Though individual stellar parameters of compact stars usually demonstrate obvious dependence on the equation of state (EOS), EOS-insensitive universal formulas relating these parameters remarkably exist. In the present paper, we explore the interrelationship between two such formulas, namely the f -I relation connecting the f -mode quadrupole oscillation frequency ω2 and the moment of inertia I, and the I-Love-Q relations relating I, the quadrupole tidal deformability λ2, and the quadrupole moment Q, which have been proposed by Lau, Leung, and Lin [Astrophys. J. 714, 1234 (2010)] and Yagi and Yunes [Science 341, 365 (2013)], respectively. A relativistic universal relation between ω l and λ l with the same angular momentum l = 2, 3, . . ., the so-called "diagonal f -Love relation" that holds for realistic compact stars and stiff polytropic stars, is unveiled here. An in-depth investigation in the Newtonian limit is further carried out to pinpoint its underlying physical mechanism and hence leads to a unified f -I-Love relation. We reach the conclusion that these EOS-insensitive formulas stem from a common physical origin -compact stars can be considered as quasiincompressible when they react to slow time variations introduced by f -mode oscillations, tidal forces and rotations.

Due to our ignorance of the equation of state (EOS) beyond nuclear density, there is still no unique theoretical model for neutron stars (NSs). It is therefore surprising that universal EOS-independent relations connecting different physical quantities of NSs can exist. Lau et al. found that the frequency of the f -mode oscillation, the mass, and the moment of inertia are connected by universal relations. More recently, Yagi and Yunes discovered the I-Love-Q universal relations among the mass, the moment of inertia, the Love number, and the quadrupole moment. In this paper, we study these universal relations in the Eddington-inspired Born-Infeld (EiBI) gravity. This theory differs from general relativity (GR) significantly only at high densities due to the nonlinear coupling between matter and gravity. It thus provides us an ideal case to test how robust the universal relations of NSs are with respect to the change of the gravity theory. Thanks to the apparent EOS formulation of EiBI gravity developed recently by Delsate and Steinhoff, we are able to study the universal relations in EiBI gravity using the same techniques as those in GR. We find that the universal relations in EiBI gravity are essentially the same as those in GR. Our work shows that, within the currently viable coupling constant, there exists at least one modified gravity theory that is indistinguishable from GR in view of the unexpected universal relations. Subject headings: dense matter -equation of state -stars: neutron

We study the hydrostatic equilibrium structure of compact stars in the Eddington-inspired BornInfeld gravity recently proposed by Bañados and Ferreira [Phys. Rev. Lett. 105, 011101 (2010)]. We also develop a framework to study the radial perturbations and stability of compact stars in this theory. We find that the standard results of stellar stability still hold in this theory. The frequency square of the fundamental oscillation mode vanishes for the maximum-mass stellar configuration. The dependence of the oscillation mode frequencies on the coupling parameter κ of the theory is also investigated. We find that the fundamental mode is insensitive to the value of κ, while higher-order modes depend more strongly on κ.