[Paper Review] Analyzing the speed of sound in neutron star with machine learning

Sagnik Chatterjee, Harsha Sudhakaran+1 more|arXiv (Cornell University)|Feb 27, 2023

Pulsars and Gravitational Waves Research4 citations

TL;DR

This study uses a neural network to predict the speed of sound and trace anomaly in neutron stars using observational mass-radius data, constructing agnostic equations of state that satisfy thermodynamic and causality constraints. The trained model reveals non-monotonic speed of sound behavior and enforces a positive trace anomaly, indicating a physically consistent, softer equation of state in intermediate-mass and massive stars compared to untrained data.

ABSTRACT

Matter properties at the intermediate densities are still unknown to us. In this work, we use a neural network approach to study matter at intermediate densities to analyze the variation of the speed of sound and the measure of trace anomaly considering astrophysical constraints of mass-radius measurement of 18 neutron stars. Our numerical results show that there is a sharp rise in the speed of sound just beyond the saturation energy density. It attains a peak around $3-4$ times the saturation energy density and, after that, decreases. This hints towards the appearance of new degrees of freedom and smooth transition from hadronic matter in massive stars. The trace anomaly is maximum at low density (surface of the stars) and decreases as we reach high density. It approaches zero and can even be slightly negative at the centre of massive stars. It has a negative trough beyond the maximal central densities of neutron stars. The change in sign of the trace anomaly hints towards a near-conformal matter at the centre of neutron stars, which may not necessarily be conformal quark matter.

Motivation & Objective

• To model the equation of state (EoS) of neutron stars in the intermediate density regime where first-principles calculations are infeasible.
• To enforce thermodynamic stability and causality constraints on the EoS using machine learning.
• To investigate whether machine learning can produce physically consistent predictions for the speed of sound and trace anomaly in neutron star interiors.
• To determine how training on observational mass-radius data affects the behavior of the speed of sound and conformal anomaly.

Proposed method

• Construct a family of agnostic equations of state by randomizing the speed of sound within physical bounds (0 ≤ c_s² ≤ 1) and ensuring thermodynamic stability.
• Filter the EoS candidates using observational mass and radius measurements of neutron stars, including their uncertainties.
• Train a feedforward neural network to map mass-radius data to the speed of sound in the intermediate density range.
• Use the trained network to extrapolate and generate a refined, physically consistent set of EoSs.
• Compute the trace anomaly as a function of energy density and analyze its behavior before and after training.
• Enforce the condition Δ ≥ 0 (positive conformal anomaly) and assess whether the neural network enforces physical consistency.

Experimental results

Research questions

• RQ1 Does a machine learning model trained on neutron star mass-radius data produce a non-monotonic speed of sound profile in the intermediate density regime?
• RQ2 How does the neural network training affect the behavior of the trace anomaly, particularly regarding the condition Δ ≥ 0?
• RQ3 Does the trained model predict a softer or stiffer equation of state compared to untrained agnostic EoSs?
• RQ4 Do massive neutron stars exhibit a softer central EoS compared to intermediate-mass stars, as suggested by the trained speed of sound profile?
• RQ5 Can the neural network enforce physical consistency in the trace anomaly, such as monotonic behavior and non-negativity?

Key findings

• The trained neural network predicts a non-monotonic speed of sound profile with a peak beyond 3ε₀ and a secondary maximum, indicating complex matter behavior in the intermediate density regime.
• The trained EoS is softer overall than the untrained EoS, with a lower peak in the speed of sound, suggesting a more constrained and physically plausible equation of state.
• The trace anomaly for untrained EoSs shows non-monotonic behavior and dips below zero (Δ < 0) in the intermediate density range, violating physical consistency.
• After neural network training, the trace anomaly exhibits monotonic behavior and remains non-negative (Δ ≥ 0) throughout the density range, satisfying the physical constraint.
• The trained EoS maintains the conformal limit (Δ → 0) at high densities, consistent with perturbative QCD predictions.
• The results are quantitatively consistent with prior studies, such as Fujimoto et al. (2022), validating the model's reliability.