Physical Attributes of Anisotropic Compact Stars in $f(R,G)$ Gravity
Modified gravity is one of the potential candidates to explain the accelerated expansion of the universe. Current study highlights the materialization of anisotropic compact stars in the context of theory of gravity. In particular, to gain insight in the physical behavior of three stars namely, Her , SAX J - and 4U -, energy density, and radial and tangential pressures are calculated. The gravity model is split into a Starobinsky like model and a power law model. The main feature of the work is a -dimensional graphical analysis in which, anisotropic measurements, energy conditions and stability attributes of these stars are discussed. It is shown that all three stars behave as usual for positive values of model parameter .
Keywords: Gravity, Spherically Symmetric Spacetime, Compact Stars.
PACS: 04.50.Kd, 98.80.-k.
The spatial behavior of constituents of the universe is a complex phenomena. Visual observation of the remote universe enlightens us about its continuous expansion, and accentuates to find alternative rationale which may be helpful to explain the phenomena of expansion of the universe . So, these alternative models are known currently as modified theories of general relativity (GR), some of which are , , , and theories of gravity, where , , and denotes Ricci scalar, the Gauss-Bonnet invariant and the trace of the energy momentum tensor respectively. The theory of GR explains the cosmological phenomena in weak field regimes while some modifications are required to address the strong fields in the scenario of continuous expansion of universe. The same thought was provoked and one of the modified theories, gravity, was proposed in by Buchdahl . Hydrostatic equilibrium and stellar structure in gravity have been investigated by considering the Lan´e-Emden equation . Some finite time future singularities in modified gravity were described with their solution using the addition of higher derivative gravitational invariants . Harko  presented modified theory in by involving both matter and curvature terms. Furthermore, the next modification to the GR was Gauss-Bonnet gravity , also known as Einstein-Gauss-Bonnet gravity, which includes the Gauss-Bonnet term.
where and are Ricci tensor and the Riemann tensor respectively.
gravity  is the further generalization of Gauss-Bonnet gravity. It has been shown that this kind of generalization may naturally approach an operative cosmological constant, quintessence and phantom cosmic acceleration . Recently, another alternative theory with the title of gravity has been proposed by Sharif and Ikram . They studied the energy conditions for the well-known Friedmann -Robertson- Walker space time. So far some interesting work has been done in this modified theory -.
Similarly, gravity is also a modified theory, which has attracted the attention of theoretical physicists gain insight of acceleration and regular thermal expansion of late-time universe by assuming an alternative of cosmological constant, where is a function of the torsion scalar . An endeavor was made by combining and in a bivariate function -. The function provides a foundation for the double inflationary scenario . It has been shown that theory is consistent with the observational data [24, 25]. Besides its stability, this theory is perfectly suitable to explain the accelerating waves of the celestial bodies as well as the phantom divide line crossing and transition from acceleration to deceleration phases. An important feature of gravity is that it reduces the risk of ghost contributions and gravitational action is regularized because of the term [26, 27]. Thus, modified theories of gravity seem interesting in explaining the universe in different cosmological contexts .
Study of compact stars has always been a strong topic of research -. Different properties like mass, radius and moment of inertia of neutron stars are studied and comparison has been developed with GR and alternative theories of gravity . Some investigations of the structure of slowly rotating neutron stars in gravity have been done using two different hadronic parameters and a strange matter equation of state (EOS) parameter . The mass-ratio for compact objects in the presence of cosmological constant has been derived . In this article, we are interested in studying the compact stars and in examining their physical behavior using standard models of spherically symmetric space time by finding exact solution of field equations. We study the stability of anisotropic compact stars using generalized modified Gauss-Bonnet gravity. The arrangement of this paper is as follows: In the second section, we find expressions for energy density, radial pressure and tangential pressure for anisotropic matter source using spherically symmetric metric in gravity. Section 3 comprises the analysis of behavior of EOS parameters, anisotropy measure, the matching conditions, the energy bounds, Tolman Oppenheimer Volkoff (TOV) equation and stability analysis of three compact stars namely, Her , SAX J - and 4U -. The last section ends with the conclusion and summery of the paper.
2 Anisotropic Matter Configuration in Gravity
We start with the action of modified Gauss-Bonnet gravity 
where and are partial derivatives with respect to and respectively,
and describes the ordinary matter. The most general spherically symmetric space time is 
The energy-momentum tensor in case of anisotropic fluid is given by
where , are four velocities. Radial pressure and tangential pressures are and respectively, and the energy density is denoted by . Using equations (5) and (6) in field equations (3), and after some manipulations we obtain,
The set of three equations (7)-(9) involves five unknown functions . Moreover, the equations are too much complicated and highly non-linear due to the involved bivariate function . So following Krori and Barua , we choose , , , , and are constants. These constants will be determined by using some physical assumptions. For the present analysis, we propose . We further consider the Starobinsky like model , where is an arbitrary constant and we take , . Also , and a prime denotes the derivatives with respect to the radial coordinate. Using these assumptions, Eqs. (7)-(9) take the form
3 Physical Analysis and Graphical Representation
This section presents the physical properties of the solutions regarding EOS, anisotropic behavior, energy conditions, TOV and matching conditions along with stability analysis of three different compact stars, Her , SAX J - and U -. Many EOS parameters have been considered in literature in different cosmological contexts. Staykov et al.  used two different hadronic parameters and a strange matter EOS parameter to study the structure of rotating neutron stars. Quadratic EOS parameters have been used to study the properties of compact stars [46, 47]. For the sake of simplicity, in this work we assume linear EOS :
It is important to note here that the EOS parameters are dependent on radius rather than a constant quantity as in the ordinary matter distribution. The non constant behavior is due to the usual matter and exotic matter contributions. Figs. and show the variation of EOS parameters and respectively. It is obvious from the figures that effective EOS in our model are the same as in normal matter distribution , i.e.
3.1 Anisotropy Measure of the Compact Stars
The variation of energy density , radial pressure and tangential pressure can be observed in Figs. -. The behavior of and in Figs. and reveals that and . It shows that with the increase in radius of the compact star, both energy density and radial pressure decrease. We analyze the variation of and at the center of the compact star and found that
Equation (15) gives the maximum value of and at center . The anisotropy measurement has been shown graphically in Fig. . The anisotropy measurement is directed outward when which results and directed inward when which results . It is depicted in Fig. that for the large values of , for all stars implying that the anisotropic force permits the construction of great massive configurations. It is worthy to mention that anisotropy measurement vanishes at the center of the star.
3.2 The Matching Conditions with Schwarzchild Exterior metric
The interior metric of the boundary surface remains same in the interior and exterior geometry of the compact star. It justifies the continuity of the metric components for the boundary surface of the star. Many choices for the matching conditions are possible and one can consider the vacuum outside a general spherically symmetric space time with suitable boundary conditions . For the present analysis, we consider Schwarzchild solution for describing the exterior geometry. Many authors have considered Schwarzchild solution for this purpose giving some interesting results -. Therefore, the exterior metric given by Schwarzchild is
The intrinsic metric (5) for the smooth match at the boundary surface with Schwarzchild exterior metric produces,
where interior solutions and exterior solutions are represented by (-) and (+). By the matching of interior and exterior metrics, we obtain
Using the approximate values of and for the compact stars under observation, the constants and are given in the following table 
3.3 The Energy Bounds
The energy bounds have gained much importance in the discussion of some important issues in cosmology. In fact, one can investigate the validity of second law of black hole thermodynamics and Hawking-Penrose singularity theorems using energy conditions . Many interesting results have been reported in cosmology using the energy bounds -. These energy conditions are defined as