Are neutron star cores thought to be stiffer than photon gases? The weak energy condition and with the dominant energy condition allow anything from w=-1 to 1 (meaning that the magnitude of the pressure can be at most equal to the total mass-energy density). Values outside this range are needed to make wormholes and other causality violations (note that the strong energy condition is violated by dark energy). Cosmic inflation and dark energy are very close to the limit of -1, but nothing seems to come close to +1 even though it is "allowed". Both photon gases and ultra-relativistic degenerate gases are +1/3, can we do better? 
Neutron stars are thought to be stiffened by a repulsion in addition to the usual degeneracy pressure. Could this repulsion stiffen the equation of state enough to exceed 1/3? It takes energy to force repelling particles together (which can be interpreted as an the energy in the virtual particles that are mediating said repulsive force). This extra energy raises the mass, behaving much like kinetic energy in degenerate matter, which would seem to prevent rising above 1/3. Is there any way around this "problem"?
 A: Repulsion between nucleons must stiffen the equation of state of a neutron star.
It was established in 1939 (well before neutron stars were discovered) by Oppenheimer & Volkoff, that ideal neutron degeneracy pressure was incapable of supporting a ball of neutrons with mass greater than $0.75 M_{\odot}$.
Since then, many neutron stars have been discovered and all of them have masses significantly greater than this, requiring stiffer equations of state.
Ultra-relativistic neutron degeneracy pressure has an equation of state of the form $P \sim \rho/3$, where $\rho$ is the total energy density. In order to get the neutron stars we see, the equation of state in the core needs to behave more like $P \sim \rho$. This stiffer equation of state is thought to be caused by the repulsion between neutrons in strongly asymmetric nuclear matter at densities greater than the nuclear saturation density.
NB: This assumes the size of $w$ indicates "stiffness".
NB2: In terms of pressure as a function of mass density, the "stiffening" of the equation of state must raise the adiabatic index from somewhere between 4/3 and 5/3 (appropriate for non-interacting fermions of an intermediate degree of "relativisticness") to $\sim 2$.
A: The idea of a short-range repulsion between nucleons was conceived to explain a paradox concerning the stability of nuclear matter, but it is not really grounded in particle physics -- neither in the meson-exchange nor in the gluon-exchange picture.  (The paradox is that purely attractive two-body forces would predict an interaction energy per nucleon scaling as $-{{n}^{1}}$, whereas the kinetic energy of a free nonrelativistic Fermi gas scales as $+{{n}^{2/3}}$, where n denotes number density.   The attraction would overwhelm the kinetic energy in the limit of infinite density, and neutron matter would collapse.)  As a snotty former particle physicist, I regard the idea of short-range repulsion as a heuristic myth.    
The meson-exchange picture predicts a bewildering variety of attractive and repulsive forces, depending on the quantum numbers of the meson involved.  Single-pion exchange predicts the longest-range part of nuclear forces correctly, because pions are isovector and pseudoscalar.  An isoscalar vector meson would mediate a consistent repulsion between nucleons, but there are many other mesons in nature’s catalog, and there is no proof of net short-range repulsion.  
The quark-gluon picture seems most appropriate at extreme densities (probably higher than in any stable neutron star) where nucleons would overlap.  The kinetic energy per light (hence ultrarelativistic) quark scales as $+{{n}^{1/3}}$,  whereas gluon exchange predicts a net attraction that scales as $-{{g}^{2}}{{n}^{1/3}}$, but the running coupling fades as ${{g}^{2}}\propto 1/\log ({{p}_{F}})$ where ${{p}_{F}}\propto {{n}^{1/3}}$ , so kinetic energy ultimately dominates.  
The bottom line is that nuclear matter at extreme densities, regarded as quark matter, should be about as “soft” as photon gas.  
