Neutron Star EoS | Yugesh Bhoge

Motivation

The internal structure of Neutron Stars (NS) remains one of the most enigmatic puzzles in modern astrophysics. While Binary Neutron Star (BNS) mergers like GW170817 have provided initial constraints, Neutron Star-Black Hole (NSBH) mergers offer a unique laboratory.

Specifically, if the NS is tidally disrupted before plunging into the BH ISCO (Innermost Stable Circular Orbit), matter remains outside the horizon, creating a cutoff frequency in the gravitational waveform. This cutoff is directly dependent on the NS radius and, consequently, the Equation of State (EoS).

Theoretical Framework

1. Dimensionless Tidal Deformability

$$ \Lambda = \frac{2}{3} \frac{c^2}{G} \left( \frac{R}{M} \right)^5 $$

Significance: $\Lambda$ quantifies the "stiffness" of the star. A stiffer EoS yields a larger radius $R$ and a significantly larger $\Lambda$. This parameter is imprinted on the phase of the gravitational wave signal during the inspiral.

2. Bayesian Model Selection

$$ \mathcal{B}_{AB} = \frac{P(d|\mathcal{M}_A)}{P(d|\mathcal{M}_B)} = \frac{\int \mathcal{L}(d|\theta_A) \pi(\theta_A) d\theta_A}{\int \mathcal{L}(d|\theta_B) \pi(\theta_B) d\theta_B} $$

Significance: The Bayes Factor $\mathcal{B}_{AB}$ compares how well two competing EoS models ($\mathcal{M}_A$, $\mathcal{M}_B$) fit the data ($d$). A value $\ln \mathcal{B} < 0$ for a model relative to the true injection indicates it is statistically disfavored.

Methodology

Injection: Synthetic signals generated using IMRPhenomNSBH with specific EoS properties (e.g., SLy, H4).
Sampler: Utilized the Bilby library with the dynesty sampler to explore the high-dimensional parameter space (chirp mass $\mathcal{M}$, mass ratio $q$, spins $\chi$, etc.).

Results: Recovering the EoS

Injection: SLy EoS

We evaluated the preference for various Equation of State models against an injected signal governed by the SLy EoS. The plot shows the natural log Bayes Factor ($\ln \mathcal{B}$) relative to the true model.

Analysis

The analysis successfully recovers the injected SLy EoS ($\ln \mathcal{B} = 0$). Notably, stiffer equations of state like H4 are strongly ruled out ($\ln \mathcal{B} \approx -8.1$).

This effectively places an upper bound on the tidal deformability, $\tilde{\Lambda}$, constraining the possible radii of neutron stars. Models predicting very large radii (stiff EoS) are inconsistent with the observed compactness required for tidal disruption effects in this regime.

Applications

This Bayesian framework is directly applicable to data from the LIGO-Virgo-KAGRA O4 and O5 observing runs. It enables robust distinction between BNS and NSBH populations based on the tidal imprint found in the high-frequency merger part of the waveform.

Implications

Constraints on $\Lambda$ map directly to the QCD phase diagram at low temperatures and supranuclear densities. Our results help rule out extreme exotic matter states that would require EoS stiffer than those observed, tightening the bounds on the nuclear symmetry energy.

Constraining Neutron Star Equation of State