Data analysis of gravitational-wave signals from spinning neutron stars. V. A narrow-band all-sky search

We present theory and algorithms to perform an all-sky coherent search for periodic signals of gravitational waves in narrow-band data of a detector. Our search is based on a statistic, commonly called the $\mathcal{F}$-statistic, derived from the maximum-likelihood principle in Paper I of this series. We briefly review the response of a ground-based detector to the gravitational-wave signal from a rotating neuron star and the derivation of the $\mathcal{F}$-statistic. We present several algorithms to calculate efficiently this statistic. In particular our algorithms are such that one can take advantage of the speed of fast Fourier transform in calculation of the $\mathcal{F}$-statistic. We construct a grid in the parameter space such that the nodes of the grid coincide with the Fourier frequencies. We present interpolation methods that approximately convert the two integrals in the $\mathcal{F}$-statistic into Fourier transforms so that the fast Fourier transform algorithm can be applied in their evaluation. We have implemented our methods and algorithms into computer codes and we present results of the Monte Carlo simulations performed to test these codes.

Figure 1

Up: Hexagonal covering generated by (4.22), with ${\mathcal{C}}_0=3/4$ and $R=1/2$ (dark points). Fourier frequencies are represented by open circles. Down: Constrained lattice, satisfying the condition (4.20). Any point in the parameter space belongs on average to 1.8171 ellipses.

Figure 2

Illustration of the nearest neighbor resampling method. The top panel shows the uniformly sampled original data and the points of the barycentric time. The middle panel shows the interpolations of the original time series at the points of the barycentric time, obtained by the nearest neighbor method. The bottom panel shows the resampled time series which is the uniformly sampled barycentric time series from the middle panel.

Figure 3

Comparison of the two interpolation methods and the perfect matched-filtering. We see that the two-step interpolation method that uses Fourier and spline interpolation very accurately reproduces the perfect matched filter.

Figure 4

An illustration of the Nelder-Mead algorithm in two dimensions. We first evaluate the function to be minimized on an initial triangle. At each step of the search, a new point in or near the current triangle is generated. The function value at the new point is compared with the function’s values at the vertices of the triangle and, usually, one of the vertices is replaced by the new point, giving a new triangle. This step is repeated until the diameter of the simplex is less than the specified tolerance. In the method the triangle adapts itself to the local landscape, elongating down long inclined planes, changing direction on encountering a valley at an angle, and contracting in the neighborhood of a minimum.

Figure 5

Standard deviations (left panels) and biases (right panels) of the ML estimators of the intrinsic parameters as functions of the SNR computed with the grid of $\mathrm{MM}=\sqrt[3]{0.9}$. The plots are made, from the top, for the following parameters: frequency, spin-down, declination, and right ascension. Each circle corresponds to a standard deviation or a bias computed from simulation of 1000 runs. The continuous lines in the left panels are the Cramér-Rao bounds for standard deviations.

Figure 6

Standard deviations (left panels) and biases (right panels) of the ML estimators of the intrinsic parameters as functions of the SNR computed with the grid of $\mathrm{MM}=\sqrt{3}/2$ for the following three cases: (i) zero padding and spline interpolation (diamonds); (ii) interbinning and spline interpolation (stars); (iii) interbinning and the nearest neighbor interpolation (circles). The continuous lines in the left panels are the Cramér-Rao bounds.

Figure 7

Standard deviations (left panels) and biases (right panels) of the ML estimators of the intrinsic parameters as functions of the SNR computed for the following three cases: (i) interbinning and spline interpolation, with the coarse grid of $\mathrm{MM}=\sqrt{3}/2$ (diamonds); (ii) interbinning with the nearest neighbor interpolation, with the fine grid of $\mathrm{MM}=\sqrt[3]{0.9}$ (stars); (iii) interbinning with the nearest neighbor interpolation, with the coarse grid of $\mathrm{MM}=\sqrt{3}/2$ (circles). The continuous lines in the left panels are the Cramér-Rao bounds.