Rapid Bayesian position reconstruction for gravitational-wave transients

Within the next few years, Advanced LIGO and Virgo should detect gravitational waves from binary neutron star and neutron star-black hole mergers. These sources are also predicted to power a broad array of electromagnetic transients. Because the electromagnetic signatures can be faint and fade rapidly, observing them hinges on rapidly inferring the sky location from the gravitational-wave observations. Markov chain Monte Carlo methods for gravitational-wave parameter estimation can take hours or more. We introduce BAYESTAR, a rapid, Bayesian, non-Markov chain Monte Carlo sky localization algorithm that takes just seconds to produce probability sky maps that are comparable in accuracy to the full analysis. Prompt localizations from BAYESTAR will make it possible to search electromagnetic counterparts of compact binary mergers.

Figure 1

The autocorrelation likelihood for a $(1.4,1.4)M_{\odot }$ binary as observed by four detector configurations: from top to bottom, the final sensitivity achieved by the LIGO Hanford, LIGO Livingston, and Virgo detectors in their “initial” configuration and the final Advanced LIGO design sensitivity. The left panels show the noise amplitude spectral densities. The middle panels show the absolute value of the autocorrelation function. The right panel shows the phase-marginalized autocorrelation likelihood for $\mathrm{S}/\mathrm{N}=1$, 2, 4, and 8. In the right panel, the time scale is normalized by the S/N so that one can see that as the S/N increases, a central parabola is approached (the logarithm of a Gaussian distribution with standard deviation given by the Fisher matrix).

Figure 2

CRLB on root mean square timing uncertainty and phase error, using the likelihood for the full GW data [Eq. (17); dashed diagonal line] or the autocorrelation likelihood [Eq. (30); solid lines] with a selection of arrival time priors.

Figure 3

Ratio between Fisher matrix elements (solid: , dashed: , dotted: ) for the autocorrelation likelihood and the full GW data. Colors correspond to different arrival time priors as in Fig. 2.

Figure 4

Relative error in the BAYESTAR integration scheme as a function of the number of Gaussian quadrature nodes. The two panels describe (a) the integral over the polarization angle $\psi$ and (b) the integral over the inclination angle $\iota$.

Figure 5

Partition of the parameter space of the distance integral into three regions for (bi)cubic interpolation.

Figure 6

Illustration of initial subdivisions for the distance integration scheme. The distance increases from left to right. In the color version, the left-hand tail, the left- and right-hand sides of the maximum likelihood peak, and the right-hand tail are colored cyan, red, green, and blue, respectively.

Figure 7

Relative error in the BAYESTAR distance integral interpolation scheme as a function of the size of the grid.

Figure 8

Illustration of the BAYESTAR adaptive HEALPix sampling scheme.

Figure 9

An example multiresolution HEALPix mesh arising from the BAYESTAR sampling scheme (plotted in a cylindrical projection). This is event 18951 from Ref. [27].

Figure 10

Cumulative histograms of sky area, broken down by observing run and detector network. The plots in the left column show the 90% credible area, and the plots in the right column show the searched area. From top bottom, the rows refer to the following observing scenarios/network configurations: 2015/HL, 2016/HL, 2016/HV, 2016/LV, and 2016/HLV. The shaded regions represent the 95% confidence bounds. The magenta lines represent BAYESTAR and the blue lines LALINFERENCE. Where relevant, dotted lines show all events in the given network configuration and solid lines show only events for which the matched-filter pipeline triggered on all operating detectors. Note that statistically significant differences in areas between the BAYESTAR and LALINFERENCE localizations occur only for events that were below the detection threshold in one or more detectors.

Figure 11

$P-P$ plots for BAYESTAR and LALINFERENCE localizations in the 2015 and 2016 configurations. The gray lozenge around the diagonal is a target 95% confidence band derived from a binomial distribution.

Figure 12

Violin plot of BAYESTAR run times as the number of OpenMP threads is varied from 1 to 32. The 2015 scenario is shown in red and the 2016 scenario in blue.