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Abstract Numerical hydrodynamics simulations of gases dominated by ideal, nondegenerate matter pressure and thermal radiation pressure in equilibrium entail finding the temperature as part of the evolution. Since the temperature is not typically a variable that is evolved independently, it must be extracted from the evolved variables (e.g., the rest-mass density and specific internal energy). This extraction requires solving a quartic equation, which, in many applications, is done numerically using an iterative root-finding method. Here we show instead how the equation can be solved analytically and provide explicit expressions for the solution. We also derive Taylor expansions in limiting regimes and discuss the respective advantages and disadvantages of the iterative versus analytic approaches to solving the quartic. 

Numerical relativity simulations provide a means by which to study the evolution and end point of strong overdensities in cosmological spacetimes. Specific applications include studies of primordial black hole formation and the robustness of inflation. Here we adopt a toy model previously used in asymptotically flat spacetimes to show that, for given values of the overdensity and the mean curvature, solutions to the Hamiltonian constraint need not exist, and if they do exist they are not unique. Specifically, pairs of solutions exist on two branches, corresponding to strong-field and weak-field solutions, that join at a maximum beyond which solutions cease to exist. As a result, there is a limit to the extent to which an overdensity can be balanced by intrinsic rather than extrinsic curvature on the initial slice. Even below this limit, iterative methods to construct initial data may converge to solutions on either one of the two branches, depending on the starting guess, leading to potentially inconsistent physical results in the evolution.