Bayesian reconstruction of primordial perturbations from induced gravitational waves

TL;DR

This work presents a Bayesian framework to infer the small-scale primordial curvature power spectrum $\mathcal{P}_{\zeta}(k)$ and the early-Universe equation of state from scalar-induced gravitational waves (SIGWs). The method represents $\mathcal{P}_{\zeta}(k)$ with interpolating splines whose number of nodes is selected via Bayesian evidence, enabling flexible, feature-rich reconstructions while guarding against overfitting. By applying the approach to mock data and Pulsar Timing Array measurements, the authors demonstrate accurate recovery of spectral features and, in favorable cases, the underlying equation of state, across standard radiation domination, early matter domination with a reheating transition, and general $w$ scenarios. The results highlight the potential of SIGW observations to probe both the small-scale primordial power spectrum and the thermal history of the early universe, with implications for primordial black hole formation and beyond, and they provide a publicly available codebase for reproducing and extending the analyses.

Abstract

The formation of primordial black holes or other dark matter relics from amplified density fluctuations in the early universe may also generate scalar-induced gravitational waves (GW), carrying vital information about the primordial power spectrum and the early expansion history of our universe. We present a Bayesian approach aimed at reconstructing both the shape of the scalar power spectrum and the universe's equation of state from GW observations, using interpolating splines to flexibly capture features in the GW data. The optimal number of spline nodes is chosen via Bayesian evidence, aiming at balancing complexity of the model and the fidelity of the reconstruction. We test our method using both representative mock data and recent Pulsar Timing Array measurements, demonstrating that it can accurately reconstruct the curvature power spectrum as well as the underlying equation of state, if different from radiation.

Figures (15)

• Figure 1: Test spectra chosen for power spectrum reconstruction assuming radiation domination. First column: $\mathcal{P}_\zeta$ and $\Omega_{\rm GW}$ for the broken power law profile of section \ref{['sec:bpl_res']}. Second column: $\mathcal{P}_\zeta$ and $\Omega_{\rm GW}$ for the template with oscillatory features of section \ref{['sec_temof']}. Third column: $\mathcal{P}_\zeta$ and $\Omega_{\rm GW}$ for the peaked profile of section \ref{['sec_shapeak']}. For each column in the second row, we also show the corresponding error bars on $\Omega_{\mathrm{GW}}$, as dictated by \ref{['eq:DOGW']}.
• Figure 2: Top: The reconstructed $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$ for the BPL model of section \ref{['sec:bpl_res']}, with parameters as in eq \ref{['eq_pabplc1']}, marginalised over the models with different number of nodes. The different shaded regions correspond to $68\%, 95\%$ and $99.7\%$ credible intervals for $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$. Bottom: The evidence $\log \mathcal{Z}$ as a function of the number of nodes.
• Figure 3: Reconstruction of the BPL power spectrum using if only the IR (top) and UV (bottom) part of the spectrum were observed by the experiment. The different shaded regions correspond to $68\%, 95\%$ and $99.7\%$ credible intervals for $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$.
• Figure 4: Top: The reconstructed $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$ for the oscillatory model of eq \ref{['eq:osc']}, marginalised over the models with different number of nodes. See section \ref{['sec_temof']}. The different shaded regions correspond to $68\%, 95\%$ and $99.7\%$ credible intervals for $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$. Bottom: The evidence $\log \mathcal{Z}$ as a function of the number of nodes.
• Figure 5: Top: The reconstructed $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$ for the peaked model of eq \ref{['eq:peaked']}, marginalised over the models with different number of nodes. See section \ref{['sec_shapeak']}. The different shaded regions correspond to $68\%, 95\%$ and $99.7\%$ credible intervals for $\mathcal{P}_\zeta$ and $\Omega_{\mathrm{GW}}$. Bottom: The evidence $\log \mathcal{Z}$ as a function of the number of nodes.