The Nanohertz Gravitational-Wave Background as an Energy Budget: Pricing the NANOGrav Signal Against the Supermassive Black Hole Binary Reservoir

The NANOGrav 15-year data set reports a Hellings-Downs-correlated stochastic background with characteristic strain amplitude A = 2.4 x 10^-15 at f = 1/yr for a power-law spectrum (Agazie et al. 2023). This paper converts that amplitude into an energy budget. Integrating the energy-density spectrum across the pulsar-timing band from 1e-9 to 1e-7 Hz gives Omega_GW = 2.45 x 10^-8 (Omega_GW h^2 = 1.12 x 10^-8), a present-day energy density of 1.88 x 10^-17 J/m^3, or 1.17 x 10^-4 eV/cm^3, formally equivalent to Delta N_eff = 0.0020, about 1.1 per cent of the integrated bound Omega_GW h^2 < 1e-6. Priced against a local supermassive black hole mass density of 4.6 x 10^5 solar masses per Mpc^3, the in-band energy density is 0.67 per cent of the reservoir rest-mass energy density, so at the canonical 5 per cent merger efficiency at least 13.5 per cent of the local mass density must have been processed through mergers. The measured amplitude is cheap for the Universe and full-price for the black hole reservoir.