#### Abstract

We have studied cosmological dynamics in gravity theory via cosmographic parameters. We have changed variables of field equations from time to red-shift and solved the achieved differential equation analytically for . Then we have used Taylor expansion to find general form of function around the present day value of scalar curvature. By introducing we would simplify our calculations; if we consider as a given function we would restrict our answers of . In this paper we offer a linear form of which leads us to a specific function, where is a constant which depends on the present day value of deceleration parameter. As an example, using Taylor expansion coefficients, we have compared our analytically calculated function with reconstructed function for Dark Energy models. To reconstruct action for Dark Energy models, we have used corresponding of each Dark Energy model for calculating Taylor expansion coefficients. As our function is linear, the Taylor expansion coefficients would be a function of present day value of deceleration parameter.

#### 1. Introduction

Recent data indicates that the Universe is in an accelerating expansion phase [1–8]. Being responsible for this expansion, several approaches such as positive cosmological constant, Dark Energy, and modified gravity have been introduced. The positive cosmological constant would be defined either geometrically as modifying the left hand side of Einstein equations or as a kinematic term on the right hand side with the equation of state parameter ; however, the fine tuning problem causes some difficulties [9–16]. There are various scalar field models of Dark Energy [16–33]. These models indeed are the outcomes of modifying the right hand side of the Einstein equations, , by considering a source term with an equation of state parameter which is recognized as Dark Energy. In modified gravity, the right hand side is left unchanged and we modify the left hand side of the Einstein field equations, so we have a large class of theories of gravity [34–42]. Particularly, we are interested in fourth order theories [43–55] based on replacing the scalar curvature in the Hilbert-Einstein action with a generic analytic function which should be reconstructed starting from data and physically motivated issues. These models fit both the cosmological data and Solar System constraints in several physically interesting cases [56–65]. Vacuum solutions of gravity theories is one of interesting subjects which are obtained for constant Ricci scalar [66–72], allowing derivation of non constant curvature scalar vacuum solutions. In this paper, following our other work [73], we have considered vacuum solutions of gravity in a different way which does not assume constant scalar curvature to obtain vacuum solutions.

Vacuum solutions of modified gravity explain the late time phase transition of cosmological parameters like deceleration parameter just by pure geometry without the need for dark energy companion of the Universe. The pioneering works on reconstruction of modified action through inverse method are done in [66–72]. They developed a general scheme for cosmological reconstruction of modified gravity in terms of red-shift without using an auxiliary scalar in intermediate calculations. Using this method, it is possible to construct the specific modified gravity which contains any requested FRW cosmology. A number of gravity examples are constructed where the following background evolutions may be realized: LCDM epoch, deceleration with subsequent transition to effective phantom super-acceleration leading to Big Rip singularity, deceleration with transition to transit phantom phase without future singularity, and oscillating Universe. It is important that all these cosmologies may be realized only by modified gravity without the use of any dark components. Then we have tried to reconstruct an appropriate action for the modified gravity through the semi-inverse solution method. We do not assume any FRW cosmology to reconstruct its related action [63–81]. Indeed, as curvature scalar itself is a function of red-shift, by introducing a linear correction which is also a function of , we can reconstruct Taylor expansion of action around present time value of red-shift . For restricting Taylor expansion, we would study the variation of its coefficients from GR.

Capozziello et al. has also studied Taylor expansion of function in a model independent way where he has shown how it is possible to relate the cosmographic parameters (i.e., the deceleration , the jerk , the snap , and the lerk parameters) to the present-day values of and its derivatives (with , 2, 3), thus offering a new tool to constrain such higher order models. Their analysis thus offers the possibility to relate the model independent results coming from cosmography to the theoretically motivated assumptions of cosmology [75]. But here we have added a new condition to the question which has limited our answers to a certain form of the function.

In Section 2, we have a briefer review of modified field equations and introduce our model as a linear function of Red-Shift; we have also tried to reconstruct the function model independently. In Section 3, we calculate the Taylor expansion of function via . In Section 4, we have discussed influence of Taylor expansion coefficients in compare with the first term of expansion. In Section 5, we have reconstructed Taylor expansion for several Dark Energy models and compared the result with our function which has been calculated in Section 3.

#### 2. Modified Field Equations

The action of modified theory of gravity is given by where is the matter action such as radiation, baryonic matter, dark matter, and so on which we do not consider in field equation. In this essay, we consider the flat Friedmann- Robertson-Walker, (FRW) background as so that the gravitational field equations for gravity are provided by the following form: where the over-dot denotes a derivative with respect to , and the prime denotes a derivative with respect to , is the scale factor, and is the Hubble parameter. Eliminating between (3) and (4) results This can be changed in the form of where . Equation (6) is a second order differential equation of with respect to time, in which both of and are undefined. The usual method to solve (6) is based on definition of . Changing the variable of the above equation from to a new variable like , the number of e-folding, was done in [70–72]. The variable is related to the red-shift by . The cosmological dynamic equation is solved by definition of Hubble parameter as a function of in a general form, then the equation is rewritten by redefinition of the variable from the number of e-folding to the Ricci scalar and solved with respect to the Ricci scalar. Thus they demonstrated that modified gravity may describe the LCDM epoch without any need for introducing the effective cosmological constant, non-phantom matter, or phantom matter. In this work we would like to replace the variable of (6) by red-shift, , directly to find the function around via Taylor expansion. Each red-shift has an associated cosmic time (the time-when objects are observed with red-shift emitting light), so we can replace all the differentials with respect to by via where we use , and we consider , in the present time. Now, we can replace the variable of (6) from to by using (7) and obtain a first order differential equation for with respect to as where depends on the definition of as Now we may solve Hubble parameter that depends on the definition of [63, 64, 73]. In this paper we consider as a linear function of : To avoid the Dolgov-Kawasaki instability [82], the second order derivative of action with respect to Ricci scalar should be positive for different values of . Solving (8), using linear correction equation (10), the Hubble parameter has been found as in which . Using Ricci scalar definition, , the scalar curvature would be in which Eliminating between (10) and (12) leads us to reach In the case of or , we would like to recover de’Sitter solution. Using (12) we obtain and then which is equal to the GR or de’Sitter model. Calculating by integrating of (14) with respect to , we reach where is the constant of integration. Since in the absence of correction term, , we should recover de’Sitter model , we compare the value of de’Sitter action with the action of (15) when or . Since the scalar curvature of de’Sitter Universe is a constant, then we can determine the value of in the presence of constant curvature. If we solve the standard algebraic equation of gravity as we can find . Then the value of de’Sitter action in the case of is which is equal to . For the case of our obtained action from (15), we have when . Since we would like to have the same solution for de’Sitter model and our obtained action in present of we force constant of integration, , to be equal to zero [73]. One of main late time cosmological constraint which is tested for redshift-based correction of gravity is SNeIa distance module with respect to redshift constraint. The result of this comparison is shown in Figure 3 [73].

#### 3. Taylor Expansion of via General Function of

As we are interested in finding appropriate action without solving field equation, here we have used Taylor expansion of function around , However, as we are using this method, our reconstructed is acceptable just around present day value of curvature scalar: and as , coefficients of Taylor expansion to the third order would be find as Via (7), (8), and definition of Ricci scalar, , we can rewrite as a function of : We can normalize to : Finding the first term of Taylor expansion, we use (3) so it also should be written with respect to : where is the deceleration parameter that after replacing all the differentials with respect to by via (7) would be as or Finally the constant term of Taylor expansion would be obtained by substituting in (20). Using the linear modification , derivatives for this function in would be as Hence, using (16) Taylor expansion coefficients would be

#### 4. Order of Taylor Expansion

In this section we keep Taylor expansion of action up to the third order term and we ignore higher order modifications. Therefore, Taylor expansion coefficients have been studied as a function with respect to present day value of deceleration parameter. For linear correction, , would be obtained by (22) as Therefore Rewriting Taylor expansion coefficients with respect to , via (26) and then introducing function [75], in which is the order of differentiation, we have: From Figure 1 it is obvious that for , the Taylor expansion coefficients have only slight effects.

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#### 5. Reconstruction of for Dark Energy Model

In this section we have related Dark Energy models to cosmographic parameters by using Taylor expansion and then compared the result with the function calculated in Section 3. For reconstructing for Dark Energy models we need to use corresponding s from Table 1, as follows:(1)Using we have calculated for each Dark Energy model.(2)Red shift must be fixed to zero.(3)Using (28) to calculate for each Dark Energy model, and replacing it in (23) and (26), we complete the Taylor expansion.

In this paper we have used GCG, LCDM, and Quiess models, since their present value of Ricci scalar is approximately the same . As errors in Dark Energy model parameters cannot be ignored, we would reach to a group of possible function for each model, which has been demonstrated as a gray band in Figure 2. For comparing results, we have plotted analytically calculated action from (15), with these diagrams, where we have replaced by that of Dark Energy according to the values of Table 2. Seeing that, for a linear correction equation (10), both analytically calculated (15) and reconstructed for Dark Energy models are in good agreement in neighboring with the present day value of curvature scalar.

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#### 6. Conclusions

Here we presented a model independent approach to calculate by relating it to the present day value of cosmographic parameters. Since we had a linear form of , the s values made us using Taylor expansion up to the third order. In higher order corrections it might be possible to keep higher terms of Taylor expansion which depend on value of s for these corrections and then we would discuss about validity of these terms in compare with desirable conditions. The Taylor expansion based functions are viable in a small range about a certain value of Ricci scalar which itself is related to cosmographic parameters; since only present day value of these parameters are derived from data, we cannot discuss behavior of Universe at high red shifts.