### Abstract

The detection of gravitational waves from astrophysical sources is probably one of the most keenly awaited events in the history of astrophysics. The paucity of gravitational wave sources and the relative difficulty in detecting such waves, as compared to those in the electromagnetic domain, necessitate the development of optimal data analysis techniques to detect the signal, as well as to extract the maximum possible information from the detected signals. Coalescing binary systems are one of the most promising sources of gravitational waves. This is due to the fact that such sources are easier to model and thus one can design detection strategies particularly tuned to such signals. A lot of attention has been devoted in the literature to studying such techniques and most of the work has revolved around the Weiner filtering and the maximum likelihood estimators of the parameters of the binary system. We investigate such techniques with the aid of differential geometry which provides geometric insight into the problem. Such a formalism allows us to explore the merits and faults of a detection scheme independent of the parameters chosen to represent the waveform. The formalism also generalizes the problem of choosing an optimal set of templates to detect a known waveform buried in noisy data. We stress the need for finding a set of convenient parameters for the waveform and show that even after the inclusion of the second-order post-Newtonian corrections, the waveform can essentially be detected by employing a one-dimensional lattice of templates. This would be very useful both for the purpose of carrying out the simulations as well as for the actual detection process. After setting up such a formalism we carry out a Monte Carlo simulation of the detection process for the initial LIGO-VIRGO configuration for the first post-Newtonian corrected coalescing binary waveform. We compare the results of our simulations with the currently available estimates of the accuracies in the determination of the parameters and the probability distribution of the maximum likelihood estimators. Our results suggest that the covariance matrix underestimates, by over a factor of 2, the actual errors in the estimation of parameters even when the signal-to-noise ratio is as high as 10. As only a tiny fraction of the events is expected to be detected with a signal-to-noise higher than this value, the covariance matrix is grossly inadequate to describe the errors in the measurement of the parameters of the waveform. It is found from our Monte Carlo simulations that the deviations from the covariance matrix are more in the case of the first post-Newtonian waveform than in the case of the Newtonian one. Inclusion of higher-order post-Newtonian corrections introduces new parameters that are correlated with those at the lower post-Newtonian waveform. Such correlations are expected to further increase the discrepancy of the covariance matrix results with those inferred from Monte Carlo simulations. Consequently, numerical simulations that take into account post-Newtonian corrections beyond the first post-Newtonian order are needed in order to get a clearer picture about the accuracy in the determination of parameters. We find that with the aid of the instant of coalescence the direction to the source can be determined more accurately than with the time of arrival.

Original language | English (US) |
---|---|

Pages (from-to) | 3033-3055 |

Number of pages | 23 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 53 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 1996 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)

### Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*53*(6), 3033-3055. https://doi.org/10.1103/PhysRevD.53.3033

}

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 53, no. 6, pp. 3033-3055. https://doi.org/10.1103/PhysRevD.53.3033

**Gravitational waves from coalescing binaries : Detection strategies and Monte Carlo estimation of parameters.** / Balasubramanian, R.; Sathyaprakash, Bangalore S.; Dhurandhar, S. V.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Gravitational waves from coalescing binaries

T2 - Detection strategies and Monte Carlo estimation of parameters

AU - Balasubramanian, R.

AU - Sathyaprakash, Bangalore S.

AU - Dhurandhar, S. V.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - The detection of gravitational waves from astrophysical sources is probably one of the most keenly awaited events in the history of astrophysics. The paucity of gravitational wave sources and the relative difficulty in detecting such waves, as compared to those in the electromagnetic domain, necessitate the development of optimal data analysis techniques to detect the signal, as well as to extract the maximum possible information from the detected signals. Coalescing binary systems are one of the most promising sources of gravitational waves. This is due to the fact that such sources are easier to model and thus one can design detection strategies particularly tuned to such signals. A lot of attention has been devoted in the literature to studying such techniques and most of the work has revolved around the Weiner filtering and the maximum likelihood estimators of the parameters of the binary system. We investigate such techniques with the aid of differential geometry which provides geometric insight into the problem. Such a formalism allows us to explore the merits and faults of a detection scheme independent of the parameters chosen to represent the waveform. The formalism also generalizes the problem of choosing an optimal set of templates to detect a known waveform buried in noisy data. We stress the need for finding a set of convenient parameters for the waveform and show that even after the inclusion of the second-order post-Newtonian corrections, the waveform can essentially be detected by employing a one-dimensional lattice of templates. This would be very useful both for the purpose of carrying out the simulations as well as for the actual detection process. After setting up such a formalism we carry out a Monte Carlo simulation of the detection process for the initial LIGO-VIRGO configuration for the first post-Newtonian corrected coalescing binary waveform. We compare the results of our simulations with the currently available estimates of the accuracies in the determination of the parameters and the probability distribution of the maximum likelihood estimators. Our results suggest that the covariance matrix underestimates, by over a factor of 2, the actual errors in the estimation of parameters even when the signal-to-noise ratio is as high as 10. As only a tiny fraction of the events is expected to be detected with a signal-to-noise higher than this value, the covariance matrix is grossly inadequate to describe the errors in the measurement of the parameters of the waveform. It is found from our Monte Carlo simulations that the deviations from the covariance matrix are more in the case of the first post-Newtonian waveform than in the case of the Newtonian one. Inclusion of higher-order post-Newtonian corrections introduces new parameters that are correlated with those at the lower post-Newtonian waveform. Such correlations are expected to further increase the discrepancy of the covariance matrix results with those inferred from Monte Carlo simulations. Consequently, numerical simulations that take into account post-Newtonian corrections beyond the first post-Newtonian order are needed in order to get a clearer picture about the accuracy in the determination of parameters. We find that with the aid of the instant of coalescence the direction to the source can be determined more accurately than with the time of arrival.

AB - The detection of gravitational waves from astrophysical sources is probably one of the most keenly awaited events in the history of astrophysics. The paucity of gravitational wave sources and the relative difficulty in detecting such waves, as compared to those in the electromagnetic domain, necessitate the development of optimal data analysis techniques to detect the signal, as well as to extract the maximum possible information from the detected signals. Coalescing binary systems are one of the most promising sources of gravitational waves. This is due to the fact that such sources are easier to model and thus one can design detection strategies particularly tuned to such signals. A lot of attention has been devoted in the literature to studying such techniques and most of the work has revolved around the Weiner filtering and the maximum likelihood estimators of the parameters of the binary system. We investigate such techniques with the aid of differential geometry which provides geometric insight into the problem. Such a formalism allows us to explore the merits and faults of a detection scheme independent of the parameters chosen to represent the waveform. The formalism also generalizes the problem of choosing an optimal set of templates to detect a known waveform buried in noisy data. We stress the need for finding a set of convenient parameters for the waveform and show that even after the inclusion of the second-order post-Newtonian corrections, the waveform can essentially be detected by employing a one-dimensional lattice of templates. This would be very useful both for the purpose of carrying out the simulations as well as for the actual detection process. After setting up such a formalism we carry out a Monte Carlo simulation of the detection process for the initial LIGO-VIRGO configuration for the first post-Newtonian corrected coalescing binary waveform. We compare the results of our simulations with the currently available estimates of the accuracies in the determination of the parameters and the probability distribution of the maximum likelihood estimators. Our results suggest that the covariance matrix underestimates, by over a factor of 2, the actual errors in the estimation of parameters even when the signal-to-noise ratio is as high as 10. As only a tiny fraction of the events is expected to be detected with a signal-to-noise higher than this value, the covariance matrix is grossly inadequate to describe the errors in the measurement of the parameters of the waveform. It is found from our Monte Carlo simulations that the deviations from the covariance matrix are more in the case of the first post-Newtonian waveform than in the case of the Newtonian one. Inclusion of higher-order post-Newtonian corrections introduces new parameters that are correlated with those at the lower post-Newtonian waveform. Such correlations are expected to further increase the discrepancy of the covariance matrix results with those inferred from Monte Carlo simulations. Consequently, numerical simulations that take into account post-Newtonian corrections beyond the first post-Newtonian order are needed in order to get a clearer picture about the accuracy in the determination of parameters. We find that with the aid of the instant of coalescence the direction to the source can be determined more accurately than with the time of arrival.

UR - http://www.scopus.com/inward/record.url?scp=0001109244&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001109244&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.53.3033

DO - 10.1103/PhysRevD.53.3033

M3 - Article

AN - SCOPUS:0001109244

VL - 53

SP - 3033

EP - 3055

JO - Physical Review D

JF - Physical Review D

SN - 0556-2821

IS - 6

ER -