Hybrid Stars with Post-Merger Rotation Profiles

This study demonstrates that differential rotation in hybrid stars with a first-order deconfinement phase transition can support unique quasi-toroidal configurations with ring-shaped quark cores, while revealing a degeneracy in rotational profiles between models with and without phase transitions that necessitates multi-messenger observations for differentiation.

The Gist

Imagine the universe's most extreme objects: neutron stars. These are the collapsed cores of dead stars, so dense that a single teaspoon of their material would weigh a billion tons on Earth. Usually, we think of them as solid, spinning balls of matter. But this paper explores what happens when these stars spin really fast and undergo a dramatic internal "phase change," similar to how ice melts into water, but with matter itself transforming.

Here is a breakdown of the paper's findings using everyday analogies:

1. The Setting: A Cosmic Dance Floor

When two neutron stars crash into each other (a merger), they don't just stop; they form a new, super-heavy object that spins incredibly fast.

• The Analogy: Imagine a figure skater pulling their arms in to spin faster. In this cosmic crash, the new star spins so fast that it can't stay a perfect sphere. It gets squashed and stretched.
• The Twist: Unlike a solid top that spins evenly, this new star spins differentially. Think of a spinning pizza dough: the center might be spinning one way, while the outer edges are spinning another. In these stars, the fastest spinning part isn't always in the dead center; sometimes it's in a ring around the middle.

2. The Ingredients: Hadrons vs. Quarks

Inside a normal neutron star, matter is made of protons and neutrons (called hadrons). But if you squeeze them hard enough, they might "melt" into a soup of their smaller parts, called quarks.

• The Analogy: Imagine a block of ice (hadrons). If you apply enough pressure, it suddenly turns into water (quark matter). This paper studies stars that have both ice and water inside them at the same time.
• The "Hybrid" Star: These are stars with a core of "quark soup" surrounded by a shell of "neutron ice." The paper looks at six different recipes for how this soup is made, changing how "stiff" or "soft" the soup is.

3. The Discovery: The "Donut" Star

The researchers used powerful computer simulations to see what happens when these hybrid stars spin with that "differential" motion (where the spin speed varies).

• The Surprising Shape: They found a configuration they call a "quasi-toroidal" star.
  • The Analogy: Imagine a donut. In this specific type of spinning star, the center is made of normal neutron matter, the outer layers are also normal neutron matter, but in the middle, there is a ring of quark soup.
  • It's like a cosmic donut where the "hole" and the "crust" are made of one material, but the "dough" in the middle is a completely different, exotic material.

4. The "Turning Point" Problem

The paper also looked at what happens as you add more and more spin (angular momentum) to these stars.

• The Analogy: Think of a spinning top. As you spin it faster, it gets more stable for a while, but eventually, it wobbles and falls.
• The Finding: As the spin increases, the "safe zone" for these hybrid stars shrinks. The point where the star becomes unstable and collapses happens at lower densities than before.
• The Consequence: For stars with this specific "ice-to-water" phase transition, spinning them faster actually makes it harder to keep the exotic quark core stable. The region where you can have a stable hybrid star gets smaller and smaller as the spin speeds up.

5. The Great Mystery: "Look-Alike" Stars

One of the most interesting findings is about how hard it is to tell these stars apart.

• The Analogy: Imagine two different cars: one is a standard sedan, and the other is a sedan with a hidden engine upgrade. If you look at them from far away, they might look identical in size and speed, even though their insides are totally different.
• The Finding: The researchers found that a star made entirely of normal matter and a star with a hidden quark core can have the exact same mass and radius. They look the same from the outside, even though their internal "ingredients" are completely different.
• The Implication: This means that just by looking at the size or weight of a post-merger star, we might not be able to tell if it has a quark core or not. We need other clues (like listening to the "sound" of the star via gravitational waves) to solve the mystery.

Summary

In short, this paper simulates what happens when two neutron stars crash and spin wildly. They discovered that:

1. These stars can form a donut shape where a ring of exotic quark matter sits between layers of normal matter.
2. Spinning them faster shrinks the safe zone for these exotic cores.
3. It is very difficult to tell a "normal" star from a "hybrid" star just by looking at their size and weight, because they can look identical from the outside.

The paper concludes that to understand the true nature of these cosmic collisions, we need to combine different types of observations (like light and gravitational waves) rather than relying on just one measurement.

Technical Summary

Technical Summary: Hybrid Stars with Post-Merger Rotation Profiles

Problem Statement
Neutron stars (NSs) formed immediately following binary mergers are characterized by extreme densities, high temperatures, and rapid differential rotation. While stationary models with differential rotation offer a computationally efficient alternative to full numerical relativity simulations for interpreting post-merger gravitational wave (GW) signals, the underlying equation of state (EoS) remains a significant source of uncertainty. Specifically, it is unclear whether the cores of such remnants reach densities sufficient to trigger a first-order deconfinement phase transition from hadronic matter to color-superconducting quark matter. Previous studies have examined uniformly rotating hybrid stars or differentially rotating stars with simplified EoSs, but a systematic investigation of hybrid stars using a realistic, four-parameter differential rotation law (capable of producing off-center angular velocity maxima) combined with tabulated hadronic and quark EoSs has been lacking. This study addresses the impact of such phase transitions on the global properties and internal structure of differentially rotating hybrid stars.

Methodology
The authors construct stationary, axisymmetric models of hybrid stars using an extended version of the RNS code, which implements the modified Komatsu-Eriguchi-Hachisu (KEH) scheme. The study employs:

• Equation of State (EoS):
  • Hadronic Phase: Modeled using the relativistic density functional DD2npY-T EoS, which includes nucleonic and hyperonic degrees of freedom and satisfies observational constraints (maximum mass, tidal deformability, NICER measurements).
  • Quark Phase: Described using a relativistic density functional (RDF) approach with medium-dependent scalar and pseudoscalar couplings, incorporating two-flavor color-superconducting (2SC) pairing.
  • Hybrid Construction: The transition is modeled via the Maxwell construction, enforcing mechanical stability and baryon chemical equilibrium. To handle numerical discontinuities in energy density associated with the first-order phase transition, a slight smoothing is introduced, creating a finite mixed-phase region rather than an infinitely thin interface.
• Rotation Law: The study utilizes the phenomenological four-parameter rotation law proposed by Uryu et al., which allows for the maximum angular velocity to be shifted away from the center. The law is parameterized by ratios $\lambda_1 = \Omega_{max}/\Omega_c$ and $\lambda_2 = \Omega_e/\Omega_c$.
• Solution Classes: The authors focus on two distinct classes of solutions identified in the literature:
  • Type A (Quasi-spherical): The maximum energy density remains at the center.
  • Type C (Quasi-toroidal): The maximum energy density shifts away from the center, potentially forming a toroidal structure.
• Parameter Space: Six hybrid configurations are generated by varying the vector coupling ($\eta_V$) and diquark pairing strength ($\eta_D$) to cover a range of quark matter onset densities and stiffness. Sequences of equilibrium solutions are computed for constant angular momentum ($J$).

Key Contributions and Results

1. Existence of Quasi-Toroidal Hybrid Configurations: The study demonstrates the existence of Type C solutions where deconfined quark matter forms a ring (torus) around the center of mass, while hadronic matter persists at the center and in the outer layers. In other configurations, if the central density exceeds the phase transition threshold, a quasi-toroidal quark core is formed.
2. Shift of Turning Points: A critical finding is that as the total angular momentum $J$ increases, the turning points (critical values defining the last stable solution) of the $J = \text{const}$ sequences shift toward lower energy densities. Consequently, the region of stable differentially rotating neutron stars that exhibit a phase transition shrinks considerably. For high $J$, solutions may only exist within the quark matter branch or a very narrow interval of the mixed phase.
3. Continuity of Angular Velocity: Despite the discontinuity (or smoothed jump) in energy density at the phase boundary, the angular velocity profile remains continuous throughout the star for both Type A and Type C solutions.
4. Degeneracy at Crossing Points: The authors analyze "crossing points" (CPs) where mass-radius curves for different EoSs intersect. They find that at these points, the rotational profiles (angular velocity distributions) of solutions with and without phase transitions are remarkably similar, despite significant differences in their internal energy density profiles.
5. Numerical Constraints: The study highlights limitations in the RNS code regarding convergence for high-$J$ sequences and specific parameter pairs, noting that the maximum reliable angular momentum for quasi-spherical configurations is reduced due to convergence issues.

Significance
The paper claims that the observed similarity in rotational profiles at mass-radius crossing points, despite differing internal compositions, reveals a degeneracy between post-merger remnant properties for models with and without phase transitions. This implies that gravitational wave observations of the post-merger spectrum alone may be insufficient to definitively distinguish between hadronic and hybrid stars. The authors emphasize the necessity of complementary multi-messenger observables to break this degeneracy and constrain the nature of the dense matter equation of state. The work provides a systematic framework for understanding how phase transitions manifest in the macroscopic properties of rapidly rotating, post-merger remnants.