Matter environments around black holes: geodesics, light rings and ultracompact configurations

TL;DR

This work analyzes how spherically symmetric dark-matter environments alter black-hole spacetimes by modeling the DM as Einstein clusters with Hernquist, NFW, and Jaffe density profiles. It derives the geodesic structure, including circular orbits, light rings, and their stability, and computes associated Lyapunov exponents, using both analytical (low-compactness) and numerical (Post-Schwarzschild) approaches. In the high-compactness regime, the study uncovers ultracompact configurations with additional light rings, marginally stable orbits, and possible secondary horizons, which imprint distinctive ringdown signatures such as long-lived trapped modes and echoes. The results establish a framework for incorporating environmental corrections into electromagnetic and gravitational-wave observables and point to important extensions to rotating backgrounds and EMRI systems relevant for LISA.

Abstract

Astrophysical black holes are invariably embedded in matter environments whose gravitational influence can alter key strong-field features of the spacetime. In this work, we investigate the impact of spherically symmetric dark-matter distributions on black hole geometry, geodesic structure, and ringdown phenomenology. Modeling the surrounding matter through Einstein clusters, we construct self-consistent spacetimes for three widely used density profiles - the Hernquist, Navarro-Frenk-White (NFW), and Jaffe models - and examine how their near-horizon behavior modifies the location and stability of circular timelike and null geodesics, including the innermost stable circular orbit (ISCO) and light rings. In the low-compactness regime, we derive analytical expressions showing that environmental effects generically shift the ISCO inward and the principal light ring outward, leading to parametric deviations in their associated orbital frequencies and Lyapunov exponents. At higher compactness, we explore the emergence of additional light rings, marginally stable orbits, and secondary horizons, identifying the regions of parameter space in which these ultracompact configurations arise. Using time-domain evolutions of scalar perturbations, we demonstrate how such structures can imprint characteristic signatures on the ringdown signal, including long-lived trapped modes and echo-like modulations associated with multiple potential barriers. Our results provide a unified framework for assessing environmental effects around black holes and highlight the importance of matter-induced corrections for interpreting upcoming electromagnetic and gravitational-wave observations.

Figures (7)

• Figure 1: We consider the parameter space characterized by $(\rho_0,a_0)$ in the first column and the parameter space characterized by $(a_0,M)$ in the second column. The Hernquist, NFW and Jaffe models are presented in each row (from top to bottom). The red regions represent the formation of a new event horizon, the lighter gray region represents the formation of two additional LRs inside the matter distribution, and the darker gray regions represent the formation of a region of unstable timelike orbits also inside the matter distribution. In all plots we are considering $r_\text{in}=6M_\text{BH}$ and in the bifurcation plots we fixed $a_0=10^3 R_S$.
• Figure 2: In the first column of the above figure we display the effective potential of geodesic motion with increasing compactness for each model. In the second column we follow the changing radial coordinates of the additional LRs and MSCOs as the total mass increases, also highlighting relevant regions between these orbits. For the bifurcation plots we fixed $r_\text{in}=R_\text{S}$ and $a_0=10^3 R_\text{S}$. We highlight that in the white region we have stable orbits.
• Figure 3: In this figure we show the relative error between the second order PS and the numerical solutions for all the models. We chose the length parameter to be $a_0=10^3R_\text{S}$, and the inner radius to be $r_\text{in}=R_\text{S}$. For the numerical solutions we chose $r_\text{cutoff}=10^5R_\text{S}$.
• Figure 4: We have the time evolution comparison between the cases with and without the DM halo. In the left panel, we have low compactness regime, and in the right panel, we have compactness $C>C^\text{LR}$ for each case.
• Figure 5: We have the potential (left panel) and the time evolution (right panel) for a BH with a DM halo for a Hernquist density profile. Increasing the DM halo compactness leads to significant modifications in the ringdown oscillation frequency and decay time.