Reconstructing phenomenological distributions of compact binaries via gravitational wave observations

Gravitational wave (GW) measurements will provide insight into the population of coalescing compact binaries throughout the universe. We describe and demonstrate a flexible parametric method to infer the event rate as a function of compact binary parameters, accounting for Poisson error and selection biases. Using synthetic data based on projections for LIGO and Virgo’s third observing run (O3), we discuss how well GW measurements could constrain the mass and spin distribution of coalescing neutron stars and black holes (BHs) in the near future, within the context of several phenomenological models described in this work. We demonstrate that only a few tens of events can enable astrophysically significant constraints on the spin magnitude and orientation distribution of BHs in merging binaries. We discuss how astrophysical priors or other measurements can inform the interpretation of future measurements. Using publicly available results, we estimate the event rate versus mass for binary black holes (BBHs). To connect to previously published work, we provide estimates including reported O2 BBH candidates, making several unwarranted but simplifying assumptions for the sensitivity of the network and completeness of the reported set of events. Consistent with prior work, we find BHs in binaries likely have low natal spin. With available results and a population favoring low spin, we cannot presently constrain the typical misalignments of the binary black hole population. All of the tools described in this work are publicly available and ready-to-use to interpret real or synthetic LIGO data, and to synthesize projected data from future observing runs.

https://git.ligo.org/daniel.wysocki/bayesian-parametric-population-models/ and https://git.ligo.org/daniel.wysocki/synthetic-PE-posteriors

Figure 1

Estimated sensitive comoving volume ($V$) versus mass and spin. Left: Sensitive comoving volume $V$ at O1 sensitivity for nonspinning BBHs, in cubic giga-parsecs. Right: Sensitive comoving volume for equal-mass, equal-spin, nonprecessing BBHs, relative to the zero-spin case. Note that $V$ is strictly increased (decreased) if ${\chi }_{i,z}>0$ ($<0$), with higher mass making the effect more pronounced.

Figure 2

Source information for our synthetic BNS population: For each synthetic signal used in the BNS population reconstruction calculation described in Sec. 3a, these two panels show the true injected source-frame parameters (as crosses) and posterior distributions (contours of their 95% highest posterior density regions). Each color corresponds to a different source. Source parameters have been inferred using full Bayesian parameter inference via RIFT, as described in the text.

Figure 3

Recovered properties of NS mass and spin distribution: For the synthetic population of BNS sources illustrated in Fig. 2, this figure shows the recovered mass distribution (top figure) and spin distribution parameters (bottom figure) derived using the Gaussian mass and $\beta$-distribution spin model described in the text. The solid line indicates the median distribution; the shaded regions indicate the 68% and 95% credible intervals. Red dashed lines denote the true underlying distribution. In the case of spin, note that the truth is a delta function at zero, so it would require an infinite number of detections to fall within the constraints on this plot.

Figure 4

Population measurement enables sharper constraints on NS tides: Cumulative posterior distribution of $\tilde{\mathrm{\Lambda }}$ for one of the synthetic sources in our BNS population model. Blue curve shows a single-event analysis, not exploiting information about the mass and spin distribution from other events; red curve shows an analysis based on Eq. (16) that employs our best estimate for the underlying mass and spin distribution, as constrained from the population of events in our BNS synthetic sample.

Figure 5

Source information for our synthetic BBH population: For each synthetic signal used in the BH population reconstruction calculation described in Sec. 3b, these two panels show the true injected source-frame parameters (as crosses) and posterior distributions (contours of their 95% highest posterior density regions). Each color corresponds to a different source.

Figure 6

Inferred merger rate versus mass: This figure shows how our estimated merger rate versus mass compares with the known distribution used to generate our synthetic source population. For a more thorough statistical test, see the $P–P$ plots in Appendix pp4. Top left: This group of figures represents the one- and two-dimensional marginal posterior distributions for $\mathcal{R}$, ${\alpha }_m$, and $m_{\max }$, with the true values overlaid as blue crosshairs. Top right: This group of figures represents the one- and two-dimensional marginal posterior distributions for $m_1^{*}$, $\mathcal{R}p(m_1^{*})$, and $\mathcal{R}p(15{\mathrm{M}}_{\odot })$, with the true values overlaid as blue crosshairs. Bottom left: In this figure, the red dashed line shows the characteristic merger rate associated with a given mass scale [$m_1\mathcal{R}p(m_1)$] versus primary mass $m_1$. The black line shows the median inferred value, and the two gray shaded regions show the symmetric 68% and 95% credible regions. Bottom right: The solid lines in this figure show our posterior predictive distribution $p(m_i|D)$: the best estimate for the probability of a future event being detected having masses $m_i$. In this figure, blue and green correspond to the primary and secondary masses. For comparison, the dotted lines show the true astrophysical distribution.

Figure 7

Recovering the true mass ratio and ${\chi }_{\mathrm{eff}}$ distribution: A comparison between the underlying truth (black solid contours) and the inferred posterior predictive (red dashed contours) for the $q$, ${\chi }_{\mathrm{eff}}$ marginal distribution. The inner (outer) contour for each denotes the 50% (90%) highest probability density credible region.

Figure 8

Inferred spin distribution derived from synthetic BBH observations: The top panel shows our inferences about the total BH spin; the bottom panel shows our inferences about BH spin-orbit misalignment. In both panels, the red dashed lines show the underlying distribution, while the black solid lines and shaded regions show the median recovered parameter distribution. To a first approximation, the constraints on spin magnitude and misalignment are as needed for the population model to reproduce the mass and ${\chi }_{\mathrm{eff}}$ distribution of the underlying population as shown in Fig. 7.

Figure 9

Inferences about astrophysical binary BH mass distribution: Inferences about the merger rate versus mass of coalescing BH-BH binaries, using only O1 observations (dashed orange) and using O1 and reported O2 observations (solid purple), for simplicity assuming the latter represents a comprehensive and fair sample. We apply an asterisk ($\mathrm{O}{2}^{*}$) to all O2 results, to highlight the nonfinal sample, simplified sensitivity model $VT$, and mocked-up posteriors used in this proof-of-concept analysis. The panels in this figure follow the format of Fig. 6 for representing one- and two-dimensional marginal posterior distributions.

Figure 10

Inferences about astrophysical binary BH spin distribution. Left: Our best estimates for the binary BH spin magnitude distribution (PPD) based on O1 (dashed orange) and O2 (solid purple) observations. We apply an asterisk ($\mathrm{O}{2}^{*}$) to all O2 results, to highlight the nonfinal sample, simplified sensitivity model $VT$, and mocked-up posteriors used in this proof-of-concept analysis. Due to the low characteristic spin and within the context of the information used in this analysis, these observations remain uninformative about BH spin-orbit orientations. Right: Our best estimates for the binary BH spin distribution, as expressed using our model hyperparameters, for O1 and O2.

Figure 11

Why ${\chi }_{\mathrm{eff}}$ measurements constrain the maximum spin: cumulative distribution function for ${\chi }_{\mathrm{eff}}$ for toy models with isotropic spins and uniform spin magnitude distributions limited by $0.1,0.2,0.3,\dots$. In the top panel, the vertical lines, corresponding to ${0.5}^{1/3}$ and ${0.5}^{1/25}$, indicate the locus of points in each cumulative distribution function we can begin to constrain with the absence of events above $X$ with 3 and 25 events, respectively. In the bottom panel, the lines have been changed to ${0.9}^{1/3}$ and ${0.9}^{1/25}$, respectively.

Figure 12

$P–P$ plots for hyperparameter recovery. Top (bottom) panel shows the $P–P$ plot for population (single synthetic event) inferences.