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

This paper presents a simple analytic framework combining local Schwarzschild geometry with cosmological dynamics to derive a relation between the apparent angular size of a black hole's shadow and the angular diameter distance, demonstrating that cosmic expansion significantly affects shadow measurements only for high-redshift sources.

The Gist

Imagine a black hole as a cosmic "hole" in space that is so heavy it swallows anything that gets too close, including light. Around this hole, there is a specific zone where light can orbit in a circle before either falling in or escaping. This creates a dark silhouette, or a "shadow," against the bright background of the universe.

For a long time, scientists have been able to calculate the size of this shadow very accurately, but they usually pretend the universe is static and empty around the black hole. This paper asks a simple question: What happens to the size of this shadow if we remember that the universe is actually expanding?

Here is the breakdown of the paper's findings using everyday analogies:

1. The "Rubber Sheet" Analogy

Think of the universe as a giant, stretching rubber sheet.

• The Black Hole: Imagine a heavy bowling ball sitting on this sheet. It creates a deep dip (the gravity well).
• The Shadow: The "shadow" is the size of the dark circle you see if you look at the bowling ball from far away.
• The Expansion: Now, imagine someone is slowly pulling the edges of the rubber sheet outward, stretching it.

The author, Debarshi Mukherjee, wanted to know: Does stretching the rubber sheet change how big the bowling ball's shadow looks to an observer standing on the sheet?

2. The "Zoom Lens" Effect

The paper finds that the expansion of the universe acts a bit like a cosmic zoom lens, but in a very specific way.

• For Nearby Objects (Like our neighbors): If you look at a black hole in our own galaxy (like the one at the center of the Milky Way, called Sgr A*), the universe is so huge compared to the black hole that the "stretching" is invisible. It's like trying to see the effect of a slow-growing tree on the size of a pebble sitting at its base. The shadow looks exactly the same as if the universe weren't expanding at all.
• For Distant Objects (Cosmic Travelers): If you look at a black hole that is billions of light-years away, the stretching of the universe matters. The paper provides a simple formula to calculate how the shadow's size changes based on how far away it is (its "redshift").

3. The "Non-Linear" Surprise

One of the most interesting findings is that the shadow doesn't just get smaller and smaller as things get farther away.

• Imagine looking at a streetlamp. As you walk away, it gets smaller.
• However, because of the specific geometry of our expanding universe, there is a point (around a certain distance) where the "lens" of the universe changes behavior.
• The paper shows that for extremely distant black holes, the shadow might actually stop shrinking and start getting slightly larger again, or at least stop shrinking as fast as you'd expect. It's a weird, non-linear effect caused by the shape of the universe itself.

4. The Bottom Line

The paper concludes with a very practical reality check:

• For us today: The effect is so tiny that for the black holes we can actually see right now (like M87* or Sgr A*), the expansion of the universe is completely negligible. We don't need to worry about it when taking pictures of them.
• For the future: If we ever build telescopes powerful enough to see black holes at the very edge of the observable universe, this "stretching" effect will become important.

In summary: The paper builds a simple mathematical bridge between the tiny world of black holes and the huge world of the expanding universe. It proves that while the universe's expansion does technically change the size of a black hole's shadow, it's a "whisper" for nearby objects and only becomes a "voice" for objects at the very edge of the cosmos. It's a way to connect the physics of the very small (black holes) with the physics of the very large (the universe) without needing a supercomputer, just a simple equation.

Technical Summary

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

Problem Statement
While the apparent shadow of a black hole is a well-established probe of strong-field general relativity in asymptotically flat spacetimes (e.g., the Event Horizon Telescope observations of M87* and Sgr A*), the influence of cosmic expansion on the shadow's apparent angular diameter remains less explored. Standard theoretical analyses typically neglect large-scale cosmological expansion, an approximation valid for nearby astrophysical black holes where the Schwarzschild radius is orders of magnitude smaller than the Hubble radius. However, as observational precision improves, it is necessary to determine how the expansion of the universe modifies the apparent size of a black hole's shadow, particularly for sources at cosmological distances or in the early universe.

Methodology
The paper presents a simple analytic framework to estimate the shadow size of a non-rotating (Schwarzschild) black hole embedded in an expanding universe. The approach avoids full general-relativistic ray-tracing simulations in favor of a perturbative analysis that connects the static Schwarzschild geometry with the dynamic, expanding background.

1. Theoretical Framework: The authors combine the local Schwarzschild geometry with large-scale cosmological dynamics using the McVittie metric, which describes a point mass in an isotropic, homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) background.
2. Derivation:
   • The critical impact parameter ($b_c = 3\sqrt{3}M$) for the photon sphere is derived from the Schwarzschild metric.
   • This local scale is projected onto the sky of a comoving observer using the standard cosmological angular-diameter distance, $D_A(z)$.
   • A compact analytic relation is derived for the shadow angular size $\alpha(z)$:
     $$\alpha_{\text{shadow}}(z) = \arcsin\left( \frac{3\sqrt{3}GM/c^2}{D_A(z)} \right)$$
   • For low redshifts ($z \ll 1$), a small-$z$ analytic approximation is developed, expanding $D_A(z)$ to show corrections proportional to the Hubble parameter $H_0$ and the deceleration parameter $q_0$.
3. Numerical Implementation: The authors perform numerical evaluations of the derived expressions using Python (NumPy, SciPy) for a flat $\Lambda$CDM universe (Planck 2018 parameters: $H_0 = 67.4$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_m = 0.315$, $\Omega_\Lambda = 0.685$). They also compare results against Einstein–de Sitter and de Sitter limiting cosmologies.

Key Contributions and Results

• Analytic Relation: The paper establishes a direct, tractable link between the local relativistic geometry of the photon sphere and global cosmological expansion, encapsulated in Equation (9). This relation demonstrates that the shadow's angular size is inversely proportional to the angular-diameter distance $D_A(z)$.
• Redshift Dependence:
  • Nearby Sources: For local black holes like Sgr A* ($z \sim 10^{-6}$) and M87*, the cosmological correction is negligible (difference $< 10^{-10}$ compared to the static estimate). The shadow size is dominated by the Schwarzschild geometry.
  • High-Redshift Sources: For hypothetical black holes at cosmological distances ($z \gtrsim 1$), the effect becomes non-negligible. The angular size decreases as $1/z$ initially but exhibits a non-monotonic trend due to the behavior of $D_A(z)$ in a $\Lambda$CDM universe. Specifically, $D_A(z)$ reaches a maximum near $z \approx 1.6$ and decreases for higher redshifts, causing the apparent shadow size to slightly increase beyond this turning point.
• Cosmological Sensitivity: Numerical comparisons show that while the absolute differences between cosmologies (e.g., $\Lambda$CDM vs. EdS vs. de Sitter) are tiny, the shadow size is qualitatively sensitive to the background expansion rate and curvature.
• Detectability: For intermediate-mass black holes ($M \sim 10^5 M_\odot$) at cosmological distances, the apparent angular diameter drops below $10^{-3}$ microarcseconds, rendering them undetectable even by future high-resolution interferometers.

Significance and Claims
The paper claims to provide a "pedagogically transparent and physically motivated extension" of black hole shadow theory to a cosmological context. Its primary significance lies in:

• Conceptual Bridge: It highlights the subtle connection between local gravitational lensing and the cosmological metric, demonstrating that the black hole shadow, often viewed as a purely local feature, encodes information about the background expansion through $D_A(z)$.
• Pedagogical Utility: The analytic framework is designed to be accessible to undergraduate and graduate students, offering a tractable method to understand how strong-gravity optics and cosmological expansion interact.
• Theoretical Foundation: While the authors acknowledge that current observational uncertainties (e.g., accretion disk emission, spin effects) dwarf cosmological corrections for known systems, the derived relations serve as a foundation for future work. They suggest that in principle, the shadow size could act as a probe of cosmological parameters, particularly for very long baseline interferometry (VLBI) observations of supermassive black holes at higher redshifts.

The authors remain modest regarding immediate observational impact, noting that the effect is "completely negligible" for currently observed systems but theoretically significant for understanding the interplay between general relativity and cosmology. Future extensions suggested include rotating black holes (Kerr–de Sitter) and inhomogeneous expansion models.