Probing the anisotropies of a stochastic gravitational-wave background using a network of ground-based laser interferometers

We present a maximum-likelihood analysis for estimating the angular distribution of power in an anisotropic stochastic gravitational-wave background using ground-based laser interferometers. The standard isotropic and gravitational-wave radiometer searches (optimal for point sources) are recovered as special limiting cases. The angular distribution can be decomposed with respect to any set of basis functions on the sky, and the single-baseline, cross-correlation analysis is easily extended to a network of three or more detectors—that is, to multiple baselines. A spherical-harmonic decomposition, which provides maximum-likelihood estimates of the multipole moments of the gravitational-wave sky, is described in detail. We also discuss (i) the covariance matrix of the estimators and its relationship to the detector response of a network of interferometers, (ii) a singular-value decomposition method for regularizing the deconvolution of the detector response from the measured sky map, (iii) the expected increase in sensitivity obtained by including multiple baselines, and (iv) the numerical results of this method when applied to simulated data consisting of both pointlike and diffuse sources. Comparisions between this general method and the standard isotropic and radiometer searches are given throughout, to make contact with the existing literature on stochastic background searches.

Figure 1

Eigenvalues of typical Fisher matrices ${\Gamma }_{\alpha \beta }$ for different baselines and the multibaseline detector network. For this analysis $l_{\max }=20$, corresponding to $(l_{\max }+1{)}^2=441$ total modes. For each individual baseline some of the SVD eigenmodes are (almost) null [see Sec. 4b]. The multibaseline network, however, has fewer null modes, illustrating the fact that a network of detectors acts as a natural regularizer—independent baselines tend to complement each other. The plot was produced using the simulated data described in Sec. 5.

Figure 2

Magnitude of the bias due to the SVD regularization scheme, for the case of point sources. A value of 0 implies that the expectation value of the corresponding pixel is equal to the point-source signal strength, while 1 implies that a point source at that location would not be seen.

Figure 3

Standard deviation for spherical-harmonics components, without SVD regularization. It illustrates how the multiple baselines (solid line) reduce the estimation error by natural regularization. (ML is maximum likelihood.)

Figure 4

Standard deviation for spherical-harmonics components, with the SVD regularization described in Sec. 4b. While the combination of network and SVD regularization leads to the minimum estimation error, the SVD regularization alone can significantly reduce the estimation error for a single-baseline.

Figure 5

The design power spectral densities used to simulate detector noise for the LIGO 4 km interferometers (H1 and L1) and the 3 km Virgo interferometer (V1) [51, 52].

Figure 6

Left column from top to bottom: dirty maps of no injection (just detector noise) produced with spherical-harmonics decomposition code for $l_{\max }=5$, $l_{\max }=10$, $l_{\max }=15$, $l_{\max }=20$. Right column: dirty maps with $l_{\max }=30$, with the radiometer search code, the difference between the $l_{\max }=30$ map and the radiometer map. By reading top to bottom, one can see how the spherical-harmonic dirty map approaches the radiometer map as $l_{\max }$ increases. Residual fluctuations on the difference map appear consistent with the angular scale set by $l_{\max }=30$.

Figure 7

From top to bottom: A clean sky map for the case of no injection; next, the associated map of uncertainty, ${\sigma }_{\mathcal{P}(\widehat{\Omega })}$; next, the associated SNR map; bottom, a histogram of SNR. The banded structure on the uncertainty map can be understood in terms of the directional sensitivity of two cross-correlated interferometers. The rotation of the Earth ensures that the uncertainty is uniform in right ascension for a fixed declination. The blue histogram bars are data, the solid dark red line is a Gaussian fit ($\mathrm{\text{sigma}}=1$, $\mathrm{\text{mean}}=1$), the solid light green line is a maximum-likelihood fit ($\mathrm{\text{sigma}}=1.07$, $\mathrm{\text{mean}}=0.06$), and the dash-dotted line is the 1-sigma error for 400 independent points. $l_{\max }=20$.

Figure 8

From top to bottom: a dirty sky map of a point-source injection at $(\mathrm{ra},\mathrm{dec})=(6\mathrm{hr},+45°)$, a clean sky map (without SVD) of the same injection, a clean sky map (with SVD) of the same injection, a map of SNR. The source has maximum SNR of 49. $l_{\max }=20$.

Figure 9

Above is a clean sky map of an isotropic injection. Below is a histogram of SNR. The average SNR across the map is 9.1. $l_{\max }=20$.

Figure 10

A clean map of a dipole injection oriented along the $z$ axis. $l_{\max }=20$.

Figure 11

A clean sky map of two point sources, one at $(\mathrm{ra},\mathrm{dec})=(6\mathrm{hr},+45°)$ ($\mathrm{SNR}=81$) and the other at $(\mathrm{ra},\mathrm{dec})=(12\mathrm{hr},-30°)$ ($\mathrm{SNR}=76$). $l_{\max }=20$.

Figure 12

Above: a toy model injection corresponding to a map measured by the WMAP satellite [42] meant to mimic a diffuse source clustered in the galactic plane ($b=0°$). The map utilizes HEALPix [48] and the injection was simulated using the Planck Simulator [47]. Below: a clean map recovered from this injection. $l_{\max }=20$.

Figure 13

Top-left: the original injection. Midleft: the regularized version of the injection ${\mathcal{P}}_{\alpha }^{\prime }$. Top-right: the recovered clean map ${\widehat{\mathcal{P}}}_{\alpha }^{\prime }$. Midright: ${\mathcal{P}}_{\alpha }^{\prime }-{\widehat{\mathcal{P}}}_{\alpha }^{\prime }$ normalized by ${\sigma }_{\mathcal{P}(\widehat{\Omega })}$. Bottom-left: a histogram of these residuals. The fluctuations in the residuals appear to be consistent with detector noise. $l_{\max }=20$.

Figure 14

Top-left: a toy model injection corresponding to a map measured by the WMAP satellite meant to mimic a diffuse source clustered around ($\mathrm{dec}=0°$). The map utilizes HEALPix [48] and the injection was simulated using the Planck Simulator [47]. Midleft: a regularized version of the injection. Top-right: a clean map recovered from this injection. Midright: the residuals ${\mathcal{P}}_{\alpha }^{\prime }-{\widehat{\mathcal{P}}}_{\alpha }^{\prime }$ normalized by ${\sigma }_{\mathcal{P}(\widehat{\Omega })}$. Bottom-left: a histogram of these residuals. Note the apparent quadrupole moment present in the midleft and top-right panels. This demonstrates the relatively low sensitivity to $l=2$ moments using the H1-L1 baseline. $l_{\max }=20$.

Figure 15

Results from a multiple baseline simulation corresponding to the injection in the top panel of Fig. 12. In the top panel is a clean map from the H1-L1 baseline. Second from the top is a clean map from the H1-V1 baseline. Third is a clean map from the L1-V1 baseline. Fourth is a clean map produced from combining all three baselines (H1-L1-V1.) The final panel is the injected source. For all maps $l_{\max }=20$.