Noble gravitational atoms: Self-gravitating black hole scalar wigs with angular momentum number

TL;DR

This work constructs self-gravitating, quasi-stationary scalar configurations around a black hole, termed noble gravitational atoms, distinguished by an angular momentum number $\ell$ and built from $2\ell+1$ classical fields. Using horizon-penetrating generalized Eddington-Finkelstein coordinates, the authors derive a nonlinear eigenvalue problem for the complex frequency $s$ and solve for $\ell=0,1,2$ to obtain density profiles and spectra. They show that away from the horizon these configurations closely reproduce $\ell$-boson-star solutions, while horizon-near regions exhibit $\ell$-dependent spikes or dips that contribute negligibly to the total mass. The results span a broad range of astrophysical regimes, including very large, dilute, long-lived structures that resemble ultralight dark matter cores, and they suggest that combining multiple angular momenta could extend galactic-halo modeling on even larger scales.

Abstract

We present new spherically symmetric solutions of the Einstein-Klein-Gordon equations in a quasi-stationary approximation that describe self-gravitating scalar field configurations around a black hole, including angular momentum number $\ell$. An approach analogous to the one which gives rise to $\ell$-boson stars is used here to construct self-gravitating ``gravitational atoms" with $\ell\ge0$. We refer to these new solutions as {\it noble gravitational atoms}, by analogy with noble atoms, which are characterized by closed electron shells. We show that, in the proper limit, noble gravitational atoms approach $\ell$-boson stars globally, displaying noticeable differences only in a region very close to the event horizon. Noble gravitational atoms with $\ell>0$ sometimes present density maxima located at relatively large radii, with small density close to the horizon for $\ell>1$. Furthermore, they do not always present the typical density spike at the event horizon if $\ell > 0$; on the contrary, they sometimes exhibit a small dip there. When $\ell=0$, a spike can appear, but its contribution to the total mass density is always negligible. The size, density, and lifetime of these objects vary significantly depending on the parameters, being in some cases as large as galaxies, as dilute as dark matter, and as long-lived as the Universe itself.

Figures (22)

• Figure 1: Left panel: Effective potentials with $\ell$ from $0$ to $3$. In each case, the value of $\mu M$ is chosen to be 75% of the maximum for which there is still a well, as reported in table \ref{['t:cl']}. The potentials have been rescaled by their asymptotic value, $V_{\rm eff}(\infty)=\mu^2$. Right panel: Effective potential with "energy levels" $\omega_{n \ell}^2$ and density profiles for $\ell=1$ and $\mu M=0.3$. The vertical dotted lines indicate the "classical turning points".
• Figure 2: Density $\rho$ of gravitational atoms with $\ell=0$, $\mu M=0.2$, and different amplitudes $A$. For large $\mu M$, the density profiles tend to be sharp and do not resemble a boson star for any value of $A$.
• Figure 3: Density $\rho$ of gravitational atoms with $\ell=0$, $\mu M=0.01$, and different amplitudes $A$. For intermediate $\mu M$, some configurations become wider, and for certain values of $A$, they start to resemble boson stars.
• Figure 4: Density $\rho$ of gravitational atoms with $\ell=0$, $\mu M=0.003$, and different amplitudes $A$. For small $\mu M$, some configurations are identical to boson stars, except for the black hole spike at their center.
• Figure 5: Spike comparison between an $\ell=0$ gravitational atom (red line) and a cold dark matter model (green line), both constructed with parameters appropriate for the Milky Way.