Quark stars with isotropic matter in Hořava gravity and Einstein–æther theory

Eur. Phys. J. C (2020) 80:537 https://doi.org/10.1140/epjc/s10052-020-8105-5 Regular Article - Theoretical Physics Quark stars with isotropic matter in Hořava gravity and Einstein–æther theory Grigorios Panotopoulos1,a , Daniele Vernieri2 , Ilidio Lopes1 1 Centro de Astrofísica e Gravitação-CENTRA, Departamento de Física, Instituto Superior Técnico-IST, Universidade de Lisboa-UL, Avenida Rovisco Pais 1, 1049-001 Lisbon, Portugal 2 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa,, Campo Grande 1749-016 Lisbon, Portugal Received: 2 March 2020 / Accepted: 1 June 2020 / Published online: 15 June 2020 © The Author(s) 2020 Abstract We study non-rotating and isotropic strange quark stars in Lorentz-violating theories of gravity, and in particular in Hořava gravity and Einstein-æther theory. For quark matter we adopt both linear and non-linear equations of state, corresponding to the MIT bag model and color flavor locked state, respectively. The new structure equations describing hydrostatic equilibrium generalize the usual Tolman–Oppenheimer–Volkoff (TOV) equations of Einstein’s general relativity. A dimensionless parameter ν measures the deviation from the standard TOV equations, which are recovered in the limit ν → 0. We compute the mass, the radius as well as the compactness of the stars, and we show graphically the impact of the parameter ν on the mass-to-radius profiles for different equations of state describing quark matter. The energy conditions and stability criteria are also considered, and they are all found to be fulfilled. 1 Introduction In 2009 Hořava gravity [1,2] was proposed as a new candidate theory for quantum gravity which explicitly breaks Lorentz invariance at any energy scale by introducing a preferred foliation of spacetime. Since then a lot of work has been done to prove, very successfully, its renormalizability by means of both power-counting arguments [3–8] and quantum field theory approaches [9–12]. Moreover, a big effort has also been made in order to unveil its phenomenological implications, e.g. concerning late-time cosmology [13,14], black holes [15–21], binary systems [22–24], and anisotropic interior solutions [25–27]. After the multiple detections, after the multiple detections of gravitational waves by the LIGOa e-mail: (corresponding author) VIRGO Collaboration, and in particular the first merger observed from a binary of neutron stars [28], a new era for gravitational-wave astronomy just got started. Very interestingly, Hořava gravity passes with flying colors all the theoretical and observational constraints which are available to date [29]. Notice that if one takes the low-energy limit of Hořava gravity and writes its action in a covariant form, the latter becomes equivalent to Einstein-æther theory [30] once the æther vector is restricted to be hypersurface-orthogonal at the level of the action [31]. In spherical symmetry it can be shown that the two theories share the same solutions [32]. Therefore, it has become even more urgent to study the implications and predictions of viable alternative gravity theories at astrophysical scales, in order to explore non-standard scenarios and the possible signatures of deviations from general relativity (GR) to be observed in the forthcoming detections. For all of these reasons in this work we will investigate some astrophysical implications of the theory. In this respect, compact objects [33–35] such as neutron stars and white dwarfs, are relativistic stars of astrophysical and astronomical interest, which are characterized by ultra dense matter densities and strong gravitational fields, and thus they serve as ideal cosmic laboratories to study and test non-standard physics as well as non-conventional theories of gravity. A new class of compact objects, that may be an alternative to neutron stars, are some as of today hypothetical objects which are supposed to be made of quark matter, and for that reason they are called strange quark stars [36–41]. Quark matter is by assumption absolutely stable [42,43], and so it could be the true ground state of hadrons. That property makes them a plausible explanation of some puzzling superluminous supernovae [44,45], which occur in about one out of every 1000 supernovae explosions, and which are more than 100 times more luminous than regular supernovae. The plan of our work is the following. In Sect. 2 we briefly review the basic ingredients of Hořava gravity and its connec- 123 537 Page 2 of 9 Eur. Phys. J. C (2020) 80:537 tion to Einstein-æther theory, while in Sect. 3 we present the field equations as well as the structure equations describing the hydrostatic equilibrium of spherically symmetric relativistic stars with isotropic matter. In Sect. 4 we obtain and discuss our numerical results for quark stars. Finally we finish our work with some conclusions in Sect. 5. We adopt the mostly negative metric signature +, −, −, −, and we work in units in which the speed of light in vacuum c as well as the reduced Planck constant h̄ are set equal to unity, h̄ = 1 = c. In those units all dimensionful quantities are measured in GeV = 1000 MeV, and we make use of the conversion rules 1 m = 5.068 × 1015 GeV−1 and 1 kg = 5.610 × 1026 GeV [46]. 2 Hořava gravity and Einstein-æther theory The action of Hořava gravity [1,2] can be written in the preferred foliation as   √ 1 dT d 3 x −g K i j K i j − λK 2 + ξ R SH = 16π G H  L4 L6 +ηai a i + 2 + 4 + Sm [gμν , ψ] , (1) M∗ M∗ where G H is the effective gravitational constant; g is the determinant of the metric gμν ; R is the Ricci scalar of the three-dimensional constant-T hypersurfaces; K i j is the extrinsic curvature and K is its trace; and ai = ∂i lnN , where N is the lapse function and Sm is the matter action where ψ collectively denotes the matter fields. The constant couplings {λ, ξ, η} are dimensionless, and GR is identically recovered when they take the values {1, 1, 0}, respectively. Moreover, L 4 and L 6 denote the fourth-order and sixth-order operators respectively, while M∗ is the characteristic mass scale which suppresses them at low-energy. In the following, we consider the covariantized version of the low-energy limit of Hořava gravity, named the khronometric model, that is obtained by keeping only the operators up to second-order derivatives, which amounts to discarding L 4 and L 6 which instead contain the higher-order operators. In order to write the action covariantly, let us first take the action of Einstein-æther theory [30]:  √ 1 d 4 x −g (−R + L æ ) + Sm [gμν , ψ], (2) Sæ = 16π G æ where ci ’s are dimensionless coupling constants. Once the æther vector is taken to be hypersurfaceorthogonal at the level of the action, that is ∂α T u α =  μν , g ∂μ T ∂ν T (5) where the preferred time T is a scalar field (the khronon) which defines the preferred foliation, then the two actions in Eqs. (1) and (2) become equivalent if the parame (...truncated)