Searching for anisotropy in the distribution of binary black hole mergers

The standard model of cosmology is underpinned by the assumption of the statistical isotropy of the Universe. Observations of the cosmic microwave background, galaxy distributions, and supernovae, among other media, support the assumption of isotropy at scales $\gtrsim 100\mathrm{Mpc}$. The recent detections of gravitational waves from merging stellar-mass binary black holes provide a new probe of anisotropy; complementary and independent of all other probes of the matter distribution in the Universe. We present an analysis using a spherical harmonic model to determine the level of anisotropy in the first LIGO/Virgo transient catalog. We find that the ten binary black hole mergers within the first transient catalog are consistent with an isotropic distribution. We carry out a study of simulated events to assess the prospects for future probes of anisotropy. Within a single year of operation, third-generation gravitational-wave observatories will probe anisotropies with an angular scale of $\sim 36°$ at the level of $\lesssim 0.1\%$.

Figure 1

Probability of detection for a number of different scenarios. (a) Detection probability as a function of sky position at 10:10:20 14th September, 2015. (b) Detection probability over the entire first observing run. (c) Detection probability over the first and second observing runs.

Figure 2

Posterior probability distribution showing the 95% credible intervals (left) and posterior predictive distribution (right) of the analysis of GWTC-1 with the ${\ell }_{\max }=2$ anisotropy model. The posterior distribution includes the real and imaginary components for the $m\ne 0$ spherical harmonic coefficients. The prior distribution (grey) is determined from the reconstruction of $a_{\ell m}$s from the uniform $b_{\ell m}$ priors. This results in notable features such as the $a_{20}$ posterior peaking away from $a_{20}=0$. These effects are suppressed with increasing ${\ell }_{\max }$. While the corner plot does not meaningfully exclude zero, the posterior predictive distribution does present regions of higher probability as Ref. [38] has also observed. However, isotropy is not excluded by this result and the Bayes factor for ${\ell }_{\max }=2$ anisotropy does not prefer either model. The white contours in the posterior predictive distribution correspond to the 95% confidence intervals for the locations of all BBH merger events in GWTC-1.

Figure 3

Injected binary black hole merger sky distribution (left) and the associated recovered posterior predictive distribution (right), for (a) an isotropic universe, (b) a universe with an $a_{11}/a_{00}=0.51$ dipolar anisotropy, and (c) a universe with an $a_{22}/a_{00}=0.51$ quadrupolar anisotropy. The sky distribution is recovered from 269 events drawn from the population prior. All events are drawn such that they exceed the detection threshold of the three detector network. We determine the detection probability as a function of signal position to remove this selection bias. We recover an approximately similar sky distribution to the injected distribution, demonstrating the accurate recovery of anisotropy.

Figure 4

Recovered Bayes factors as a function of the number of binary black hole events observed. The plus markers correspond to the injection study of simulated binary black hole sky positions from full parameter estimation of the source. The bounded regions correspond to the 68% confidence intervals for the log Bayes factors recovered from point source estimates. We observe the $\ln {\mathrm{BF}}_{\mathrm{iso}}^{\mathrm{ani}}$ trend for the simulated binary injections to approximately follow the point source estimates. Since isotropy is a compact subset of anisotropic models, the Bayes factor will prefer isotropy until enough events are observed. This is observed in the behavior of the $a\ell m/a_{00}=0.10$ results.

Figure 5

Inferred maximum 99.7% upper limits on the anisotropy present in the simulated point source analyses given a number of events. The maximum anisotropy is presented for five different scenarios analyzed for 100 different realizations with ${\ell }_{\max }=5$ anisotropic models. In the regime of a high number of detected events, the maximum anisotropy follows a $\sim N^{-1/2}$ relation. As the number of events increases, the maximum anisotropy measured plateaus towards the true anisotropy.

Figure 6

Posterior distribution and posterior predictive distribution for GWTC-1 using a ${\ell }_{\max }=1$ anisotropy model.

Figure 7

Posterior distribution and posterior predictive distribution for GWTC-1 using a ${\ell }_{\max }=3$ anisotropy model.

Figure 8

Posterior distribution and posterior predictive distribution for GWTC-1 using a ${\ell }_{\max }=4$ anisotropy model.

Figure 9

Posterior distribution and posterior predictive distribution for GWTC-1 using a ${\ell }_{\max }=5$ anisotropy model.