Microlensing of non-singular black holes at finite size: a ray tracing approach

TL;DR

The paper tackles microlensing by static, spherically symmetric non-singular black holes and horizonless compact objects. It combines a fourth-order parametrized post-Newtonian (PPN) lensing framework for point sources with a ray-tracing approach and simple radiative transfer for extended sources to generate time-dependent lightcurves as a moving lens crosses the line of sight, validating the methods by recovering point-source results in the appropriate limit. The study finds that finite-size sources tend to exhibit larger magnifications for non-singular models at finite size, contrasting with the point-source Schwarzschild prediction, and provides detailed near-zone and far-zone analyses for the Hayward, Minkowski core, and Simpson--Visser metrics. The results offer observably testable signatures for regular black holes and horizonless counterparts, informing future microlensing surveys and constraining regulator-length scenarios in quantum-gravity-inspired spacetimes.

Abstract

We study the gravitational microlensing of various static and spherically symmetric non-singular black holes (and horizonless, non-singular compact objects of similar size). For pointlike sources we extend the parametrized post-Newtonian lensing framework to fourth order, whereas for extended sources we develop a ray tracing approach via a simple radiative transfer model. Modelling non-relativistic proper motion of the lens in front of a background star we record the apparent brightness as a function of time, resulting in a photometric lightcurve. Taking the star radius to smaller values, our numerical results approach the theoretical predictions for point-like sources. Compared to the Schwarzschild metric in an otherwise unmodified lensing geometry, we find that non-singular black hole models (and their horizonless, non-singular counterparts) at finite size tend to feature larger magnifications in microlensing lightcurves, contrary to the point-source prediction.

Figures (14)

• Figure 1: Visualization of the lensing setup utilized in this paper. For $\mathcal{B} \not=0$ the upper (lower) image is denoted by a plus (minus). Generally, we utilize $\vartheta$ for the image positions, $\mathcal{B}$ for the source positions, and $\alpha$ for the total deflection angle.
• Figure 2: The microlensing event OGLE-2011-BLG-0462 / MOA-2011-BLG-191, expressed in the conventions of this article. The errors are shown in orange, the curve between measured intensities is the solid line in blue, and the green area corresponds to the naive fit \ref{['eq:naive-fit']}.
• Figure 3: The parametrized post-Newtonian lightcurves (zeroth-order contributions: solid line; second-order contributions: dashed lines; third-order contributions: dotted lines) for the parameters described in the text. Note that the zeroth-order curve is rescaled by a factor of $1/20$. Higher order PPN corrections reduce the magnification, in line with the intuition from the fact that non-singular black holes generally have smaller horizons than their equal-mass Schwarzschild counterparts.
• Figure 4: Visualization of the ray tracing setup. Instead of a moving lens, it is computationally easier to model the lens in the origin and consider instead a moving star and screen that share the same $x$-coordinate for their center point. For the purposes of this paper we restrict the motion of the lens to be purely along the $x$-axis.
• Figure 5: Luminosity of a lensed star under passage of a black hole described by the scenarios $S$, $H_-$ and $H_+$ at $T=0.25$. The Einstein angle is visualized at the bottom left, the dashed cyan line is the projected outline of the black hole horizon (if present), and the yellow dashed line is the projected Einstein radius. The images are individually normalized to their brightest spot and then amplified by a factor of 2.5 to enhance contrast. The subcritical Hayward image features a smaller black hole horizon, and novel features appear in the center of the horizonless supercritical Hayward case on the rightmost panel.