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Generalized Self-similar Scalar-Tensor Theories

We study through symmetry principles the form of the functions in the generalizated scalar-tensor theories under the self-similar hypothesis. The results obtained are absolutely general and valid for all the Bianchi models and the flat FRW one. We study the concrete example of the Kantowsky-Sach model finding some exact self-similar solutions.


Introduction
Current observations of the large scale cosmic microwave background suggest to us that our physical universe is expanding in an accelerated way, isotropic and homogeneous models with a positive cosmological constant. The analysis of Cosmic Microwave Background (CMB) fluctuations could confirm this picture. But other analyses reveal some inconsistencies. Analysis of WMAP data sets shows us that the universe might have a preferred direction. For this reason, it may be interesting to study Bianchi models since these models may describe such anisotropies.
The observed location of the first acoustic peak of the temperature fluctuations on the CMB corroborated by the data obtained in different experiments [1], indicates that the universe is dominated by an unidentified "dark energy" and suggests that this unidentified dark energy has a negative pressure [2]. This last characteristic of the dark energy points to the vacuum energy or cosmological constant as a possible candidate for dark energy. Although, it is a general belief that the current curvature of the universe is negligible and mostly the universe is considered with a flat geometry, recent observations support the possibility of a non-flat universe and detect a small deviation from k = 0 [3]. For example, evidence from CMB and also supernova measurements of the cubic correction to the luminosity distance favour a positively curved universe [4,5].
In order to explain the current acceleration of the universe, within General Relativity (GR), it is necessary to introduce a new type of energy with a negative pressure. Between the different possible approaches is one which consists of considering so-called dark energy (DE). There are several candidates for DE, where the simplest one is the cosmological term Λ. However, this choice has several drawbacks such as coincidence and fine tuning problems. For this reason other models have been proposed. Examples of such models are quintessence [6,7], K-essence [8], or chameleonic fields in which a scalar field is coupled to matter [9], etc. Hence it is natural and important to consider a variable Λ−term in more general frameworks where, furthermore, other quantities, such as the Newton gravitational constant may be considered as dynamical. One such class of theories are the scalar-tensor theories (STT) of gravity. This class of models has received a renewed interest in recent times, for two main reasons: First, the new inflationary scenario as the extended inflation has a scalar field that solves several problems present in the old theories. Secondly, string theories and other unified theories contain a scalar field which plays a similar role to the scalar field of the STT. The scalar-tensor theories started with the work of P. Jordan in 1950 [10]. A prototype of such models was proposed by Brans and Dicke in 1961 [11]. Their aim for presenting this model was to modify Einstein's theory in such a way as to incorporate the so called "Mach's principle". These theories have been generalized by P.G. Bergmann [12], K. Nordtverdt [13] and R. T. Wagoner [14]. For a recent review of this class of theories we refer to [15] and [16].
In this paper we want to consider a family of scalar-tensor theories with a dynamical cosmological constant [17] and with a potential [15], that is equivalent to a time dependent cosmological constant. Recently several authors have considered the cosmological consequences of a time varying cosmological constant. Most of them introduce the time dependence in an ad hoc manner. In this work we consider an equivalent problem in the well known general scalar-tensor theory of gravity where the time dependence can occur in a natural way, without any new assumption or modification of the theory and provide an explanation for the acceleration of the universe expansion [18].
Many authors have studied these general scalar-tensor theories. They use the observational data in order to obtain restrictions or constraint between the functions that appear in the action to obtain an accelerated model [19]- [22]. Our approach is different: We want to derive these functions from symmetry principles as self-similarity. We shall carry out our study under this assumption and state some general theorems that are valid for all the Bianchi models and of course for the flat FRW one. It is most appropriate for us to work in the Jordan frame (JF), in which the physical quantities are those that are being measured in experiments, even though the Einstein frame (EF) often provides a better mathematical insight.
The study of self-similar (SS) models is quite important since a large class of orthogonal spatially homogeneous models are asymptotically self-similar at the initial singularity and are approximated by exact perfect fluid or vacuum self-similar power-law models. Exact self-similar power-law models can also approximate general Bianchi models at intermediate stages of their evolution. This last point is of particular importance in relating Bianchi models to the real Universe. At the same time, self-similar solutions can describe the behaviour of Bianchi models at late times i.e. as t → ∞ [23]. This paper is organized as follows. In section two we start by considering a particular formulation of the theory. In this case the cosmological constant is introduced directly by the function Λ (φ ) [17]. We study this model through two different approaches. The first consists of studying the effective stress-energy tensor under the matter collineation approach. This method allows us to obtain relationships between the physical quantities as well as to determine the exact form of the scalar field φ . The second approach consists of studying the wave equation under the Lie group method. By imposing a particular symmetry we are able to determine the exact form for each of the functions that appear in this equation. We summarize all the results by stating a very general theorem. In section three we study a very general scalar-tensor theory. In this case the cosmological constant is introduced by the potential. Following the same exposed procedure as in the above section, we are able to determine the exact form that all of the unknowns involved in this model must have. We show, from the stated theorem, how different versions of this theory arise, which are the standard Brans-Dicke theory, the induced gravity model [24] and a very particular solution where the effective gravitational function is constant. In section four we study a chameleon Jordan-Brans-Dicke model. In order to show how all the obtained results work, in section five we study a particular example which is the Kantowski-Sach model. We start this section by showing that this metric admits a homothetic vector fields and then we study several models. We put special emphasis on comparing the solutions obtained in each case. In section six we end by summarizing all the results. We have added an appendix where we study in detail one of the equations obtained in section 2 in order to show with out any doubt that the Brans-Dicke parameter ω (φ ) must be constant in this framework of self-similar solutions.

Cosmological models with dynamical Λ in scalar-tensor theories
Following to Will (see [17]) we start with the action for the most general scalar-tensor theory of gravitation where g = det(g i j ), G * is Newton's constant, S NG is the action for the nongravitational matter. We use the signature (−, +, +, +). The arbitrary functions ω (φ ) and Λ (φ ) distinguish the different scalar-tensor theories of gravitation, Λ (φ ) is a potential function and plays the role of a cosmological constant, and ω (φ ) is the coupling function of the particular theory.
The explicit field equations are where T = T i i is the trace of the stress-energy tensor. The gravitational coupling G eff (t) is given by

Matter collineations
We may calculate the relationship between the quantities (in a SS approach) by calculating the matter collineations. Therefore we have to compute (we use unit where 8π = c = 1) where HO stands for a homothetic vector field. For simplicity we have used a flat FRW metric but we would like to emphasize that all the obtained results are absolutely valid for all the Bianchi models, since we only look for the behaviour of the physical quantities instead of restriction on the scale factors. Therefore the homothetic vector field (HFV) is Note the non-singular character of the HVF, nevertheless for simplicity in the calculations we use the singular case.
We find the next solution for Eq. (11) For example this is the unique solution mathematically possible (compatible with the SS hypothesis, see the appendix for an explanation).
If ω = const. then and therefore we get the following ODE i.e.
note that H = ht −1 , h ∈ R + . These equation are different for each Bianchi model and we only obtain restriction on the scale factors.

Lie groups
We are going to study the Eq. (3) through the LG method, i.e. we study the kind of functions Λ (φ ) and ω (φ ) such that this equation is integrable. We start by rewriting it in an appropriate way where h = const., h ∈ R + , B = 2, and C = 8π (1 − 3γ) ρ 0 . Note that we are taking into account the conservation equation div T = 0, i.e. ρ = ρ 0 t −α , where α = (1 + γ)h, and H = ht −1 .
We need to solve the following system of PDE where The symmetry ξ = t, η = nφ , brings us to obtain the following restriction on Λ (φ ) . From Eq. (27) we get so and where we have taken into account Eq. (30) therefore we have obtained the following ODE for Λ (φ ) and whose general solution is we choose since from our result from the MC approach we already know that, Λ (t) = t −2 , so this means that n + α = 2.
In an alternative way, from Eq. (25) we get and therefore since 2 = n + α. In the appendix we shall give and alternative and detailed proof of this result.

Theorem 1
The scaling and in particular the self-similar solution admitted for the FE (2-3) have the following form

The General case
We start by defining the action [21] and therefore the FE read and where R is the scalar curvature.
The effective gravitational constant G eff between two test masses measured in laboratory Cavendish-type experiments is given by

Matter collienations
We may define and therefore, as above: 3. T 3 = 1 F(φ ) F ;µ F ;ν − g µν F depends on the metric so we only obtain restriction on the scale factors.
and therefore

Lie groups
For this model we have to solve the following equation that we may rewrite in the following form where we have assumed φ = φ (t), and the derivatives respect t are denoted by a comma. Note that R ≈ Ct −2 , with C ∈ R.
The standard procedure brings us to outline the following system of PDE: The symmetry ξ = t, η = nφ , brings us to obtain the following restrictions. From Eq. (54) where m ∈ R. From Eq. (56) we get and therefore so while obtaining This theorem state that we have a set of theories which admit scaling and in particular self-similar solutions if the involved functions take this particular form. The action reads So setting different values for the constant m we obtain different theories. For example if m = 0 (induced gravity case) then the action (40) yields and if m = 1 (usual JBD theory with a potential) then the action (40) yields and to end if m = 2, then the action (40) yields we get: This particular case is very similar to the scalar field cosmological model with G eff ≈ const. Notice that this model is quite different from the Barker's theory [25].

Chameleon cosmology
We begin with the BD chameleon theory in which the scalar field is coupled non-minimally to the matter field via the action where R is the Ricci scalar curvature, φ is the BD scalar field with a potential U(φ ). The chameleon field φ is non-minimally coupled to gravity, ω is the dimensionless BD parameter. The last term in the action indicates the interaction between the matter Lagrangian L m and some arbitrary function J(φ ) of the BD scalar field. In the limiting case J(φ ) = 1, we obtain the standard BD theory.
The gravitational field equations derived from the action (68) with respect to the metric is The Klein-Gordon equation (or the wave equation) for the scalar field is Similarly the energy conservation for the cosmic fluid iṡ We shall use the equation of state (EoS) for the fluid p = γρ, thus (71) yields ρ = ρ 0 (t + t 0 ) −α .

Matter collineations
By defining as it is observed it is only necessary to calculate the first component, the rest of them have been already calculated in the above sections. If yields algebra brings us to get: whereJ ′ = J φ φ ′ , but we do not obtain more information about the behaviour of some of the functions φ or J (φ ) .

Lie Groups
We need to study the following ODE where, Therefore the standard procedure brings us to outline the following system of PDE As we already know, from the MC approach, it must be verified the relationship algebra brings us to get For example the symmetry ξ = t, η = nφ , brings us to obtain the following restriction on U (φ ) . From Eq. (83) we obtain the ODE whose solution has been obtained in section 3, i.e. U (φ ) = U 0 φ 1 n (n−2) . From Eq. (82) we get the next ODE for J : finding that the most general solution is

The Kantowski-Sach model
We start by considering the Killing vector fields (KVF) such that in such a way that the metric takes the following form We find that the metric (91) admits the following HVF where the scale factors behave as Tanking into account these result we shall calculate some exact cosmological solutions. It is a straightforward task to carry out the calculations for this reason we only show the results. The detailed exposition of the followed method may be found for example in [27].

Vacuum solution
There is no SS vacuum solution for this model

Perfect fluid model
The stress-energy tensor is defined by, T i j = (ρ + p)u i u j − pg i j , and where we are taking into account the conservation principle so, T ; j i j = 0, and the equation of state p = γρ, (γ = const.). We find that and this solution is only valid for Note that our solution accelerates since

Time-varying constant model
In this model we consider the constants G and Λ as time varying functions in such a way that the FE are: with the constrain T ; j i j = 0.
We have found the next results where where γ c is given by Eq. (95). This solution is valid for all EoS γ, i.e. there is no restrictions on the γ parameter. As above q < 0.

JBD model with CC
For this model (this corresponds to the exposed one in section 2) the FE are as follows where T = T i i is the trace of the stress-energy tensor. The gravitational coupling G eff (t) is given by From the stated theorem of section 2 we already know that the physical quantities behave as follows: We have found the next solution. The scale factors behave as where note that γ c is given by Eq. (95).

JBD model with potential
For this model (which corresponds to the particular model, m = 1, exposed in section 3) the FE are as follows where T = T i i is the trace of the stress-energy tensor. The gravitational coupling G eff (t) is given by We have found the following solution. The scale factors behave as where note that γ c is given by Eq. (95), while Therefore this solution is only valid if γ ∈ −1, 1 9 and γ = 1.

Chameleon JBD model
This model corresponds to the exposed one in section 4. In this case FE read: where the main quantities behave as follows We have obtained the following solution. n = n(m); in such a way that n = 0, iff m c = ±1.414213562 = ± √ 2, we only consider the positive solution, so, m ∈ R + \ {1} . n < 0 if m ∈ [0, 1) ∪ (m c+ , ∞) , and n > 0 if m ∈ (1, m c+ ) .
The rest of the quantities depend on (m, γ, ω) , so in order to carry out the numerical analysis it is necessary to fix the value of ω and γ. Setting ω = 4 · 10 4 , we have studied some equation of state, γ = 1, 1/3, 0, −1/3, i.e. the usual ones.
In order to clarify and to compare (with the following results) the results we have write in the following table the roots of the constants (n, φ 0 ,U 0 , ρ 0 ) :   In fig. 4 we have plotted in detail the interval m ∈ (m 1 , , m c ).

Induced Gravity model
In this model the FE read where R is the scalar curvature and F (φ ) = φ 2 /4ω. The gravitational coupling G eff (t) is given by For the metric (91) the scalar curvature is given by: As we already know, the physical quantities behave as follows: We have found the following solution. The scale factors behave as where, as it is observed if n = 0, then m = √ 2. φ 0 = φ 0 (n, γ) behaves as follows note that φ 0 = 0, iff γ = −1,and n = 0. With regard to the constant U 0 = U 0 (n, γ, ω) we have found the next value The constant for the energy density, ρ 0 = ρ 0 (n, γ, ω) , behaves as follows: Since the solutions depend on the parameters (n, γ, ω) then we need to fix them. For example, if we set For ω ≈ 4 ·10 4 , then the solutions only depend on (n, γ) in such a way that given different values to n then we shall may to study their behaviour. For n = −1, ω ≈ 4 · 10 4 we get φ 0 > 0 and ρ 0 > 0 iff γ < −1 while U 0 < 0 if γ < −1. We arrive at the same conclusion if n = −2. Note that the cases n = 0 and n = 1 are forbidden. Therefore, the obtained solution is only valid if γ < −1.

Conclusions
We have studied under the self-similar hypothesis the admitted form of the different unknown functions in several scalartensor theories. By employing the matter collineation (MC) approach, i.e. calculating the Lie derivate of the effective stressenergy tensor with respect to an HVF, and the Lie group method, we have been able to state theorems valid for all the Bianchi geometries as well as for a flat FRW metric. We have used both tactics, because with the MC in some of the models studied we are only able to obtain relationships between the physical quantities. With this tactic we only obtain self-similar solutions. Nevertheless, with the Lie group method we are able to obtain the exact form for each of the physical quantities. Furthermore, with this approach we obtain scaling solutions which are more general than the self-similar one.
In the first of the models studied we arrive to the conclusion that φ ≈ (t + t 0 ) n , and therefore G eff ≈ (t + t 0 ) −n , while the dynamical cosmological constant behaves as Λ ≈ (t + t 0 ) −2 . In the same way we have deduced that the Brans-Dicke parameter ω BD (φ ) must be constant. In the second of the models studied, the generalized scalar-tensor model, the dynamical cosmological constant is mimicked by the potential U (φ ) , and therefore we have three unknown functions, F (φ ) , Z (φ ) and the potential U (φ ) . We arrive at a very general result which allows us to outline different scalar-tensor models that admit self-similar solutions. Actually this result is in agreement with the fact that we may pass from one model to another through conformal transformations. As an example we have emphasized three relevant models, the standard scalar-tensor model, the induced gravity model and a specific model which is very similar to the scalar cosmological model where G eff ≈ const. In the third of the models studied, following the same procedure, we calculate the admitted form for the unknown functions J (φ ) and the potential in order to obtain scaling and self-similar solutions.
Once we have established all these results then we study a particular example, the Kantowski-Sach model. The same procedure may be applied to other Bianchi models. We begin by showing that this metric admits and HVF. We explore some cosmological models and applying the stated theorems we find exact solutions. We show that there is no vacuum solution. For the perfect fluid case, within the general relativity framework, we find that the solution obtained is only valid for a particular equation of state, γ c , which is not strange in this class of solutions, while the exponent of one of the scale factors is irrational. This fact is odd since these constants usually are rational ones. Note that the solution is inflationary since q < 0 without any necessity to appeal to a DM component. In the third model, where G and Λ are considered as time-varying within the general relativity framework we have shown that the obtained solution is valid for all value of γ. In this case G may be a growing or decreasing function finding that it behaves as a true constant only when γ = γ c . The dynamical cosmological constant is always a decreasing time function but it may be positive or negative and it vanishes if γ = γ c . Once we know how each physical quantity works in the general relativity framework then we explore some solutions for the scalar-tensor models.
We have not been able to find a solution in the case of a Brans-Dicke model with a dynamical Λ. This result is quite surprising since following the same procedure we have obtained solutions for several Bianchi models. Nevertheless for the BD model with a potential we obtain an exact self-similar solution. The numerical analysis carried out shows us that the solution obtained is only valid in a small neighbourhood, E (γ c ) , of γ c , the critical value of γ, where the energy density and the scalar function are positive. The solution shows analogies with that obtained in the subsection where G and Λ are considered as time-varying within the general relativity framework. For example, G eff ≈ (t + t 0 ) −n may be a growing or decreasing function finding that it behaves as a true constant only when γ = γ c . The dynamical cosmological constant, the potential U (φ ) , is always a decreasing time function but it may be positive in the interval of definition or vanish if γ = γ c . The model also accelerates since q < 0. Trying to generalize this scenario we also consider a chameleon BD model. We show that the obtained solution is valid for all values of γ instead of only in E (γ c ) . In the last model studied, the induced gravity case, we find that the solution is only valid if γ < −1 (a phantom scenario). In the appendix we emphasize the fact that the only form compatible with self-similar solutions for the BD parameter ω BD (φ ) is ω BD (φ ) = const. Therefore none of the scaling solutions admit a variable ω BD (φ ) . this means that ω (φ ) = const.
as we already know.