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

TL;DR

The paper addresses whether the GR-predicted compactness gap between neutron stars and black holes persists in deformed Hořava-Lifshitz gravity. By solving the Tolman-Oppenheimer-Volkoff equations for fermionic matter within this HL framework and exploring a deformation parameter $q$, fermion mass $m_f$, and interaction strength $y$, it demonstrates that a minimum fermion mass $m_f^{\text{(min)}}(q,y)$ exists above which the gap vanishes, allowing BH-like and NS-like configurations to merge in the mass-radius landscape. Consequently, objects residing in the GR lower-mass gap could be misclassified without horizons, and HL gravity offers potential sub-solar mass compact objects as dark matter candidates, including near-EBH branches for certain parameters. These findings provide a distinctive strong-field signature of HL gravity with implications for gravitational-wave observations and dark matter phenomenology, motivating further exploration of the parameter space and observational tests. $M/R$-based classifications may thus be modified in HL gravity, reshaping interpretations of compact-object signals.

Abstract

We show that the gap in the compactness between black holes and neutron stars witnessed in general relativity may be vanishing in Hořava-Lifshitz (HL) gravity. Assuming a fermion equation-of-state for simplicity, and solving the Tolman-Oppenheimer-Volkoff equation within the HL gravity framework, we see that there exists a minimum fermion mass $m_f^\text{(min)}(q,y)$, above which the gap of the compactness between black hole and fermionic compact object vanishes, for a given deformation parameter $q$ of HL and interaction strength $y$ between fermions. Thus, in HL gravity, the mass and radius of an object found in the lower mass gap by LIGO-Virgo-KAGRA observations might not be able to classify it as a black hole or a neutron star. It is interesting to note that a fermion of mass $\sim 40\ \text{GeV}$ can form a highly compact object of mass $\sim 10^{-4}\ \msun$ and radius $\sim 1\ \text{m}$ that may play the role of the cold dark matter. In addition, we find the possible existence of another class of compact objects whose compactness is comparable to that of a black hole.

Figures (4)

• Figure 1: Comparison of fermionic compact objects of $m_f = m_n$ with and without interaction term and compact object whose EOS is apr4 in (a) gr and (b,c) hl. The blue solid curves represent compact objects made of free fermions, the orange dotted curves represent fermionic compact objects with two-body interactions, and the green dashed curves are the compact objects with the apr4 EOS. The horizons of the ks and Schwarzschild bh are also plotted as the dash-dotted and dash-double dotted curves, respectively. The sound speed limit (causal limit) of ks and Schwarzschild vacua are depicted as dotted curves, respectively.
• Figure 2: Comparison of fermionic compact objects with and without two-body interaction term in EOS for some interaction strengths $y$ and various fermion masses $m_f$. For a given $y$, the point of smaller $M/|q|$ and $R/|q|$ indicates the case of more massive $m_f$. All the most massive stable objects are below the sound speed limit (causal limit) in ks vacuum.
• Figure 3: Compactness of the most massive fermionic compact objects in hl gravity for several interaction strengths $y$ are depicted with respect to (a) various fermion masses $m_f$ and (b) various central densities $\rho_c$. Compactness of several fermionic compact objects in gr with interaction energy $m_I \sim 100\ \text{MeV}$ are also shown in dimension (a) [GeV] and (b) [g/cm$^3$] for comparison. The compactness of a ks bh is between that of a Schwarzschild bh, 1/2, and that of a minimal bh, 1, according to the ratio of $|q|/M_\mathrm{BH}$, and it is represented by the shaded region. The compactness of sound speed limit of ks vacuum is also between that of causal limit of Schwarzschild vacuum in gr and that of a minimal bh, according to the ratio of $|q|/R$.
• Figure 4: Mass-radius profile of fermionic compact objects with fermion mass $m_f = 20 (|q|/1\ \textrm{m})^{-1/2}\ \textrm{GeV}$ near the minimal bh suggests a possible new state of compact object that is more compact than a ns. Sound speed limit is also depicted to show that this new compact objects does not exceed the sound speed limit.