I-Love-Q anisotropically: Universal relations for compact stars with scalar pressure anisotropy

Certain physical quantities that characterize neutron stars and quark stars (e.g. their mass, spin angular momentum, and quadrupole moment) have recently been found to be interrelated in a manner that is approximately insensitive to their internal structure. Such approximately universal relations are useful to break degeneracies in data analysis and model selection for future radio, x-ray, and gravitational wave observations. Although the pressure inside compact stars is most likely nearly isotropic, certain scenarios have been put forth that suggest otherwise, for example due to magnetic fields or phase transitions in their interior. We investigate here whether pressure anisotropy affects the approximate universal relations and, if so, whether it prevents their use in future astrophysical observations. We achieve this by numerically constructing slowly rotating and tidally deformed, anisotropic, compact stars in general relativity to third order in stellar rotation relative to the mass shedding limit. We adopt simple models for pressure anisotropy where the matter stress-energy tensor is diagonal for a spherically symmetric spacetime but the tangential pressure differs from the radial one. We find that the equation-of-state variation increases as one increases the amount of anisotropy, but within the anisotropy range studied in this paper (motivated from anisotropy due to crystallization of the core and pion condensation), anisotropy affects the universal relations only weakly. The relations become less universal by a factor of 1.5-3 relative to the isotropic case when anisotropy is maximal, but even then they remain approximately universal to 10%. We find evidence that this increase in variability is strongly correlated to an increase in the eccentricity variation of isodensity contours, which provides further support for the emergent approximate symmetry explanation of universality. Whether one can use universal relations in actual observations ultimately depends on the currently unknown amount of anisotropy inside stars, but within the range studied in this paper, anisotropy does not prevent the use of universal relations in gravitational wave astrophysics or in experimental relativity. We provide an explicit example of the latter by simulating a binary pulsar/gravitational wave test of dynamical Chern-Simons gravity with anisotropic neutron stars. The increase in variability of the universal relations due to pressure anisotropy could affect their use in future x-ray observations of hot spots on rotating compact stars. Given expected observational uncertainties, however, the relations remain sufficiently universal for use in such observations if the anisotropic modifications to the moment of inertia and the quadrupole moment are less than 10% of their isotropic values.