Hybrid stars among mass gap objects are excluded by twin stars at $1.4\,M_\odot$

This paper argues that hybrid stars are unlikely candidates for observed mass-gap compact objects because Bayesian analysis of modern mass-radius constraints favors equations of state with deconfinement occurring around $1.4\,M_\odot$, which would produce twin stars that effectively rule out hybrid stars in the mass gap.

The Gist

The Big Mystery: The "Mass Gap"

Imagine the universe has a strict weight limit for different types of heavy objects.

• Neutron Stars: These are the heaviest "normal" stars we know. They are like giant, super-dense sugar cubes made of atomic matter. The paper says the heaviest one can get is about 2.1 times the weight of our Sun.
• Black Holes: These are objects so heavy that gravity crushes them completely. They usually start appearing at about 5 times the weight of our Sun.

The Problem: Astronomers have found some mysterious objects that weigh between 2.5 and 5 suns. This is a "no-man's-land" or a Mass Gap. We don't know what they are. Are they broken neutron stars? Are they small black holes? Or is there something else?

The Hypothesis: The "Hybrid Star"

The authors ask: Could these heavy objects be "Hybrid Stars"?

Think of a neutron star like a layered cake. Usually, it's made of one type of frosting (normal atomic matter). But a Hybrid Star is a cake where the bottom layer suddenly turns into a completely different, super-dense substance (quark matter) before the cake gets too heavy to hold together.

If this "layer change" happens, the star might be able to hold more weight than a normal star, potentially reaching into that mysterious Mass Gap.

The "Twin Star" Twist

The paper introduces a fascinating concept called "Mass Twins."

Imagine you have two identical twins. They weigh exactly the same (say, 1.4 suns).

• Twin A is tall and fluffy (a large radius).
• Twin B is short and compact (a small radius).

In the world of stars, this means two stars can have the exact same weight but be different sizes because one has turned into that super-dense "quark" material while the other hasn't. The paper suggests that if we find these "Twins" in the universe, it changes everything.

The Investigation: Testing the Cake

The scientists used a computer model to test if Hybrid Stars could explain the Mass Gap objects. They looked at two main things:

1. When does the layer change happen? (Does it happen when the star is light, or only when it gets very heavy?)
2. How stiff is the new material? (Is the quark matter like jelly or like steel?)

They drew a map (called a Seidov diagram) to see which combinations of "when" and "how stiff" allow a star to survive in the Mass Gap.

The Findings: Two Possible Worlds

The paper found two very different scenarios, and they cannot both be true at the same time:

Scenario A: The Mass Gap Stars are Hybrid Stars

• The Condition: For a Hybrid Star to get heavy enough to enter the Mass Gap, the "layer change" must happen extremely early (when the star is still very light) and the new material must be incredibly stiff (almost like a solid block).
• The Result: If this is true, the "Twin Stars" (the 1.4 sun examples) would have to be very rare or non-existent in the way we currently observe them.

Scenario B: The Mass Gap Stars are Black Holes

• The Condition: The paper looks at real data from telescopes (like NICER) that measure the size and weight of known stars. The data strongly suggests that "Twin Stars" exist around 1.4 suns.
• The Result: If Twin Stars exist at 1.4 suns, the physics of the universe prevents Hybrid Stars from getting heavy enough to reach the Mass Gap.
• The Conclusion: If the Twin Star theory is correct, then the mysterious objects in the Mass Gap cannot be Hybrid Stars. They must be Black Holes.

The Final Verdict

The authors conclude that while Hybrid Stars theoretically could exist in the Mass Gap, the evidence points to a different reality.

If we confirm that we have found "Twin Stars" (stars with the same weight but different sizes) at the standard 1.4-sun mark, then Hybrid Stars are ruled out as candidates for the heavy Mass Gap objects. Those heavy objects are almost certainly Black Holes.

In short: The universe seems to have a rulebook. If the "Twin Star" rule is confirmed, the "Hybrid Star" rulebook is closed for the heavy objects, leaving Black Holes as the only explanation for the Mass Gap.

Technical Summary

Technical Summary: Hybrid stars among mass gap objects are excluded by twin stars at 1.4 M⊙

Problem Statement
The nature of compact objects observed in the "mass gap" ($2.5 \le M/M_\odot \le 5.0$) remains ambiguous. While gravitational wave events such as GW170817, GW190814, and GW230529 have confirmed the existence of objects in this mass range, it is unclear whether they are black holes or stable neutron stars. Purely hadronic neutron stars are theoretically limited to a maximum mass of approximately $2.1 M_\odot$ due to the softening of the equation of state (EoS) from the appearance of hyperons and other baryonic resonances. Consequently, hybrid neutron stars (containing a quark matter core) are the primary candidates for non-black-hole mass-gap objects. A critical question arises: Can these mass-gap objects belong to a "third family" of compact stars—disconnected from the standard hadronic branch via a first-order phase transition—potentially forming a distinct population of pulsars? Furthermore, the existence of such a third family is linked to the "mass-twin" phenomenon, where two stable configurations (hadronic and hybrid) share the same gravitational mass but different radii. This phenomenon has been proposed as a mechanism to explain the eccentric orbits of certain millisecond pulsars (MSPs) in low-mass X-ray binaries via accretion-induced phase transitions.

Methodology
The authors employ a generic hybrid EoS featuring a first-order deconfinement transition. The hadronic phase is modeled using a relativistic density functional of the Walecka type (Walecka+qPauli), incorporating repulsive terms from quark Pauli blocking. The quark matter phase is characterized by a constant squared sound speed ($c_s^2$), with representative values of 0.50, 0.75, and 1.00 chosen to reflect color superconducting matter and theoretical bounds.

The study maps the parameter space of the EoS using a modified Seidov diagram, plotting the density jump at the phase transition ($\Delta n$) against the onset density ($n_{\text{onset}}$). This analysis utilizes the Seidov criterion to identify regions where a stable third family (disconnected branch) exists. The authors solve the Tolman-Oppenheimer-Volkoff (TOV) equations to generate mass-radius ($M-R$) diagrams and calculate the maximum mass ($M_{\text{max}}$) and the mass defect ($\Delta M$) associated with twin-star transitions.

To confront theoretical models with observation, the authors apply a Bayesian analysis scheme. This analysis incorporates modern multi-messenger constraints, including mass and radius measurements from NICER (for pulsars PSR J0030+0451, PSR J0740+6620, PSR J0437-4715, and PSR J0614-3329), HESS J1731-347, and mass constraints from gravitational wave events (GW170817, GW190814, GW230529). The Bayesian posterior probability is evaluated over the EoS parameter space ($n_{\text{onset}}$, $\Delta n$).

Key Contributions and Results

1. Theoretical Possibility of Mass-Gap Hybrid Stars: The study confirms that hybrid stars can theoretically reach the mass gap ($M_{\text{max}} > 2.5 M_\odot$). This requires an extremely early onset of deconfinement (at densities $n_{\text{onset}} \lesssim 1.2 n_0$, corresponding to masses $M_{\text{onset}} \lesssim 0.7 M_\odot$) and very stiff quark matter (high $c_s^2$).
2. Third Family Constraints: For mass-gap objects to be members of a third family (disconnected branch), the density jump must exceed the modified Seidov criterion. However, the parameter space allowing for both a third family and mass-gap masses is highly restricted.
   • For $c_s^2 = 0.50$, third-family members in the mass gap are marginal cases occurring only at very low onset densities.
   • For $c_s^2 = 0.75$ (consistent with hydrodynamic transport limits), third-family members exist for onset densities $n_{\text{onset}} \le 1.5 n_0$. However, the associated mass defects for transitions to these twins are extremely small ($\sim 10^{-4} M_\odot$), insufficient to generate the gravitational kicks required to explain the eccentric orbits of MSPs.
   • To explain eccentric MSP orbits, the onset mass must exceed $\sim 1.2 M_\odot$ (implying $n_{\text{onset}} \gtrsim 1.7 n_0$). In this regime, mass-gap objects are not members of the third family.
3. Bayesian Analysis and Twin Stars: The Bayesian analysis favors EoS models where deconfinement occurs at typical neutron star masses around $1.4 M_\odot$. These models produce mass-twin stars (e.g., potentially PSR J0437-4715 and PSR J0614-3329) but limit the maximum mass of the hybrid sequence to just beyond $2.0 M_\odot$. Consequently, these favored models cannot support hybrid stars in the mass gap.
4. Exclusion of Hybrid Mass-Gap Candidates: The study finds a tension between two scenarios:
   • If mass-gap objects are hybrid stars, they require early deconfinement and stiff quark matter, which contradicts the Bayesian-favored models that predict twin stars at $1.4 M_\odot$.
   • If the existence of mass-twin stars at $1.4 M_\odot$ is confirmed (supporting the eccentric MSP scenario), the maximum mass of such hybrid EoS is capped below $2.2 M_\odot$.

Significance and Claims
The paper concludes that the existence of mass-twin stars at $1.4 M_\odot$ effectively rules out hybrid stars as candidates for the observed mass-gap compact objects. If the pulsars PSR J0614-3329 and PSR J0437-4715 are confirmed as mass twins, the maximum mass for the corresponding hybrid EoS is limited to $M_{\text{max}} < 2.2 M_\odot$. Under this constraint, any compact object observed in the mass gap ($M > 2.5 M_\odot$) cannot be a hybrid neutron star and must be a black hole. The work highlights that while hybrid stars can theoretically populate the mass gap, the specific conditions required (early deconfinement) are incompatible with the observational evidence supporting the twin-star hypothesis at standard neutron star masses.