Bridging the $\mu \mathrm{Hz}$ Gap in the Gravitational-Wave Landscape with Binary Resonances

Gravitational-wave (GW) astronomy is transforming our understanding of the Universe by probing phenomena invisible to electromagnetic observatories. A comprehensive exploration of the GW frequency spectrum is essential to fully harness this potential. Remarkably, current methods have left the $\mu \mathrm{Hz}$ frequency band almost untouched. Here, we show that this $\mu \mathrm{Hz}$ gap can be filled by searching for deviations in the orbits of binary systems caused by their resonant interaction with GWs. In particular, we show that laser ranging of the Moon and artificial satellites around the Earth, as well as timing of binary pulsars, may discover the first GW signals in this band, or otherwise set stringent new constraints. To illustrate the discovery potential of these binary resonance searches, we consider the GW signal from a cosmological first-order phase transition, showing that our methods will probe models of the early Universe that are inaccessible to any other near-future GW mission. We also discuss how our methods can shed light on the possible GW signal detected by NANOGrav, either constraining its spectral properties or even giving an independent confirmation.

Figure 1

SGWB sensitivity curves of current and future GW experiments, as well as our forecasts. Each curve is a 95% confidence upper limit ($\mathrm{SNR}=2$), with shaded regions extending up to $\mathrm{SNR}=20$. Solid curves indicate existing results from the LIGO/Virgo/KAGRA Collaboration [72, 73], gravimeter monitoring of the Earth’s normal modes [74], Doppler tracking of the Cassini spacecraft [75], pulsar timing by the Parkes PTA [2], and indirect constraints from $N_{\mathrm{eff}}$ [71], as well as our forecast present-day sensitivities for binary resonance searches with binary MSPs, LLR, and SLR, which are presented for the first time here. Hatching indicates the new region probed by our present-day forecasts. Dashed curves indicate our binary resonance forecast sensitivities for 2038, along with expected bounds from ET [54], LISA [5], SKA [4], and the proposed km-scale atom interferometer AION [55], as well as improved $N_{\mathrm{eff}}$ constraints [71]. Dotted curves show various potential SGWB signals in the $\mu \mathrm{Hz}$ band. The purple curves indicate a possible signal associated with the common process identified by NANOGrav (NANOGrav CP) [3], while the overlaid pink curves show the inferred amplitude for the NANOGrav CP when assuming a ${\mathrm{\Omega }}_{\mathrm{gw}}\sim f^{2/3}$ spectrum, as expected for supermassive binary black holes (SMBBHs). The yellow curves show two FOPT spectra at temperatures $T_{*}=2\mathrm{GeV}$ and 200 GeV, peaking at $f\approx 1\mu \mathrm{Hz}$ and $\approx 100\mu \mathrm{Hz}$, respectively. The orange curve shows the predicted spectrum from a population of horizonless supermassive black hole (SMBH) mimickers [76]. The pale green curves show the predicted spectra from ultralight bosonic condensates around SMBHs [77], with boson masses varying from ${10}^{-20}\mathrm{eV}$ (leftmost curve) to ${10}^{-15}\mathrm{eV}$ (rightmost curve).

Figure 2

Forecast exclusion regions of the FOPT parameter space for various SGWB searches at 2038 sensitivity. Here, $T_{*}$ is the temperature at which the FOPT occurs, $\alpha$ is the energy density released by the FOPT in units of the radiation density at the transition epoch, $\beta /H_{*}$ is the inverse duration of the transition in units of the Hubble rate at the transition epoch, and $v_w$ is the bubble wall velocity.