Approximate universal relations for neutron stars and quark stars Authors: Kent Yagi and Nicolas Yunes NOTICE: this is the author’s version of a work that was accepted for publication in Physics Reports. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physics Reports, VOL# 681, (March 2017). DOI# 10.1016/j.physrep.2017.03.002 Yagi, Kent , and Nicolas Yunes. "Approximate universal relations for neutron stars and quark stars." Physics Reports (March 2017). DOI: 10.1016/j.physrep.2017.03.002. Made available through Montana State University’s ScholarWorks scholarworks.montana.edu Approximate universal relations for neutron stars and quark stars Kent Yagi a,b,*, Nicolás Yunes b a Department of Physics, Princeton University, Princeton, NJ 08544, USA b eXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, MT 59717, USA Abstract Neutron stars and quark stars are ideal laboratories to study fundamental physics at supra nuclear densities and strong gravitational fields. Astrophysical observables, however, depend strongly on the star's internal structure, which is currently unknown due to uncertainties in the equation of state. Universal relations, however, exist among certain stellar observables that do not depend sensitively on the star's internal structure. One such set of relations is between the star's moment of inertia (I), its tidal Love number (Love) and its quadrupole moment (Q ), the so-called I–Love–Q relations. Similar relations hold among the star's multipole moments, which resemble the well-known black hole no-hair theo- rems. Universal relations break degeneracies among astrophysical observables, leading to a variety of applications: (i) X-ray measurements of the nuclear matter equation of state, (ii) gravitational wave measurements of the intrinsic spin of inspiraling compact objects, and (iii) gravitational and astrophysical tests of General Relativity that are independent of the equation of state. We here review how the universal relations come about and all the applications that have been devised to date. Contents 1. The ubiquity of universality ..................................................................................................................................................................... 2 1.1. Universality in black holes........................................................................................................................................................... 3 1.2. Universality in non-vacuum spacetimes .................................................................................................................................... 4 1.3. Why universality matters ............................................................................................................................................................ 4 1.4. Layout and conventions ............................................................................................................................................................... 5 2. Universal relations for neutron stars and quark stars in general relativity .......................................................................................... 6 2.1. Neutron stars, quark stars and approximation schemes ........................................................................................................... 6 2.1.1. The structure of compact stars .................................................................................................................................... 6 2.1.2. Useful approximations ................................................................................................................................................. 7 2.2. I–Love–Q ...................................................................................................................................................................................... 9 2.2.1. Newtonian results......................................................................................................................................................... 9 2.2.2. Relativistic results......................................................................................................................................................... 10 2.3. No-hair relations .......................................................................................................................................................................... 11 2.3.1. Newtonian results......................................................................................................................................................... 12 2.3.2. Relativistic results......................................................................................................................................................... 14 3. Extensions of universality in general relativity ...................................................................................................................................... 15 3.1. Different normalizations to strengthen universality ................................................................................................................. 15 3.2. Rotation......................................................................................................................................................................................... 16 3.2.1. Rapid uniform rotation................................................................................................................................................. 16 3.2.2. Differential rotation...................................................................................................................................................... 18 3.3. Magnetic fields ............................................................................................................................................................................. 19 3.4. Anisotropic pressure .................................................................................................................................................................... 20 3.5. Non-barotropic equations of state .............................................................................................................................................. 22 3.6. Dynamical tides ............................................................................................................................................................................ 22 4. Connection to other universal relations .................................................................................................................................................. 23 4.1. Multipole Love relations .............................................................................................................................................................. 23 4.1.1. No rotation .................................................................................................................................................................... 23 4.1.2. Slow rotation................................................................................................................................................................. 26 4.2. Binary love relations .................................................................................................................................................................... 27 4.3. I–Love–Q and oscillation frequencies ........................................................................................................................................ 29 4.4. I–Love–Q and compactness......................................................................................................................................................... 29 4.4.1. Numerical analysis........................................................................................................................................................ 30 4.4.2. Analytic investigation................................................................................................................................................... 31 4.5. Darwin–Radau relation ................................................................................................................................................................ 32 5. Why universality holds............................................................................................................................................................................. 33 5.1. The Newtonian limit .................................................................................................................................................................... 34 5.2. The black hole limit...................................................................................................................................................................... 35 5.3. An approximate emergent symmetry......................................................................................................................................... 36 6. I–Love–Q in modified gravity and exotic compact objects ..................................................................................................................