The Dark Side of the Moon: Listening to Scalar-Induced Gravitational Waves

TL;DR

The paper addresses whether a stochastic background of scalar-induced gravitational waves (SIGWs) in the μHz band, produced by large primordial curvature perturbations that form planetary-mass PBHs, can be used to constrain PBH abundance. It combines a Gaussian, log-normal curvature-power spectrum with two PBH-formation formalisms (threshold statistics and peak theory) and computes the resulting SIGW spectrum, including electroweak-transition effects. It then forecasts LLR, eLO, and eSLR sensitivities to SIGWs using a Fisher approach and translates non-detections into bounds on the curvature power spectrum and PBH abundance, noting dominance of eSLR at the low-mass end and eLO at higher masses. The results imply strong constraints on planetary-mass PBHs and offer a novel probe of early-Universe physics, especially around the EW transition, while also intersecting with microlensing hints.

Abstract

The collapse of large-amplitude primordial curvature perturbations into planetary-mass primordial black holes generates a scalar-induced gravitational wave background in the $μ$Hz frequency range that may be detectable by future Lunar Laser Ranging and Satellite Laser Ranging data. We derive projected constraints on the primordial black hole population from a null detection of stochastic gravitational wave background by these experiments, including the impact of the electroweak phase transition on the abundance of planetary-mass primordial black holes. We also discuss the connection between the obtained projected constraints and the recent microlensing observations by the HSC collaboration of the Andromeda Galaxy.

Figures (5)

• Figure 1: Left Panel: Threshold $\mathcal{C}_{\rm th}$ (red solid) and shape parameter $\alpha_{\rm s}$ (blue dashed) values for different widths $\Delta$ of a log-normal power spectrum. Right Panel: The corresponding evolution of the threshold $\mathcal{C}^{\rm ew}_{\rm th}$ during the EW phase transition for different values of the shape parameter $\alpha_{\rm s}$, normalised with respect to the value of $\mathcal{C}_{\rm th}$ in the standard radiation dominated era.
• Figure 2: Power-law-integrated sensitivity curves (solid lines) of LLR, eLO and eSLR Foster:2025nzf. We also show the first 14 bins of NANOGrav 15 yrs experiment NANOGrav:2023gorNANOGrav:2023hvm and future sensitivity (dashed lines) for planned experiments like SKA Zhao:2013bbaBabak:2024yhu and LISA LISA:2022kgy and AION-km Badurina:2019hstBadurina:2021rgtAbdalla:2024sst. For completeness, we also show in black the SIGW spectra for two benchmark scenarios, with $A=10^{-2}$, $k_{\star}=10^9$ Mpc$^{-1}$, $\Delta=0.1$ (solid) and $\Delta=1$ (dashed).
• Figure 3: Prospects from LLR, eLO and eSLR experiments based on the predicted sensitivity, which provide an upper bound on the power spectrum amplitude $A$ in case no signal is observed for different spectral shape $\Delta$ in the case of a log-normal power spectrum.
• Figure 4: Left Panel: The implied upper limits on PBH abundance using the threshold statistics formalism. The orange and pink regions are respectively the 95% CL allowed region of PBH abundance, obtained by microlensing events in the OGLE+HSC Niikura:2017zjdMroz:2017mvfNiikura:2019kqi (orange) (see also Sugiyama:2021xqg) and in the HSC Sugiyama:2026kpv (pink) data due to the potential existence of PBHs. Right Panel: As in the left panel, but with PBH abundances calculated via the peak theory formalism.
• Figure 5: Evolution of the squared sound speed, $c_s^2$, and the equation of state parameter $w(T)\equiv p/\rho$ during the EW phase transition as a function of the cosmological horizon mass $M_{ k}$ in units of a solar mass.