Radiating Isotropic Homogeneous f(R,G) Gravity Model with Quark and Strange Quark Matter

Present analysis devoted to the dynamical investigation of radiating quark and strange quark matter in the framework of f(R,G) theory of gravity towards spatially homogeneous isotropic Friedman-Robertson-Walker line element filled with isotropic fluid. We govern the features of the derived cosmological model by considering the well known relation between Hubble parameter H and scale factor. Some kinematical and physical parameters for quark and strange quark matter towards isotropic model is discussed in detail. PACS numbers: 95.36.+x, 04.50.kd, 98.80.Jk.


INTRODUCTION
The present universe is spatially flat and at late-time it has an accelerated expansion (the strength of this acceleration is a remarkable question in recent year) confirmed by the recent observations like SNe-Ia Supernova, Cosmic Microwave Background Radiation (CMBR), Large Scale Structure (LSS) and Wilkinson Microwave Anisotropy Probe (WMAP). The recommendations that have been put forward to explain this observed phenomenon can be classified into one either an exotic component with negative pressure so called dark energy which introduce into General Theory of Relativity (GTR) or second the change in the gravity law through the modification of action in GTR called Modified Gravity (MG). The MG has become one of the most popular candidates to understand the idea of dark energy. Various modification in the action of GTR is present, out of which one where the Hilbert-Einstein action is generalized to a general function of the Ricci scalar R called f (R) theory of gravity, these theory not only are able to simulator the behavior of the cosmological constant but also can reproduce the complete cosmological history (see Ref. ; Elizalde and Sez-Gmez (2009);Faraoni (2008); Cruz-Dombriz and Dobado (2006)). Appleby et al. (2010) studied f (R) MG models by showing how the problem of a weak curvature singularity with can get up generically in viable f (R) model of present dark energy indicating an internal incompleteness be cured by adding a quadratic correction with a sufficiently small coefficient to the f (R) function at large curvatures. Felice and Tsujikawa (2010) study the f (R) theory to cosmology and gravity such as inflation,local gravity constraints, dark energy, cosmological perturbations and spherically symmetric solutions in weak and strong gravitational backgrounds. Bamba et al. (2012) presented the ΛCDM cosmology, Little Rip and Pseudo-Rip universes, the quintessence cosmolo-gies and phantom with Type I, II, III and IV finite-time future singularities and non-singular dark energy universes as well as equivalent class of dark energy models which includes f (R) gravity. Azadi et al. (2008), Miranda et al. (2009), Sharif andYousaf (2014), Bhoyar et al. (2016), Chirde and Shekh (2016;2017) along with many authors have discussed some features of f (R) gravity using different space-time.
Another is f (R, T ) gravity presented by Harko et al. (2011) with several aspects of this theory including FRW dust universe. Sharif and Zubair (2012), Chaubey and Shukla (2013), Sahoo et al. (2014), Shekh (2015, 2016a), Bhoyar et al. (2015) are some of the authors who have analyzed the same gravity for different context of use.. Next, the gravity which is obtained by adding a function of Gauss Bonnet invariant invariant G = R 2 − 4R µν R µν + R µνσρ R µνσρ (R µν is the Ricci tensor and R µνσρ is the Riemann tensor) in the Einstein-Hilbert action, the gravity so called f (G) gravity ). The motivation for this theory comes from string theory by low energy effective scale which is consistent with the global universe evolution and local gravity restrictions (Cognola et al. 2007;Felice and Tsujikawa (2009). The detailed analysis of f (G) gravity is presented in ref. Easson (2005), Nojiri et al. (2005Nojiri et al. ( , 2007; Carter and Neupane (2006); Koivisto and Mota (2007); Sadjadi (2014); Sharif and Fatima (2014), Sadeghi et al. In 1984, Witten (1984) validated that at a critical temperature T c = 100 − 200 MeV which could have led to the establishment of quarks at larger density than normal nuclear matter density. Strange quark matter is settled with an equation of state (EoS) p = (ρ−4Bc) 3 , based on the phenomenological bag model of quark matter where B c is known as Bag constant. Nowadays, the study of quark and strange quark matter is an exciting area of exploration. Quark matter is considered in strange stars (Drake et al. (2002)), even as small pieces of strange matter (Weber (2005)) or to exist at the centre of neutron stars (Perez-Garcia (2010)). This paper is organized as follows: In sec. 2 some basic concepts of f (R, G) gravity with the source containing quark and strange quark matter. Metric, components of field equation of f (R, G) gravity with their solution is presented in sec. 3. Sec. 4 contains some physical and kinematical parameters of quark and strange quark matter using matter dominated power law expansion. Finally, Sec. 5 deals with some concluding remarks.

BASICS OF f (R, G) GRAVITY WITH STRANGE QUARK MATTER
The most general action for f (R, G) gravity is given as (Dombriz and Gmez 2013), where S M (g ij , ϕ) is the matter action, R is Ricci scalar and G is Gauss-Bonnet invariant defined by, where, the notations R αβ and R αβσν are occupied for the Ricci and Riemann tensors respectively.
Variation of the standard action (1) with respect to the metric gives us the following gravitational field equation, where, Here, ∇ µ represents the covariant derivative and f R ≡ ∂f (R,G) gives the partial derivatives of f (R, G) with respect to R and G respectively.
In this work, we obtain the solution of field equations and the behavior of the universe using some kinematical and physical quantities for the f (R, G) gravity model i.e.
where f 0 > 0 be any constant.
Without loss of generality take in to account that f 0 = 1, above f (R, G) gravity model becomes, For the values of constant m, two types of gravity models are recovered: Let us consider the matter content energy-momentum tensor T ν µ for strange quark matter is of the form, For strange quark matter, the linear equation of state (EoS) is given by, here ω is a constant and ρ 0 is the energy density at zero pressure. If the model is rediating towards EoS ω = 1/3 and ρ 0 = 4B c the above linear equation of state is reduced to the following EoS for strange quark matter in the bag model.

METRIC, FIELD EQUATION AND THEIR SOLUTIONS
We consider the spatially homogeneous and isotropic Friedman-Robertson-Walker (FRW) line element in the form, ds 2 = dt 2 − a 2 (t) dr 2 1 − kr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 , where a be the scale factor of the universe.
In this work, we deliberate on the flat universe taken after k = 0 with infinite radius. In the co-moving co-ordinate system, the equation of motion (4) for the spatially homogeneous and isotropic Friedman-Robertson-Walker (FRW) line element (10) with the fluid of stress-energy tensor (9) can be written as, 2f Rȧ The overhead dot represents the differentiation with respect to cosmic time t measure in Gyr. Now, we consider some of the kinematical parameters for the FRW cosmological model that are important in cosmological observations.
The spatial volume, The generalized mean Hubble parameter, The expansion scalar, Deceleration parameter,

MATTER DOMINATED POWER LAW EXPANSION
After the discovery of the late time acceleration of the universe, many authors who have used constant deceleration parameter to obtain cosmological models in the context of different fluids as well as dark energy in general theory of relativity and also in modified theories of gravitation. The sign of deceleration parameter (q) indicates whether the model accelerate or decelerate. The negative sign of q indicates acceleration whereas positive value stands for deceleration. Also, recent observations of SNe-Ia expose that the present universe is accelerating for H > 0 and q < 0.
In this article, we use the well known relation between Hubble parameter H and scale factor a (sing et al. (2008a, 2008b)) as, where b > 0 and η ≥ 0 are constants. The above relation yields a constant value of deceleration parameter.
Using the definition of Hubble parameter (14), we can get, Integration of equation (19) yields, provided γ = 1 η = 1 1+q where q = −1 and a 1 = ηb = 0 , b 1 are constants of integration. From the equation (20), it is observed that the scale factor of the model is the functions of cosmic time, which increase with time at q > −1, decreases with time at q < −1, and does not exist at q = −1.

Kinematical parameters of isotropic FRW line element
Expression of deceleration parameter given in equation (16) together with the value of scale factor provided in equation (20). The value of deceleration parameter is obtained as, Equation (21) designates that the deceleration parameter comes out to be independent of cosmic time. The sign of q indicates whether the universe accelerate or not. The recent observations like CMB and WMAP, as well as with the high redshirts of type-Ia supernova expected the sign of q become negative some where in range −1 ≤ q ≤ 0 (q = −0.45) which correspond to the standard accelerating behavior of the universe. Hence, our model resumbles with the value got from recent observations for η = 0.55 or γ = 2.2.
The spatial volume and scale factor both are obtained as, We observed that the spatial volume along with scale factor of the model starts with constant value (big-bang) at t → 0 and with the increase of cosmic time it always expands. Thus, inflation is possible in this model. This shows that the universe starts evolving with zero volume and expands with cosmic time t. The behavior of spatial volume of the model versus cosmic time with an appropriate choice of constant is clearly shown in Fig.1 The Hubble parameter is obtained as, The scalar expansion is obtained as, Hubble parameter and scalar expansion both are constant. These are diverse when the model is in acceleration phase and approaches to zero monotonically at t → ∞. This suggested that at initial stage of the universe, the expansion of the model is much more faster and then slow down for later cosmic time this shows that the evolution of the universe starts with finite constant rate and with the expansion it declines.

Physical parameters of isotropic FRW line element
In this section we discussed the physical parameters such as pressure for strange quark matte, energy density strange for quark matter, pressure for quark matter and energy density for quark matter.
Strange quark matter in isotropic FRW line element: From the equations (11) and (12) by applying scale factor provided in equation (20) with the help of linear EoS (9) for ω = 1/3, pressure and the energy density for strange quark matter can be extracted.
Pressure for strange quark matter is defined as, By considering V = (a 1 t + b 1 ) 3γ and rewriting the above equation we can obtain the equation of state P − V for strange quark matter as, Energy density for strange quark matter is given by, Equations (26) and (27) represents strange quark pressure and the energy density of the model. We conclude that pressure and the energy density for strange quark matter both are function of cosmic time with inverse relation.
Initially when the model start to expand at t = 0 both are approaches to some spacific constant value but with the expansion when t > 0 these are decreases and at infinitely large expansion when t → ∞, then p, ρ → 0 as shown in the plot of Fig. 3 and Fig.4. Hence, pressure and the energy density for strange quark matter initially observed in constant amount while at infinite expansion of the model it declines and diverge to zero. Quark matter in isotropic FRW line element: Pressure for quark matter is obtained as, By considering V = (a 1 t + b 1 ) 3γ and rewriting the above equation we can obtain the equation of state P − V for quark matter as, Energy density for quark matter can be expressed as, Equation (28) and (30) gives the expression of pressure and energy density for quark matter. The behavior of pressure and energy density for quark matter is clearly depicted in Fig.5 and Fig.6 respectively with the appropriate choice of constants and observed that both are same as that of strange quark matter, except that it is shifted by the bag constant as shown in the right side of the equations (28) and (30). The right side view of energy density and pressure for quark and strange quark matter are identical except for the additional bag constant. For energy density of strange quark we add the bag constant whereas for pressure we take away (subtract) it. From equations (28) and (30), one can notice that, ρ q → B c and p q → −B c when t → ∞. . Hence our results resembles with the work investigated by those of Sahoo at al. (2018), Chirde and Shekh (2018).
Using equations (26) and (29), the P − V diagrams for the strange quark and quark matter are drawn in Fig.7 and Fig.8. These figures show that in both cases, as isotherms diagrams of an ideal gas, the pressure decreases with increasing volume.

CONCLUDING REMARK
The spatial volume and scale factor of the model both are starts with constant value, with the increase of cosmic time also increases and always expands which indicates that tha inflation is possible in this our derived model. This shows that the universe starts evolving with constant volume and expands with cosmic time. The Hubble parameter and scalar expansion in the derived model both are the functions of cosmic time and decreases as cosmic time increases and approaches to zero at later infinite cosmic time. At an initial epoch when the model start to expand, the Hubble parameter and scalar expansion both are constant and these are diverse when the model is in acceleration phase and approaches to zero monotonically at infinite expansion. This suggested that at initial stage of the universe, the expansion of the model is much more faster and then slow down for later cosmic time this shows that the evolution of the universe starts with finite constant rate and with the expansion it declines.
The pressure and energy density for quark and strange quark matter both are same as except that of strange quark matter shifted by the bag constant and observed that quark pressure and quark energy density both are approches to Bag constant with positive and negative sign respectively i.e. ρ q → B c and p q → −B c when t → ∞. The value of