Gravitational wave radiometry: Mapping a stochastic gravitational wave background

The problem of the detection and mapping of a stochastic gravitational wave background (SGWB), either cosmological or astrophysical, bears a strong semblance to the analysis of the cosmic microwave background (CMB) anisotropy and polarization, which too is a stochastic field, statistically described in terms of its correlation properties. An astrophysical gravitational wave background (AGWB) will likely arise from an incoherent superposition of unmodelled and/or unresolved sources and cosmological gravitational wave backgrounds (CGWB) are also predicted in certain scenarios. The basic statistic we use is the cross correlation between the data from a pair of detectors. In order to “point” the pair of detectors at different locations one must suitably delay the signal by the amount it takes for the gravitational waves (GW) to travel to both detectors corresponding to a source direction. Then the raw (observed) sky map of the SGWB is the signal convolved with a beam response function that varies with location in the sky. We first present a thorough analytic understanding of the structure of the beam response function using an analytic approach employing the stationary phase approximation. The true sky map is obtained by numerically deconvolving the beam function in the integral (convolution) equation. We adopt the maximum likelihood framework to estimate the true sky map using the conjugate gradient method that has been successfully used in the broadly similar, well-studied CMB map-making problem. We numerically implement and demonstrate the method on signal generated by simulated (unpolarized) SGWB for the GW radiometer consisting of the LIGO pair of detectors at Hanford and Livingston. We include “realistic” additive Gaussian noise in each data stream based on the LIGO-I noise power spectral density. The extension of the method to multiple baselines and polarized GWB is outlined. In the near future the network of GW detectors, including the Advanced LIGO and Virgo detectors that will be sensitive to sources within a thousand times larger spatial volume, could provide promising data sets for GW radiometry.

Figure 1

Geometry of an elementary radiometer. Above, $\mathbf{\Delta }\mathbf{x}(t)$ is the separation or baseline vector between the two detectors; as the Earth rotates, its direction changes, but its magnitude remains fixed. The direction to the source $\widehat{\mathbf{\Omega }}$ is also fixed in the barycentric frame. The phase difference between signals arriving at two detector sites from the same direction is also shown.

Figure 2

Illustration of the agreement between numerical and theoretical GW radiometer beam patterns at declination $+12°$ for the LIGO detectors at Livingston and Hanford (with white noise, upper cutoff frequency of 1024 Hz, $H(f)=\mathrm{\text{constant}}$ and observation time of one sidereal day). Same color map has been used for both panels. Note that, the shape of the beam does not depend on the right ascension of the pointing direction for observation time of full sidereal day(s).

Figure 3

The baseline formed by two detectors, $\mathbf{\Delta }\mathbf{x}(t)$, traces out a cone as the earth rotates. A schematic diagram is shown here. The vector ${\widehat{\mathbf{n}}}_{\mathrm{cone}}(t)$ is normal to the baseline as well as the cone.

Figure 4

Contour plot of GW radiometer beam pattern at declination $+12°$ revealing the approximate size and the (negative) side lobes of the beam for the LIGO detectors with white noise, upper cutoff frequency $f_u=1024\mathrm{Hz}$, $H(f)=\mathrm{\text{constant}}$, and observation time of one sidereal day. The contours are drawn starting from the highest level of 0.9 with a difference in levels of 0.1. The beam falls by $1/e$ in between the 5th and the 6th contour (from the highest value), which is a good indicator for the beam size. Clearly, the narrowest beam size (along the “minor axis”) is in reasonable agreement with the theoretical prediction of $6°\sim 0.1\mathrm{rad}$. The beam becomes broader (not shown in the figure) for real detector noise and negative source spectral index, e.g., $H(f)\propto f^{-3}$.

Figure 5

The solid line is a plot of the singular values of the kernel for the LIGO pair of detectors with white noise, upper cutoff frequency of $f_u=1024\mathrm{Hz}$, and observation time of one sidereal day. From the plot it is evident that the singular values are negligible after $\sim 1000$. These results agree with the numerical and SPA results which give the size of the central spot $\sim 0.1\mathrm{radian}$. The dotted line shows the singular values for the same detector pair, but with LIGO-I noise PSD and upper cutoff frequency of $f_u=512\mathrm{Hz}$. Clearly, the latter curve indicates a loss of angular resolution due to relatively poorer higher frequency response of the detectors.

Figure 6

A typical beam matrix for the LIGO Hanford-Livingston baseline at a low resolution (192 pixels) is shown in this figure. Each row of the matrix is the beam response function for the pointing direction that corresponds to the row index. The matrix is diagonal dominated as the radiometer receives maximum contribution from the pointing direction. The stripes are related to the isoLatitude pixelization scheme—the indices of the neighboring pixels at different latitudes differ by the total number of pixels on that latitude. The possibility of making the beam matrix more diagonal dominated using a nested pixelization scheme, where the indices of the neighboring pixels are close, is being explored.

Figure 7

Illustration of deconvolution for localized sources: A 4-pixel wide localized source was injected near the location of the Virgo cluster—a potential source of SGWB.

Figure 8

Illustration of deconvolution for broad sources: Maps similar to CMB temperature anisotropy sky with the galactic foreground were injected as toy maps. The left panels correspond to a map measured by the WMAP satellite [52], as seen in the galactic coordinates, as a toy model for an equatorial source and the right panels correspond to a map generated by the Planck simulator [53], as a toy model for a multideclination extended source.