Simple Analytic Estimate of Black Hole Shadow Size in an Expanding Universe

TL;DR

The paper investigates how cosmic expansion alters the apparent size of a non-rotating black hole shadow by embedding Schwarzschild geometry in expanding backgrounds and linking the shadow angle to the cosmological angular-diameter distance via $\alpha_{\text{shadow}}(z)=\arcsin\left[\frac{3\sqrt{3}\,GM/c^2}{D_A(z)}\right]$. It derives top-level relations from the Schwarzschild photon sphere, extends them with the McVittie embedding, and provides a small-$z$ analytic approximation that exposes the dependence on $H_0$ and $q_0$. Numerical results using Planck 2018 cosmology show that cosmological corrections are negligible for nearby sources but become nontrivial at high redshift, with a non-monotonic behavior driven by $D_A(z)$. The work highlights a conceptual bridge between strong-field gravitational lensing and cosmological geometry, offering a compact framework that can be extended to more general spacetimes such as Kerr–de Sitter and non-$\Lambda$CDM cosmologies to map additional cosmological imprints on black-hole imaging.

Abstract

The apparent shadow of a black hole provides one of the most direct probes of strong-field general relativity. While the shadow size in asymptotically flat spacetimes is well understood, the influence of cosmic expansion on its apparent angular diameter remains less explored. In this work, we present a simple analytic framework to estimate the shadow size of a non-rotating black hole embedded in an expanding universe. By combining the local Schwarzschild geometry with large-scale cosmological dynamics through the McVittie and Kottler metrics, we derive a compact relation between the shadow angular size and the angular diameter distance $D_A(z)$. This approach captures the essential dependence on cosmological parameters such as the Hubble constant $H_0$ and the cosmological constant $Λ$, while remaining analytically tractable. We further perform numerical estimates to quantify the redshift dependence of the apparent shadow size, showing that the effect of cosmic expansion is negligible for nearby sources but becomes relevant for high-redshift black holes. Our results demonstrate a clear conceptual connection between strong-gravity optics and cosmological expansion, providing a pedagogically transparent and physically motivated extension of black hole shadow theory to a cosmological context.

Figures (1)

• Figure 1: Numerical computation of the apparent shadow angular radius $\alpha_{\text{shadow}}(z)$ for black holes of different masses in a $\Lambda$CDM universe with $H_0=67.4~\text{km s}^{-1}\text{Mpc}^{-1}$, $\Omega_m=0.315$, and $\Omega_\Lambda=0.685$. The shaded region marks the redshift range where $D_A(z)$ reaches its maximum value, producing a turning point in $\alpha(z)$.