Vanishing Compactness Gap and Fermionic Compact Dark Matter in Hořava-Lifshitz Gravity

This paper demonstrates that in Hořava-Lifshitz gravity, the compactness gap between black holes and neutron stars can vanish for fermionic objects above a certain mass threshold, potentially blurring the classification of LIGO-Virgo-KAGRA detections and suggesting that fermions with a mass of approximately 40 GeV could constitute compact dark matter.

The Gist

Imagine the universe as a giant construction site where gravity is the foreman. For over a century, we've believed in one specific set of blueprints: General Relativity. According to these blueprints, there's a strict "No-Go Zone" between two types of heavy cosmic objects: Neutron Stars (super-dense city blocks) and Black Holes (bottomless pits).

In our current understanding, you can have a heavy object that is a bit squishy (a Neutron Star), or one so heavy it collapses into a pit (a Black Hole). But there is a gap in the middle. You can't build a stable object that is just a little bit denser than a Neutron Star but not quite a Black Hole. It's like trying to build a house that is slightly taller than a two-story building but shorter than a skyscraper; the laws of physics say it would just collapse into a skyscraper or fall apart.

This paper, written by Edwin J. Son, Kyungmin Kim, and John J. Oh, suggests that if we swap the blueprints to a newer, more complex design called Hořava-Lifshitz (HL) Gravity, that "No-Go Zone" disappears.

Here is the breakdown of their findings in simple terms:

1. The "Gap" Vanishes

In standard physics, there is a "compactness gap." Compactness is a measure of how much mass is squeezed into a specific size.

• Neutron Stars have a low compactness (they are big and heavy, but not too heavy for their size).
• Black Holes have a high compactness (they are incredibly heavy for their tiny size).
• The Gap: Nothing can exist in the middle.

The authors used a computer to solve the equations for HL gravity (a theory that changes how gravity behaves at very high energies, like near the Big Bang or inside black holes). They found that in this new framework, you can build a stable object right in the middle of that gap. It's like discovering that the "No-Go Zone" was actually just a construction error in the old blueprints, and the new blueprints allow for a perfectly stable "mid-rise" building that fits right between the city block and the skyscraper.

2. The Secret Ingredient: Heavy Fermions

How do you build these mysterious "mid-rise" objects? The paper suggests using fermions (a type of fundamental particle, like electrons or neutrons) that are much heavier than the ones we usually see.

• The Analogy: Imagine trying to stack bricks. If the bricks are light (like standard neutrons), the stack collapses if you try to make it too tall. But if you use "super-bricks" (fermions with a mass around 40 GeV, which is much heavier than a neutron), you can stack them incredibly high and tight without them collapsing into a black hole.
• The Result: These heavy fermions can form objects that are as dense as black holes but don't have an event horizon (the point of no return). To an outside observer, they look almost exactly like black holes, but they are actually solid, stable objects.

3. A New Kind of "Dark Matter"

The paper also explores what happens if we make these objects very small.

• If we tweak the parameters of the HL gravity theory, these heavy fermion objects can become tiny—about the size of a house (1 meter) but with the mass of a small asteroid.
• The Dark Matter Connection: The universe is full of invisible "Dark Matter" that holds galaxies together. We don't know what it is. The authors suggest that these tiny, ultra-compact fermion objects could be the missing pieces of the Dark Matter puzzle. They are small enough to hide in plain sight and dense enough to have gravity, but they don't emit light, making them perfect candidates for the "ghost" matter we can't see.

4. Why This Matters for Real Observations

The paper mentions a real-world mystery: Astronomers using the LIGO and Virgo detectors have found some objects that weigh between 2.5 and 5 times the mass of our Sun.

• The Problem: In standard physics, objects this heavy should be Black Holes. But they are too light to be the "typical" Black Holes we expect, and too heavy to be Neutron Stars. They sit right in that "No-Go Zone."
• The Paper's Take: If HL gravity is correct, these mysterious objects might not be Black Holes at all. They could be these new, stable "fermionic compact objects" that fill the gap. This would explain why they exist and why they are so hard to classify.

Summary

The paper argues that the universe might be more flexible than we thought. If the laws of gravity change at high energies (as Hořava-Lifshitz gravity suggests), the strict barrier between Neutron Stars and Black Holes vanishes. This allows for:

1. New types of stars that are denser than neutron stars but not black holes.
2. A potential explanation for the mysterious objects found in the "mass gap" by gravitational wave detectors.
3. A new candidate for Dark Matter: Tiny, invisible, ultra-dense balls of heavy particles that could make up the missing mass of the universe.

In short, the authors are saying: "If you change the rules of gravity slightly, the universe allows for a whole new class of objects that we previously thought were impossible."

Technical Summary

Technical Summary: Vanishing Compactness Gap and Fermionic Compact Dark Matter in Hořava-Lifshitz Gravity

Problem Statement
In General Relativity (GR), a distinct "compactness gap" exists between neutron stars (NS) and black holes (BHs). The compactness parameter, defined as $M/R$ (with $c=G=1$), for a Schwarzschild BH is strictly 0.5, while typical NSs exhibit compactness values of approximately 0.1–0.2. Theoretical limits suggest a maximum NS compactness of roughly 0.3, leaving a forbidden region between 0.3 and 0.5 where no stable compact objects can form in GR. This theoretical gap parallels the observational "lower mass gap" (typically 3–5 $M_\odot$) detected by LIGO-Virgo-KAGRA, where the nature of compact objects remains ambiguous. The central question addressed is whether this compactness gap is a universal feature of gravity or a theory-dependent phenomenon that might vanish in modified gravity theories.

Methodology
The authors investigate this problem within the framework of deformed Hořava-Lifshitz (HL) gravity, a theory that introduces anisotropic scaling between time and space to achieve power-counting renormalizability in the ultraviolet (UV) regime while recovering GR in the infrared (IR) limit. Specifically, they utilize the deformed HL model which possesses an asymptotically Minkowskian structure.

The study proceeds through the following steps:

1. Field Equations: The authors derive the Tolman-Oppenheimer-Volkoff (TOV) equations for a static, spherically symmetric metric in deformed HL gravity. The theory is characterized by a single free deformation parameter, $q$, which determines the asymptotic behavior and the existence of horizons.
2. Equation of State (EOS): To model the matter content, the authors assume a fermionic gas at zero temperature. They consider two scenarios:
   • A free fermion gas.
   • A fermion gas with two-body interactions, parameterized by an interaction strength $y = m_f/m_I$, where $m_f$ is the fermion mass and $m_I$ is the interaction energy scale.
3. Numerical Solution: The TOV equations are solved numerically using an eighth-order Runge-Kutta method. The authors vary the central energy density ($\rho_c$), the deformation parameter $q$, the fermion mass ($m_f$), and the interaction strength $y$ to map out the mass-radius ($M-R$) relationships and the resulting compactness ($M/R$).
4. Constraints: The solutions are constrained by the causal limit (sound speed $v_s \leq c$) and the requirement for stable equilibrium configurations.

Key Contributions and Results

• Vanishing of the Compactness Gap: The primary result is that the compactness gap between neutron stars and black holes is not a universal prediction but can vanish in deformed HL gravity. The authors demonstrate that for specific values of the deformation parameter $q$ and fermion mass $m_f$, stable fermionic compact objects can exist with compactness values ($M/R$) that continuously approach and fill the gap between 0.3 and 0.5, and even extend up to the compactness of the minimal Kehagias-Sfetsos (KS) black hole (where $M/R \to 1$).
• Mass-Radius Convergence to Minimal Black Holes: In the limit of large fermion masses and high central densities, the mass and radius of the most massive stable fermionic objects converge to those of the minimal KS black hole ($M \sim |q| \sim R$). This suggests that in HL gravity, objects with BH-like compactness can exist without forming an event horizon, potentially appearing as "exotic compact objects" indistinguishable from black holes with current observational capabilities.
• Dependence on Interaction Strength: The study finds that the fermion mass scale required to achieve high compactness depends on the interaction strength $y$. For non-zero interaction strengths, the required fermion masses are higher than for free fermions. However, the qualitative behavior of the mass-radius profiles remains consistent: higher fermion masses lead to more compact configurations.
• Fermionic Compact Dark Matter Candidates: The paper identifies a potential class of compact dark matter (DM) candidates. For small values of the deformation parameter (e.g., $q \sim 1$ m or $q \sim 1$ nm), the minimal KS black hole has a mass of $\sim 10^{-4} M_\odot$ or $\sim 10^{-13} M_\odot$, respectively. The authors show that fermionic compact objects can form with similar masses and radii.
  • Specifically, a fermion of mass $\sim 40$ GeV can form a highly compact object with mass $\sim 10^{-4} M_\odot$ and radius $\sim 1$ m.
  • These objects are stable, do not evaporate (as the Hawking temperature vanishes for minimal BHs and near-minimal configurations), and their small size allows them to survive without fatal interactions, making them viable sub-solar mass DM candidates.
• New Compact Object Branch: The numerical results suggest the existence of a new branch of compact objects smaller than neutron stars in both mass and radius. For instance, with $q \sim 1$ m and $m_f \approx 20$ GeV (without interaction), the most massive object has a mass of $\sim 10^{-3} M_\odot$ and radius $\sim 20$ m. An additional peak in the mass-radius curve indicates configurations even closer to the minimal BH limit.

Significance and Claims
The paper claims that the existence of a compactness gap is a theory-dependent effective phenomenon rather than a fundamental law of gravity. By demonstrating that deformed HL gravity allows for stable, highly compact fermionic objects that bridge the gap between neutron stars and black holes, the authors propose a strong-field astrophysical signature of modified gravity.

The significance of these findings is twofold:

1. Astrophysical Interpretation: It offers a theoretical explanation for objects observed in the "lower mass gap" (such as the GW230529 181500 event) that might not be standard black holes or neutron stars but rather exotic compact objects supported by modified gravity.
2. Dark Matter Phenomenology: It suggests that fermionic compact objects in HL gravity could constitute a viable class of cold dark matter, particularly in the sub-solar mass regime, which is difficult to detect via current microlensing techniques but consistent with the constraints on primordial black holes.

The authors conclude that comparing fermionic compact dark matter in GR with its counterpart in deformed HL gravity provides a pathway to test the viability of modified gravity as a dark matter candidate, while noting that the specific fermion mass scales required (ranging from GeV to PeV depending on $q$) may be beyond current particle physics experimental reach.