Shadows of Giants: Constraints on Stupendously Large Black Holes from Negative Sources against the Cosmic Microwave Background

This paper proposes that the shadows of hypothetical stupendously large black holes (SLABs) against the cosmic microwave background serve as a powerful observational constraint, effectively ruling out SLABs with masses exceeding $10^{17}\ M_{\odot}$ within the last scattering surface and limiting their contribution to the cosmic energy density.

The Gist

Imagine the universe as a giant, glowing stage curtain made of the Cosmic Microwave Background (CMB). This curtain is the oldest light in existence, a faint, warm glow that fills every corner of space. Now, imagine a black hole not as a vacuum cleaner sucking things in, but as a giant, invisible hole punched through that curtain.

This paper, titled "Shadows of Giants," is about hunting for a specific type of monster: the Stupendously Large Black Hole (SLAB).

What is a SLAB?

We know about stellar black holes (the size of a city) and supermassive black holes (the size of a solar system, like the one in the center of our galaxy). But a SLAB is a hypothetical beast with a mass of one trillion suns or more. To put that in perspective, a SLAB is heavier than an entire galaxy.

For a long time, scientists thought these things were just science fiction. But this paper asks: If they exist, why haven't we seen them yet?

The "Negative Light" Trick

Usually, when we look for black holes, we look for bright spots. We see gas swirling around them, heating up, and glowing like a cosmic lighthouse.

But this paper proposes a different way to look: look for the dark spots.

Because a SLAB is so massive, its "shadow" (the area where light cannot escape) is enormous. If a SLAB is floating in space, it blocks the background glow of the CMB curtain behind it.

• The Analogy: Imagine holding a giant, pitch-black bowling ball in front of a bright, glowing wall. From your perspective, the ball doesn't just look black; it looks like a hole in the wall where the light is missing.
• In astronomy terms, this is a "negative source." It's a spot that is darker than the empty space around it.

The Time-Travel Advantage

Here is the clever part of the paper. Usually, things look smaller the farther away they are. But the universe is weird. Because of how space expands, objects at a certain distance (about 1.6 billion light-years away and further) actually appear larger on the sky the farther back in time you look.

• The Metaphor: Imagine looking through a telescope that acts like a funhouse mirror. As you look deeper into the past (higher redshift), the "funhouse" makes the distant objects appear bigger, not smaller.
• The Result: A SLAB that is 100 times farther away might cast a shadow that looks bigger than a closer one. This means the universe is actually easier to search for these monsters the further back in time we look. The "early universe" is the best place to find them because their shadows are huge and easy to spot against the bright CMB.

The "Growth" Problem

The author, Brian Lacki, knows there's a catch. Black holes usually eat matter. If a SLAB is eating gas and stars, it might:

1. Grow: If it eats a lot, it gets bigger, but it also gets brighter.
2. Glow: The gas falling in gets hot and shines like a blinding spotlight. This "positive light" could hide the "negative shadow."
3. Obscure: The gas around it might be thick and foggy, blocking our view of the shadow entirely.

However, the paper argues that even if they are glowing, they would be so bright that we would have seen them by now as "positive sources." If they are not glowing (perhaps because they formed in a special way that prevents them from eating), then their shadows are the only thing we see. And we haven't seen any shadows.

The Verdict: The Giants Are Missing

By scanning the sky with powerful telescopes (like Planck and the South Pole Telescope) and looking for these "negative spots" or "dark holes" in the cosmic background, the author sets strict rules:

• If SLABs exist: They must be incredibly rare.
• The Limit: We can rule out the existence of SLABs with masses over $10^{17}$ suns within the observable universe. If they were there, we would have seen their giant shadows blocking the cosmic background light.
• The Density: Even if they exist, they can't make up more than a tiny fraction (less than one hundred-thousandth) of the universe's total mass.

Summary

Think of the universe as a giant, glowing whiteboard. If there were giant black holes (SLABs) floating around, they would leave massive, dark fingerprints on the board.

This paper says: "We've looked at the whiteboard with our best microscopes, and we don't see any fingerprints."

Therefore, these stupendously large black holes are either non-existent, or they are so rare that they are practically ghosts. The universe is safe from these invisible giants, at least for now.

Technical Summary

1. Problem Statement

Stupendously Large Astrophysical Black Holes (SLABs) are hypothetical black holes with masses exceeding $10^{12} M_\odot$ (a trillion solar masses), potentially reaching up to $10^{18} M_\odot$ or higher. While Primordial Black Holes (PBHs) are a candidate for dark matter, SLABs are generally considered too massive to constitute the bulk of dark matter but could exist in low numbers across the Universe.

Previous constraints on SLABs have relied on:

• Energy release from early accretion distorting the Cosmic Microwave Background (CMB).
• High-energy signatures from dark matter halos.
• Isocurvature perturbations in the CMB.
• Gravitational lensing effects.

However, these constraints are often model-dependent or limited to specific redshift ranges. The paper addresses a gap in the literature: the direct visibility of SLAB shadows against the CMB. Unlike standard black holes, SLABs have Schwarzschild radii large enough (comparable to or larger than galaxies) to cast detectable "shadows" (negative flux sources) on the CMB, even at cosmological distances.

2. Methodology

The author employs a multi-faceted approach combining standard cosmology, black hole physics, and observational data analysis.

A. Theoretical Framework

• Shadow Geometry: The paper calculates the projected cross-section ($A_{proj}$) of a Schwarzschild black hole, which is $27\pi (GM/c^2)^2$. This area acts as an absorber, blocking CMB photons.
• Negative Luminosity: The shadow creates a "negative luminosity" ($L_{BH}$) relative to the CMB background. Using the Stefan-Boltzmann law, the deficit is calculated as $L_{BH} \propto -M_{BH}^2 (1+z)^4$. Crucially, because the CMB temperature scales as $(1+z)$, the negative luminosity increases dramatically with redshift.
• Angular Diameter Distance ($D_A$): A key cosmological insight utilized is that in $\Lambda$CDM cosmology, $D_A$ peaks at $z \approx 1.6$ and decreases for higher redshifts. This means a fixed-mass object appears larger on the sky at very high redshifts ($z > 1.6$).
• Volume Integration: The method integrates the comoving volume where $D_A$ is small enough for a shadow of a given mass to be resolved by current instruments. This includes the "early volume" ($z > 1.6$) where $D_A$ is small, offering a vast volume for detection.

B. Accretion Effects

The paper rigorously analyzes how accretion might obscure or outshine the shadow:

• Growth Models: It considers the Bertschinger (B85) self-similar infall model, where SLABs grow as $M_{BH}(z) \propto (1+z)^{-1}$.
• Luminosity vs. Shadow: It compares the negative flux of the shadow against the positive flux from accretion luminosity ($L_{acc}$). It determines regimes where the shadow dominates (massive black holes, high redshift) versus where accretion dominates.
• Obscuration: It calculates the Thomson optical depth ($\tau_T$) of infalling ionized gas to determine if the shadow is hidden. It finds that for certain parameters, the gas may be optically thick, but for others (or if the gas is neutral/cooling), the shadow remains visible.

C. Observational Constraints

The author utilizes data from three major microwave surveys to set upper limits on SLAB abundance:

1. Planck PCCS2: Covers nearly the entire sky but has lower sensitivity.
2. SPT-SZ: Smaller sky coverage but deeper sensitivity.
3. SPT-3G: Very deep, small-field survey.

The methodology assumes a null detection of negative point sources (shadows) in these surveys. Using Poisson statistics, a 95% confidence upper limit is set on the number density of SLABs ($n_{max}$) based on the survey volume where a shadow of a specific mass would be detectable.

3. Key Contributions

1. Identification of the "Negative Source" Signature: The paper establishes that SLABs should appear as distinct "cold spots" (negative flux) against the CMB, distinct from the positive flux of active galactic nuclei.
2. High-Redshift Detectability: It highlights the counter-intuitive fact that SLAB shadows are easier to detect at high redshifts ($z > 1.6$) due to the decreasing angular diameter distance, which makes the shadow angular size larger, combined with the higher CMB temperature increasing the flux deficit.
3. Comprehensive Accretion Analysis: Unlike previous works that might ignore accretion, this paper models the interplay between shadow dimming and accretion brightening, concluding that even if accretion occurs, the net flux is likely dominated by one or the other, making detection probable.
4. New Density Limits: It derives the tightest constraints to date on the abundance of SLABs across a wide mass range ($10^{13} - 10^{22} M_\odot$).

4. Results

• Mass Limits:
  • SLABs with masses $\gtrsim 10^{17} M_\odot$ are ruled out within the last scattering surface ($z \approx 1100$).
  • For masses between $10^{15} M_\odot$ and $10^{18} M_\odot$, the fraction of the critical density contributed by SLABs is constrained to $\Omega_{BH} \lesssim 10^{-5}$.
  • Specifically, for $M_{BH} \sim 10^{17} M_\odot$, the constraints reach $\Omega_{BH} \sim 10^{-8}$ (Planck) to $10^{-7}$ (SPT-SZ).
• Redshift Sensitivity: The constraints are most powerful for primordial SLABs (constant mass) because they benefit from the vast "early volume" where $D_A$ is small. For growing SLABs (B85 model), the constraints are weaker at high redshift (as they were smaller then) but still effective at lower redshifts.
• Comparison to Other Methods:
  • For constant-mass SLABs with $M_{BH} \gtrsim 10^{14} M_\odot$, the shadow method is roughly an order of magnitude more powerful than constraints derived from CMB isocurvature perturbations.
  • The method is sensitive to intermediate redshifts ($z \sim 10-100$), a regime where other PBH constraints are often weak.
• Accretion Implications:
  • If accretion is efficient, the SLABs would appear as extremely bright positive sources, which are also not observed, implying even stricter limits.
  • If the accretion flow is optically thick, it could hide the shadow, but the paper argues that the physical conditions (temperature, ionization) required for total obscuration are uncertain and likely not universal.

5. Significance

This work provides a robust, model-independent constraint on the existence of stupendously large black holes. By leveraging the fundamental property that black holes are "black" and the unique geometry of the expanding universe, it rules out the existence of SLABs as a significant component of the Universe's mass density in the mass range of $10^{15} - 10^{18} M_\odot$.

The study demonstrates that current CMB data (Planck, SPT) is already sufficient to probe physics at scales previously thought to require future, more sensitive instruments. It also opens a new avenue for searching for exotic black hole populations formed by late-time mechanisms or in compensated density fluctuations, suggesting that if such objects exist, they must be exceedingly rare. The paper concludes that the "shadows of giants" are a powerful, yet previously underutilized, tool for testing cosmological models and the nature of dark matter.