# Branes constrictions with White Dwarfs

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## Abstract

We consider here a robust study of stellar dynamics for white dwarf stars with polytropic matter in the weak-field approximation using the Lane–Emden equation from the brane-world scenario. We also derive an analytical solution to the nonlocal energy density and show the behavior and sensitivity of these stars to the presence of extra dimensions. Similarly, we analyze stability and compactness, in order to show whether it is possible to agree with the conventional wisdom of white dwarfs dynamics. Our results predict an average value of the brane tension of \(\langle \lambda \rangle \gtrsim 84.818\;\mathrm{MeV^4}\), with a standard deviation \(\sigma \simeq 82.021\;\mathrm{MeV^4}\), which comes from a sample of dwarf stars, being weaker than other astrophysical observations but remaining higher than cosmological results provided by nucleosynthesis among others.

## Keywords

Neutron Star Brane Tension Polytropic Index Dwarf Star Emden Equation## 1 Introduction

Stellar astrophysics has been a cornerstone to demonstration of the predictive capabilities of the General Theory of Relativity (GR), describing high energy astrophysical phenomena such as white dwarfs and neutron stars, with unprecedented success [1, 2, 3]. One of the most important results in this vein is the Lane–Emden (LE) equation [4, 5], which is a Newtonian approach to GR, under the assumption that the dwarf star is formed by polytropic matter; we remark that these types of stars are excellent high energy laboratories with which it is possible to test the phenomena described by GR and even to corroborate or refute our most plausible extensions [6, 7, 8, 9, 10, 11, 12].

Moreover, brane-world theory (for a good review see [13, 14]) has been one of the most captivating extensions to GR, due to its theoretical predictions and its ability to solve fundamental phenomena such as the hierarchy problem, among others [15, 16, 17]. It is worth mentioning that the brane-world models have a very long tradition in the specialized literature and their properties have been extensively studied under diverse circumstances, ranging from the cosmological scenarios [18, 19, 20, 21, 22, 23, 24, 25] to the study of astrophysical models [6].

Following the conventional wisdom, it is possible to extend the classical astrophysics for polytropic stellar systems with the brane-worlds frame work. In this vein, many authors have been given the task of showing the different stellar behavior, studying stability, collapse [6, 7, 8, 9, 10, 11, 12] or the stellar dynamics in general [6, 7, 8, 9, 10, 11, 12].

With a view of this scenario, this paper is devoted to a study of the modifications of LE equation caused by the brane in the cases of a star with polytropic matter, it being our main goal to produce observational verifications in these systems. It is important to remark that here we have one of the most suitable signatures is the sensitivity of these kinds of stars to the corrections provided by brane theory, producing a new dynamics in energy density (or pressure) and in the effective mass; as well as the implementation of a new range of exclusion, where the star is dynamically unstable. From this new range, it is possible to propose a bound to the brane tension in order to avoid an unstable stellar configuration among other pathologies.

Before starting, we would like to mention here some experimental constraints on brane-world models, most of them concerning the so-called brane tension \(\lambda \), which appears explicitly as a free parameter in the corrections of the gravitational equations mentioned above. As a first example we have the measurements on the deviations from Newton’s law of the gravitational interaction at small distances. It is reported that no deviation is observed for distances \(l \gtrsim 0.1 \, \mathrm{mm}\), which then implies a lower limit on the brane tension in the model Randall–Sundrum II (RSII): \(\lambda > 1 \, \mathrm{TeV}^{4}\) [26, 27]; it is important to mention that these limits do not apply to the two-branes case of the model Randall–Sundrum I (RSI) (see [14] for details). Astrophysical studies related with gravitational waves and stellar stability constrain the brane tension as \(\lambda > 5\times 10^{8} \, \mathrm{MeV}^{4}\) [6, 7, 8, 9, 10, 11, 12, 28, 29], whereas the existence of black hole X-ray binaries suggests that \(l\lesssim 10^{-2}\, \mathrm{mm}\) [14, 30, 31]. Finally, from cosmological observations, the requirement of successful nucleosynthesis provides the lower limit \(\lambda > 1\, \mathrm{MeV}^{4}\), which is a much weaker limit as compared to other experiments (other cosmological tests can be found in [18, 19, 20, 21, 22, 23, 24, 25, 32, 33]).

We divide this paper in the following sections: Sect. 2 is dedicated to showing the equations of motion for a stellar structure, showing the modified Tolman–Oppenheimer–Volkoff (TOV) equation and the respective conservation equations; considering always the regularity of the functions and maintaining a Schwarzschild stellar exterior [6, 7, 8, 9, 10, 11, 12]. In Sect. 3 we derive the LE and mass equations, based on a set of minimal assumptions which are in concordance with the current studies of stellar dynamics. Also, an analytical form of the nonlocal energy density is derived which essentially is a function of the polytropic constant and the interior central energy density of the star. In Sect. 4 the initial conditions are imposed and we generate numerical solutions to the LE and mass equations for the case with a polytropic index of \(n=3\), related with white dwarf stars. Finally in Sect. 5 we give some conclusions and make important remarks.

Henceforth we will use units in which \(\hbar =c=1\), unless explicitly stated otherwise.

## 2 Equations of motion

*p*and \(\rho \) are, respectively, the pressure and energy density of the stellar matter of interest, \(\mathcal {U}\) is the nonlocal energy density, and \(\mathcal {P}\) is the nonlocal anisotropic stress. Also, \(u_{\alpha }\) is the four-velocity (which also satisfies the condition \(g_{\mu \nu }u^{\mu }u^{\nu }=-1\)), \(r_{\mu }\) is an unit radial vector, and \(h_{\mu \nu } = g_{\mu \nu } + u_{\mu } u_{\nu }\) is the projection operator orthogonal to \(u_{\mu }\).

*r*, \(A(r)=[1-2G_{N}\mathcal {M}(r)/r]^{-1}\), and the effective energy density and pressure, respectively, are given by

*Schwarzschild exterior*, which can easily be accomplished under the boundary conditions \(\mathcal {V}^+(R) = 0 =\mathcal {N}^+(R)\), as for them the simplest solution that arises from Eq. (4c) is the trivial one: \(\mathcal {V}(r \ge R) = 0 =\mathcal {N}(r \ge R)\). Thus, for the purposes of this paper, we will refer hereafter to the restricted ID matching condition given by

*the only interior solution of the nonlocal anisotropic stress under the conditions of a Schwarzschild exterior, and for a non-constant density with*\(\rho (R) = 0\),

*which are the conditions we expect to have in realistic stars, is the trivial one:*\(\mathcal {N}(r) \equiv 0\) (see [6, 7, 8, 9, 10, 11, 12] for details). This implies that Eq. (7) can be written

## 3 The modified Lane–Emden equation

*minimal conditions*:

- (a)
- (b)
The pressure vanishes at the surface and in the exterior of the star, and the \(p(r)=0\) for \(r\ge R\) [6, 7, 8, 9, 10, 11, 12].

- (c)
The star is described by the polytropic equation \(p=K\rho ^{(1+n)/n}\), where

*n*is the polytropic index with \(n\ge 0\). - (d)
The pressure is negligible compared with the energy density \(p\ll \rho \).

- (e)
We assume the relation \(4\pi r^{3}p_\mathrm{eff}\ll \mathcal {M}\) between the effective variables.

- (f)
The gravitational potential in terms of the effective mass is negligible, \(2G_{N}\mathcal {M}/r\ll 1\).

*r*. The solution of the previous differential equation, without loss of generality, can be written as

Particularly, *low energy* stars like dwarf stars can be modeled in this context and now we are in a position to determine how the brane effects provide the interior of a star with extra dynamics. It is important to mention that white dwarfs can be modeled by the polytropic index \(n=3\), and neutron stars by polytropes with an index in the range \(n=0.5\)–1. However, in the case of neutron stars the weak-field approximation is not sufficient to make a general description of these stars; it is necessary to add the corrections provided by GR with the full modified TOV equation.

## 4 Numerical solutions for dwarf stars

Let us start studying a dwarf star using the modified LE equation; we observe from Eq. (12) that the free parameters are \(\bar{\rho }\) and \(\chi _{n}\), related with the central energy density of the star, the brane tension, and the polytropic constant.

Our analysis shows that the central energy density and the polytropic constant are redundant, because in particular they depend on the characteristics of each star; then we fix by hand the values of \(\chi _{n}\), where are encoded by both parameters. In this case, we board the region \(\chi _{3}=10\), due to the orders of magnitude being greater, causing divergences which imply non-compact configurations. This results in the dependence \(K=5 \rho (0)^{-1/3}/8\). Therefore, we only explore the limit case, when the minimal requirements (a)–(f) are fulfilled.

### 4.1 Physical initial conditions

*k*and \(k+\mathrm{d}k\), \(k_{F}\) being the maximum momentum and \(m_{e}\) being related with the electron mass. From Eqs. (21) and (22), we obtain for the dwarf stars with index \(n=3\) the following conditions:

### 4.2 Results of the numerical solutions

To begin with, we show the numerical solutions implemented for dwarf stars showing the behavior of energy density and mass profiles in Fig. 1, top and bottom. We implement the usual initial conditions: \(\theta (0)=1\), \(\mathrm{d}\theta (0)/\mathrm{d}\zeta =0\) and \(\bar{\mathcal {M}}(0)=0\), for \(n=3\), as in the textbook case [4, 5] and \(\bar{\mathcal {V}}(0)=0\) considering an inward integration.

We start showing the non-brane case as a benchmark, adding first only the quadratic part of the energy-momentum tensor. Under this assumption, we predict a lower energy density compared with the non-brane case (see Fig. 1, top). Clearly, the stellar configuration is more massive for a similar radius to the previous case (see Fig. 1, bottom). Also, we present the compactness plot (see Fig. 2), which shows the different behaviors with different values of the brane terms. It is obvious how we have a most compact configuration when the presence of the quadratic terms predicted by branes plays an important role. Clearly, this is an incomplete analysis due to lack of Weyl terms, however, in the following, we took on the task of presenting the nonlocal terms.

When we *turn on* the Weyl terms, these cause higher energy densities and smaller masses in comparison with the case of non-branes, while we increase the presence of extra terms; the effects are accentuated, causing a non-compact configuration, i.e., conditions (a) and (b) are not fulfilled. In this sense, one may note that \(\bar{\rho }=0.016\) is the higher bound to a stable stellar configuration; we notice that when we exceed this bound we have an unstable star, implying a non-real stellar structure (see Fig. 1, top and bottom, and Fig. 2).

From left to right the columns read: name of the star, mass in solar units \(M_{\odot }\), radius in \(R_{\odot }\), density as \(\rho (0) = 3M/4\pi R^3\) in \(\mathrm{MeV}^4\), and brane tension in \(\mathrm{MeV}^4\) deduced from the constraint mentioned above; using a catalog of several white dwarfs reported in [36, 37, 38]

White Dwarf | Mass (\(M_{\odot }\)) | Radius (\(R_{\odot }\)) | \( \rho (0)\) (\(\mathrm{MeV}^4\)) | \(\lambda \) (\(\mathrm{MeV}^4\)) |
---|---|---|---|---|

Sirius B | 1.034 | 0.0084 | 10.5993 | 313.588 |

Procyon B | 0.604 | 0.0096 | 4.1478 | 122.715 |

40 Eri B | 0.501 | 0.0136 | 1.21009 | 35.801 |

EG 50 | 0.50 | 0.0104 | 2.70063 | 79.900 |

GD 140 | 0.79 | 0.0085 | 7.81565 | 231.232 |

CD-38 10980 | 0.74 | 0.01245 | 2.3298 | 68.928 |

W485A | 0.59 | 0.0150 | 1.06212 | 31.423 |

G154-B5B | 0.46 | 0.0129 | 1.3006 | 38.4793 |

LP 347-6 | 0.56 | 0.0124 | 1.7827 | 52.7426 |

G181-B5B | 0.54 | 0.0125 | 1.6781 | 49.6479 |

WD1550+130 | 0.535 | 0.0211 | 0.3456 | 10.2266 |

Stein 2051B | 0.48 | 0.0111 | 2.13023 | 63.0229 |

G107-70AB | 0.65 | 0.0127 | 1.926 | 56.9807 |

L268-92 | 0.70 | 0.0149 | 1.28438 | 37.9984 |

G156-64 | 0.59 | 0.0110 | 2.69047 | 79.5976 |

## 5 Conclusions and remarks

The presented analysis of a weak field maintaining the brane terms, conducted by using the LE equation, shows the new behavior of the density, the mass, and the compactness of stars with polytropic matter. The research developed in this paper shows how dwarf stars are sensitive to the Weyl terms, causing a non-compact configuration under particular conditions, implying a non-real star. It should be mentioned that only the existence of quadratic terms in the energy momentum tensor shows a less dense and more massive star compared to the non-brane case. In general, when we *turn on* also the Weyl contributions, the star’s model rather suggests a behavior of higher energy density and lower mass, beyond standard GR, which is discussed in 4.2. Significantly, there is a physical limit to the parameters \(\bar{\rho }\) and \(\chi _{n}\) (see Figs. 1, 2) such that one meets the minimum requirements for a stable star, which in this case must be \(\lambda \gtrsim 29.585 \; \rho (0)\) and \(K=5/(8\rho (0)^{1/3})\). Taking astrophysical data of a sample of white dwarfs, it is possible to establish an average bound of the brane tension as shown in the previous section: \(\langle \lambda \rangle \gtrsim 84.818\;\mathrm{MeV^4}\), with a standard deviation \(\sigma \simeq 82.021\;\mathrm{MeV^4}\) and the average of the polytropic constant must be constrained as \(\langle K\rangle \simeq 0.508 \; \mathrm{MeV^{-4/3}}\) with a standard deviation \(\sigma \simeq 0.142;\mathrm{MeV^{-4/3}}\). It is important to remark how the previous values are necessary to fulfill the minimal requirements to obtain a stable star; i.e. a real stellar configuration.

It is important to clarify that nonlocal terms caused by Weyl terms are gravitons that escape to the fifth dimension, causing stars not to have a compact configuration as they begin to dominate. An excess of Weyl terms is the cause that it does not satisfy the conditions (a) and (b), as we show. In fact, the Weyl terms eventually generate a divergence for a given radius. However, we can use this disadvantage to quantify the minimum value required for the brane tension, which is shown in our conclusions.

In addition, it is worth mentioning that modifications to the LE equation *prohibit* the case \(n=0\) (at least for the case where \(\mathcal {C}_{1}\ne 0\) or \(\mathcal {C}_{2}\ne 0\)), for a stable stellar configuration unlike that predicted by the non-brane limit. This is due to the divergence in the \(\chi _{n}\) term in the central energy density, causing conditions (a) and (b) not to be fulfilled.

Despite the fact that we are treating a weak gravitational limit and the brane effects are not accentuated strongly in the dynamics, it is possible to extract relevant information as regards the constraint of the brane tension, establishing an exclusion limit of the theory, taking as a premise the stability of the dwarf star. Strong evidence of branes can be found in the direct observation of the compactness of a dwarf star, when comparing the predictions of GR and branes, bearing in mind the technical challenges of this endeavor due to the subtle brane effects.

Finally, we suggest that studies of neutron stars can give us better constraints, and even evidence of the existence of extra dynamics which comes from brane theories. In the case of neutron stars, part of the machinery has been studied in Refs. [6, 7, 8, 9, 10, 11, 12], still as the most general way to treat this type of stars in a strong gravitational field. However, this is work that will be presented elsewhere.

## Notes

### Acknowledgments

The author acknowledge the suggestions of the anonymous referee and the enlightening conversation with F. Linares. Also the author is grateful for the support provided by SNI-México, CONACyT research fellowship and Instituto Avanzado de Cosmología (IAC) collaborations.

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