𝑓(𝑅,𝜙) and 𝑓(𝑅) Models: A Self-Similar Approach

Abstract

We study the 𝑓(𝑅,𝜙) models and their particular case, the so-called 𝑓(𝑅)-models under the self-similarity hypothesis. In particular, we calculate the exact form that each quantity may take in order that field equations (FEs) admit self-similar solutions. The methods employed allow us to obtain general results that are valid not only for the FRW metric, but also for all the Bianchi types as well as for the Kantowski-Sachs model (under the self-similarity hypothesis and the power-law hypothesis for the scale factors).

1. Introduction

The physical and mathematical importance of the modified gravitational models has been recently pointed out for several authors (see, e.g., [1–4]), since this kind of theories explains in a better way the dynamics of the very early universe as well as its current acceleration. In a recent paper [5] (for a review see [6]), we have studied some particular cases of the so-called generalized scalar-tensor theories (STTs or 𝐹(𝜙) models). Obviously, although such class of theories is more general than the usual Jordan-Brans-Dicke models, they may be generalized in order to incorporate corrections to the Ricci scalar term as the 𝑓(𝑅) models (see, e.g., [7–9] and for a review [10]). Therefore, the purpose of this brief paper is to study, under the self-similarity hypothesis, the generalized 𝑓(𝑅,𝜙) theories, where the 𝑓(𝑅) models (without 𝜙) result as subclasses and the generalized SST with 𝑓=𝐹(𝜙)𝑅 studied in [5]. In particular, we study the form that the different quantities may take in order that the field equations (FEs) admit self-similar solutions. To carry out our study, we employ two techniques: the Lie group method (LG) [11] and the matter collineation approach (MC) [12]. In this way, we shall be able to state and proof general theorems that are valid not only for the FRW symmetries, otherwise for all kind of Bianchi types as well as for the Kantowski-Sach models.

The self-similarity property of a space-time is characterized by the existence of a homothetic vector field ℋ (HVF). For a metric, 𝑔, and for a vector field ℋ∈𝔛(ℳ), (ℋ=ℋ_𝑖(𝑡,𝑥,𝑦,𝑧)𝜕_𝑥𝑖)⁴_𝑖=1 the homothetic equation reads: ℒ_ℋ𝑔=2𝑔 [13]. We study self-similar solutions since, as it has been pointed out by Rosquist and Jantzen [14], they correspond to equilibrium points. In the same way, Coley [15] has stressed the fact that this kind of solutions describes asymptotic behaviour of the Bianchi models playing a dominant role in the dynamics of Bianchi cosmological models (see, e.g., [16] for a review of results in the context of the general relativity).

The paper is organized as follows. In Section 2 we study through the LG method the 𝑓(𝑅,𝜙) models while Section 3 is devoted to study the 𝑓(𝑅) models. We summarize the main results in Section 4.

2. 𝑓(𝑅,𝜙) Models

Our purpose is to describe a class of scalar-tensor theories of gravity represented by the action [17, 18]: 𝑆=𝑑⁴𝑥√−𝑔𝛽12𝑓−1𝜙,𝑅2𝑍𝜙𝜙^;𝜇𝜙_;𝜇−𝑈𝜙+𝛽𝐿_𝑚, (1) where 𝑅 is the Ricci scalar, and 𝜙 stands for the scalar field, where we assume that the scalar field is homogeneous, that is, 𝜙=𝜙(𝑡). We introduce the constant 𝛽 in order to fix units, and 𝐿_𝑚 is a classical matter Lagrangian. The gravitational field equations derived from the action (1) are 𝐹𝐺_𝜇𝜈=𝛽𝑇^𝑚_𝜇𝜈+𝑍𝜙_,𝜇𝜙_,𝜈−12𝑔_𝜇𝜈𝜙_,𝜎𝜙^;𝜎+𝐹_,𝜇;𝜈−𝑔_𝜇𝜈□𝐹−𝑈𝑔_𝜇𝜈+12𝑓−𝐹𝑅𝑔_𝜇𝜈, (2) where 𝐺_𝜇𝜈 is the Einstein tensor, 𝑇^𝑚_𝜇𝜈 is the energy-momentum tensor, 𝑍=𝑍(𝜙),𝑈=𝑈(𝜙) and 𝑓=𝑓𝑅,𝜙,𝐹=𝜕_𝑅𝑓𝑅,𝜙. (3)

The wave equation reads 2𝑍□𝜙=𝑓_,𝜙−𝑍_𝜙𝜙^′2−2𝑈_,𝜙, (4) where a prime denotes derivation with respect to time.

We will work under the assumptions 𝑓=𝐾(𝑅)𝑔𝜙,𝐾=(𝑅)^𝑟,𝑟∈ℝ, (5) that is, 𝑓 may be split in separable variables and the function 𝐾 takes this particular form, which is appropriate under the self-similarity hypothesis.

Theorem 1. The scaling and in particular the self-similar solutions admitted for the FEs (2) and (4) under the assumptions (5) have the following form: 𝜙=𝜙₀𝑡^𝑛,𝑍𝜙=𝑍₀𝜙^−𝑚,𝑈𝜙=𝑈₀𝜙^{−(𝑚𝑛−2𝑛+2)},𝑔𝜙=𝐶₁+𝐶₂𝜙^𝐴, (6) with 𝐴=(2−𝑚−(2/𝑛)(1−𝑟)), and therefore 𝑓=𝐾(𝑅)⋅𝑔𝜙=𝑓₀𝑡^{2𝑛−𝑚𝑛−2},𝐹=𝐾_𝑅(𝑅)⋅𝑔𝜙=𝐹₀𝑡^{𝑛(2−𝑚)}, (7) with 𝐾=𝑅^𝑟=(𝑎𝑡⁻²)^𝑟,   𝑎∈𝑅.

Proof. We are going to study (4) through the LG method, that is, we study the kind of functions 𝑓=𝑓(𝑅,𝜙),𝑍(𝜙), 𝑈(𝜙), and 𝜙 such that the FEs (2) and (4) admit power-law and in particular self-similar solutions.
We start by rewriting it in an appropriate way: 2𝑍𝜙^′′+ℎ𝑡⁻¹𝜙^′=−𝐾𝑔_𝜙+𝑍_𝜙𝜙′²+2𝑈_𝜙, (8) since 𝐻=ℎ𝑡⁻¹, ℎ=∑³_𝑖=1𝑎_𝑖∈ℝ⁺, being 𝑎_𝑖 the exponents of the scale factors, that is, 𝑎(𝑡)=𝑎₀𝑡^𝑎1, and so forth, since we are considering not only the FRW case, but also the Bianchi types as well. We also assume that the scalar field is homogeneous, that is, 𝜙=𝜙(𝑡); therefore, a prime denotes a derivative with respect to time. Note that in the self-similar framework, the scalar curvature always behaves as 𝑅∼𝑡⁻² [14]; therefore, we may assume that 𝐾(𝑅)∼𝐾(𝑡).
Roughly speaking, a symmetry, 𝑋=𝜉(𝑥,𝑦)(𝜕/𝜕𝑥)+𝜂(𝑥,𝑦)(𝜕/𝜕𝑥), of a differential equation is an invertible transformation that leaves it form-invariant. By applying the standard Lie procedure (see, e.g., [11]), we need to solve the following overdetermined system of linear partial differential equations for 𝜂 and 𝜉 (from the extended infinitesimal or prolonged transformations), which allows us to determine the set of the symmetries admitted by (8): 𝑍_𝜙𝜉_𝜙−2𝑍𝜉_𝜙𝜙=0,𝑍𝑍_𝜙𝜙−𝑍²_𝜙𝜂+𝑍𝑍_𝜙𝜂_𝜙+4𝑍²ℎ𝑡⁻¹𝜉_𝜙+2𝑍²𝜂_𝜙𝜙−4𝑍²𝜉_𝑡𝜙=0,32𝑉_𝜙−𝐾(𝑡)𝑔_𝜙𝜉_𝜙+2ℎ𝑡⁻²𝑍𝑡𝜉_𝑡−𝜉+4𝑍𝜂_𝑡𝜙−2𝑍𝜉_𝑡𝑡+2𝑍_𝜙𝜂_𝑡=0,𝐾𝑍_𝜙𝑔_𝜙−𝑍𝑔_𝜙𝜙+2𝑍𝑈_𝜙𝜙−𝑍_𝜙𝑈_𝜙𝜂+𝑍𝐾𝑔_𝜙−2𝑈_𝜙𝜂_𝜙+2𝑍2𝑈_𝜙−𝐾𝑔_𝜙𝜉_𝑡−𝐾^′𝑍𝑔_𝜙𝜉+2𝑍²ℎ𝑡⁻¹𝜂_𝑡+𝜂_𝑡𝑡=0. (9)(10)(11)(12) The knowledge of one symmetry 𝑋 might suggests the form of a particular solution as an invariant of the operator 𝑋, that is, a solution of 𝑑𝑥/(𝜉(𝑥,𝑦))=𝑑𝑦/(𝜂(𝑥,𝑦)). This particular solution is known as an invariant solution (generalization of similarity solution). The symmetry, 𝜉=𝑡,𝜂=𝑛𝜙, which induces the invariant solution, 𝜙=𝑡^𝑛, leads us to obtain the following restrictions. From (10), 𝑍_𝜙𝜙=𝑍²_𝜙𝑍−𝑍_𝜙𝜙⇒𝑍𝜙=𝑍₀𝜙^−𝑚, (13) with, 𝑚∈ℝ. From (12), we get 2𝑍𝑈_𝜙𝜙−𝑍_𝜙𝑈_𝜙𝜂−2𝑍𝑈_𝜙𝜂_𝜙+4𝑍𝑈_𝜙𝜉_𝑡=0,𝐾𝑍_𝜙𝑔_𝜙−𝑍𝑔_𝜙𝜙𝜂+𝐾𝑍𝑔_𝜙𝜂_𝜙−2𝐾𝑍𝑔_𝜙𝜉_𝑡−𝐾^′𝑍𝑔_𝜙𝜉=0. (14) Therefore, 𝑈_𝜙𝜙=𝑛−2𝑛𝑈−𝑚^𝜙𝜙, (15) where we have taken into account that 𝑍(𝜙)=𝑍₀𝜙^−𝑚. In this way, 𝑈𝜙=𝑈₀𝜙^{−(1/𝑛)(𝑚𝑛−2𝑛+2)}, (16) while 𝑔_𝜙𝜙=11−𝑚−𝑛𝐾2+𝑡^′𝐾𝑔_𝜙𝜙, (17) obtaining the following solution 𝑔𝜙=𝐶₁+𝐶₂𝜙^𝐴1,𝐴=2−𝑚−𝑛𝐾2+𝑡^′𝐾. (18) Compare this result with the obtained one in [5]. Note that if 𝐾=𝑡⁻² (in the self-similar framework the Ricci scalar always varies as 𝑡⁻² [14]), that is, 𝑟=1, then 𝐾2+𝑡^′𝐾=0, (19) and therefore 𝑔𝜙=𝐶₁+𝐶₂𝜙^2−𝑚. (20)
In any case, when 𝐾=𝑅^𝑟=𝑡^−2𝑟, then 𝑔𝜙=𝐶₁+𝐶₂𝜙^{2−𝑚−(2/𝑛)(1−𝑟)}, (21)
thus 𝑓=𝐾(𝑅)⋅𝑔𝜙≈𝑅^𝑟𝑡^{𝑛(2−𝑚−(2/𝑛)(1−𝑟))}=𝑓₀𝑡^{2𝑛−𝑚𝑛−2},𝐹=𝐾_𝑅(𝑅)⋅𝑔𝜙≈𝑅^𝑟−1𝑡^{𝑛(2−𝑚−(2/𝑛)(1−𝑟))}=𝐹₀𝑡^{𝑛(2−𝑚)}. (22) Therefore, the so-called 𝑓(𝑅,𝜙) models admit power-law and in particular self-similar solutions.

As we have shown, for different values of 𝑚 and 𝑟 we obtain different models. Note that the case 𝑟=1 (the usual one) has been already studied in [5].

3. 𝑓(𝑅) Models

As a particular case of the above theories we may study the so-called 𝑓(𝑅) models. This kind of models is described by the action: 𝑆=𝑑⁴𝑥√−𝑔𝑓(𝑅)+𝐿_𝑚, (23) where 𝑅 is the Ricci scalar, and 𝐿_𝑚 is a matter Lagrangian as in the above model. By defining ≡𝜕𝑓/𝜕𝑅(𝑓=𝑓(𝑅)), the gravitational field equations derived from the action (23) are [1]: 𝐹𝐺_𝜇𝜈=𝑇^𝑚_𝜇𝜈+𝐹_;𝜇;𝜈−𝑔_𝜇𝜈1□𝐹+2𝑓−𝐹𝑅𝑔_𝜇𝜈, (24) where 𝑅 is the Ricci scalar, and 𝐺_𝜇𝜈 is the Einstein tensor. The trace of the field equations read 3□𝐹+𝐹𝑅−2𝑓=𝑇, (25) where 𝑇 is the trace of the stress-energy tensor.

We deduce the form that can acquire the function 𝑓(𝑅) in order that the FE admit self-similar and power-law solutions (for the scale factors). For example, in the framework of FRW symmetries it is already known that 𝑓(𝑅)=𝑅^𝑛 if the scale factor follow a power-law (see, e.g., [19]). In this section, we extend this result to all the Bianchi types as well as for the Kantowski-Sachs model through two methods. The first one, the matter collineation approach allows us to determine that the unique admitted form for the function 𝑓(𝑅) is precisely 𝑓(𝑅)=𝑅^𝑛 compatible with the self-similar hypothesis. With the second one, by studying the trace equation with the Lie group method we proof that 𝑓(𝑅)=𝑅^𝑛 is the unique possible form for the function 𝑓(𝑅) under the power-law hypothesis for the scale factors.

3.1. Matter Collineation Approach

Theorem 2. Under the hypothesis of self-similarity, the unique form admitted for the function 𝑓(𝑅) is 𝑓(𝑅)=𝑅^𝑛.

Proof. We may define an effective stress-energy tensor as follows: 𝑇^eff_𝜇𝜈=𝐹⁻¹𝑇^𝑚_𝜇𝜈+𝐹⁻¹𝐹_;𝜇;𝜈−𝑔_𝜇𝜈+1□𝐹2𝑓𝐹⁻¹−𝑅𝑔_𝜇𝜈, (26) which allows to use the standard Einstein equations by simply replacing the fluid quantities with the effective ones, so we have to calculate ℒ_ℋ𝑇^eff_𝜇𝜈=0, (27) where ℋ stands for a homothetic vector field. We calculate the Lie derivative of each component of the effective stress-energy tensor.(1)Thus, ℒ_ℋ(𝐹⁻¹𝑇^𝑚_𝜇𝜈)=0, in components, ℒ_ℋ⋅(𝐹⁻¹𝑇^𝑚₁₁)=0, 𝜌′𝜌−𝐹^′𝐹2=−𝑡⟺𝜌𝐹⁻¹=𝜌₀𝑡⁻². (28) As it is noted, we are only able to obtain a relationship between the two quantities 𝐹 and 𝜌, but we have not determine their particular form.(2)ℒ_ℋ𝑇^𝑓_𝜇𝜈=ℒ_ℋ((1/2)(𝑓𝐹⁽⁻¹⁾−𝑅)𝑔_𝜇𝜈)=0. The first term is ℒ_ℋ(𝑇^𝑓₁₁)=0, that is, 𝑡𝑓^′−𝑅^′𝐹−𝐹^′𝑓𝐹⁻¹=−2𝑓−𝐹𝑅, (29) that we may rewrite in the following form: 𝑡𝐹𝑅^′−𝐹𝑅^′−𝑓_𝑅𝑅𝑅^′𝑓𝐹⁻¹=−2𝑓−𝐹𝑅, (30) that we may write as 𝑡−𝑓_𝑅⁻¹𝑓_𝑅𝑅𝑅^′𝑓+2𝑓−𝑓_𝑅𝑅=0 (31) since 𝑓^′=(𝑑𝑓/𝑑𝑅)𝑅^′=𝐹𝑅^′, and 𝐹^′=𝑓_𝑅𝑅𝑅^′. Now taking into account the fact that 𝑡𝑅^′≈−2𝑅, then the equation yields 𝑓_𝑅𝑅=𝑓²_𝑅𝑓−𝑓𝑅, (32) whose solution is 𝑓=𝐶₁𝑅^𝑛𝑛∈ℝ. (33) With the other components, ℒ_ℋ(𝑇^𝑓_1𝑖)=0, we only obtain restriction on the scale factors.(3)ℒ_ℋ𝑇^𝐹_𝜇𝜈=0, that is, ℒ_ℋ𝐹⁻¹𝐹_;𝜇;𝜈−𝑔_𝜇𝜈□𝐹=0. (34)  We would like to emphasize that this term depends on the metric, that is, on the scale factors so, we mainly obtain restriction on the scale factors. Nevertheless, the first component is ℒ_ℋ(𝑇^𝐹₁₁)=0, 𝑡𝐹^′′𝐻+𝐹^′𝐻^′−𝐹^′2𝐹𝐻=−2𝐻𝐹^′, (35) now, taking into account that 𝐻=ℎ𝑡⁻¹, then we may rewrite it as follows: 𝐹^′′=𝐹^′2𝐹−𝐹^′𝑡, (36) and whose solution is 𝐹=𝐹₀𝑡^𝐶1,𝐶₁∈ℝ. (37) If, for example, we set 𝐶₁=2−2𝑛, from the above result (𝑓(𝑅)=𝑅^𝑛), therefore, we arrive to the conclusion that 𝐹=𝐹₀𝑡^2−2𝑛=𝑛𝑅^𝑛−1. (38) Now if we take into account that 𝜌=𝜌₀𝑡^−𝛼 (from the conservation equation for the matter field) being 𝛼=ℎ(𝛾+1) and the result from (28), we arrive to the conclusion that 𝑛=𝛼/2.
This concludes the proof of the theorem as it is required.

Since the hypothesis of self-similarity could be strong constraining the form of the function 𝑓(𝑅), in the next section we study if under the hypothesis of power-law solutions for the scale factors the function 𝑓(𝑅) could take a different form. For this purpose, we go next to study the trace equation through the Lie group method.

3.2. Lie Group Method

Theorem 3. Under the hypothesis of power-law solutions for the scale factors, the unique form admitted for the function 𝑓(𝑅) is 𝑓(𝑅)=𝑅^𝑛.

Proof. By studying the trace equation (25), that we rewrite as follows: 𝐹^′′+ℎ𝑡⁻¹𝐹^′=𝑟3𝑡⁻²𝐹−3𝛾−13𝜌₀𝑡^−𝛼−23𝑓, (39) where we have set 𝐻=ℎ𝑡⁻¹,𝑅=𝑟𝑡⁻²,𝑇=3𝛾−1𝜌₀𝑡^−𝛼,𝑓=𝑓(𝑡), (40) with ℎ,𝑟∈ℝ, that is, they are constants, 𝛼=ℎ(𝛾+1).
The general solution of (39) is 𝐹=𝐶₂𝑡^𝐴++𝐶₁𝑡^𝐴−+𝐷⁻¹𝑡^1/2×𝑡^{𝐴−−1/2}2𝑡^𝐵+𝑓+3𝛾−1𝜌₀𝑡^𝐵+−𝛼𝑑𝑡−𝑡^𝐴+−1/22𝑡^𝐵−𝑓+3𝛾−1𝜌₀𝑡^{𝐵−−𝛼}𝑑𝑡, (41) where 𝐷=9(ℎ−1)²+12𝑟,𝐶₁,𝐶₂𝐴∈ℝ,_±=16(3(1−ℎ)±𝐷),𝐵_±=16(3(1+ℎ)±𝐷). (42)
Note that if in (39) we set 𝐹=𝑎=const., then 3𝑓=2𝑟𝑎𝑡⁻²−3𝛾−1𝜌₀𝑡^−𝛼⟺𝑓(𝑅)≈𝑅≈𝑡⁻², (43) which implies that 𝛼=2 (as in the standard case).
We will determine the form of the unknown function 𝑓 in order that the FEs (24)-(25) admit power-law solutions. By studying (39) through the Lie group method, then the standard Lie procedure brings us to get following system of PDEs: 2𝜉_𝐹𝐹𝑡²=0,2𝜉_𝐹ℎ𝑡+𝜂_𝐹𝐹𝑡²−2𝜉_𝑡𝐹𝑡²=0,𝜉_𝑡ℎ𝑡−𝜉_𝐹𝑟𝐹−𝐶𝑡^−𝛼+2−2𝑓𝑡²+2𝜂_𝑡𝐹𝑡²−𝜉_𝑡𝑡𝑡²−𝜉ℎ=0,13𝜉2𝑓^′𝑡²+2𝑟𝐹𝑡⁻¹−𝐶𝛼𝑡^−𝛼+1+ℎ𝑡𝜂_𝑡+23𝜉_𝑡2𝑡²𝑓+𝐶𝑡^−𝛼+2−𝑟𝐹+𝑡²𝜂_𝑡𝑡+13𝜂_𝐹𝑟𝐹−2𝑓𝑡²−𝐶𝑡^−𝛼+2−13𝑟𝜂=0. (44)(45)(46)(47)
For example, the symmetry =𝑡/(2−2𝑛),  𝜂=𝐹, which induces the invariant solution 𝐹=𝑡^2−2𝑛, brings us to obtain the following restriction on the function 𝑓. From (47), we obtain 𝐶(2𝑛−𝛼)+4𝑛𝑓𝑡^𝛼+2𝑡𝑡^𝛼𝑓^′=0, (48) whose solution is 𝐶𝑓=¹𝑡^2𝑛−12𝐶𝑡^𝛼𝑓=𝑡^−𝛼𝛼⟺𝑛=2, (49) and which is consistent with the previous result 𝐹⁻¹𝜌=𝑡⁻².
To finish this proof, we would like to emphasize that if one tries to find a symmetry (𝜉,𝜂) whose invariant solution coincides with the solution 𝐹=𝑛𝑅^𝑛−1=𝑛(𝑟𝑡⁻²)^𝑛−1, from the assumption 𝑓=𝑅^𝑛=(𝑟𝑡⁻²)^𝑛, as for example, 𝜉=1/(−2𝑛(𝑛−1)𝑟𝑡⁻³(𝑟𝑡⁻²)^𝑛−2),𝜂=1, then it is easily checked that it does not verify (46).

We may come to the same conclusion, simply rewriting the trace equation in an appropriate way, by considering the relationship between the functions 𝑓 and 𝐹=𝑓_𝑅 (i.e., 𝑓_𝑅=𝑑𝑓/𝑑𝑅). In this way, (39) yields 𝑓_𝑅𝑅𝑅𝑅^′2+𝑓_𝑅𝑅𝑅^′′+𝑓_𝑅𝑅𝑅^′1𝐻=31𝑅𝐹−32𝑇−3𝑓, (50) and taking into account that 𝑅=𝑟𝑡⁻²,𝑟∈ℝ, and 𝐻=ℎ𝑡⁻¹, then it may be rewritten in the following way: 4𝑟𝑓_𝑅𝑅𝑅𝑅³+6𝑟−2ℎ𝑟𝑓_𝑅𝑅𝑅²=13𝜌𝑅𝐹−1−3𝛾₀3𝑟^𝛼/2𝑅^𝛼/2−23𝑓, (51) where we have identified 𝑇=(1−3𝛾)𝜌=(1−3𝛾)𝜌₀𝑡^−𝛼, and, therefore, we can write 𝑇=(((1−3𝛾)𝜌₀)/𝑟^(𝛼/2))𝑅^𝛼/2=𝐶𝑅^𝛼/2. We have found that a particular solution of this third-order ODE is the next one: 𝑓=𝑓₀𝑅^𝛼/2, (52) induced by the symmetry 𝜉=𝑅,𝜂=(𝛼/2)𝑓, which is a consistent solution with the previous one. Note that 𝑓₀=6𝑟𝐶/(−4𝑟+𝑟𝛼−3𝛼²+9𝛼²−6𝛼+3𝛼ℎ²−6𝛼ℎ).

4. Conclusions

We have studied under the self-similar hypothesis the modified gravitational 𝑓(𝑅,𝜙) models as well as its particular case 𝑓(𝑅) models arriving at the conclusion that the 𝑓(𝑅,𝜙) models admit power-law and in particular self-similar solutions if 𝑓=𝐾(𝑅)𝑔(𝜙), and 𝐾=(𝑅)^𝑟 (see Theorem 1). In the same way, we have shown that the so-called 𝑓(𝑅)-models, admit power-law and self-similar solutions (see Theorems 2 and 3) if 𝑓(𝑅)=𝑅^𝑛. This result coincides with the previous one only obtained for FRW symmetries (studying the Friedmann equation). In this work, we have shown that this result is also valid for any Bianchi type under the self-similarity hypothesis as well as under the power-law solution for the scale factors.

Acknowledgments

The author would like to thank F. Navarro-Lérida for reading and criticizing this work. The author is also extremely grateful to all referees for their valuable comments and suggestions.