Implementation of the frequency-modulated sideband search method for gravitational waves from low mass x-ray binaries

L. Sammut, C. Messenger, A. Melatos, and B. J. Owen

Phys. Rev. D 89, 043001 – Published 4 February 2014

Abstract

We describe the practical implementation of the sideband search, a search for periodic gravitational waves from neutron stars in binary systems. The orbital motion of the source in its binary system causes frequency modulation in the combination of matched filters known as the $F$ -statistic. The sideband search is based on the incoherent summation of these frequency-modulated $F$ -statistic sidebands. It provides a new detection statistic for sources in binary systems, called the $C$ -statistic. The search is well suited to low-mass x-ray binaries, the brightest of which, called Sco X-1, is an ideal target candidate. For sources like Sco X-1, with well-constrained orbital parameters, a slight variation on the search is possible. The extra orbital information can be used to approximately demodulate the data from the binary orbital motion in the coherent stage, before incoherently summing the now reduced number of sidebands. We investigate this approach and show that it improves the sensitivity of the standard Sco X-1 directed sideband search. Prior information on the neutron star inclination and gravitational wave polarization can also be used to improve upper limit sensitivity. We estimate the sensitivity of a Sco X-1 directed sideband search on ten days of LIGO data and show that it can beat previous upper limits in current LIGO data, with a possibility of constraining theoretical upper limits using future advanced instruments.

Received 6 November 2013

DOI:https://doi.org/10.1103/PhysRevD.89.043001

Authors & Affiliations

L. Sammut^1,2,*, C. Messenger^3,2, A. Melatos¹, and B. J. Owen⁴

¹University Of Melbourne, Victoria 3052, Australia
²Albert Einstein Institute, Callinstraße 38, D-30167 Hannover, Germany
³University of Glasgow, Glasgow G12 8QQ, United Kingdom
⁴The Pennsylvania State University, University Park, Pennsylvania 16802, USA

^*lsammut@student.unimelb.edu.au

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Vol. 89, Iss. 4 — 15 February 2014

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Images

Figure 1
Graphical illustration of the sideband search pipeline, showing the frequency-modulated $F$ -statistic (left, red), the sideband template (middle, black), and their convolution, known as the $C$ -statistic (right, blue). In this noise-free case, a signal of $f_{0} = 200 Hz$ with amplitude $h_{0} = 1$ in a system with $a = 0.005 s$ s, $P = 7912.7 s$ , sky position $α = 2.0 rad$ , $δ = 1.0 rad$ , and phase parameters $ψ = 0.2 rad$ , $\cos ι = 0.5$ , $φ_{0} = 1 rad$ , was simulated for a ten-day observation span. Frequency increases along the horizontal axis, which ranges from 199.998 to 200.002 Hz on each plot. In each case, the location of the injected signal at 200 Hz is indicated by the vertical dashed black line.
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Figure 2
ROC curves comparing performance of the flat (blue) and Bessel-function-weighted (red) templates, described by Eqs. (41) and (45), respectively. The theoretical (fine black) curve is constructed from a noncentral $χ_{4 M}^{2} (λ)$ distributed statistic with noncentrality parameter $λ = ρ_{0}^{2} = 20$ and represents the expected performance of the flat template. Dashed and dotted curves represent templates with positive and negative 10% errors on the semimajor axis, respectively. Here, the signal parameters were chosen such that the number of sidebands was $M = 2001$ , and curves were constructed using $10^{6}$ realizations of noise.
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Figure 3
ROC curves showing how the performance of the flat template $C$ -statistic is affected by an offset in the orbital semimajor axis, assuming it is measured exactly (i.e. $a = a_{0}$ ). The thick black curve represents a zero offset ( $ξ = 0$ ). Thick colored curves represent a positive offset in the semimajor axis ( $ξ > 0$ ). Dashed colored curves represent negative offsets ( $ξ < 0$ ). The fainter black curve is constructed from a statistic governed by a noncentral $χ_{4 M}^{2} (λ)$ distribution with a noncentrality parameter $λ = ρ_{0}^{2}$ and represents the theoretical expected behavior of a perfectly matched template. Signal parameters are the same as described in Fig. 2, with $ρ_{0} = 20$ , $M = 2001$ , and using $10^{6}$ realizations of noise.
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Figure 4
Performance of the $C$ -statistic with respect to offsets in the semimajor axis at given false-alarm probability $F_{a}$ . The offset parameter $ξ$ quantifies the semimajor axis error in terms of known parameters $a$ and $Δ a$ . The color bar represents the log of the false-alarm probability, ranging from $10^{- 3}$ (bottom, blue) to 0 (top, red). Signal parameters are the same as in Figs. 2 and 3, with $ρ_{0} = 20$ , $M = 2001$ , and a $Δ a / a = 10 %$ fractional uncertainty on the semimajor axis, using $10^{6}$ realizations of noise.
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Figure 5
Sensitivity estimate for a ten-day sideband search in various modes—standard, approximate demodulation, and approximate demodulation with known priors (fine, medium, and bold solid curves, respectively)—using LIGO (H1L1) S5 data (upper, purple group), and using the three-detector (H1L1V) advanced LIGO configuration (lower, red group). Also shown are results from the the previous coherent search for Sco X-1. in S2 data (solid black dashes) [53] and the maximum upper limits for each Hz band of the directed stochastic (radiometer) search in S4 and S5 data (light and dark blue dashed curves, respectively) [55, 56]. The theoretical torque-balance gravitational wave strain upper limit [ $h_{0}^{EQ}$ from Eq. (7)] for Sco X-1 is indicated by the thick gray straight line.
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