The First Upper Bound on the Non-Stationary Gravitational Wave Background and its Implication on the High Redshift Binary Black Hole Merger Rate

TL;DR

This work targets non-stationary features of the stochastic gravitational-wave background (SGWB) by introducing and constraining its spectral covariance, a frequency–frequency correlation in $\Omega_{\rm GW}(f)$. Using cross-correlation of LVK detectors in the O3 and O4a runs, the authors construct a spectral covariance estimator and compare it to model templates derived from astrophysical (ABH) and primordial (PBH) black-hole populations, with mass distributions and redshift evolutions encoded in $P_{\rm ABH}(m)$, $P_{\rm PBH}(m)$, $\mathcal{R}_{\rm ABH}(z)$, and $\mathcal{R}_{\rm PBH}(z)$. No significant spectral covariance is detected; the analysis yields the first upper limits on the covariance amplitude between 25 Hz and 30 Hz, translating into upper bounds on the high-redshift merger rate $\mathcal{R}(z=1)$ that depend on the assumed mass distribution (PBH vs ABH). These results constrain PBH formation scenarios even with substantial spatial clustering and establish a benchmark for future, more sensitive SGWB searches with next-generation detectors. The methodology—template-based spectral covariance estimation from cross-correlated detector data—promises enhanced power to distinguish ABH and PBH contributions as detector sensitivity and frequency coverage improve.

Abstract

The high redshift merger rate and mass distribution of black hole binaries (BHBs) is a direct probe to distinguish astrophysical black holes (ABHs) and primordial black holes (PBHs), which can be studied using the Stochastic Gravitational-Wave Background (SGWB). The conventional analyses solely based on the power spectrum are limited in constraining the properties of the underlying source population under the assumption of a non-sporadic Gaussian distribution. However, recent studies have shown that SGWB will be sporadic and non-Gaussian in nature, which will cause non-zero 'spectral correlation' depending on the high redshift merger rate and mass distribution of the compact objects. In this work, we present the first spectral covariance analysis of the SGWB using data from the LIGO-Virgo-KAGRA collaboration during the third and the first part of the fourth observing runs. Our analysis indicates that the current spectral correlation is consistent with non-stationary noise, yielding no detection from the current data and providing only upper bounds between frequencies in the range 25 Hz to 100 Hz. This upper bound on the spectral correlation translates to the upper bounds on the mass-dependent merger rate of PBHs between $2.4\times10^{4}$ and $2.3\times10^{2} \rm ~Gpc^{-3}yr^{-1}$ (at ${\rm z} = 1 $ ) with a log-normal mass distribution with median masses between $20 ~M_{\odot}$ and $120 ~M_{\odot}$. This provides a stringent upper bound on the PBH merger rate at high redshift and hence puts constraints on the PBH formation scenario even in the presence of large spatial clustering. In the future, detection of this signal will lead to direct evidence of the high-redshift black hole population using gravitational waves.

Figures (9)

• Figure 1: Schematic overview of the analysis pipeline for estimating the spectral covariance in the GW strain data. The procedure begins with the strain data from the LIGO detectors, from which the spectral covariance signal is computed. Model-dependent templates of the spectral covariance matrix are then generated using simulated populations of BHBs. An optimal statistic is constructed by performing a weighted average of the covariance structure with these templates, which is finally used to derive model-dependent upper limits on the spectral covariance amplitude of the SGWB.
• Figure 2: Covariance matrix $\mathcal{C}_{S}(f_1, f_2)$ of $\Omega_{\rm GW}(f)$ (see Eq. \ref{['Covs']}), computed for a short time bin of $\Delta T = 192$ seconds. The panels show the covariance between different frequency modes for the ABH population with (a) $M_{\mu}=40\,M_{\odot}$ and (b) $M_{\mu}=120\,M_{\odot}$. The color scale indicates $\log_{10}[\mathcal{C}_{S}(f_1, f_2)]$.
• Figure 3: Covariance matrix $\mathcal{C}_{S}(f_1, f_2)$ of $\Omega_{\rm GW}(f)$ (see Eq. \ref{['Covs']}), computed for a short time bin of $\Delta T = 192$ seconds. The panels show the covariance between different frequency modes for the PBH population with (a) $M_{\rm c}=40\, M_{\odot}$ and (b) $M_{\rm c}=120\, M_{\odot}$. The color scale indicates $\log_{10}[\mathcal{C}_{S}(f_1, f_2)]$.
• Figure 4: The signal spectral covariance matrix $\hat{\mathcal{C}}_{S,w}^{IJ}(f_1, f_2)$ (see eq. \ref{['Cov_weight_t']}) of the strain cross-correlation obtained by cross-correlating the strain data from H1 and L1 during O3 and O4a of the LVK observing runs. The covariance is obtained over the segments of the short-time Fourier transform signal (coarse-grained at a frequency resolution of 1/32 Hz) with $\Delta T = 192$ seconds. A symmetric logarithmic (symlog) color scale is used to display the wide dynamic range of $\hat{\mathcal{C}}_{S,w}^{IJ}(f_1, f_2)$ values, capturing both positive and negative correlations.
• Figure 5: Most likely value of the spectral covariance amplitude, $\mathcal{A}_{\rm min}\,\mathcal{C}^{\rm model}_{S}(25,30)$, corresponding to the value of $\mathcal{A}$ that minimizes $\chi^{2}_{\rm model}$, shown as a function of $M_{\mu}$ for the ABH population and $M_{c}$ for the PBH population. This figure illustrates how the best-fit spectral covariance amplitude varies with the characteristic mass scale of the binary population. For the ABH models, the inferred amplitude remains approximately constant over the range of $M_{\mu}$ considered, whereas for PBH models, it exhibits a systematic increase with $M_{c}$.