Hierarchical search strategy for the detection of gravitational waves from coalescing binaries: Extension to post-Newtonian waveforms

The detection of gravitational waves from coalescing compact binaries would be a computationally intensive process if a single bank of template wave forms (one step search) is used. In an earlier paper we presented a detection strategy, called a two step search, that utilizes a hierarchy of template banks. It was shown that in the simple case of a family of Newtonian signals, an on-line two step search was $\simeq 8$ times faster than an on-line one step search (for the initial LIGO). In this paper we extend the two step search to the more realistic case of zero spin ${\mathrm{post}}^{1.5}$-Newtonian wave forms. We also present formulas for detection and false alarm probabilities which take statistical correlations into account. We find that for the case of a ${\mathrm{post}}^{1.5}$-Newtonian family of templates and signals, an on-line two step search requires $\sim 1/21$ the computing power that would be required for the corresponding on-line one step search. This reduction is achieved when signals having a strength $S=10.34\mathrm{}$ are required to be detected with a probability of 0.95, at an average of one false event per year, and the noise power spectral density used is that of the advanced LIGO. For the initial LIGO, the reduction achieved in computing power is $\sim 1/27$ for $S=9.98\mathrm{}$ and the same probabilities for detection and false alarm as above. The increase in the efficacy of a two step search in the ${\mathrm{post}}^{1.5}$-Newtonian case comes about chiefly because of an increase in the number of signal parameters since the ${\mathrm{post}}^{1.5}$-Newtonian signal depends on the binary masses $m_1$ and $m_2$ separately unlike the Newtonian case where only a combination of these masses enters the signal parametrization. The shift to ${\mathrm{post}}^{1.5}$-Newtonian signals also gives rise to some new problems which are not encountered in the analysis of Newtonian wave forms. We describe these problems and take them into account in our analysis.