Gravitational wave radiometry: Mapping a stochastic gravitational wave background

Abstract

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.

Phase-coherent mapping of gravitational-wave backgrounds using ground-based laser interferometers

“…This is in contrast to the case for a pulsar timing array, which is completely insensitive to the curl modes. Also, by mapping both the amplitude and phase of h + (f,k) and h × (f,k) as functions of direction on the sky (as referenced from the SSB), our method extends previous approaches [3][4][5][6][7][8][9] for anisotropic backgrounds, which map the distribution of gravitational-wave power, |h + | 2 + |h × | 2 . Our formalism can be cast in terms of either the traditional + and × polarization modes of the background {h + (f,k), h × (f,k)}, or the gradient and curl modes {a G (lm) (f ), a C (lm) (f )}, with respect to a decomposition of the metric perturbations in terms of spin-weighted or tensor (gradient and curl) spherical harmonics [1].…”

“…Searches for anisotropic gravitational-wave backgrounds have typically been formulated in terms of the distribution of gravitational-wave power on the sky (see, e.g., [3][4][5][6][7][8][9]). The basic idea underlying these approaches is to use to cross-correlation measurements from two or more detectors to estimate the power in the gravitational-wave background as a function of sky position.…”

Multibaseline gravitational wave radiometry

“…The filter that maximizes the signal-to-noise ratio (SNR) associated with this statistic is a scalar, squareintegrable function on the sky [2] and, hence, can be resolved linearly in an appropriate basis, such as a pixel basis or the spherical harmonic basis. In the former case, k is the pixel index.…”

“…Therefore, it does not suffer from errors inherent to the deconvolution procedure and is especially useful for detecting weak sources. In the limit of a single baseline, it reduces to the detection statistic studied by Ballmer [1] and Mitra et al [2]. Unlike past studies, here the MLR statistic enables us to compare quantitatively the performances of a variety of baselines searching for a SGWB signal in (simulated) data.…”

“…[12] and was implemented in the pixel basis on data from LIGO's fourth science run [13]. An elaborate study of this method, including the maximumlikelihood (ML) estimation of the true anisotropy of GW background by deconvolving the observed sky map, was presented in Mitra et al [2]. Even though the pixel-based search is promising and simpler to understand, it is not the best basis for probing sources with angular spreads greater than the angular resolution of the GW radiometer.…”