Stability of Einstein and de Sitter Universes in the Quadratic Theory of Modified Gravity

We consider the Einstein static and the de Sitter universe solutions and examine their instabilities in a subclass of quadratic modified theories for gravity. This modification proposed by Nash is an attempt to generalize general relativity. Interestingly, we discover that the Einstein static universe is unstable in the context of the modified gravity. In contrast to Einstein static universe, the de Sitter universe remains stable under metric perturbation up to the second order.


Introduction
According to Wilkinson Microwave anisotropy probe [1,2], the BICEP2 experiment [3,4], Sloan Digital Sky Surveys [5] and Planck satellite [6][7][8], it turns out that less than 5% of the Universe is composed of ordinary matter, 68% is of dark energy and 27% of Universe is composed of the dark matter. On the other hand it has been shown also that the observed Universe is undergoing an accelerated expansion [9]. The late-time cosmic acceleration can be explained by two promising explanations, at least. One of them is the dark energy component in the Universe [10] deduced from the abovementioned detectors although the nature of the dark energy is not known yet. The other explanation is to tackle the problem using a geometrical picture modifying Einstein theory of gravity. The approach is known as the modified gravity which have several motivations in high-energy physics, cosmology and astrophysics [11]. Modified theories of gravity can be achieved from different contexts. In modifying gravity theories f (R), f (R, T ), f (G), f (R, G), f (R, ϕ) and f (R, R ab R ab , ϕ) are some attractive choices. Here, R, T , G, R ab R ab and ϕ are Ricci scalar, trace of energy momentum tensor, Gauss-Bonnet invariant, Ricci invariant and scalar field respectively. In f (R) theories of gravity the Lagrangian density f is an arbitrary function of the scalar curvature R [12,13]. One of the earlier modifications to Einstein's general relativity (GR) is known as the Brans-Dicke gravity. This theory introduced a dynamical scalar field using a variable gravitational constant [14]. Later, there was a study of a scalar-tensor theory of gravity in which the metric is coupled to a scalar field where, a 'missing-mass problem' can be successfully described [15]. This approach can be applied to the Bianchi cosmological models.
These theories, also, include higher order curvature invariants [16,17]. For example, some modified gravity theory models describe flat rotation of galaxies without taking into account the cold dark matter particles [19]. Other models which describe accelerating Universe expansion with a quadratic term of R in the Lagrangian density, proposed by Starobinsky [18].
In these theories it is an important task to show and prove that these Universe models are stable against small perturbations in the Hubble parameters. Searches have been performed showing that Einstein Universe [20], for example, is stable against such perturbations in vector or tensor field or scalar density instabilities. John Nash has developed Einstein's theory of gravity alternatively. Quantum gravity theories can make a benefit using this theory. Also, many cosmological models have a problem of divergence whereas this theory is divergence free.
In this paper, we examine the Einstein static and the de Sitter Universe solutions and quantify their stabilities for the Bianchi type I model using Nash modified theory of gravity in the context of Bianchi Universe geometry.
In Section II we introduce the Nash theory for gravity. In Section III we explain the Bianchi-I Universe and in Section IV the Lagrangian used in the paper is stated. In Section V we apply the Lagrangian with respect to Einstein static Universe and study its corresponding stability. In Section IV we repeat the analysis in Section V but with the de Sitter Universe. Finally, we conclude our findings in the last section.
2 Nash's theory for gravity J. Nash developed an alternative theory of modified gravity in which he modified the GR. This is a way to ultimately consider GR as renormalizable theory [21]. More recent works are presented in [22,23]. The original Nash gravity action proposed without Einstein-Hilbert general relativity term is: There are several Lagrangians used to develop theories of quantum gravity. The one written above is one of them. Using the above action and taking into account the metric g µν as a dynamical field, the gravitational field equations are directly derived as: Nash theory have been investigated in the context of Noether thery and its cosmological implication [24]. In this project we examine the cosmological solutions for homogeneous anistropic Universe.

Bianchi-I Universe
Due to the diversity and non-locality of subregions in the Universe in terms of galactic distributions and internal galactic structures we study Nash model in the context of Bianchi Type I model which can considers the homogeneity and non-isotroposity as well as the non-rotation of cosmos. We recall that in FLRW model the scale factor is unique. However, because of the non-isotropic feature of the Bianchi Type I different scale factors are encountered in each direction. Consequently, in x µ = (t, x, y, z) coordinates, the line-element is, Here, A(t), B(t) and C(t) are scale factors and they are all functions of the cosmic time, t. It is notable that the FLRW has been generalized to the above metric function. The mean of the three directional Hubble parameters in the Bianchi Type I Universe is given by
We define the Hubble constants as new parameters: where the unknown functions which must be obtained by this symmetry methods are {H 1 , H 2 , H 3 }.
We consider the metric (3) and substitute it into the action (1). We then perform the integration by parts and eliminate the second derivative terms with respect to time (Ä i ). We, then, obtain the following point-like Lagrangian, which enable us to investigate the symmetry properties of the system:

Einstein static solutions and their (in)stability
The viability of cosmological solutions is plagued by the issue of stability. This is so because of the existence of varieties of perturbations, e.g. the quantum fluctuations. In order to study the instability problem, the Einstein static solutions must be retrieved in the context of different theories of modified gravity [25][26][27][28][29][30][31][32][33][34][35]. In this section, we examine the stability of the solutions by using a local stability method. Using the metric (3), the field equations for Nash gravity can be written in the following system of second order differential equations: dB(t)/dH 2 and C ′ = dC(t)/dH 3 and A, B and C are functions of t. The Einstein static universe is a closed universe and the scale factors A(t), B(t), C(t) are constant, implying that, where i = 1, 2, 3. Thus, we introduce perturbation in the Hubble parameter which depends only on time: Substituting the perturbations (10) into Eqs.(6)-(8), we find that the perpetuated equations can be written as for which the solutions take the form where H (0) i and H (1) i are constants of integration. It is apparent from Eq.(11) that we have no oscillation equations. Consequently, the Einstein static universe is unstable against the perturbations and hence this concludes that Einstein universe is unstable in the context of the Nash gravity.

Stability of the de Sitter Universe
In the previous section we investigated the linear stability of the Einstein universe. The analysis was done quickly because in the Einstein Universe all Hubble terms and their time derivatives vanish. In this section we will investigate stability of the de Sitter solution. This Einstenian anisotropic metric is defined as the anisotropic expansion of the flat FLRW metric. The directional Hubble parameters H i = h i are considered as stationary values at the beginning, when the space time metric is stationary. Plugging these constant Hubble parameters and A = ae h 1 t , B = be h 2 t , C = ce h 3 t in the field equations (6-8) we obtain: Now we make the perturbation H i → h i +δξ i (t) in equations (6)(7)(8). Using the constraint of the zero order equations (13-15) we obtain a system of the nonlinear differential equations for δξ i (t) where we omit terms of high orders of δξ i (t). Hence, we yield to the following system of the differential equations: These set of equations can be reduced further by taking the derivative of the A, B and C where the derivative is required. Hence we obtain: Collecting the perturbation terms together yields: In the above equations, provided that A 1,2 , C 1,2 , E 1,2 > 0, if we substitute very large numbers for t; (i.e t → ∞) we find out that δξ i → 0 for any values of h 1 , h 2 or h 3 since the power of the exponent is negative. Consequently, we can deduce that the de Sitter universe is stable against the perturbations. Therefore, it is stable in the context of the Nash gravity. Otherwise, the de Sitter universe will become unstable as the perturbations become very large in the context of Nash gravity.

Conclusion
As an alternative theory for gravity Einstein's theory of gravity was developed and modified by John Nash. The theory is divergence free and considered to be of interest in constructing theories of quantum gravity. In this paper, we examine the Einstein static and the de Sitter Universe solutions and quantify their stabilities for the Bianchi type I model using Nash modified theory of gravity in the context of Bianchi universe geometry. We searched for the stability of Einstein and de Sitter Universes using Nash gravity Model. We used the metric of Bianch type I model in the Nash field equation. Then we substituted the appropriate scale factors in the field equations depending on the type of universe under study. We have seen that the Einstein static universe leads to instabilities against perturbation in the Hubble constant. On the other hand the de Sitter universe shows stability in the perturbation if the parameters, which combine different Hubble constants for an anisotrpic universe is A 1,2 , C 1,2 , E 1,2 > 0.