BNS mergers and inclination angles

Multimessenger detections of binary neutron star mergers and their inclination angle distributions

M. Saleem
Chennai Mathematical Institute, Siruseri, 603103 Tamilnadu.
Accepted XXX. Received YYY; in original form ZZZ

For binary neutron star (BNS) mergers detectable from the nearby Universe, the distribution of orbital inclination angles (), as measured from gravitational wave (GW) observations follows Schutz distribution (Schutz, 2011). If the BNSs produce short GRBs which are independently detected, they can trigger BNS detections and then the resulting inclination angle distribution would also depend on the jet structure of the SGRBs and hence deviate from the Schutz distribution. We demonstrate this by obtaining the inclination angle distribution by assuming all the BNS mergers to have SGRB counterparts with structured jets of Gaussian-like profiles, with the half opening angle of the jet core and the total ray energy in the rest frame erg, consistent with what was inferred for GRB170817A. With the new distribution, we find that the detection probability of low inclination angle events significantly increases depending on the GW network configuration/sensitivity. Assuming a 50% duty cycle for GW detectors and a 60% sky coverage for Fermi GBM, we find that almost 52% of the total BNS detections in O3 with LHV network will have coincident prompt -ray detections. With the five-detector network LHVKI with designed sensitivity and with Aplus sensitivity, the total BNS detections increases, however the coincident detection fraction decreases as 28% and as 19%.

Gravitational waves – keyword2 – keyword3
pubyear: 2019pagerange: Multimessenger detections of binary neutron star mergers and their inclination angle distributionsLABEL:lastpage

1 Introduction

Historic detection of a BNS merger GW170817 by LIGO/Virgo detectors (Abbott et al., 2017a) and the consequent follow up of the event in various electromagnetic (EM) frequency bands and by neutrino observatories dawned a new era in multi-messenger astronomy (Abbott et al., 2017c; Goldstein et al., 2017; Savchenko et al., 2017; Valenti et al., 2017; Margutti et al., 2018; D’Avanzo et al., 2018; Troja et al., 2018; Ruan et al., 2018; Lyman et al., 2018; Resmi et al., 2018; Lazzati et al., 2018). Multimessenger observations provide tremendous opportunities to investigate astrophysics, cosmology and fundamental physics, for example, the estimation of Hubble constant (Abbott et al., 2017b; Hotokezaka et al., 2018), testing the speed of gravity (Abbott et al., 2017d), estimating the neutron star EOS (Coughlin et al., 2018a; Coughlin et al., 2018b; Radice & Dai, 2018) are a few among them. With more GW detectors expected to be operational (Collaboration et al., 2015; Acernese et al., 2015; Aso et al., 2013; Iyer et al., 2011) with the improved sensitivities, we expect tens to hundreds of BNS mergers and tens of joint BNS-EM detections over next few years (Abbott et al., 2018b; Coward et al., 2012; Clark et al., 2015; Regimbau et al., 2015; Howell et al., 2018).

One of the common parameters which has important role in the GW detectability of BNS mergers as well as the EM detectability of their counterparts is the inclination angle of the binary which is the angle between the orbital angular momentum axis and the line of sight. The inclination angle is also known as the viewing angle of the observer, as commonly referred to in GRB literature and is a key parameter in understanding the physics of EM counterparts (Arun et al., 2014; Lamb & Kobayashi, 2017a) especially when they are observed off-axis (Granot et al., 2002; Donaghy, 2006; Lazzati et al., 2016; Kathirgamaraju et al., 2018; Eichler, 2018). For example, the fact that GRB170817A was viewed off-axis has been central in understanding the underlying jet structure models (Lamb & Kobayashi, 2018; Ioka & Nakamura, 2018; Granot et al., 2017; Resmi et al., 2018; Lazzati et al., 2018; Ghirlanda et al., 2019). The inclination angle of GW170817 was estimated independently from GW and EM observations as well as from the combined GW+EM analyses (Mandel, 2018; Finstad et al., 2018; Mooley et al., 2018). From the observation of the superluminal motion of the radio counterpart, the inclination angle of GW170817 was constrained to (Mooley et al., 2018).

The dependence of GW detectability of BNS on the inclination angle has been well formulated in literature. Bernard Schutz showed that the detected inclinations follow the probability distribution (Schutz, 2011) which applies well for low redshift events (say )111Essentially those redshifts for which the luminosity distance can be approximated to the comoving distance (Hogg, 1999).. Seto (2015) also carried out a similar and detailed study on inclination angle distributions with broadly similar conclusions. As far as the EM counterparts are concerned, due to the collimated jets and relativistic beaming (Rhoads, 1999; Sari et al., 1999; Harrison et al., 1999), their detectabilities at various frequency bands (short-GRB (SGRB) prompt ray emission, afterglows etc. ) have even stronger dependence on the inclination angle (Granot et al., 2002; Donaghy, 2006; Saleem et al., 2018). Unlike GW, the EM detectability dependence on inclination angle is only known subject to the assumption of an underlying model of jet structure (Donaghy, 2006; Salafia et al., 2015) while our knowledge about the structure is still developing. The accurate knowledge of the statistical distribution of inclination angle is important in forecasting the future multimessenger detection rates and detection scenarios.

The probability distributions given by Schutz (2011) and Seto (2015) are excellent representations for the detected BNS inclinations from an independent GW search campaign. However, in a multimessenger observing scenario, due to the BNS-SGRB association, it is possible to have SGRB triggered BNS detections which can be made at a lower detection threshold than needed for an independent GW detection. This brings in several additional BNS detections. Due to the strong inclination angle dependence of SGRB detectability, these additional detections also depend on the inclinations. Consequently, the distribution of overall (independent plus SGRB triggered ) detected BNS inclinations will be different from the ones given by (Schutz, 2011; Seto, 2015).

The subject of this work is to demonstrate these new inclination distributions and discuss their implications. We use a Monte Carlo simulated population of BNS mergers and associated SGRBs to compute the detected inclination angle distributions for independent BNS searches as well as SGRB-triggered BNS searches. We investigate how the fraction of SGRB-triggered BNS detections varies if the GW detector network sensitivity increases. We demonstrate how this will reflect in the overall BNS detection rates and the joint BNS-SGRB detection rates and estimate the fraction of on-axis to off-axis SGRBs expected to detect in the upcoming observing runs.

In section 2, we simulate the population and obtain the inclination angle distributions. In section 3, we discuss the implications and in section 4 we conclude the work and discuss the caveats.

Figure 1: Distribution of inclination angles for untriggered and total (triggered plus untriggered) detections for various network and sensitivity configurations. The curves are normalized such that the area under the solid curves are unity while the area under the dashed curves are larger by the fraction of additional detections which comes from the SGRB-triggered searches.

2 Simulated BNS-SGRB populations and Inclination angle distributions

In this section, we numerically compute the distributions of detected inclination angles of BNS mergers for multimessenger observing scenarios. By multimessenger observations, we precisely mean the GW detections of BNS mergers and the EM detections of their prompt ray emissions. EM counterparts in relatively lower frequencies such as afterglows and Kilonovae emissions are not considered (we explain the reason later). This is because the prompt emission provides the best opportunity for an EM-triggered BNS detection due to their strong temporal coincidence with BNS merger (delay at the order of seconds) while afterglow or Kilonovae emission appears with a delay of the order of hours to days (Metzger & Berger, 2012).

2.1 Simulation of BNS merger population

We simulate a population of BNS mergers and associated SGRB prompt emission counterparts. The component NS masses are distributed uniformly between and component spins are taken to be zeros as spins have little effects on BNS detectability. Sources are distributed in comoving volume with constant number density and the source orientation is taken as uniform over the polarisation sphere, ie, the polarisation angle is distributed uniformly in [0,] and the cosine of inclination () is distributed uniformly in [0,1].

For all the sources, we computed coherent network SNR (Schutz, 2011) for the three network configurations/sensitivities discussed below.

  1. O3 LHV - the three-detector LIGO-Virgo network with the best reported O2 sensitivities (Abbott et al., 2018a) which we consider as (conservative) representative O3 sensitivity.

  2. Designed LHVKI - the five-detector network including KAGRA(Aso et al., 2013) and LIGO India (Iyer et al., 2011) with all of them at their designed sensitivities (Harry, 2010; Collaboration et al., 2015; Acernese et al., 2015).

  3. Aplus LHVKI - same as case 2 but the three LIGO detectors at the Aplus sensitivities (Barsotti, 2018).

Note that for each network configuration, the sources are distributed up to a distance which is the farthest detectable distance (Horizon distance) for that configuration. In obtaining the total detected fraction of events of our population, we have assumed a 50% duty cycle for each of the detectors. In practice, this is achieved by setting to zero the SNRs of 50% of the randomly chosen sources for each detector.

2.2 SGRB counterparts with Gaussian structured jets

For each BNS in the population, its SGRB counterpart is considered to have structured jets (Rossi et al., 2002; Zhang & Meszaros, 2002; Nakar et al., 2004; Lamb & Kobayashi, 2017b) with Gaussian-like profiles. In a Gaussian structured jet (Zhang & Meszaros, 2002; Salafia et al., 2015; Resmi et al., 2018), the energy and bulk Lorentz factor in the source frame follow Gaussian-like angular variations, , where is the angle away from the jet axis and is 1- width of the Gaussian profile which is also referred to as the semi-opening angle of the core of the Gaussian jet.

It was found that the observed features of EM counterparts of GW170817 (prompt rays and afterglows) were consistent with Gaussian structured jets (Resmi et al., 2018; Lamb & Kobayashi, 2018; Howell et al., 2018) with and viewed far off-axis, ie, (Recall that the best estimate of obtained in Mooley et al. (2018) is around ). In our simulated population, we assume that all the BNS mergers have jet counterparts with properties consistent with what is inferred for GRB170817A, with . This is consistent with a recent study by Beniamini et al. (2019) which argues that most neutron star mergers result in tightly collimated successful jets. For all the sources in the population, we also assume the initial bulk Lorentz factor at the axis of the jet and erg, where is the rest frame total -ray energy.

To estimate the detectability of prompt ray emission, it also requires to compute the observable quantities such as ray flux (in units of erg cm s) or fluence (in units of erg cm) in the sensitive bandwidth of the instrument under consideration. In a most recent work, Mohan et al. (2019) has numerically computed the prompt ray fluence for Fermi GBM assuming Gaussian structured jet with parameters as described above, where the fluence is obtained for a range of viewing angles from 0 to 90, at a luminosity distance Mpc which is the inferred luminosity distance of GW170817. In order to estimate the detectability of SGRB counterparts of the BNS mergers in our population, we use this fluence data by scaling appropriately for desired distances. If the fluence obtained in Mohan et al. (2019) is denoted as , then the fluence from a source at an arbitrary distance at viewing angle can be schematically written by the scaling relation,


where the effect of redshift has been ignored which is expected to have minimal effects for the distance ranges we deal with.

2.3 Detectabilities

Given the population of BNS mergers along with their SGRB counterparts described above, We now asses their detectabilities by imposing appropriate detection thresholds on their SNR and fluence respectively. We consider the following two scenarios:

  1. BNS mergers are detected due to a GW-only search. For this case, we use an SNR threshold as the detection criterion. Henceforth, these are referred to as untriggered detections.

  2. BNS mergers are detected from the SGRB-triggered search. For this, we use as the detection criterion.

The lowered SNR threshold () for the SGRB-triggered searches follows from the fact that the presence of a detected SGRB counterpart significantly reduces the fraction of sky as well as the time window to be searched over. This significantly reduces the number of search templates which in turn reduces the false alarm rates (FAR). A low FAR allows to use a lower SNR threshold without increasing the number of false detections. This has been comprehensively discussed in a number of cases in literature (Kelley et al., 2013; Mandel et al., 2012; Baret et al., 2012; Chen & Holz, 2013; Dietz et al., 2013; Bartos & Márka, 2015; Patricelli et al., 2016; Howell et al., 2018). Our choice of thresholds with a 20% reduction for triggered cases as opposed to untriggered cases is consistent with the previous studies mentioned above (for example Chen & Holz (2013); Dietz et al. (2013)). In recent times, there were triggered searches performed on LIGO-Virgo data following the detections of short and long GRBs (Aasi et al., 2014a, b, 2013).

Whether a given BNS in the population belongs to the SGRB-triggered category or the untriggered category is determined based on whether or not the prompt ray fluence obtained by equation 1 is detectable by Fermi GBM space telescope. We use Fermi GBM 64-ms fluence threshold f erg cm (Goldstein et al., 2017). We consider 60% sky coverage (time-averaged) for Fermi GBM telescope following Burns et al. (2016). In practice, this is achieved by setting the fluence to zero for randomly picked 40% of the sources in our populations.

2.4 Distribution of detected inclinations

Applying the detectability criterion above, we obtain the following two distributions of detected BNS mergers:

  1. Untriggered BNS merger detections obtained by applying the SNR threshold .

  2. Total BNS merger detections which is the union of untriggered and triggered detections (with ).

Figure-1 shows the distribution of detected inclination angles corresponding to untriggered detections (solid curves) and total detections (dashed curves). The curves are normalized such that the area under solid curves are unity while the area under the dashed curves is larger by the fraction of additional detections which comes from the SGRB-triggered searches. The three panels correspond to the three different network configurations/sensitivities. The solid curve in the left panel is the same as the one predicted by Schutz (2011) which has been discussed above, while the solid curves in the middle and right panels slightly differ from Schutz distribution. This is because for the latter cases, the sensitivity is higher (especially in the right panel) and hence sources are detectable up to larger distances for which the redshift effects cannot be ignored as assumed in deriving Schutz distribution.

It is observed that the two curves are substantially different at lower inclinations and as the inclination increases, the difference between them decreases and join together. This implies that improvement in rates due to the SGRB-triggered detections are prominent only at lower inclination angles while there are hardly any additions at higher inclinations. This is a consequence of the fact that the prompt emission detection and hence the ‘triggering’ is possible favorably at lower inclination angles which in turn is a consequence of the narrowly collimated jets and the Lorentz boosted ray emissions (see Mohan et al. (2019) for a detailed picture of the fluence profile).

We notice that, among the three panels in figure-1, for Aplus LHVKI (rightmost), the curves join together at a lower value of inclination compared to the relatively less sensitive ones. We understand this as follows: due to the fluence dependency on , SGRBs at larger (larger than aka off-axis sources) can be detected only if they are at sufficiently close distances whereas sources with can be detected up to much larger distances (). For ‘O3 LHV’, a good fraction of detectable off-axis SGRBs will have their SNR of the BNS counterpart between 8 and 10 and hence will become SGRB-triggered detections. On the other hand, with ‘LHVKI Aplus’, due to its high sensitivity, the BNS counterparts of almost all the detectable off-axis SGRBs will be already detectable (SNR > 10) in the untriggered search. Therefore, the improvement from SGRB-triggered detections reduces at larger inclinations as the sensitivity increases. However, at lower inclinations, the SGRB can be detected up to much larger distances than the reach of second-generation GW detectors. This implies that at lower inclinations, ‘triggering’ is possible at any distance within the BNS distance reach and hence there is always a significant gain in the number of detections.

3 Detection rates and other implications

In this section, we compute the detection rates of BNS events as well as BNS-SGRB coincident events for various network configurations. Motivated by the features observed in Figure 1, we compute detection rates for different inclination ranges, particularly for (on-core) and (off-core). To compute the rates using our simulated population, we follow the method described below.

The BNS detection rates is obtained as the product of the intrinsic BNS merger rate and the detection volume ,


where represents the non-evolving BNS merger rate density (in units of ). In this work, we use the merger rate density estimated from O1 and O2 observing runs using the GstLAL search pipeline (Abbott et al., 2018a), is the detection volume of the detector network ie, the volume (in ) which our detector network is sensitive to for BNS mergers. depends on the network configuration as well as their sensitivity and the properties of the population which is assumed (for example, mass distribution). Given our simulated population, can be estimated as,


where is the number of detected sources out of the total sources which are distributed in a volume . In Table-1, we have shown the detection rates (per year) for untriggered BNS, total BNS and the joint BNS-SGRB detections in all three network configurations. Different columns correspond to the overall rates, rates for and the rates for respectively.

Case Any
Untriggered BNS 0.2- 5.0 0.0- 0.1 0.2- 4.9
Total BNS 0.3- 7.3 0.0- 0.1 0.3- 7.1
Joint BNS-SGRB 0.2- 5.6 0.0- 0.1 0.2- 5.4
Designed LHVKI
Untriggered BNS 9.2- 214.7 0.1- 3.1 9.0- 211.7
Total BNS 12.1- 282.6 0.3- 6.3 11.8- 276.3
Joint BNS-SGRB 5.3- 124.7 0.3- 6.3 5.1- 118.4
Untriggered BNS 59.9- 1402.7 0.8- 17.7 59.2- 1385.0
Total BNS 68.2- 1595.9 1.4- 34.0 66.7- 1561.9
Joint BNS-SGRB 20.8- 486.7 1.4- 34.0 19.3- 452.7
Table 1: Detection rates (per year) in three different network configurations. Three columns correspond to all the detections, detections with and , where we have fixed in our simulations. The rows corresponding to Untriggered BNS are the independent BNS detections, Total corresponds to Untriggered plus the SGRB-triggered detections while the joint BNS-SGRB corresponds to those cases in which both are simultaneously detected irrespective of whether triggered or untriggered.

We make the following observations from Table 1:

  • The fraction of total BNS detections which are coincident with its SGRB counterpart is 52% in O3 LHV configuration, 28% in Designed LHVKI, and 19% in Aplus LHVKI. The decrease in joint detection fraction at higher GW sensitivity is due to the several BNS detections at large inclinations for which the SGRB counterpart is undetected.

  • Due to the SGRB-triggered BNS detections, the overall BNS rate increases by 29% (O3 LHV), 26% (Designed LHVKI) and 8% (Aplus LHVKI). This shows that the contribution from triggered detections becomes less relevant as the sensitivity of GW network increases, or more precisely, their relevance gets restricted to the low inclination angle cases. This is a direct consequence of the previous point.

  • The number of joint detections with inclination larger than the core opening angle ie, (off-core) is several times larger than the number of joint detections with (on-core) where we have assumed . For the O3 LHV, Designed LHVKI and the Aplus LHVKI configurations considered in this paper, the off-core joint detection rates are approximately times, 21 times and 14 times more than the on-core detection rates respectively.

  • For , the joint detection rates are very close to the total BNS detection rates, for all the three networks. The small differences found in the table are due to the limited sky coverage of Fermi GBM, rather than sensitivity. For a 100% sky-coverage instrument, the joint detection rates will be exactly the total BNS detection rates when . In other words, in the era of the second-generation GW detectors, all the BNS detections with are SGRB-detectable too, unless restricted by instrument’s sky coverage.

4 Conclusion and Outlook

In this work, using simulated BNS-SGRB populations, we have demonstrated the inclination angle distributions of multimessenger BNS detections in which we consider SGRB triggered detections in addition to the independent untriggered detections. We find that the narrowly collimated nature of SGRB jet structures plays an important role in the SGRB detectability and hence in the SGRB-triggered BNS detection rates. We assumed that all BNS mergers have SGRB counterparts with Gaussian structured jets and their properties are similar to the inferred properties of GRB170817A which differs from the earlier works in the literature on SGRB-triggered detection rates (Williamson et al., 2014; Chen & Holz, 2013; Dietz et al., 2013) which assumed uniform top-hat jets.

Given that the SGRB triggered detections are influenced by the jet structure properties, one can argue that the resulting inclination angle distributions (dashed lines in figure 1) carry the signatures of jet structure. Though the structure properties such as energy, Lorentz factor, opening angle etc. can all vary from source to source, the underlying structure model (for example, the Gaussian, power-law, top-hat etc. ) may be treated as a universal property of SGRBs, at least for those which originate from a given progenitor class (say BNS). Therefore the inclination distributions from accumulated sources detected over the years can be used to distinguish between different structured jet models. In Miller et al. (2015), the authors explored a similar idea using the observed distribution of viewing angles of the SGRB’s where only the independent SGRB detections are considered. They find that around 300 GRBs are sufficient for distinguishing their structure if their viewing angles are reconstructed well. Viewing angle distributions from multimessenger observations (as shown in this work) carries the added advantage of being more accurately estimated due to the GW detection with a multi-detector network (Arun et al., 2014; Tagoshi et al., 2014; Chen et al., 2018) as obtained for GW170817 (Mandel, 2018; Finstad et al., 2018).

As mentioned before, we have not considered the X-ray, optical and radio counterparts for triggering BNS searches and detections, primarily due to their large time window for the coincidence w.r.t the merger which makes it difficult to establish a GW-EM temporal coincidence. Moreover, independent detections of such delayed counterparts are challenging too which may be attributed to the limited field of view of many present EM instruments as well as the contamination from other EM transients such as supernovae. This scenario is expected to improve in the near future when there are several current and upcoming wide-field EM instruments such as ZTF, Pan-STARRS, LSST, SKA, ATLAST, THESEUS etc. search for transients. Also once joint detections become routine as we predict in Table-1, our theoretical understanding of various afterglows or Kilonovae counterparts (lightcurves) will tremendously improve which might enable us to establish the temporal coincidence between GW and EM sources despite hours or days of delay in between. This will enable triggered GW search following the detections of orphan X-ray, optical or radio counterparts bringing more BNS detections. In short, in the era of wide field EM instruments, we expect the EM-triggered BNS detections to increase significantly compared to the present era.

5 Acknowledgements

The author is partially supported by a grant from the Infosys Foundation. The author is extremely thankful to K.G. Arun for many useful discussions and inputs over the course of this entire work. The author thanks and L. Resmi for useful discussions regarding the SGRB models and providing the fluence data, The author also thanks K. Haris, B. Sathyaprakash and N.V. Krishnendu for useful comments and inputs. This document has LIGO preprint number P1900134.


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