LIGO Document G1301187-v2

New Coordinates for the Amplitude Parameter Space of Continuous Gravitational Waves

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The parameter space for continuous gravitational waves can be divided into amplitude parameters (signal amplitude, inclination and polarization angles describing the orientation of the source, and an initial phase) and phase-evolution parameters (signal frequency and frequency derivatives, and parameters such as sky position which determine the Doppler modulation of the signal). The division is useful in part because of the existence of a set of functions known as the Jaranowski-Krolak-Schutz (JKS) coordinates, which are a set of four coordinates on the amplitude parameter space such that the gravitational-wave signal can be written as a linear combination of four template waveforms (which depend on the phase-evolution parameters) with the JKS coordinates as coefficients. We define a new set of coordinates on the amplitude parameter space, with the same properties, which can be more closely connected to the physical amplitude parameters. These naturally divide into two pairs of Cartesian-like coordinates on two-dimensional subspaces, one corresponding to left- and the other to right-circular polarization. We thus refer to these as CPF (circular polarization factored) coordinates. We illustrate several applications of these new coordinates, and related coordinate systems, in particular to calculation of the B-statistic. This is a Bayesian detection statistic for directed gravitational wave search, which marginalizes the likelihood ratio over the amplitude parameter space, as opposed to the maximum likelihood method used in the standard F-statistic.
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