Searching for General Binary Inspirals with Gravitational Waves
SSearching for General Binary Inspiralswith Gravitational Waves
Horng Sheng Chia and Thomas D. P. Edwards , , Institute for Theoretical Physics, University of Amsterdam,Science Park 904, Amsterdam, 1098 XH, The Netherlands Gravitation Astroparticle Physics Amsterdam (GRAPPA), University of Amsterdam,Science Park 904, Amsterdam, 1098 XH, The Netherlands The Oskar Klein Centre, Department of Physics,Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden
Abstract
We study whether binary black hole template banks can be used to search for the gravitationalwaves emitted by general binary coalescences. To recover binary signals from noisy data, matched-filtering techniques are typically required. This is especially true for low-mass systems, with totalmass M (cid:46) M (cid:12) , which can inspiral in the LIGO and Virgo frequency bands for thousands ofcycles. In this paper, we focus on the detectability of low-mass binary systems whose individ-ual components can have large spin-induced quadrupole moments and small compactness. Thequadrupole contributes to the phase evolution of the waveform whereas the compactness affectsthe merger frequency of the binary. We find that binary black hole templates (with dimensionlessquadrupole κ = 1) cannot be reliably used to search for objects with large quadrupoles ( κ (cid:38) a r X i v : . [ a s t r o - ph . H E ] A p r ontents Introduction
The direct detection of gravitational waves [1, 2] has opened up a unique way to view the darkside of our Universe. By virtue of Einstein’s equivalence principle, all forms of matter and energydensity must interact gravitationally, making gravitational waves universal probes of new physicsin regimes which are typically inaccessible by other experimental means. This new observationalwindow has come at a time when challenges in fundamental physics, cosmology, and astrophysicsremain abound, for instance: we still do not know what 95% of the energy budget of our universeis [3]; there is a significant discrepancy between early and late time measurements of the Hubbleconstant [4]; and the origin of supermassive black holes in the early universe is still unknown[5]. The current network of gravitational-wave detectors allows us to explore large volumes ofour dark universe, hopefully helping to answer some of these questions [6, 7]. Next generationdetectors will see further and over a greater range of frequencies, revealing even more — thefuture of gravitational-wave astrophysics is bright [8, 9].Compact binary systems are one of the loudest and most important sources of gravitationalwaves. These systems are unique in that accurate computations of their gravitational waveforms,especially in the early inspiraling regime, are attainable [10–13]. This makes an observed wave-form a rich source of information about the binary’s dynamics and the physics of its components.In fact, accurate waveform models are essential for detecting these signals, as they are most reli-ably extracted from noisy data streams through matching with template waveforms [14–16]. Ourreliance on this matched-filtering technique, however, also implies that we are bound to only de-tect signals that we can predict. In particular, an order-one mismatch between the phases of thesignal and template waveforms can easily degrade the detectability of a signal [16], thereby result-ing in a missed event. It is hence crucial that we develop increasingly precise template banks thatalso cover a wider range of the parameter space. This strategy would certainly broaden existingsearches for binary black hole and neutron star systems. At the same time it could potentiallydiscover new types of compact objects that have been proposed in various beyond the StandardModel (BSM) scenarios, such as: primordial black holes [23, 24], gravitational atoms [25, 26],boson stars [27–31], soliton stars [32–35], oscillons [36, 37], and dark matter spikes [38, 39].Modeling accurate waveforms for general binary inspirals is a laborious task. Fortunately, thecomplicated microphysics of a general object are integrated over astrophysical length scales, andaffects their binary waveforms through various universal finite-size effects . For instance, differentmicrophysics often result in distinctive mass-radius relations for the object. The radius of theobject, or equivalently its compactness, in turn dictates the merging frequency of the binary [40].In addition, these finite-size effects induce subtle yet important phase imprints in the binary’swaveform. For a non-spinning object, the dominant effect arises from the object’s induced tidaldeformation, which first enters the waveform at five post-Newtonian (PN) order [41, 42]. While Coherent burst search methods [17–19] have been developed to detect transient gravitational waves in a model-independent way. Nevertheless, they only capture loud and short-duration events, such as the near-merger regimeof binary coalescences, and are insensitive to weak and long-duration binary inspirals [20–22]. In this paper, we use the word ‘general’ to refer to any binary system involving at least one non-standardastrophysical compact object. In contrast, we use the word ‘standard’ to refer to binary systems involving onlyblack holes and neutron stars. atched FilteringCoherent Burst Focus of this PaperHigh-massBinary Mergers Low-massBinary InspiralsDetectorNoiseSignal Amplitude Signal Duration(seconds)10 − − − Figure 1 : Overview of the different methods used to search for gravitational waves emitted bya general binary system with the LIGO and Virgo detectors. The signals of low-mass binarysystems are typically weaker and last longer than those of high-mass systems. The horizontaldashed line schematically illustrates the detector noise level, below which coherent burst searchesbecome quickly insensitive. In this paper we focus on low-mass binary inspirals which are only detectable with matched filtering.this tidal effect offers a useful way of testing the nature of the binary components [42–45], itwould only become appreciable near the merger of the binary, where strong gravity dynamicsmust be taken into account. This modeling challenge can be circumvented when consideringspinning objects, which generate a series of spin-induced multipole moments [46–48] that perturbthe dynamics of the binary in its early inspiraling regime. Specifically, the dominant spin-inducedquadrupole moment first enters the phase evolution of the waveform at 2PN order [49], and hasbeen incorporated into existing templates for binary black hole systems. This quadrupolar term isespecially significant for the BSM objects described above, as it can be orders-of-magnitude largerthan those of black holes and neutron stars [50–52]. In certain examples, the time dependenceof the quadrupole can provide further unique fingerprints of the masses and intrinsic spins of theboson fields that constitute the object [53, 54]. Since analytic predictions of the waveform in theearly inspiral regime are known in detail, this spin-induced finite-size effect is a much cleanerprobe of new physics than the tidal deformability.In this paper, we attempt to address the following question: to what extent can existingtemplate banks be used to search for gravitational-wave signals emitted by general binary coa-lescences? We do so by computing the effectualness [55, 56] of existing template banks to generalwaveforms. The effectualness describes how much signal-to-noise ratio is retained when comput-ing the overlap between a signal and the best-fitting template waveform in a bank. In additionto the usual mass and spin parameters, these general waveforms incorporate the effects of the2pin-induced quadrupole moment and the compactness of the binary components. A detectionof these general signals would therefore represent a discovery of new physics in binary systems.Our template bank is designed to resemble those used by the LIGO/Virgo collaboration [57–60],demonstrating that, if these new signals exist, they could remain undetected. When constructingour template bank, we follow closely the geometric-placement method presented in Refs. [61].Our work is complementary to Ref. [62, 63] where they further optimized the LIGO/Virgo searchpipelines. Instead, we hope to broaden the searches beyond these standard binary black hole andbinary neutron star signatures.We emphasize that our work is in contrast to several proposed tests of new physics in binarysystems [64–67], which seek to measure or constrain plausible parametric deviations in observedwaveforms. This a priori assumes a successful detection of the new binary system. Detectionis typically achieved through matched filtering with current template banks, which necessarilymeans that the waveform deviations are small. Our focus is instead on the detectability of theseplausible new binary signals, including those that incur large deviations from the binary blackhole template waveforms.The ability to generate accurate template waveforms is a crucial prerequisite to achieving ourgoal. We therefore restrict ourselves to general low-mass binary systems, where the total massof the binary is M (cid:46) M (cid:12) , for the following reasons: • In the LIGO and Virgo detectors, the inspiral regime only dominates for low-mass binarysystems. The binary inspiral is an interesting regime because analytic results of the PNdynamics are readily available. This provides us with a well-defined framework to constructprecise waveforms that incorporate additional physics, such as finite-size effects. • The inspiraling signals of these systems last up to several minutes (corresponding to hun-dreds or thousands of cycles) and are typically very weak. They are therefore hard to detectwith coherent burst searches. Matched filtering is the optimal and only realistic avenue tosearch for them (cf. Fig. 1 for a comparison of these different search techniques).By assuming that the PN dynamics are valid up until the merger regime, we have implicitlyignored other plausible effects that may occur even in the early inspiraling regime such as: Roche-lobe mass transfer [68]; third-body perturbation [69–71]; floating, sinking, or kicked orbits [53];dark matter environmental effects [72–74]; new fifth forces [75–77]; and strong gravitationaldynamics [78–80]. Despite these limitations, our general waveforms still capture a wide class ofnew types of binary systems which have been overlooked in the literature. Crucially, we believethat this work represents a fundamental step towards realistically searching for new physics inthe nascent field of gravitational-wave astronomy.
Outline
The structure of this paper is as follows: in Section 2, we construct the waveform fora general binary inspiral. Specifically, the impact of various finite-size effects on the waveformwill be incorporated. In Section 3, we construct a template bank that is representative of thoseused in standard search pipelines. We then compute the effectualness of our general waveform tothis template bank. Finally, we summarize and present an outlook in Section 4.
Convention
We work in geometric units, G = c = 1.3 General Inspiral Waveforms
In this section, we construct the template waveforms for general binary inspirals. We first reviewhow an astrophysical object’s multipole structure imprints itself on the phase of the gravitationalwaves emitted by the binary system ( § § The shape of a general astrophysical object, when viewed at large distances, can be describedthrough a series of source multipole moments [46–48]. In this paper, we only consider objectsthat are spherically symmetric when they are not spinning — in this limit only the monopolecontributes. Birkhoff’s theorem [81, 82] then implies that, regardless of the underlying micro-physical properties of the object, its exterior metric is given by the Schwarzschild solution andis therefore only described by its mass, m . In this case, it is challenging to distinguish betweendifferent types of non-spinning objects (though we will shortly discuss how induced tidal effectscan help to alleviate this degeneracy).Fortunately, the nature of an astrophysical object can be readily probed when it has non-vanishing spin. In particular, its spinning motion generates a hierarchy of axisymmetric multipolemoments, whose precise values do depend on the object’s microscopic properties. The dominantmoment is given by the axisymmetric quadrupole, Q , which is often parameterized through [49] Q ≡ − κ m χ , (2.1)where χ ≡ S/m is the dimensionless spin parameter, with S being the spin angular momentum,and κ is the dimensionless quadrupole parameter, which quantifies the amount of shape defor-mation due to the spinning motion. In particular, the larger the (positive) value of κ , the moreoblate the object is. Crucially, the value of κ depends sensitively on the detailed properties ofthe object. For instance, it is known that κ = 1 for Kerr black holes [47, 48], while 2 (cid:46) κ (cid:46) κ can be as large as ∼ − . It can even develop oscillatory time-dependences or vary significantly with χ [50–53].Absent a specific compact-object model in mind, we will henceforth treat κ as a free constantparameter, with the requirement that κ ≥ Q on the gravitational-wavesignal is known in the early-inspiral, post-Newtonian regime of the coalescence [49, 64, 85–91]. Tosimplify our analysis, we restrict ourselves to binary orbits which are quasi-circular, and assumethat the binary components’ spins are parallel to the (Newtonian) orbital angular momentumvector of the binary. In the Fourier domain, the gravitational wave strain is [92]˜ h ( f ; p ) = A ( p ) f − / e iψ ( f ; p ) θ (cid:0) f cut ( p ) − f (cid:1) , (2.2) This need not be the case, as a general astrophysical object can inherit higher-order permanent multipolemoments, which are present even when the object is not spinning f is the gravitational wave frequency, p is the set of intrinsic parameters of the binary, A is the strain amplitude, ψ is the phase, θ is the Heaviside theta function, and f cut is the cutofffrequency of our general waveform. Schematically, the phase evolution reads ψ ( f ; p ) = 2 πf t c − φ c − π ν v (cid:16) ψ NS + ψ S (cid:17) , (2.3)where t c is the time of coalescence, φ c is the phase of coalescence, ν = m m /M is the symmetricmass ratio, M = m + m is the total mass of the binary, and v = ( πM f ) / is the circularorbital velocity. The quantities ψ NS and ψ S represent the non-spinning and spinning phasecontribution, respectively. Because the overlap between waveforms are especially sensitive tophase coherence [16], we will retain terms in the phase up to 3.5PN order — these terms are fullyknown in the literature; see for example Refs. [64, 93–95]. The quadrupole parameter κ in (2.1)contributes to ψ S through the interaction between Q and the tidal field sourced by the binarycompanion. It first appears at 2PN order [49, 94] ψ S ⊃ − (cid:88) i =1 (cid:16) m i M (cid:17) κ i χ i v , (2.4)where the subscript i = 1 , ψ S ⊃ (cid:88) i =1 (cid:88) j (cid:54) = i (cid:20) (cid:16) m i M (cid:17) + 27032 m i m j M + 9407 (cid:16) m j M (cid:17) (cid:21) (cid:16) m i M (cid:17) κ i χ i v . (2.5)We have simplified (2.4) and (2.5) by enforcing χ i to be aligned with the orbital angular momen-tum (0 < χ i ≤
1) or anti-aligned with it ( − ≤ χ i < M (cid:46) M (cid:12) , whose merger frequenciestypically lie above the upper bound of the observational windows of ground-based detectors (see § The amplitude is independent of f at leading Newtonian order. We will ignore higher-order PN correctionsto A , as they do not substantially affect the overlap between different waveforms. As a result, the constant A disappears in the normalized inner product (see Section 3 later). .0 1.0 2.0Time [s] − . − . . . . S tr a i n × − as C decreases ←− κ = 1 (Black Hole Inspiral) κ = 100 (General Inspiral) Figure 2 : Comparison between the waveforms of a binary black hole inspiral and a generalbinary inspiral. In both cases, the binary components’ masses and spins are m = m = 2 M (cid:12) and χ = χ = 0 .
8. No amplitude modulation is observed as we assume that the components’spins are aligned with the orbital angular momentum of the binary. One of the binary componentsis assumed to be a black hole ( κ = 1), while the other can be a more general object ( κ ≡ κ ≥ C decreases).While we have only focused on the object’s quadrupole moment so far, other types of finite-sizeeffects, such as the object’s higher-order spin-induced moments and tidal deformabilities [96–99],can also contribute to the phase (2.3). Nevertheless, these additional terms only start to appearat 3 . By focusing on the binary’s early inspiral regime we can therefore ignore these higher-order effects. For concreteness, we will set these quantities to their corresponding values for blackholes [47, 48, 98, 99]. While the parameter κ , described in § C ≡ mr , (2.6)where r is the equatorial radius of the object. Black holes, which are the most compact knownastrophysical objects, have 0 . ≤ C ≤
1, where the lower and upper bound corresponds to thecompactness of a Schwarzschild and an extremal Kerr black hole respectively. For neutron stars, Since the leading-order term in (2.3) scales as ∼ v − , roughly speaking, orbital parameters that appear at (cid:46) . (cid:38) . Note however that we retain the κ -dependence in the 3.5PN phasing term [64] in Section 3. can range between ∼ . − . C (cid:28) . R , is smaller than the binary’s innermost stable circular orbit (ISCO), r ISCO ≈ M [10, 108]. However, if the binary components have sufficiently small C ’s, the binary wouldhave already merged at R < r
ISCO . In this case, the strong-gravity regime is not reached atmerger, making the analytic PN approximation still a valid description of the dynamics.Pledging ignorance to the merger dynamics of binary systems with small-compactness objects,we will terminate our waveform (2.2) when the binary touches, i.e. when R = r + r . UsingKepler’s third law, our cutoff frequency is therefore f cut = 1 π (cid:115) m + m ( m / C + m / C ) , (2.7)where we have ignored PN corrections to (2.7). Furthermore, we have neglected the influenceof the binary components’ quadrupoles and higher-order multipole moments on (2.7), which canbe important when the binary separation is small. These additional corrections would deepenthe gravitational potential of the binary, thereby increasing the cutoff frequency towards largervalues [10, 108]. In other words, (2.7) underestimates the actual touching frequency of the binary,and may thus be viewed as a conservative cutoff of our waveform frequency. The impact of thiscutoff on our waveform is schematically illustrated in Fig. 2. In the special case where both ofthe binary components have the same compactness, C = C = C , (2.7) becomes f cut (cid:39) (cid:18) C . (cid:19) / (cid:18) M (cid:12) M (cid:19) . (2.8)For low-mass binary systems ( M (cid:46) M (cid:12) ), the observational lower bound of ground-baseddetectors, f (cid:38)
10 Hz, implies that we can probe binary inspirals with C (cid:38) − . Although thetouching condition (2.8) is inaccurate for binaries with C (cid:38) / ≈ .
17 (see discussion above),their actual merging frequencies are greater than f (cid:38) Hz, which is beyond the upper bound ofthe detector sensitivity bands. The precise values of f cut in these cases are therefore immaterial,as the inspiral signal in the observational band remains unchanged.Finally, we note that the parameters χ, κ and C of a given astrophysical object are in principlerelated to each other. For instance, by requiring the speed of a test mass on the equatorial surface Some papers use R = r ISCO as a merger condition for low-compactness binary systems. However, the notionof an ISCO ceases to exist for objects with C (cid:46) / ≈ .
17, as this fictitious ISCO would be located in the interiorof the object. This leads to a factor of ≈ f cut , which can substantially reduce the frequencyrange over which the SNR could be accummulated.
7o be smaller than its escape velocity, we can obtain a mass-shedding bound that relates C withthe maximum value of χ . Furthermore, through simple dimensional analysis Q ∝ mr , we findthat κχ ∝ C − . These imply that the dephasing from (2.4) and (2.5) are actually correlatedwith the compactness of the object for a given equation of state. Absent a detailed astrophysicalmodel in mind, we will treat χ, κ , and C as independent parameters, although we emphasizethat their implicit correlation can perhaps be exploited in future work, similar to how universalrelations [109] are used to simplify analyses of binary neutron star signals. In this section, we study the detectability of our newly proposed waveforms through currentmatched-filtering searches. We first construct a binary black hole template bank that is repre-sentative of those used by the LIGO/Virgo collaboration ( § effectualness ( § h and h , defined as [92]( h | h ) ≡ (cid:90) ∞ d f ˜ h ( f )˜ h ∗ ( f ) S n ( f ) , (3.1)where ˜ h , ˜ h are their Fourier representations, and S n is the (one-sided) noise spectral density.For future convenience, we denote the normalized inner product by[ h | h ] ≡ ( h | h ) (cid:112) ( h | h ) ( h | h ) . (3.2)Throughout this work, we use the aLIGO MID LOW [110] detector specification for S n , which isrepresentative of the first LIGO observational run, O1. When evaluating the frequency integral(3.1), we use the lower and upper cutoff frequencies f l = 30 Hz and f u = 512 Hz. These choicesreflect the fact that low mass binary inspirals accumulate a minimum of 95% of their signal-to-noise ratio (SNR) within this frequency range. Finally, although represented as a continuousintegral, (3.1) is in practice evaluated discretely in frequency. We therefore specify a samplingrate of 1024 Hz and take the maximum time spent in band to be 94 s. These choices give us thegrid of possible coalescence times to maximize over when computing the effectualness later.
We now construct a template bank that is representative of those used by the LIGO/VirgoCollaboration [57–60]. In order to do so, we use the Taylor F2 waveform model for binaryblack holes [93–95] with intrinsic parameters p bbh = { m , m , χ , χ } , where the spins χ , χ areparallel to the orbital angular momentum of the binary. This waveform model is exactly the sameas that in (2.2), except we now specify the κ parameters to be unity [47, 48] and neglect the cutofffrequency introduced by the C ’s for black holes. Crucially, since we are only interested in thesignals emitted during the inspiraling regime, we do not have to use the phenomenological [111] This corresponds to the time it takes a binary with component masses m = m = 1 M (cid:12) , which is the smallestmass we consider in this paper, to inspiral between the stated frequency cutoffs.
8r the effective-one-body waveform [107] models, which include numerical relativity waveformsnear the binary merger.Matched-filtering searches involve computing the inner product (3.2) between the data anda set of template waveforms. Practically, this requires a discretized sampling of the parameterspace p bbh in the form of a template bank. We adopt the geometric-placement technique describedin Ref. [61], although many other methods exist; see for example Refs. [57, 112, 113]. We onlysketch this method here and refer the reader to the original work for further details. The keyfeature of this formalism lies in the following decomposition of the waveform phase ψ ( f ; p bbh ) = ψ ( f ) + N (cid:88) α =0 c α ( p bbh ) ψ α ( f ) , (3.3)where ψ is an average behaviour of the phase, which is chosen for convenience, c α is a setof basis coefficients that only depend on p bbh , and ψ α is a set of orthonormal basis functionsthat satisfy (cid:104) ψ α | ψ β (cid:105) = δ αβ , with (cid:104)·|·(cid:105) being an inner product that is slightly modified from(3.1) [61]. Crucially, this new inner product is designed such that the mismatch distance betweenneighbouring waveforms in parameter space translates to a Euclidean distance in the space of c α . After the phases of a random sample of waveforms are extracted, the functions ψ α can thenbe computed through a singular-value decomposition of a matrix that consists of these extractedphases [61]. The template bank is finally constructed by grid sampling the set of basis coefficients c α . Because the metric is Euclidean in the space of c α (at least for neighbouring waveforms), anoptimal bank is therefore easily obtained by sampling a regular grid in this space. The dimensionof the basis, N , and the grid spacing, ∆ c α , therefore dictate the resolution of our template bank.To create the basis functions ψ α , we randomly sample 4 × parameter combinations andgenerate waveforms for each. We use the parameter ranges 1 . M (cid:12) ≤ m i ≤ . M (cid:12) and | χ i | ≤ . c α . This mass range is motivated by the fact that the resulting binarysystems have relatively large chirp masses, and at the same time, would inspiral over long periodsof time in the detector band (e.g. (cid:38)
10 s). A study which includes a wider mass range, especiallysubsolar-mass objects, is certainly possible, though it would not alter the qualitative conclusionsof this work (see § N = 3 in (3.3) issufficient to describe the behaviour of the phases of these low-mass binary systems (cf. Fig. 5).Finally, we take the grid spacing ∆ c α = 0 . , which leads to a total of 572 ,
558 templates inour bank. As we shall see in § The match between two waveforms h and h is obtained by maximizing (3.2) over the time and phase ofcoalescence, i.e. m ≡ max t c ,φ c [ h | h ]. The mismatch distance is then defined as √ − m . Not all combinations of c α give physically realizable waveforms. We therefore use a fudge factor [61] of ζ = 0 . .2 Effectualness to Inspiral Waveforms We are now ready to test how well a binary black hole template bank can be used to detect thegeneral inspiral waveforms that we constructed in Section 2. For concreteness, we assume thatone of the binary components is a black hole, κ = 1 and 0 . ≤ C ≤
1, while the other is ageneral compact object, whose finite-size parameters are labeled by κ ≡ κ and C ≡ C . Sinceit is important that we distinguish the intrinsic parameters of the binary black hole templatewaveforms from those of the general waveforms, we will denote them by p bbh and p g respectively,with the latter being p g = { m , m , χ , χ , κ, C} .The primary tool for assessing the template bank’s effectiveness at recovering our generalwaveforms is its effectualness [55, 56]. More precisely, this is obtained by maximizing the innerproduct (3.2) between the template and general waveforms over their relative time of coalescence t c , phase of coalescence φ c , and the intrinsic parameters p bbh of every template in the bank: ε ( p bbh , p g ) ≡ max t c ,φ c , { p bbh } [ h ( p bbh ) | h ( p g )] , (3.4)where { p bbh } denotes the list of template parameter combinations. In other words, (3.4) quantifiesthe overlap between the general waveform and the best-fitting template waveform in the bank.To obtain a rough idea of how a reduced effectualness translates into a loss in signal detectability,we note that existing searches adopt an SNR detection threshold of 8, while a typical binarysystem detected thus far has an SNR ranging from 10 to 15 [22]. For a signal with true SNRof 12 .
5, a template bank with effectualness ε < / . ≈ .
64 would reduce the observed SNRto values below the detection threshold, thereby leading to missed events. This can instead bephrased as a reduced sensitive volume of 1 − ε (cid:38) .
74 [114]. A commonly adopted target inLIGO and Virgo searches is ε > .
97, which leads to a 10% loss in sensitive volume.While the maximization of (3.4) over t c and φ c can be performed efficiently (through fastFourier transform), the iterative computation over the list { p bbh } is much more computationallyexpensive. One of the benefits of the geometric-placement method described in § best-fitting point, { c α ( p bbh ) } best , inthe bank [61]. This is achieved by projecting c α ( p g ) = (cid:104) ψ ( p g ) − ψ | ψ α (cid:105) , where ψ ( p g ) is the phaseof the general waveform, while ψ and ψ α are the average phase and basis functions constructedfor our bank in (3.3), respectively. The best-fitting point { c α ( p bbh ) } best is the closest c α ( p bbh )to c α ( p g ), as measured by their Euclidean distance. The effectualness can then be evaluatedstraight-forwardly using these nearby parameters and maximizing over t c and φ c .To test the validity of this procedure, we randomly sample an independent set of 10 binaryblack hole waveforms within the same parameter ranges used to generate ψ α in § left panel of Fig. 3, where the effectualness is plotted as a function of thebinary total mass M and the effective mass-weighted spin, χ eff = ( m χ + m χ ) /M . We findthat 99% of the random templates have ε > . ε > .
97. We also find that theeffectualness decreases slightly as χ eff increases, indicating that the basis functions are less ableto capture the high-spin behaviour. In essence, this figure demonstrates both the validity of ourmethod of evaluating (3.4) and our construction of a highly effectual bank for detecting binaryblack hole signals. 10 M (cid:12) ]0 . . . . . χ e ff A/C BD 2 3 4 5 6Total Mass [ M (cid:12) ] 0.00.20.40.60.81.0 E ff ec t u a l n e ss Figure 3 : Effectualness of our template bank for binary black hole inspiral waveforms ( left ) andgeneral inspiral waveforms with κ = 500 and C = 0 . right ). The effectualness is the ratio ofthe observed SNR to the true SNR of a signal. Comparing these panels we see a drastic loss ineffectualness from the finite-size effects. For convenience, we have indicated the masses and spinsof scenarios A − D that are considered in Table 1 and Fig. 4 in the left panel. Note that all pointsare plotted from least to most effectual — the points with the highest effectualness are thereforethe most visible.Taking the left panel of Fig. 3 as a baseline (optimal) effectualness of our template bank, we cancompare the bank’s effectualness to a general inspiral waveform. For concreteness, we generate10 general inspiral waveforms within the same mass and spin ranges, fixing κ = 500 and C = 0 . right panel of Fig. 3, where we see that a large spin-inducedquadrupole moment can significantly decrease the effectualness of the bank. This is especiallytrue in the large-spin limit, since the phase contributions (2.4) and (2.5) are proportional to κ i χ i .Statistically, we find that only 6% of the random templates have ε > . ε > . C = 0 . f u , thereby having no effect on our analysis.Scenario m [M (cid:12) ] m [M (cid:12) ] χ χ C DescriptionA 2.0 2.0 0.7 0.7 0.1 Fiducial caseB 2.8 2.8 0.7 0.7 0.1 Heavier total massC 3.0 1.0 0.7 0.7 0.1 Lighter general objectD 2.0 2.0 0.2 -0.2 0.1 Reduced and anti-aligned spinsE 2.0 2.0 0 0 0.01 Reduced compactness
Table 1 : List of representative binary configurations for Fig. 4. The parameters { m , χ } describethe black hole, while { m , χ , κ, C} are the parameters of the general astrophysical object.11
10 10 κ . . . . . . E ff ec t u a l n e ss NeutronStars Boson Stars,Superradiant Clouds, . . . . . . κ = 500 ε = 0 . Figure 4 : Effectualness of the various scenarios listed in Table 1 as a function of κ . The verticaldashed line denotes systems with κ = 500 and is included for comparison with the right panel ofFig. 3. We also include the reference line ε = 0 .
97, which is a commonly adopted requirement fortemplate banks in actual searches. For convenience, we indicate the approximate ranges of κ forneutron stars and various BSM objects in green and purple respectively. Kerr black holes have κ = 1.Instead of fixing κ and C , it is also interesting to examine the bank’s effectualness as a functionof these parameters. For concreteness, we consider five qualitatively-distinct scenarios, which arelisted in Table 1. For each case, we treat κ as a free parameter and calculate the effectualnesswithin the range 1 ≤ κ ≤ . Note that scenario E has no spins and is therefore unaffected byvarying κ ; this scenario is meant to show the effect of reducing the compactness of an object,which terminates the waveform in the detector sensitivity band. The results are presented inFig. 4, where we find that each scenario with non-vanishing spins (A − D) shows a universalbehaviour with the following three distinct regions as a function of κ :1. No loss in effectualness at low values of κ . For scenario A this occurs for 1 ≤ κ (cid:46) κ , possiblyindicating the varying rates of importance of the various higher-order PN contributionssuch as (2.4) and (2.5). For scenario A we see this at 20 (cid:46) κ (cid:46) × .3. Finally we see a flattening of the effectualness. This flattening occurs when κ is so largethat the 2PN term (2.4) becomes the dominant contribution to the phase evolution. Atthis point, maximising over t c and φ c always finds a small region of frequency space wherethe overlap between the two waveforms is nearly vanishing.We tested many additional scenarios and found this overall behaviour to be universal, showingthe three distinct regions described above. Note that the value of κ at which each scenario enters12he three regions and the length spent there differs greatly as can be seen in Fig. 4. Below wegive some qualitative arguments to compare the differences between the scenarios A − E.Importantly, in the first region we see no noticeable reduction in the effectualness for κ (cid:46) § κ for neutron stars [83, 84]. It is for this reason that κ can be safely ignoredwhen searching for black hole - neutron star systems with binary black hole templates, althoughvariations from κ = 1 must be accounted for during parameter estimation [103]. Binary neutronstar systems would have additional contributions to their phase evolution since both objects nowcontribute to the phase with κ (cid:38)
1. LIGO and Virgo typically only consider slowly spinningneutron stars, χ ≤ . < κ (cid:46)
10 is small — binary black holetemplates are therefore still suitable at the search level.For larger values of κ , the effectualness quickly drops below the normal requirement of ε ≥ . κ ≈
200 at which pointthey start to diverge. To understand this behaviour, we first note that the prefactors of the v − dependence in (2.4) and (2.5) are unchanged for equal-mass-ratio binary systems, regardless oftheir total mass. However, v retains some dependence on the total mass — v ∝ M / . The variousPN terms therefore scale differently with M , causing a different rate of loss of effectualness. Thedifference between scenarios A and C can again be easily understood by examining the prefactorsof (2.4) and (2.5). Since our general object is the lighter of the two components in scenario C(see Table 1), the mass dependencies of these terms dictate that the general object contributesless to the phase evolution. This reduced contribution can be compensated for by a larger valueof κ , producing an overall shift to the right in Fig. 4 from scenarios A to C. Similarly, if we wereto fix the total mass and choose the heavier component to be our general object, we would seean overall shift from scenario A to the left. Finally, scenario D has significantly smaller spins,reducing the overall phase contribution from the spin-induced quadrupole.For scenario E we see a significant reduction in effectualness, even for small values of κ (thehorizontal line merely reflects our choice of vanishing spins). This is because the frequency cutoffset by (2.7) is f cut ≈
44 Hz, which lies inside the sensitivity band of ground-based detectors. Forcomparison, the cutoffs for scenarios A − D lie above f u = 512 Hz in our analysis. More generally,we find that C (cid:38) .
05 gives a cutoff frequency of f cut (cid:38) f u . This loss in effectualness cannotbe compensated for by adding additional waveforms to the bank, unlike for scenarios A − D.Instead it represents a truncation of the waveform and therefore a reduction in SNR. Since theintroduction of f cut merely represents our ignorance of the actual merger dynamics of the binary,the effectualness for objects with small values of C can potentially be improved. For instance, asdiscussed in § f cut towards higher values. Alternatively,model-dependent numerical relativity simulations can be performed to fully extract the mergerwaveforms. The precise range of C depends on the mass of the binary components, and can be calculated more accuratelythrough (2.7). . . . δ ψ m = m = 2 . M (cid:12) χ = χ = 0 . κ = 1 κ = 10 κ = 100 κ = 1000
100 200 300 400 500Frequency [Hz] − δ ψ b e s t Figure 5 : The residual phases of scenario A at various values of κ . In the top panel, themagnitudes of δψ are smaller than unity, indicating that the basis functions ψ α can sufficientlydescribe the phases of the general waveforms. The gray band in the bottom panel indicates therange − ≤ δψ best ≤
1. The phase residual δψ best for κ (cid:38)
100 clearly exceeds order unity,resulting in a reduced effectualness of the binary black hole template bank.It is important to assess whether the loss of effectualness in Fig. 4 is due to the limited range ofcomponent masses considered in the bank. We study this by repeating the procedure outlined in § . M (cid:12) ≤ m i ≤ . M (cid:12) . We find no differencein the initial reductions of effectualness for all scenarios (this corresponds to the κ (cid:46)
100 and ε (cid:38) .
85 region in scenario A). As κ increases to larger values, the effectualness still drops rapidly(as observed for κ (cid:38)
100 in scenario A) although the rates of reduction slightly decrease. Thisslight improvement occurs because, in this large- κ region, the best-fitting points { c α ( p bbh ) } best are located inside and outside the bounds of c α ( p bbh ) in the bigger and smaller template bank,respectively. In contrast, the earlier reduction is robust to the increased component mass rangebecause { c α ( p bbh ) } best lies within the bounds of c α ( p bbh ) for both banks. Despite this slightdependence on the size of the template bank parameter space, the effectualness maintains itsmonotonically-decreasing trend with increasing κ for all scenarios. This robust behaviour suggeststhat waveforms with large κ cannot be mimicked by binary black hole waveforms with vastlywrong intrinsic parameters.While the effectualness is a good measure of the differences between the signal and templatewaveforms, it is an integrated quantity with a less clear interpretation. We therefore examinetwo phase residuals: δψ = ψ ( p g ) − (cid:34) ψ + N =3 (cid:88) α =0 c α ( p g ) ψ α (cid:35) , δψ best = ψ ( p g ) − (cid:34) ψ + N =3 (cid:88) α =0 { c α ( p bbh ) } best ψ α (cid:35) , (3.5)14here c α ( p g ) = (cid:104) ψ ( p g ) − ψ | ψ α (cid:105) (see § δψ is a measure of the basisfunctions ψ α ’s ability to capture the phase evolution of a general waveform. Instead, δψ best quantifies the phase deviation between the general waveform and its best-fitting waveform in ourbank.These residual phases are shown in Fig. 5 at various values of κ for scenario A. In the top panel, the fact that | δψ | (cid:28) κ indicates that our basis functions, with N = 3 in(3.3), are able to describe the phase of the general waveform to a high degree of accuracy. Thereduction in effectualness in Fig. 4 is therefore a consequence of the large separations between c α ( p g ) and { c α ( p bbh ) } best . In the bottom panel, we see that | δψ best | (cid:28) κ = 1 and κ = 10— it is for this reason that we can still detect these binary systems with standard binary blackhole waveforms, in agreement with Fig. 4. For κ = 100, we start to see deviations exceedingorder unity, | δψ best | (cid:38) O (1), resulting in a reduced effectualness. Crucially, the bottom panel ofFig. 5 illustrates that our general inspiral waveforms are not degenerate with a binary black holetemplate waveform with the wrong intrinsic parameters; if this were the case, we would see zerophase residuals.In a nutshell, Figs. 3, 4, and 5 clearly indicate that, for values of κ (cid:46)
20, binary black holetemplate banks are still able to detect general astrophysical objects. On the other hand, for κ (cid:38)
20, there are large parts of parameter space where the binary black hole templates cannotbe used to recover these general signals. This is especially true if the general object is highlyspinning and has a larger mass when compared with its binary companion. Moving forward,new template banks must be constructed with κ as a one-parameter extension to the standardwaveforms. Since C is simply a truncation of the waveform, it is not necessary to include it as anadditional parameter. In this paper, we examined whether binary black hole template banks can be used to search for thegravitational waves emitted by a general binary coalescence. We focused on binary systems withcomponents that can have large spin-induced quadrupole moments and/or small compactness.Figure 4 clearly demonstrates that as the quadrupole term becomes large, its phase contributionto the waveform becomes significant. Binary black hole template banks are thus insufficient forsearching for these general astrophysical objects. More precisely, we find that the effectualnessof these template banks are quickly reduced for κ (cid:38)
20 for highly-spinning objects (for examplescenario A in Table 1). This range of κ coincides with an interesting part of the parameter spacewhere compact objects in various BSM scenarios may exist [50–53]. Figure 5 further shows thatthese signatures are not degenerate with binary black hole template waveforms with the wrongintrinsic parameters. It is therefore essential that extended template banks are created in orderto search for these novel signatures. As a byproduct of our analysis, we recovered the result thatthe effectualness remains high for smaller values of κ . Binary black hole waveforms can thereforebe used to search for binary systems with neutron stars, as is currently done by the LIGO/Virgocollaboration [2]. 15n addition, we considered the impact of an object’s compactness on the merger frequency ofthe binary. Since a detailed description of the merger dynamics is model-dependent, we truncatethe waveform through a frequency cutoff. For objects with small-compactness, this cutoff is setby the point at which the binary components touch. This truncation only has a significant effecton the effectualness when the cutoff frequency is within the sensitivity bands of ground-baseddetectors. As a fiducial guide, our estimate shows that this is the case for C (cid:46) .
05 in low-massbinary systems. This loss in effectualness can be compensated for through more detailed modelingof the binary merger dynamics.Throughout this paper, we focused on the early inspiral regime of low-mass binary systems.This restriction had multiple benefits. Firstly, the inspiral is a regime where analytic resultsof the PN dynamics are readily available. This provided us with a well-defined framework toconstruct our general waveforms, where the physics contributing to waveform deformations canbe clearly interpreted. Secondly, ground-based detectors are able to probe these inspirals overhundreds or thousands of cycles, thereby allowing for a precise characterization of the physicsat play. Inspiral signals in LIGO/Virgo observations therefore represent a new avenue to probeBSM physics and novel astrophysical phenomena.Our findings show that many new signatures could be missed by current search pipelines.Although we focused on finite-size effects, many other types of physical phenomena can affectthe frequency evolution of a binary. We hope to incorporate these additional dynamics into ourgeneral waveforms in future work. Furthermore, we aim to search for these novel signatures inthe data collected in the O1 − O3 observation runs. Using the same procedure and grid spacingas in § Acknowledgements
We thank Liang Dai, Tanja Hinderer, Cody Messick, Samaya Nissanke, Rafael Porto, JohnStout, Tejaswi Venumadhav, Christoph Weniger, Matias Zaldarriaga and Aaron Zimmermanfor enlightening discussions. The work of HSC is supported by the Netherlands Organisation forScientific Research (NWO). TE acknowledges support by the Vetenskapsr˚adet (Swedish ResearchCouncil) through contract No. 638-2013-8993 and NWO through the VIDI research program“Probing the Genesis of Dark Matter” (680-47-532; TE). HSC thanks the Oskar Klein Centre atStockholm University for its hospitality while some of this work was completed. TE thanks theWeinberg Theory Group at the University of Texas at Austin for its hospitality while some ofthis work was completed. We also thank the authors of Ref. [61] for sharing their template bankgeneration code. Numerical computations and plots in this paper are produced with the Pythonscientific computing packages NumPy [115] and SciPy [116]. This research utilised the HPCfacility supported by the Technical Division at the Department of Physics, Stockholm University.16 eferences [1] B. Abbott et al. (LIGO/Virgo Collaboration), “Observation of Gravitational Waves from a BinaryBlack Hole Merger,”
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