Asteroid-mass Primordial Black Holes as Dark Matter from Supersymmetry

This paper demonstrates that supersymmetric extensions of the Standard Model can facilitate the formation of asteroid-mass Primordial Black Holes as a viable dark matter candidate by transiently softening the early Universe's equation of state, a mechanism that remains consistent with observational constraints for heavy supersymmetric masses but fails within the Standard Model.

The Gist

The Big Mystery: What is Dark Matter?

Imagine the universe is a giant party. We can see the guests (stars, planets, gas), but there are many more invisible guests (Dark Matter) holding the party together. We know they are there because the visible guests wouldn't stay in their seats without them. But nobody knows what these invisible guests are made of.

Scientists have looked for "particles" (tiny, invisible marbles) that could be Dark Matter, but they haven't found any yet. So, this paper asks: What if Dark Matter isn't made of particles at all, but of tiny, ancient black holes?

The Cast of Characters: Primordial Black Holes (PBHs)

Usually, black holes are the "end of the line" for massive stars that explode. But Primordial Black Holes are different. They are like "cosmic babies" that formed right at the very beginning of the universe, just after the Big Bang, from clumps of energy collapsing under their own gravity.

The authors are specifically interested in Asteroid-Mass PBHs. These are black holes that are tiny—about the size of an asteroid or a small mountain—but heavy enough to be Dark Matter.

The Problem: Why Don't We See Them?

There is a "Goldilocks" problem with these tiny black holes:

1. Too Light: If they are too small, they evaporate (disappear) like steam from a hot cup of coffee before today.
2. Too Heavy: If they are too big, we would see them "winking" stars out of existence as they pass in front of them (a phenomenon called microlensing).
3. Just Right: There is a specific "window" (the asteroid-mass range) where they could exist without being detected yet.

However, in our standard model of physics (the Standard Model), the conditions in the early universe just didn't seem right to make enough of these black holes to fill the Dark Matter quota.

The New Idea: Supersymmetry (The "Heavy Hitters")

The authors propose a twist. They look at a theory called Supersymmetry (SUSY). You can think of the Standard Model as a band with a specific set of instruments. Supersymmetry says, "Actually, every instrument has a heavier, twin version that we haven't heard yet."

These "twin" particles are very heavy. The paper suggests that when the universe was extremely hot, these heavy twins were active. As the universe cooled down, they suddenly "turned off" (became non-relativistic).

The Analogy: The Traffic Jam
Imagine the early universe as a highway where cars (particles) are zooming along at the speed of light. This is the "radiation" state.

• Suddenly, a bunch of heavy trucks (the Supersymmetric particles) enter the highway and stop moving fast.
• This causes a temporary traffic jam. The flow of the universe slows down and gets "softer."
• In physics terms, this is a softening of the Equation of State.

The Result: A Cosmic Squeeze

When the universe gets "softer" (like that traffic jam), it becomes much easier for gravity to win.

• In the Standard Model: It's like trying to squeeze a sponge that is very stiff. It's hard to make a black hole.
• In the Supersymmetric Model: The "traffic jam" makes the sponge soft and squishy. Now, gravity can easily squeeze the universe into tiny black holes.

The authors calculated that if these heavy Supersymmetric particles exist with masses above a certain threshold (around 100,000 times the mass of a proton), this "softening" effect creates a resonant boost. It's like pushing a child on a swing at exactly the right moment; the swing goes much higher.

The Findings

1. The Sweet Spot: If the heavy particles are heavy enough (above ~100,000 GeV), the "softening" happens at just the right time to create a massive amount of asteroid-mass black holes.
2. Filling the Void: With this boost, these black holes could account for 100% of Dark Matter without breaking any of the current rules (like microlensing or evaporation limits).
3. The Contrast: If we stick to the Standard Model (no heavy twins), the same conditions produce almost no black holes in this size range. They would be too rare to be Dark Matter.
4. The Warning: If the heavy particles are too light, the black holes formed would be too big, and we would have already seen them. So, the theory only works if the particles are very heavy.

The Conclusion

The paper concludes that if Supersymmetry is real and the particles are heavy enough, the early universe had a "soft moment" that acted like a factory, churning out just the right amount of tiny black holes to explain all the Dark Matter we see today.

It's a clever solution: instead of looking for a new particle to be the Dark Matter, the heavy particles of Supersymmetry act as the factory that builds the Dark Matter (the black holes) for us.

Technical Summary

1. Problem Statement

The fundamental nature of Dark Matter (DM) remains unknown despite decades of searching for particle candidates (e.g., WIMPs, axions). Primordial Black Holes (PBHs) formed in the early Universe offer a compelling alternative. However, observational constraints (evaporation, microlensing, CMB) have largely ruled out PBHs as the dominant DM component across most mass ranges, leaving only a narrow "asteroid-mass window" ($10^{18} \text{ g} \lesssim M_{\text{PBH}} \lesssim 10^{22} \text{ g}$) as a viable candidate.

The formation of PBHs is governed by two main factors:

1. The primordial curvature power spectrum ($\mathcal{P}_\zeta$).
2. The Equation of State (EoS) of the primordial plasma ($w = p/\rho$) at the time of horizon re-entry.

In the Standard Model (SM), the EoS is $w \approx 1/3$ during radiation domination, with minor softening during phase transitions (e.g., QCD). This paper investigates whether Supersymmetry (SUSY), specifically the Minimal Supersymmetric Standard Model (MSSM), can induce a significant, transient softening of the EoS at high temperatures ($T \gtrsim 1 \text{ TeV}$). The authors ask: Can the presence of heavy supersymmetric particles enhance PBH formation specifically within the asteroid-mass window, allowing them to constitute 100% of Dark Matter without violating current bounds?

2. Methodology

A. Equation of State in the MSSM

The authors compute the EoS parameter $w(T)$ for the MSSM, considering a general mass spectrum characterized by three independent scales rather than a single degenerate scale:

• $m_f$: Mass scale for fermionic superpartners (gauginos).
• $m_b$: Mass scale for bosonic superpartners (squarks, sleptons).
• $m_h$: Mass scale for the second Higgs doublet.

They calculate the effective degrees of freedom for energy ($g_*$) and entropy ($g_{*s}$) as particles transition from relativistic to non-relativistic states. The EoS is given by:
$$w(T) = \frac{4}{3} \frac{g_{*s}(T)}{g_*(T)} - 1$$
When heavy SUSY particles become non-relativistic, $g_*$ drops, causing a temporary softening of $w$ (decreasing below $1/3$).

B. PBH Formation Framework

The study employs the critical collapse formalism:

• Threshold: The collapse threshold $\delta_c$ depends on $w$. A lower $w$ reduces $\delta_c$, exponentially enhancing the probability of collapse for a given density perturbation amplitude.
• Mass Function: They assume a broad, nearly scale-invariant primordial curvature power spectrum $\mathcal{P}_\zeta(k)$ (truncated power law) to cover the relevant mass scales.
• Mass Relation: The PBH mass is related to the horizon mass $M_H$ at formation: $M_{\text{PBH}} \approx \kappa (\delta - \delta_c)^\gamma M_H$.
• Constraints: They compare the resulting extended PBH mass functions against observational bounds (evaporation and microlensing) using the "monochromatic constraint" prescription (integrating the mass function against the inverse of the maximum allowed fraction $F_i^{\text{max}}$).

C. Parameter Space Exploration

The authors scan the parameter space of $(m_f, m_b, m_h)$ ranging from $10^3$ to $10^7$ GeV. They normalize the primordial power spectrum amplitude ($A_*$) such that the total PBH fraction $f_{\text{PBH}} \approx 1$ (accounting for all DM) and then check if this configuration violates observational limits.

3. Key Contributions

1. Multi-Scale MSSM Analysis: Unlike previous studies assuming a single SUSY breaking scale, this work systematically explores a non-degenerate MSSM spectrum with three distinct mass scales ($m_f, m_b, m_h$).
2. Quantification of EoS Softening: They demonstrate that the transition of heavy MSSM particles to non-relativistic states causes a significant drop in $w$ (up to $\sim 10\%$ reduction from $1/3$), particularly when bosonic and fermionic sectors decouple simultaneously (degenerate masses).
3. Resonant Enhancement Mechanism: The paper establishes that this EoS softening lowers the collapse threshold $\delta_c$, leading to a resonant enhancement of PBH production at masses corresponding to the horizon mass at the time of the SUSY particle decoupling.
4. Viability of Asteroid-Mass PBHs: They identify specific regions in the MSSM parameter space where the enhanced PBH mass function peaks precisely within the asteroid-mass window ($10^{18} - 10^{22}$ g), evading current microlensing and evaporation constraints.

4. Results

• EoS Behavior:
  • Degenerate Spectrum ($m_f = m_b = m_h$): Leads to a single, deep dip in $w$ (down to $\approx 0.289$), causing a massive enhancement in PBH production.
  • Hierarchical Spectrum: Results in a series of smaller, staggered dips in $w$, creating multiple peaks in the PBH mass function.
• PBH Mass Function Peaks:
  • The peak mass of the PBH distribution is primarily determined by the bosonic mass scale $m_b$.
  • The relationship is approximated by: $M_{\text{peak}} \approx 5 \times 10^{22} \text{ g} \times (10^5 \text{ GeV} / m_b)^2$.
• Comparison: SM vs. MSSM:
  • Standard Model: For the same primordial power spectrum, SM configurations produce PBH mass functions that are observationally excluded (either too light and evaporated, or too heavy and ruled out by microlensing).
  • MSSM: For supersymmetric masses $m_b \gtrsim 10^5$ GeV, the PBH mass function shifts and narrows into the asteroid-mass window. In these scenarios, PBHs can account for 100% of the Dark Matter ($f_{\text{PBH}} \approx 1$) without violating constraints.
• Spectral Index Dependence:
  • The results are robust for spectral indices close to scale-invariance ($n_s \approx 0.99 - 0.995$).
  • For lower $n_s$, the mass function develops a high-mass tail that is strongly constrained by microlensing, reducing the allowed DM fraction.

5. Significance

• New DM Paradigm: The paper provides a concrete, theoretically motivated mechanism (SUSY-induced EoS softening) to populate the "asteroid-mass window," a region previously difficult to access with standard inflationary models.
• Interplay of Particle Physics and Cosmology: It demonstrates that the high-energy particle content of the Universe (specifically the SUSY mass spectrum) leaves an observable imprint on small-scale structure formation via PBHs.
• Constraints on SUSY: If PBHs constitute all Dark Matter, the underlying particle physics model must satisfy specific conditions:
  • SUSY masses must be heavy ($m_b \gtrsim 10^5$ GeV).
  • The contribution of particle Dark Matter (e.g., the lightest supersymmetric particle) must be negligible or absent, implying strong constraints on the SUSY breaking mechanism and relic density calculations.
• Future Directions: The authors note that a complete assessment requires embedding the power spectrum in a specific inflationary model and investigating the interplay between PBH and particle DM components.

In conclusion, the study argues that Supersymmetry naturally facilitates the formation of asteroid-mass Primordial Black Holes as the sole component of Dark Matter, provided the supersymmetric mass spectrum is sufficiently heavy and non-degenerate in a specific way to shift the PBH peak into the observationally allowed window.