Are black holes a serious threat to scalar field dark matter models?

TL;DR

The paper addresses whether ultra-light scalar field dark matter can form enduring halos around central black holes. By treating the scalar as a test field on Schwarzschild spacetime and solving the Klein-Gordon equation, it shows that true stationary, localized configurations cannot exist due to horizon-related energy issues, but resonant quasi-stationary states can persist for long times when a potential well appears in the effective Schrödinger-like potential. Numerical evolutions of carefully constructed initial data reveal exponential energy decay with discrete resonant frequencies, indicating long-lived clouds that leak energy into the horizon very slowly. These findings suggest that scalar-field dark matter halos around super-massive black holes could survive on cosmological timescales for realistic parameters, though extensions to self-gravity and dynamical black-hole growth are needed for fuller astrophysical applicability.

Abstract

Classical scalar fields have been proposed as possible candidates for the dark matter component of the universe. Given the fact that super-massive black holes seem to exist at the center of most galaxies, in order to be a viable candidate for the dark matter halo a scalar field configuration should be stable in the presence of a central black hole, or at least be able to survive for cosmological time-scales. In the present work we consider a scalar field as a test field on a Schwarzschild background, and study under which conditions one can obtain long-lived configurations. We present a detailed study of the Klein-Gordon equation in the Schwarzschild spacetime, both from an analytical and numerical point of view, and show that indeed there exist quasi-stationary solutions that can remain surrounding a black hole for large time-scales.

Figures (13)

• Figure 1: The effective potential $M^2 V_{\textrm{eff}}$ for $\ell=1$, and different values of the parameter $M\mu$. The shaded regions represent the "resonance bands" where the potential has a local minimum (see Section \ref{['sec:resonant']} below). Notice that for $M\mu=0$ and $M\mu \gtrsim 0.466$ the local minimum does not appear. For the case $M\mu=0$ we recover the usual Regge-Wheeler potential. The case with $M\mu=0.3$ will be considered later on in our future examples and it appears as a solid line to emphasize it.
• Figure 2: Resonance band in parameter space. The cut-off in $M\mu$ is reached for values so large that condition \ref{['condition_real']} is no longer satisfied. For $\ell=0$ this condition is $M\mu < 1/4$, while for $\ell=1$ one finds $(M\mu)^2<-9/32+\sqrt{20577}/288 \approx 0.217$ , which implies $M \mu \lesssim 0.466$.
• Figure 3: Numerical solutions of the Schrödinger equation \ref{['Eq:TimeIndependentSchrodinger']} for different values of $M\omega$. The upper panel corresponds to $M\omega=0.295$, while the lower panel corresponds to $M\omega=0.29619$. Both solutions have been scaled so that the maximum overall amplitude is equal to $1$.
• Figure 4: Resonant frequencies. We show a log plot of the ratio $A_{\rm out}/A_{\rm in}$ between the amplitude $A_{\rm out}$ of the solution for $r^*/M<0$, and the amplitude $A_{\rm in}$ for the solution inside the potential well $r^*/M>10$.
• Figure 5: Top panel: Effective potential $V_{\rm eff}(r)$ for $\ell=1$ and $M\mu=0.3$; and $(M\omega_i)^2$ for each of the solutions shown in the bottom panel. Bottom panel: Radial energy density $\rho_E(r)$ for the first ($n_1$) pseudo-resonant mode (type \ref{['type.resonant']}) and a solution (labeled $n_x$) with eigenvalue (labeled $w_x^2$) below the local minimum of $V_{\rm eff}$ (type \ref{['type.nonresonant.w<min']}). An arbitrary rescaling was chosen for each $\rho_E$ in this plot in order to improve visualization. The vertical lines intersect the roots of $V_{\rm eff}=w^2$, a total of three for the pseudo-resonant mode ($n_1$), and only one for the non-resonant mode (note that the root of the non-resonant mode is very close to the first root of the resonant mode, so the two corresponding vertical lines are indistinguishable in the figure).