{'fraction_non_alphanumeric': 0.03393338769365866, 'fraction_numerical': 0.014802734742520229, 'mean_word_length': 4.432367972742759, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Mass radius relation of compact stars in the braneworld

The braneworld scenario, based on the fact that the four dimension space-time is a hyper-surface of a five dimensional manifold, was shown to deal in a satisfactory way with the hierarchy problem. In this work we study macroscopic stellar properties of compact stars from the braneworld point of view. Using neutron star equations of state, we test the possibility of extra dimensions by solving the brane Tolman-Oppenheimer-Volkoff equations obtained for three kinds of possible compact objects: hadronic, hybrid and quark stars. By comparing the macroscopic solutions with observational constraints, we establish a brane tension lower limit and the value for which the Tolman-Oppenheimer-Volkoff equations in the braneworld converge to the usual Tolman-Oppenheimer-Volkoff equations.


I. INTRODUCTION
The four fundamental nature interactions (gravity, electromagnetism, strong and weak nuclear forces) were investigated separately during their initial studies and developments. At some point, it became obvious that an unified theory could help the understanding of the primordial universe and its evolution. Two of these fundamental forces (electromagnetism and weak nuclear forces) were then joined in an unique formalism known as the electroweak interaction by Abdus Salam, Sheldon Glashow and Steven Weinberg, but the unification of the other two is still pursued nowadays. Around 1990's, string theory came up as a possible candidate for the theory of everything or M-theory because it presented the advantage of conciliating quantum mechanics with gravity by putting aside the idea of particles as the elementary bricks of the universe and explaining them as particular quantum states of strings. String theory started with bosons only and later incorporated fermions and the requirement of extra dimensions was a key point in its development. M-theory assumes the existence of 11 dimensions, some of them treated as hidden dimensions. These extra dimensions of the space-time have since then been used as an attempt to explain several open problems like the existence of dark matter, dark energy, the accelerated expansion of the universe, etc.
The braneworld scenario is based on the assumption that the four dimensional space-time is a hyper-surface of a five dimensional manifold. The gravity in the braneworld is a five dimensional phenomenon and the other fundamental interactions are confined to the brane. The braneworld scenario has attracted a lot of interest, because it tackles the hierarchy problem [1] in an effective way. Another attractive property is that the Newtonian law of gravity with a correction is also given in this braneworld scenario [2]. In the Randall-Sundrum (RS) model [1], one can further add scalar fields [3] with usual dynamics and allow them to interact with gravity in the standard way. In this scenario, the smooth character of the solutions generate thick branes with a diversity of structures [4]. In the braneworld scenario an import issue is how gravity and different observable matter fields of the Standard Model of particle physics are localized on the brane. Recently, the localization of fermions and bosons on the brane have received considerable attention in the literature [5]- [12].
On the other hand, in [13,14], the idea that neutron stars could be a laboratory for testing extra dimensions was first explored. It was found that the star in the braneworld is less compact that in general relativity and an astrophysical lower limit for the brane tension was established. In [15], brane theory was utilized to estimate the correction in the waves radiated by a binary system and the observational masses of the PSR B1913+16 were then used to constrain the brane tension.
Original neutron star models assume that the dense matter in its interior is composed of hadrons (protons and neutrons only or the complete lowest lying baryonic octet) and leptons, responsible for ensuring charge neutrality and β-equilibrium [16]. On the other hand, the Bodmer-Witten conjecture [17] states that quarks can be deconfined from the hadrons, forming a stable quark matter under certain conditions. Hence, compact stars can be constituted of pure quark matter or perhaps of hy-brid matter, containing in their core a pure quark phase or a mixed phase of quarks and hadrons [16,[18][19][20][21]. In order to rule out improbable stellar configurations, observational constraints have been used. While most neutron stars have masses of the order of 1.4 M ⊙ , at least two pulsars, PSR J1614-2230 [22] and PSR J0348+0432 [23] were confirmed to be very massive objects, with masses of the order of 2M ⊙ . The theoretical calculation of macroscopic stellar properties, as the masses and radii, is done by solving the Tolman-Oppenheimer-Volkoff (TOV) [24] equations, which use equations of state (EOS) as input.
In this work, we study the effects of the braneworld scenario on the macroscopic stellar properties of compact stars. In that spirit, we revisit the idea of solving a TOV-like system of equations in the braneworld (brane-TOV) and see if the resulting mass and radius results survive the known observational constraints. We also check weather, with more realistic equations of state than the perfect fluid one used in [13,15], limits for the brane tension can be established. Furthermore, we show that the star becomes more compact and that the radii can be adjusted to smaller values depending of the brane-TOV parameters.
The paper is organized as follows: In Sec. II, we present the standard TOV formalism. In Sec. III, we give a brief review of the brane-TOV formalism. In Sec. IV, we use different EOS as input to the brane-TOV equations, present and discuss our results. Finally, in Sec. V, we draw our final conclusions.

II. TOLMAN-OPPENHEIMER-VOLKOFF EQUATIONS
The Tolman-Oppenheimer-Volkoff (TOV) equations for static and spherical stars are derived from general relativity [24] where κ 2 = 8πG, g αβ is the metric, R αβ is the Ricci tensor, R = g αβ R αβ is the scalar curvature and T αβ is the energy-momentum tensor. The differential element for a spherical relativistic star is given by Solving Einstein's field equations for a perfect fluid matter, we obtain where m(r) and p(r) are respectively the gravitation mass and pressure both defined as star radius functions and ǫ is the energy density. To solve these equations, we need an equation of state p(ǫ) and boundary conditions, which we choose as: where p c is central pressure, R is the star radius and M is the total gravitation mass of the star.

III. TOLMAN-OPPENHEIMER-VOLKOFF EQUATIONS IN THE BRANEWORLD
The local and non-local extra-dimensional modifications to Einstein equations on the brane can be consolidated into an effective total energy-momentum tensor [25] where G αβ is the usual Einstein tensor and where λ is the brane tension. The local correction is carried via the tensor S αβ (matter corrections), while the non-local correction is carried via the projection of the bulk Weyl tensor E αβ . For a perfect fluid or minimally coupled scalar field the expressions for T αβ and S αβ are given by [26] T αβ = ρu α u β + ph αβ , where u α is the four-velocity and h αβ = g αβ + u α u β is the projection orthogonal to u α . Further, assuming static spherical symmetry the expression for E αβ becomes [13]: where r α is a unit radial vector, U and P are the nonlocal energy density and non-local pressure, respectively. From (7), we see that E αβ → 0 as λ −1 → 0. Applying this limit in eq. (4), we obtain that T eff αβ = T αβ , therefore the general relativity is regained.
Solving Einstein's equations and assuming that the Weyl terms obey the relation P = wU (an equation of state-like relation), we obtain the modified TOV equations where A = 3w r U + (2 + w)U dφ dr . To solve the system of differential equations above we need an equation of state p = p(ǫ) and boundary conditions given by To solve the Tolman-Oppenheimer-Volkoff (TOV) equations numerically in the brane world we use mixed units [16], which are useful in stellar calculations, where pressure and energy density are given in km −2 units and w is a dimensionless quantity. The unit of the brane tension λ is also km −2 , but for convenience we present our results in dyn/cm 2 , the unit generally used in the literature (see [15], for instance).
The solutions from equations (8), (9), (10) and (11) can be separated into internal and external solutions. The internal solution is the result obtained by integrating the system from the core of the star until the null pressure point, at its surface. The external solution is obtained from the null pressure until the Weyl term U becomes approximately null. In the present work we restrict ourselves to the study of the internal solutions.

IV. TESTING THE EQUATIONS OF STATE
We next choose some EOS as input to the brane-TOV equations. As mentioned in the Introduction, depending on their possible interior composition, neutron stars can be classified as hadronic stars with or without hyperons [16,27], hybrid stars containing hadronic and quark phases [18,19,28] or hadronic and pion or kaon condensates [29] and quark (also known as strange) stars [30]. In the following all three possibilities are analyzed. We start from the case in which just the solution related to the interior of the star is considered. For hadronic and hybrid stars, a crust is always expected and hence, we add the Baym-Pethick-Sutherland (BPS) [31] equation of state for very low densities. Due to the recent discoveries of massive stars [22,23] we start our analysis from EOS that we know are capable of generating high maximum masses when used as input to the usual TOV equations.
In Fig. 1 we display the brane-TOV solutions for hadronic stars whose equations of state include only nucleons and leptons, necessary to enforce β-equilibrium and charge neutrality conditions. The equation of state was obtained with the relativistic non-linear Walecka model as in [19,32]. We have fixed both the brane tension λ = 10 37 dyn/cm 2 and λ = 10 38 dyn/cm 2 and varied w. A first analysis of the mass-radius curves for λ = 10 37 dyn/cm 2 , displayed in Fig. 1a) shows that the stellar masses vary in an oscillatory way with the change of w and for some values (w = −0.6 in the figure), the solutions may become non-physical. As already stated in the Introduction, the star becomes more compact, in the sense that the masses and radii decrease for all w values as compared with the usual TOV solution. Our re-sults of stellar properties for hadronic stars obtained with λ = 10 37 dyn/cm 2 are summarized in Table I. From Fig.  1b), we see that the central energy density is practically the same for all w values until the maximum mass star is reached. Only after this point, the solutions deviate from each other. For the sake of completeness the radii in function of the central energy density is depicted in Fig. 1c). We then analyze the situation when λ = 10 38 dyn/cm 2 shown in Fig. 1d), Fig. 1e) and Fig. 1f). All brane-TOV solutions still represent slightly more compact stars than the TOV one, but for this value of the brane tension, the stellar properties for a family of stars are almost independent of w. Moreover, for this value of the brane tension, the results are very similar to the ones obtained with the standard TOV solution. One has to bear in mind that these solutions are not 100% precise due to small numerical uncertainties. Our results of stellar properties for hadronic stars obtained with λ = 10 38 dyn/cm 2 are summarized in Table II. We then vary λ from 10 35 to 10 39 dyn/cm 2 for two fixed values of w, one negative (w = −3) and one positive (w = 2). The mass-radius curves are shown in Fig.  2, from where we see that the stellar masses increase with the increase of λ and tend to converge to their maximum values around λ = 10 38 dyn/cm 2 , which is approximately the limit tension that the brane can support. On the other hand, when this limit is achieved, the re-    sults depend only slightly on the w values, as already seen in Figs. 1d), 1e) and 1f). An interesting aspect related to these solutions is the fact that λ clearly controls the values of the maximum star masses, while w influences the corresponding radii. For small values of the brane tension, the brane-TOV solutions become either non-physical or produce very low maximum masses, what is not expected from astronomical observations. Thus, we can obtain both a lower and a higher limit for the brane tension, as already discussed in [13,15], in a simplified context, where a perfect fluid was used instead of a more realistic equation of state. In the present work, we establish a range for λ in between 3.89 × 10 36 < λ < 10 38 dyn/cm 2 , obtained in such a way that at least a 1.44 M ⊙ star can be achieved. Of course, as discussed above and seen in Fig. 3, these tensions depend on the w values. Next we move to the study of hybrid stars. The equation of state we use as input to the brane-TOV equations is obtained from the non-linear Walecka model for the hadron phase, the Nambu-Jona-Lasinio model for the quark phase and a Gibbs construction for the mixed phase as in [18]. In face of the results obtained for hadronic stars, we have chosen λ = 10 38 dyn/cm 2 . Our results of stellar properties for hybrid stars are summarized in Table III. The general qualitative behavior is the same as for hadronic stars, except for the kinks in all curves related to the transitions from one phase to the other inside the star structure. However, for certain values of w (w = −0.1, for instance), there are still sensitive deviations from the standard TOV results, which means that, even for the limit brane tension, the results still depend on the value of w if hybrid stars are considered.
To finish our analysis, we look at an equation of state that is used to describe quark stars. A model for quark matter that yields maximum masses of the order of 2 M ⊙ is the quark mass density dependent model [33] and we use an equation of state taken from [34]. Once again we see, from Fig. 5, that for the chosen value of λ = 10 38 dyn/cm 2 , the TOV solutions are practically reproduced for a certain range of w values, as in the case of hybrid stars. Our results of stellar properties for quark stars are summarized in Table IV. A point worth mentioning  is the behavior of the quark star radius as a function of the energy density shown in Fig. 5c), very similar to the mass-energy density, due to the fact that quark stars present a finite density at their surface contrary to hadronic and hybrid stars, which have zero densities at zero pressure surface points. These different physical structures are seen when one compares Fig. 1a) and Fig.  4a) with Fig. 5a).

V. FINAL CONCLUSIONS
In the present work we have revisited the brane-TOV solutions discussed in [13,15] for realistic equations of state normally used in the literature and compared the results with the ones obtained from the standard TOV solutions. We have confirmed that there is a range of validity for the brane tension and checked that, once the maximum brane tension value is attained, the results are practically independent of the value of w for hadronic stars and very little dependend for hybrid and quark stars.
We have also seen that the solutions encountered with the maximum brane tension value reproduce the standard TOV solutions. This fact means that, as far as the equations of state survive the observational constraints when the macroscopic properties are computed from the usual TOV equations, they are also suitable as input to the brane-TOV equations. One advantage of using the brane-TOV equations is that the radii can be adjusted to smaller values, as seen, for instance in Fig. 1a). This can be useful since small radii are in accordance with some recent estimates [35].