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Certain physical quantities that characterize neutron stars and quark stars(e.g. their mass, spin angular momentum and quadrupole moment) are interrelatedin a way that is approximately insensitive to their internal structure. Suchapproximately universal relations are useful to break degeneracies in dataanalysis for future radio, X-ray and gravitational wave observations. Althoughthe pressure inside compact stars is most likely nearly isotropic, certainscenarios have been put forth that suggest otherwise, for example due to phasetransitions. We here investigate whether pressure anisotropy affects theapproximate universal relations and whether it prevents their use in futureobservations. We achieve this by numerically constructing slowly-rotating andtidally-deformed, anisotropic, compact stars in General Relativity to thirdorder in spin. We find that anisotropy affects the universal relations onlyweakly; the relations become less universal by a factor of 1.5-3 relative tothe isotropic case, but remain approximately universal to 10%. We succeed inexplaining this increase in variability as an increase in the eccentricityvariation of isodensity contours, which provides further support for theemergent approximate symmetry explanation of universality. Anisotropy does notaffect the universal relations to a sufficient level to prevent their use ingravitational wave astrophysics or in experimental relativity. We provide anexplicit example of the latter in dynamical Chern-Simons gravity. The increasein variability of the universal relations due to pressure anisotropy couldaffect their use in future X-ray observations. Given expected observationaluncertainties, however, the relations remain sufficiently universal for use insuch observations if the anisotropic modifications to the moment of inertia andthe quadrupole moment are less than 10% of their isotropic values.

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