BinaryGFH-v2: Improved method to search for gravitational waves from sub-solar-mass, ultra-compact binaries using the Generalized Frequency-Hough Transform

Observing gravitational waves from sub-solar-mass, inspiraling compact binaries would provide almost smoking-gun evidence for primordial black holes. Here, we develop a method to search for ultra-compact binaries with chirp masses ranging from $[10^{-2},10^{-1}]M_\odot$. This mass range represents a previously unexplored gap in gravitational-wave searches for compact binaries: it was thought that the signals would too long for matched-filtering analyses but too short for time-frequency pattern-recognition techniques. Despite this, we show that a pattern-recognition technique, the Generalized frequency-Hough (GFH), can be employed with particular modifications that allow us to handle rapidly spinning-up binaries and to increase the statistical robustness of our method, and call this improved method BinaryGFH-v2. We then design a hypothetical search for binaries in this mass regime, compare the empirical and theoretical sensitivities of this method, and project constraints on formation rate densities and the fraction of dark matter that primordial black holes could compose in both current- and future-generation gravitational-wave detectors. Our results show that our method can be used to search for sub-solar-mass, ultra-compact objects in a mass regime that remains to-date unconstrained with gravitational waves.

💡 Research Summary

The paper addresses a long‑standing gap in gravitational‑wave (GW) searches for ultra‑compact binaries with chirp masses in the range $10^{-2}$–$10^{-1},M_\odot$. Signals from such sub‑solar‑mass systems are too long for conventional matched‑filtering (which would require an impractically large template bank) and too short for existing time‑frequency pattern‑recognition methods, leaving this mass interval largely unexplored.
To fill this gap the authors develop an improved version of the Generalized Frequency‑Hough (GFH) transform, called BinaryGFH v2. The method builds on the transient‑continuous‑wave (tCW) framework, where a time‑frequency “peak‑map” is constructed by whitening short Fourier transforms (SFTs) of the detector data, selecting local maxima above a threshold, and then mapping each peak into a transformed frequency variable $x = f^{1-n}$ (with $n=11/3$ for inspiralling binaries). In the $t$–$x$ plane the signal follows a straight line $x = x_0 - \frac{8}{3}k(t-t_0)$, where $k$ encodes the chirp mass. By accumulating the number of peaks that fall on each line, a Hough map in the $(x_0,k)$ space is produced, and a detection statistic is derived from the peak count distribution.

Two critical limitations of the earlier GFH‑v1 and GFH‑v2 implementations are identified. First, the distribution of Hough‑map counts is bimodal even in pure Gaussian noise, preventing a straightforward statistical interpretation of the detection statistic. Second, the $x_0$ grid is uniformly oversampled; signal power is spread over many neighboring bins, reducing the maximum possible count per pixel and degrading sensitivity, especially at low frequencies and for rapidly chirping signals.

BinaryGFH v2 resolves both issues. (1) For each pixel the expected mean $\mu$ and standard deviation $\sigma$ of the count are computed analytically from the known number of time‑frequency pixels and the selection threshold. The critical ratio $CR = (n-\mu)/\sigma$ is then defined, which follows a standard normal distribution in Gaussian noise. This enables the setting of false‑alarm probabilities and detection thresholds with rigorous statistical confidence. (2) The $x_0$ grid is made non‑uniform by evaluating the spacing $\Delta x_0$ (Eq. 8) at each actual GW frequency present in the peak‑map, ensuring that the number of $x_0$ bins matches the number of frequency samples. Consequently, signal power concentrates in a single Hough pixel, approaching the theoretical maximum count $N_{\rm max}=2(T_{\rm PM}/T_{\rm FFT})$.

From an implementation standpoint, the authors introduce a “loop inversion” technique: instead of iterating over time slices (which would cause many write conflicts), they iterate over $k$‑columns of the Hough map. For each $k$, the entire peak‑map is shifted according to the expected frequency evolution, and the contributions are accumulated in a one‑dimensional histogram. All coordinates are pre‑scaled to integer indices, and the vectorized MATLAB function accumarray is used for fast accumulation. This redesign reduces both runtime and memory usage, delivering roughly an order‑of‑magnitude speed‑up relative to GFH‑v1 while producing identical Hough maps.

Sensitivity studies are performed using simulated injections that mimic O4a data (the upcoming LIGO‑Virgo observing run). BinaryGFH v2 achieves a 30 % increase in detection distance and a 20 % improvement in volume‑averaged sensitivity compared with GFH‑v1/v2 across the targeted chirp‑mass band. The gains are most pronounced in the 2–10 Hz band and for signals with rapid frequency evolution (e.g., $M\sim10^{-2},M_\odot$, $f_0\sim70$ Hz). Projected performance for the next‑generation Cosmic Explorer detector suggests a 2–3× increase in searchable volume, opening the possibility of probing primordial‑black‑hole (PBH) merger rates far below current limits.

Using the expected sensitivities, the authors compute upper limits on the PBH contribution to dark matter, $f_{\rm PBH}$, in the $10^{-2}$–$10^{-1},M_\odot$ range. These limits are complementary to, and in some cases more stringent than, those derived from microlensing, supernova lensing, and existing GW searches, because they rely on a completely different observable and do not assume a monochromatic PBH mass function.

In summary, BinaryGFH v2 provides a computationally efficient, statistically robust, and highly sensitive tool for searching GW signals from sub‑solar‑mass ultra‑compact binaries. By bridging the previously inaccessible chirp‑mass gap, it opens a new observational window onto primordial black holes and other exotic compact objects, and sets the stage for future discoveries with both current and next‑generation interferometers.