbit () for a spinning compact binary and use the fits obtained in Husa et al. (2016)111Here we only consider contributions from the second harmonic as discussed in Sec. II.. The lower frequency cut-off in the integral of Eq. (4) is fixed by the sensitivity of the detector given by the function . In this work we intend to explore the parameter estimation analysis for two different third-generation gravitational wave detector configurations: Cosmic Explorer (CE)Abbott et al. (2016e); Hild et al. (2011b) and Einstein Telescope (ET) Hild et al. (2011b).Since the two have comparable sensitivities and we choose to work with just one of them (in our case CE psd) for the most part of the paper. However, we compare the performance of CE and ET for a few representative cases. The low frequency cut-off for CE (ET) configuration is chosen to be Hz (Hz) which defines the value we use in the integral given in Eq. (4). We also discuss the improvements one expect due to the use of third-generation detector sensitivities over advanced LIGO and choose low frequency cut-off as 20 Hz for advanced LIGO.

Iv Results and discussions

In this section, we present the results of our analyses. We choose to perform the parameter estimation analysis for a set of prototypical (stellar mass) compact binary systems. We assume that the binaries are optimally oriented and are located at a fiducial distance of Mpc. The component spin magnitudes are represented by the dimensionless spin parameter, , where subscripts 1(2) represents primary (secondary) binary component. We also follow the convention to assign higher mass and spin values to the primary component. As discussed above we choose to work with the Cosmic Explorer noise PSD as a representative noise sensitivity of a third-generation detector configuration Regimbau et al. (2012); Hild et al. (2011a, 2008). The lower (upper) frequency cut-offs appearing in Eq. (4) are chosen to be 5Hz ( for spinning BBHs). These results are compared with the corresponding ones for advanced LIGO and Einstein Telescope for a selected set of binary configurations.

iv.1 Bounds on binary’s spin-induced quadrupole moment

If we assume the two objects in the binary system suffer equal deformation due to their individual spins ( i.e., ), the symmetric combination of the coefficient of spin-induced quadrupole moments, , will be the suitable parameter to constrain the binary BH nature  (Krishnendu et al., 2017). Any deviation from the BBH value of can be interpreted as possible constraints on the BBH nature of the compact binary system. The parameter space considered here is the following,

(6)

where and are the time and phase at coalescence, is the chirp mass, is the symmetric mass ratio, is the total mass and and , are the masses and dimensionless spin parameters of the binary constituents. Note that, here is the only spin-induced parameter that is considered free in the analysis; other combinations, (, , ), are set to their BBH values of , and .

Figure 2 shows the variation of the errors in the measurement of the parameter , as a function of the total mass of the binary. Three different set of markers in the top panel plot correspond to three different mass ratio cases (q=1.2, 3, 5) while the component spins are fixed to the values of , . On the other hand, the bottom panel assumes a binary with fixed mass ratio () and displays the errors for four different spin configurations. Each set of markers in both panels suggest that errors decrease as the binary’s mass increases. This is largely due to larger signal-to-noise ratios associated with heavier binaries with fixed mass ratio and component spins. In addition, the trends displayed in the top panel suggest improved estimates for larger mass ratio cases (though the improvement is very minor) while those in the bottom panel show that the best estimates correspond to the case when the two objects have component spins aligned to the orbital angular momentum. Both trends can be attributed to the enhancement in the magnitude of the contributions from terms involving in the waveform for the cases where we observe the maximum improvements.

Figure 3 explores error estimates in full component spin parameter space for a binary with total mass of 30 and a mass ratio of 1.2 (top panel) and 3 (bottom panel). Solid (dashed) contours represent errors on in the context of CE (advanced LIGO) detector. Clearly, improved sensitivity of Cosmic Explorer compared to advanced LIGO leads to better estimates of and enhance the accessible parameter space in the case of former. We also note minor improvement in accessible parameter space for larger mass ratio case.

Finally, Fig. 4 compares estimates obtained using two different third-generation detector configurations, Cosmic Explorer (CE) and Einstein Telescope (ET-D). In this case, errors on as a function of total mass for a fixed mass ratio of 1.2 is shown. We consider two spin orientations here, both the black holes aligned and both the black holes anti-aligned to the orbital angular momentum axis. As we expect the performance of CE and ET detectors are not much different in general. Cosmic Explorer error estimates are marginally better than ET-D for all cases except low mass, aligned spin configuration. This should be a reflection of the improved low frequency sensitivity of ET-D at frequencies less than 5 Hz.

Figure 6: Errors on spin-induced quadrupole and octupole moment parameters – (left panel) and (right panel) in the - plane for a binary system with total mass . Solid contours represent mass ratio of 1.2 and dashed ones represent mass ratio 3.

iv.2 Simultaneous bounds on binary’s spin-induced quadrupole and octupole moment

Below we discuss the measurability of both the quadrupolar and octupolar spin-induced deformations due to individual BH spins, simultaneously. This time we intend to measure a symmetric combination of coefficients characterizing the spin-induced octupole moment of the compact binary system: along with the parameter . Again the anti-symmetric combinations and are set to their BBH value of zero. Formally, simultaneous bounds on and are more stringent than the alone as we are sensitive to octupolar contributions and not just the quadrupole. However, the errors on slightly increase due to the inclusion of another free parameter in the problem. As discussed in Sec. II spin-induced octupole moment terms start to appear at 3.5 PN order in the PN phasing formula while the leading spin-induced quadrupole moment contributes at the 2PN order and hence is a dominant effect in PN dynamics. Hence, among and the better constrained parameter is always . The parameter space considered for this analysis is,

(7)

where all the parameters have their usual meaning. Figure 5 shows variations in estimating bounds on (filled markers) and (unfilled markers) as a function of the total mass of the binary for three different mass ratios () and for fixed spin magnitudes of 0.9 and 0.8. Spin orientations chosen are those where both the black hole spins aligned, heavier black hole spin aligned and other anti-aligned, heavier black hole spin anti-aligned other aligned and both the spins anti-aligned to the orbital angular momentum axis, respectively from left to right of Fig. 5 .

Figure 5 shows that the bounds on both and are tightly constrained for cases where the spin of the heavier black hole aligned to the orbital angular momentum axis and if the binary is more asymmetric. When both spins are aligned w.r.t. the orbital angular momentum the effect of mass ratio is marginal (similar to the case presented in Sec. IV.1 where only is measured). On the other hand, having the lighter component anti-aligned w. r. t. the orbital angular momentum vector only marginally affects the measurements with the near equal mass case is most affected. We also note that the trends are not clear when we deal with cases where heavier or both components are anti-aligned. In any case, we don’t expect the best results when heavier or both components are anti-aligned.

The effect of spin magnitudes on the error estimates for simultaneous (left panel) and (right panel) measurements are shown in Fig. 6. We choose a total mass of and mass ratio of (solid contours) and (dotted contours). Broadly the features seen here resemble those of Fig. 3 where only was estimated. For nearly equal mass systems, the contours are less circular which may be due to the degeneracies brought in by the estimation of . Effect of increasing mass ratio is to tilt the semi-major axis of the ellipse towards the vertical similar to Fig. 3.

iv.3 Bounding the black hole nature of the compact binary constituents

In this section we turn to our third and final analysis item – measuring both that characterize the quadrupolar spin-induced deformations of each binary component. Recall that simultaneous measurement of both provides a much stronger test compared to earlier cases where we assumed the spin-induced multipole coefficients to be same for both the components of the binary (, ). The parameter space explored in this case is as follows,

(8)

where the parameters have usual meaning. Figure 7 shows variations in errors on (filled markers) and (empty markers) as a function total mass of then binary for three different mass ratios () and four different spin configurations (each with fixed spin magnitudes of 0.9 and 0.8 for heavier and lighter component, respectively). Here again, the spin orientations chosen are those where both the black hole spins aligned, heavier component aligned and other anti-aligned, heavier component anti-aligned other aligned and both the spins anti-aligned to the orbital angular momentum axis, respectively from left to right of Fig. 7. One of the first things we observe is that estimates of (which characterizes spin-induced deformations of the heavier BH) is consistently better than those of (which characterizes spin-induced deformations of the lighter BH) for all mass-ratio and spin configurations. This is not surprising since we assign higher spins to heavier black holes which induce larger deformations and leads to better estimates for parameters characterizing those deformations. We also note that is measured with smaller errors for systems which are more asymmetric and if the heavier BH is aligned with the orbital angular momentum axis. Note that there are no clear trends when the heavier BH is anti-aligned to the binary’s orbital angular momentum axis. On the other hand, is measured better when the binary is close to symmetric and either heavier BH or both black holes are anti-aligned to the orbital angular momentum axis.

Figure 7: Figure displays variation of errors on (filled markers) and (unfilled markers) as a function of the binary’s total mass for three representative mass ratio cases and four representative spin-orientations with fixed component spin magnitudes () of . The four panels (left to right) represent binaries where spins of the two BH are aligned, heavier one aligned and the other anti-aligned, heavier one anti-aligned and the other aligned and both the spins are anti-aligned to the orbital angular momentum axis. Also, note the (upscaled) y-axes in last two panels.

V Conclusion

In three different set of numerical experiments discussed above, we find that improved sensitivities of third-generation detectors (Cosmic Explorer or Einstein Telescope) over the current advanced LIGO detectors not only allow us to significantly constrain the leading spin-induced effects in gravitational waveforms but also enable us to explore a much wider mass and spin parameter space (Sec. IV.1). As expected, estimated bounds using the two third-generation (Cosmic Explorer or Einstein Telescope) are comparable with a slight favor towards Cosmic Explorer configuration for high mass systems whereas the low mass, aligned spins systems benefit the most from the improved low frequency sensitivity of ET-D. We also showed that at least for a narrower parameter space it would be possible to put stringent bounds on the first two spin-induced multipole moments (quadrupolar and octupolar) simultaneously to assess the nature of the involved compact binary (see Sec. IV.2 above). This also means one would be able to constrain four multipole moments of compact binary system facilitating a thorough probe of their BBH nature. Finally, the possibility of bounding the leading spin-induced moment for each binary component was explored in Sec. IV.3. We find that although this could be used to rule out a class of exotic compact objects constituting the binary one would not be able to confirm the BH nature of the involved binary components due to larger error estimates.

Acknowledgement: The authors thank M. Saleem for useful discussions. We thank Anuradha Gupta for useful comments on the manuscript. K. G. A. and N. V. K were partially supported by a grant from Infosys foundation. K. G. A. acknowledges the Indo-US Science and Technology Forum through the Indo-US Centre for the Exploration of Extreme Gravity, grant IUSSTF/JC-029/2016. K. G. A also acknowledges partial support by the grant EMR/2016/005594. This document has LIGO preprint number P1800325.

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