[Paper Review] Determination of the symmetry energy from the neutron star equation of state

Pedro Barata de Tovar, Márcio Ferreira+1 more|arXiv (Cornell University)|Dec 10, 2021

Pulsars and Gravitational Waves Research55 references4 citations

TL;DR

This paper investigates uncertainties in extracting the symmetry energy and proton fraction from neutron star $β$-equilibrium equations of state (EOS), showing that neglecting muon contributions or higher-order asymmetry terms (beyond parabolic approximation) introduces significant errors—up to 15% in proton fraction and overestimation of symmetry energy. The study demonstrates that accurate symmetry energy extraction requires including muons and higher-order asymmetry terms, especially near saturation density.

ABSTRACT

We analyze the uncertainties introduced in the determination of the neutron star matter proton fraction, in a range of densities close to the saturation density, if the cold $β$-equilibrium neutron star matter equation of state (EoS) is known. In particular, we discuss the effect of neglecting the muon contribution and of considering that the energy density of nuclear matter is well described by taking only terms until second order in the proton-neutron asymmetry. It is shown that two types of uncertainties may be associated with the extraction of the symmetry energy from the $β$-equilibrium equation of state: an overestimation if terms above the parabolic approximation on the asymmetry parameter are neglected, or an underestimation if the muon contribution is not considered. The effect of the uncertainty on the symmetric nuclear matter EoS on the determination of the proton fraction is discussed. It could be shown that the neutron star mass-radius curve is sensitive to the parabolic approximation on the asymmetry parameter.

Motivation & Objective

• To assess how approximations in the neutron star matter equation of state affect the determination of the symmetry energy and proton fraction.
• To quantify the impact of neglecting muon contributions on the proton fraction and symmetry energy in $β$-equilibrium neutron star matter.
• To evaluate the validity of the parabolic approximation in the asymmetry parameter and its effect on symmetry energy extraction.
• To examine the sensitivity of the neutron star mass-radius relation to higher-order terms in the symmetry energy and symmetric nuclear matter EOS.
• To determine whether Taylor expansions of the EOS can reliably reproduce key neutron star properties when fitted to reference models like DD2.

Proposed method

• Uses the DD2 relativistic mean-field EOS as a reference to compute exact $β$-equilibrium proton fractions and symmetry energy.
• Compares results from full $β$-equilibrium calculations (including muons) with approximations neglecting muons or higher-order asymmetry terms.
• Applies a Taylor expansion of the symmetric nuclear matter energy per nucleon and symmetry energy around saturation density, testing up to fourth and sixth order.
• Fits the Taylor expansion coefficients to the DD2 EOS to assess their physical interpretation and accuracy in reproducing the mass-radius relation.
• Analyzes the proton fraction and symmetry energy as functions of density, comparing exact and approximate treatments.
• Evaluates the sensitivity of the mass-radius curve to higher-order terms in the asymmetry parameter and in the symmetric nuclear matter expansion.

Experimental results

Research questions

• RQ1 How does neglecting muon contributions in $β$-equilibrium matter affect the extracted proton fraction and symmetry energy near saturation density?
• RQ2 To what extent does the parabolic approximation in the asymmetry parameter distort the symmetry energy and proton fraction in neutron star matter?
• RQ3 Can a Taylor expansion of the symmetric nuclear matter energy and symmetry energy around saturation density accurately reproduce the neutron star mass-radius relation?
• RQ4 What is the impact of higher-order terms (beyond quadratic) in the asymmetry parameter on the symmetry energy and proton fraction?
• RQ5 How do uncertainties in the symmetric nuclear matter EOS propagate into the determination of the symmetry energy from $β$-equilibrium conditions?

Key findings

• Neglecting muons leads to a 7% underestimation of the proton fraction at densities near saturation, increasing to 10–15% above muon onset.
• The parabolic approximation overestimates the symmetry energy by incorporating positive higher-order asymmetry terms (e.g., $S_4$) into the effective asymmetry.
• The proton fraction is underestimated by ~7% when higher-order asymmetry terms (beyond quadratic) are neglected, due to an effective increase in asymmetry.
• The mass-radius curve is sensitive to fourth-order terms in the EOS but not to sixth-order terms, indicating diminishing returns beyond fourth order.
• A fourth-order Taylor expansion of both symmetric nuclear matter energy and symmetry energy can reproduce the DD2 mass-radius curve within 10% uncertainty.
• Fitting Taylor expansion coefficients to the DD2 EOS allows recovery of the symmetry energy at saturation and below within 10%, provided sufficient terms are included.