Cosmological reconstruction and energy bounds in $f(R,R_{\alpha \beta}R^{\alpha\beta},\phi)$ gravity

We discuss the cosmological reconstruction of $f(R,R_{\alpha\beta}R^{\alpha\beta},\phi)$ (where $R$, $R_{\alpha\beta}R^{\alpha\beta}$ and $\phi$ represents the Ricci scalar, Ricci invariant and scalar field) corresponding to power law and de Sitter evolution in the framework of FRW universe model. We derive the energy conditions for this modified theory which seem to be more general and can be reduced to some known forms of these conditions in general relativity, $f(R)$ and $f(R,\phi)$ theories. We have presented the general constraints in terms of recent values of snap, jerk, deceleration and Hubble parameters. The energy bounds are analyzed for reconstructed as well as known models in this theory. Finally, the free parameters are analyzed comprehensively.


Introduction
In current cosmic picture dark energy (DE) is introduced as an effective characteristic which tends to accelerate the expansion in universe.Modified theories have achieved significant attention to explore the effect of cosmic acceleration [1].These models have been developed to distinguish the source of DE as modification to the Einstein Hilbert action.Some modified theories of gravity are f (R) gravity with Ricci scalar R [2], f (T ) gravity with torsion scalar T [3], Gauss-Bonnet gravity with G invariant [4], f (R, T ) gravity with T as the trace of stress-energy tensor [5], f (R, T , R µν )T µν [6] and f (R, G) gravity that contains both R and G [7] etc.The acceleration of the expanding universe can be explored by these theories through their comparable invariants.
To generalize Einsteins theory of general relativity (GR), there is a vast literature on relativistic theories that reduce to GR in the proper limitations.An especially attractive class of these generalizations are the fourth-order theories.These theories were initially considered by Eddington in early 1920's [8].Whatever the inspiration to examine the generalized fourth-order theories, it is necessary to understand their weak-field limit, and these limits confirm the increasing behavior of these theories in observational data.
Generally a fourth order theory of gravity is obtained by adding R ab R ab and R abcd R abcd in the standard Einstein Hilbert action [9,10].However, it is now established that we can ignore the R abcd R abcd term if we use the Gauss Bonnet theorem [11].About a half century back, Brans and Dicke (BD) [12] presented the scalar-tensor theory of gravitation, which is still popular and have received great interest in cosmological dynamics as a replacement to dark matter and dark energy.The motivation behind the BD theory was Mach idea [13] to present a varying gravitational constant in general relativity.Amongst the alternative theories to Einstein's gravity, the simplest and well known is Brans-Dicke theory.In this theory, the gravitational constant has been taken to be inversely proportional to the scalar field φ.The BD theory may be presented as a generalization of f (R) theory with f ′ (R) = F (R) = φR [2].
In modified theories, cosmological reconstruction is one of the important prospects in cosmology.In f (R) gravity, the reconstruction scheme has been used in different contexts to explain the conversion of matter dominated era to DE phase.This can ne examined by considering the known cosmic evolution and the field equations are used to calculate particular form of Lagrangian which can reproduce the given evolution background.In these theories the existence of exact power law solutions for FRW spacetime has been examined.In [14,15,16] people have reconstructed f (R, T ) gravity models by employing various cosmological scenarios.Nojiri et al. developed f (R) gravity models [17], which were further applied to f (R, G) and modified Gauss-Bonnet theories [18].To reconstruct f (R) gravity models, Carloni et al. [19] has established a new technique by using the cosmic parameters instead of using scale factor.
Energy conditions are necessary to study the singularity theorems moreover the theorems related to black hole thermodynamics.For example, the well known Hawking-Penrose singularity theorems [20] invoke the null energy condition (NEC) as well as strong energy condition (SEC).The violation of (SEC) allows to observe the accelerating expansion, and null energy conditions (NEC) are involved in proof of second law of black hole thermodynamics.
Energy conditions have been explored in different contexts like f (T ) theory [21,22], f (R) gravity [23] and f (G) theory [24], Brans-Dicke theory [25].Further the energy conditions of a very generalized second order scalar tensor gravity have been discussed by Sharif and Saira [26].Sharif and Zubair have examined these conditions for f (R, T ) gravity [14] and for f (R, T, R µν T µν ) gravity [27] which involves the nonminimal coupling between the Ricci tensor and energy-momentum tensor.Saira and Zubair [28] have discussed these conditions for F (T, T G ) having term T torsion invariant along with T G , equivalence of Gauss-Bonnet term and teleparallel.
In this paper we are interested to develop some cosmic models coherent with the recent observational data in the vicinity of generalized scalar tensor theories.We present the energy conditions in f (R, R αβ R αβ , φ) gravity utilizing FRW universe model with perfect fluid matter and developed some constraints on free parameters on reconstructed as well as well known models.The paper is arranged in the following pattern: In next section, we are providing a general introduction of f (R, R αβ R αβ , φ) gravity.In section 3 we have defined the basic expressions of energy conditions and then derive the energy conditions of f (R, R αβ R αβ , φ) gravity using deceleration, jerk and snap parameters.Section 4 is devoted to the reconstruction of models in f (R, R αβ R αβ , φ) gravity and energy bounds of these models and in section 5 we have derived the energy conditions of some known f (R, φ) models.In section 6, we sum up our conclusion.

Scalar Tensor fourth Order Gravity
The f (R, R αβ R αβ , φ) gravity has an interesting prospect among the more general scalar tensor theories and its action is of the form [29], where f is an unspecified function of the Ricci scalar, the curvature invariant and the scalar field denoted by R, R αβ R αβ ≡ Y and φ (where R αβ is the Ricci tensor).The L m is the matter Lagrangian density, ω is a generic function of the scalar field φ, g is the determinant of the metric tensor g µν .
In the metric approach, by varying the action (1) with respect to g µν the field equations are obtained as where = g µν ∇ µ ∇ ν and κ 2 ≡ 8πG.We consider the flat FRW universe model with a(t) as scale factor given by The gravitational field equations corresponding to perfect fluid as matter content, are given by The field equation ( 2) can be rearranged in the following form which is similar to the standard field equations in GR.Here T ef f µν , the effective energy-momentum tensor in f (R, Y, φ) gravity is defined as One can define the effective energy density and pressure of the form and

Energy Conditions
Energy conditions have an important role in GR, and also have useful applications in modified theories of gravity.In the context of GR, these constraints help to constrain the possible choices of matter contents.Four types of energy conditions are developed in GR by applying a geometrical result known as Raychaudhuri equation [20].These conditions are known as null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and dominant energy condition (DEC).In a spacetime manifold, the temporal evolution of expansion scalar is described as Raychaudhuri equation given by, where R µν , σ µν , ω µν are Ricci tensor, shear tensor and rotation, and the tangent vectors to timelike and null-like curves in the congruence are represented by u µ and k µ .The interesting aspect of gravity makes the congruence geodesic convergent and leads to the condition dθ dτ < 0. By ignoring the second -order terms and integrating, the Raychaudhuri equation implies that θ = −τ R µν u µ u ν and θ = −τ R µν k µ k ν .It further leads to the inequalities These inequalities can be written as a linear combination of energy-momentum tensor and its trace by the inversion of the gravitational field equations as follows: In case of perfect fluid with density ρ and pressure p, these inequalities gives NEC, WEC, SEC, and DEC defined by: In modified theories of gravity, assuming that the total matter contents act like perfect fluid, these conditions can be determined by interchanging ρ with ρ ef f and p with p ef f .Energy conditions for scalar tensor fourth order gravity are: WEC: SEC: DEC: Inequalities ( 14)-( 17) represent the null, weak, strong and dominant energy conditions in the context of f (R, Y, φ) gravity for FRW spacetime.We define the Ricci scalar and its derivatives in terms of deceleration, jerk and snap parameters as [30,31] (18) where and express Hubble parameter and its time derivatives in terms of these parameters as [27,28] Using the above definitions, the energy conditions ( 14)-( 17) can be rewritten as WEC: 4 Reconstruction of f (R, Y, φ) gravity In this section, we are presenting the reconstruction of f (R, Y, φ) gravity by using well-known cosmological solutions namely de-Sitter (dS) and power law cosmologies.

de-Sitter Universe Models
The dS solutions are very importance in cosmology to explain the current cosmic epoch.The dS model is described by the exponential scale factor, Hubble parameter and Ricci tensor as, In this reconstruction, we consider the matter source with constant EoS parameter w = p ρ so that Here we are using [32] Using these quantities along with Eqs.( 25) and (26) in Eq.( 4), we obtain This is a second order partial differential equation which can be converted in canonical form whose solution yields where α ′ i s are constants of integration and Introducing model (29) in the energy conditions ( 14)-( 17) it follows, Table 1: Validity regions of WEC and NEC for dS f (R, Y, φ) model.The inequalities ( 31)-( 34) depend on six parameters α 1 , α 2 , α 3 , β, m and t.In this approach, we fix two parameters and find the viable region by exploring the possible ranges of other parameters.We prefer to fix integration constants and show the results for WEC and NEC.Herein, we set the present day values of Hubble parameter, fractional energy density and cosmographic parameters as H 0 = 67.3,Ω m0 = 0.315 [33] q = −0.81,j = 2.16, s = −0.22,[14].The viability regions for all the possible cases for dS f (R, Y, φ) model are presented in Table 1.

Power Law Solutions
It would be very useful to discuss power solutions in this modified theory according to different phases of cosmic evolution.These solutions are helpful to explain all cosmic evolutions such as dark energy, matter and radiation dominated eras.We are discussing power law solutions for two models of f (R, Y, φ) gravity.The scale factor for this model is defined as [14,34] where n > 0. For decelerated universe we have 0 < n < 1, which leads to dust dominated (n = 2 3 ) or radiation dominated (n = 1 2 ) while n > 1 leads to accelerating picture of the universe.
• Power Law Solution independent of R Here, we are taking function f (Y, φ), inserting Eqs.( 26), ( 27) and (49) in Eq.( 4) we obtain whose solution results in following f (Y, φ) model where α ′ i s are constants of integration and Introducing (51) in the energy constraints ( 14)-( 17 • Power Law Solution independent of Y Now we are taking function f (R, φ), inserting Eq.( 26) along with Eqs.( 27) and (49) in Eq.( 4) yields Solving this we have, where α ′ i s are constants of integration and Inserting (54) in the energy conditions ( 14)-( 17

Energy Conditions for Some known Models
To present how these energy conditions apply limits on f (R, Y, φ) gravity, we have also considered some well-known functions in the following discussion.

f (R, φ) Models
Here, we present f (R, Y, φ) gravity models which does not involve variation with respect to Y and corresponds to f (R, φ) gravity.We present the energy constraints for the following models For these models we explore the energy constraints in the background of power law solutions with n > 1 favoring the current accelerated cosmic expansion.

Model-I
In [36], Myrzakulov et al. discussed the inflation in f (R, φ) theories by analyzing the spectral index and tensor-to-scalar ratio and found results in agreement with the recent observational data.In our paper, we have selected the following f (R, φ) model [36] where κ 3 is introduced for dimensional reasons and b is a dimensionless number of order unity.Introducing this model in the energy conditions ( 14)-( 17) along with Eqs.( 26), ( 27) and (49), we find the following constraints Here, we are left with four parameters b, β, n and t and we constrain these according to WEC and NEC.Starting with b ≥ 0, NEC is valid for n > 1 with β ≤ −1.5 whereas WEC is only valid for b = 0 with n > 1, β ≤ 0 and t ≥ 1.1.Moreover, for b < 0 with n > 1, NEC and WEC are valid for all values of β.In Figure 2, we show the plot of NEC for this model verses the parameters m, β and t by fixing n > 1.

Model-II
Here, we have formulated a specific model in this theory using the form f (R, φ) = Rf (φ).We have calculated f (φ) from Klein-Gordon equation by using ω(φ) = ω 0 φ m and φ = a(t) β given in [29], In this regard, we find the following expression where ω 0 and a 0 are constants.Using this model in the energy conditions ( 14)-( 17) along with Eqs.( 26), ( 27) and (49) we have energy conditions, We examine the NEC and WEC against the parameters β, n, m and t.We find that WEC can be satisfied for all values of m and β only if t ≥ 1.3 while the validity of NEC requires;

Model-III
In this case we present the energy constraints for the following model [37] where ξ is the coupling constant.Recently, this model has been employed to discuss the cosmological perturbations for non-minimally coupled scalar field dark energy in both metric and Palatini formalisms.The interaction has been analyzed depending on the coupling constant.Using this model in the energy conditions ( 14)-( 17) along with Eqs.( 26), ( 27) and (49) we get, We intend to discuss the NEC, WEC and constrain the parameters like β, ξ, n, m and t.Here, we develop three cases depending on choice of scalar field power m.Starting with m > 0 with n > where α is a constant with suitable dimensions.This gravitational action is very familiar in the text as it is able to reproduce inflation.Inserting this model in the energy conditions ( 14)-( 17) along with Eqs.( 26), ( 27) and (49) we have energy conditions, We are considering here NEC and WEC and check their validity for different values of β, α, n, m and t.Following the previous case we vary the coupling parameter α and set the other parameters for the validity of WEC and NEC.If α > 0 with n > 1, then WEC can be met in two regions namely, (β ≥ 0, m ≤ −1 with t ≥ 1) and (for all values of m with β ≤ −9 and t ≥ 6).Now taking α < 0 with n > 1, WEC is valid for all m with β ≤ 0, and NEC is valid if β ≤ −0.7 with m ≥ 0 and t ≥ 1 and for β ≥ 0.85 with m ≤ −1 and t ≥ 1. Taking α = 0 with n > 1, for β ≥ 0 NEC is valid for m ≤ −1.05 with t > 1.01 and for β ≤ 0 it is valid for m ≥ 0 with t ≥ 1. WEC is valid for all values of m with β ≤ 0.

Conclusion
Scalar tensor theories of gravity are very useful to discuss accelerated cosmic expansion and to predict the universe destiny.One of more general modified gravity is, f (R, R µν R µν , φ) which include the contraction of Ricci tensors Y = R µν R µν and scalar field φ.In this paper, we have applied the reconstruction programme to f (R, R µν R µν , φ).The action (1) in original and specific forms f (R, φ), f (Y, φ) is reconstructed for some well-known solutions in FRW background.The existence of dS solutions has been investigated in modified theories [39].Here, we have developed multiple dS solutions which can handy in explaining the differenr cosmic phenomena.In de-Sitter universe, we have constructed the more general case f (R, Y, φ) and establish f (R, φ) considering the function independent of Y and f (Y, φ) by taking function independent of R. The power law expansion history has also been reconstructed in this modified theory for both general as well as particular form of the action (1).These solutions explain the matter/radiation dominated phase that connects with the accelerating epoch.The f (R, R µν R µν , φ) model can also be reconstructed which will reproduce the crossing of phantom divide exhibiting the superaccelerated expansion of the universe.
Lagrangian of f (R, R µν R µν , φ) gravity is more comprehensive implying that different functional forms of f can be suggested.The versatility in Lagrangian raises the question how to constrain such theory on physical grounds.In this paper, we have developed some constraints on general as well as specific forms of f (R, T, R µν T µν ) gravity by examining the respective energy conditions.The energy conditions are also developed in terms of deceleration q, jerk j, and snap s parameters.To illustrate how these conditions can constrain the f (R, R µν R µν , φ) gravity, we have explored the free parameters in reconstructed and well known models.In general dS case f (R, Y, φ) energy conditions are depending on six parameters β, m, t and α i 's where i = 1, 2, 3.In this procedure we have fixed the α i 's and observe the feasible region by varying the other parameters.
In dS f (R, φ) and f (Y, φ) models, the NEC depend on five parameters α 1 , α 2 , β, m & t and WEC depend only on three parameters α 1 , α 2 & t.In case of NEC we have fixed α 1 and α 2 and find the constraints on the other parameters.In WEC we are changing α 1 and explore the possible ranges on α 2 and t.For power law f (R, φ) and f (Y, φ) models, functions depend on six parameters α 1 , α 2 , β, m, n and t.In power law case we have n > 1, and varying α 1 , α 2 we have analyzed the viable constraints on β, m and t.Further more we have considered three particular forms of f (R, Y, φ) gravity taking function independent of Y , i.e., f (R, φ), Rf (φ), φf (R) from which we can deeply understand the applications of energy conditions.Model-I is a function of four parameters b, β, n and t, we have checked the validity of NEC and WEC by varying b.Model-II is depending on β, m, n and t, for n > 1 we have explored the viability of other parameters.Next in model-III we have five parameters β, ξ, n, m and t, for n > 1 we have find the feasible constraints on other parameters by fixing m.In model-IV the conditions are depending on five parameters β, α, n, m and t. we have n > 1 and varying β we examined the possible regions for the other parameters.
Finally, we generally discuss the variations of parameters involved in power law solutions and scalar field coupling function, denoted by m and n respectively.In de-Sitter models we have examined that the more general case f (R, Y, φ) is more effective as compared to f (R, φ) and f (Y, φ) models since in general case one can specify the parameters in more comprehensive way.In all cases of de-Sitter models, WEC is valid for all m and NEC is valid if (m ≥ 1 & m ≤ −5).In power law case f (R, φ), for both NEC and WEC n has a fixed value n = 3 and m has variations (m ≥ 0 & m ≤ −5.5).For f (Y, φ) case we have (n ≥ 2.3 with m ≥ 4, m ≤ −1) for WEC and for NEC we have n ≥ 2 with (m ≥ 0, m ≤ −4).In other known f (R, φ) models, the validity of these conditions require n > 1 with (m ≥ 0, m ≤ −2).

Figure 1 :
Figure 1: Variation of energy constraints for dS f (R, Y, φ) model with α 1 > 0 and α 2 > 0. In left plot we set m = −10 (one can set any value since results are valid for all m) and show the variation for all α 3 and β.Right plot shows the validity regions of NEC for α 3 = 0.
) we can find the energy conditions for this model.Here we are discussing the validity of NEC and WEC for different values of β, m and t by fixing n and α i 's where i = 1, 2. Starting with α 1 and α 2 both as positive, WEC is valid for all values of m and β = 0 with n = 3 while NEC is valid for n = 3 with β ≤ −2, m ≥ 0 and t ≥ 1.03.Now taking α 1 as negative and α 2 as positive, WEC is valid for m ≥ 0 with n = 3, β ≥ 2.6 & t ≥ 0.65 and for m ≤ −2 it is valid for n = 3 with β ≥ 22.5.For this choice of α i 's NEC is valid for n > 1 with β > 1, m ≤ −5 & t ≥ 1.05.Next we are taking α 1 as positive and α 2 as negative, here WEC is valid for n = 3 with (i) β ≥ 2.7 , m ≥ 0 & t ≥ 0.65 and with (ii) β ≤ −2, m ≤ −5.5 and t ≥ 0.65.NEC is valid for n = 3 and for all values of m and β except β = 0.If we take α 1 and α 2 both as negative, both WEC and NEC are valid for all values of m and β = 0 with n = 3.

Figure 2 :
Figure 2: Plot of NEC for Model-II versus the parameters m, β and t with n = 1.1.
), one can find the inequalities for this model depend on six parameters α 1 , α 2 , β, m, n and t.We will only discuss the WEC and NEC for different values of β and m by fixing n and α i 's where i = 1, 2. Starting with α 1 and α 2 both as positive, WEC is valid for n > 1 with β ≤ −0.1, m ≥ 0, t ≥ 1.1 and NEC is valid for all values of m with n > 1 & β ≥ 0. Now taking α 1 as negative and α 2 as positive, WEC is valid for 1 < n ≤ 1.8 with β ≤ −3, m ≥ 0 and for n ≥ 2.3 withβ ≥ 2 & m ≤ −1.Similarly, NEC is valid for all values of m with n > 1, β ≤ −0.12 and t ≥ 1.01.Now taking α 1 as positive and α 2 as negative, WEC is valid for n ≥ 1.7 with β ≥ 0.1 & m ≤ −10 and NEC is valid for all values of m with n > 1, β ≥ 0 & t ≥ 1.07.Taking α 1 and α 2 both as negative, WEC is valid for 1 < n ≤ 1.9 with β > 0, m ≤ −6.5 & t > 1 and for n ≥ 2 WEC is valid for β ≤ 0 with m ≥ 4. In this case NEC is valid for 1 < n ≤ 1.5 with β ≥ 0, m ≤ −2.6 & t ≥ 1.9 and for n ≥ 2 it is valid for β < 0 with m ≥ 0, t ≥ 1.05 and for β ≥ 0 with m ≤ −4, t ≥ 1.08.