Distinguishing double neutron star from neutron star-black hole binary populations with gravitational wave observations

Gravitational waves from the merger of two neutron stars cannot be easily distinguished from those produced by a comparable-mass mixed binary in which one of the companions is a black hole. Low-mass black holes are interesting because they could form in the aftermath of the coalescence of two neutron stars, from the collapse of massive stars, from matter overdensities in the primordial Universe, or as the outcome of the interaction between neutron stars and dark matter. Gravitational waves carry the imprint of the internal composition of neutron stars via the so-called tidal deformability parameter, which depends on the neutron star equation of state and is equal to zero for black holes. We present a new data analysis strategy powered by Bayesian inference and machine learning to identify mixed binaries, hence low-mass black holes, using the distribution of the tidal deformability parameter inferred from gravitational-wave observations.

Figure 1

Left panel: Mass-radius relations for selected EOS models. Left to right: APR4 [89] (thick green), SLY4 [97] (dashed blue), APR3 [89] (dashed red), MPA1 [98] (dashed orange), ALF2 [90] (thick brown), and H4 [99] (dashed cyan). The gray lines represent the 90% confidence regions for the companion masses and their radii for the LIGO/Virgo event GW170817, assuming a parametrized EOS and imposing a lower limit on the maximum mass of $1.97M_{\odot }$ (cf. Fig. 3 of Ref. [13]). Right panel: Dimensionless tidal deformability $\mathrm{\Lambda }$ (on a log scale) as a function of the NS mass for the same EOS models considered in the left panel.

Figure 2

Bottom: Conditional probability distributions $\mathcal{P}(\tilde{\mathrm{\Lambda }}|i)$, where $i=\mathtt{BHNS}-\mathtt{DG}$ (solid red), BHNS-U (dashed red), BNS-DG (solid blue) or BNS-U (dashed blue), for EOS ALF2 (left panel) and APR4 (right panel). Vertical lines identify the 68% confidence intervals for the three distributions (cf. Table 1). Top: ${\log }_{10}(r)$, where $r$ is the probability ratio defined in Eq. (3) for different combinations of mass distribution models, as indicated in the legend. When this ratio is above the shaded region the binary is likely to be a BHNS. Below the shaded region it is likely to be a BNS. In the grey shaded region, the binary’s origin is uncertain.

Figure 3

Violin plot showing the probability distribution of $\mathcal{P}(\tilde{\mathrm{\Lambda }})$ defined in Eq. (4) for the ALF2 (left) and APR4 (right) EOS and for selected values of $\mathcal{F}=[0,0.1,0.2,\dots 1]$. The population is dominated by BNSs when $\mathcal{F}\to 0$, and by BHNSs when $\mathcal{F}\to 1$.

Figure 4

Reconstructed probability density functions of the parameter $\mathcal{F}$ assuming $N_{\mathrm{obs}}=10$ observations (solid lines) or $N_{\mathrm{obs}}=60$ observations (dashed lines). The left panel refers to a second-generation detector network (HLV), and the right panel to a third-generation network composed of two CEs and ET. We focus on three extreme cases: a pure BNS population ($\mathcal{F}=0$, blue), a “perfectly mixed” population ($\mathcal{F}=0.5$, green), and a pure BHNS population ($\mathcal{F}=1$, red). For concreteness here we focus on EOS APR4 and we compare the mass distribution models BHNS-U and BNS-DG, but results are qualitatively similar for other EOS models and mass distributions.

Figure 5

Reconstructed probability density functions of the parameter $\mathcal{F}$ assuming $N_{\mathrm{obs}}=60$ observations with a second-generation detector network (HLV). Here we look at the most pessimistic scenario where we inject an ALF2 population into a Bayesian framework trained with the APR4 EOS, and we focus on the two extreme cases: a pure BNS-DG population ($\mathcal{F}=0$, blue) and a pure BHNS-U population ($\mathcal{F}=1$, red).

Figure 6

Probability distribution of $\mathcal{F}$ obtained by injecting 10 observations where tidal disruption occurs, as determined by imposing $q<Q_D(\mathcal{C},\mathcal{\chi })$ (cf. Eq. (2) of Ref. [25]). We simulate our binaries using either the IMRPhenomPv2_NRTidal (red) or the PhenomNSBH (black) waveform models, using the ALF2 EOS and the $\mathrm{CE}+\mathrm{ET}$ detector network. Allowing for tidal disruption does not significantly change the distributions inferred in the right panel of Fig. 4.