Model-agnostic gravitational-wave background characterization algorithm

TL;DR

This paper tackles the problem that fixed power-law models of the gravitational-wave background may be inadequate as detector sensitivity improves. It introduces a transdimensional, knot-based spline interpolation framework implemented via reversible-jump MCMC to flexibly model the GWB spectrum $\Omega_{\rm GW}(f)$ and, in a hierarchical setup, the merger-rate evolution $R(z)$; knots can be added or removed to adapt model complexity, with multiple interpolation schemes (linear, cubic, Akima) to match different signal shapes. Through injections that mimic a BBH background, subtraction residuals, and first-order phase-transition–inspired spectra, the method demonstrates robust recovery of spectral features and turnover frequencies, and shows how next-generation detectors like CE enable precise spectral and population inferences. The approach also enables joint, multi-parameter analyses (e.g., $R(z)$) from the stochastic background, offering a flexible, model-agnostic path toward understanding both astrophysical and cosmological GW sources as detector networks evolve. This transdimensional framework reduces prior-volume penalties and provides a principled means to compare models while remaining adaptable to diverse potential signals in $\Omega_{\rm GW}(f)$ and beyond.

Abstract

As ground-based gravitational-wave (GW) detectors improve in sensitivity, gravitational-wave background (GWB) signals will progressively become detectable. Currently, searches for the GWB model the signal as a power law; however, deviations from this model will be relevant at increased sensitivity. Therefore, to prepare for the range of potentially detectable GWB signals, we propose an interpolation model implemented through a transdimensional reversible-jump Markov chain Monte Carlo algorithm. This interpolation model foregoes a specific physics-informed model (of which there are a great many) in favor of a flexible model that can accurately recover a broad range of potential signals. In this paper, we employ this framework for an array of GWB applications. We present three dimensionless fractional GW energy density injections and recoveries as examples of the capabilities of this spline interpolation model. We further demonstrate how our model can be implemented for hierarchical GW analysis on $Ω_{\rm GW}$.

Figures (13)

• Figure 1: A visualization of the RJMCMC implemented in this work. In both panels, we show the injected signal (black dashed line) and the injected signal plus Gaussian white noise (blue points). The error bars on the blue points correspond to one standard deviation of the Gaussian white noise. In the top panel, the gray boxes correspond to the uniform priors that the interpolating knots (yellow stars) are allows to move within [see Eq. \ref{['eq:priors']}]. The knots are interpolating using cubic splines (red lines). The bottom panel shows the death proposal (see Sec. \ref{['sec:proposals']}) where we have turned off one knot (gray star) and interpolated the remaining knots to fit the signal. This schematic highlights the transdimensional nature of the RJMCMC that we implement to fit injected GW signals with various detector sensitivities.
• Figure 2: Cumulative signal-to-noise ratio (SNR) curves for the three different injections considered in this work: BBH background (purple), subtraction background (red), and first-order phase transition (FOPT) signal (brown). For reference, we include the scaling of the cumulative SNR on the observation time, for the BBH signal case: 1 month (dotted line), 1 year (full line), and 2 years (dashed line). This dependence scales identically for the different signals. The SNR for stochastic signals employed here is defined as the optimal filter SNR in Allen:1999. Vertical lines mark the frequency values that correspond to where 99% of the SNR is accumulated.
• Figure 3: Results for the injection and recovery of a BBH-informed $\Omega_{\rm GW}$, given by Eq. \ref{['eq:omgw_astrophys']}, assuming A+ sensitivity with a range of observation times. On the left, we show the 95% confidence recovery envelopes for the posteriors of the injected signal (solid black line). On the right, we show the number of knots in each posterior draw for each observation time as a heat map. We find that at two years of observing time, we can confirm a detection of the injected $\Omega_{\rm GW}$. However, the range in number of knots marginally decreases as the observation time increases, which signifies that the A+ sensitivity remains low enough for the posterior distributions to fit for noise.
• Figure 4: The same injection and recovery as Fig. \ref{['fig:cbc_injection']} but with a CE sensitivity curve. On the top left, we show the 95% confidence recovery envelopes for the posteriors of the injected signal (solid black line). The bottom left panel shows the residuals, $\Omega_{\rm GW, res}$, which are obtained by subtracting the $95$% confidence intervals in the top left panel from the injected signal. On the right, we show the number of knots in each posterior draw for each observation time as a heat map. We are able to recover the injection well up to $f\sim 100$ Hz for all observing times with the number of knots converging as observation time increases.
• Figure 5: Recovered posteriors of the unresolved BNS signal $\Omega_{\rm BNS, residual}$ after subtracting resolved signals with SNR$>8$ using CE noise. The injected signal we use is from Sachdev:2020bkk. The left plots shows the recovered posteriors for $N_{\rm max} = 30$ and the top right plots shows the recovered posteriors for $N_{\rm max} = 2$. Each color envelope represents 95% confidence for the recovered posteriors for various observation times for the simulated CE noise (legend). The bottom plots provide the residuals $\Omega_{\rm BNS, residual;res}$ for the 95% confidence envelopes subtracted from the injected signal, $\Omega_{\rm BNS, residual}$, from the top row. Limiting $N_{\rm max} = 2$ reveals PL posteriors that track the injection at later frequencies, while allowing $N_{\rm max} = 30$ detects a low-frequency subtraction feature. Mostly positive residuals $\Omega_{\rm BNS, residual;res}$ indicate that fitting a non-PL signal with a PL biases the fit, causing posterior draws miss sections of the injection and consequentially underfit the PL portion.