Gravitational Wave Memory Imprints on the CMB from Populations of Massive Black Hole Mergers

Aims: To showcase and characterise the rich phenomenology of temperature fluctuation patterns that are imprinted on the CMB by the gravitational wave memory (GWM) of massive black hole (BH) mergers. Methods: We analyse both individual binaries as well as populations of binaries, distributed in local cosmological boxes at a given redshift. Results: The magnitude of the temperature fluctuations scales primarily as a function of binary total mass and pattern angular scale, and accumulates as a random-walk process when populations of mergers are considered. Fluctuations of order $\sim {10}^{-10}$ K are easily reached across scales of $\sim 1^{\mathrm{\prime }}$ to $\sim 1^{\circ }$ for realistic volumetric merger rates of 10${}^{-3}$ Mpc${}^{-3}$ Gyr${}^{-1}$, as appropriate for massive galaxies at $z=1$. We determine numerically that GWM temperature fluctuations result in a universal power spectrum with a scaling of $P(k)\propto k^{-2.7}$. Conclusion: While not detectable given the limitations of current all-sky CMB surveys, our work explicitly shows how every black hole merger in the Universe left us its unique faint signature.