Primordial black hole formation from transient $f(T)$ cosmology

This paper demonstrates that a transient departure from standard radiation domination in minimally coupled $f(T)$ teleparallel cosmology can exponentially enhance primordial black hole formation and produce asteroid-mass black holes that account for all dark matter, without requiring ad hoc modifications to the radiation sector.

The Gist

The Big Idea: A Temporary "Soft Spot" in the Universe

Imagine the early universe as a giant, expanding balloon filled with a super-hot, super-dense gas (radiation). Usually, this gas is very "stiff." If you try to squeeze a clump of this gas together to make something heavy (like a black hole), the gas pushes back hard, like a spring resisting compression. It's very hard to make a black hole in this environment because the pressure fights against gravity.

This paper proposes a new idea: What if, for a very short time, the universe got a little bit "softer"?

The authors suggest that a specific type of modified gravity (called $f(T)$ gravity) caused the universe to temporarily lose some of its stiffness. During this brief window, the gas didn't push back as hard. Because the resistance was lower, gravity could win the fight more easily, collapsing clumps of matter into Primordial Black Holes (PBHs) much more efficiently than usual.

The Cast of Characters

1. The Standard Model (The Stiff Spring): In our normal understanding of physics, the early universe is dominated by radiation. It has a specific "stiffness" (equation of state) that makes it very hard for black holes to form. You need a massive, rare fluctuation to overcome this pressure.
2. The $f(T)$ Gravity (The Magic Softener): The authors introduce a tweak to Einstein's gravity. Instead of just curving space, gravity also involves "twisting" space (torsion). They designed a model where this twisting effect is invisible at the very beginning and the very end of time, but it "switches on" for a short middle period.
3. The Transient Epoch (The Soft Spot): During this middle period, the "twist" acts like a negative pressure fluid. It counteracts the stiffness of the radiation. Think of it like adding a little bit of oil to a squeaky, stiff hinge. The hinge (the universe) suddenly becomes much easier to bend.

How It Works: The Analogy of the Crowd

Imagine a crowded dance floor (the early universe).

• Normal Scenario: Everyone is dancing energetically and pushing against each other. If one person tries to huddle with a few others to form a tight group (a black hole), the crowd shoves them apart. It's very difficult to form a group.
• The Paper's Scenario: Suddenly, for a few seconds, the music changes, and everyone stops pushing so hard. They become "soft." Now, if a few people try to huddle, the crowd doesn't push them apart as hard. A tight group forms easily.
• The Result: Because the crowd was soft for just a moment, a huge number of groups formed during that specific time. Once the music changed back to the "hard" rhythm, the groups stayed formed.

The Results: Asteroid-Sized Black Holes

The paper calculates that this "softening" effect would create a very specific type of black hole population:

• Mass: They would be roughly the size of an asteroid (very small for a black hole, but heavy enough to be dark matter).
• Abundance: Because the "collapse threshold" (the amount of effort needed to make a black hole) dropped slightly, the number of black holes formed would explode exponentially.
• Dark Matter: The authors show that with the right settings, these asteroid-sized black holes could make up 100% of the Dark Matter in the universe.

Why Is This Cool?

Usually, to explain why we have so much Dark Matter, scientists have to invent weird new particles or change the laws of physics for the entire history of the universe.

This paper is elegant because:

1. It's Temporary: The weird physics only happens for a short time. Before and after that moment, the universe behaves exactly as we expect it to.
2. It's Pure Gravity: It doesn't require changing the "radiation" (the gas) itself. It just changes the "container" (gravity) holding the gas.
3. It Fits the Data: The resulting black holes would be small enough to avoid detection by current telescopes (which look for bigger black holes) but heavy enough to explain the missing mass in the universe.

The Bottom Line

The authors found a mathematical "loophole" in the laws of gravity. For a brief moment in the early universe, gravity acted like a softener, making it easy for tiny black holes to form. These tiny black holes survived until today, potentially hiding in plain sight as the mysterious Dark Matter that holds our galaxies together.

It's like finding out that for one specific minute in history, the universe was made of Jell-O instead of steel, allowing a million tiny marbles to form instantly, which are now floating around us as the invisible glue of the cosmos.

Technical Summary

1. Problem Statement

Primordial Black Holes (PBHs) are hypothetical compact objects formed in the early Universe from the gravitational collapse of large density perturbations. In the standard radiation-dominated (RD) scenario, the equation of state (EoS) is $w = 1/3$. High pressure gradients in this regime efficiently counteract gravitational collapse, requiring a very high threshold amplitude ($\delta_c \approx 0.4$) for perturbations to form PBHs. Consequently, generating a significant abundance of PBHs (e.g., to account for Dark Matter) typically requires either extreme fine-tuning of primordial power spectra or specific physical mechanisms that temporarily soften the EoS (such as the QCD phase transition).

The authors investigate whether modified gravity, specifically $f(T)$ teleparallel gravity, can naturally induce a transient softening of the cosmic background equation of state without requiring ad hoc modifications to the radiation fluid sector. The goal is to determine if such a gravitational mechanism can lower the collapse threshold sufficiently to enhance PBH formation to observable levels while remaining consistent with early and late-time cosmological constraints.

2. Methodology

Theoretical Framework

The study utilizes $f(T)$ gravity, an extension of General Relativity (GR) based on the torsion scalar $T$ rather than the curvature scalar $R$. The action is given by:
$$S = \frac{1}{16\pi G} \int d^4x \, e [T + f(T)] + S_m$$
where $S_m$ represents standard matter/radiation coupled minimally to the metric. The authors employ an effective fluid description, rewriting the modified Friedmann equations as a system of standard radiation plus an effective "torsion fluid" with energy density $\rho_f$ and pressure $p_f$.

Model Construction

The authors propose a specific functional form for $f(T)$ designed to be transient:
$$f(T) = \lambda T_\star \left(\frac{T}{T_\star}\right)^3 e^{-T/T_\star}$$
where $T_\star$ is a reference torsion scale and $\lambda$ is a dimensionless coupling.

• Asymptotic Behavior: As $T \to 0$ (late times) and $T \to \infty$ (early times), $f(T)$ and its derivatives vanish, recovering standard GR (TEGR).
• Transient Epoch: At intermediate epochs, the torsion term becomes dynamically relevant, acting as an effective fluid with a time-dependent EoS parameter $w_f$.

Dynamics and Parameters

• The model is calibrated such that at a reference epoch $t_{ref}$ (defined by $x = T/T_\star = 3/2$), the torsion fluid contributes $\Omega_f = 0.1$ to the total energy density.
• At this epoch, the torsion fluid has $w_f = -1$, causing the total effective EoS ($w_{tot}$) to drop from the standard radiation value of $1/3$ to $0.2$.
• The authors map this evolution to physical temperature and horizon mass, focusing on the asteroid-mass scale ($M \sim 10^{-15} M_\odot$) where observational constraints are currently weak.

PBH Formation Calculation

• Collapse Threshold: The critical density contrast $\delta_c$ required for collapse is calculated using the analytic estimate by Harada, Kohri, and Yoo, which depends on $w_{tot}$.
• Abundance: The PBH mass fraction $\beta$ is calculated using the Press-Schechter formalism assuming Gaussian statistics. The authors incorporate the critical collapse scaling law ($M_{PBH} \propto (\delta - \delta_c)^\gamma$) to derive the extended PBH mass function.
• Constraints: The resulting PBH mass functions are compared against current observational limits (microlensing, CMB distortions, gravitational waves, and Hawking evaporation).

3. Key Contributions

1. Transient Gravitational Softening: The paper demonstrates that $f(T)$ gravity can naturally generate a transient epoch where the total EoS drops below $1/3$ purely through gravitational effects, without altering the microphysics of the radiation fluid.
2. Threshold Reduction: The reduction of $w_{tot}$ from $0.33$ to $0.2$ lowers the collapse threshold $\delta_c$ from $\approx 0.41$ to $\approx 0.38$. While seemingly small, this reduction leads to an exponential enhancement in the PBH formation probability for Gaussian perturbations.
3. Localized Mass Function: Unlike standard scenarios that often produce broad spectra, the transient nature of the $f(T)$ modification creates a sharply peaked PBH mass function centered around the horizon mass at the transient epoch.
4. Dark Matter Viability: The authors show that for perturbation amplitudes $\sigma^2 \sim \mathcal{O}(10^{-3})$, this mechanism can generate enough PBHs to account for a significant fraction, or potentially the entirety, of the Cold Dark Matter (CDM) in the asteroid-mass window ($10^{-16} - 10^{-14} M_\odot$), while satisfying all current observational constraints.

4. Results

• Background Evolution: The model successfully exhibits a "bump" in the torsion fraction $\Omega_f$ around the reference time, causing $w_{tot}$ to dip to $0.2$ before returning to $1/3$. The radiation component remains dominant ($\Omega_r \approx 0.9$), ensuring the universe does not enter an accelerated expansion phase during this epoch.
• Collapse Threshold: Figure 5 and 6 illustrate that the threshold $\delta_c$ drops significantly in the vicinity of the transient epoch. The minimum threshold reaches $\delta_c \approx 0.12$ (due to the interplay of the scaling law and the specific epoch), though the value at the reference point is $\approx 0.38$.
• PBH Abundance:
  • For $\sigma^2 \approx 0.0016$, the model produces a PBH population peaked at $M_{peak} \sim 10^{-15} M_\odot$.
  • The resulting fraction of Dark Matter in PBHs, $f_{PBH}$, can reach unity ($f_{PBH} \approx 1$) within the asteroid-mass window.
  • The mass function is narrow, avoiding constraints from other mass ranges (e.g., solar mass or supermassive scales).
• Consistency: The model remains consistent with Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background (CMB) constraints because the modification is negligible at very early and very late times.

5. Significance

• Alternative to QCD Softening: While the QCD phase transition is a known mechanism for PBH enhancement, it is limited to a specific mass scale (solar mass). This work provides a gravitational alternative that can be tuned to different mass scales (specifically asteroid masses) by adjusting the model parameters ($T_\star$).
• No Ad Hoc Fluid Modifications: The enhancement arises strictly from the geometric sector (torsion), avoiding the need to artificially modify the equation of state of the radiation fluid.
• Observational Testability: The prediction of a sharply peaked PBH mass function in the asteroid-mass range offers a specific target for future observations, such as microlensing surveys (e.g., Roman Space Telescope) or gravitational wave detectors sensitive to PBH mergers.
• Theoretical Robustness: The mechanism is robust against fine-tuning; varying the coupling $\lambda$ simply shifts the enhancement level without requiring complex adjustments to the perturbation spectrum.

In conclusion, the paper establishes that transient modifications in teleparallel gravity provide a viable, natural, and efficient mechanism for generating Primordial Black Holes, potentially explaining the entirety of Dark Matter in specific mass windows without violating current cosmological bounds.