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This paper reviews the development of
*f(R)* gravity theory and Phantom and Quintessence fields. Specifically, we present a new general action of
*f(R)* gravity and Phantom and Quintessence fields coupled to scalar curvature. Then, we deduce Euler-Lagrange Equations of different fields, matter tensor and effective matter tensor. Additionally, this paper obtains the general pressure, density and speed sound of the new general field action, and investigates different cosmological evolutions with inflation. Further, this paper investigates a general
*f(R)* gravity theory with a general matter action and obtains the different field equations, general matter tensor and effective matter tensor. Besides, this paper obtains the effective Strong Energy Condition (SEC) and effective Null Energy Condition (NEC). Then, we prove that when
* f(R)* approaches to R, the effective SEC and the effective NEC approach to the usual SEC and the usual NEC, respectively. Finally, this paper presents a general action of
*f(R)* gravity, Quintessence and Phantom fields and their applications.

Researchers began to question the theory of gravity after the advent of the theory of general relativity (GR). Weyl (1919) and Eddington (1923) considered modifications to the theory by including higher-order invariants in the action [

GR correction is not an easy task. First, there are many naive GR corrections, which are unrealistic [

where

There are two motivations for GR correction: (1) adding higher-order gravitational action in high-energy physics, and (2) applying the GR correction to cosmology and astrophysics.

However, there are two problems. The first problem is why specifically

When the

The second problem is that it is related to a possible loophole. First, how can high-energy corrections of the gravitational action have nothing to do with late- time cosmological phenomenology? Would not effective field theory considerations require that the coefficients in Equation (3) be such, as to make any modifications to the standard Einstein-Hilbert term important only near the Planck scale? [

The observed late-time acceleration of the Universe poses one challenge to theoretical physics. In principle, this phenomenon may be the result of unknown physical processes. For instance, it involves either the correction of gravitational theory or the existence of new fields in high-energy physics. Although the latter one is most commonly used, the correction of gravitational theory is an attractive and complementary approach to explain this phenomenon, known as

In [

Several different forms for

In this paper, we introduce a new action, the new action effective amount of inclusion of Quintessence and Phantom can solve the problem. Thus, this model is a more general model of dark energy. We obtained the general sound speed in the evolutions of the Universe, and give the exact expressions for the exact energy- momentum tensor, pressure, energy, and different

The rest of this paper is organized as follows. Section 2 investigates a general action of

The first idea is to combine the actions for Phantom and Quintessence fields into one action, by adding a parameter α into the general action of

(i) When

(ii) When

Further, the scalar curvature R can be coupled to the Phantom and Quintessence fields. Thus, the second idea is to add scalar curvature R into Equation (4)

where

The variance of Equation (5) is

For the deducing details see Appendix A.

To study Equation (6), we discuss

and

We have

Using

We rewrite Equation (9) as

Substituting Equation (11) into Equation (8), we have

The deducing details of Equation (12) are in Appendix B.

Substituting Equation (12) into Equation (7), we obtain

Substituting Equation (13) into Equation (6), we have

Considering a general partial integration, we have

The deducing details of Equation (15) are in Appendix C. Thus, we have

Substituting Equation (16) into Equation (14), we have

Using Equation (17), we deduce Euler-Lagrange Equations

where

i.e.,

where

Therefore, a general action of

From Equation (5) and Equations (20)-(23), we can obtain the exact energy- momentum tensor.

To simplify the exact energy-momentum tensor, we can rewrite Equation (24) as

where

Substituting Equation (26) into Equation (25), we obtain a general energy- momentum tensor

where

In general, using

we have

i.e.,

and

i.e.,

Due to

Furthermore, we can obtain the general sound speed

Using Equations (30) and (31), we obtain

Thus, we can rewrite Equations (33) and (34) as

where

where

In conclusion, we obtain the general energy-momentum tensor, pressure, density and speed sound of the new general single field action of

The most hopeful models for the different evolutions of the Universe are the cosmological models of the initial evolution and subsequent development. They are supported by the most comprehensive and accurate explanations based on the current scientific evidences and observations. According to the observations on the current Universe, the dark matter accounts for 24% of the mass-energy density of the observable Universe, the dark energy amounts for nearly 72% and the ordinary matter only accounts for about 4%.

The equation of state (EOS) is a powerful way to describe the matter and the evolutions of the Universe. In cosmology, the EOS of a perfect fluid is characterized by a dimensionless number that is equal to the ratio of its pressure to its energy density. It is closely related to the thermodynamic EOS and ideal gas law.

Therefore, with EOSs of matter, dark energy and dark matter, the new general action can be used to explain the different cosmological evolutions: (I) Big Rip Universe; (II) De Sitter Universe; (III) Harmonic Universe.

In the case of cosmological inflation, using Equations (28), (30) and (5), we deduce the EOS

Substituting

To discuss the different evolutions of the Universe, we use the Friedman equations [

and then we have

Now accelerating expansion

thus

Using Equation (33), we rewrite Equation (35) as

Equation (42) can describe the different evolution characteristics of the Universe:

(i) Big Rip Universe, i.e., the case where

Generally, the expansion of the Universe is accelerating when

When EOS for Phantom energy is

(ii) De Sitter Universe, i.e. the case where

The de Sitter Universe is a solution to Einstein’s field equations of General Relativity, which is named after Willem de Sitter. When one considers the Universe as spatially flat and neglects ordinary matter, the dynamics of the Universe would be dominated by the cosmological constant, corresponding to dark energy. If the current acceleration of our Universe is due to a cosmological constant, then the universe would continue to expand. All the matter and radiation will be diluted. Eventually, there will be almost nothing left except the cosmological constant, and the Universe will become a de Sitter Universe.

(iii) Harmonic Universe, i.e., the case where

The EOS for ordinary non-relativistic matter is

The EOS of radiation and matter is

then the Universe is diluted as

Substituting Equation (26) into Equation (42), we can further deduce a concrete expression for Equation (42)

which shows their concrete evolution details. For example, we focus on the case of different potentials as follows:

(a) When

(b) When

(c) When

Thus, we deduce the EOS, i.e., Equations (35), (36), (42), (47)-(49), of

We begin with a general action of

where

where a prime denotes differentiation with respect to

i.e.,

Multiplying Equation (53) by

i.e.,

Then, we get

In addition, we can have

Using Equations (56) and (57), we deduce

Combining Equation (56) with Equation (58), we have

The deducing details are in Appendix E.

Equations (56) and (59) are consistent. This is because by multiplying Equation (59) with

The deducing details are in Appendix F. Equation (60) is just Equation (56), so Equations (56) and (59) are consistent.

In addition, we have

When

where

Using Equation (61), we similarly have

where

For the homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker (FLRW) metric with scale factor

where

The evolution equation for the expansion of a null geodesic congruence is defined by a vector field

where

Using Equation (59), we concretely have

and

Substituting Equations (68) and (69) into Equations (65) and (67), we deduce inequalities

For the deducing details see Appendix I.

Thus, we have

where

For

and

where

In sum, we investigate a general

To investigate more general cases and extend the applications of the new action proposed in Section 2, the single scalar field

We now generalize Equation (4) to a general action of

When

i.e.,

Equation (76) is a generic Lagrangian of

When

i.e.,

Equation (78) is a generic Lagrangian of

Using Equation (74) or Equation (77), we can calculate and obtain all corresponding results similar to all investigations above Equation (74).

This paper introduces the development process of

Then, this paper obtains the general energy-momentum tensor, pressure, density and speed sound of the new general single field action of

Further, this paper deduces the equations of states of

In addition, this paper investigates a general

The Hawking-Penrose singularity theorems invoke the weak and strong energy conditions, whereas the proof of the second law of black hole thermodynamics requires the null energy condition. More recently, several researchers used the classical energy conditions of GR to investigate cosmological issues.

In the cosmology, these theories provide an alternative way to explain the cosmic speed-up. The freedom in building different functional forms of

Finally, we generalize Equation (4) to a general action of

We would like to thank Prof. Rong-Gen Cai for the helpful discussions and comments.

The authors declare no conflict of interest regarding the publication of this paper.

The work is supported by National Natural Science Foundation of China (Grant Nos. 11275017 and 11173028).

Zhang, X.Y. and Huang, Y.C. (2017) Phantom and Quintessence Fields Coupled to Scalar Curvature in General f(R) Gravity Theory. Journal of Modern Physics, 8, 1234-1256. https://doi.org/10.4236/jmp.2017.88081