Accelerated inference of binary black-hole populations from the stochastic gravitational-wave background

TL;DR

This work addresses the challenge of inferring binary black hole population parameters from the stochastic gravitational-wave background (SGWB) when individual CBC events cannot be resolved. It combines importance sampling with neural-emulator interpolation to rapidly compute the SGWB mean spectrum and explicitly includes the intrinsic variance due to finite source realizations, enabling robust population inference with 3G detectors. The approach demonstrates accurate constraints on redshift evolution and mass-distribution parameters across Madau-Dickinson, PP+MD, and BPL+MD models, using CE and ET sensitivities and a one-year observing period. The results highlight the practical impact of accounting for SGWB intrinsic variance and point to future enhancements, such as treatment of the variance as a free parameter and more advanced uncertainty quantification techniques.

Abstract

Third-generation ground-based gravitational wave detectors are expected to observe $\mathcal{O}(10^5)$ of overlapping signals per year from a multitude of astrophysical sources that will be computationally challenging to resolve individually. On the other hand, the stochastic background resulting from the entire population of sources encodes information about the underlying population, allowing for population parameter inference independent and complementary to that obtained with individually resolved events. Parameter estimation in this case is still computationally challenging, as computing the power spectrum involves sampling $\sim 10^5$ sources for each set of hyperparameters describing the binary population. In this work, we build on recently developed importance sampling techniques to compute the SGWB efficiently and train neural networks to interpolate the resulting background. We show that a multi-layer perceptron can encode the model information, allowing for significantly faster inference. We test the network assuming an observing setup with CE and ET sensitivities, where for the first time we include the intrinsic variance of the SGWB in the inference, as in this setup it presents a dominant source of measurement noise.

Figures (16)

• Figure 1: Black-hole binary population models. Left: Primary-mass distributions for both the PP (blue) and BPL (green) models. For this example, we set $\alpha = 3.5$, $\delta_m = 4.5$, $\lambda_{\mathrm{peak}} = 0.04$, $m_{\mathrm{max}} = 100$, $m_{\mathrm{min}} = 4$, $m_{\mathrm{peak}} = 34$, $\sigma_{\mathrm{peak}} = 4$ for PP, and $\alpha_1 = 2.5$, $\alpha_2 = 7.5$, $b = 0.5$, $\delta_m = 4.5$, $m_{\mathrm{max}} = 100$, $m_{\mathrm{min}} = 4$ for BPL. Right: Redshift distribution for the MD model (pink) assuming $\gamma = 3.2$, $\kappa = 5.9$, and $z_{\mathrm{peak}} = 1.9$.
• Figure 2: Mean (top panels) and variance (bottom panels) of the SGWB power spectra $\Omega_{\mathrm{GW}}(f)$ computed over 100 realizations for each hyperparameter configuration in the training set only. Results are shown separately for the PP+MD (left panels, blue) and BPL+MD (right panels, blue) models. The spectra span 400 logarithmically spaced frequencies between 10 Hz and 2000 Hz, capturing the sensitivity band of ground-based 3G detectors.
• Figure 3: Mean relative error of the MLP predictions on the test set as a function of frequency. Relative error as a function of frequency for the MLP-based models. The PP+MD model (red) and the BPL+MD model (grey) are compared across the frequency range of interest. Both axes are on a logarithmic scale. In the frequency range relevant to inference, the relative error remains well below 1%.
• Figure 4: Relative standard deviation $\sigma_{\mathrm{rel}}(f; \mathbf{\Lambda})$, as defined in Eq. \ref{['eq::rel_std']}, computed across 100 realizations for the PP+MD (blue) and BPL+MD (green) models. Shaded bands show the standard deviation of $\sigma_{\mathrm{rel}}(f; \mathbf{\Lambda})$ across different hyperparameter configurations within each model. The curves are shown up to 750 Hz, which corresponds to the frequency range where the SGWB signal is strongest relative to the expected detector noise (see Sec. \ref{['sec:inference']}). The relative variance is lowest at lower frequencies, which is favorable for inference. At higher frequencies, the variance increases as the number of contributing sources drops and the spectrum signal becomes weaker.
• Figure 5: Instrumental uncertainties $\sigma_{\rm detector}$ converted to dimensionless SGWB power spectrum units. Red and blue curves correspond to ET and CE, respectively. The black curve shows an example SGWB realization ${\overline{\Omega}_{\rm GW}}$ from the training set, while the red curve represents its interpolated intrinsic standard deviation $\sigma_{\Omega_{\rm GW}}$. Note the relative amplitude between these quantities: $\sigma_{\Omega_{\rm GW}}$ is orders of magnitude larger than $\sigma_{\rm detector}$ in the lower portion of the frequency spectrum, thus dominating the measurement below $\sim 100$ Hz, while the opposite is true for frequencies higher than $\sim 300$ Hz, for either detector setup.