Neutron stars (NSs) in a simple textbook-like setup where they are non-rotating, have no magnetic fields, and are in isolation in their equilibrium configuration are almost perfect spheres due to their strong self-gravity. In real astrophysical environments, however, they typically have a non-zero angular momentum causing their shape to be flattened along the spin axis. Many NSa are also part of a binary system where they are subject to the companion’s tidal force causing them to develop tidal bulges. The amounts of the spin- or tidally-induced deformations as well as the neutron stars’ moment of inertia characterizing how fast it rotates for a given angular momentum value, have two important features: (i) they depend sensitively on the properties of neutron star matter (which has been a longstanding frontier in science, see the article “What are neutron stars made out of?”), and (ii) their influence on observables is likely to be measured in the next few years. Recently, an additional, highly intriguing feature was discovered: there exist universal relations between these quantities that are to a very good approximation independent of the NS’s internal structure. Aside from the theoretical interest (why do these universal relations exist? are they connected with the no-hair properties of black holes? under which circumstances do they break down?) they also yield immediate practical uses where applying these relations can substantially improve the measurement accuracy of NS properties or provide NS-structure-independent tests of General Relativity.
Characteristic parameters for rotating and tidally deformed NSs
The moment of inertia of a NS is the ratio of its spin angular momentum to its angular frequency and characterizes how fast it rotates for a fixed value of its angular momentum. When a NS is spinning its shape deforms away from a sphere: it flattens along the spin axis and bulges along the orthogonal direction. Similarly, when a NS is in a binary, the companion’s tidal force causes it to develop tidal bulges along the line connecting the two bodies. These tidal and rotational deformation are imprinted in the NS’s external gravitational potential in terms of which the multipole moments are defined. An important property of the rotationally / tidally induced multipole moments is that they are directly proportional to the disturbing potential (the centrifugal and tidal potentials respectively) with proportionality coefficients that depend sensitively on the NS’s internal structure and are related to the dimensionless rotational / tidal Love numbers  named after the British mathematician A. E. H. Love¹
¹In the case of tidal deformation the Love numberscharacterize the dominant effect, however, the NS’s response to the tidal field becomes more complex at small orbital separation [2,3].
Recently, an intriguing feature of the characteristic NS parameters described above was revealed : suitably non-dimensionalized versions of the moment of inertia (“I”), tidal Love number (“Love”), and rotational quadrupole moment (“Q”) are related in a universal fashion that is independent of the NS’s internal structure. These relations hold to a high, percent-level, accuracy, and have been tested to hold in a variety of contexts for cold, rigidly rotating NSs . Additional approximately universal relations involving other NS properties have also been found, some of which hold to a lesser accuracy, e.g. [5-8]. The I-Love-Q universality is spoiled only in very limited astrophysical relevant contexts, with the most interesting being when a slowly rotating NS has a strong magnetic field in a twisted-torus configuration . The particular values describing the universality curve in the parameter space may be used to test General Relativity since it can differ in some alternative theories of gravity .
As recent controversies have highlighted, it is important to recognize that the universality relations hold only for appropriately normalized parameters rather thandirectly measured observables, and that conclusions about the existence of universality are highly sensitive to the choice of normalization.
Detailed investigations have elucidated several reasons which, when taken together, may explain the universality. The first is the consideration of which part of the NS interior dominates the contributions: this turns out to be the region between (50-95)% of the total radius, which corresponds to the density range (10^14-10^15)g/cm^3, where various proposed models for NS interiors have smaller differences than they do in the ultra-high density region. Second, the universality can be viewed as the emergence of an approximate symmetry such that the eccentricity profile of isodensity contours becomes self-similar, which occurs for compact stars and is lost when considering ordinary, non-compact stars. Third, NSs exhibit a feature reminiscent of the no-hair property of black holes: all of their multipole moments can (approximately) be determined from only the first three: their mass, spin, and quadrupole moment .
The universality relations have a wide range of uses in breaking observational degeneracies and hence improving measurement accuracies [4,7,8,13], distinguishing neutron stars from quark stars, and testing general relativity in a NS structure–independent fashion. The NS’s moment of inertia is likely to be measured in the next few years from pulsar timing observations, where it manifests in precessional effects . The NS quadrupole moment as well as the shape of its surface play an important role in the profiles of lines that originate near NS surfaces and are used to infer the masses and radii from X-ray spectroscopy . The NS’s tidal deformability and quadrupole moment are imprinted in the gravitational wave signal from inspiralling binaries , which are a major source for advanced LIGO , with their first detection expected to occur within the next few years.
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