GW170817: Measurements of neutron star radii and equation of state
On August 17, 2017, the LIGO and Virgo observatories made the first direct detection of gravitational waves from the coalescence of a neutron star binary system. The detection of this gravitational wave signal, GW170817, offers a novel opportunity to directly probe the properties of matter at the extreme conditions found in the interior of these stars. The initial, minimal-assumption analysis of the LIGO and Virgo data placed constraints on the tidal effects of the coalescing bodies, which were then translated to constraints on neutron star radii. Here, we expand upon previous analyses by working under the hypothesis that both bodies were neutron stars that are described by the same equation of state and have spins within the range observed in Galactic binary neutron stars. Our analysis employs two methods: the use of equation-of-state-insensitive relations between various macroscopic properties of the neutron stars and the use of an efficient parameterization of the defining function of the equation of state itself. From the LIGO and Virgo data alone and the first method, we measure the two neutron star radii as km for the heavier star and km for the lighter star at the 90% credible level. If we additionally require that the equation of state supports neutron stars with masses larger than as required from electromagnetic observations and employ the equation of state parametrization, we further constrain km and km at the 90% credible level. Finally, we obtain constraints on at supranuclear densities, with pressure at twice nuclear saturation density measured at at the 90% level.
pacs:04.80.Nn, 97.60.Jd, 95.85.Sz, 97.80.–d
] compiled 13 July 2019
Since September 2015, the Advanced LIGO Aasi et al. (2015) and Advanced Virgo Acernese et al. (2015) observatories have opened a window on the gravitational-wave (GW) Universe Abbott et al. (2016, 2018a). A new type of astrophysical source of GWs was detected on the 17th of August 2017, when the GW signal emitted by a low-mass coalescing compact binary was observed Abbott et al. (2017a). This observation coincided with the detection of a gamma ray burst, GRB 170817A Abbott et al. (2017b); Goldstein et al. (2017), verifying that the source binary contained matter, which was further corroborated by a series of observations that followed across the electromagnetic spectrum, e.g. Abbott et al. (2017a); Coulter et al. (2017); Troja et al. (2017); Haggard et al. (2017); Hallinan et al. (2017). The measured masses of the bodies and the variety of electromagnetic observations are consistent with neutron stars (NSs).
Neutron stars are unique natural laboratories for studying the behavior of cold, high-density nuclear matter. Such behavior is governed by the equation of state (EOS), which prescribes a relationship between pressure and density. This determines the relation between NS mass and radius, as well as other macroscopic properties such as the stellar moment of inertia and the tidal deformability (see e.g. Steiner et al. (2015)). While terrestrial experiments are able to test and constrain the EOS at densities below the saturation density of nuclei g cm (see e.g. Tsang et al. (2012); Baldo and Burgio (2016); Lattimer and Prakash (2016); Oertel et al. (2017) for a review), currently they cannot probe the extreme conditions in the core of NSs. Astrophysical measurements of NS masses, radii, moments of inertia and tidal effects, on the other hand, have the potential to offer information about whether the EOS is soft or stiff and what the pressure is at several times the nuclear saturation density Hebeler et al. (2013); Lattimer and Prakash (2016); Özel and Freire (2016); Steiner et al. (2018).
GWs offer another opportunity for such astrophysical measurements to be performed, as the GW signal emitted by merging NS binaries differs from that of two merging black holes (BHs). The most prominent effect of matter during the observed binary inspiral comes from the tidal deformation that each star’s gravitational field induces on its companion. This deformation enhances GW emission and thus accelerates the decay of the quasi-circular inspiral Flanagan and Hinderer (2008); Vines et al. (2011); Damour et al. (2012). In the post-Newtonian (PN) expansion of the inspiral dynamics Buonanno et al. (2009); Blanchet (2014); Goldberger and Rothstein (2006); Goldberger (2007); Damour et al. (2001a); Blanchet et al. (1995, 2004); Damour et al. (2001b); Flanagan (1998), this effect causes the phase of the GW signal to differ from that of a binary BH from the fifth PN order onwards Damour (1982); Flanagan and Hinderer (2008); Gralla (2018). The leading-order contribution is proportional to each star’s tidal deformability parameter, , an EOS-sensitive quantity that describes how much a star is deformed in the presence of a tidal field. Here is the relativistic Love number Damour (1983); Hinderer (2008); T. Hinderer, B. D. Lackey, R. N. Lang and J. S. Read (2010); Binnington and Poisson (2009); Damour and Nagar (2009), is the compactness, is the areal radius, and is the mass of the NS. The deformation of each NS due to its own spin also modifies the waveform and depends on the EOS. This effect enters the post-Newtonian expansion as a contribution to the (lowest order) spin-spin term at the second order in the GW phase Poisson (1998); Bohé et al. (2015). The EOS also affects the waveform at merger, the merger outcome and its lifetime, as well as the post-merger emission (see e.g. Baiotti and Rezzolla (2017)). Finally, other stellar modes can couple to the tidal field and affect the GW signal Lai (1994); Flanagan and Hinderer (2008); Hinderer et al. (2016); Andersson and Ho (2018).
Among the various EOS-dependent effects, the tidal deformation is the one most readily measurable with GW170817. The spin-induced quadrupole has a larger effect on the orbital evolution for systems with large NS spin Laarakkers and Poisson (1999); Pappas and Apostolatos (2012); Agathos et al. (2015); Harry and Hinderer (2018) but is also largely degenerate with the mass ratio and the NS spins, making it difficult to measure independently Krishnendu et al. (2017). The post-merger signal, while rich in content, is also difficult to observe, with current detector sensitivities being limited due to photon shot noise Aasi et al. (2015) at the high frequencies of interest. The merger and post-merger signal make a negligible contribution to our inference for GW170817 Abbott et al. (2017b, 2018b).
In Abbott et al. (2017a), we presented the first measurements of the properties of GW170817, including a first set of constraints on the tidal deformabilities of the two compact objects, from which inferences about the EOS can be made. An independent analysis further exploring how well the gravitational-wave data can be used to constrain the tidal deformabilites, and, from that, the NS radii, has also been performed recently De et al. (2018). Our initial bounds have facilitated a large number of studies, e.g. Margalit and Metzger (2017); Bauswein et al. (2017); Zhou et al. (2018); Rezzolla et al. (2018); Fattoyev et al. (2018); Nandi and Char (2018); Paschalidis et al. (2018); Ruiz et al. (2018); Annala et al. (2018); Raithel et al. (2018); Most et al. (2018), aiming to translate the measurements of masses and tidal deformabilities into constraints on the EOS of NS matter. In a companion paper Abbott et al. (2018b), we perform a more detailed analysis focusing on the source properties, improving upon the original analysis of Abbott et al. (2017a) by using Virgo data with reduced calibration uncertainty, extending the analysis to lower frequencies, employing more accurate waveform models, and fixing the location of the source in the sky to the one identified by the electromagnetic observations.
Here we complement the analysis of Abbott et al. (2018b), and work under the hypothesis that GW170817 was the result of a coalescence of two NSs whose masses and spins are consistent with astrophysical observations and expectations. Moreover since NSs represent equilibrium ground-state configurations, we assume that their properties are described by the same EOS. By making these additional assumptions, we are able to further improve our measurements of the tidal deformabilities of GW170817, and constrain the radii of the two NSs. Moreover, we use an efficient parametrization of the EOS to place constraints on the pressure of cold matter at supranuclear densities using GW observations. This direct measurement of the pressure takes into account physical and observational constraints on the NS EOS, namely causality, thermodynamic stability, and a lower limit on the maximum NS mass supported by the EOS to be . The latter is chosen as a - conservative estimate, based on the observation of PSR J0348+0432 with Antoniadis et al. (2013), the heaviest NS known to date.
The radii measurements presented here improve upon existing results (e.g. Annala et al. (2018); Fattoyev et al. (2018); De et al. (2018)) which had used the initial tidal measurements reported in Abbott et al. (2017a). We also verify that our radii measurements are consistent with the result of the methodologies presented in these studies when applied to our improved tidal measurements. Moreover, we obtain a more precise estimate of the NS radius than De et al. (2018) by almost a factor of 2.
In this section we describe the details of the analysis. We use the same LIGO and Virgo data and calibration model analyzed in Abbott et al. (2018b). The data can be dowloaded from the LIGO Open Science Center (LOSC) LIGO Open Science Center (LOSC) (). The data include the subtraction of an instrumental artifact occurring at LIGO-Livingston within 2 s of the GW170817 merger Abbott et al. (2017a); Pankow et al. (2018), as well as the subtraction of independently measurable noise sources Driggers et al. (2018, 2012); Meadors et al. (2014); Tiwari et al. (2015).
ii.1 Bayesian methods
We employ a coherent Bayesian analysis to estimate the source parameters as described in Veitch et al. (2015); Abbott et al. (2016). The goal is to determine the posterior probability density function (PDF), , given the LIGO and Virgo data . Given a prior PDF on the parameter space (quantifying our prior belief in observing a source with properties ), the posterior PDF is given by Bayes’ Theorem , where is the likelihood of obtaining the data given that a signal with parameters is present in the data. Evaluating the multi-dimensional analytically is computationally prohibitive so we resort to sampling techniques to efficiently draw samples from the underlying distribution. We make use of the Markov-chain Monte Carlo algorithm as implemented in the LALInference package Veitch et al. (2015), which is part of the publicly available LSC Algorithm Library (LAL) LIGO Scientific Collaboration and Virgo Collaboration (2017). For the likelihood calculation, we use s of data around GW170817 and consider a frequency range of 23–2048 Hz covering both the time and frequency ranges where there was appreciable signal above the detector noise. The power spectral density (PSD) of the noise is computed on-source Littenberg and Cornish (2015); Cornish and Littenberg (2015); Abbott et al. (2018b), and we marginalize over the detectors’ calibration uncertainties as described in Farr et al. (2015); Abbott et al. (2018b, 2016).
In the analysis of a GW signal from a binary NS coalescence, the source parameters on which the signal depends can be decomposed as , into parameters that would be present if the two bodies behaved like point-masses , and EOS-sensitive parameters that arise due to matter effects of the two finite-sized bodies (e.g. tidal deformabilities). The priors on the point-mass parameters that we use are described in Abbott et al. (2018b), and we do not repeat them here. We only consider the “low-spin” prior of Abbott et al. (2018b) where the dimensionless NS spin parameter is restricted to , in agreement with expectations from Galactic binary NS spin measurements Tauris et al. (2017), and we fix the location of the source in the sky to the one given by EM observations. Regarding the EOS-related part of the parameter space and the corresponding priors, we consider two physically motivated parameterizations of different dimensionalities, which we describe in detail in the following sections. The first method requires the sampling of tidal deformability parameters, whereas the second method directly samples the EOS function from a 4-dimensional family of functions. In both cases, the assumption that the binary consists of two NSs that are described by the same EOS is implicit in the parametrization of matter effects (in contrast with the analysis of Abbott et al. (2018b), where minimal assumptions are made about the nature of the source).
ii.2 Waveform models and matter effects
The measurement process described above requires a waveform model that maps the source parameters to a signal that would be observed in the detector. The publicly available LALSimulation software package of LAL LIGO Scientific Collaboration and Virgo Collaboration (2017) contains several such waveform models obtained with different theoretical approaches. The impact of varying the models among several choices Schmidt et al. (2012); Hannam et al. (2014); Schmidt et al. (2015); Husa et al. (2016); Khan et al. (2016); Dietrich et al. (2017); Bohé et al. (2017); Bernuzzi et al. (2015); Lackey et al. (2017); Hinderer et al. (2016); Steinhoff et al. (2016); Sathyaprakash and Dhurandhar (1991); Bohé et al. (2013); Arun et al. (2009); Mikoczi et al. (2005); Bohé et al. (2015); Mishra et al. (2016); Flanagan and Hinderer (2008); Vines et al. (2011) is analyzed in detail in Abbott et al. (2018b), showing that for GW170817 the systematic uncertainties due to the modeling of matter effects are smaller than the statistical errors in the measurement. We perform a similar analysis here and find consistent results with Abbott et al. (2018b) when using different waveform models to determine the radius and EOS. Since the net effect of varying waveform models is very different for each of the source properties, we refer to the tables and figures in Abbott et al. (2018b) for quantitative statements to assess the impact of modeling uncertainties. In the GW170817 discovery paper Abbott et al. (2017a) the results for the inferred tidal deformabilities were obtained with the model TaylorF2 that is based solely on post-Newtonian results for both the BBH baseline model Sathyaprakash and Dhurandhar (1991); Bohé et al. (2013); Arun et al. (2009); Mikoczi et al. (2005); Bohé et al. (2015); Mishra et al. (2016) and for tidal effects Flanagan and Hinderer (2008); Vines et al. (2011), as this model led to the conservatively largest bounds. In this paper, we use a more realistic waveform model PhenomPNRT Schmidt et al. (2012); Hannam et al. (2014); Schmidt et al. (2015); Husa et al. (2016); Khan et al. (2016) whose BBH baseline is calibrated to numerical relativity data. The model incorporates point-mass, spin, and the dominant precession effects based on Taracchini et al. (2014); Bohé et al. (2013); Buonanno et al. (2009); Arun et al. (2009); Mikoczi et al. (2005); Barausse and Buonanno (2010); Buonanno and Damour (2000, 1999) and tidal effects in the phase from combining analytical information Bernuzzi et al. (2015); Wade et al. (2014); Damour et al. (2012); Vines et al. (2011) with results from numerical-relativity simulations of binary NSs as described in Dietrich et al. (2017, 2018). Matter effects in the spin-induced quadrupole are included in the phase using post-Newtonian results Poisson (1998); Bohé et al. (2015); Arun et al. (2009); Mikoczi et al. (2005); Mishra et al. (2016), with the characteristic quadrupole deformation parameters computed from through EOS-insensitive relations Yagi and Yunes (2013a, b) as described in Agathos et al. (2015); Chatziioannou et al. (2015). PhenomPNRT is also used as the reference model in our detailed analysis of the properties of GW170817 Abbott et al. (2018b).
ii.3 EOS-insensitive relations
Despite the microscopic complexity of NSs, some of their macroscopic properties are linked by EOS-insensitive relations that depend only weakly on the EOS Yagi and Yunes (2017). We use two such relations to ensure that the two NSs obey the same EOS and to translate NS tidal deformabilities to NS radii.
The first such relation we employ was constructed in Yagi and Yunes (2016) and studied in the context of realistic GW inference in Chatziioannou et al. (2018). It combines the mass ratio of the binary , the symmetric tidal deformability and the antisymmetric tidal deformability in a relation of the form . Fitting coefficients and an estimate of the relation’s intrinsic error were obtained by tuning to a large set of EOS models Yagi and Yunes (2017), ensuring that the relation gives pairs of tidal deformabilities that correspond to realistic EOS models. We sample uniformly in the symmetric tidal deformability , use the EOS-insensitive relation to compute , and then obtain and , which are used to generate a waveform template. The sampling of tidal parameters also involves a marginalization over the intrinsic error in the relation, which is also a function of and . This procedure leads to unbiased estimation of the tidal parameters for a wide range of EOSs and mass ratios Chatziioannou et al. (2018).
The second relation we employ is between NS tidal deformability and NS compactness Maselli et al. (2013); Urbanec et al. (2013). We employ this – relation with the coefficients given in Sec. (4.4) of Yagi and Yunes (2017) to compute the posterior for the radius and the mass of each binary component. Reference Yagi and Yunes (2017) reports a maximum relative error in the relation when compared to a large set of EOS models. We assume that the relative error is constant across the parameter space and distributed according to a zero-mean Gaussian with a standard deviation of and marginalize over it. We verified that our results are not sensitive to this choice of error estimate by comparing to the more conservative choice of a uniform distribution in .
ii.4 Parameterized EOS
Instead of sampling macroscopic EOS-related parameters such as tidal deformabilities, one may instead sample the defining function of the EOS directly. A number of parametrizations of different degrees of complexity and fidelity to realistic EOS models have been proposed (see Lindblom (2018) for a review), and here we employ the spectral parametrization constructed and validated in Lindblom (2010); Lindblom and Indik (2012, 2014). This parameterization expresses the logarithm of the adiabatic index of the EOS , as a polynomial of the pressure , where are the free EOS parameters. The adiabatic index is then used to compute the energy and rest-mass density , which are inverted to give the EOS. The parameterized high-density EOS is then stitched to the SLy EOS Douchin and Haensel (2001) below about half the nuclear saturation density. This is chosen because such low densities do not significantly impact the global properties of the NS Lattimer and Prakash (2001). Though use of a specific parametrization makes our results model-dependent, we have checked that they are consistent with another common EOS parametrization, the piecewise polytropic one Read et al. (2009); Raaijmakers et al. (2018), as also found in Carney et al. (2018).
In this analysis, we follow the methodology detailed in Carney et al. (2018), developed from the work of Lackey and Wade (2015), to sample directly in an EOS parameter space. We sample uniformly in all EOS parameters within the following ranges: , , , and and additionally impose that the adiabatic index . This choice of prior ranges for the EOS parameters was chosen such that our parametrization encompasses a wide range of candidate EOSs Lindblom (2010). Then for each sample, the four EOS parameters and the masses are mapped to a pair through the Tolman-Oppenheimer-Volkoff (TOV) equations describing the equilibrium configuration of a spherical star Shapiro and Teukolsky (1983). The two tidal deformabilities are then used to compute the waveform template.
Sampling directly in the EOS parameter space allows for certain prior constraints to be conveniently incorporated in the analysis. In our analysis, we impose the following criteria on all EOS and mass samples: (i) causality, the speed of sound in the NS must be less than the speed of light (plus 10% to allow for imperfect parameterization) up to the central pressure of the heaviest star supported by the EOS; (ii) internal consistency, the EOS must support the proposed masses of each component; and (iii) observational consistency, the EOS must have a maximum mass at least as high as previously observed NS masses, specifically . Another condition the EOS must obey is that of thermodynamic stability; the EOS must be monotonically increasing (). This condition is built into the parametrization Lindblom (2010), so we do not need to explicitly impose it.
We begin by demonstrating the improvement in the measurement of the tidal deformability parameters due to imposing a common but unknown EOS for the two NSs. In Fig. 1 we show the marginalized joint posterior PDF for the individual tidal deformabilities. We show results from our analysis using the relation in green and the parametrized EOS without a maximum mass constraint in blue. These are compared to results from Abbott et al. (2018b), where the two tidal deformability parameters are sampled independently, in orange. The shaded region marks the region that is naturally excluded when a common realistic EOS is assumed, but is not excluded from the analysis of Abbott et al. (2018b). In both cases imposing a common EOS leads to a smaller uncertainty in the tidal deformability measurement. The area of the 90% credible region for the – posterior shrinks by a factor of , which is consistent with the results of Chatziioannou et al. (2018) for soft EOSs and NSs with similar masses. The tidal deformability of a NS can be estimated through a linear expansion of around as in Del Pozzo et al. (2013); Agathos et al. (2015); Abbott et al. (2017a) to be at the level when a common EOS is imposed (here and throughout this paper we quote symmetric credible intervals). Our results suggest that “soft” EOSs such as APR4, which predict smaller values of the tidal deformability parameter, are favored over “stiff” EOSs such as H4 or MS1, which predict larger values of the tidal deformability parameter and lie outside the credible region.
We next explore what inferences we can make about the structure of NSs. We do this using the spectral EOS parameterization described above in combination with the requirement that the EOS must support NSs up to at least , a conservative estimate based on the heaviest known pulsar Antoniadis et al. (2013). From this we obtain a posterior for the NS interior pressure as a function of rest-mass density. The result is shown in Fig. 2, along with predictions of the pressure-density relationship from various EOS models. The pressure posterior is shifted from the 90% credible prior region (marked by the orange lines) and towards the soft floor of the parameterized family of EOS. This means that the posterior is indicating more support for softer EOS than the prior. The vertical lines denote the nuclear saturation density and two more density values that are known to approximately correlate with bulk macroscopic properties of NSs Özel and Freire (2016). The pressure at twice (six times) the nuclear saturation density is measured to be () at the 90% level.
The pressure posterior appears to show minor signs of a bend above a density of . Evidence of such behavior at high densities would be an indication of extra degrees of freedom, though this is not an outcome of the GW data alone. Indeed the horizontal lines denote the 90% intervals for the central pressure of the two stars, suggesting that our data are not informative for pressures above that. The bend is an outcome of two competing effects: the GW data point toward a lower pressure, while the requirement that the EOS supports masses above demands a high pressure at large densities. The result is a precise pressure estimate at around and a broadening above that, giving the impression of a bend in the pressure. We have verified that the bend is absent if we remove the maximum mass constraint from our analysis.
Finally we place constraints in the 2-dimensional parameter space of the NS mass and areal radius for each binary component. This posterior is shown in Fig. 3. The left panel is obtained by first using the relation to obtain tidal deformability samples assuming a common EOS and then using the – relation to compute the NS radii. The right panel is computed by integrating the TOV equation to compute the radius for each sample in the spectral EOS parametrization after imposing a maximum mass of at least . At the 90% level, the radii of the two NSs are km and km from the left panel and km and km from the right panel.
The difference between the two radii estimates is mainly due to different physical information included in each analysis. The EOS-insensitive-relations analysis (left panel) is based on GW data alone, while the parametrized-EOS analysis (right panel) imposes an additional observational constraint, namely that the EOS must support NSs of at least . This has a large effect on the radii priors as shown in the 1-dimensional plots of Fig. 3, since small radii are typically predicted by soft EOSs, which cannot support large NS masses. In the case of EOS-insensitive relations (left panel), the prior allows for smaller values of the radius than in the parametrized-EOS case (right panel), something that is reflected in the posteriors since the GW data alone cannot rule out radii below km. Therefore the lower radius limit in the EOS-insensitive-relations analysis is determined by the GW measurement, while in the case of the parametrized-EOS analysis it is determined by the mass of the heaviest observed pulsar and its implications for NS radii Antoniadis et al. (2013). Additionally, we verified that the parametrized-EOS analysis without the maximum mass constraint leads to similar results to the EOS-insensitive-relations analysis.
To quantify the improvement from assuming that both NSs obey the same EOS, we apply the – relation to tidal deformability samples calculated without assuming the relation (the orange posterior of Fig. 1) and obtain km and km at the 90% level. This suggests that imposing a common EOS for the two binary components leads to a reduction of the 90% credible interval width for the radius measurement of almost a factor of two from km to km.
In this letter, we complement our analysis of the tidal effects of GW170817 in Abbott et al. (2018b) with a targeted analysis that assumes astrophysically plausible NS spins and tidal parameters, as well as the same EOS for both NSs. This additional prior information enables us to measure NS radii with an uncertainty less than km if consistency with observed pulsar masses is enforced, and km using GW data alone at the 90% credible level. Simultaneously, the pressure at twice the nuclear saturation density is measured to be . Our results are consistent with X-ray binary observations (e.g. Steiner et al. (2010); Özel and Freire (2016); Steiner et al. (2018); Nattila et al. (2017)) and suggest that NS radii are not large. Additionally, our results can be compared to tidal inference based on the electromagnetic emission of GW170817 Radice et al. (2018); Coughlin et al. (2018).
Our results are comparable and consistent with studies that use the tidal measurement from Abbott et al. (2017a) to obtain bounds on NS radii. Using our bound of (the only tidal parameter in Abbott et al. (2017a), which assumed a common EOS for both NSs) and different EOS parametrizations, several studies found km Annala et al. (2018); Zhou et al. (2018); Fattoyev et al. (2018); Most et al. (2018). Reference Raithel et al. (2018) arrives at a similar conclusion using our constraint Abbott et al. (2017a) (though see Abbott et al. (2018b) for an amended bound) and the observation that is almost insensitive to the binary mass ratio Wade et al. (2014). Our improved estimate of , and km and km for the EOS-insensitive-relation analysis is roughly consistent with these estimates (see for example Fig. 1 of Annala et al. (2018) and Fattoyev et al. (2018)). If we additionally enforce the heaviest observed pulsar to be supported by placing direct constraints on the EOS parameter space, we get further improvement in the radius measurement, with km and km.
A recent analysis of the GW170817 data was performed in De et al. De et al. (2018) using the TaylorF2 model, imposing that the two NSs have the same radii which, under the additional assumption that (an alternative to the – relation used here Yagi and Yunes (2017)), directly relates the two tidal deformabilities as . De et al. constrain the common NS radius to a 90% credible interval , corresponding to a width of km, which is wider than the uncertainties on radii presented in this paper by a factor of about two. There are differences in several details of the set-up of the two analyses (most notably, frequency range, data calibration, the noise PSD estimation, waveform model, parameter priors, assumed relations between radii and s and treatment of corresponding uncertainties), each of which may be responsible for part of the observed discrepancies. The analysis of De et al. reproduces the initial tidal deformability results of Abbott et al. (2017a), but improvements detailed in Abbott et al. (2018b) and used in this work improved our tidal constraints by -%. Here, in contrast to De et al, we found that enforcing a common EOS additionally restricts the recovered tidal parameters, as shown in Fig 1. We note, however, that while our resulting posteriors for the two NS radii are similar to each other, a fraction of the posterior samples gives pairs with significantly different NS radii, up to km. Therefore, the De et al. analysis makes considerably different assumptions when enforcing a common EOS than us.
Our results, and specifically the lower radius limit, do not constitute observational proof of tidal effects in GW170817, as our analysis has explicitly assumed that the coalescing bodies were NSs both in terms of their spins and tidal deformabilities. In particular, the spins are restricted to small values typical for galactic NSs in binaries, and the tidal deformabilites are calculated consistently assuming a common typical NS EoS. Moreover, the – map diverges as approaches zero (BH), and therefore the lower bounds obtained for the radii do not imply lower bounds on the tidal deformabilities. Meanwhile, the analysis of Abbott et al. (2018b) assumes independent tidal parameters and finds a lower bound on only under the small-spin assumption but not if spins larger than are allowed.
The detection of GW170817 has opened new avenues in astrophysics and in the study of matter at conditions currently unattainable in terrestrial laboratories. As the network of GW observatories expands and improves in sensitivity, we expect many more observations of BNS mergers Abbott et al. (2018a). Each new observation will yield additional information about the properties of NSs, and the increasing precision of our measurements will simultaneously raise new challenges. As statistical uncertainties shrink, systematic uncertainties that are naturally introduced by our models and the underlying assumptions of our methods may begin to dominate. Improved waveform models and data analysis techniques are an area of active research for the GW community, and will be required to achieve our most complete understanding of these extreme systems.
The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Conselleria d’Educació i Universitat del Govern de les Illes Balears, the Conselleria d’Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources.
Data associated with the figures in this article, including posterior samples generated using the PhenomPNRT model, can be found at dcc.ligo.org/LIGO-P1800115/public. The GW strain data for this event are available at the LIGO Open Science Center LIGO Open Science Center (LOSC) (2017). This article has been assigned the document number LIGO-P1800115.
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