Test a dense-matter equation of state

For a cold one-parameter equilibrium family, let n be baryon number density and epsilon total energy density, including baryon rest energy. At zero temperature the first law and the pressure relation connect these quantities. Pressure cannot be chosen independently from an unrelated energy table without checking that connection. A table for fixed proton fraction is not automatically the same barotrope as one in beta equilibrium, even if both are described informally as cold nuclear matter.

At finite temperature, pressure depends on entropy and composition as well as density. A small-amplitude perturbation may be faster than weak reactions, freezing composition instead of following the equilibrium composition curve. Its sound speed then uses that constrained thermodynamic derivative. The simple derivative below describes the specified cold barotrope; applying it to oscillations or merger matter needs an explicit choice of reaction and thermal time scales. The equation of state and the perturbation closure are related but distinct inputs.

Positive compressibility requires a nonnegative pressure derivative in the stable homogeneous region. Relativistic causality requires its sound speed not exceed c, giving $0\le dP/d\varepsilon\le1$ for this simple perfect-fluid description. Phase coexistence can have a zero equilibrium slope. These conditions do not guarantee a stable star: gravity can destabilize a sequence whose local matter remains causal. Conversely, the inequality $P<\varepsilon$ by itself does not constrain a sharply increasing derivative.

A thermodynamically consistent polytrope in rest density illustrates the danger. For constant exponent Gamma greater than one, integrate the cold first law to obtain the internal energy contribution P divided by Gamma minus one. The resulting sound-speed formula uses total energy plus pressure in its denominator. At very high pressure, an exponent larger than two eventually becomes acausal, even though it may provide an adequate fit over a restricted lower-density interval. An extrapolation must be tested over its actual use domain.

As a dated empirical benchmark, Fonseca and collaborators reported a gravitational mass $2.08\pm0.07M_\odot$ for PSR J0740+6620 in 2021, at 68.3 percent credibility from relativistic Shapiro delay. A viable stellar model must accommodate the measured rotating star within the relevant uncertainties and assumptions. Translating that observation into a constraint on the nonrotating maximum mass requires assessing rotational support; the observed central value is not itself a universal maximum-mass theorem.

Radius and tidal measurements probe integrated structure and typically emphasize pressures below the highest central density of the maximum-mass star. Their inference includes priors and emission or waveform models. The 2024 analysis by Rutherford and collaborators explicitly combines new NICER estimates, nuclear-theory inputs, masses, and tides under chosen parameterizations. Such dated analyses are evidence for a restricted family of equations of state, rather than a unique measured pressure curve or a proof that a proposed exotic component is present.