Addressing the spin question in gravitational-wave searches: Waveform templates for inspiralling compact binaries with nonprecessing spins

This paper presents a post-Newtonian (PN) template family of gravitational waveforms from inspiralling compact binaries with nonprecessing spins, where the spin effects are described by a single “reduced-spin” parameter. This template family, which reparametrizes all the spin-dependent PN terms in terms of the leading-order (1.5PN) spin-orbit coupling term in an approximate way, has very high overlaps (fitting factor $>0.99$) with nonprecessing binaries with arbitrary mass ratios and spins. We also show that this template family is “effectual” for the detection of a significant fraction of generic spinning binaries in the comparable-mass regime ($m_2/m_1\lesssim 10$), providing an attractive and feasible way of searching for gravitational waves from spinning low-mass binaries. We also show that the secular (nonoscillatory) spin-dependent effects in the phase evolution (which are taken into account by the nonprecessing templates) are more important than the oscillatory effects of precession in the comparable-mass ($m_1\simeq m_2$) regime. Hence the effectualness of nonspinning templates is particularly poor in this case, as compared to non-precessing-spin templates. For the case of binary neutron stars observable by Advanced LIGO, even moderate spins (${\widehat{\mathbf{L}}}_N·\mathbf{S}/m^2\simeq 0.015–0.1$) will cause considerable mismatches ($\sim 3\%–25\%$) with nonspinning templates. This is contrary to the expectation that neutron-star spins may not be relevant for gravitational wave detection.

Figure 1

Source frame in the Finn-Chernoff convention. The $z$ axis points to the direction of the initial total angular momentum vector $\mathbf{J}$, the detector lies in the $x-z$ plane, and the angles $\alpha$ and $i$ describe the evolution of the orbital angular momentum vector ${\mathbf{L}}_N$ in the source frame.

Figure 2

Relative significance of the spin terms neglected in the reduced-spin templates, as a function of the symmetric mass ratio $\eta$. The plot shows the ratios of the coefficients of the spin terms ${\chi }_s$ and ${\chi }_a$ (which are neglected in reduced-spin templates) with the coefficients of $\chi$ (which are included). It can be seen that the relative contribution of the neglected terms is small ($\le 10\%$), and the dominant spin effects are described by the single parameter $\chi$. See Eq. (5.10) for the definition of ${\sigma }_i$, ${\gamma }_i$, and ${ϵ}_i$.

Figure 3

Contribution to the phase $\Psi (f)$ at ISCO from the spin terms neglected in the reduced-spin template family. The plots show the contours of $|\Delta {\Psi }_{\mathrm{ISCO}}|$ [see Eq. (5.13) for definition] as a function of the spin parameters ${\chi }_1$ and ${\chi }_2$ of the two compact objects (${\chi }_2$ being the spin of the more massive object), for the case of six different mass ratios. Darker shades correspond to lower values.

Figure 4

Match of the reduced-spin template family $\tilde{h}(f;\chi )$ with nonprecessing-spin signals $\tilde{h}(f;{\chi }_1,{\chi }_2)$ in the Advanced LIGO noise spectrum. Horizontal axes report the spin ${\chi }_1$ of the lighter object and vertical axes report the spin ${\chi }_2$ of the more massive object. The title of each plot reports the component masses $(m_1,m_2)$ in units of $M_{\odot }$.

Figure 5

Fitting factor (match maximized over $M$, $\eta$, and $\chi$) of the reduced-spin template family $\tilde{h}(f;\chi )$ with nonprecessing-spin signals $\tilde{h}(f;{\chi }_1,{\chi }_2)$. See Fig. 4 for a full description.

Figure 6

Fitting factor (match maximized over $M$, $\eta$) of the nonspinning PN template family $\tilde{h}(f;{\chi }_1={\chi }_2=0)$ with nonprecessing-spin signals $\tilde{h}(f;{\chi }_1,{\chi }_2)$. See Fig. 4 for a full description.

Figure 7

Mismatch (1-FF) of the nonspinning PN template family with nonprecessing-spin signals with equal spins. The horizontal axis reports the spins of the target binary (${\chi }_1={\chi }_2$). Component masses of the target binaries (in units of $M_{\odot }$) are shown in the legend. Mismatch values of 3% and 10% are indicated by horizontal dashed (black) lines.

Figure 8

Fitting factor (match maximized over $M$, $\eta$, $\beta$) of the PN template family considering only the leading-order spin effect (described by $\beta$) with nonprecessing-spin signals in Advanced LIGO. See Fig. 4 for a full description.

Figure 9

Fraction of precessing PN binaries producing a fitting factor $\ge \mathrm{FF}$ with the reduced-spin PN templates (thick red solid lines) and nonspinning PN templates (thick red dashed lines). The total mass $m/M_{\odot }$ and mass ratio $q$ are shown in the titles. Thin black lines plot the same, except that the spin distribution of neutron stars is restricted to the interval (0, 0.3).

Figure 10

Scatter plots of the fitting factor (indicated by the color of the dots) of the reduced-spin template family with precessing PN binaries as a function of different parameters describing the initial configuration of the binary. The angles $\Theta$ and $\psi$ describe the initial orientation of the binary with respect to the detector, $||{\chi }_2||$ is the spin magnitude of the more massive compact object, ${\theta }_{S2}$ describes its orientation with respect to the orbital angular momentum (see Sec. 3a for a complete description), and $\chi \equiv (1-76\eta /113){\mathit{\chi }}_s·{\widehat{\mathbf{L}}}_N+\delta {\mathit{\chi }}_a·{\widehat{\mathbf{L}}}_N$ is the initial value of the reduced-spin parameter. The total mass $m/M_{\odot }$ and mass ratio $q$ of the target population in each column are shown in the title.

Figure 11

Same as Fig. 10, except that here the fitting factors are calculated with respect to nonspinning PN templates.