Cosmological dynamics of fourth order gravity with a GaussBonnet term
Abstract
We consider cosmological dynamics in fourth order gravity with both and correction to the Einstein gravity ( is the GaussBonnet term). The particular case for which both terms are equally important on powerlaw solutions is described. These solutions and their stability are studied using the dynamical system approach. We also discuss the condition of existence and stability of a de Sitter solution in a more general situation of powerlaw and .
Physical Faculty, Moscow State University, Moscow, 119991, Russia
Sternberg Astronomical Institute, Moscow State University, Universitetsky prospect, 13, Moscow, Russia
1 Introduction
Theories of modified gravity become very popular in the beginning of our century in attempts to explain the accelerated expansion of our Universe which is proved observationally [1, 2]. Another motivation is incorporating quantum corrections to General relativity (GR) which are believed to be present in the form of terms of higher order in the curvature.
It is known that introducing additional terms, proportional to the Kretchmann invariant into the action often leads to solution with increasing spatial anisotropy [3, 4] incompatible with the known picture of our Universe. This is the reason why modified gravity theories often deal with only corrections to HilbertEinstein action. However, special properties of the GaussBonnet combination make correction terms reasonable to study. Since is a topological invariant in 4 dimensions, terms linear in do not modify the equations of motion (if enters into the action in a product with function of another dynamical variable, like in stringgravity corrections, it modifies equations of motion and gives rise to interesting effects [5, 6, 7]). Cosmological effects of correction terms have been studied recently for a number of different functions .
In the present paper we consider both and corrections to the Einstein gravity, find a situation where they are equally important and outline differences of cosmological behaviour in such combined model from models with only or terms in the action.
Viable cosmological models have been constructed for each theory separately (see, e.g., [8, 9, 10, 11] for ). Furthermore, finitetime singularities ([12]), cosmological perturbations ([13]), stability around a spherically symmetric static spacetime ([14]), and the number of degrees of freedom and ghost problems ([15]) have already been investigated. Recently more general theories with have become a subject of investigation [14, 16]. The review [17] considers some more complicated versions of modified gravity including nonlocal theories and solutions which can be of interest from the viewpoint of cosmic acceleration have been described. Our goal is different: we try to consider the general picture of cosmological dynamics in modified gravity theories applying the Dinamical system approach, used actively in gravity [18, 19, 20]. We make the next step taking into account the GaussBonnet term restricting ourself in the present paper to a particular case where the function is a sum of a function of and a function of . We find asymptotic regimes and studying their stability in such a modified GaussBonnet gravity theory.
2 Basic equations
The action we study is given by^{1}^{1}1The signature of the metric is assumed to be and the speed of light is taken to be equal to 1.
(1) 
where and are general differentiable functions of the Ricci scalar and the GaussBonnet invariant , is the Newton constant. Variation of this action with respect to leads to the following field equations:
(2) 
where
(3) 
The stressenergy tensor is
(4) 
Note that here and henceforth we use the notations , .
As the attempt to consider the general even without the GaussBonnet term [18] appears to be not fully successful and was criticized in [20], we consider a particular simplest and mostly natural family of powerlaw functions in the Lagrangian. Namely, we study the case of and because it is important to reproduce GR results in a lowcurvature limit.
In the highcurvature limit, the Einstein contribution can be neglected, so we can choose . The GaussBonnet term is a total derivative in spacetime, therefore, a linear function of in the action will not contribute to the field equations. We consider the FriedmanLemaitreRobertsonWalker metric for a flat isotropic universe:
(5) 
using the Hubble parameter , we rewrite the field equations in the following form:
(6a)  
(6b) 
Moreover, we will need the continuity equation, the cosmological equation of state and expressions for the Ricci scalar and the GaussBonnet invariant in terms of the Hubble parameter:
(7)  
(8)  
(9)  
(10) 
3 The existence of de Sitter solutions
The existence of de Sitter solutions, corresponding to an exponentially increasing scale factor (), is an important opportunity in modified gravity theories.
Taking the trace of Eq.(2), and assuming we find:
(11) 
Consider the cases, mentioned in the previous section:
1
:
(12) 
Using the relations , we get:
(13) 
We can see that de Sitter is absent for the following combinations of and :

for any

for any

,
Thus de Sitter solutions exist for all other combinations of and .
2
:
(14) 
and
(15) 
We can see that de Sitter is absent for the following combinations of and :

,

and
Thus, de Sitter solutions exist for all other combinations of and .
We can see that adding function into the modified gravity action allows us to obtain new de Sitter solutions which do not exist in and pure GaussBonnet gravity.
4 Cosmological dynamics
Since the field equations in Modified Gravity theories are nonlinear fourth order differential equations, which are very difficult to solve, we use the Dynamical system approach. It provides a powerful and relativity simple scheme for obtaining asymptotic solutions and investigating their stability.
The main goal of this approach is to get some autonomous system of first order differential equations, as a consequence of the cosmological equations of motion. In this approach, the dynamics of the Universe corresponds to motion along a phase curve, and stationary points represent some asymptotic regimes of the Universe evolution.
Dividing Eq. (6a) by , we find:
(16) 
Now we introduce the following normalized expansion variables:
(17a)  
(17b)  
(17c)  
(17d)  
(17e)  
(17f)  
(17g) 
We take derivative of these variables with respect to the dimensionless time :
(18a)  
(18b)  
(18c)  
(18d)  
(18e)  
(18f)  
(18g) 
Using the new variables, Eq.(16) can be rewritten as
(19) 
We have obtained a set of seven equations, but our theory is of the fourth order. As we will see further, our dimensionless variables are not independent, consequently, this system is overdetermined, and our next purpose is to reduce its dimensionality.
To complete transformation, we need to express all righthand side terms in Eqs. (18) through the variables defined in Eqs.(17).
Using Eqs. (8), (7), (9), and (10), we find:
(20)  
(21)  
(22)  
(23)  
(24)  
where  
(25) 
If the functions and are chosen, we can close the system due to additional relations between our variables.
The case of . Using Eqs. (8) and (9), we get:
(26)  
(27) 
Thus, using these relations with the constraint (19), we exclude , and from our autonomous system. As a result, we obtain a set of four equations:
(28a)  
(28b)  
(28c)  
(28d) 
where
(29)  
(30) 
We would get additional relations between the variables for every particular function .
5 The highcurvature case of
First of all, we can easily find in this special case
(31)  
(32) 
We can substitute these relations into (28) and exclude from our system:
(33) 
Note that the factor is a denominator in our system. It means that the selected set of variables is useless for GaussBonnet gravity with the Einstein term, where .
Solving the system with vanishing lefthand sides, we find the following fixed points :
,
,
,
,
.
The first four points are precisely the same as those obtained by Carloni, Dunsby, Capozziello and Troisi for gravity [19].^{1}^{1}1 Note that they have used a bit different set of variables and signature from our paper. The GaussBonnet term does not contribute here.
Except the gravity solutions, here are two new stationary points:

. It is the de Sitter point.

. The scale factor is , so this point represents the wellknown scale factor evolution dynamics of gravity, though the set of coordinates for this point is different from the first and second points, which also correspond to .
It was shown in [21] (where combined effects of and corrections were studied) that using the dynamical system approach we can obtain some new nontrivial solutions mainly where the correction terms in the action are equally important for a powerlaw evolution of the scale factor. Also there exist some additional relations between variables in this case. As we see further, the similar picture exists in our system.
The case in models. There is an additional relation between and in this special case:
(34) 
Consequently, one can reduce the system to two equations:
(35) 
It is useful to describe our future strategy. We have started from the model and consider its highcurvature limit, where can be neglected. We have two coupling constants, and , however, when neglecting the Einstein term in the action, effectively our result depends only on their ratio /. Thus we can assume . Next, we select the model by fixing and find stationary points for some values of . Having this information we will discuss the situation in the general case.
We mostly investigate the case of positive integers and , so is chosen to be even. If , so the GaussBonnet term does not contribute. Thus, we study the case of in detail.
The case of ,. Stationary points in this model, which have been found numerically are, by the scheme (, , , ):
:(, , , ) ,
:(, , , ) ,
The corresponding eigenvalues are (for the last point eigenvalues depends on equationofstate parameter , so the values are presented for several particular ):
These points are summarized in the Table 1.
Point  Coordinates  Stability  The scale 
of stationary point  type  factor,  
Repulsive node  
Attractive node  
Saddle  
The case of , . Stationary points in this model are, by the scheme :
(, , , ) ,
(, , , ,
The stability types for all this points did not change as compared with the case . The situation in all these cases is the same as has been found in gravity – only one stationary point (which corresponds to a phantom solution) is stable, others are unstable.
The general case of . Now we try to describe what happens for an arbitrary . Substituting into Eq. (6b), we get for () an algebraic instead of differential equation:
(36) 
Neglecting matter and the Einstein term, assuming , we find the equation for as a function of :
(37) 
A Stability analysis provided for a set of particular reveals the following picture:

A0. In this area there are only phantom branches, which are always stable. For they are solutions . Note that in the coupling constant interval we do not find any phantom solution at all.

A1. Branches of this region are always unstable.

A. For this interval [,] the branch stability depends on the parameter (e.g., for the eigenvalues are []). Considering also the matterdominated solution, we get the following picture: there is some critical value
(38) In the case of , the matterdominated solution is stable while the vacuum powerlaw solution is unstable. In the opposite case , the solution with is stable while is unstable, i.e. solution with a larger power index is always stable. Thus, there is always one stable nonphantom solution for any in the range of the coupling constants . In the range (between the local maximum and minimum of function , see Fig. 2) we can find two stable vacuum solutions for some values of .

A. Here the vacuum powerlaw solution becomes unstable for any , while the matterdominated solution is still stable for some values close to . Furthermore, it becomes also unstable for .
However, it does not mean that there are no stable solutions for this range of coupling constants. In this section we considered models without the Einstein term, i.e., highcurvature regimes. Actually, earlier we noted that we should add contribution for obtaining a latetime standard cosmology. This contribution gives us a de Sitter solution, which is stable in this range of coupling constants (see below).
The case of general n. Note, that Eq.(36) can be simplified only when the powerlaw ansatz terms originating from and contribution are equally important. In this section we considered the case for and discussed it in detail. Our studies show that the picture obtained does not change qualitatively for any . In all these models we can consider as a function of the power index in the time dependence of the scale factor ,
(39) 
and admit some common features:

There is a stable nonphantom solution in the coupling constants range , where is the value of the function at the local minimum point
. Moreover, there is some critical value , determining stability in this range:
(40) Solution with a larger power index (it is a matter dominated point or a vacuum powerlaw solution, depending on ) is stable.

After reaches the value , this branch becomes unstable.

There also exists a range of the coupling constants
where we have no phantom solutions at all.
The only qualitative difference for the case of odd and noninteger is the absence of type solution in the range .
We should conclude that for there is a stable solution with . It means that this solution cannot be of powerlaw type. Using Eq. (13) we easily find that it cannot be a de Sitter point. Since in Quantum field theory coupling constants are usually running, we do not consider in detail this solution existing only for a particular value of the coupling constant.
It is interesting that in gravity with we always have a stable phantom solution. By using the powerlaw ansatz it is simple to obtain that in pure GaussBonnet gravity with there is a phantom solution when . So, a combined effect of both and correction terms may lead to solutions that are qualitatively different from those in pure or theories.
6 Dynamics in theories
Now we consider cosmological dynamics in a theory where the only correction to Einstein gravity is a function of the GaussBonnet term. It was shown in Sec.5 that the set of dimensionless variables (17) does not allow us to study this case, so we choose another set of variables. Substituting into (6), we find the field equations:
(41)  
(42) 
We rewrite these equations in a dimensionless form:
(43)  
(44) 
Substituting , , we get:
(45) 
The new dimensionless variables are
(46a)  
(46b)  
(46c)  
(46d) 
Taking derivative of the introduced variables with respect to , we get the following set of equations:
(47a)  
(47b)  
(47c)  
(47d) 
Taking into account the relations
(48)  
(49)  
(50)  
(51) 
we find the resulting autonomous system:
(52a)  
(52b)  
(52c) 
Equating the lefthand side of our system to zero, we find the stationary point , which corresponds to a de Sitter solution.
As in the previous case, nontrivial results have been obtained for a particular value of the power index .
The case . There is an additional relation in the case of :
(53) 
We can substitute this relation into our system and find
(54a)  
(54b) 
Using the technique applied in the previous section, we find stationary points for particular values of . For they are, by the scheme ():
,
) ,
The corresponding eigenvalues are:
Now turn to the case of . We get the fixed points by the scheme ():
) ,
with the eigenvalues
We summarize our results in Table 2.
Point  Coordinates  Stability  The scale 
of stationary points  type  factor,  
Repulsive node  
Attractive node  
Saddle  
Repulsive node  
Saddle 