Effectual template bank for the detection of gravitational waves from inspiralling compact binaries with generic spins

We report the construction of a three-dimensional template bank for the search for gravitational waves from inspiralling binaries consisting of spinning compact objects. The parameter space consists of two dimensions describing the mass parameters and one “reduced-spin” parameter, which describes the secular (nonprecessing) spin effects in the waveform. The template placement is based on an efficient stochastic algorithm and makes use of the semianalytical computation of a metric in the parameter space. We demonstrate that for “low-mass” ($m_1+m_2\lesssim 12M_{\odot }$) binaries, this template bank achieves effective fitting factors $\sim 0.92–0.99$ towards signals from generic spinning binaries in the advanced detector era over the entire parameter space of interest (including binary neutron stars, binary black holes, and black-hole neutron-star binaries). This provides a powerful and viable method for searching for gravitational waves from generic spinning low-mass compact binaries. Under the assumption that spin magnitudes of black holes (neutron stars) are uniformly distributed between 0–0.98 [0–0.4] and spin angles are isotropically distributed, the expected improvement in the average detection volume (at a fixed signal-to-noise-ratio threshold) of a search using this reduced-spin bank is $\sim 20\%–52\%$, as compared to a search using a nonspinning bank.

Figure 1

Comparison of the match ellipses computed from the semianalytical calculation of the metric (black ellipses) with contours of the match function computed numerically (color contours). The component masses and reduced-spin parameter $(m_1,m_2,\chi )$ corresponding to point in the parameter space relative to which the match function is computed is shown on the top of each column (masses in units of $M_{\odot }$). The different rows correspond to two-dimensional slices of these contours in the $\mathrm{\Delta }{\theta }_0-\mathrm{\Delta }{\theta }_3$ plane (top row), $\mathrm{\Delta }{\theta }_0-\mathrm{\Delta }{\theta }_{3\mathrm{S}}$ plane (middle row) and $\mathrm{\Delta }{\theta }_3-\mathrm{\Delta }{\theta }_{3\mathrm{S}}$ plane (bottom row). The solid black ellipses correspond to a match of 0.97 and dashed black ellipses correspond to a match of 0.99.

Figure 2

The colored contours show the log of the square root of the determinant of the template-space metric ($\log \sqrt{|\mathbf{g}|}$) over the range of template bank parameters chosen in Table 1. The number density of templates in each region of the bank is roughly determined by $\sqrt{|\mathbf{g}|}$. The top panel corresponds to the metric computed in the dimensionless chirp-time coordinate system $({\theta }_0,{\theta }_3,{\theta }_{3\mathrm{S}})$. The three subplots in the top row corresponds to three different slices in the three-dimensional template bank, corresponding to $\chi =-0.3$, 0, 0.3. The black dots correspond to the templates placed in each slice (of thickness $\mathrm{\Delta }\chi ={10}^{-3}$) by the stochastic algorithm described in Sec. 4. Notice that the template density in different regions generally agree with the expectation from the metric. In the $({\theta }_0,{\theta }_3,{\theta }_{3\mathrm{S}})$ coordinate system, the density has a maximum variation of $\sim 10$ over the entire template bank. In contrast, we show in the bottom panels the same quantity computed in the coordinate system described by $(M_{\mathrm{c}},\eta ,\chi )$. Notice that, in this coordinate system, the template density has considerable variation ($\sim {10}^5$) over the parameter space. This illustrates the advantage of using the $({\theta }_0,{\theta }_3,{\theta }_{3\mathrm{S}})$ coordinate system.

Figure 3

Histogram of the achieved mismatch ($1-\text{fitting factor}$) by the reduced-spin template bank in detecting reduced-spin injections. The plot is a demonstration of the achieved “coverage” of the bank. The vertical dashed line corresponds to a mismatch of 5%. Only $\sim 0.7\%$ of the $\sim 100,000$ injections have mismatch $>5\%$. The effective fitting factor towards this population of injections is $\simeq 0.98$.

Figure 4

The left plots show the effective fitting factor ${\mathrm{FF}}_{\text{eff}}$ of the reduced-spin template bank at different regions in the component mass plane. Each filled circle corresponds to 5000 injections (with fixed component masses, corresponding to the center of each circle) of generic spinning binaries with parameters reported in Table 3. The gray line ($m_1+m_2=12M_{\odot }$) shows the expected boundary above which the contribution of merger-ringdown becomes non-negligible, and where the inspiral template bank needs to be replaced by an inspiral-merger-ringdown template bank. The black dashed lines correspond to the assumed boundary between the neutron-star and black-hole mass. The top panel assumes that the maximum spin of neutron stars is 0.4, while the bottom panel assumes a maximum spin of 0.05 for neutron stars. The right plot shows the same for a nonspinning template bank.

Figure 5

Average increase in the detection volume of a search employing the reduced-spin template bank as compared to one employing a nonspinning template bank (corresponding to a fixed SNR threshold). The reduced-spin template bank is expected to bring about $\sim 20\%–52\%$ increase in the average detection volume (top panel), assuming that the maximum spin of neutron stars is 0.4. The bottom panel shows the result of the same calculation assuming a maximum spin of 0.05 for neutron stars.

Figure 6

Fitting factor (indicated by the color of the dots) of the reduced-spin template bank in detecting generic spinning binaries with component masses $(6M_{\odot },6M_{\odot })$ (left plot) and $(10M_{\odot },1.4M_{\odot })$ (right plot). The $x$ axis corresponds to the spin magnitude of the more massive compact object, while the $y$ axis corresponds to the cosine of the angle between the spin and initial Newtonian orbital angular momentum. In the left plot (equal-mass binary) fitting factors are $\sim 1$ irrespective of the magnitude and orientation of the spin vector, while in the right plot (highly unequal-mass binary) fitting factors can be as low as $\sim 0.7$ for binaries with large, misaligned spins.

Figure 7

Normalized SNR (such that the maximum SNR is (1) of generic spinning binaries plotted against the FF of the reduced-spin template bank in detecting them. It can be seen that fitting factors are high towards binaries with large SNR. The color of the dots corresponds to the sine of the inclination of the total angular momentum vector with respect to the line of sight (darker shades correspond to binaries whose total angular momentum is along the line of sight). The left plot corresponds to binaries with component masses $(6M_{\odot },6M_{\odot })$ and the right plot to binaries with component masses $(10M_{\odot },1.4M_{\odot })$.

Figure 8

Same as Fig. 4, except that in this plot, the target waveforms are generated using the TaylorT4 approximation. The difference in the effectualness between Fig. 4 and this figure is due to the difference between the two different PN approximants, and is a reflection of the current uncertainty in the PN waveforms. This could be improved by computing the higher order (spin-dependent) PN terms.

Figure 9

Same as the left plot of Fig. 4, except that in this plot, the target waveforms are TaylorT4 with nonprecessing spins. It can be seen that the mismatch of the template bank at high-mass ratios (in the low-mass regime) can be as large as 5%, which cannot be attributed to the effects of precession.