In this work we provide a new, well-controlled expansion of the equation of state of dense matter from zero to finite temperatures $\left(T\right)$ while covering a wide range of charge fractions $\left(Y_Q\right)$, from pure neutron to isospin symmetric nuclear matter. Our expansion can be used to describe neutron star mergers using the equation of state inferred from neutron star observations. We discuss how knowledge from low-energy nuclear experiments and heavy-ion collisions can be directly incorporated into the expansion. We also suggest new thermodynamic quantities of interest that can be calculated from theoretical models or directly inferred by experimental data that can be used to infer the finite temperature equation of state. With our new method, we can quantify the uncertainty in our finite $T$ and $Y_Q$ expansions without making assumptions about the underlying degrees of freedom. We can reproduce results from a microscopic equation of state up to $T=100$ MeV for baryon chemical potential ${\mu }_B\gtrsim 1100$ MeV $[\approx (1–2)n_{\mathrm{sat}}]$ within $5\%$ error, with even better results for larger ${\mu }_B$ and/or lower $T$. We investigate the sources of numerical and theoretical uncertainty and discuss future directions of study.

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• Equations of state
• Nuclear matter in neutron stars