### The Neutron Star Connection

Collisions of heavy nuclei at RHIC (and soon LHC) energies create a state of matter similar to that of the early universe a few microseconds after the big bang, characterized by a temperature of T ~200-400 MeV. Temperature is one useful intensive thermodynamical variable for depicting the phase diagram of nuclear matter. Another is the net baryon density ρ_{B} (or alternatively, the baryon chemical potential μ_{B}), reflecting the fact that baryon number is a conserved quantity. The central region of a collision at RHIC is nearly (net!) baryon free, with a value of μ_{B}/T ~10% << 1, and therefore crudely approximates the early universe, where this ratio was of order 10^{-9}.

As exciting as it is to explore the early universe in the laboratory, very relativistic nuclear collisions access only a ‘small’ region of the phase diagram near μ_{B }=0. There is also enormous astrophysical relevance *along* the μ_{B} axis very near T=0. This is the domain that describes nuclear matter as found in *neutron stars.*

A brief glance at this nuclear phase diagram cartoon gallery suggests that there is a great deal of room for improving our quantitative understanding the nuclear phase diagram everywhere, but particularly at very high baryon density. One method to achieve this will be special low energy runs at RHIC (the so-called critRHIC program), and ultimately via the dedicated GSI-FAIR facility. Until then, neutron star observations coupled with theoretical modeling are our best ‘laboratory’ for understanding the equation of state of nuclear matter at high baryon density. An excellent review of this topic by Page and Reddy

describes the state of the art circa 2006. However, re-analysis of a previously measured neutron star radius may require rethinking our models for the interior of the largest neutron stars (see below).

This recent preprint

provides a very readable introduction to the physics of neutron stars. The emphasis in this paper is on ‘conventional’ nuclear physics extrapolated to neutron star densities , but even modest extrapolations beyond normal nuclear density require modeling the nuclear interaction in regimes well beyond laboratory measurements. A common approach is based on extensions to the venerable Walecka model, tuned to reproduce the known features of nuclear saturation, binding energy and compressibility. The equation of state is obtained via mean-field theory, and used to determine the all-important mass-radius relation.

At very large densities, an alternative description starting from the QCD Lagrangian rather than an effective theory of baryons and mesons becomes necessary. In fact, at asymptotically high densities (at T=0) QCD becomes weakly-coupled, and rigorous calculations become possible. A rich phase structure emerges (due to the extra degrees of freedom of flavor and color as compared to the electrons in ordinary superconductors, exhibiting color superconductivity (CSC) and various forms of Cooper-pairing up to and including completely flavor-color correlated “color-flavor locking” (CFL). Complete reviews of these fascinating possibilities may be found in

- The Condensed Matter Physics of QCD
- The Quark Gluon Plasma in Equilibrium
- Color Superconductivity in Dense Nuclear Matter

and references therein. While there has been an enormous theoretical literature devoted to the study of CSC and CFL, ambiguities remain in the precise determination of the equation of state and its connection to that of baryonic nuclear matter.

The available astrophysical data on neutron star masses and radii constrain but by no means determine the equation of state. While the observed distributions of masses clusters around 1.5 solar masses, only the famous Hulse-Taylor system, with its value of 1.44 solar masses, and PSR B1534+12 are particularly well-measured. The situation in 2006 is nicely illustrated in Figure 15 of the Page-Reddy review article, which, based on the (then) value of 2.1 +/- 0.2 solar masses for PSR J0751+1807, excluded entire classes of “hybrid models” which exhibited soft equations of state due to condensation and/or phase transitions.

Therefore, it is of very considerable interest that a re-analysis of the PSR J0751+1807 data has led to a very different value of ≈1.3 +/- 0.2 solar masses. If true, this changes everything. In particular, the 1994 prediction by Bethe and Brown of a maximum neutron star mass of 1.5 solar masses

would no longer be invalidated. Stimulated by this development, Brown, Lee and Rho returned to the Bethe-Brown prediction of a maximum neutron star mass of 1.5 solar masses and investigate its consequences up to and including the implications for “cosmological natural selection”. More relevant to this discussion is their conclusion that if kaon condensation occurs at roughly 3 times normal nuclear density, “*then quark matter – with or without color superconductivity – that can appear at higher densities would be buried in the ‘nothingness’ of black holes*.” This scenario can be falsified by the discovery of a neutron star of mass greater than 2 solar masses. Its contrapositive, i.e, that kaon condensation up to the limit of gravitational collapse prevents the appearance of the exotic CSC and CFL states ‘present’ in the QCD lagrangian is perhaps the more striking conclusion- it would imply a sort of “gravitational censorhip” that would prevent these otherwise-plausible states from being realized in nature.