Exact cosmological solutions with nonminimal derivative coupling
We consider a gravitational theory of a scalar field with nonminimal derivative coupling to curvature. The coupling terms have the form and where and are coupling parameters with dimensions of length-squared. In general, field equations of the theory contain third derivatives of and . However, in the case the derivative coupling term reads and the order of corresponding field equations is reduced up to second one. Assuming , we study the spatially-flat Friedman-Robertson-Walker model with a scale factor and find new exact cosmological solutions. It is shown that properties of the model at early stages crucially depends on the sign of . For negative the model has an initial cosmological singularity, i.e. in the limit ; and for positive the universe at early stages has the quasi-de Sitter behavior, i.e. in the limit , where . The corresponding scalar field is exponentially growing at , i.e. . At late stages the universe evolution does not depend on at all; namely, for any one has at . Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.
For many years scalar fields have been an object of great interest for physicists. The reasons for this are manifold. One of them is quite pragmatic: models with scalar fields are relatively simple, and therefore it appeared possible to study them in detail and then extrapolate the results to more realistic and complicated models. More physical motivations include such important topics as the idea about variable “fundamental” constants, the Jordan-Brans-Dicke theory initially elaborated to solve the Mach problem, the Kaluza-Klein compactification scheme, the low-energy limit of the superstring theory, and others. Scalar fields play an especially important role in cosmology. As a bright example, one may mention numerous inflationary models in which inflation in the early Universe is typically driven by a fundamental scalar field called an inflaton. Furthermore, a recent discovery of cosmic acceleration has only refreshed the interest to scalar fields which began to be considered as candidates to explain dark energy phenomena.
The rather general form of action for a scalar-tensor theory of gravity with a single scalar field can be given as111Throughout this paper we use units such that . The metric signature is and the conventions for curvature tensors are and .
where is a metric, , and is the scalar curvature. Functions and are varying from theory to theory. The function is responsible for the sign of kinetic energy of the scalar field. For example, the choice leads to a wide class of theories with the negative kinetic term. The function , being in general nonlinear, provides a nonminimal coupling between a scalar field and curvature. Though a freedom in choosing of leads to an unlimited variety of scalar-tensor theories, it is known (see, for example, Mae (); FarGunNar (); CroFra ()) that there exist conformal transformations transforming this kind of theories to Einstein’s theory with a new minimally coupled scalar field and an effective potential describing its self-interaction. The potential is a very important ingredient of various cosmological models: a slowly varying potential behaves like an effective cosmological constat providing one or more than one inflationary phases. An appropriate choice of is known as a problem of fine tuning of the cosmological constant.
A further extension of scalar-tensor theories can be represented by models with nonminimal couplings between derivatives of a scalar field and curvature. This kind of couplings may appear in some Kaluza-Klein theories KK1 (); KK2 () (see also Lindebook (), Section 9.5). In 1993, Amendola Ame () has been considered the most general gravity Lagrangian linear in the curvature scalar , quadratic in , and containing terms with four derivatives including all of the following terms (see also CapLamSch () for details):
where coefficients are coupling parameters with dimensions of length-squared. Using the divergencies
one may conclude that, without loss of generality, , , and are not necessary to be considered. Also one may rule out because it contains itself, while coupling term of main interest are those, where only the gradient of is included. Thus, a general scalar-tensor theory with nonminimal derivative couplings may include only two terms and .
As was shown by Amendola Ame (), a theory with derivative couplings cannot be recasting into Einsteinian form by a conformal rescaling . He also supposed that an effective cosmological constant, and then the inflationary phase can be recovered without considering any effective potential if a nonminimal derivative coupling is introduced. Amendola himself Ame () has considered a cosmological model in the theory with the only derivative coupling term and, by using a generalized slow-rolling approximation (i.e., neglecting all terms of order higher than the second one), he has obtained some analytical inflationary solutions. A general model containing both and has been discussed in CapLamSch () (see also CapLam ()); it was shown that the de Sitter spacetime is an attractor solution of the model if . Recently Daniel and Caldwell DanCal () have considered a theory with the derivative coupling term ; in particular, they studied constraints which precision tests of general relativity impose on the coupling parameter . It is also worth mentioning a series of papers devoted to a nonminimal modification of the Einstein-Yang-Mills-Higgs theory BalDehZay:07 () (see also a review BalDehZay () and references therein).
In this paper we continue studying a scalar-tensor theory with nonminimal derivative couplings and construct new exact cosmological solutions of the theory.
Ii Field equations
Let us consider a gravitational theory of a scalar field with nonminimal derivative couplings to curvature described by the action
Here coefficients and are derivative coupling parameters with dimensions of length-squared. Note that the action 2 does not include any potential. Varying the action 2 with respect to the metric gives the gravitational field equations
where is the Einstein tensor. Then, varying the action 2 with respect to gives the scalar field equation of motion:
Note also that because of the Bianchi identity, , the scalar field and order- terms form a conserved system, hence the scalar field equation 4 can be obtained as a consequence of the generalized conservation law .
Generally, the gravitational field equations 3 contain third derivatives of , whilst the scalar field equation 4 contains third derivatives of the metric. However, an important feature of the theory 2 is the fact that the order of field equations can be reduced for a specific choice of and . To show this we rewrite, after some algebra, the expressions for and as follows:
It is seen that both expressions contain similar third-order terms, and , which are cancelled in the combination provided the coupling parameters are chosen as follows
The gravitational field equations 3 now read
and the scalar field equation of motion 4 yields
Iii Cosmological models
Consider a spatially-flat cosmological model with a metric
where , and are constants of integration. Without loss of generality one may put and , then the corresponding metric reads
The spacetime with the metric 17 has an initial singularity at .
Consider now a general case . In this case the constraint 12 can be rewritten as follows
From here it follows that and should obey the following conditions:
Let us analyze an asymptotical behavior of and for large times. Suppose that tends to some nonzero constant at ; respectively, this means that should go to zero. However, as it follows from Eq. 22, is not zero in this limit. Thus, we face with a contradiction and, therefore, should conclude that if . By using this asymptotical property, we obtain the following asymptotic form of Eq. 22:
The corresponding asymptotical solution is
where and are constants of integration. An asymptotic for can be found straightforwardly from the constraint 18:
To characterize an asymptotical behavior of and for small times, we consider separately two cases.
First, let be negative, . In this case the condition 21 gives the following bound for :
while the condition 20 is fulfilled for all values of . Since is monotonically decreasing with time, its value should be growing with decreasing time. Let be some initial moment of time (possibly ). Suppose that tends to some constant value in the limit ; respectively, this means that should go to zero. However, as follows from Eq. 22, is not zero in this limit. This is a contradiction, and hence we should conclude that is boundlessly increasing in the limit . Assuming at gives the following asymptotical form of Eq. 22:
with the asymptotic solution
The asymptotic for is found from Eq. 18 as
where , and are constants of integration. The corresponding asymptotical form of the metric 11 is
A spacetime with this metric is singular at . This singularity is analogous to initial cosmological singularities in models with usual scalar fields. However, a new interesting feature of the examined model is that the scalar field with negative derivative coupling has the regular behavior 30 near the singularity.
Then, let be positive, . In this case the condition 20 gives the following bound for :
with the asymptotic solution
where is a constant of integration which without loss of generality can be set zero. We see that in the limit the scalar is exponentially growing, and is exponentially approximating to its asymptotic
Hence, in the limit the spacetime metric 11 takes asymptotically the de Sitter-like form:
with . Thus, in the case an universe at early stages has the quasi-de Sitter behavior corresponding to the cosmological constant .
Results of numerical study in the case are shown in Fig. 2. Note that all solutions represent an interesting feature. Namely, they describe two phases in evolution of the universe. First, for an infinitely long time the universe is living in the quasi-de Sitter or inflationary phase. Then, during a relatively short time the universe exits from the inflationary stage and goes to a power-law expansion with (it is worth noting that this law corresponds to the equation of state ).
We have considered the gravitational theory of a scalar field with nonminimal derivative coupling to curvature and studied cosmological models in this theory. The main results obtained are as follows:
1. The Lagrangian of the theory includes two derivative coupling terms and , where and are coupling parameters with dimensions of length-squared. In general, field equations of the theory are of third order, i.e., contain third derivatives of and , but in the particular case the order of equations is reduced up to the second one. This case corresponds to the choice , then a combination of derivative coupling terms turn into . It is worth noting that Capozziello et al CapLamSch () , at pages 43 and 47, have mentioned the case to play a special role, because it represents a singular point of the differential equation. In this paper, we have supposed that the theory with is more preferable with the physical point of view, since the corresponding field equations do not contain derivatives of dynamical variables of order higher than the second.
2. Assuming , we have studied a cosmological model with the spatially-flat Friedman-Robertson-Walker metric. It was shown that a behavior of the scale factor and the scalar field at large times is the same for all values of including zero, that is the late evolution of universe does not depend on . Namely, one has and at . Note this asymptotical behavior coincides with that of the exact solution 15, 16 obtained for (no coupling).
3. General properties of the model crucially depends on a sign of . For an asymptotical form of the cosmological metric for small times is given by Eq. 31. A corresponding scale factor is ; it describes the universe with an initial singularity at . A new interesting feature of the model with derivative coupling is that a behavior of the scalar field near the cosmological singularity is regular, (see Eq. 30). For the law of universe evolution is qualitatively distinct from that for . Now at early stages the universe has the quasi-de Sitter behavior 37 corresponding to the cosmological constant . In the limit the scale factor has the following asymptotical form (see Eq. 35), hence exponentially fast goes to the de-Sitter form with . At the same time, the scalar field is exponentially growing at , namely .
In conclusion, let us summarize the most essential features of cosmology with nonminimal derivative coupling. First of all, we should emphasize that cosmological solutions with the quasi-de Sitter phase are typical solutions of the gravitational theory of a scalar field with derivative coupling of the form with positive . So, in order to obtain an inflationary phase, one need no fine-tuned potential, and so one do not face with the problem of fine-tuning. Another important feature of the model consists in the fact that an exact cosmological solution with describes in a unique manner both a quasi-de Sitter phase and an exit from it. Thus, the problem of graceful exit from inflation in cosmology with the derivative coupling term has a natural solution without any fine-tuned potential.
This work was supported in part by the Russian Foundation for Basic Research grants No. 08-02-91307, 08-02-00325.
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