Gravitational decoupling and superfluid stars

The gravitational decoupling is applied to studying minimal geometric deformed (MGD) compact superfluid stars, in covariant logarithmic scalar gravity on fluid branes. The brane finite tension is shown to provide more realistic values for the asymptotic value of the mass function of MGD superfluid stars, besides constraining the range of the self-interacting scalar field, minimally coupled to gravity. Several other physical features of MGD superfluid stars, regulated by the finite brane tension and a decoupling parameter, are derived and discussed, with important corrections to the general-relativistic limit that corroborate to current observational data.


I. INTRODUCTION
The investigation of stellar distributions, and black holes of stellar origin, currently comprises one of the most relevant topics in gravitational physics. At extremely strong gravitational fields, gravity can be probed by the framework of Einstein's theory of general relativity (GR) and beyond, also encompassing other models that generalize it. Current observations in the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the Evolved Laser Interferometer Space Antenna However (eLISA) can precisely approach any existing theory or model of gravity, by the observation of ring-down phases of binary coalescing black hole mergers [1]. The minimal geometric deformation (MGD) procedure and other MGD-decoupling methods have been playing an important role in describing a plethora of compact stellar distributions in a wide range of regimes. The MGD-decoupling also includes models with anisotropy and engender analytical solutions of Einstein's effective field equations [2][3][4][5]. In the MGD-decoupling setup, the universe can be modeled by a finite tension fluid brane of codimension-1 embedded in a bulk, with GR occupying the single limit when the fluid brane is rigid, or equivalently, when the brane tension attains an infinite value [6,7]. A running MGD parameter, which is inversely proportional to the fluid brane tension, tunes the modifications of MGDdecoupled models to GR. Refs. [8][9][10][11][12][13] posed the most precise bounds on parameters of several MGD-decoupled solutions, also using LIGO and eLISA data.
In this work, gravitational decoupled superfluid stars are studied in the covariant logarithmic scalar gravity setup, taking into account strongly-interacting and extremely dense superfluids. The Klein-Gordon equation with logarithmic potential was proposed in the 1960s, in the context of QFT [75]. Thereafter, several contexts have used it to study baryogenesis, inflationary cosmology, and phase transitions in this context, with density inhomogeneities [75,76]. Since inflationary universe models driven by scalar fields with logarithmic potential shed new light on cosmology [75][76][77], it is now natural to argue how the setup of the Klein-Gordon equation with logarithmic potential, coupled to GR, can be used to study MGD-decoupled superfluid stars.
The recently observed gravitational wave emitted from the binary neutron star merger GW170817 was a landmark in the analysis of the extremely dense matter at strong gravity. As relic stellar distributions, their core has a lower temperature than the transition temperature for the emergence of Bardeen-Cooper-Schrieffer-like pairs. It becomes an appropriate condition to superfluid neutrons states in the elastic inner crust [78,79]. Pulsar glitches, the thermal X-ray radiation, and the observation that the Cassiopeia A neutron star has been fastly cooling down, are unequivocal and prominent observational signatures of superfluid stars in nature [80][81][82][83]. We will analyze the finite brane tension influence on the superfluid compact stars, in the gravitational MGD-decoupling setup. It is indeed natural to ascertain, in a realistic model that complies with recent observational data, how the finite tension of the brane can drive new gravitational MGD-decoupled solutions of the Einstein-Klein-Gordon coupled system with logarithmic scalar potential, that encodes compact strongly-interacting and extremely dense superfluids.
This work is structured in the following way: Sec. II is dedicated to appraise the MGD as a particular case of the gravitational decoupling algorithm and to describe stars on fluid branes with finite tension. Sec. III is devoted to scrutinizing the MGD-decoupling in covariant logarithmic scalar gravity, describing superfluid stars. The resulting Einstein-Klein-Gordon system with self-interacting logarithmic potential is then numerically solved, discussed, and analyzed in detail. The finite brane tension setup, complying with current observational data, is shown to improve the Schwarzschild GR analysis, paving new ways to describe realistic models for superfluid stars. Sect. IV is reserved for drawing the concluding remarks.

II. THE MGD AND MGD-DECOUPLING PROTOCOLS
The MGD algorithm has been comprehensively used to construct new solutions of Einstein's effective field equations, also encoding improvements to classical generalrelativistic solutions due to a fluid brane scenario [3,5,17,20]. In the MGD-decoupling setup, the most accurate brane tension bound γ 2.8131×10 −6 GeV [11] was predicted. Since the 4-dimensional fluid brane that describes our universe is immersed into a bulk, the Gauss-Codazzi-Mainardi equations compels the bulk Riemann tensor to split into two geometrically distinct terms. The first one is the Riemann tensor on the brane and the second term consists of a sum of quadratic expressions involving the extrinsic curvature, that encrypts the part of the curvature due to the embedding of the brane in the bulk. The effective Einstein's equations on the brane read where R is the Ricci scalar, 8πG 4 = 1 is adopted, being G 4 the Newton's constant on the brane, R µν represents the Ricci curvature tensor, and Λ 4 is the Einstein's cosmological constant on the brane. One can split the stress-energy-momentum tensor in (1) into [84] T υτ = T υτ + E υτ + γ −1 S υτ + L υτ + P υτ .
The first term, T υτ , is the stress-energy-momentum tensor that represents ordinary and dark matter, also encoding dark energy; the purely electric component of the Weyl tensor, E υτ , is non-local and depends on the inverse of the brane tension; the S υτ tensor consists of quadratic combinations of the stress-energy-momentum tensor 1 , where T = g υτ T υτ ; the tensor L υτ dictates the way that the fluid brane is immersed and bent into the codimension-1 bulk; and P υτ describes eventual Kaluza-Klein bosons, fermions, and other stringy fields living in the bulk [84][85][86]. The purely electric component of the Weyl tensor, describes a Weyl fluid, where h υτ emulates an induced metric which projects quantities in the orthogonal direction to the Weyl fluid flow velocity field, u υ . In addition, dark radiation, the non-local anisotropic stressmomentum-tensor, and energy flux, have the following respective expressions [24] U = −γE υτ u υ u τ , Denote hereon by p = G 4 /c 3 the Planck length and by G 5 the 5-dimensional Newton's gravitational constant, that can be related to G 4 by G 5 = G 4 p . A fine tuning involves the 4-dimensional gravitational constant, the bulk cosmological constant, Λ 5 , and the brane tension, by [87] Adopting Λ 4 = 0 yields no exchange of energy between the bulk and the brane, complying to the conservation law ∇ ν T µν = 0 [14]. Therefore, Eq. (8), together with the fact that the 4-dimensional coupling constant κ 4 = 8πG 4 /c 2 1.866 × 10 −26 m · kg −1 , and κ 5 = 8πG 5 /c 2 , are related through the brane tension by κ 2 4 = 1 6 γκ 4 5 [85], imply that what complies to an AdS bulk. Besides, the brane tension is related to the 5-dimensional Planck mass by γ ≈ πr c −Λ 5 /24M 3 5 , where r c ≈ 1 µm as stated by the original Randall-Sundrum model [87].
Compact stars are solutions of the Einstein's effective field equations (1), with static and spherically symmetric metric where dΩ 2 stands for the metric solid angle. It implies that the tensor fields in Eqs. (6, 7) take a simplified expression, respectively given by P υτ = P(u υ u τ + h υτ /3) and Q υ = 0, where P = g υτ P υτ . Correspondingly, the brane stress-energy-momentum tensor can be emulated by a perfect fluid, with u υ = b(r)δ υ 0 . The rest of this section is dedicated to show that the MGD and the MGD-decoupling method produce analytical solutions of the Einstein's effective field equations (1, 2), modeling realistic compact stars [2][3][4][5].

A. MGD algorithm
The Einstein's effective field equations on the brane, (1), denoting by G µν = R µν − 1 2 Rg µν the Einstein's tensor, are equally expressed as where dr .
Eqs. (12a -12c) induce anisotropic effects. The effective density (ˇ ), the radial pressure (p r ), and also the tangential pressure (p ), are respectively expressed as [5] = + 1 γ Eq. (13) illustrates how the MGD stellar density can be split into a sum that involves the GR-like central stellar density and another term regulated by the inverse of the brane tension. This additional term, beyond GR, is a sum of a numerical factor of the dark radiation and a term involving the quadratic GR-like star density. It is worth mentioning that such last terms vanish in the GR case, where the brane tension takes the limit γ → ∞.
The coefficient of anisotropy on MGD stars is defined by Gravity in the bulk produces the minimal geometric deformation, ξ(r), in the radial component of the metric, where for R denoting the stellar radius The effective mass function can be alternatively written down as [4,24], where M 0 is the GR mass function, owing to the brane tension range bound γ 2.8131 × 10 −6 GeV [11]. Therefore Eq. (18) can be condensed into the Misner-Sharp-Hernandez function, The solutions of the coupled system (12a -12d) can be derived, when one substitutes Eq. (12c) into (12a), implying that 1 b The metric component b(r) thus has the expression b(r) = e −I r 0 e I ( da dr r + 4a) 4ar where the function β = β(γ) is inversely proportional to the fluid brane tension γ, with GR regime lim γ→∞ β(γ) = 0, and dr . (24) The MGD kernel (17) must be compatible to Eq. (23), yielding ξ(r) = e −I(r) β+ r 0 2rae I r da dr + 4a impels anisotropic effects onto the brane. The geometric deformation ξ(r) in the vacuum, ξ (r), is minimal and is calculated by Eq. (25) restricted to L(r) = 0 [3], The radial metric component in Eq. (17) then becomes Using Israel-Darmois boundary matching conditions and Eqs. (18,21), in the interior star region, r < R, the MGD metric solution reads with mass function To match the MGD metric solution in the star interior region to the exterior one, the trace of the anisotropic stress-energy-momentum tensor and the Weyl dark radiation have the following respective expressions [3,5], Since p(r) = 0 = (r) for r > R, the exterior MGD stellar line element is given by [4] At the star surface, r = R, Israel-Darmois matching conditions imply that [4] log(a - The Misner-Sharp-Hernandez, Schwarzschild-like solution, is now put back in Eq. (27), implying the MGD Also, at the MGD star surface, the function (37) is negative. Therefore the MGD star horizon, r MGD = 2M, is nearer the star core than the GR Schwarzschild horizon r Schw = 2M 0 , as stated by Eq. (20). The explicit formula of the function β(γ) was derived in Ref. [3], The MGD star exterior metric components can be then written as [3,4] is the brane tension-dependent MGD parameter. The GR limit of a rigid brane, γ → ∞, thus recovers the Schwarzschild metric. Using Solar system classical tests of GR, the lower bound range | l | 6.261 × 10 −4 m was established [8]. Also, strong gravitational lensing effects applied to Sagittarius A set up the bound range |l| 6.370×10 −2 m [88]. In the limit l → 0, corresponding to a rigid fluid brane of infinite tension, the GR Schwarzschild results are then recovered.

B. MGD-decoupling method
The gravitational decoupling method states that the Einstein's effective field equations can be written as [2] In (41), the MGD-decoupled stress-energy-momentum tensor reads for representing the stress-energy-momentum tensor of a core solution of Einstein's effective field equations (EFEs), whereas the term T s µν evokes a source of anisotropy regulated by a coupling parameter, α, as and Θ µν denotes ancillary sources of gravity, coupled to the perfect fluid described by T EFE µν . The MGD-decoupling procedure can be described by considering the metric satisfying Einstein's effective field equations (41), that can be written as the coupled system The conservation law derived out the Bianchi identity, can be also expressed as dp dr Moreover, the effective density, the tangential and radial pressures, respectively read The anisotropic parameter is expressed by From the coupled system (46 -53), the MGD setup can be finally recovered when one observes Eqs. (11,45) and (52), using the notation in the metric (10), with the identifications (55d)

III. MGD SUPERFLUID STARS
Long before the experimental confirmation of the Higgs boson stated that scalar fields do exist in nature, fundamental scalar fields have been employed in several setups, including quantum field theory, gravitation, and in describing dark matter, inflation, and dark energy as well. Scalar fields can model realistic matter, emulating hydrodynamical fluids governed by some equation of state [89]. To describe MGD superfluid stars, one can emulate the paradigm of Bose-Einstein condensates, considering a MGD model coupled to a self-interacting scalar field with covariant logarithmic nonlinearity that arises from dense superfluid models. In the GR setup, the Einstein-Klein-Gordon coupled system of field equations has no static solutions for real scalar fields. As GR is a limiting case of the MGD solutions, for γ → ∞, it is natural to assume a spherically symmetric minimally-coupled, selfinteracting, complex scalar field φ hereon. The Einstein-Klein-Gordon system is defined by the Lagrangian To describe MGD superfluid stars, potential is adopted, where a 1 and κ are coupling parameters [90,91]. In superfluid Bose-Einstein condensates models, a 1 increases linearly as a function of the wave-mechanical temperature taken as a thermodynamic conjugate of the Everett-Hirschman's quantum information entropy [91]. The Einstein's effective field equations, can be derived from the Lagrangian (56), as with stress-energy-momentum tensor The Euler-Lagrange equations of motion that govern the complex scalar field, yield The Einstein-Klein-Gordon system has the MGD metric (10) as solutions, as long as stationary scalar fields are constrained to the field equations (58,59) and (62), that can be rewritten as the coupled system where one denotes and ( ) ≡ d/dx. Besides, the mass function now reads The coupled system of equations (64a -64c) and (66) is compatible to the MGD metric (10, 39a, 39b) and can be solved to constrain the scalar field λ, with Dirichlet and Neumann conditions λ(0) = λ 0 , λ (0) = 0, and M(0) = 0.
The value κ c = 0.06918 plays a prominent role as a critical value of κ. We will see in detail that at κ c , the behavior of the quantities to be analyzed, in most cases, splits into two distinct cases, namely, for κ ≶ κ c . When one analyzes the asymptotic value of the mass function, an absolute maximum at is reached at κ = κ c , where just here we must explicitly adopt the Newton's gravitational constant G 4 . This profile is presented in Fig. 1 Therefore, the analysis hereon takes into account the impact of different values of the brane tension on the physical quantities to be investigated. The higher the value of the brane tension, the more the results approximate the GR limit of an infinitely rigid brane, corresponding to γ → ∞. Fig. 1 shows the maximum asymptotic value of the mass function, M max , as a function of the parameter κ.
It illustrates a sharp change of behavior of M max at the critical value κ c , when κ varies. At κ c , the quantity M max has a discontinuity of the first derivative ∂M max /∂κ in the graphic, irrespectively the value of the brane tension is. For all values of κ lower [higher] than κ c , the first derivative is non-negative [non-positive]. Besides, Fig. 1 and the numerical analysis involved show that In what follows the value Ω = 9 will be employed, following Ref. [91] that implements the GR case. The  Fig. 3, where l = 10 −6 m and is blunter in Fig. 4, where l = 10 −4 m is taken into account. It means that lower values of the brane tension can decrease the effective range of the scalar field λ(x). Independently of the brane tension value, the scalar field presents a fast decay, going to zero at the spatial infinity, also exhibiting neither singular points nor nodes.
Figs. 5 -7 also illustrate the scalar field λ(x) surrounding a MGD superfluid star, with respect to the dimensionless radius x, for different values of the brane tension, encrypted into the MGD parameter l. Again one can realize a Gaussian-like profile of the scalar field. However, the effect of the brane tension is exactly opposed for values κ > κ c , when comparing to the respective Figs. 2 -4 that show the range κ < κ c . In Fig. 5, the MGD parameter l = 10 −8 m is employed and the scalar field decreases more bluntly with respect to x. This decrement is sharpened in Fig. 6, where l = 10 −6 m and is even sharper in Fig. 7, for l = 10 −4 m. It shows that for values of κ above the critical value, κ c , higher values of the brane tension decrease the reach of the scalar field λ(x) along the radial coordinate, going to zero at the spatial infinity as well. Also, whatever the brane tension value is, the regular scalar field fastly decays, with no nodes.     Figs. 14 -19 show the relativisticity function (2M(x)/x) of a MGD superfluid star, for distinct values of κ. Figs. 14 -16 regard the range κ > κ c , wherein the function 2M(x)/x has a negative second derivative and a steep increment from zero, attains an absolute maximum and smoothly decays, after an inflection point. For each fixed value of κ, the higher the brane tension, the higher the peak of 2M(x)/x is. This can be realized by the data in Table I. Now, for fixed values of the fluid brane tension, the lower the value of κ, the lower the peak of 2M(x)/x is and, in general, the higher the value of the radial coordinate where it occurs. Table I summarizes the 2-tuples corresponding to the (dimensionless) radial coordinate and the respective peaks of the function 2M(x)/x, in Figs. 14 -16.  derivative and an abrupt increment from zero, attaining an absolute maximum, and smoothly decays, after an inflection point. Also, for each fixed value of κ, the higher the brane tension, the higher the peak of 2M(x)/x is. This can be also noted in the data in Table II. But the analogies to the analysis of Figs. 14 -16 end here. Contrary to that behavior, for fixed values of the fluid brane tension, the lower the value of κ, the higher the peak of 2M(x)/x is, and the higher the value of the radial coordinate where it occurs. Besides, after the peak, the quantity 2M(x)/x fades away, and the limit with the GR limit of MGD-decoupling. Despite this general behavior, the rate that that 2M(x)/x decays varies according to the value of the brane tension.     Figs. 20 -22 show the asymptotic value of the mass function M ∞ , as a function of the central scalar field λ 0 , for several values of the fluid brane tension, with κ < κ c . They are qualitatively similar to the GR limit (see, e. g., Fig. 4 of Ref. [91]), although in the MGD case here analyzed M ∞ increases around one order of magnitude slower, as a function of the scalar field λ 0 . Besides, the absolute values of M ∞ , for each fixed κ, are slightly higher than the GR limit. Still regarding Figs. 20 -22, it is also worth emphasizing that the higher the value of the brane tension, the higher the values of M ∞ are, for each fixed κ. One can observe that the in-   Table III.
On the other hand, the case κ > κ c is shown in Figs. 23 -25 and maintain quite few qualitative similarities to the GR limit. In the MGD case here discussed the M ∞ increases slower, with respect to the scalar field λ 0 , than the GR limit. Also, the absolute values of M ∞ , for each fixed κ, are slightly higher than the GR limit. The higher the value of the brane tension, the higher the values of M ∞ are, for each fixed κ. One can observe that the increment of the brane tension increases the value of the mass function M ∞ . However, in the GR limit, M ∞ presents: a) for all values of κ 0.078, absolute max-     (Fig. 25). The function M ∞ presents one absolute maximum with non-null derivative for κ 0.514. In the range κ 0.514, there is one absolute maximum of M ∞ with null derivative. Table IV shows the 2-tuples corresponding to the central scalar field λ 0 and the respective maxima of the M ∞ function, in Figs. 23 -25. Again, the central field λ 0 also constrains the asymptotic value of the mass function M ∞ , for each value of κ. Far away this range of λ 0 , no real solutions for M ∞ exist. Consequently, for each value of κ, there is a maximum value of λ 0 above which M ∞ breaks off. The suprema of the scaled scalar field λ 0 characterizing a superior limiting bound to M ∞ are presented as abcissae of the 2-tuples, in Table IV.  In Figs. 26 -28, the asymptotic value of the mass function M ∞ is displayed as a function of the squared frequency Ω 2 , for values of κ above the critical value κ c .  Although the graphics in Figs. 26 -28 keep a relative qualitative similarity to the GR case, the role played by the brane tension-dependent MGD parameter (40) makes profound quantitative differences. For each fixed value of κ, the higher the brane tension, the lower the peak of the asymptotic value of the mass function M ∞ is, as a function of the squared frequency Ω 2 . For fixed values of the fluid brane tension, the higher the value of κ, the lower the peak of the asymptotic value of the asymptotic mass function M ∞ is. Besides, an important feature resides in the fact that there is a maximal frequency, for each value of κ, above which M ∞ ceases to exist. Now, Figs. 29 -31 display the asymptotic value of the mass function M ∞ , as a function of the squared frequency Ω 2 , for values of κ below the critical value κ c . Again these graphics have a relative qualitative analogy to the GR case, but quantitative aspects make them quite different, due to the finite brane tension in the MGD solutions. For each fixed value of κ, the higher the brane tension, the lower the peak of M ∞ is, as a function of the squared frequency Ω 2 . Contrary to the case when κ > κ c , for fixed values of the brane tension, the higher the value of κ, the higher the peak of the asymptotic value of M ∞ is. Just for values close to the critical κ c , there are non-null values of the frequency which correspond to a minimal value of M ∞ , in Figs. 29 -31, illustrated for κ = 0.068. Below these values of the frequency, M ∞ ceases to exist.  To end this section, let us provide a concrete estimative of MGD-decoupling modifications with respect to GR. Ref. [91] showed that for Solar size astrophysical stellar objects with mass M ∼ 1.99 × 10 30 kg in the GR regime, a Schwarzschild superfluid star has radius R ∼ 1/ √ a 1 16.92 km, where a 1 = 2.86 × 10 −8 m −2 . Here, using Eq. (51), MGD superfluid stars have radii that are slightly altered by a term involving the parameter α that governs the source term of MGD-decoupling (Eq. (44)). Besides, the Einstein-Klein-Gordon system involves the mass function (66), which is also function of the brane tension and the coupling parameter a 1 as well, as it includes terms that depends on the MGD metric coefficients a(r) and b(r). Assuming the physically realistic values for the finite fluid brane tension value, γ ∼ 3 × 10 −6 GeV, and for the MGD parameter, l = 10 −4 m, MGD superfluid stars have effective radius R 17.01 km. It is a considerable difference of 0.53%, susceptible to observation. When l = 10 −8 m, one has R 16.95 km, a difference of 0.23% regarding the GR case.

IV. CONCLUSIONS
Some results regarding superfluid stars in the GR limit [91] are qualitatively similar to the ones obtained in this work, scrutinizing MGD superfluid stars. However, most of our results show that the finite fluid brane tension and the MGD parameter bring new results that comply with a more realistic and physically feasible model, whose eventual gravitational-wave observations can be more reliable in a finite brane tension setup. The maximal masses of the MGD superfluid stars increase at a slower pace, with respect to the minimally-coupled scalar field, when compared to the GR limit. Besides, higher values of the brane tension can increase the effective range of the scalar field that satisfies the Schrödinger-like equation with logarithmic potential, coupled to gravity. It illustrates the effect of a finite brane tension, that attenuates the dependence of the MGD superfluid stars with the scalar field. On the other hand, the results analyzed show that the absolute values of the mass function, for each fixed value of the coupling parameter κ, are slightly higher than the GR limit. This time, the finite fluid brane tension plays the role of heightening the maximal values of the mass function, M ∞ are, for each fixed κ. This difference may be further probed by observations of precise signatures of the MGD-decoupling. MGD superfluid stars, driven by self-interacting logarithmic scalar fields, can generate regular equilibria stellar configurations. This setup can be employed to report astrophysical MGD stellar distributions. The study here completely differs from the Einstein-Klein-Gordon system studied in Ref. [10], which derives self-gravitating compact stellar systems surrounded by a coupled scalar field. The scalar field in this context defined a fluid that dilutes the gravitational field at the nuclear source of the Schwarzschild-like metric. Here, the scalar field satisfies the covariant equation (61), that mimics a Schrödingerlike equation with logarithmic potential, that precisely models MGD superfluid stars.