Analytical Representation for Equations of State of Dense Matter

Abstract: We present an analytical unified representation for 22 equations of state (EoS) of dense matter in neutron stars. Such analytical representations can be useful for modeling neutron star structure in modified theories of gravity with high order derivatives.

• The paper introduces a unified, differentiable analytical representation for 22 dense matter EoSs, significantly improving accuracy in neutron star modeling.
• It employs a composite function with switching functions and explicit parametrizations to seamlessly bridge low- and high-density regimes.
• The approach yields a maximum 0.05% relative error in mass-radius relations, ensuring robust application in GR and extended gravity simulations.

Analytical Representation for Equations of State of Dense Matter

Introduction

The accurate modeling of neutron star (NS) internal structure is fundamentally dependent on the Equation of State (EoS) of dense matter. Traditionally, the EoSs are provided as tabulated data, which are then interpolated during numerical solution of the Tolman-Oppenheimer-Volkoff (TOV) equations in General Relativity (GR). While spline or piecewise polynomial interpolations offer sufficient accuracy for second-order ODEs like the TOV equations, this approach is inadequate for modified gravity theories involving higher-order differential equations, where continuity of higher derivatives is required. "Analytical Representation for Equations of State of Dense Matter" (1108.2166) systematically addresses this gap by introducing a unified, differentiable analytical parametrization for 22 commonly used EoSs, facilitating their application in both GR and extended gravity frameworks.

Methodology

This work generalizes and extends the analytical formalism previously developed by Haensel & Potekhin (2004) for the FPS and SLY EoSs. The authors construct a composite function for $\zeta = \log_{10}(P/ \mathrm{dyn\,cm^{-2}})$ as a function of $\xi = \log_{10}(\rho/ \mathrm{g\,cm^{-3}})$, using a combination of matching functions and explicit parametrizations for low- and high-density regimes. The analytical form is:

$$\zeta = \zeta_{\mathrm{low}} f_0[a_1 (\xi - c_{11})] + f_0[a_2 (c_{12} - \xi)] \zeta_{\mathrm{high}},$$

with $f_0(x) = 1/(1+e^x)$ serving as the switching function for seamless interpolation between domains. The low-density part is anchored to the well-established BPS and NV EoSs, fixing 12 parameters. The high-density sector incorporates 11 adjustable parameters per EoS, which are optimized to fit the dense matter region relevant for NS interiors. All fit coefficients are provided for 22 standard models, including AP4, MPA1, GM3nph, WFF2, and MS2.

Numerical Precision and Validation

The analytical representation achieves a maximum relative error in the mass-radius ($M$-$R$) relation of approximately $0.05\%$ near the maximum mass, compared to the outcome using direct tabulated EoS interpolation. This level of precision is sufficient for state-of-the-art neutron star structure calculations, including those where higher-order continuity is required due to the presence of high derivatives—such as in hydrostatic equilibrium equations in $f(R)$ or other scalar-tensor gravity theories. The adiabatic index $\Gamma = d\zeta/d\xi$ obtained from the fit remains smooth across the parameter space, further confirming the differentiability and physical fidelity of the representation.

Implications and Future Applications

The provided parametrization enables robust modeling of NSs in modified theories of gravity, specifically supporting fourth-order structural equations where interpolation artifacts would otherwise introduce unphysical discontinuities and numerical instabilities. The analytical form allows for direct computation of all required derivatives with respect to density, which is critical when incorporating EoS effects into extended gravity theories, numerical relativity codes, or in studies of NS cooling and oscillations where thermodynamic derivatives play a central role.

In practical terms, the framework drastically simplifies the incorporation of a wide array of EoS models into simulations, eliminating the need for cumbersome tabular handling while guaranteeing smoothness and accuracy. These developments also facilitate automated model comparison and Bayesian inference analyses in high-density astrophysics, as all derivatives and function evaluations are explicit and analytic.

Theoretical Significance and Future Directions

Analytical representations of EoS, as exemplified by this framework, shift the paradigm for incorporating dense matter microphysics in astrophysical and gravitational studies. The approach is agnostic to the underlying microphysical details but flexible enough to encode the diversity of published EoS models. This enables studies of populations of neutron stars under different high-density physics scenarios and systematic sensitivity studies with respect to microphysical uncertainties.

Looking forward, this methodology could be extended to accommodate additional physical effects, such as temperature dependence, composition gradients, phase transitions (e.g., hadron-quark), and magnetic field influences, all provided the availability of reference data for parameter fitting. Automation of the fitting process and connections to nuclear effective theory predictions could further support multi-messenger constraints on the EoS.

Conclusion

By providing a highly accurate, differentiable analytical representation for 22 equations of state of dense matter, this work resolves a critical bottleneck in neutron star modeling, particularly in the context of theories of gravity beyond GR, where high-order derivatives of EoS are required. The formalism guarantees minimal deviation from tabular results while ensuring the mathematical properties necessary for advanced theoretical modeling. This contribution constitutes a foundational tool for future research spanning neutron star interiors, gravitational theory, and dense matter astrophysics.