Observing Gravitational Waves From The Post-Merger Phase Of Binary Neutron Star Coalescence
We present an effective, low-dimensionality frequency-domain template for the gravitational wave signal from the stellar remnants from binary neutron star coalescence. A principal component decomposition of a suite of numerical simulations of binary neutron star mergers is used to construct orthogonal basis functions for the amplitude and phase spectra of the waveforms for a variety of neutron star equations of state and binary mass configurations. We review the phenomenology of late merger / post-merger gravitational wave emission in binary neutron star coalescence and demonstrate how an understanding of the dynamics during and after the merger leads to the construction of a universal spectrum. We also provide a discussion of the prospects for detecting the post-merger signal in future gravitational wave detectors as a potential contribution to the science case for third generation instruments. The template derived in our analysis achieves match across a wide variety of merger waveforms and strain sensitivity spectra for current and potential gravitational wave detectors. A Fisher matrix analysis yields a preliminary estimate of the typical uncertainty in the determination of the dominant post-merger oscillation frequency as Hz. Using recently derived correlations between and the neutron star radii, this suggests potential constraints on the radius of a fiducial neutron star of m. Such measurements would only be possible for nearby ( Mpc) sources with advanced LIGO but become more feasible for planned upgrades to advanced LIGO and other future instruments, leading to constraints on the high density neutron star equation of state which are independent and complementary to those inferred from the pre-merger inspiral gravitational wave signal. We study the ability of a selection of future gravitational wave instruments to provide constraints on the neutron star equation of state via the postmerger phase of binary neutron star mergers.
pacs:04.80.Nn, 07.05.Kf, 97.60.Jd, 04.25.dk
The second generation of gravitational wave (GW) observatories has now become operational with the first observations by advanced LIGO (aLIGO) underway . Instruments such as advanced Virgo (advVirgo)  and KAGRA  will soon come online, eventually culminating in a world-wide network of GW observatories. The GW signal from the inspiral stage of binary neutron star (BNS) coalescence is amongst the most promising sources for this second generation of GW detectors. Observations of BNS GW inspiral signals from relatively nearby events (a few tens of Mpc) can lead to strong constraints on the supranuclear equation of state (EoS) via the impact on the phase evolution of the signal from tidal interactions during the latter stages of the merger [4, 5, 6, 7, 8, 9, 10] 111Reviews of the subject may also be found in [11, 12, 13, 14, 15].
For example, Read et al  showed that neutron star radii could be constrained with an uncertainty of for a single nearby (100 Mpc, assuming optimal orientation and sky-location) source, based on Fisher matrix estimates. More recently, a number of full Bayesian analyses have been carried out which have used astrophysically-motivated simulated populations of BNS merger events to develop and probe EoS constraints in the low signal-to-noise ratio (SNR) regime. In  it was found that only a few tens of inspiral events are required to measure neutron star (NS) tidal deformability to . Similar results are found in [17, 18], where the NS radius is determined to km and tidal deformability is determined to % after a few tens of GW detections.
The focus of this work, however, is on the independent and complementary constraints on the EoS which may be obtained from the post-merger signal. Depending on the mass configuration of the system and the EoS, a BNS merger may result in prompt collapse to a black hole (high-mass, soft EoS) or the formation of a stable or quasi-stable NS remnant which again, may or may not collapse to a black hole depending on its mass and the EoS, while transient non-axisymmetric deformations and quadrupolar oscillations in this remnant typically give rise to a richly structured, high-frequency (1–4 kHz) GW spectrum and a signal lasting ms, [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Characterising the frequency content of GW signals from the post-merger system provides unique opportunities for GW asteroseismology: the dominant post-merger oscillation frequency exhibits a tight correlation with the radius of nonrotating neutron stars, with an overall uncertainty of a few hundred meters, depending on the total binary mass. For example, for a total binary mass of 2.7 the uncertainty in the radius of a cold, non-rotating NS of mass 1.6 (denoted as ) is about 220m [34, 40]. Similar relationships between the dominant spectral features and stellar parameters have been confirmed elsewhere [36, 38]. A deeper understanding of the features of postmerger GW spectra has been provided in [29, 41, 42], where it was shown that the spectrum is dominated by a linear feature (quadrupolar oscillations), a quasi-linear feature (a coupling between quadrupolar and quasi-radial oscillations) and a fully nonlinear feature (a transient spiral deformation), leading to a classification scheme of the postmerger GW emission depending on the EoS and binary mass. More recently, efforts have been made to find correlations between the pre- and post-merger signals. In Ref.  the authors derive a relation between the tidal coupling constant that determines the tidal interactions before and during the merger and the peak frequency in the post-merger spectrum. Thus, measurements of the inspiral signal (which determine ) could be used to constrain by restricting its range of possible values and by combining measurements with those of the post-merger signal.
This connection between the tidal interactions and the post-merger oscillations highlights the complementarity of pre- and post-merger GW observations. The constraints arising from inspiral observations may be subject to systematic biases induced by errors in the phase due to missing high PN-order terms or insufficiently accurate descriptions of spin or tidal effects. These systematic errors can be as large as the statistical uncertainty in characterising the inspiral signal [44, 18]. While inspiral waveform models will continue to improve and incorporate such effects, we note that analyses of the post-merger signal are subject to a completely independent source of systematic error (e.g., the precise relationship). Moreover, since the majority of pre-merger NSs are likely to have masses in the range , the pre-merger waveforms are limited to probing the structure of NSs in that mass-range, while the post-merger signal allows us to probe the regime of higher masses (this is because, e.g. the central density of the remnant of a merger is close to the central density of a nonrotating star).
This high frequency component of the merger signal, however, will be somewhat challenging to observe in the upcoming generation of GW detectors. Typically, the most sensitive frequencies of ground-based GW instruments lie around 10–1000 Hz, with a rapidly diminishing sensitivity in the kHz regime. Additionally, the absence of a complete waveform model for the full pre- and post-merger signal, or even for the post-merger signal alone, currently prohibits the use of matched filtering and one must turn to more robust, but ultimately less sensitive unmodelled burst searches. For example, the study in  revealed that a typical realistic burst analysis yielded an effective range approximately 30–40% of that which could be possible with an optimal matched filter.
Clearly then, there is great motivation and opportunity to develop more sensitive, more targeted analysis techniques and effective models which will bring us closer to the sensitivity offered by a matched filtering analysis. It is the goal of this work to explore a principal component analysis (PCA) based approach to constructing precisely such an effective model for the high-frequency component of the BNS merger signal. We construct a catalogue of 50 numerical waveforms from the merger and post-merger evolution of a variety of BNS systems with various EoSs and mass configurations. The magnitude and phase spectra of the waveforms in the catalogue are then decomposed into orthogonal bases using PCA. These basis functions can then be used to construct a frequency-domain waveform template which provides, on average, a 93% match for the waveforms in the catalogue for both aLIGO and a variety of potential upgrades and new GW instruments.
The structure of this paper is as follows: in section 2 we provide a detailed review of BNS merger and post-merger phenomenology, focussing on the resulting features in the GW spectrum and hence how one may constrain the NS EoS. Section 3 summarises the expected detectability of the high-frequency BNS waveforms used in this study, assuming a matched-filtering approach and a variety of current and potential future GW instruments. In section 4 we describe and characterise our PCA-based frequency-domain waveform template in terms of waveform match and provide Fisher-matrix estimates of the uncertainties in and based on this template as a preliminary guide to its potential. Finally, section 5 provides a summary and some concluding remarks relating to the planned applications of this model and the potential for similar approaches to enhance unmodelled burst analyses.
2 Properties of Postmerger GW Spectra and Constraining the Neutron Star EOS
2.1 Types of merger dynamics and GW spectra
For symmetric (i.e., equal component masses) and mildly asymmetric binaries the GW postmerger spectra of NS mergers (see e.g. right panel of Fig. 1 or Fig. 1 in Ref. ) show a generic behavior in the sense that certain features of the spectrum depend in a particular way on the total binary mass and the high-density EoS . Specifically, distinct peaks in the spectrum can be associated with distinct mechanisms generating those features, and the frequency and strength of the different GW peaks are determined by the total binary mass and EoS. The presence or absence of certain secondary peaks in the spectrum, together with their relative strengths is determined by the quasi-linear coupling between the quasi-radial and quadrupolar oscillation modes and by the orbital motion of antipodal bulges of a spiral deformation in the remnant. The characteristics of these distinct spectral features can be used to classify the post-merger dynamics of the system .
The most striking feature of the postmerger spectrum is a major peak generated by the dominant quadrupolar oscillation of the remnant, which is present in all models that form a NS merger remnant. The determination of the frequency of this peak in a GW measurement is the focus of this work because the peak frequency scales tightly with the radii of nonrotating NSs (see Fig. 4 below and discussion in [33, 46]) and thus provides strong constraints on the only incompletely known EoS of NS matter. Apart from this main peak, there can be up to two pronounced secondary peaks at frequencies below .
One of the secondary features is a peak generated by the quasi-linear interaction between the dominant quadrupolar oscillation and the quasi-radial mode of the remnant (the latter does not appear strongly in the GW spectrum on its own) . The corresponding peak of the quasi-linear mode coupling has an amplitude proportional to the product of the amplitudes of the quadrupolar mode and of the quasi-radial mode, while its frequency, which we denote as , is equal to the difference of the frequencies of these two modes, i.e. , where is the frequency of the quasi-radial mode. The feature is particularly pronounced for relatively high total binary masses and soft EoSs. Another secondary spectral peak is produced by the orbital motion of antipodal bulges, which form during the merging as a spiral deformation and then orbit around the inner remnant for a few milliseconds  (see Fig. 2). This dynamical feature is present in addition to the main emission at for the first few milliseconds after merging. Bulges moving with an orbital frequency result in a peak in the GW spectrum at . This finding receives further support by the time-frequency map of the GW signal shown in Fig. 1 for the 1.35-1.35 merger with the TM1 EoS [47, 48]. One can clearly recognize that in the early postmerger phase there are two distinct frequencies simultaneously contributing to the GW signal. The frequency of the dominant remnant oscillation is present for many milliseconds. The secondary peak at is generated within the first few milliseconds, when the antipodal bulges are pronounced (see Fig. 2). There is no evidence for a strong time variation of the frequencies, especially of the dominant frequency, which was suggested as an explanation for the structure of the GW spectrum in [49, 50].
The information in the time-frequency map of the GW signal can be related to the dynamical behavior of the remnant, which we illustrate by the evolution of the rest-mass density in the equatorial plane for the same simulation (see Fig. 2). The time step of the different snapshots are marked in the time-frequency map (Fig. 1) by vertical lines. Evidently, the presence of antipodal bulges at the outer remnant coincides with the presence of power at in the time-frequency map. It is apparent that the feature is initially particularly strong exceeding even the emission at ; the antipodal bulges are strongest during and immediately after merging and the spiral deformation forming the bulges initially comprises large parts of the remnant (see upper right panel in Fig. 2). In Fig. 2, the antipodal bulges complete approximately one orbit from the top right to the bottom left panel in about 1.2 ms. Thus, the orbital frequency kHz is expected to produce a peak at kHz, where a peak is found in the spectrum (see Fig. 1). For comparison we also show the time-frequency analysis for the SFHO EoS and component masses 1.35-1.35 in figure 3. Here the secondary peak at 2.2 kHz likely arises from the feature. An examination of the hydrodynamical data for this model reveals an of about 1.25 kHz (resulting in kHz), whereas the frequency of the quasi-radial mode is kHz, and thus the peak is expected to occur at about 2.2 kHz.
The above findings on the time-frequency characteristics of the peak are consistent with the explanations of its origin presented in . The feature is particularly strong for mergers with relatively low binary masses and stiff EoSs because less compact NSs favor the spiral deformation and the formation of the antipodal bulges during merging. In contrast, binaries with more massive components, i.e. very compact stars, merge with a higher impact velocity, which favours a strong excitation of the quasi-radial mode of the remnant, leading to a strong feature, while the spiral deformation becomes less pronounced. For intermediate cases, i.e. moderately high binary masses, both secondary peaks are clearly present with comparable strength and distinguishable in frequency. Overall, this implies that for a given EoS the binary mass determines the presence and strength of the different secondary features. According to the classification scheme introduced in , one can identify three different types of spectra: High-mass/soft EOS binaries produce spectra where the dominant secondary peak is (Type I mergers). For intermediate binary masses and EOS stiffness, both the and features are present with roughly comparable amplitude (Type II mergers). Low-mass/stiff EOS binaries produce spectra with a strong peak and an absent feature (Type III mergers). See [41, 42] for further discussion.
2.2 Universal Post-merger Spectra & Measuring The Neutron Star Radius
For a fixed total binary mass the frequencies of the three different peaks depend in a particular way on the EoS, which can be characterized by the radius or compactness of nonrotating NSs [33, 46, 41] (see also [51, 50] for the dependence of the strongest secondary feature on compactness, without distinguishing the different nature of secondary peaks). The two secondary frequencies show a tight correlation with the dominant postmerger frequency . This is shown in Fig. 4 for 1.35-1.35 mergers.
The existence of generic spectral features with predictable behavior suggests that the construction of a universal spectrum should be feasible through the appropriate alignment of the main peaks from spectra for various EoSs. Figure 5 shows the GW spectra for equal mass binaries (1.35-1.35) with different EoSs. The waveforms have been normalised such that the root-sum-squared amplitude is unity,
In the right panel, we rescale the frequency axis such that the dominant quadrupolar oscillation peak feature is located at a common reference value (2.6 kHz) for all models. Apart from small variations of the secondary features a remarkable universality of the spectra is found, which can be explained as follows: We choose a reference peak frequency of kHz. Thus, for a spectrum with the main peak at the factor for rescaling the frequency is . This factor is also applied to the frequencies of the secondary peaks. Therefore, a rescaled secondary peak (i.e. or ) is located at . Since the fraction is approximately constant and similar for both secondary features (see Fig. 4), a rescaled secondary feature occurs at approximately the same frequency for all models.
This universality of the scaled spectra suggests that it should be possible to produce a model from the mean spectrum, computed over a number of numerical simulations, plus some small deviations. In Sec. 4, we demonstrate that PCA provides an approach to solve exactly this problem by producing an orthornomal basis constructed from a superposition of the mean-centered spectra. Furthermore, we find that the perturbations from the mean spectrum are generally well described by a small number of basis functions.
It is important to stress here that, for distances which allow for the detection of postmerger GW emission, the individual masses of the binary components can be determined with an accuracy of a few per cent [52, 53, 54]. Furthermore, current observations suggest that BNS mass configurations will not be dramatically asymmetric (see e.g. the compilation of NS masses in ). Peak frequencies which are recovered within this data analysis study, are converted to NS radii via
describing the empirical relation between NS radii and the dominant postmerger oscillation frequency for symmetric mergers for total binary masses of 2.7 222We note that, even in the abseence of a measurement of mass ratio, the relation is quite robust for a constant chirp mass, which is generally recovered to high precision. See e.g., . Here we adopt the coefficients , and from previous work . For this particular total binary mass the maximum uncertainty in the empirical relation is 175 m. In addition to this systematic error, the measurement of in noisy GW data will introduce a statistical error, whose determination is one of the major goals of this work and is quantified in Sect. 4.
2.3 Binary Neutron Star Merger Waveforms Used In This Study
The numerical waveforms used in this study rely mostly on the calculations discussed in [33, 46, 40, 45, 41], where further information can be found. Additional waveform models employed here are obtained within the same physical and numerical model, for which further details are provided in [56, 23, 57, 34]. The EoS models for the hydrodynamical simulations are chosen to cover a large variety including very stiff and very soft EoSs (see Table 1) and a variety of binary mass configurations are used. All EoSs are compatible with a maximum NS mass of [58, 59]. Figure 6 illustrates the selection of waveforms used in this study in terms of their EoSs and mass configuration.
We now discuss the expected detectability of the post-merger GW signal in current and planned GW instruments. A natural, preliminary, measure of detectability is the matched-filter SNR one would obtain given a perfect model, or template, for the signal waveform in GW detector data . The matched-filter SNR is defined as,
The optimal SNR, where the template exactly matches detector output is then simply,
where is the usual inner product :
and is the noise spectrum of a given GW detector and the asterix indicates complex conjugation. Note that we impose a lower bound on the frequency over which the inner product is evaluated in order to target the detectability of the high-frequency part of the signal. In this study we use kHz. The inner product is evaluated up to the Nyquist frequency of the spectrum, 8192,Hz in this study. We also characterise detectability in terms of horizon distance : the distance at which an optimally oriented source yields an SNR at least as large as some nominal threshold, . For GW searches in which the time of arrival of the signal and the source sky-location are unknown, it is typical to evaluate horizon distances with . In our application, however, we envisage a hierachical ‘triggered’ analysis, similar to that described in , wherein the earlier, lower-frequency inspiral portion of the coalescence signal has already been detected at high confidence. It is likely then that the time of coalescence has been determined to an accuracy of a few or a few 10’s of milliseconds and we can significantly reduce the threshold used to define the horizon distance. Following , we choose . Finally, we can determine the rate with which we will obtain signals with from the expected number of BNS mergers which are accessible to a search with a given horizon distance . For the purposes of this study, we assume the ‘realistic’ rate of BNS coalescence from : MWEGMyr.
We now compute each of the figures of merit (the SNR for an optimally oriented source at 50 Mpc; the horizon distance assuming an SNR threshold and the expected detection rate ) for aLIGO [73, 1], as well as the following selection of proposed upgrades to aLIGO and new facilities. The following descriptions emphasise the expected increases in sensitivity relative to aLIGO only over 1–4 kHz; the band of interest for the post-merger signal. Comparisons with the increased range and sensitivity to the earlier inspiral part of the signal are left to future studies. Note also that we take aLIGO to be the most sensitive of the second generation GW detectors; instruments such as advVirgo and Kagra offer comparable or reduced sensitivity in the frequency regime of interest to this study. It should be noted, however, that a network of detectors with comparable sensitivity could improve the range of an search by a factor of up to with respect to the single detector expectation, assuming stationary Gaussian noise and an optimal analysis. We restrict our estimates to single detector ranges and rates in the interests of conservatism and simplicity.
- LIGO A+ [74, 75]
a set of upgrades to the existing LIGO facilities, including frequency-dependent squeezed light, improved mirror coatings and potentially increased laser beam sizes. Noise amplitude spectral sensitivity would be improved by a factor of over 1–4 kHz. A+ could begin operation as early as 2017–18.
- LIGO Voyager (LV) 
a major upgrade to the existing LIGO facilities, including higher laser power, changes to materials used for suspensions and mirror substrates and, possibly, low temperature operation. LV would become operational around 2027–28 and offer noise amplitude spectral sensitivity improvements of over 1–4 kHz.
- LIGO Cosmic Explorer (CE) 
a new LIGO facility rather than an upgrade, with operation envisioned to commence after 2035, probably as part of a network with LIGO Voyager. In its simplest incarnation, Cosmic Explorer would be a straightforward extrapolation of A+ technology to a much longer arm length of 40 km, referred to as CE1 which would be more sensitive than aLIGO over 1–4 kHz. An alternative extrapolation is that of Voyager technology to the 40 km arm length, referred to as CE2. CE2 is only more sensitive than aLIGO for the frequency range of interest in this study. For simplicity, we consider only CE1.
- Einstein Telescope (ET-D) [76, 77]
the European third-generation GW detector. In this work, we consider the ET-D configuration which is comprised of two individual inteferometers where one targets low frequency sensitivity and the other high frequency sensitivity. Both interferometers will be of 10 km arm length and housed in an underground facility. Furthermore, the full observatory will consist of three such detectors in a triangle arrangement. ET-D is more sensitive than aLIGO over 1–4 kHz. Due to the network configuration (i.e., the alignment of the component instruments) the effective sensitivity of ET-D is % higher than that for a single ET-D detector.
Figure 7 shows the design sensitivity spectra for each of these instruments, again focussing on the 1–4 kHz range of interest for the post-merger BNS GW signal. For comparison, we also show the amplitude spectrum of a typical BNS waveform (the TM1 1.35+1.35 example discussed in section 2) for an optimally oriented source at 50 Mpc. Finally, the figures of merit describing the detectability of the post-merger signal for each instrument are summarised in table 2. Note that, we compute two measures of SNR: SNR, where we simply evaluate equation 5 over 1–4 kHz for the full merger waveform; as well as SNR, where the time-domain waveform has been windowed to suppress power prior to the merger (taken to occur at the peak strain amplitude), in order to yield an estimate of the contribution to the SNR from the post-merger oscillations. Since there are 5 instruments, 50 waveforms and 4 figures of merit, we choose to summarise the results for each instrument in terms of the 10, 50 and 90 percentiles, evaluated over the 50 waveforms used in the study.
4 A Waveform Model Using Principal Component Analysis
The optimal data analysis method for the identification and characterisation of a GW signal in noisy data is matched-filtering, wherein an exact analytic model, or template, for the waveform is convolved with data stream from a network of GW detectors. Unfortunately, the physical complexity of the merging binary neutron star system is such that detailed numerical simulations are required to produce even an approximate waveform. Furthermore, since the physical parameters of the system are essentially unknown, many such simulations would be required in order to build a template bank to maximise the likelihood of signal detection.
We are, therefore, confronted with a similar data analysis problem to that in the analysis of GWs from core collapse supernovae: the absence of an accurate analytic waveform template, a limited number of computationally expensive and approximate simulations and a requirement to significantly reduce the complexity of the modelling problem to faciliate the use of an approximate matched-filter. Motivated by the work in [78, 79], we find that we can construct an effective waveform model from a basis constructed using PCA of a suite of merger simulations comprised of systems with different equations of state, masses and mass ratios.
Our goal is to reduce the complexity of the modelling problem from a high-dimensional physical parameter space, where the waveforms are modelled directly through numerical simulation, to a lower-dimensional problem to model the dominant features of the waveform. PCA of a catalogue of simulated waveforms provides a solution to precisely this problem. Denoting the time-domain merger waveform as , its complex Fourier spectrum is given by,
where and are the magnitude and phase spectra of signal , respectively, In a similar spirit to the approach described in  we construct orthnormal bases for the amplitude and phase spectra separately, using similar a PCA decomposition to that described in [78, 79]. Principal component analysis forms a basis from the eigenvectors of the covariance matrix of some set of data. The procedure is as follows:
Collate a representative sample of binary merger GW waveforms, sampled at 16384 Hz. This sample of waveforms is hereafter referred to as our training catalogue. Each waveform is normalised to unit root-sum-squared amplitude (equation 1) to reduce catalogue variance from different amplitude scales and emphasise morphological differences.
Compute the complex Fourier spectra of the time-domain waveforms in our catalogue. A Tukey window is initially applied to the time-domain signals to minimise spectral ringing and the the waveforms are zero-padded to a uniform samples. The complex spectra are computed using the fast Fourier transform. The amplitude and phase spectra are computed from the absolute values and arguments of the complex frequency series and the phase spectra are unwrapped to yield smooth functions, each of samples.
The unique feature to the analysis presented in this work is our choice of feature alignment, an absolutely key component to PCA. In , for example, the GW waveforms are aligned such that the peak amplitudes lie at a common reference time removing the need for the PCA to account for trivial variance in the catalogue. The analogous procedure in our application is to align features in the frequency domain. Each amplitude spectrum is rescaled such that the dominant post-merger peak, labelled , is aligned to a common reference value . This alignment is achieved by computing a set of frequencies , where are the angular frequencies of the original spectrum. We then interpolate the original spectrum to the new frequencies where the dominant spectral feature (the post-merger oscillation peak) is aligned. Although it is not perfect, this geometric scaling (as opposed to a simple linear shift) also helps to align the sub-dominant and features. Three examples of original and aligned amplitude spectra are shown in figure 8.
Figure 8: Left: Example magnitude spectra; Right: Example spectra after alignment to a common peak frequency .
Next, we construct an matrix where each row correponds to the -sample feature-aligned amplitude spectrum of each waveform after subtracting the mean spectrum (averaged over the waveforms)333Note that this matrix is transposed relative to the descriptions in [78, 79] for more straightforward comparison with other PCA literature and use with software packages such as that offered in .. The mean amplitude spectrum, evaluated over our waveforms is shown in the left panel of figure 9.
Finally, we perform the PCA decomposition in which we compute the eigenvectors of the empirical covariance matrix . Following  and noting the change in row/column convention for the data matrix, the centered data matrix , of dimension , can be factorised using singular value decomposition,
where and are orthonormal matrices with dimensions and , respectively; is a diagonal matrix of singular values of in descending order and . The columns of , , contain the eigenvectors of the covariance matrix , our principal components, and the singular values in are the square roots of the eigenvalues of the covariance matrix . Finally, the columns of contain the eigenvectors of . The principal components , comprise an orthonomal basis of the rows (i.e., the aligned and centered waveforms) in so that each of the aligned waveform amplitude (or phase) spectra can be represented as a linear sum of principal components and the mean. For example, the aligned amplitude spectrum of the first waveform can be constructed as:
where is the mean amplitude spectrum over the aligned waveform catalogue and are weighting coefficients given by the projection of the centered onto the principal component basis,
and are the elements of . The original waveform magnitude spectrum is, at last, obtained by applying the inverse of the alignment procedure in step (3). Figure 9 shows the mean aligned magnitude spectrum and the first principal component as computed for the 50 waveforms described in section 2.3. Note that we have chosen to align the dominant post-merger peak to a value of 2710 Hz; this choice is essentially arbitrary and simply corresponds to the mean of the peak frequencies in the catalogue. It is important to note here that the value of is a free parameter in the spectral model; in practice, its value must be inferred from GW observations.
Figure 9: Left: Mean magnitude spectrum; Right: First principal component for the magnitude spectrum
Ultimately, our goal is to construct a reduced basis from which any post-merger waveform can be reconstructed to some accuracy. It is, therefore, helpful to understand the relative importance of each principal component. A measure of the total variance in the centered catalogue is given by the trace of the covariance matrix . The variance explained by principal components is, then, the sum of the first eigenvalues:
where are the singular values from equation 7. The post-merger waveforms can then be approximated by using a reduced basis with , with the choice of based on capturing a reasonable degree of variance in the catalogue, and equation 8. Figure 10 shows the cumulative explained variance for both the magnitude and phase spectra of the waveforms in our catalogue (i.e., equation 10) as a function of the number of principal components. One can immediately see that the variation between the waveforms is dominated by the rich and varied structure in the magnitude spectra; only of the total variance is explained by the first principal component of the magnitude spectra, while of the variance in the phase spectra is described by the first component.
4.1 PCA Templates: Characterisation & Expected Performance
Remembering that our goal is to build an approximate waveform template for matched filtering, a useful figure of merit to characterise PCA–based model is the waveform match , which describes the fraction of the optimal signal-to-noise ratio for a given signal which is captured by the waveform template :
where is the inner-product, defined by equation 5, maximised over the start time and initial phase offset of the signal. The match is normalised such that for a perfect template and zero for an template which is orthogonal to the target signal. In the following examples, the match is computed assuming the aLIGO noise curve. Figure 11 shows an example of the reconstructed time series and magnitude spectrum for the TM1 1.35+1.35 system considered earlier in section 2. The time series is given by the inverse Fourier transform of the complex spectrum constructed from the separate amplitude and phase PCA. As expected, when we use the full PCA basis with and include the waveform in the data matrix , we obtain a complete basis which allows a perfect reconstruction such that .
It is unlikely that nature will provide us with a signal which exactly matches one of those contained in the set of training data . The right panel of figure 11 again shows the original and reconstructed TM1 1.35+1.35 waveform, except now this waveform has been excluded from the training data. In addition, we use only the first principal components in amplitude and phase. With this more realistic example and a much smaller parameter space, we are still able to reconstruct the target signal with a match 444Again, the value of is assumed known here; the match here represents the the best case scenario.
We now compute similar matches for all of the waveforms in our catalogue and for the different instruments described in § 3. To begin, we compute match using the aLIGO noise curve as a function of the number of principal components used and compare the results of including and excluding each waveform from the data used to compute the PCA. These results are summarised in figure 12 with the mean, minimum, maximum and the tenth and ninetieth percentiles over the matches computed for each of the fifty waveforms. The left panel shows the results when all of the waveforms are used while the right panel summarises the matches when each waveform is removed from the catalogue prior to computing the PCA. We see that, as before, perfect reconstruction fidelity is attained using the full basis when all waveforms are used. In contrast, the match remains approximately constant with respect to the number of principal components used when each waveform being matched is excluded from the training data. This is a reflection of the fact that the lower-order principal components represent the most common generic features in the catalogue, while the higher-order components are essentially minor corrections to the mean which may not be present in the waveform which is excluded. Given the quite respectable matches obtained with just the first principal component and its apparent robustness, we propose modelling the high-frequency GW spectrum for binary neutron star mergers using equation 8 with .
We now repeat the match calculation for each of the instrument noise curves described in § 3, using just the first principal component. The 10, 50 (i.e., the median) and 90 percentiles, computed over the matches for different waveforms, are summarised in figure 13 and listed explicitly in table 3. We find that the PCA templates yield a match of across all of the instruments considered. Variations in the match arise from differences in the shapes of the noise curve, i.e., the denominator in equation 5; in the kHz regime, where sensitivity is limited by photon shot-noise, the noise curves mostly only differ in their overall amplitude scale and we do not expect significant variations in match quality.
4.2 Implications For Parameter Estimation
Given this approximate waveform template it is useful to determine its effectiveness in parameter estimation and, ultimately, the extraction of astrophysics. Recall from § 2 that the single most robust feature of the GW spectrum for these signals is the presence of a dominant spectral peak due to excitation of the post-merger remnant’s quadrupolar -mode oscillation. Recall also that the peak frequency of this excitation in systems with total binary masses of 2.7 correlates strongly with the radius of a fiducial non-spinning NS across a wide variety of equations of state. Our goal then, is to determine how accurately we might expect to measure using our PCA–based waveform model. In this section, we review the waveform template and derive Fisher-matrix estimates for the accuracy with which we may measure and hence given current and planned GW observatories.
The aligned magnitude spectrum at frequency of our waveform model is given by
where the spectrum is aligned such that the dominant peak lies at frequency , is the mean magnitude spectrum of the catalogue, is the first principal component and is the coefficient an arbitrary waveform’s projection onto . The final magnitude spectrum is given by interpolating the aligned spectrum to a set of new frequencies :
and an identical procedure is applied to the phase spectrum . In this prescription then, the peak frequency is a direct parameter of the model.
We estimate the expected accuracy of the estimation from the Fisher matrix and by considering a one-parameter family of waveform templates in which only varies. We assume for simplicity that other important parameters, such as the start time of the signal, have already been determined we hold the value of at its nominal value. While does indeed play a role in determining the detailed shape of the spectrum and, particularly, the degree of asymmetry in the main spectral peak, its will not have a strong correlation with the location of the maximum. A more detailed and realistic Bayesian analysis will be conducted in the near future to account for possible correlations.
The expected error in some parameter can be determined from the Fisher matrix :
Following the procedure in , we estimate the error in to first order from,
where and are our waveform templates evaluated at peak frequencies and , which lie below and above the true , respectively and Hz. We find that this expansion is quite stable to the choice of , such that varies by only a few percent up to Hz.
Figure 14 and table 3 summarise the expected frequency errors obtained across waveforms and instruments using the Fisher matrix estimate. Errors are evaluated at SNR=5, corresponding to a source at the horizon distance. As with the match summary from earlier the results for each instrument are summarised with the 10, 50 and 90 percentiles computed over the different waveforms. Again, the expected frequency error is fairly consistent between the different instruments and we find Hz. We can propagate the expected error in the determination to that in the NS radius using equation 2:
The errors thus obtained represent the statistical uncertainty in the radius, arising from the measurement of a signal in noisy data. The fit given by equation 2 is also subject to a systematic error which, as described in section 2.2, we take to be the maximum deviation in the relationship across a variety of EoSs, m. To arrive at a total expected error in the determination of the radius then, we quote the quadrature sum of the statistical and systematic errors:
5 Summary & Outlook
The wealth of information contained in the high-frequency spectrum of BNS mergers means that there is a strong motivation to develop effective models for the merger and post-merger phase of the coalescence GW signal. Through consideration of the general morphology of the post-merger spectrum and the phenomenology during and after the merger, we have determined that the high-frequency complex spectrum is remarkably well modelled by an orthogonal basis constructed from a catalogue of numerical simulations using a PCA decomposition.
Typically, the waveform templates thus constructed yield a match of , over the frequency range 1–4 kHz, with the majority of the waveforms used in this study. While the typical desideratum in most matched-filtering analyses is , it is worth noting that the only other systematic and well-quantified estimate of GW search effectiveness to date has been the burst analysis reported in . While the burst analysis is robust to uncertainties in the waveform it was found that its effective range was only that of an optimal matched filter analysis. The PCA model presented in this work therefore holds the potential to double or even triple the sensitivity offered by existing analyses. Furthermore, a preliminary Fisher matrix analysis reveals that the uncertainty in the determination of the peak post-merger oscillation frequency is Hz, implying a statistical uncertainty on the radius of a 1.6 NS of m for sources with sufficient power at 1–4 kHz or proximity to Earth to produce SNR=5. Assuming a conservative estimate of for the systematic error in the relation when the binary masses are known, we find the total error in the radius is m. For comparison, the analysis in  found m. Both cases assume an relationship appropriate for a symmetric mass configuration with total mass . Note that a) the estimate in  included only the statistical error and b) the burst analysis requires a relatively large SNR before sufficient GW signal power is acquired to generate a detection candidate; by this time, the peak frequency itself can be quite easily resolved. Furthermore, our estimates here are based on a single-detector analysis; those in  considered a three-detector network operating with comparable sensitivity in each instrument.
For aLIGO the horizon distance with an optimal template for SNR=5 is generally Mpc with a plausible signal rate of approximately 1 event per 100 years, comparable to the rate of Galactic supernovae (see Table 2). Our template, however, will lose % of this SNR, resulting in a proportional decrease in the horizon distance. In fact, thanks to the local over-density of galaxies, the impact on signal rate from this mismatch is rather negligible (%) until we consider the ET-D or CE sensitivities.
In addition to the obvious benefit of potentially yielding a greater detection horizon than an unmodelled search, it is worth highlighting a number of other advantages of using even a rather ad hoc waveform template such as ours. The strain sensitivity spectrum of GW detectors and the short duration of post-merger GW signals suggest that the pre-merger inspiral signal will always be observed at high SNR for any source which is sufficiently close to observe the post-merger signal. This will lead to a quite precise determination of the time of coalescence, potentially a constraint on the sky-location and constraints on the binary masses. If we also assume that the merger does indeed result in the formation of a stable or quasi-stable NS remnant then the analysis need only consist of inferring the parameters of our waveform model and does not necessarily require a signal to produce an SNR above some detection threshold. Instead, our estimate of the parameters (e.g., ) simply have greater uncertainty for low SNR signals. This is in contrast to typical burst analyses which require signals to be sufficiently loud that they produce statistically significant loud pixels in the time-frequency plane. We note, however, that a time-frequency PCA could very easily be used to ‘inform’ burst clustering algorithms to better target signals such as these where there is a rich time-frequency structure.
For the purposes of this study, we adopted an SNR threshold as a fiducial point of reference; in practice, however, it may be possible to determine at larger distances (although with correspondingly greater uncertainty) than suggested in this work. Since the Fisher matrix approximation is not valid in the low SNR regime, this point will be investigated via a full Bayesian analysis using our PCA templates in a future study.
Finally, it is worth mentioning that the construction of PCA templates leads to an intriguing and natural way to feed GW observations back to the numerical modelling community to produce a feedback loop for the estimation of the NS EoS and refinement of our waveform models. For now we simply sketch the basic algorithm as follows, leaving an example implementation to future work:
Construct a PCA template from a coarsely-sampled catalogue of merger waveforms, whose span the full frequency space permitted by allowed EoSs and masses.
Following a nearby BNS detection, determine the probable component masses from the inspiral signal and the best estimate of from the PCA template constructed in (i).
Produce a refined, more finely-sampled catalogue of merger waveforms (with new simulations if necessary) which correspond only to those EoSs and mass configurations which are compatible with the observations in (ii). Construct a new PCA template from this catalogue.
This process could then be iterated until some desirable stopping criterion, such as reaching some critical value of the minimum match between the PCA templates and waveforms used, is reached. This approach may provide an avenue to go beyond simply determining and allow an accurate reconstruction of the full spectrum of the underlying signal. We see then that the application of PCA to construct robust and simple phenomenological templates for the characterisation of post-merger BNS signals holds great promise on its own, and may provide a useful tool to augment other approaches such as those in [45, 43, 42].
The authors thank Tjonnie Li for helpful input and careful reading of this manuscript. J. C. and D. S. gratefully acknowledge support from NSF grants PHY-0955825, PHY-1212433, PHY-1333360, PHY-1505824 and PHY-1505524. A. B. is a Marie Curie Intra-European Fellow within the 7th European Community Framework Programme (IEF 331873). Partial support came from “NewCompStar”, COST Action MP1304. The computations were performed at the Max Planck Computing and Data Facility (MPCDF), the Max Planck Institute for Astrophysics, and the Cyprus Institute under the LinkSCEEM/Cy-Tera project.
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