Constraints on cosmological coupling from the accretion history of supermassive black holes

Coupling of black hole mass to the cosmic expansion has been suggested as a possible path to understanding the dark energy content of the Universe. We test this hypothesis by comparing the supermassive black hole (SMBH) mass density at $z=0$ to the total mass accreted in AGN since $z=6$, to constrain how much of the SMBH mass density can arise from cosmologically-coupled growth, as opposed to growth by accretion. Using an estimate of the local SMBH mass density of $\approx 1.0\times10^{6}\,$M$_{\odot}\,$Mpc$^{-1}$, a radiative accretion efficiency, $\eta$: $0.05<\eta<0.3$, and the observed AGN luminosity density at $z\approx 4$, we constrain the value of the coupling constant between the scale size of the Universe and the black hole mass, $k$, to lie in the range $0<k\stackrel{<}{_{\sim}}2$, below the value of $k=3$ needed for black holes to be the source term for dark energy. Initial estimates of the gravitational wave background using pulsar timing arrays, however, favor a higher SMBH mass density at $z=0$. We show that if we adopt such a mass density at $z=0$ of $\approx 7.4\times 10^{6}\,$M$_{\odot}\,$Mpc$^{-1}$, this makes $k=3$ viable even for low radiative efficiencies, and may exclude non-zero cosmological coupling. We conclude that, although current estimates of the SMBH mass density based on the black hole mass -- bulge mass relation probably exclude $k=3$, the possibility remains open that, if the GWB is due to SMBH mergers, $k>2$ is preferred.