Gravitational atoms beyond the test field limit: The case of Sgr A* and ultralight dark matter

TL;DR

This work extends the gravitational-atom paradigm beyond the test-field limit by solving the Einstein–Klein–Gordon equations for a self-gravitating scalar field around a non-rotating black hole in a quasi-stationary regime using horizon-penetrating coordinates. It formulates a nonlinear eigenvalue problem for the complex frequency $s$ and imposes horizon and asymptotic boundary conditions to obtain self-gravitating configurations parameterized by $(M_{BH}, m_\phi, A)$, with the gravitational fine-structure constant $\alpha_G$ guiding the regime $\alpha_G<1/4$. The results show the total mass remains bounded similarly to boson stars, density spikes near the horizon can occur but contribute negligibly to the mass, and an inner BH-dominated region decouples from an outer SF-dominated region, allowing long-lived cores for ultralight DM around SMBHs such as Sgr A*. The study illustrates both nearly test-field-like and strongly self-gravitating regimes for Sgr A*, highlighting a broad parameter space and challenging traditional CDM spike predictions while providing a practical framework for realistic gravitational-atom configurations in galactic contexts.

Abstract

We construct gravitational atoms including self-gravity, obtaining solutions of the Einstein-Klein-Gordon equations for a scalar field surrounding a non-rotating black hole in a quasi-stationary approximation. We resolve the region near the horizon as well as the far field region. Our results are relevant in a wide range of masses, from ultralight to MeV scalar fields and for black holes ranging from primordial to supermassive. For instance, a system with a scalar field consistent with ultralight dark matter and a black hole mass comparable to that of Sagittarius A* can be modeled. A density spike near the event horizon, although present, is negligible, contrasting with the prediction in [P. Gondolo and Silk, Phys. Rev. Lett., 83:1719-1722, 1999] for cold dark matter.

Figures (2)

• Figure 1: Energy density and mass function at $t=0$ for solutions with various values of $\alpha_G$, together with those for BSs with the same central amplitude in each case. The amplitude is $A=10^{-3}$ in all cases except the one indicated as $A=1.7\times 10^{-7}$. Color dotted lines in the density plots indicate each apparent horizon's location, color dashed lines in the mass plots indicate each BH mass, and black dotted lines indicate the pericenter distance of stars S1 and S2 in the case of Sgr A*. The curves belonging to $\alpha_G$ from $3{\times}10^{-4}$ to $3{\times}10^{-7}$ in the first plot are not clearly seen in most of the region because they are superposed almost perfectly with the BS curve.
• Figure 2: $M_T$ vs $R_{99}$ (left panel) and $M_T$ vs $\omega$ (right panel) of solutions with various values of $\alpha_G$, varying $A$. The color map indicates the characteristic time $t_0$. Black dots indicate, for each $\alpha_G$, the points where $M_T=M_{BH}$, $R_{99}=R_{\rm Sch}$ and $(\hbar/m_{\phi}c^2)\omega_n=\sqrt{1-\alpha_G^2/n^2}$ (with $n=1$). Short vertical lines indicate the locations of the test field's effective potential minima.