Enabling multimessenger astronomy with continuous gravitational waves: Early warning and sky localization of binary neutron stars in the Einstein Telescope

Next-generation gravitational-wave detectors will provide unprecedented sensitivity to inspiraling binary neutron stars and black holes, enabling detections at the peak of star formation and beyond. However, the signals from these systems will last much longer than those in current detectors, and overlap in both time and frequency, leading to increased computational cost to search for them with standard matched filtering analyses, and a higher probability that they are observed in the presence of non-Gaussian noise. We therefore present a method to search for gravitational waves from compact binary inspirals in next-generation detectors that is computationally efficient and robust against gaps in data collection and noise nonstationarities. Our method, based on the Hough transform, finds tracks in the time/frequency plane of the detector that uniquely describe specific inspiraling systems. We find that we could detect $\sim 5$ overlapping, intermediate-strength signals (matched-filter signal-to-noise ratio $\rho \approx 58$) without a sensitivity loss. Additionally, we demonstrate that our method can enable multimessenger astronomy: using only low frequencies (2–20 Hz), we could warn astronomers $\sim 2.5$ hours before a GW170817-like merger at 40 Mpc and provide a sky localization of $\sim 20{\mathrm{deg}}^2$ using only one “L” of Einstein Telescope. Additionally, assuming that primordial black holes (PBHs) exist, we derive projected constraints on the fraction of dark matter they could compose, $f_{\mathrm{PBH}}\sim {10}^{-6}–{10}^{-4}$, for $\sim 1–0.1M_{\odot }$ equal-mass systems, respectively, using a rate suppression factor $f_{\sup }=2.5\times {10}^{-3}$. Comparing matched filtering searches to our proposed method at a fixed sensitivity, we find a factor of $\sim 10–50$ speedup when we begin an analysis at a frequency of 5 Hz up to 12 Hz for a system with a chirp mass between $\mathcal{M}\in [1,2]M_{\odot }$.

Figure 1

Chirp signals that show frequency evolution as a function of time to merger, with $\dot{f}$ colored. This figure indicates that we can observe for long periods of time at small $\dot{f}$ at the lower end of this frequency band. From left to right, each curve corresponds to: $\mathcal{M}=[0.4353,0.8706,1.3058,1.7411]M_{\odot }$.

Figure 2

$m_1/m_2$ space describing how long signals would last in the low-frequency (2–20 Hz) band.

Figure 3

Left: normalized sensitivity as a function of $T_{\mathrm{FFT}}$ length, for different chirp-mass systems, when analyzing 2–20 Hz, when varying $T_{\mathrm{obs}}$ and $T_{\mathrm{FFT}}$ in Eq. (9). The peak corresponds to the $T_{\mathrm{FFT}}$ that maximizes our sensitivity to different compact binary coalescences. Right: normalized sensitivity vs observation time, optimized by amount of frequency band analyzed and $T_{\mathrm{FFT}}$, for a signal starting at 2 Hz. The chirp mass is indicated for each curve, and matches those in Fig. 1.

Figure 4

Luminosity distance reach and sensitivity loss with respect to matched filtering colored as a function of chirp mass. Observing for longer periods of time (corresponding to smaller chirp masses) increases the sensitivity loss compared to larger chirp masses, since matched filtering can accumulate more signal power over time compared to the semicoherent generalized frequency Hough, due to reducing the $T_{\mathrm{FFT}}$ length [see Fig. 3]. This is valid for signals that span 2–20 Hz.

Figure 5

Left: minimum detectable amplitude with matched filtering as a function of the frequency at which an analysis begins $f_{\min ,\mathrm{MF}}$. The colored dashed lines represent $h_{0,\min }^{\mathrm{GFH}}$ computed using a threshold $C{R}_{\mathrm{thr}}=5$ when beginning an analysis of a $\mathcal{M}=2M_{\odot }$ system at 5 Hz (red) or 8 Hz (magenta). The intersection of these two horizontal lines with the curve represents when matched filtering and the generalized frequency Hough have equal sensitivities. Right: the computational speedup as a function of $f_{\min ,\mathrm{MF}}$ for an analysis using the generalized frequency Hough starting at 5 Hz (red) and 8 Hz (magenta), with the factor of sensitivity loss in strain with respect to matched filtering colored.

Figure 6

The speedup of our generalized frequency-Hough search with respect to matched filtering as a function of the minimum matched-filtering frequency and sensitivity loss, if we start analyzing at $f_{\min }=5\mathrm{Hz}$ for systems with different chirp masses.

Figure 7

Ten signals have been injected in white noise between 2–4 Hz, and are recovered by the Hough transform. Signals overlapping in time and in frequency can be easily isolated in the Hough plane. $\sqrt{S_n}\sim 8\times {10}^{-24}{\mathrm{Hz}}^{-1/2}$. The amplitude for all signals is the same at $t=0$ ($h_0={10}^{-23}$), but over time, each changes in different ways based on the frequency evolution and the chirp mass of the injected signal. The matched-filter signal-to-noise ratio $\rho \approx 58$. In the case of non-Gaussian, nonstationary noise, this method has been tested extensively in [92, 104, 105].

Figure 8

Efficiency as a function of number of injected signals per peakmap, where 50 simulations at each $N_{\mathrm{inj}}$ were performed. Even for strong signals, efficiency degrades rapidly past $N_{\mathrm{inj}}\sim 30$. “Efficiency” is defined as the fraction of the total number of signals injected across all simulations that are recovered by the generalized frequency Hough.

Figure 9

Normalized sensitivity as a function of the beginning analysis frequency. At $f_{\min }=5\mathrm{Hz}$, we see that we would only be 35% as sensitive to inspiraling systems compared to if we observed them starting from $f_{\min }=2\mathrm{Hz}$.

Figure 10

Finest possible sky localization based on a single-interferometer analysis as a function of chirp mass, assuming that the signal frequency evolution can be completely demodulated after estimating the chirp mass and initial frequency of the system, resulting in a purely monochromatic signal. For an equal-mass system of $1.4M_{\odot }$ ($\mathcal{M}=1.22M_{\odot }$), this plot is consistent with the estimations present in [60] for a single Einstein Telescope (ET) interferometer, though we consider only the 2–20 Hz band.

Figure 11

Left: maximum time available to warn astronomers that a merger of two compact objects will occur at different distances. Right: fraction of coalescence time necessary to observe a signal at a particular chirp mass is colored. At higher distances (and lower masses), more time is needed to obtain a given distance reach. If the source is closer, then we do not have to observe for as long as if the source were farther, at a fixed chirp mass.

Figure 12

Projected merger rate densities, treating each chirp mass as a separate population of compact objects.

Figure 13

Projected constraints on subsolar and solar-mass PBH binaries, for both equal-mass and asymmetric mass ratio binaries, in terms of $f_{\mathrm{PBH}}$ (blue) and $\tilde{f}$ (black). Blue and black axes correspond to the blue and the black curve, respectively. $f_{\sup }=2\times {10}^{-3}$, and $m_1=2.5M_{\odot }$.

Figure 14

The number of templates in a matched filtering search for $\mathcal{M}=[1,3]M_{\odot }$, with a minimal match of 0.97, a minimum frequency of $f_{\min ,\mathrm{MF}}$, and a maximum frequency $f_{\max }=12\mathrm{Hz}$ at one post-Newtonian order.