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Journal of Mathematics
Volume 2013 (2013), Article ID 169020, 5 pages
and Models: A Self-Similar Approach
Departamento Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, 28040 Madrid, Spain
Received 15 September 2012; Revised 28 November 2012; Accepted 12 December 2012
Academic Editor: Jianming Zhan
Copyright © 2013 José Antonio Belinchón. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the models and their particular case, the so-called -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).
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  (for a review see ), we have studied some particular cases of the so-called generalized scalar-tensor theories (STTs or 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 models (see, e.g., [7–9] and for a review ). Therefore, the purpose of this brief paper is to study, under the self-similarity hypothesis, the generalized theories, where the models (without ) result as subclasses and the generalized SST with studied in . 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)  and the matter collineation approach (MC) . 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 (HVF). For a metric, , and for a vector field , the homothetic equation reads: . We study self-similar solutions since, as it has been pointed out by Rosquist and Jantzen , they correspond to equilibrium points. In the same way, Coley  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.,  for a review of results in the context of the general relativity).
Our purpose is to describe a class of scalar-tensor theories of gravity represented by the action [17, 18]: where is the Ricci scalar, and stands for the scalar field, where we assume that the scalar field is homogeneous, that is, . We introduce the constant in order to fix units, and is a classical matter Lagrangian. The gravitational field equations derived from the action (1) are where is the Einstein tensor, is the energy-momentum tensor, and
The wave equation reads where a prime denotes derivation with respect to time.
We will work under the assumptions that is, may be split in separable variables and the function takes this particular form, which is appropriate under the self-similarity hypothesis.
Proof. We are going to study (4) through the LG method, that is, we study the kind of functions ,, , and 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: since , , being the exponents of the scale factors, that is, , 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, ; therefore, a prime denotes a derivative with respect to time. Note that in the self-similar framework, the scalar curvature always behaves as ; therefore, we may assume that .
Roughly speaking, a symmetry, , of a differential equation is an invertible transformation that leaves it form-invariant. By applying the standard Lie procedure (see, e.g., ), we need to solve the following overdetermined system of linear partial differential equations for and (from the extended infinitesimal or prolonged transformations), which allows us to determine the set of the symmetries admitted by (8): The knowledge of one symmetry might suggests the form of a particular solution as an invariant of the operator , that is, a solution of . This particular solution is known as an invariant solution (generalization of similarity solution). The symmetry, , which induces the invariant solution, , leads us to obtain the following restrictions. From (10), with, . From (12), we get Therefore, where we have taken into account that . In this way, while obtaining the following solution Compare this result with the obtained one in . Note that if (in the self-similar framework the Ricci scalar always varies as ), that is, , then and therefore
In any case, when , then
thus Therefore, the so-called models admit power-law and in particular self-similar solutions.
As we have shown, for different values of and we obtain different models. Note that the case (the usual one) has been already studied in .
As a particular case of the above theories we may study the so-called models. This kind of models is described by the action: where is the Ricci scalar, and is a matter Lagrangian as in the above model. By defining , the gravitational field equations derived from the action (23) are : where is the Ricci scalar, and is the Einstein tensor. The trace of the field equations read where is the trace of the stress-energy tensor.
We deduce the form that can acquire the function 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 if the scale factor follow a power-law (see, e.g., ). 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 is precisely compatible with the self-similar hypothesis. With the second one, by studying the trace equation with the Lie group method we proof that is the unique possible form for the function 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 is .
Proof. We may define an effective stress-energy tensor as follows:
which allows to use the standard Einstein equations by simply replacing the fluid quantities with the effective ones, so we have to calculate
where stands for a homothetic vector field. We calculate the Lie derivative of each component of the effective stress-energy tensor.(1)Thus, , in components, ,
As it is noted, we are only able to obtain a relationship between the two quantities and , but we have not determine their particular form.(2). The first term is , that is,
that we may rewrite in the following form:
that we may write as
since , and . Now taking into account the fact that , then the equation yields
whose solution is
With the other components, , we only obtain restriction on the scale factors.(3), that is,
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 ,
now, taking into account that , then we may rewrite it as follows:
and whose solution is
If, for example, we set , from the above result (), therefore, we arrive to the conclusion that
Now if we take into account that (from the conservation equation for the matter field) being and the result from (28), we arrive to the conclusion that .
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 , in the next section we study if under the hypothesis of power-law solutions for the scale factors the function 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 is .
Proof. By studying the trace equation (25), that we rewrite as follows:
where we have set
with , that is, they are constants, .
The general solution of (39) is where
Note that if in (39) we set ., then which implies that (as in the standard case).
We will determine the form of the unknown function 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:
For example, the symmetry , , which induces the invariant solution , brings us to obtain the following restriction on the function . From (47), we obtain whose solution is and which is consistent with the previous result .
To finish this proof, we would like to emphasize that if one tries to find a symmetry whose invariant solution coincides with the solution , from the assumption , as for example, , 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 and (i.e., . In this way, (39) yields and taking into account that , and , then it may be rewritten in the following way: where we have identified , and, therefore, we can write . We have found that a particular solution of this third-order ODE is the next one: induced by the symmetry , which is a consistent solution with the previous one. Note that .
We have studied under the self-similar hypothesis the modified gravitational models as well as its particular case models arriving at the conclusion that the models admit power-law and in particular self-similar solutions if , and (see Theorem 1). In the same way, we have shown that the so-called -models, admit power-law and self-similar solutions (see Theorems 2 and 3) if . 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.
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.
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