Reconstruction of Scalar Potentials in f(R,R_{\alpha\beta}R^{\alpha\beta},\phi) theory of gravity

Reconstruction of Scalar Potentials in theory of gravity

M. Zubair mzubairkk@gmail.com; drmzubair@ciitlahore.edu.pk Department of Mathematics, COMSATS Institute of Information Technology Lahore, Pakistan    Farzana Kousar farzana.kouser83@gmail.com Department of Mathematics, COMSATS Institute of Information Technology Lahore, Pakistan    Saira Waheed swaheed@pmu.edu.sa Prince Mohammad Bin Fahd University, Al Khobar, 31952 Kingdom of Saudi Arabia.
July 16, 2019
Abstract

In this paper, we explore the nature of scalar field potential in gravity using a well-motivated reconstruction scheme for flat FRW geometry. The beauty of this scheme lies in the assumption that the Hubble parameter can be expressed in terms of scalar field and vice versa. Firstly, we develop field equations in this gravity and present some general explicit forms of scalar field potential via this technique. In the first case, we take De Sitter universe model and construct some field potentials by taking different cases for coupling function. In the second case, we derive some field potentials using power law model in the presence of different matter sources like barotropic fluid, cosmological constant and Chaplygin gas for some coupling functions. From graphical analysis, it is concluded that using some specific values of the involved parameters, the reconstructed scalar field potentials are cosmologically viable in both cases.

Keywords: Scalar-tensor theory; Scalar field; Field potentials.

PACS: 98.80.-k; 04.50.Kd.

I Introduction

The investigation about the possible causes of accelerated expansion of cosmos and nature of its missing mass and energy are some leading topics of this century. Numerous researchers are working over these lines and usually they proposed two ways to describe this accelerating expansion 1 (). Some consider, general relativity (GR), as the right theory of gravity which present dark energy (DE) 2 () as an easily conveyed, gradually changing cosmic fluid with negative pressure. This technique is referred as GR with modified matter sources. Other way is to modify the gravitational sector of GR 3 ()-6 (). The gravity, is one of the most attracting examples of modified gravity theories, that can be mapped into GR with extra scalar fields using an appropriate conformal transformation of metric 5 ()-7 (). In modern cosmology, scalar fields play an important role in explaining the nature of DE 2 () and to drive inflation in the beginning of universe 8 (); 9 (). In literature like 10 (), it is concluded that the history of cosmic expansion can be successfully discussed using scalar-tensor theories.

Many cosmological models and modified gravity theories with scalar fields, involve some general functions which cannot be derived easily from the basic theory. Then some questions frequently arise like how these particular functions should be chosen and what are physical reasons behind those particular choices. In this respect, the reconstruction technique is not a new concept, it has a long history to reconstruct the DE models. This technique allows one to find the form of scalar field potential as well as of scalar field for a specific choice of Hubble parameter in terms of scale factor or cosmic time. For a better understanding of this technique, we refer the readers to see the literature 38 ().

In scalar tensor theories, it is very necessary to investigate the nature of scalar field potential and its role in explaining DE and cosmic expansion history. In 39 (), the nature of scalar potential has been discussed for a minimally coupled scalar tensor theory using reconstruction technique. The reconstruction technique to explore the nature of field potentials for the models involving minimally coupling to scalar fields, two-field models and tachyon models has been given in literature 11 ()-28 (). The reconstruction of field potentials in the models involving non-minimally coupling of scalar fields to gravity was studied in 29 ()-31 (). Furthermore, the applications of this reconstruction technique in different gravity models and theories like the models containing non-minimally coupling of Yang-Mills fields 32 (), in the framework of local 37 () and nonlocal gravity 36 (), in and Gauss-Bonnet gravity theories 21 (); 33 (); 34 () and the gravity theory that involves torsion scalar as a basic ingredient 35 () are available in literature.

The models of gravity that are non-minimally coupled to scalar fields are of great interest in cosmology 40 ()-47 (). Particularly, the models having Hilbert-Einstein term in addition to the term relative to the Ricci scalar with squared scalar field were considered in quantum and inflationary cosmology 48 (); 49 (). In 30 (), the reconstruction process has been studied for induced gravity (, where is an arbitrary constant). It is shown that for these cases, linearizing the differential equations to solve in reconstruction process, to derive potentials according to committed cosmological evolution. It is interesting to mention here that from this process, one can get explicit potentials which can reproduce the dynamics of flat (FRW) universe derived by different matter sources like barotropic and perfect fluids, Chaplygin gas 50 (), and modified Chaplygin gas 30 ().

In this regard, Kamenshchik et al. 30 () has used this approach to reconstruct the field potential in terms of scalar field for FRW universe in the framework of induced gravity. They discussed this procedure for different matter distributions and concluded that the corresponding cosmic evolution can be reproduced in these cases. In 52 (), the same authors used another technique known as super potential reconstruction technique for FRW model to reconstruct scalar field potentials in a non-minimally coupled scalar tensor gravity. They examined its nature for de-Sitter and barotropic models and discussed their cosmic evolution. Sharif and Waheed 53 () studied the nature of scalar field potential for locally rotationally symmetric (LRS) Bianchi type I (BI) universe model in a general scalar-tensor theory via reconstruction technique and they concluded that the reconstructed potentials are viable on cosmological grounds. In a recent paper 54a (), we have discussed cosmological reconstruction and energy bounds in a new general gravity. In this gravity, we have also studied the first and second laws of black hole thermodynamics for both equilibrium and non-equilibrium descriptions 54b (). Being motivated from the literature, in the present paper, we examine the nature of field potential for flat FRW universe in gravity by applying reconstruction procedure.

This paper is arranged in the following pattern. In the next section, we give a general overview of this procedure and derive the general form of scalar field potential for this theory. In section III, we derive field potentials for de-Sitter model by taking different choices for function . Section IV is devoted to explore the form of field potential for a power law model with matter sources as barotropic fluid, cosmological constant and Chaplygin gas in separate cases. In the last section, we present a summary of all sections by highlighting the major achievements.

Ii Basic Field Equations and General Scalar Field Potential

In this section, we present the basic formulations of the most general scalar-tensor gravity namely theory. The gravitational action for this theory is given as follows 54 (),

(1)

where is a general function depending upon the Ricci scalar , the curvature invariant (where is the Ricci tensor) and the scalar field symbolized by . Further, is a coupling function of scalar field , the symbol corresponds to the scalar field potential and is the determinant of metric tensor whereas represents the gravitational coupling constant.

The flat FRW spacetime with cosmic radius is given by the following metric

(2)

For which, the quantities like scalar curvature and Ricci invariant turn out to be

(3)

while . The Friedmann equations constructed in 54b () are

(4)

and

(5)

The Klein-Gordon equation is

(6)

From Eq. (II), we have

(7)

It is more appropriate to consider all the functions dependent on instead of cosmic time ,

(8)

By substituting the derivative of Eq. (7) in Eq. (6), we eliminate and the resulting equation can be written as

(10)

In terms of , the above equation is written as Appendix A given in the Appendix. The field equations involve five unknowns namely and . Now we evaluate the scalar potential for de-Sitter and power law models (in barotropic fluid, cosmological constant and in Chaplygin gas) by taking different choices of for the remaining unknowns.

Iii de-Sitter Models

In cosmology, the dS-solutions are of great significance to explain the current cosmic epoch. In dS-model, the scale factor, the Hubble parameter and the Ricci tensor take the following form 55 ()

(11)

Here we are using and 55 ().

iii.1 Model

We have derived the general form of for dS-model in 54a (), here we use this form to evaluate the scalar field potential. For this model, function is defined as

(12)

where are constants of integration and

(13)
  • Case-I:  

Substituting model (12) into (Appendix A) and choosing , we have

(14)

For the sake of simplicity, we introduce a new variable , thus (14) takes the following form:

(15)

For further simplification, by introducing a new function

(16)

we get a differential equation for of the form:

(17)

where . It is easy to see, on comparing that

(18)

Equations (II) and (12) lead to the following form of scalar potential:

(19)

If , all the assumptions regarding derivative of (7) are justifiable. The constant field must be discussed separately. First consider, if is a non-zero constant, then and are also all independent of time. This leads to the cosmological evolution which is occurred due to cosmological constant. Now we can rewrite Eq.(II) as follows

(20)

Then, on substituting and into Eq.(6), we have

(21)

Multiplying Eq.(20) by Eq.(21) and then integrating, we have

(22)

Now for the previous case, i.e., time dependent scalar field, the basic equation (17) is given by

(23)

whose general solution is

(24)

where and are integration constants. From (24), it can be written as

(25)

and from (18), we have

(26)

and inversely, it takes the form

(27)

Another useful formula, in this respect, is

(28)

Inserting (26) and (28) into Eq.(19), we get

If we choose in (26), we have and using this in the above equation, we get the scalar potential in terms of scale factor as follows

(30)
  • Case-II:   

If we choose then we have and we get potential of the form

(31)

iii.2 Model

Now we are utilizing model, independent of which we have already constructed in the paper 54a () and it is given by

(32)

where are constants of integration and

  • Case-I:   

Substituting model (32) into (Appendix A) and choosing , we have

(33)

Introducing the variable for simplification, we can write the above equation as follows

(34)

With the help of this new function

(35)

we get the differential equation for in the following form

(36)

Clearly, we have

(37)

Equation (II) in case of (32) becomes

(38)

If , all the assumptions regarding derivative of (7) are justifiable. The constant field must be discussed separately. First consider, if is constant, then and are also independent of time. This leads to the cosmological evolution which is occurred due to cosmological constant. Now we can rewrite Eq. (II) as

(39)

Then, on substituting and into Eq. (6), we obtain

(40)

Multiplying Eq. (39) by Eq. (40) and then by integration, we have

(41)

The basic equation (36) is now given by

(42)

whose general solution is

(43)

where and are integration constants. From (43), we can write

(44)

and from (37), we have

(45)

and inversely, we can write

(46)

Further,

(47)

Inserting (45) and (47) into (38), we get