Optimal Search for an Astrophysical Gravitational-Wave Background

Roughly every 2–10 min, a pair of stellar-mass black holes merge somewhere in the Universe. A small fraction of these mergers are detected as individually resolvable gravitational-wave events by advanced detectors such as LIGO and Virgo. The rest contribute to a stochastic background. We derive the statistically optimal search strategy (producing minimum credible intervals) for a background of unresolved binaries. Our method applies Bayesian parameter estimation to all available data. Using Monte Carlo simulations, we demonstrate that the search is both “safe” and effective: it is not fooled by instrumental artifacts such as glitches and it recovers simulated stochastic signals without bias. Given realistic assumptions, we estimate that the search can detect the binary black hole background with about 1 day of design sensitivity data versus $\approx 40$ months using the traditional cross-correlation search. This framework independently constrains the merger rate and black hole mass distribution, breaking a degeneracy present in the cross-correlation approach. The search provides a unified framework for population studies of compact binaries, which is cast in terms of hyperparameter estimation. We discuss a number of extensions and generalizations, including application to other sources (such as binary neutron stars and continuous-wave sources), simultaneous estimation of a continuous Gaussian background, and applications to pulsar timing.

Popular Summary

Over the past few years, astronomy has been revolutionized by the detection of gravitational waves—ripples in the fabric of spacetime—created by merging black holes and neutron stars. However, for every gravitational-wave event that is detected, there are many more that are too far away to be seen with current observatories. We think that every few minutes, a pair of black holes merges somewhere in the Universe. All these mergers contribute to a background hum of gravitational waves, which researchers have sought to detect for years. We have come up with a new, more sensitive way of searching for this background, which may allow researchers to detect it much sooner than previously thought.

We derive an optimal search strategy based on Bayesian parameter estimation. Using simulations, we show that our approach is robust, that it is not fooled by instrumental glitches, and that it can reliably recover simulated signals. When current observatories reach their designed sensitivity, this technique should be able to detect the gravitational-wave background with about one day of data—as opposed to roughly 40 months of data using traditional search methods. As a bonus, this search strategy reveals details about the population of binary black holes. It may eventually be applied to other sources of gravitational waves such as merging neutron stars.

Measuring the gravitational-wave background will allow us to study populations of black holes at vast distances. Someday, the technique may even enable us to see gravitational waves from the big bang, hidden behind gravitational waves from black holes and neutron stars.

Figure 1

Histogram of the coherent network matched-filter signal-to-noise ratio [Eq. (31)] for a data set containing simulated signals plus Gaussian noise. The network signal-to-noise ratio is calculated by maximizing over all astrophysical parameters. This is why the distribution falls off sharply near ${\rho }_{\text{network}}=1$; one of the many available templates tends to produce a reasonable fit to the data such that ${\rho }_{\text{network}}\gtrsim 1$, even for weak signals. The signal distribution is generated from a population of mergers uniform in volume between (0.50 Gpc, 5 Gpc). Gold-plated detections with ${\rho }_{\text{network}}\ge 12$ are excluded.

Figure 2

Duty cycle posteriors $p(\xi |h)$ [Eq. (11)] for Monte Carlo data sets. The orange data are pure Gaussian noise, the blue data are a population of subthreshold binary black hole events added to Gaussian noise, and the green data correspond to a mixture with a true duty cycle of ${\xi }_{\mathrm{true}}=0.05$. The fact that each posterior peaks at the appropriate value shows that the method is safe, effective, and unbiased. (a) Safe and effective duty cycle posteriors. (b) Unbiased duty cycle posterior for Gaussian noise containing software injections with a duty cycle ${\xi }_{\mathrm{true}}=0.05$. The vertical line shows the true duty cycle.

Figure 3

Probability density functions for signal and noise evidences. (a) Fit for the signal data set and (b) fit for the noise data set. The horizontal axis is the log signal evidence while the vertical axis is the log noise evidence.

Figure 4

Contours of average lnBF as a function of the simulated true duty cycle ${\xi }_{\mathrm{true}}$ and number of 4-s segments $N_{\mathrm{segs}}$. The blue contour corresponds to $⟨\ln \mathrm{BF}⟩=8$, where an astrophysical background is detectable. The color bar saturates at $\ln \mathrm{BF}=20$. The horizontal and vertical lines correspond to ${\xi }_{\mathrm{true}}=4\times {10}^{-4}$ and $N_{\mathrm{segs}}=17500$, respectively.

Figure 5

Biased duty cycle posterior, generated with O1 background data and computed using the Gaussian likelihood function, Eq. (5).

Figure 6

Probability tree for Eq. (34) assuming a two-detector network. The left-hand branches correspond to “yes” and the right-hand branches to “no.” The probability of each branch is labeled accordingly. The “leaves” at the base of the tree show the evidence associated with each path.

Figure 7

Duty cycle posteriors computing using the “glitchy” likelihood function Eq. (35). (a) Safe and effective duty cycle posteriors for O1 background. (b) Unbiased duty cycle posterior for O1 background containing software injections with a duty cycle ${\xi }_{\mathrm{true}}=0.09$. The vertical line shows the true duty cycle.

Figure 8

A comparison of likelihood and marginalized likelihood functions for stochastic power spectral density $S_h$. We carry out a numerical experiment with toy-model data. We plot likelihood functions $\mathcal{L}(s|S_h)$ using cross- and autopower [Eq. (47), “Unmarginalized,” green], using cross-power only [Eq. (50), “Cross correlation,” hatched purple], and using cross- and autopower, but marginalizing over large uncertainty in the autopower [Eq. (52), “Marginalized,” hatched red]. The purple is nearly indistinguishable from the red. The green unmarginalized distribution is narrower, but it is also biased because it does not take into account uncertainty in $P$. (If we do not include a systematic error in $P$, the posterior peaks in the correct place.) The horizontal axis is strain power in arbitrary units and the vertical axis is the likelihood. The black vertical line indicates the injected value.

Figure 9

(a) Amplitude spectral density $[S_h(f){]}^{1/2}$. Blue is for the data set used in our mock data set ($z<0.77$). The drop at 20 Hz is an artifact from our cutoff frequency. The orange curve is the LIGO design sensitivity noise curve. (b) Energy density ${\mathrm{\Omega }}_{\mathrm{gw}}(f)$. Solid blue is for the data set used in our mock data set ($z<0.77$) while dashed red is the same as solid blue, but including binaries out to arbitrarily high redshifts. The dashed red background is 3 times higher than the solid blue. The black curve is the $1\sigma$ power-law sensitivity curve [61], indicating the sensitivity of the stochastic cross-correlation search assuming a LIGO Hanford-Livingston network operating at design sensitivity for 1 year. Since the red dashed curve is below the black sensitivity curve, the background is not detectable with cross-correlation after 1 year.