Schwarzschild black holes as low-pass filters of fuzzy dark matter granularity

Abstract

We study the relaxation of a complex massive scalar field around a supermassive black hole, modeling the near-horizon dynamics of Fuzzy Dark Matter (FDM) with spatial granularity. The field is initialized as a broadband, anisotropic multi-mode random field in momentum space with various spectral widths $σ$, decomposed into spherical harmonics up to $\ell_{\max}=100$, and numerically evolved on a finite domain using a specialized radial basis adapted to the background spatial geometry. We track the evolution of each multipole and calculate its Noether flux across both the event horizon and the outer boundary. Since angular momentum barriers suppress the horizon absorption of high-$\ell$ modes, while high-frequency radial components and fine-scale angular structures dissipate via outward radiation and horizon accretion, the Schwarzschild spacetime acts as an effective low-pass filter. Regardless of the initial spectral width, all configurations converge toward a universal late-time relaxation state dominated almost exclusively by low multipoles ($\ell \le 2$) and small radial wavenumbers, with the initial bandwidth $σ$ determining the timescale of global charge depletion. For concrete physical estimates, we set the boson mass to $m_b=10^{-22}\,\mathrm{eV}/c^2$ and the black hole mass to $M_{\mathrm{BH}}=6.5\times10^9 M_\odot$.