Role of the symmetry energy on hybrid stars

This paper analyzes the role of symmetry energy in neutron and hybrid stars, demonstrating that a low-density onset of stiff quark matter in hybrid stars helps reconcile GW170817 and NICER observations while suggesting the binary system may consist of such hybrid stars or exhibit a quarkyonic crossover.

The Gist

Imagine the universe as a giant kitchen, and inside it, there are the densest, most extreme "cakes" imaginable: Neutron Stars. These are the leftover cores of massive stars that have collapsed. They are so heavy that a single teaspoon of their material would weigh a billion tons on Earth.

For a long time, scientists have been trying to figure out exactly what these stars are made of and how they behave under such crushing pressure. This paper is like a team of detectives (physicists) trying to solve a mystery: What is the recipe for these cosmic cakes, and does the recipe change if we add a secret ingredient called "Quark Matter"?

Here is a breakdown of their investigation using simple analogies:

1. The Two Competing Recipes (The Equation of State)

To understand a neutron star, scientists need a "recipe" called an Equation of State (EoS). This recipe tells us how the material inside the star reacts when you squeeze it.

• The "Soft" Recipe (SLy5): Imagine a sponge. If you squeeze it, it squishes down easily. This model suggests the star is made of normal nuclear matter that is relatively easy to compress.
• The "Stiff" Recipe (PKDD): Imagine a steel beam. If you try to squeeze it, it barely moves. This model suggests the star is made of matter that is very hard to compress.

The Problem:

• The "Soft" recipe fits well with data from gravitational waves (ripples in space-time from colliding stars, like the famous GW170817 event).
• The "Stiff" recipe fits well with data from telescopes that weigh heavy pulsars (stars spinning fast, like NICER observations).
• The Conflict: You can't have a recipe that is both soft enough to fit the wave data and stiff enough to hold up the heavy stars. It's like trying to build a bridge that is both made of jelly and made of steel at the same time.

2. The Secret Ingredient: Symmetry Energy

The paper focuses on a specific property of nuclear matter called Symmetry Energy. Think of this as the "balance" between neutrons and protons.

• In normal matter, neutrons and protons are balanced.
• In a neutron star, there are way more neutrons (it's "neutron-rich").
• The Symmetry Energy is like a "tension meter" that measures how much energy it takes to create that imbalance.
• The authors show that whether your recipe is "Soft" or "Stiff" depends entirely on how you tune this tension meter.

3. The Plot Twist: The Phase Transition

The authors propose a solution: What if the star isn't made of just one thing? What if, deep inside, the pressure gets so high that the "normal" nuclear matter melts into something else?

• The Phase Transition: Imagine an ice cube (solid) suddenly turning into water (liquid) because it got too hot. In the star, the "ice" is normal nuclear matter, and the "water" is Quark Matter (a soup of even smaller particles called quarks).
• This transition happens at a specific depth. The paper uses a mathematical model to describe this "melting point."

4. The Investigation: Testing the Scenarios

The team ran thousands of simulations (like running a cooking show 400,000 times with slightly different ingredients) to see which scenarios could explain the real-world data from GW170817 and NICER. They looked at three possible outcomes for the colliding stars in GW170817:

1. BNS (Two Normal Stars): Both stars are made of normal matter.
2. HSNS (One Hybrid, One Normal): One star has a quark core, the other doesn't.
3. BHS (Two Hybrid Stars): Both stars have quark cores.

The Findings:

• If the "Tension Meter" (Symmetry Energy) is High (Stiff Recipe): The star is very hard to compress. To match the gravitational wave data, the star must have a quark core. In this case, GW170817 was likely a collision of two Hybrid Stars (BHS). The transition to quark matter softens the star just enough to fit the data.
• If the "Tension Meter" is Low (Soft Recipe): The star is easier to compress. In this case, GW170817 could have been two normal stars, or a mix, or two hybrid stars. The data doesn't rule out normal stars as easily.
• The Best Fit: The data from the gravitational waves fits best if the stars were Hybrid Stars built on the Stiff recipe. This suggests that even if the outer layers are stiff, the core must turn into quark matter to explain the observations.

5. The "Quarkyonic" Mask

The paper mentions a fascinating possibility: The "Phase Transition" (melting) might actually be a "Quarkyonic" crossover.

• Analogy: Imagine a magician's trick. You think you are seeing a rabbit (normal matter) turn into a dove (quark matter). But maybe the rabbit was actually a dove in a rabbit costume all along.
• The authors suggest that what looks like a sharp "melting point" in their math might actually be a smooth transition (crossover) predicted by other theories. Their model can "mask" this smooth transition as a sharp one, making it hard to tell the difference without more precise data.

Summary

The paper concludes that Symmetry Energy is the key to unlocking the mystery of neutron stars.

• It determines whether a star is "soft" or "stiff."
• It dictates whether a star can exist as a normal neutron star or if it must have a quark core to survive the collision data we see.
• The evidence points toward the idea that the stars involved in the GW170817 event were likely Hybrid Stars (with quark cores), especially if the nuclear matter is "stiff."

In short, the universe's densest objects might be cosmic layer cakes: a crust of normal matter, but with a gooey, exotic quark center that changes the whole recipe.

Technical Summary

Technical Summary: Role of the Symmetry Energy on Hybrid Stars

Problem Statement
The internal composition of compact stars (CSs), particularly neutron stars (NSs), remains a subject of significant uncertainty due to tensions between terrestrial nuclear physics constraints and astrophysical observations. Specifically, there is a conflict between:

1. Tidal Deformability: The gravitational wave event GW170817 suggests a soft equation of state (EoS) with low tidal deformability ($70 < \tilde{\Lambda} \lesssim 500-720$), implying smaller stellar radii.
2. Mass Constraints: Radio astronomy observations of massive pulsars ($M > 2M_\odot$) require a stiff high-density EoS to support such gravitational fields.
3. Symmetry Energy Uncertainty: The density dependence of the nuclear symmetry energy ($E_{sym}$) is not precisely known. Different nuclear models (soft vs. stiff) yield different predictions for the symmetry energy slope ($L_{sym}$), leading to divergent predictions for CS properties.

The paper investigates how the nuclear symmetry energy influences the properties of hybrid stars (HSs)—stars containing a quark matter (QM) core formed via a first-order phase transition (FOPT)—and whether this transition can resolve the tension between GW170817 data and massive pulsar constraints.

Methodology
The authors employ a Bayesian statistical framework to explore the parameter space of nuclear and hybrid EoSs, constrained by nuclear physics and multi-messenger astrophysical data.

• Nuclear EoS Models: Two representative nucleonic EoSs are utilized:
  • SLy5 (Soft): Based on the Skyrme Hamiltonian, compatible with chiral effective field theory and finite nuclei properties, yielding a low symmetry energy slope ($L_{sym,2} \approx 48.3$ MeV).
  • PKDD (Stiff): Based on a relativistic Lagrangian, compatible with PREX-II experiment results, yielding a high symmetry energy slope ($L_{sym,2} \approx 79.5$ MeV).
• Phase Transition Model: The transition to quark matter is modeled using a parameterized first-order phase transition (FOPT) approach (Zdunik et al. model). The QM EoS is linear (constant sound speed), defined by three parameters:
  • $p_{PT}$: Pressure at the onset of the phase transition.
  • $\Delta\epsilon_{PT}$: Latent heat (energy density jump).
  • $c_{s,PT}$: Sound speed in quark matter (controlled by parameter $\alpha_{PT}$).
• Bayesian Analysis:
  • Priors: Approximately 374,000 prior models were generated, varying FOPT parameters and considering the two nuclear EoSs.
  • Constraints: Models were filtered based on causality, stability, the existence of $2M_\odot$ stars, and the effective tidal deformability ($\tilde{\Lambda}$) distribution from GW170817.
  • Binary Configurations: Three scenarios for the GW170817 binary system were tested: Binary Neutron Stars (BNS), Hybrid Star + Neutron Star (HSNS), and Binary Hybrid Stars (BHS).
  • Note: While NICER mass-radius data are discussed for comparison, they were not used as hard constraints in the Bayesian selection to avoid bias from conflicting analyses of specific sources.

Key Contributions and Results

1. Resolution of Tension via Hybrid Stars:
   The study confirms that introducing a first-order phase transition to quark matter can resolve the tension between the soft EoS required by GW170817 and the stiff EoS required by $2M_\odot$ pulsars. Hybrid EoSs successfully populate the parameter space compatible with both NICER contours and GW170817 tidal deformability, whereas pure nucleonic EoSs struggle to satisfy both simultaneously.

2. Impact of Symmetry Energy on Binary Nature:
   The nature of the nuclear symmetry energy dictates the possible configurations of the GW170817 binary system:

   • Stiff Nuclear EoS (PKDD): If the symmetry energy is stiff (high $L_{sym}$), the GW170817 event must be a Binary Hybrid Star (BHS). Pure BNS or HSNS configurations are excluded because the stiff nucleonic branch produces tidal deformabilities too high to match GW170817, even with a phase transition at high densities.
   • Soft Nuclear EoS (SLy5): If the symmetry energy is soft (low $L_{sym}$), the GW170817 event can be a BNS, HSNS, or BHS. The soft nucleonic branch is naturally more compatible with the observed tidal deformability.

3. Optimal Description of GW170817:
   The posterior probability distributions indicate that the GW170817 waveform is best described by a Binary Hybrid Star (BHS) system composed of stars with a stiff nuclear EoS (PKDD) that undergoes a phase transition to quark matter at low densities. This implies a low-density onset of stiff quark matter. The authors note this could also be interpreted as a quarkyonic crossover masquerading as a FOPT.

4. Parameter Correlations:
   The study quantifies correlations between FOPT parameters ($p_{PT}$, $\mu^*_{QM}$, $n_{PT}$) and the symmetry energy:

   • For a fixed transition pressure ($p_{PT}$), a stiff nuclear EoS (PKDD) favors a lower onset density ($n_{PT}$) and a higher chemical potential ($\mu^*_{QM}$) compared to a soft EoS.
   • The BHS configuration is weakly dependent on the nuclear EoS regarding the transition pressure, whereas BNS and HSNS configurations strictly require a soft nuclear EoS.

Significance
The paper claims that the nuclear symmetry energy plays a critical role in determining the nature of compact stars and the interpretation of gravitational wave events. Specifically:

• It demonstrates that the symmetry energy is not just a property of nuclear matter but a decisive factor in whether a binary merger like GW170817 involves quark matter.
• It provides a statistical framework showing that if the symmetry energy is stiff (as suggested by PREX-II), the presence of quark matter in GW170817 is a necessity, not just a possibility.
• It highlights that the "nature" of the binary system (BNS vs. BHS) is inextricably linked to the density dependence of the symmetry energy, suggesting that future precise measurements of the symmetry energy (e.g., via its curvature $K_{sym}$) are essential for constraining the dense matter phase diagram.

The authors conclude that while their findings support the BHS interpretation for GW170817 under stiff symmetry energy assumptions, further work involving a broader set of nuclear EoSs and the inclusion of NICER data in Bayesian constraints is necessary to refine these conclusions.