Primordial Black Hole Formation in Rastall Gravity: Shifted Collapse Threshold and Exponential Abundance Sensitivity

This paper demonstrates that in Rastall gravity, the non-minimal coupling between matter and geometry significantly alters the collapse threshold and exponential abundance of primordial black holes, establishing them as sensitive probes for testing modified gravity theories within current cosmological constraints.

The Gist

Here is an explanation of the paper "Primordial Black Hole Formation in Rastall Gravity," translated into simple, everyday language with creative analogies.

The Big Picture: A Cosmic "What If?"

Imagine the universe as a giant, expanding balloon. In the very beginning, it was hot, dense, and filled with a chaotic soup of energy. Sometimes, in this soup, a little clump of energy gets so heavy and dense that it collapses in on itself, forming a Primordial Black Hole (PBH). These are tiny, ancient black holes that could make up the "Dark Matter" that holds galaxies together.

Usually, scientists use General Relativity (Einstein's theory of gravity) to calculate how these black holes form. But this paper asks: What if Einstein's rules are slightly tweaked?

The author, Mayukh Gangopadhyay, explores a theory called Rastall Gravity. Think of Rastall Gravity as a "modified version" of Einstein's rules where matter and space don't quite play by the standard conservation laws. It's like if, in a game of billiards, the balls didn't just bounce off each other but occasionally exchanged a little bit of their "mass" with the table itself.

The Main Discovery: The "Hidden" Effect

Here is the surprising twist the paper found:

1. The Background is Normal: If you look at the universe as a whole (the big picture), Rastall Gravity looks exactly like Einstein's General Relativity. The universe expands at the same speed. It's like driving a car on a highway; if you only look at the speedometer, you can't tell if the engine is standard or slightly modified.
2. The Ripples are Different: However, if you look at the tiny ripples and waves in the universe's early soup (density fluctuations), the modified rules do change things. It's like the car's engine is humming a slightly different tune, even if the speed is the same.

The "Goldilocks" Threshold

For a black hole to form, a clump of matter needs to be heavy enough to overcome the pressure pushing it apart.

• In Einstein's World: There is a specific "tipping point" (a critical density). If a clump is 41.4% denser than average, it collapses. If it's 41.3%, it survives.
• In Rastall's World: This tipping point shifts slightly. The paper shows that the Rastall parameter (let's call it $\lambda$) acts like a tiny screwdriver turning the dial on this threshold.
  • If $\lambda$ is positive, it's harder to collapse (the threshold goes up).
  • If $\lambda$ is negative, it's easier to collapse (the threshold goes down).

The "Snowball" Effect: Why Small Changes Matter Huge

This is the most exciting part of the paper. The author explains that the number of black holes formed depends on this threshold in a super-sensitive way.

The Analogy: The Avalanche
Imagine standing on a snowy mountain.

• General Relativity is a mountain where the snow is just stable enough that you need a huge shout to trigger an avalanche.
• Rastall Gravity is a mountain where the snow is almost the same, but the angle is shifted by a fraction of a degree.

Because the formation of black holes is an "exponential" process (like an avalanche), that tiny fraction of a degree doesn't just change the result a little bit. It changes it by orders of magnitude.

• A tiny tweak in the rules might mean zero black holes form.
• A tiny tweak in the other direction might mean billions of black holes form.

The paper shows that even if the Rastall parameter is incredibly small (something we can't measure directly yet), it could change the number of primordial black holes in the universe by a factor of 1,000 or 1,000,000.

Why This Matters

1. Dark Matter Detective: If we find that there are way more (or way fewer) primordial black holes than Einstein's theory predicts, it might not mean our inflation models are wrong. It might mean gravity itself is slightly different (Rastall Gravity).
2. A New Tool: Since the "background" of the universe looks the same in both theories, we can't tell them apart by just watching the universe expand. But by counting black holes, we can detect the "hidden" difference in how gravity works on small scales.
3. The "Nearly Conformal" Plasma: The paper notes that in a perfectly ideal gas, Rastall gravity wouldn't do anything. But the early universe wasn't perfect; it had tiny quantum quirks (like the QCD trace anomaly). These tiny imperfections are what "wake up" the Rastall effects, allowing the theory to leave its fingerprint on the black holes.

The Bottom Line

This paper is like finding a new dial on a radio. Even if the station sounds mostly the same, turning that dial just a tiny bit changes the volume from a whisper to a scream.

The author concludes that Primordial Black Holes are the perfect "microphones" to listen for these tiny changes in gravity. If we can measure how many of these ancient black holes exist, we might finally prove that Einstein's gravity needs a slight update, or at least that the universe is playing by slightly different rules than we thought.

Technical Summary

Here is a detailed technical summary of the paper "Primordial Black Hole Formation in Rastall Gravity: Shifted Collapse Threshold and Exponential Abundance Sensitivity" by Mayukh R. Gangopadhyay.

1. Problem Statement

Primordial Black Holes (PBHs) are compelling candidates for dark matter, formed from the gravitational collapse of large-amplitude density perturbations during the radiation-dominated era of the early universe. In standard General Relativity (GR), generating a sufficient PBH abundance to account for dark matter typically requires significant fine-tuning of the primordial power spectrum (enhancing it by orders of magnitude compared to CMB scales).

The paper investigates whether Rastall gravity, a modification of GR where the energy-momentum tensor is not covariantly conserved ($\nabla_\mu T^{\mu\nu} = \lambda \nabla^\nu R$), can alter the conditions for PBH formation. A key challenge is that for a strictly conformal radiation fluid ($p = \rho/3$), the background expansion in Rastall gravity is identical to GR, and the trace of the energy-momentum tensor vanishes, rendering the theory dynamically equivalent to GR at the background level. The problem is to determine if PBH formation can serve as a probe for Rastall gravity by analyzing perturbation dynamics in a "nearly conformal" plasma where small deviations from ideal radiation behavior activate the Rastall parameter ($\lambda$).

2. Methodology

The author employs a systematic analytical approach combining cosmological perturbation theory and phenomenological modeling:

• Theoretical Framework: The study utilizes the Rastall field equations, which modify the coupling between geometry and matter via the parameter $\lambda$. The energy-momentum tensor is assumed to represent a physical radiation fluid satisfying the modified conservation law.
• Background Evolution: The author demonstrates that for a pure radiation fluid, the background Friedmann equations remain identical to GR ($H^2 \propto \rho$). However, to activate Rastall effects, the paper introduces a nearly conformal plasma model where the equation of state is $p = (1/3 - \delta)\rho$ with $\delta \ll 1$. This non-zero trace ($T \neq 0$) allows the Rastall modification to become active at the perturbative level.
• Perturbation Analysis:
  • The study works in the Newtonian (longitudinal) gauge.
  • It derives the modified evolution equation for the density contrast $\delta \equiv \delta\rho/\rho$ by combining the perturbed Rastall field equations and the modified conservation laws.
  • The analysis is performed to linear order in $\lambda$, consistent with current observational constraints.
• PBH Formation Criteria:
  • Critical Threshold ($\delta_c$): The paper derives the critical density contrast required for collapse. While the Jeans scale remains unchanged due to a cancellation between effective gravity and sound speed modifications, the nonlinear collapse threshold is parameterized phenomenologically as $\delta_c(\lambda) = \delta_c^{GR}(1 + \gamma\lambda)$.
  • Abundance Calculation: The PBH mass fraction $\beta(M)$ is calculated using the Press-Schechter formalism assuming Gaussian statistics. The variance of density fluctuations $\sigma(M)$ is modified by the Rastall parameter at horizon crossing.

3. Key Contributions

• Derivation of the Master Equation: The paper derives a new second-order differential equation for the density contrast in Rastall gravity (Eq. 42), revealing modifications to both the driving gravitational term and the effective sound speed.
• Jeans Scale Invariance: A significant theoretical finding is that, at linear order, the Jeans scale remains identical to GR. The modifications to the effective gravitational constant and the sound speed exactly cancel each other out in the dispersion relation for radiation domination.
• Shifted Horizon Crossing Amplitude: The amplitude of density fluctuations at horizon crossing, $\delta_H$, is shown to acquire a linear correction dependent on $\lambda$, altering the variance $\sigma(M)$.
• Exponential Sensitivity Mechanism: The paper establishes that the PBH abundance depends exponentially on the square of the peak significance parameter $\nu_c = \delta_c / \sigma$. Consequently, even minute linear corrections to $\delta_c$ and $\sigma$ induced by $\lambda$ result in exponentially large changes in the predicted PBH abundance.

4. Key Results

• Modified Threshold and Variance:
  • The critical threshold shifts as: $\delta_c(\lambda) \approx \delta_c^{GR}(1 + \gamma\lambda)$.
  • The variance shifts as: $\sigma(M, \lambda) \approx \sigma^{GR}(M)(1 + \beta\lambda)$.
  • Here, $\gamma$ encodes the nonlinear collapse response, and $\beta$ encodes the linear perturbation matching.
• Abundance Enhancement/Suppression:
  The ratio of PBH abundance in Rastall gravity to GR is given by:
  $$\frac{\beta(\lambda)}{\beta_{GR}} \approx \exp\left[ -(\nu_c^{GR})^2 (\gamma - \beta)\lambda \right]$$
  • Since typical rare-peak scenarios involve $\nu_c^{GR} \sim 5-10$, the term $(\nu_c^{GR})^2$ is large ($\sim 25-100$).
  • This implies that a percent-level deviation in the dynamics (small $\lambda$) can alter the PBH abundance by orders of magnitude.
• Directionality: The effect is not universally enhancing.
  • If $(\gamma - \beta)\lambda < 0$, the effective peak significance decreases, leading to exponential enhancement of PBHs.
  • If $(\gamma - \beta)\lambda > 0$, PBH formation is suppressed.
• Constraints: The results suggest that current constraints on PBH abundances can be reinterpreted as constraints on the combination $(\gamma - \beta)\lambda$, offering a new way to test modified gravity even when background cosmology is indistinguishable from GR.

5. Significance

• Novel Probe for Modified Gravity: The paper demonstrates that PBHs act as highly sensitive probes for Rastall gravity. Unlike standard cosmological tests (like CMB or Large Scale Structure) which might be degenerate with background expansion effects, PBH formation isolates perturbation-level dynamics.
• Relaxation of Fine-Tuning: In GR, explaining PBH dark matter often requires extreme fine-tuning of the inflationary power spectrum. Rastall gravity offers a mechanism where small deviations in the gravitational coupling ($\lambda$) can naturally produce the necessary PBH abundance without requiring such extreme enhancements in the primordial power spectrum.
• Theoretical Insight: The finding that the Jeans scale remains invariant while the collapse threshold and fluctuation amplitude shift provides a unique signature of Rastall gravity, distinguishing it from other modified gravity theories where the Jeans scale might change directly.
• Future Outlook: The paper highlights the need for full nonlinear numerical simulations in Rastall gravity to determine the precise value of the parameter $\gamma$ from first principles, which would solidify the predictive power of this framework.

In conclusion, this work establishes that Primordial Black Holes are not just dark matter candidates but also powerful "gravitational laboratories" capable of detecting minute deviations from General Relativity through their exponential sensitivity to the underlying gravitational theory.