anisotropic neutron and hybrid stars

You can find further details of this work in the following papers: Zuraiq, Mukhopadhyay & Weber, Phys. Rev. D. (2024), Zuraiq et al., Universe 12 5, 130 (2026) and Zuraiq & Mukhopadhyay (submitted, 2026), arXiv: 2604.06308

The first neutron star (a pulsar) was discovered accidentally by then graduate student Jocelyn Bell Burnell more than fifty years ago. Ever since, everything we learn about these objects confirms the fact that neutron stars are some of the most extreme objects in our Universe. A typical neutron star contains something like the mass of the sun within just a 10 km radius. At their cores, neutron stars can reach densities several times the nuclear saturation density (the densities within atoms). These stars are probably the only laboratory that we know of at present that exhibit matter existing in such extreme conditions. In fact, the conditions are so extreme within neutron stars that we still do not know the exact equation(s) of state (EoS) that governs these objects. Strong interactions particularly are poorly constrained at these high densities. This is why there is no fixed “Chandrasekhar” type exact limiting mass for neutron stars.

Different high density EoS have been proposed over the years. A very popular class of phenomenological EoS are the relativistic mean field models. Dense matter EoS are constrained in two ways - astrophysically, from neturon star observations; and from laboratory measurements of the properties of nuclear matter at saturation density. Each of these EoS result in a specific mass-raidus relationship for the neutron star, with each supporting a certain maximum mass. Additionally, at the high densities in neutron star cores, exotic particles, which do not exist in stable form terrestrially, can further be present leading to further complexity.

Massive neutron stars are then of interest in multiple ways. Observationally detecting more massive neutron stars allows us to constrain and rule out more and more of this unknown high density paradigm of nuclear matter. On the other hand, there exists a well known “mass gap” in the literature between ~ 2.5 and 5 solar masses. With gravitational wave observations such as GW190814, this gap is slowly shrinking and it could be that massive neutron stars are what end up populating it. It all then comes back to the question - How heavy can a neutron star get?

On the other hand, as with any compact star, it is not only the EoS which determines mass but also additional effects that serve to push against gravity. One effect in particular, that serves to be extremely relevant, both observations and in the interest of “massive” neutron stars is the underlying magnetic field. High magnetic fields can have many effects - in fact, a high enough field can lead to microscopic changes in the EoS itself. Magnetic fields can also lead to the star being generally anisotropic and deviating from spherical symmetry.

We bring all the above considerations together to investigate massive, magnetized neutron stars. We consdier a few phenomenological EoS, which we felt best fit both the constraints mentioned above. We assume approximate spherical symmetry and introduce an analytical model for both the magnetic field (considered with two different orientations - radial and transverse) and a general model for the anisotropy (adapted from Bowers and Liang). We have also included the exotic hyperon and delta particles in our EoS considered. It appears that under all these physical effects working together, mass gap type massive neutron stars could very well exist.

Another interesting effect that can show up in the high densities of NS cores is the possibility of a deconfinement phase transition from hadronic matter to quark matter. We find that the possibility of quark cores does not prevent NSs (hybrid stars, in this case) from reaching mass gap range masses. Once again, the presence of magnetic fields and anisotropy can overcome the softening effects of quark matter. Quark cores are associated with new and interesting physics, including color superconductivity. We have proposed new, phenomenological, physics-informed anisotropy profiles within the star, such that the deformation of the star is linked directly to its internal physics, i.e., superconductivity and the magnetic field. Anisotropy-induced deformation can lead to the detection of these objects through gravitational wave (GW) observations, thereby giving us a way to probe hybrid star cores and constrain the relevant physics. The interactive plots of our predicted GW strain as a function of various quark matter parameters for planned future GW detectors are avilable here: Einstein Telescope and the Indian deci-hertz detector, IndIGO-D.