Unraveling Dense Nuclear Matter Through a Three Dimensional Equation of State

May 29, 2024

Nanxi Yao

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The nuclear equation of state (EOS) is a fundamental guidebook, showing how the microscopic interactions of nuclear matter result in an emergent macroscopic behavior that can be described by just a few collective properties. An important example of this macroscopic behavior occurs in neutron stars.  They represent the density frontier of nuclear physics - a single sugar cube of neutron star material weighs a billion tons. Obtaining the nuclear EOS under such conditions is difficult, but recent strides in data collection from both nuclear experiments and astrophysical observations have significantly enhanced our understanding. In our recent work, we provide a new technique to obtain a three dimensional EOS for nuclear matter across density, charge fraction, and temperature by expanding from a one dimensional EOS.

Currently, our theoretical models of neutron stars are limited to a very restricted part of the phase diagram: the models assume a neutron star is at zero temperature and in what we call beta-equilibrium, a state where the various generalized nuclear charges sum to zero. This simplification, while helpful, doesn't capture the full complexity of neutron stars. The shortcoming becomes particularly apparent in a neutron star merger, when two neutron stars spiral towards each other and collide in a cosmic event. This collision releases a burst of energy which is carried away by gravitational waves, providing valuable insights into extreme astrophysical phenomena. By expanding the theoretical model to three dimension, non-zero charge fractions and non-zero temperatures, we can better understand what the nuclear equation of state tells us about the physics of these events.

An illustration of two neutron stars merging, along with the resulting gravitational waves.<br><em><br>Image credit: <a href="https://svs.gsfc.nasa.gov/10543">NASA Goddard Flight Center</a></em>
An illustration of two neutron stars merging, along with the resulting gravitational waves.

Image credit: NASA Goddard Flight Center

 Previously, numerical simulations of merging binary neutron stars have explored the impact of temperature variations in different ways. While these approaches offer insights, they often reflect a limited range of microscopic interactions or degrees of freedom. What is needed is a more flexible framework that can systematically analyze both temperature and charge effects without making specific assumptions about particle types or phase transitions. In addition, we need to ensure thermodynamic consistency, causality and stability.

To do this, we need to ensure that the models we create are consistent with what we know from experiments. Our method involves two main steps: first, we expand our model to cover a range of charge fractions, using a pre-existing expansion called the symmetry energy expansion, which is essentially a Taylor expansion. We make sure our model satisfies properties from heavy-ion collisions up to this point. Second, we expand our model to include a finite range of temperatures. We propose a new, well-controlled expansion of the pressure to finite temperatures to estimate how temperature affects pressure and heat capacity. We then used numerical simulations to investigate how accurate our expansion is for a known model. Doing expansions like this usually results in a trade-off between applicability and accuracy. The further away we go from the restricted limit, the wider the range of applicability of the resulting NS EOS, but the higher the error is likely to be. By studying different sources of uncertainty, we were able to expand into the range needed to model neutron star merges, whilst keeping the error within 5%.

We have developed open-source code that will be released on publication, and is ready to be used in relativistic simulations of neutron star mergers. This will provide a framework for extending our understanding of nuclear matter under extreme densities. While our inital focus has been on studying neutron star mergers, these methods could also be useful for understanding heavy-ion collisions and supernova events.  Additionally, strange particles play a big role in heavy-ion collisions and needs to be considered in our models. Finally we are interested in understanding how approximations in the symmetry energy expansion might affect nuclear reactions, especially in supernovae simulations where neutrino transport effects are crucial.


This piece was based on the paper: D. Mroczek, N Yao, Katherine Zine, J. Noronha-Hostler, V Dexheimer, A Haber, E  Most, “Finite-temperature expansion of the dense-matter equation of state,” to appear in Physical Review D.