Energetic Ceilings of Astrophysical Gravitational-Wave Backgrounds

Abstract

Every astrophysical stochastic gravitational wave (GWB) is limited by the amount of rest mass available to be converted into gravitational radiation. Here we derive a population-agnostic scaling law that places an absolute energetic ceiling on stochastic backgrounds across the entire GW frequency spectrum, from nanoHertz to kilohertz. We apply this framework to bound the backgrounds from supermassive black hole binaries, intermediate-mass black hole captures by supermassive black holes in AGN disks, extreme mass-ratio inspirals, binary neutron stars, Population III remnants, and stellar-mass binary black holes. We find that the energetic ceiling for supermassive black hole binaries is $A \leq 1.6^{+0.3}_{-0.3} \times 10^{-15}$ at a reference frequency of $1\,{\rm yr}^{-1}$. This astrophysical GWB ceiling is within $1σ$ with the GWB amplitude reported by NANOGrav, EPTA, and PPTA, implying that the current observed signal is consistent with being powered by a population of ultramassive black holes ($M_\bullet \gtrsim 10^{10}\,M_\odot$). Finally, we demonstrate that the total astrophysical background from all channels combined cannot exceed $Ω_{\rm gw} \sim 10^{-7}$.

Figures (2)

• Figure 1: Comparison of the energetic ceiling on the GWB amplitude from SMBHBs with PTA measurements. The blue histogram shows the probability density of the predicted ceiling amplitude using the Liepold2024 local SMBH mass density, yielding $A_{\rm bench} = 1.6^{+0.3}_{-0.3} \times 10^{-15}$ at $f_{\rm ref} = 1\,{\rm yr}^{-1}$ (solid vertical line with shaded $1\sigma$ region). Horizontal error bars show GWB amplitude measurements from PTAs, all referenced to $f = 1\,{\rm yr}^{-1}$ assuming $\gamma = 13/3$: PPTA DR3, $A = 2.04^{+0.25}_{-0.22} \times 10^{-15}$Reardon2023; NANOGrav 15-year, $A = 2.4^{+0.7}_{-0.6} \times 10^{-15}$Agazie2023; EPTA DR2, $A = (2.5 \pm 0.7) \times 10^{-15}$Antoniadis2023; MPTA, $A = 4.8^{+0.8}_{-0.9} \times 10^{-15}$Miles2025; and CPTA, $\log_{10} A = -14.7^{+0.9}_{-1.9}$Xu2023, where the arrow indicates the upper uncertainty extends beyond the plotted range.
• Figure 2: Energetic ceilings on astrophysical GWBs. Solid colored curves show the maximum isotropic energy density $\Omega_{\rm gw}(f)$ allowed by conservation of mass for key populations, truncated at their respective ISCO frequencies: SMBHBs (Sec. \ref{['sec:SMBHB-PTA-Band']}), IMBH--SMBH captures (Sec. \ref{['sec:IMBH-SMBHcapture']}), EMRIs (Sec. \ref{['sec:EMRIs']}), BNS (Sec. \ref{['sec:BNS']}), Population III remnants (Sec. \ref{['sec:POPIII']}), and BBH (Sec. \ref{['sec:stellarBBH']}). The horizontal red line at $\Omega_{\rm gw} \sim 10^{-7}$ marks the absolute limit imposed by the cosmic baryon budget (Sec. \ref{['sec:astroCeilingOmegaGW']}). Note that this is a constraint on the log-frequency integral of $\Omega_{\rm gw}$, effectively a reference budget, rather than a pointwise ceiling at every frequency. The Galactic double white dwarf foreground (gray fill) is shown in the millihertz band (Sec. \ref{['sec:DWD']}), and violins display free-spectral parameters from the NANOGrav 15-year dataset Agazie2023, adjusted to $H_0=67.4$ km s$^{-1}$ Mpc$^{-1}$Planck2020. Sensitivity curves represent 4-year missions for LISA and $\mu$Ares, 5-year BBO, Advanced LIGO design, and 1-year Cosmic Explorer Reitze2019. The apparent convergence of EMRI and IMBH ceilings with the PTA band in $\Omega_{\rm gw}$ reflects the $f^2$ weighting in the conversion from characteristic strain, $\Omega_{\rm gw} \propto f^2 h_c^2$; at fixed energy density, millihertz sources require characteristic strains $\sim 10^5$–$10^6$ times smaller than nanoHertz sources.