Towards low-latency real-time detection of gravitational waves from compact binary coalescences in the era of advanced detectors

Electromagnetic (EM) follow-up observations of gravitational wave events will help shed light on the nature of the sources, and more can be learned if the EM follow-ups can start as soon as the gravitational wave event becomes observable. In this paper, we propose a computationally efficient time-domain algorithm capable of detecting inspiral gravitational waves from coalescing binaries of compact objects with nearly no further delay in addition to the time required to condition the data into a time series of calibrated gravitational-wave strain. Our algorithm, if can be expanded to include sky localization, will serve as the first step towards triggering EM observation before the merger. The key to the efficiency of our algorithm arises from the use of chains of so-called infinite impulse response filters, which filter time-series data recursively. Computational cost is further reduced by a template interpolation technique that requires filtering only done for a “coarse bank”, much sparser than the “fine bank” normally required to sufficiently recover the optimal signal-to-noise ratio: the filter chain of each coarse-bank template is divided into several sections, filtering output from these sections are combined appropriately to reconstruct the output of each of the nearby fine-bank templates. The filter construction and interpolation techniques are illustrated in this paper using Newtonian-chirp waveforms, although these will be generalizable to more accurate post-Newtonian waveforms. Towards future detectors with sensitivity extending to lower frequencies, our algorithm’s computational cost is shown to increase rather insignificantly compared to the conventional time-domain correlation method using finite impulse response filters.

Figure 1

Theoretical (dashed curves) and numerical (labeled by “$+$”’s) scaling of the computational cost with $f_{\min }$ for the FIR (red upper curve) and IIR (blue lower curve) method for one template, fixing $f_{\max }=2000\mathrm{Hz}$. The theoretical scaling is based on Eqs. (52, 54) [see Sec. 3e2], numerical values are taken from Table , column 15.

Figure 2

Illustration of the phase function $\Phi (f)$ vs frequency for the presumed parameter $A$ (blue lower solid curve) and its neighboring parameter $A+\Delta A$ (red upper solid curve), the linear shift of the blue line to match the red line (green dashed curve), and a piecewise approximation (black segmented dashed lines) of the red line by shifting segments from the blue line. It shows that with smaller frequency intervals, it is easier to match phases arising from different intrinsic parameters.

Figure 3

Schematic diagram of the IIR filtering process for a template with parameter $A+p\Delta A_{\mathrm{cb}}+q\Delta A_{\mathrm{fb}}$. The first part is the IIR filtering for a member of the coarse bank, $A+p\Delta A_{\mathrm{cb}}$, which produces a range of filter outputs, labeled by $U_1\dots U_{l_M}$. These are grouped into $M$ groups of summed IIR results $V_1,\dots ,V_M$. The result for $A+p{\Delta }_{\mathrm{cb}}+q\Delta A_{\mathrm{fb}}$ is obtained by combining these $V_J$’s after each one is multiplied by $d_J(q\Delta A_{\mathrm{fb}})$ and shifted by $\Delta t_c^J(q\Delta A_{\mathrm{fb}})$. The entire data analysis process still computes $N_{\mathrm{fb}}$ filter results, by including $N_{\mathrm{cb}}$ possible $p$’s and $N_{\mathrm{fb}}/N_{\mathrm{cb}}$ possible $q$’s for each $p$. [In the special case of $q=0$, the $V_J$’s are directly summed without having to go through multiplications and time shifts.] The down-sampling or up-sampling process is not shown.

Figure 4

Matches achievable with a Newtonian-chirp signal at $A+\Delta A$, by various templates built for $A$, using initial LIGO noise spectral density for a $(1.4+1.4)M_{\odot }$ NS-NS binary. Black solid curves corresponds to the result for a Newtonian-chirp template, therefore the match is equal to unity at $\Delta A=0$. Red solid curve corresponds to that using IIR filters, while red dashed curve corresponds to the interpolated match that can be recovered by using 6 filter subgroups.

Figure 5

Analysis with overlapping data segments. The two horizontal lines represent two adjacent data stretches used for FFT. The lower data segment starts data accumulation with a delay of $T_{\mathrm{latency}}$ relative to the upper one. The duration of the overlap between the two stretches is that of the longest signal in the template bank (see text in Sec. 5b).

Figure 6

Computational cost as a function of $T_{\mathrm{latency}}$ for a straightforward FFT analysis with overlapping data segments (solid line) and for the IIR filter method with down-sampling technique (“$+$” symbols) and without (“$\times$” symbols) for real-time filtering with one template of a $(1.4+1.4)M_{\odot }$ binary. The upper panel shows the cost for aLIGO and the lower one for ET. The dotted lines illustrate the equal cost between the FFT and IIR method and the corresponding latencies. The computational cost of the FFT method is calculated from Eq. (109) with the longest template taken to be that of $(1+1)M_{\odot }$ binary and sampling rate $S=4096\mathrm{Hz}$. The IIR data are from Table  (column 15) [with down sampling] and Eq. (112) [without down sampling].