In this paper we investigate the properties of relativistic stars made of isotropic matter within the framework of the minimal Standard Model Extension where a bumblebee field coupled to spacetime induces spontaneous Lorentz symmetry breaking. We adopt analytic equationsofstate describing either condensate dark stars or strange quark stars. We solve the structure equations numerically and we compute the masstoradius relationships. The influence of the bumblebee parameter l is examined in detail and an upper bound is obtained using the massive pulsar PSR J07406620 and the strangely light HESS J1731347 compact object.
['General Relativity and Quantum Cosmology', 'Astrophysics - Solar and Stellar Astrophysics']
Strange Quark Stars and Condensate Dark Stars in Bumblebee Gravity
{'Strange Quark Stars and Condensate Dark Stars in Bumblebee Gravity': "Grigoris Panotopoulos 1, ∗ and Ali Ovgun 2, †  \n1  \nDepartamento de Ciencias F'ısicas, Universidad de la Frontera, Casilla 54-D, 4811186 Temuco, Chile.  \n2 Physics Department, Eastern Mediterranean University, Famagusta, Cyprus.", 'Abstract': 'In this paper, we investigate the properties of relativistic stars made of isotropic matter within the framework of the minimal Standard Model Extension, where a bumblebee field coupled to spacetime induces spontaneous Lorentz symmetry breaking. We adopt analytic equations-of-state describing either condensate dark stars or strange quark stars. We solve the structure equations numerically, and we compute the mass-to-radius relationships. The influence of the bumblebee parameter l is examined in detail, and an upper bound is obtained using the massive pulsar PSR J0740+6620 and the strangely light HESS J1731-347 compact object.  \nKeywords: Modified gravity; Relativistic stars; Stellar composition; Lorentz symmetry breaking; Bumblebee gravity.', 'I. INTRODUCTION': "General Relativity and the Standard Model respectively describe gravity classically and particles/interactions quantum mechanically. Unifying these theories remains a fundamental challenge. While proposed quantum gravity theories offer potential solutions, direct experimental verification is currently unattainable due to the requisite Planck scale ( 10 19 GeV) energies. Nonetheless, subtle quantum gravity effects might manifest at lower energy scales, opening avenues for experimental investigation. One such potential signal is the breaking of Lorentz symmetry, rooted in special relativity, ensures that physical laws are consistent for all observers in inertial frames. This symmetry, encompassing rotational and boost transformations, is fundamental to general relativity and the standard model of particle physics . In curved spacetimes, local Lorentz symmetry is maintained due to the Lorentzian nature of the background. However, violating Lorentz invariance introduces directional or velocity dependencies in physical variables, altering the dynamics of particles and waves. In 1989, Kosteleck'y and Samuel introduced bumblebee gravity [1], a model for spontaneous Lorentz violation. In this model, a bumblebee field with a vacuum expectation value (VEV) breaks Lorentz symmetry through the dynamics of a single vector field, B µ . This approach suggests that investigating Lorentz symmetry violation could provide insights into Planck-scale physics, combining aspects of both General Relativity and the Standard Model within the framework of effective field theories known as the Standard Model Extension (SME), which offers a framework that includes all possible coefficients for Lorentz/CPT violation, with its gravitational sector defined on a Riemann-Cartan manifold, treating torsion and the metric as dynamic geometric quantities. While the SME allows for non-Riemannian terms, research has primarily focused on the metric approach, using the metric as the main dynamic field. There are certain effects of Lorentz violation within the gravitational sector have been explored in [2-18].  \nNew discoveries of super-dense objects are pushing the boundaries of our understanding in astrophysics. While countless compact objects have been found, the exact laws governing their behavior remain a mystery. Recent observations have challenged the traditional theories, particularly general relativity and the standard model for neutron stars. One key example is the secondary object in the GW190814 merger [20-22]. Some scientists believe it might be a quark star, with its properties described by a single parameter ( λ ) within the framework of general relativity. This theory aligns with recent research suggesting non-strange quark matter as the most stable form of baryonic matter under extreme conditions [23]. Promising findings have also been obtained  \nby similar investigations using the quark matter phase or using other models for matter behaviour (equation of state) [15-19].  \nStrange stars: Strange quark stars are hypothetical compact objects proposed as an alternative to neutron stars [24-30]. Composed of quark matter, which is theorized to be the absolutely stable ground state of hadrons [31, 32], these stars could explain the puzzling nature of super-luminous supernovae [33, 34]. Unlike ordinary supernovae, these exceptionally bright explosions are 100 times more luminous and occur in approximately one out of every thousand cases.  \nWhile strange quark stars remain theoretical at present, their existence cannot be definitively excluded. Some observed compact objects exhibit anomalous properties, such as unusually small radii, that standard neutron star models cannot fully explain [35-37]. These discrepancies have fueled speculation about the potential presence of strange matter in these objects. While some researchers propose that strange matter could constitute the core of hybrid neutron stars [38-40], others argue that such stars would be virtually indistinguishable from ordinary neutron stars [41].  \nBoson stars: The standard cosmological model posits that dark matter consists of weakly interacting massive particles (WIMPs). While this assumption aligns well with large-scale cosmic observations (scales of megaparsecs and larger) ( ≥ Mpc ), it faces challenges at smaller galactic scales. Discrepancies such as the core-cusp problem, the diversity problem, the missing satellites problem, and the too-big-to-fail problem arise [42]. Self-interacting dark matter has been proposed as a potential solution [43, 44], as particle collisions within this model can mitigate the formation of sharp density cusps. Furthermore, if dark matter is composed of ultralight scalar particles with a mass below one electronvolt m ≤ eV and a weak repulsive self-interaction, it could form a BoseEinstein condensate (BEC) with long-range correlations. This BEC scenario has been proposed as a potential explanation for the aforementioned galactic scale discrepancies [45-47].  \nRegarding the composition and inner structure of compact objects, the most massive pulsars [48-50] observed over the last 15 years or so are putting constraints on different equations-ofstate, since any mass-to-radius relationship that predicts a highest mass lower than the observed ones must be ruled out. In order to model a certain star, it would be advantageous to know both its mass and its radius, which is not always the case as measuring the radius is way more difficult. There are some good strange quark star candidates, see e.g. Table 5 of [51] or Table 1 of [37], and also the recently discovered massive pulsar PSR J0740+6620 [52-54] and the strangely light object HESS [55], where both the stellar mass and radius are known observationally.  \nOur current understanding of ultra-dense stars, particularly those exceeding the limitations of traditional neutron stars, remains a fascinating mystery in astrophysics. This work explores this field by examining the characteristics of isotropically-composed stars.  \nOur key objective is to explore the impact of a phenomenon known as spontaneous Lorentz symmetry breaking within these exotic stars. This breaking is induced by a hypothetical field, the bumblebee field, coupled to spacetime in the framework of the minimal Standard Model Extension.  \nIn the present article we propose to study non-rotating relativistic stars made of isotropic matter within bumblebee gravity adopting for the matter content two analytic equations-of-state. Our work is organized as follows: After this introductory section, we briefly review how to obtain interior solutions describing hydrostatic equilibrium within GR in the next section, while in section 3 we present the Bumblebee model as well as the corresponding modified structure equations. In the fourth section we comment on the equations-of-state assumed here, and we discuss our numerical results. Finally, we summarize and conclude our work in section 5. Throughout the manuscript we work in geometrical units setting the universal constants (Newton's constant and speed of light in vacuum) to unity, G = 1 = c , and we adopt the mostly positive metric signature {-, + , + , + } .", 'II. RELATIVISTIC STARS IN GENERAL RELATIVITY: HYDROSTATIC EQUILIBRIUM AND STRUCTURE EQUATIONS': "Due to their intense gravitational fields, the structure and dynamics of neutron stars (NSs) are governed by Einstein's equations of General Relativity (GR) [56], given by:  \nG µν = R µν -1 2 Rg µν = 8 πT µν , (1)  \nwhere R µν and R denote the Ricci tensor and Ricci scalar, respectively. The energy-momentum tensor for isotropic matter, T µν , is expressed as:  \nT µν = Pg µν +( P + ρ ) u µ u ν , (2)  \nwhere g µν is the metric tensor, P is the pressure, E is the energy density, and u µ is the fourvelocity. For static, spherically symmetric stars, the line element in Schwarzschild-like coordinates { t, r, θ, ϕ } is described as:  \nds 2 = -e ν ( r ) dt 2 + e λ ( r ) dr 2 + r 2 ( dθ 2 + sin 2 θdϕ 2 ) , (3)  \nwhere e ν ( r ) and e λ ( r ) are the metric functions. One obtains the Tolman-Oppenheimer-Volkoff (TOV) equations [57, 58] for the equilibrium structure of NSs by solving the Einstein field equation with the above-defined metric,  \ndP ( r ) dr = -[ ρ ( r ) + P ( r )][ m ( r ) + 4 πr 3 P ( r )] r 2 (1 -2 m ( r ) /r ) , (4)  \ndm ( r ) dr = 4 πr 2 ρ ( r ) . (5)  \nThe metric functions become  \ne λ ( r ) = (1 -2 m/r ) -1 , (6)  \nν ( r ) = log (1 -2 M/R ) + 2 ∫ r R dr ' e λ ( r ' ) r ' 2 [ m ( r ' ) + 4 πr ' 3 P ( r ' )] . (7)  \nCombined with the given EoS-i.e., P ( ρ ) of the matter-, TOV equations can be solved with the initial conditions at the center of the star, m ( r = 0) = 0 and ρ ( r = 0) = ρ c , where ρ c is the central energy density. The stellar mass and radius are determined using the matching conditions at the surface of the star upon comparison to the Schwarzschild exterior vacuum solution [59]  \nds 2 = -(1 -2 M/r ) dt 2 + 1 1 -2 M/r dr 2 + r 2 d Ω 2 . (8)  \nThus, the radius of the star is determined by requiring that the energy density vanishes at the surface, P ( R ) = 0, and the stellar mass is then given by M = m ( R ) .", 'III. BUMBLEBEE GRAVITY: MODIFIED TOV EQUATIONS': "In this section, we briefly review the bumblebee gravity, a theory that expands on General Relativity (GR). Inspired by the works of Kostelecky and collaborators [1], bumblebee gravity introduces a twist: it breaks a fundamental symmetry (Lorentz symmetry) within the realm of gravity. This twist manifests as a special value (nontrivial vacuum expectation value) that influences how other fields behave around a mysterious 'bumblebee field'. Interestingly, even with these extra interactions, bumblebee gravity preserves the geometrical framework and basic laws established by GR in curved spaces.  \nAmong various models capable of breaking Lorentz symmetry, one of the simplest approaches involves a vector field B µ , known as the bumblebee field, within a torsion-free spacetime. This can be expressed as follows [15]:  \nS = ∫ d 4 x √ | g | ( 1 16 π ( R + ξB µ B ν R µν ) -1 4 B µν B µν -V ( B µ B µ ± b 2 ) + L M ) , (9)  \nwhich encapsulates the interaction between the bumblebee vector field ( B µ ) and gravity, represented by the Ricci tensor ( R µν ). The strength of this interaction is modulated by a coupling constant ( ξ ).  \nThe bumblebee field B µν ≡ ∂ µ B ν -∂ ν B µ also interacts with matter, as outlined by the matter Lagrangian density ( L m ). The potential function V ( B µ B ν ± b 2 ) , where b 2 is a positive constant, dictates the behavior of the bumblebee field. Notably, this potential enables the field to achieve a non-zero vacuum expectation value, specifically B µ = b µ = (0 , b r ( r ) , 0 , 0) , where b r ( r ) is a function of the radial coordinate.  \nTo break a specific symmetry, such as U (1) , the potential must reach a minimum at V = 0 , with its derivative V ' = 0 also being zero at that point. This condition allows the bumblebee field to acquire a constant, non-zero vacuum expectation value, denoted by b µ , which satisfies B µ ≡ b µ with b µ b µ = ∓ b 2 ≡ constant (note that b 2 is a real positive constant).  \nBy varying the action, as given in Eq. (9), we derive the equations governing both the gravitational field and the dynamics of the bumblebee field, including the corresponding equations of motion.  \nR µν -1 2 g µν R = 8 π ( T B µν + T M µν ) , (10)  \n∇ µ B µν = J B ν + J M ν . (11)  \nWithin the bumblebee gravity framework, two key terms come into play: the source term for the BF (represented by J M ν ) and the current due to the BF's self-interaction (denoted by J B ν = 2 V ' B ν -ϱB µ R µν / 8 π ). Note that the prime symbol ( ' ) denotes differentiation with respect to the potential argument, specifically V ' ≡ ∂V ( y ) /∂y , where y = B µ B ν ± b 2 .  \nAdditionally, T B µν represents the energy-momentum tensor associated with the bumblebee field, which can be expressed as  \nT B µν = B µα B α ν -1 4 g µν B αβ B αβ -g µν V +2 B µ B ν V ' + ϱ 8 π [ 1 2 g µν B α B β R αβ -B µ B α R αν -B ν B α R αµ + 1 2 ∇ α ∇ µ ( B α B ν ) + 1 2 ∇ α ∇ ν ( B α B µ ) -1 2 ∇ 2 ( B µ B ν ) -1 2 g µν ∇ α ∇ β ( B α B β ) ] . (12)  \nRemember that T M µν represents the energy and momentum distribution of matter, and we have a constant value denoted by b µ b µ = ∓ b 2 ≡ constant related to the bumblebee field (where b µ is the VEVand b is its magnitude). These factors influence the radial component b r ( r ) of the BF when it reaches its minimum energy state (VEV). Interestingly, research by Casana et al. [15] shows that this radial component can be expressed as b r ( r ) = | b | e ζ ( r ) .  \nThe next step is to understand how this VEV of the BF affects the geometry of spacetime itself. This is described by the metric, which will be expressed in the following spacetime  \nds 2 = -e 2 χ ( r ) dt 2 + e 2 ζ ( r ) dr 2 + q 2 r 2 d Ω 2 , (13)  \nwhere χ ( r ) and ζ ( r ) , valid in the region 0 ≤ r ≤ R , represent the sought metric functions. It is noteworthy that choosing e -2 ζ ( r ) = g ( r ) is always possible, resulting in:  \ng ( r ) = 1 -2 M( r ) r , (14)  \nin which the collective mass function denoted by M( r ) can be viewed as the combination of the BH mass.  \nThus, for an interacting system of this nature, the energy-momentum tensor associated with the spacetime metric can be represented as T µ ν = diag [ -ρ ( r ) , P r ( r ) , P θ ( r ) , P ϕ ( r )] . Consequently, these considerations facilitate the derivation of the Einstein field equations G µν = 8 πT µν as  \n8 πρ ( r ) = 1 -q 2 +2 q 2 M ' ( r ) 2 2 ,  \n8 πP θ ( r ) = 8 πP φ ( r ) = ( 1 r + χ ' ( r ) )( M ( r ) r 2 -M ( r ) r + ( 1 -2 M ( r ) r ) χ ' ( r ) ) + ( 1 -2 M ( r ) r ) χ '' ( r ) .  \nq r (15a) 8 πP r ( r ) = 1 r 2 ( 1 -1 q 2 ) -2 M ( r ) r 3 + 2 r ( 1 -2 M ( r ) r ) χ ' ( r ) , (15b) ' (15c)  \nWithin the bumblebee gravity framework, the parameter called q 2 = 1 / ( l +1) plays a crucial role. It depends on a constant known as parameter of Lorentz symmetry breaking l , which itself is linked to the coupling strength ξ of the BF and a constant related to the BF b as with l = ξb 2 . The prime symbol ( ' ) denotes differentiation with respect to the radial position (r). Interestingly, when we take the limit as l approaches zero, q approaches 1 , has a significant implication. In this specific limit ( l → 0 , q 2 → 1 ), the Einstein field equations (refer to Eq. (15) for reference) simplify back to their standard form.  \nNow, let us bring in another piece of the puzzle: the equation of state P r ( r ) = ωρ ( r ) , which relates pseudo-pressure ( P r ( r ) ) and energy density ( ρ ( r ) ) with a constant factor ( ω ). By combining this equation of state with the modified Einstein field equations we obtained earlier (Eq. (15a)), and considering a well-known law in physics (conservation of energy-momentum) T ν µ ; ν = 0 , we can arrive at a modified version of the TOV equations. These modified TOV equations are crucial for understanding how bumblebee gravity affects the structure and stability of stars and other selfgravitating objects.  \ndχ ( r ) dr = M ( r ) -( 1 2 -1 2 q 2 ) r +4 πr 3 P r ( r ) r ( r -2 M ( r )) , (16a)  \ndP r ( r ) dr = -( ρ ( r ) + P r ( r )) dχ ( r ) dr , (16b)  \nHence, in the exterior (vacuum) region where T µν = 0 , the solution simplifies to the bumblebee black hole (BH) solution [15]  \nds 2 = -(1 -2 M/r ) dt 2 + l +1 1 -2 M/r dr 2 + r 2 d Ω 2 , (17)  \nor, after a coordinate transformation and a mass redefinition as follows  \nt → t, r → √ 1 + l r, M → √ 1 + l M, (18)  \nit may be recast in the following form [17]  \nds 2 = -(1 -2 M/r ) dt 2 + 1 1 -2 M/r dr 2 + q 2 r 2 d Ω 2 . (19)  \n̸  \nWhen studying a star in bumblebee gravity, we need to ensure our solutions for the different regions (interior and exterior) seamlessly match at the star's surface, located at a specific radius ( R ). To achieve this perfect match, we impose three matching conditions when q = 1 :", '· Matching Mass:': 'M = M ( R ) (20)  \nthe first condition states that the value of the mass function ( M ( r ) ) at the surface ( r = R ) must be equal to the total mass ( M ) of the star.', '· Zero Pressure at the Surface:': 'P r ( R ) = 0 , (21)  \nthe second condition requires the pressure at the surface to be zero. This condition allows us to compute the radius of the star.', '· Setting the Spacetime Geometry:': "e 2 χ ( R ) = 1 -2 M/R, (22)  \nthe third condition involves the exponential term in a metric function ( e 2 χ ( R ) ). Here, we stipulate that its value at the surface ( r = R ) is related to the star's mass ( M ) and radius ( R ) through a specific relationship ( 1 -2 M/R ). This essentially sets the initial condition for this metric function, which helps describe the geometry of spacetime around the star.  \nIn summary, the aforementioned matching conditions act as bridge equations, ensuring that our solutions for the star's internal structure and the surrounding vacuum smoothly connect at the boundary, providing a physically consistent description of a bumblebee gravity star.", 'IV. PROPERTIES OF STARS: NUMERICAL ANALYSIS AND DISCUSSION OF MAIN RESULTS': 'In this section, we explore the properties of relativistic stars made of isotropic matter, P r = P θ = P ϕ = P , within the framework of Lorentz-violating gravity theories. Through numerical integration of the relevant structure equations, we subsequently present and analyze our key findings', 'A. Equation-of-state': 'In the discussion to follow we shall consider i) a polytropic EoS and ii) MIT bag model [60, 61] for quark matter. Polytropes of the form  \nP = kρ γ , γ = 1 + 1 n (23)  \nwith n being the polytropic index, are used to describe interior solutions of white dwarfs (index n = 3 or n = 3 / 2 ) as well as condensate dark stars (index n = 1 ). Here we shall consider the case  \n2.5  \nFIG. 1: Mass-to-radius relationships for the two EoSs considered here. Upper panel corresponds to polytrope ( l = 0 , 0 . 15 , 0 . 30 from top to bottom), lower panel to the extreme MIT bag model ( l = 0 , 0 . 030 , 0 . 055 from left to right). The contours in magenta color indicate the allowed mass and radius range of the massive pulsar PSR J0740+6620 and the light HESS J1731-347 compact object.  \n<!-- image -->  \nn = 1 assuming k = 4 . 012 × 10 -4 fm 3 /MeV . On the other hand, the extreme SQSB40 MIT bag  \nFIG. 2: Stellar mass versus normalized central energy density for the two EoSs considered here. Upper panel corresponds to polytrope, lower panel to the extreme MIT bag model.  \n<!-- image -->  \nmodel is given by a linear analytic function [62]  \nP = a ( ρ -ρ 0 ) (24)  \nwith a = 0 . 324 and ρ 0 = 3 . 0563 × 10 14 g/cm 3 [62]. The numerical values of the three parameters of the MIT bag model are as follows: The bag constant B = 40 MeV/fm 3 , while the mass of the  \nFIG. 3: Factor of compactness versus stellar mass for the two EoSs considered here. Upper panel corresponds to polytrope, lower panel to the extreme MIT bag model.  \n<!-- image -->', 'B. Numerical results': 'In Fig. 1, we show the mass-to-radius relationships for the two EoSs considered here, i.e. a polytrope with n = 1 (upper panel) and the extreme MIT bag model (lower panel). The dashed curves represent the standard mass-to-radius relationship within GR. Those curves serve as a reference for how quark stars behave under GR without modifications from the bumblebee gravity theory. The curves in blue color correspond to the bumblebee parameter l = 0 . 15 for the polytrope and l = 0 . 03 for the MIT bag model. In this case, the mass-to-radius relationship for quark stars shows slight deviations from the GR case. The introduction of a small bumblebee parameter indicates minor changes in the structure and stability of the quark star, reflecting how quark stars might behave under weak modifications to gravity. The curves in red color represent the bumblebee parameter l = 0 . 30 for the polytrope and l = 0 . 055 for the MIT bag model. With a larger bumblebee parameter, the mass-to-radius relationship exhibits more significant deviations from the GR case. This indicates that stronger modifications to gravity have a notable impact on the structure, stability, and maximum mass of quark stars.  \nIn Fig. 2 the stellar mass versus central (normalized and dimensionless) energy density is shown. As in previous figure, the upper panel corresponds to the polytrope, while the lower panel to the extreme MIT bag model. The dashed curves correspond to GR, the blue curves to the lowest value of l , and the red curves to the highest value of l . The curves increase, reach a maximum value and then they decrease. The maximum stellar mass is the one that is shown in Fig. 1. The Harrison-Zeldovich-Novikov criterion [63, 64] for stability  \ndM dρ c > 0 (25)  \nstates that a stellar model is a stable configuration only if the mass of the star grows with the central energy density. Therefore according to the criterion, only the first part of the function M ( ρ c ) is physical, namely up to the maximum stellar mass.  \nNext, in Fig. 3 we display the factor of compactness, M/R , as a function of the stellar mass for the two EoS assumed in this work. Panels and colors are the same as before. The factor of compactness increases with the stellar mass, which acquires a maximum value, and which is precisely the same that is shown in Figures 1 and 2. Notice that the factor of compactness satisfies the Buchdahl limit of GR, ( M/R ) ≤ 4 / 9 ≈ 0 . 444 [65].', 'V. CONCLUSION': 'To summarize our work, in the present article we investigated the properties of non-rotating relativistic stars made of isotropic matter within the bumblebee gravity in four-dimensional spacetime. For the matter content of the stars we considered analytic equations-of-state corresponding to quark matter (extreme MIT bag model) and condensate dark stars (polytropic EoS with index n = 1 ). In the first part of the article we presented the exterior vacuum solution, the structure equations describing hydrostatic equilibrium of interior solutions as well as the appropriate conditions both at the center and at the surface (matching conditions) of the stars.  \nNext, we integrated numerically the generalized TOV equations imposing the initial conditions at the center of the star, and then making use of the matching conditions we computed the stellar mass and radius as well as the factor of compactness. The impact of the bumblebee parameter l on the properties of the stars was shown in three figures, where we displayed i) the stellar mass versus stellar radius, ii) the factor of compactness as a function of the stellar mass, and iii) stellar mass versus central energy density.  \nOur results show that a) as we increase l the deviation from GR becomes more significant, as expected, b) the factor of compactness satisfies the Buchdahl limit of GR, ( M/R ) ≤ 4 / 9 ≈ 0 . 444 [65], c) regarding the MIT bag mode, increasing l shifts the M-R relationships to the left, which implies that at a certain point the EoS cannot model the HESS compact object, d) regarding the polytrope, increasing l shifts the M-R profiles downwards, which implies that at a certain point the EoS cannot support stars at two solar masses. Therefore, an upper limit on l was obtained, although the bound coming from the MIT bag model EoS is the strongest one.', 'VI. ACKNOWLEDGMENTS': "A. 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