Charged Compact Stars in $f(\mathcal{G})$ Gravity

This work is devoted to investigate some of the interior configuration of static anisotropic spherical stellar charged structures in the regime of $f(\mathcal{G})$ gravity, where $\mathcal{G}$ is the Gauss Bonnet invariant. The structure of particular charged stars is analyzed with the help of solution obtained by Krori and Barua under different viable models in $f(\mathcal{G})$ gravity theory. The behavior of some physical aspects is investigated with the help of plots and the viability of our modeling is analyzed through different energy conditions. We have also studied some behavior of these realistic charged compact stars and discuss some aspects like density variation, evolution of stresses, different forces, stability of these stars, measure of anisotropy, equation of state parameters and the distribution of charges.


Introduction
Despite of the great, well established and successful theory, the general theory of relativity, in the past century, numerous valuable modifications are being suggested by researchers. In these modification, the Ricci scalar is replace by some arbitrary function, like f (R) in which R is ricci scalar, f (G), where G in Gauss-Bonnet invariant and many others as discussed in ref [1][2][3][4][5][6][7][8][9]. The expansion of the universe is the remarkable phenomenon which is being addressed by these modified theories [10]. The well-established fact is, the accelerated expansion of universe cannot be explaining by GR alone in its regular arrangement without adding extra term in the gravitational Lagrangian or exotic matter [11,12]. The simplest modification was given by Buchdal in 1970 [13] with the help of replacement of R by f (R) arbitrary function of the different energy conditions, for different observational data of compact stars. This paper is design as, in a very next section, we discuss the modified f (G) gravity with charged anisotropic matter distribution of the static spherically symmetric geometry. In section 3, we demonstrate some of viable f (G) gravity models. Section 4 is dedicated to check the physical analysis and viability of different well known compact stars through plots. And finally, we summarize the main results in last section.

f (G) gravity
This section is to provide the extended version of Gauss-Bonnet gravity with its equations of motion. For f (G) gravity, the usual Einstein-Hilbert action is modified as follows where κ 2 = 8πG ≡ 1, R, f, S m (g µν , ψ), S e (g µν , ψ) are the Ricci scalar, arbitrary function of Gauss-Bonnet invariant, the matter action and the charged action ,respectively. The Gauss-Bonnet invariant quantity is where R µν , R µναβ are the Ricci and the Riemannian tensors. Upon varying the above action with respect to g µν , we get the modified field equations for f (G) gravity as where T eff µν is named as effective stress-energy tensor with its expression as follows where subscript G defines the derivation of the corresponding term with the GB term, while T µν is the usual stress energy momentum tensor and

Anisotropic matter distribution in f (G) gravity
Here, we wish to examine the effects of anisotropic stresses over the stability of compact charged stars. For this purpose, we take the distribution of matter content source to be anisotropic having the following mathematical formulation where ρ is fluid energy density, P t is tangential pressure component, P r is radial pressure component and Π is equal to P r − P t . Furthermore, V γ and U γ are four velocity and four vector of the fluid, respectively. These quantities obey V γ V γ = 1 and U α U α = −1 relation under the comoving coordinate system,. Now, we suppose the interior relativistic structure to be static and spherical symmetric everywhere. In this direction, we take the general line element static spherical symmetric geometry as following Where a and b are arbitrary constant. Now by solving the field equations (3), we get Here, E 2 = Q 2 8πr 4 . We suppose a = r 2 B +C and b = r 2 A as suggested by Krori and Barua [45], here A, B and C are the arbitrary constant. Using these definitions, we reach at We will use these equations with different models. Here, we see charge contribute in ρ, p r and p t . Now consider the quark matter EoS where B g is bag constant. Using this equation, we find the expression for charge, read as The expression for Q contain a square-root which means both sign for charge are acceptable but we will consider the positive sign of charge for further investigation.

Matching condition and Different Models
In this section, we consider a hypersurface Σ that is a boundary of both exterior and interior regions. Furthermore, we suppose Reissner-Nordstrm metric for the description of exterior geometry, written as Where m, r and Q is the mass, radius and charge, respectively. The interior of given metric in Eq. (7) for the charged fluid distribution join smoothly with the above exterior of Reissner-Nordstrm metric. By matching the these two geometries at r = R and m(R) = M, we get We find the numerical values of these constat for three different strange compact physical stars as shown in Table: 1. Furthermore, we will take some of the viable models for the study of different compact star properties like the stability analysis energy conditions etc As where we will take three different models i = 1, 2, 3

Model 1
First, we assume the power-law model with the additional logarithmic correction term [46] where α 1 , n 1 and β 1 are arbitrary constants. This model could provide observationally wellconsistent cosmic results because of its extra degrees of freedom allowed in the dynamics.

Model 2
Next, we take another model having the form [47] where α 2 , β 2 and m are any constant number, while n 2 > 0. This model is very helpful for the treatment of finite time future singularities.

Model 3
Further, we assume another viable model of the form here a 1 , a 2 , b 1 , b 2 and n 3 are arbitrary constants with n 3 > 0. From the condition p r (R) = 0, We find the value of B g as shown in Table: 2.
Models B g for CS1 B g for CS2 B g for CS3  Using these model with eq. (11-13), we get ρ, p r and p t from which we check the different aspect of compact stars as shown in Table 1. We will discuss these aspect one by one in the following section.

Aspects of f (G) Gravity Models
In this section, we discuss some of physical aspects of the above charged stars from the interior solution. We present the anisotropic behavior and stability of these charged stars under consideration of three different f (G) viable models. We discuss these aspects one by one in following

Variation of Energy Density and anisotropic stresses
We study the influence of quark matter EoS with the anisotropic stresses at the center with modified f (G) gravity models. The corresponding variations in the vicinity of energy density along with anisotropic stresses are shown in Figs. 1, 2 and 3, respectively. The evolution of the density for the strange star candidate V elaX − 1, SAXJ1808.43658, and 4U1820 − 30 are shown in Fig. 1. Here, for r → 0, the density goes to its maximum value. In fact, this indicates the high compactness of the core of these stars and validating  Table 3.
Similarly, the variation of the radial and traverser pressure,are shown in Fig. 2 The variation of radial derivative of density, dρ dr , radial derivative of radial pressure, dpr dr , and radial derivative of transverse pressure, dpt dr are shown in Fig. 4, 5 and 6 respectively. We see that these variations are negative and for r = 0, we get dρ dr r=0 = 0 dp r dr r=0 = 0

Energy conditions
To deal with a physically viable and acceptable matter field, there are some mathematical constraints which should be obeyed by stress-energy tensor, these constraints are known as energy conditions. These energy conditions are coordinate invariant and can be written as following. Figure 5: Evolution of dp r /dr for charged stars, Vela X -1, SAX J 1808.4-3658, and 4U 1820-30, under different viable f (G) models. Figure 6: Evolution of dp t /dr for charged stars, Vela X -1, SAX J 1808.4-3658, and 4U 1820-30, under different viable f (G) models.
Here i = r, t and ρ, p r and p t include electric charge contributions as well.

Equilibrium condition
To investigate the equilibrium of inner structure of these charged compact stars, we use the generalized Tolman-Oppenheimer-Volko (TOV) equation. For charged spherical anisotropic stellar interior geometry, this equation is written dp r dr + ν ′ (ρ + p r ) 2 + 2(p r − p t ) r + σQ r 2 e λ/2 = 0 Where σ is charge density. Furthermore, the above Eq. (23) may be written as a sum of different forces e.g. gravitational, hydrostatic, anisotropic and electric forces which yields By using these definitions with the values of different parameters from Table 1, we check the variations of these forces and their hydrostatic equilibrium, as shown in Fig. 10. The left plot shows the evolution of these forces in background of first model, the middle one is for second model and the right plot describe the variation of these forces because of third model. It is clear from Fig. 10, that the electric force has a very negligible effect in this balancing mechanism.

Stability Analysis
In this section, we investigate the stability of the interior of stars under modified f (G) theory. For the mathematical modeling of compact stellar structures, it is to be noted that only those stellar models are significant which are stable against the variations. Hence, the role of stability is very crucial and burning issue in the modeling of compact objects. The stability of stellar structure has been studied by many researches. Here we adopt the techniques which is based on the concept of overturning (or cracking) [48]. According to this, the radial speed of sound v 2 sr as well as transverse speed of sound v 2 st must be in the range of a closed interval [0, 1] to preserve the causality condition and for stability the necessary condition 0 ≤ v 2 sr − v 2 sr ≤ 1 should be obeyed. The the radial and transverse speeds is defined as In our case, v 2 sr ∼ 1/3 and v 2 sr is plotted in fig. 11, which obey the condition 0 ≤ v 2 sr ≤ 1 and 0 ≤ v 2 st ≤ 1 which is the indication of causality preservation within these charged compact stars. Similarly, for stability, we plot v 2 st − v 2 sr as shown in Fig. 12. It is to be noted that all of our charged stellar structures under consideration of different viable f (G) models obey   the constraint: 0 < |v 2 st − v 2 sr | < 1 We concluded that the stability is attained in f (G) gravity models for three considered strange candidate stars, V elaX − 1, SAXJ1808.43658, and 4U1820 − 30.

EoS Parameter
Now for anisotropic stresses, there are two equation of state parameters, written as For a radiation dominant era, equation of state parameters must lie between 0 and 1. More precisely, 0 < w r < 1 and 0 < w t < 1. Here, we check the evolution of EoS parameters for three different charged stars and their behavior are shown graphically in Fig. 13 and 14.
We can see that both w r and w t lies in given range.

Mass Radius Relationship, Compactness and Redshift Analysis
The mass of charged compact stars can be written as Here we know that mass m is function of r and m(r = 0) = 0 while m(r = R) = M. The variation in masses of charged compact stars are shown in Fig. 15. We see that the mass is regular at core because it is directly proportional to radial distance e.g. m(r) → 0 for r → 0. The maximum mass is attained at r = R, as shown in fig. 15. The mass radius relation is also compatible with the study of neutron stars under f (G) gravity [49]. Furthermore, the compactness, µ can be define as The compactness for three different strange stars are shown in Fig. 16. similarly, the Redshift, Z s can be define for compact object The bound over Z s ≤ 2. In our case, wee check the variation in redshift from the core to surface of stars. These evolution are shown with the help of plots, as given in Fig. 17.

The Measurement of Anisotropy
In modeling of relativistic stellar interior structures, it is important to discuss the anisotropicity or anisotropy which is defined as We check the anisotropy for three different charged strange stars under consideration of three viable models in f (G) gravity. After plugging the constant values with these models, we plot the anisotropy and get that ∆ > 0 e.g. p t > p r . This implies that the anisotropy is directed outward for all three stars. These plots are shown in Fig. 18. It is important to note that ∆ → 0 at r → 0 and becomes monotonically increasing outwards with the increase of r near the surface of the star.

Electric field and Charge
We observed that the electric charge on the boundary for star 1 is 6.459234 × 10 20 C, for star 2 6.7510838 × 10 20 C and for star 3 6.7460087 × 10 20 C and zero at the core of these stars under consideration of model 1. The charge profile is monotonically increasing away from the center, as shown in Fig. 19. Furthermore, the electric charge density is monotonically decreasing outward and is maximum at the center of these stars as shown in Fig. 20.
Similarly, the behavior of electric field intensity E 2 is also discuss and their variation for different stars are shown in Fig. 21.    From these plots, we conclude that the core of these stars contain Q(r = 0) = 0, σ(r = 0) = σ 0 and E 2 (r = 0) = 0 while at surface of these stars Q(r = R) = Q m σ(r = R) = 0 and E 2 (r = R) = E 2 m where σ 0 , Q m and E 2 m is the maximum charge density, charge and electric field intensity. The stellar structure formation in the background of modified gravity are comparatively higher contraction in the collapsing rate of spherical systems at its initial stages unlike GR. The extra curvature (non-gravitational fluid) on the existence of compact structure could lead arena of having relatively more compact stars than in GR. Similarly, the influences of these additional dark source terms on mass radius relationships for compact stars predict more massive relativistic systems with comparatively smaller radii than in GR. Perhaps, the calculated apparent masses of neutron star models in modified gravity are more massive star with smaller radii than in GR. Such type of investigations could provide theoretical well-consistent way to handle and study classes of massive and super massive structures at large scales.

Summary
It has been attracting challenge to find the correct model for charged realistic geometry of interior compact objects not only in general relativity but also in extended theories of gravity like f (G) gravity. For this purpose, we have considered the three-different observed compact stars, labeled as Vela X -1, SAX J 1808.4-3658, and 4U 1820-30. Our desire is to study the real composition of these compact objects in their central regions under consideration of three different viable models. We have investigated several aspects of compact stars in the regime of f (G) gravity with the anisotropic matter content under Einstein Maxwell spacetime. We have utilized the solutions for the metric function suggested by Krori-Barua for a spherical compact object whose arbitrary constants are calculated across the boundary of interior and exterior geometry. The values of these arbitrary constants are determined with the help of charge, mass, and radius of any compact object. We have used three different strange candidate stars with their experimental observational data to study the effects of additional degree of freedom coming from modified gravity theories. For this purpose, we used three different viable models in f (G) gravity. By using these models along with calculated values for three different stars, we have plotted the relevant quantities like variation of anisotropic stress and energy density against radial distance. It is found that the energy density is very high at core of these stars and gradually decreases with the increasing radius, thereby indicating the high compactness structures of these stellar interiors.
We concluded our discussion as: • The variation in energy density and both radial as well as transverse stress are positive throughout these charged stars configurations.
• The radial derivative of density and anisotropic pressures (dρ/dr, dP r /dr, dP t /dr) remains negative and for r = 0, these values vanish which confirm the density and anisotropic stress maximum value at core.
• All the energy conditions are well satisfied which show the realistic matter content.
• Both the radial and transverse sound speed remains within the bounds, which mean the causality condition is obeyed.
• All these stars are stable.
• Both radial and transverse EoS parameters lie in the range of 0 and 1.
• The isotropy remains positive throughout these charged stars.
• The distribution of charges increase from central to surface of stars.
• Electric field intensity is maximum at the surface of these stars.