Observational appearance and photon rings of non-singular black holes from anisotropic fluids

TL;DR

The study develops the optical appearance of a non-singular black hole in EiBI gravity with an anisotropic fluid, showing a horizon near $r_h\approx 2M$ and a modified photon sphere that yields a smaller shadow and altered photon rings compared to Schwarzschild. Through backward ray-tracing of optically thin disks and GLM-based emission models, the authors quantify photon-ring positions and Lyapunov dynamics, finding $\gamma_{ps}\approx 2.972$ and a radius sequence that converges to the critical curve, yet current observational and disk-model uncertainties obscure a clear distinction from Schwarzschild. They estimate $\gamma_{ps, width}\approx 3.2$, a ~8% deviation from the theoretical value, and $t_{ps, width}\approx 4.5M$, which is modestly shorter than Schwarzschild’s $5.196M$, implying degeneracy in static images. The results underscore the need for dynamical probes (e.g., hot-spots or GW ringdowns) to break degeneracies and robustly test strong-field gravity with black-hole imaging.

Abstract

We consider the optical appearance of a non-singular, spherically symmetric black hole from Eddington-inspired Born-Infeld gravity coupled to anisotropic fluids. Such a black hole has a single (external) horizon located very near the Schwarzschild radius, $r_h=2M$, while its surface of unstable bound geodesics (photon sphere) is located at a moderately shortened radius than its Schwarzschild counterpart. Relying on a geometrically and optically thin accretion disk with a monochromatic emission described by suitable adaptations of Standard Unbound profiles previously employed in the literature, we generate images of this solution, which displays relevant modifications to the typical photon ring and central brightness depression features found in black hole images. In this sense, we fit the width of the two first photon rings in order to reconstruct the Lyapunov exponent of nearly-bound geodesics characterizing the theoretical ratio of successive rings. Such an exponent is tightly attached to observational features of photon rings such as their relative intensities in time-averaged images and the time-scale of hot-spots. Our results point out that non-singular black holes of this type are hard to distinguish from their Schwarzschild counterparts using this method alone, since the theoretical, numerical, disk-modeling, and observational uncertainties are too entangled with one another to allowing a neat distinction of such an exponent. It also points out to the need of incorporating dynamical settings such as hot-spots or quasi-normal modes from gravitational wave ringdowns as a way to circumvent such difficulties.

Figures (6)

• Figure 1: Behavior of the metric components $g_{tt}$ (blue) and $g_{rr}^{-1}$ (red) as a function of $r/M$ for the EiBI non-singular black hole with the choice of parameters of Sec. \ref{['sec:cop']} as compared to the Schwarzschild one (dashed black).
• Figure 2: Ray-tracing of null geodesics from a screen at $r_o=100M$ to a non-singular EiBI black hole with the choice of parameters of Sec. \ref{['sec:cop']}. We draw the rays that correspond to the shadow, i.e. those corresponding to $b<b_{ps}\simeq 4.366$, in shade of gray. The rest of the rays find a turning point and go back to infinity, and are colored using the intrinsic impact parameter of the trajectory.
• Figure 3: Transfer functions $r_n$ for $n=0,1,2$, which correspond to the disk's direct image and the first and second photon ring emissions, respectively, for the non-singular EiBI black hole and the Schwarzschild solution. We plot as a vertical line the theoretical position of the shadow as determined by the respective critical impact parameters.
• Figure 4: Optical appearance of a geometrically and optically thin emission disk, with a face-on orientation, near a non-singular EiBI black hole with one horizon supported by an anisotropic fluid. The emitted $I_{em}$ (left plots) and observed $I_{obs}$ (middle plots) intensities are normalized using the maximum value of the emitted intensity $I_{0}$. The emission models employed are the GLM ones, introduced as suitable adaptations of those introduced in Gralla:2020srx via Eq.(\ref{['eq_SUem']}), with the parameters of \ref{['eq:parameters']}. The optical appearances plots (right figures) display strong differences in the visibility of their photon rings according to whether the emission extends to the event horizon (top and middle rows) or remains beyond the ISCO (bottom row), and furthermore show moderate differences in the effective region of emission of their photon rings between the non-singular black hole and the Schwarzschild one (solid and dashed curves in the $I_{obs}$ plots.
• Figure 6: Main peaks of emission fitted to the GLM 3 profile