Stability of Einstein and de Sitter Universes in the Quadratic Theory of Modified Gravity
aa r X i v : . [ phy s i c s . g e n - ph ] S e p SciPost Physics Proceedings Submission
Linear Stability of Einstein and de Sitter Universes in theQuadratic Theory of Modified Gravity
Mudhahir Al Ajmi , Department of Physics, College of Science, Sultan Qaboos University,P.O. Box 36, P.C. 123, Muscat, Sultanate of Oman* [email protected] 15, 2020
Abstract
We consider the Einstein static and the de Sitter universe solutions and exam-ine their instabilities in a subclass of quadratic modified theories for gravity.This modification proposed by Nash is an attempt to generalize general relativ-ity. Interestingly, we discover that the Einstein static universe is unstable in thecontext of the modified gravity. In contrast to Einstein static universe, the deSitter universe remains stable under metric perturbation up to the second order.
International Conference on Holography, String Theory and Discrete Approaches in HanoiPhenikaa UniversityAugust 3-8, 2020Hanoi, Vietnam
Contents ciPost Physics Proceedings Submission According to Wilkinson Microwave anisotropy probe [1, 2], the BICEP2 experiment [3, 4],Sloan Digital Sky Surveys [5] and Planck satellite [6–8], it turns out that less than 5% ofthe Universe is composed of ordinary matter, 68% is of dark energy and 27% of Universeis composed of the dark matter. On the other hand it has been shown also that the ob-served Universe is undergoing an accelerated expansion [9]. The late-time cosmic accelerationcan be explained by two promising explanations, at least. One of them is the dark energycomponent in the Universe [10] deduced from the abovementioned detectors although thenature of the dark energy is not known yet. The other explanation is to tackle the problemusing a geometrical picture modifying Einstein theory of gravity. The approach is knownas the modified gravity which have several motivations in high-energy physics, cosmologyand astrophysics [11]. Modified theories of gravity can be achieved from different contexts. Inmodifying gravity theories f ( R ) , f ( R, T ) , f ( G ) , f ( R, G ) , f ( R, ϕ ) and f ( R, R ab R ab , ϕ ) are someattractive choices. Here, R , T , G , R ab R ab and ϕ are Ricci scalar, trace of energy momentumtensor, Gauss-Bonnet invariant, Ricci invariant and scalar field respectively. In f ( R ) theoriesof 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 gravitationalconstant [14]. Later, there was a study of a scalar-tensor theory of gravity in which the metricis 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, somemodified gravity theory models describe flat rotation of galaxies without taking into accountthe cold dark matter particles [19]. Other models which describe accelerating Universe ex-pansion 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 arestable against small perturbations in the Hubble parameters. Searches have been performedshowing that Einstein Universe [20], for example, is stable against such perturbations in vectoror tensor field or scalar density instabilities.John Nash has developed Einstein’s theory of gravity alternatively. Quantum gravitytheories can make a benefit using this theory. Also, many cosmological models have a problemof divergence whereas this theory is divergence free.In this paper, we examine the Einstein static and the de Sitter Universe solutions andquantify their stabilities for the Bianchi type I model using Nash modified theory of gravityin the context of Bianchi Universe geometry.In Section II we introduce the Nash theory for gravity. In Section III we explain theBianchi-I Universe and in Section IV the Lagrangian used in the paper is stated. In Section Vwe apply the Lagrangian with respect to Einstein static Universe and study its correspondingstability. 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 ciPost Physics Proceedings Submission 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 worksare presented in [22, 23]. The original Nash gravity action proposed without Einstein-Hilbertgeneral relativity term is: S = Z d x L = Z d x √− g (cid:16) R µν R µν − R (cid:17) . (1)There are several Lagrangians used to develop theories of quantum gravity. The one writtenabove is one of them. Using the above action and taking into account the metric g µν as adynamical field, the gravitational field equations are directly derived as: ✷ G µν + G αβ (cid:16) R µα R νβ − g µν R αβ (cid:17) = 0 . (2)Nash theory have been investigated in the context of Noether thery and its cosmologicalimplication [24]. In this project we examine the cosmological solutions for homogeneousanistropic Universe. Due to the diversity and non-locality of subregions in the Universe in terms of galactic distri-butions and internal galactic structures we study Nash model in the context of Bianchi TypeI model which can considers the homogeneity and non-isotroposity as well as the non-rotationof cosmos. We recall that in FLRW model the scale factor is unique. However, because ofthe non-isotropic feature of the Bianchi Type I different scale factors are encountered in eachdirection. Consequently, in x µ = ( t, x, y, z ) coordinates, the line-element is, ds = − dt + A ( t ) dx + B ( t ) dy + C ( t ) dz . (3)Here, A ( t ) , B ( t ) and C ( t ) are scale factors and they are all functions of the cosmic time, t . Itis notable that the FLRW has been generalized to the above metric function. The mean of thethree directional Hubble parameters in the Bianchi Type I Universe is given by H = P H i where H i = d ln( A i ) dt , A i = { A, B, C } . Here, the Lagrangian can be formalized as point-like parameters characterized by the config-uration space, i.e. L = L ( A, B, C, H , H , H , ˙ H , ˙ H , ˙ H ).We define the Hubble constants as new parameters: H = ˙ AA = d ln( A ( t )) dt , H = ˙ BB = d ln( B ( t )) dt , H = ˙ CC = d ln( C ( t )) dt , (4)where the unknown functions which must be obtained by this symmetry methods are { H , H , H } .3 ciPost Physics Proceedings Submission We consider the metric (3) and substitute it into the action (1). We then perform theintegration by parts and eliminate the second derivative terms with respect to time ( ¨ A i ). We,then, obtain the following point-like Lagrangian, which enable us to investigate the symmetryproperties of the system: L = − ABC (cid:16) H H + H H + H H + 3 H H H + H H + H H (5)+3 H H H + 3 H H H + H H + H H + H H + H H + H ˙ H + H ˙ H + H H ˙ H + H H ˙ H + 2 H H ˙ H + H H ˙ H + 2 H H ˙ H + H H ˙ H + H ˙ H + H ˙ H + 2 H H ˙ H + H H ˙ H + H H ˙ H + H ˙ H + H ˙ H + ˙ H ˙ H + ˙ H ˙ H + ˙ H ˙ H (cid:17) . The viability of cosmological solutions is plagued by the issue of stability. This is so becauseof the existence of varieties of perturbations, e.g. the quantum fluctuations. In order tostudy the instability problem, the Einstein static solutions must be retrieved in the contextof different theories of modified gravity [25–35]. In this section, we examine the stability ofthe solutions by using a local stability method. Using the metric (3), the field equations forNash gravity can be written in the following system of second order differential equations: − H H A ′ BC − H H A ′ BC − H H A ′ BC − H H H A ′ BC − H H A ′ BC (6) − H H A ′ BC − H H H A ′ BC − H H H A ′ BC − H H A ′ BC − H H A ′ BC − H H A ′ BC − H H A ′ BC + 3 H H ABC + 3 H H ABC + 2 H H ABC +6 H H H ABC + 2 H H ABC + H ABC + 3 H H ABC + H ABC + ABC ′ H H ˙ H + ABC ′ H H ˙ H + ABC ′ H ˙ H + 2 ABC ′ H H ˙ H + ABC ′ H ˙ H + AB ′ CH H ˙ H + AB ′ CH H ˙ H + AB ′ CH ˙ H + 2 AB ′ CH H ˙ H + AB ′ CH ˙ H − ˙ H H A ′ BC − ˙ H H A ′ BC − ˙ H H H A ′ BC − H H H A ′ BC − H H H A ′ BC − ˙ H H H A ′ BC − ˙ H H A ′ BC − ˙ H H H A ′ BC − ˙ H H H A ′ BC − H ˙ H A ′ BC + ˙ H H ABC + ˙ H H ABC − ˙ H H ABC − ABCH ˙ H − ˙ H H ABC + ABC ′ ˙ H ˙ H + ABC ′ ˙ H + AB ′ C ˙ H + AB ′ C ˙ H ˙ H − ˙ H ˙ H A ′ BC − ¨ H ABC − ¨ H ABC = 0 , ciPost Physics Proceedings Submission and − H H B ′ AC − H H B ′ AC − H H B ′ AC − H H H B ′ AC − H H B ′ AC (7) − H H B ′ AC − H H H B ′ AC − H H H B ′ AC − H H B ′ AC − H H B ′ AC − H H B ′ AC − H H B ′ AC + H ABC + 2 H H ABC + 3 H H ABC +3 H H ABC + 6 H H H ABC + 3 H H ABC + 3 H H ABC + 2 H H ABC + H ABC + ABC ′ H ˙ H + ABC ′ H H ˙ H + 2 ABC ′ H H ˙ H + ABC ′ H H ˙ H + ABC ′ H ˙ H − ˙ H H B ′ AC − H H ˙ H B ′ AC − H H H B ′ AC − H H ˙ H B ′ AC − ˙ H H H B ′ AC − H ˙ H B ′ AC − ˙ H H B ′ AC − H H ˙ H B ′ AC − ˙ H H H B ′ AC − H ˙ H B ′ AC + A ′ BCH ˙ H + A ′ BCH H ˙ H +2 A ′ BCH H ˙ H + A ′ BCH H ˙ H + A ′ BCH ˙ H − H ˙ H ABC + H ˙ H ABC + ˙ H H ABC − ˙ H H ABC + ABC ′ ˙ H ˙ H + ABC ′ ˙ H − ˙ H ˙ H B ′ AC + A ′ BC ˙ H + A ′ BC ˙ H ˙ H − ¨ H ABC − ¨ H ABC = 0 , as well as − H H C ′ AB − H H C ′ AB − H H C ′ AB − H H H C ′ AB − H H C ′ AB (8) − H H C ′ AB − H H H C ′ AB − H H H C ′ AB − H H C ′ AB − H H C ′ AB − H H C ′ AB − H H C ′ AB + H ABC + 3 H H ABC + 2 H H ABC +3 H H ABC + 6 H H H ABC + 3 H H ABC + H ABC + 2 H H ABC + 3 H H ABC − ˙ H H C ′ AB − H H ˙ H C ′ AB − ˙ H H H C ′ AB − H H ˙ H C ′ AB − H H H C ′ AB − H ˙ H C ′ AB − H H ˙ H C ′ AB − ˙ H H H C ′ AB − H ˙ H C ′ AB − H ˙ H C ′ AB + AB ′ CH ˙ H + 2 AB ′ CH H ˙ H + AB ′ CH H ˙ H + AB ′ CH ˙ H + AB ′ CH H ˙ H + A ′ BCH ˙ H + 2 A ′ BCH H ˙ H + A ′ BCH H ˙ H + A ′ BCH ˙ H + A ′ BCH H ˙ H − H ˙ H ABC − ˙ H H ABC + H ˙ H ABC + ˙ H H ABC − ˙ H ˙ H C ′ AB + AB ′ C ˙ H ˙ H + AB ′ C ˙ H + A ′ BC ˙ H + A ′ BC ˙ H ˙ H − ¨ H ABC − ¨ H ABC = 0 , where we have defined A ′ = dA ( t ) /dH , B ′ = dB ( t ) /dH and C ′ = dC ( t ) /dH 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, H i = ˙ H i = 0 , (9)where i = 1 , ,
3. Thus, we introduce perturbation in the Hubble parameter which dependsonly on time: H i → δH i , ˙ H i → δ ˙ H i , ¨ H i → δ ¨ H i . (10)Substituting the perturbations (10) into Eqs.(6)-(8), we find that the perpetuated equationscan be written as δ ¨ H + δ ¨ H = 0 , δ ¨ H + δ ¨ H = 0 , δ ¨ H + δ ¨ H = 0 . (11)for which the solutions take the form δH i = H (0) i (cid:18) tt (cid:19) + H (1) i , (12)5 ciPost Physics Proceedings Submission where H (0) i and H (1) i are constants of integration. It is apparent from Eq.(11) that we haveno oscillation equations. Consequently, the Einstein static universe is unstable against theperturbations and hence this concludes that Einstein universe is unstable in the context ofthe Nash gravity. In the previous section we investigated the linear stability of the Einstein universe. Theanalysis was done quickly because in the Einstein Universe all Hubble terms and their timederivatives vanish. In this section we will investigate stability of the de Sitter solution. ThisEinstenian 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 begin-ning, when the space time metric is stationary. Plugging these constant Hubble parametersand A = ae h t , B = be h t , C = ce h t in the field equations (6-8) we obtain: − h h A ′ BC − h h A ′ BC − h h A ′ BC − h h h A ′ BC − h h A ′ BC (13) − h h A ′ BC − h h h A ′ BC − h h h A ′ BC − h h A ′ BC − h h A ′ BC − h h A ′ BC − h h A ′ BC + 3 h h ABC + 3 h h ABC + 2 h h ABC +6 h h h ABC + 2 h h ABC + h ABC + 3 h h ABC + h ABC = 0 , − h h B ′ AC − h h B ′ AC − h h B ′ AC − h h h B ′ AC − h h B ′ AC (14) − h h B ′ AC − h h h B ′ AC − h h h B ′ AC − h h B ′ AC − h h B ′ AC − h h B ′ AC − h h B ′ AC + h ABC + 2 h h ABC + 3 h h ABC +3 h h ABC + 6 h h h ABC + 3 h h ABC + 3 h h ABC + 2 h h ABC + h ABC = 0 , − h h C ′ AB − h h C ′ AB − h h C ′ AB − h h h C ′ AB − h h C ′ AB (15) − h h C ′ AB − h h h C ′ AB − h h h C ′ AB − h h C ′ AB − h h C ′ AB − h h C ′ AB − h h C ′ AB + h ABC + 3 h h ABC + 2 h h ABC +3 h h ABC + 6 h h h ABC + 3 h h ABC + h ABC + 2 h h ABC + 3 h h ABC = 0 , Now we make the perturbation H i → h i + δξ i ( t ) in equations (6-8). Using the constraint ofthe 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 systemof the differential equations: 6 ciPost Physics Proceedings Submission − ( h h + 3 h h δξ + h δξ ) A ′ BC − ( h h + 3 h h δξ + h δξ ) A ′ BC (16) − ( h h + 2 h h δξ + 2 h h δξ ) A ′ BC − h h h + 2 h h h δξ + h h δξ + h h δξ ) A ′ BC − ( h h + 2 h h δξ + 2 h h δξ ) A ′ BC − ( h h + 3 h h δξ + h δξ ) A ′ BC − h h h + 2 h h h δξ + h h δξ + h h δξ ) A ′ BC − h h h + 2 h h h δξ + h h δξ + h h δξ ) A ′ BC − ( h h + 2 h h δξ + h δξ ) A ′ BC − ( h h + 3 h h δξ + h δξ ) A ′ BC − ( h h + 2 h h δξ + 2 h h δξ ) A ′ BC − ( h h + 3 h h δξ + h δξ ) A ′ BC + 3 ( h h + 2 h h δξ + h δξ ) ABC +3 ( h h + 2 h h δξ + h δξ ) ABC + 2 ( h h + 2 h h δξ + h δξ ) ABC +6 ( h h h + h h δξ + h h δξ + h h δξ ) ABC + 2 ( h h + 2 h h δξ + h δξ ) ABC +( h + 3 h δξ ) ABC + 3 ( h h + 2 h h δξ + h δξ ) ABC + ( h + 3 h δξ ) ABC + ABC ′ ( h h δ ˙ ξ ) + ABC ′ ( h h δ ˙ ξ ) + ABC ′ ( h δ ˙ ξ )+2 ABC ′ ( h h δ ˙ ξ ) + ABC ′ ( h δ ˙ ξ ) + AB ′ C ( h h δ ˙ ξ )+ AB ′ C ( h h δ ˙ ξ ) + AB ′ C ( h δ ˙ ξ )+2 AB ′ C ( h h δ ˙ ξ ) + AB ′ C ( h δ ˙ ξ ) − ( h δ ˙ ξ ) A ′ BC − ( h δ ˙ ξ ) A ′ BC − ( ˙ h h δξ + h h δ ˙ ξ ) A ′ BC − h h δ ˙ ξ ) A ′ BC − h h δ ˙ ξ ) A ′ BC − ( h h δ ˙ ξ ) A ′ BC − ( h δ ˙ ξ ) A ′ BC − ( h h δ ˙ ξ ) A ′ BC − ( h h δ ˙ ξ ) A ′ BC − ( h δ ˙ ξ ) A ′ BC + ( h δ ˙ ξ ) ABC + ( h δ ˙ ξ ) ABC − ( h δ ˙ ξ ) ABC − ABC ( h δ ˙ ξ ) − ( h δ ˙ ξ ) ABC − ( δ ¨ ξ ) ABC − ( δ ¨ ξ ) ABC = 0 , ciPost Physics Proceedings Submission and − ( h h + 3 h h δξ + h δξ ) B ′ AC − ( h h + 3 h h δξ + h δξ ) B ′ AC (17) − ( h h + 2 h h δξ + 2 h h δξ ) B ′ AC − h h h + 2 h h h δξ + h h δξ + h h δξ ) B ′ AC − ( h h + 2 h h δξ + 2 h h δξ ) B ′ AC − ( h h + 3 h h δξ + h δξ ) B ′ AC − h h h + 2 h h h δξ + h h δξ + h h δξ ) B ′ AC − h h h + 2 h h h δξ + h h δξ + h h δξ ) B ′ AC − ( h h + 2 h h δξ + h δξ ) B ′ AC − ( h h + 3 h h δξ + h δξ ) B ′ AC − ( h h + 2 h h δξ + 2 h h δξ ) B ′ AC − ( h h + 3 h h δξ + h δξ ) B ′ AC + ( h + 3 h δξ ) ABC +2 ( h h + 2 h h δξ + h δξ ) ABC + 3 ( h h + 2 h h δξ + h δξ ) ABC +3 ( h h + 2 h h δξ + h δξ ) ABC + 6 ( h h h + h h δξ + h h δξ + h h δξ ) ABC +3 ( h h + 2 h h δξ + h δξ ) ABC + 3 ( h h + 2 h h δξ + h δξ ) ABC +2 ( h h + 2 h h δξ + h δξ ) ABC + ( h + 2 h δξ ) ABC + ABC ′ h δ ˙ ξ + ABC ′ h h δ ˙ ξ + 2 ABC ′ h h δ ˙ ξ + ABC ′ h h δ ˙ ξ + ABC ′ h δ ˙ ξ − h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h δ ˙ ξ B ′ AC − h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h δ ˙ ξ B ′ AC + A ′ BCh δ ˙ ξ + A ′ BCh h δ ˙ ξ +2 A ′ BCh h δ ˙ ξ + A ′ BCh h δ ˙ ξ + A ′ BCh δ ˙ ξ − h δ ˙ ξ ABC + h δ ˙ ξ ABC + h δ ˙ ξ ABC − h δ ˙ ξ ABC − δ ¨ ξ ABC − δ ¨ ξ ABC = 0 , as well as − ( h h + 3 h h δξ + h δξ ) C ′ AB − ( h h + 3 h h δξ + h δξ ) C ′ AB (18) − ( h h + 2 h h δξ + 2 h h δξ ) C ′ AB − h h h + 2 h h h δξ + h h δξ + h h δξ ) C ′ AB − ( h h + 2 h h δξ + 2 h h δξ ) C ′ AB − ( h h + 3 h h δξ + h δξ ) C ′ AB − h h h + 2 h h h δξ + h h δξ + h h δξ ) C ′ AB − h h h + 2 h h h δξ + h h δξ + h h δξ ) C ′ AB − ( h h + 2 h h δξ + h δξ ) C ′ AB − ( h h + 3 h h δξ + h δξ ) C ′ AB − ( h h + 2 h h δξ + 2 h h δξ ) C ′ AB − ( h h + 3 h h δξ + h δξ ) C ′ AB + ( h + 3 h δξ ) ABC + 3 ( h h + 2 h h δξ + h δξ ) ABC + 2 ( h h + 2 h h δξ + h δξ ) ABC +3 ( h h + 2 h h δξ + h δξ ) ABC + 6 ( h h h + h h δξ + h h δξ + h h δξ ) ABC +3 ( h h + 2 h h δξ + h δξ ) ABC + ( h + 3 h δξ ) ABC + 2 ( h h + 2 h h δξ + h δξ ) ABC +3 ( h h + 2 h h δξ + h δξ ) ABC − h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h δ ˙ ξ C ′ AB − h δ ˙ ξ C ′ AB + AB ′ Ch δ ˙ ξ + 2 AB ′ Ch h δ ˙ ξ + AB ′ Ch h δ ˙ ξ + AB ′ Ch δ ˙ ξ + AB ′ Ch h δ ˙ ξ + A ′ BCh δ ˙ ξ + 2 A ′ BCh h δ ˙ ξ + A ′ BCh h δ ˙ ξ + A ′ BCh δ ˙ ξ + A ′ BCh h δ ˙ ξ − h δ ˙ ξ ABC − h δ ˙ ξ ABC + h δ ˙ ξ ABC + h δ ˙ ξ ABC − δ ¨ ξ ABC − δ ¨ ξ ABC = 0 , ciPost Physics Proceedings Submission − (3 h h δξ + h δξ ) A ′ BC − (3 h h δξ + h δξ ) A ′ BC (19) − (2 h h δξ + 2 h h δξ ) A ′ BC − h h h δξ + h h δξ + h h δξ ) A ′ BC − (2 h h δξ + 2 h h δξ ) A ′ BC − (3 h h δξ + h δξ ) A ′ BC − h h h δξ + h h δξ + h h δξ ) A ′ BC − h h h δξ + h h δξ + h h δξ ) A ′ BC − (2 h h δξ + h δξ ) A ′ BC − (3 h h δξ + h δξ ) A ′ BC − (2 h h δξ + 2 h h δξ ) A ′ BC − (3 h h δξ + h δξ ) A ′ BC +3 (2 h h δξ + h δξ ) ABC + 3 (2 h h δξ + h δξ ) ABC + 2 (2 h h δξ + h δξ ) ABC +6 ( h h δξ + h h δξ + h h δξ ) ABC + 2 (2 h h δξ + h δξ ) ABC +(3 h δξ ) ABC + 3 (2 h h δξ + h δξ ) ABC + (3 h δξ ) ABC + ABC ′ ( h h δ ˙ ξ ) + ABC ′ ( h h δ ˙ ξ ) + ABC ′ ( h δ ˙ ξ )+2 ABC ′ ( h h δ ˙ ξ ) + ABC ′ ( h δ ˙ ξ ) + AB ′ C ( h h δ ˙ ξ )+ AB ′ C ( h h δ ˙ ξ ) + AB ′ C ( h δ ˙ ξ )+2 AB ′ C ( h h δ ˙ ξ ) + AB ′ C ( h δ ˙ ξ ) − ( h δ ˙ ξ ) A ′ BC − ( h δ ˙ ξ ) A ′ BC − ( ˙ h h δξ + h h δ ˙ ξ ) A ′ BC − h h δ ˙ ξ ) A ′ BC − h h δ ˙ ξ ) A ′ BC − ( h h δ ˙ ξ ) A ′ BC − ( h δ ˙ ξ ) A ′ BC − ( h h δ ˙ ξ ) A ′ BC − ( h h δ ˙ ξ ) A ′ BC − ( h δ ˙ ξ ) A ′ BC + ( h δ ˙ ξ ) ABC + ( h δ ˙ ξ ) ABC − ( h δ ˙ ξ ) ABC − ABC ( h δ ˙ ξ ) − ( h δ ˙ ξ ) ABC − ( δ ¨ ξ ) ABC − ( δ ¨ ξ ) ABC = 0 , − (3 h h δξ + h δξ ) B ′ AC − (3 h h δξ + h δξ ) B ′ AC (20) − (2 h h δξ + 2 h h δξ ) B ′ AC − h h h δξ + h h δξ + h h δξ ) B ′ AC − (2 h h δξ + 2 h h δξ ) B ′ AC − (3 h h δξ + h δξ ) B ′ AC − h h h δξ + h h δξ + h h δξ ) B ′ AC − h h h δξ + h h δξ + h h δξ ) B ′ AC − (2 h h δξ + h δξ ) B ′ AC − (3 h h δξ + h δξ ) B ′ AC − (2 h h δξ + 2 h h δξ ) B ′ AC − (3 h h δξ + h δξ ) B ′ AC +(3 h δξ ) ABC + 2 (2 h h δξ + h δξ ) ABC + 3 (2 h h δξ + h δξ ) ABC +3 (2 h h δξ + h δξ ) ABC + 6 ( h h δξ + h h δξ + h h δξ ) ABC +3 (2 h h δξ + h δξ ) ABC + 3 (2 h h δξ + h δξ ) ABC +2 (2 h h δξ + h δξ ) ABC + (2 h δξ ) ABC + ABC ′ h δ ˙ ξ + ABC ′ h h δ ˙ ξ + 2 ABC ′ h h δ ˙ ξ + ABC ′ h h δ ˙ ξ + ABC ′ h δ ˙ ξ − h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h δ ˙ ξ B ′ AC − h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h h δ ˙ ξ B ′ AC − h δ ˙ ξ B ′ AC + A ′ BCh δ ˙ ξ + A ′ BCh h δ ˙ ξ +2 A ′ BCh h δ ˙ ξ + A ′ BCh h δ ˙ ξ + A ′ BCh δ ˙ ξ − h δ ˙ ξ ABC + h δ ˙ ξ ABC + h δ ˙ ξ ABC − h δ ˙ ξ ABC − δ ¨ ξ ABC − δ ¨ ξ ABC = 0 , ciPost Physics Proceedings Submission and − (3 h h δξ + h δξ ) C ′ AB − (3 h h δξ + h δξ ) C ′ AB (21) − (2 h h δξ + 2 h h δξ ) C ′ AB − h h h δξ + h h δξ + h h δξ ) C ′ AB − (2 h h δξ + 2 h h δξ ) C ′ AB − (3 h h δξ + h δξ ) C ′ AB − h h h δξ + h h δξ + h h δξ ) C ′ AB − h h h δξ + h h δξ + h h δξ ) C ′ AB − (2 h h δξ + h δξ ) C ′ AB − (3 h h δξ + h δξ ) C ′ AB − (2 h h δξ + 2 h h δξ ) C ′ AB − (3 h h δξ + h δξ ) C ′ AB +(3 h δξ ) ABC + 3 (2 h h δξ + h δξ ) ABC + 2 (2 h h δξ + h δξ ) ABC +3 (2 h h δξ + h δξ ) ABC + 6 ( h h δξ + h h δξ + h h δξ ) ABC +3 (2 h h δξ + h δξ ) ABC + (3 h δξ ) ABC + 2 (2 h h δξ + h δξ ) ABC +3 (2 h h δξ + h δξ ) ABC − h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h h δ ˙ ξ C ′ AB − h δ ˙ ξ C ′ AB − h δ ˙ ξ C ′ AB + AB ′ Ch δ ˙ ξ + 2 AB ′ Ch h δ ˙ ξ + AB ′ Ch h δ ˙ ξ + AB ′ Ch δ ˙ ξ + AB ′ Ch h δ ˙ ξ + A ′ BCh δ ˙ ξ + 2 A ′ BCh h δ ˙ ξ + A ′ BCh h δ ˙ ξ + A ′ BCh δ ˙ ξ + A ′ BCh h δ ˙ ξ − h δ ˙ ξ ABC − h δ ˙ ξ ABC + h δ ˙ ξ ABC + h δ ˙ ξ ABC − δ ¨ ξ ABC − δ ¨ ξ ABC = 0 , 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:( − (3 h h δξ + h δξ ) − (3 h h δξ + h δξ ) (22) − (2 h h δξ + 2 h h δξ ) − h h h δξ + h h δξ + h h δξ ) − (2 h h δξ + 2 h h δξ ) − (3 h h δξ + h h δξ ) − h h h δξ + h h h δξ + h h δξ ) − h h h δξ + h h h δξ + h h δξ ) − (2 h h δξ + h h δξ ) − (3 h h h δξ + h h δξ ) − (2 h h h δξ + 2 h h h δξ ) − (3 h h h δξ + h h δξ )+3 (2 h h δξ + h δξ ) + 3 (2 h h δξ + h δξ ) + 2 (2 h h δξ + h δξ )+6 ( h h δξ + h h δξ + h h δξ ) + 2 (2 h h δξ + h δξ )+(3 h δξ ) + 3 (2 h h δξ + h δξ ) + (3 h δξ )+( h h h δ ˙ ξ ) + ( h h δ ˙ ξ ) + ( h h δ ˙ ξ ) + 2( h h δ ˙ ξ ) + ( h δ ˙ ξ ) + ( h h δ ˙ ξ )+( h h h δ ˙ ξ ) + ( h δ ˙ ξ ) + 2( h h δ ˙ ξ ) + ( h h δ ˙ ξ ) − ( h δ ˙ ξ ) − ( h δ ˙ ξ ) − ( h h δ ˙ ξ ) − h h δ ˙ ξ ) − h h δ ˙ ξ ) − ( h h δ ˙ ξ ) − ( h δ ˙ ξ ) − ( h h h δ ˙ ξ ) − ( h h h δ ˙ ξ ) − ( h h δ ˙ ξ ) + ( h δ ˙ ξ ) + ( h δ ˙ ξ ) − ( h δ ˙ ξ ) − h δ ˙ ξ ) − ( h δ ˙ ξ ) − ( δ ¨ ξ ) − ( δ ¨ ξ ))( abc ) e ( h + h + h ) t = 0 , ciPost Physics Proceedings Submission ( − (3 h h δξ + h h δξ ) − (3 h h h δξ + h h δξ ) (23) − (2 h h δξ + 2 h h δξ ) − h h h δξ + h h h δξ + h h δξ ) − (2 h h h δξ + 2 h h h δξ ) − (3 h h δξ + h δξ ) − h h h δξ + h h δξ + h h δξ ) − h h h δξ + h h δξ + h h h δξ ) − (2 h h h δξ + h h δξ ) − (3 h h δξ + h δξ ) − (2 h h δξ + 2 h h δξ ) − (3 h h δξ + h h δξ )+(3 h δξ ) + 2 (2 h h δξ + h δξ ) + 3 (2 h h δξ + h δξ )+3 (2 h h δξ + h δξ ) + 6 ( h h δξ + h h δξ + h h δξ )+3 (2 h h δξ + h δξ ) + 3 (2 h h δξ + h δξ )+2 (2 h h δξ + h δξ ) + (2 h δξ )+ h h δ ˙ ξ + h h h δ ˙ ξ + 2 h h δ ˙ ξ + h h δ ˙ ξ + h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h h h δ ˙ ξ − h h h δ ˙ ξ − h δ ˙ ξ − h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ + h δ ˙ ξ + h h δ ˙ ξ +2 h h δ ˙ ξ + h h h δ ˙ ξ + h h δ ˙ ξ − h δ ˙ ξ + h δ ˙ ξ + h δ ˙ ξ − h δ ˙ ξ − δ ¨ ξ − δ ¨ ξ )( abc ) e ( h + h + h ) t = 0 , and ( − (3 h h h δξ + h h δξ ) − (3 h h δξ + h h δξ ) (24) − (2 h h h δξ + 2 h h h δξ ) − h h h δξ + h h δξ + h h h δξ ) − (2 h h δξ + 2 h h δξ ) − (3 h h h δξ + h h δξ ) − h h h δξ + h h δξ + h h h δξ ) − h h h δξ + h h δξ + h h δξ ) − (2 h h δξ + h δξ ) − (3 h h δξ + h h δξ ) − (2 h h δξ + 2 h h δξ ) − (3 h h δξ + h δξ )+(3 h δξ ) + 3 (2 h h δξ + h δξ ) + 2 (2 h h δξ + h δξ )+3 (2 h h δξ + h δξ ) + 6 ( h h δξ + h h δξ + h h δξ )+3 (2 h h δξ + h δξ ) + (3 h δξ ) + 2 (2 h h δξ + h δξ )+3 (2 h h δξ + h δξ ) − h h δ ˙ ξ − h h h δ ˙ ξ − h h h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h h δ ˙ ξ − h δ ˙ ξ − h δ ˙ ξ + h h δ ˙ ξ + 2 h h δ ˙ ξ + h h h δ ˙ ξ + h δ ˙ ξ + h h δ ˙ ξ + h δ ˙ ξ + 2 h h δ ˙ ξ + h h δ ˙ ξ + h h δ ˙ ξ + h h h δ ˙ ξ − h δ ˙ ξ − h δ ˙ ξ + h δ ˙ ξ + h δ ˙ ξ − δ ¨ ξ − δ ¨ ξ )( abc ) e ( h + h + h ) t = 0 , Collecting the perturbation terms together yields:11 ciPost Physics Proceedings Submission ( − δ ¨ ξ − δ ¨ ξ + δ ˙ ξ ( − h − h h − h h − h h h + h h + h + h h h + h + h h + 2 h h ) (25)+ δ ˙ ξ (cid:16) − h − h h − h h + h + h − h + 2 h h − h + h h − h (cid:17) + δξ ( − h h − h h − h h − h h − h h h − h h − h h − h h h + 6 h h − h h h + 6 h h + 2 h + 2 h + 6 h h )+ δξ ( − h − h h − h h − h h − h h − h h h + 3 h − h h − h h h + 4 h h − h h h +6 h h + 3 h + 6 h h ) + δξ ( − h − h h − h h − h h − h h − h h h + 3 h − h h − h h h + 6 h h − h h h + 4 h h + 3 h + 3 h ))( abc ) e ( h + h + h ) t = 0( − δ ¨ ξ − δ ¨ ξ + δ ˙ ξ (cid:16) h + h h + 2 h h − h h + h h − h − h − h h + h − h h (cid:17) (26)+ δ ˙ ξ (cid:16) − h − h h − h h − h h + h h + h + h + 2 h h + h h − h (cid:17) + δξ ( − h − h h − h h − h h − h h − h h h + 3 h − h h − h h h + 4 h h − h h h +6 h h + 3 h + 3 h + 6 h h )+ δξ ( − h h − h h − h h h + 2 h − h h − h h h + 6 h h − h h h + 6 h h − h h − h h + 2 h − h h + 6 h h )+ δξ ( − h − h h − h h − h h − h h − h h h + 3 h − h h − h h h +6 h h − h h h + 4 h h + 3 h + 2 h + 6 h h ))( abc ) e ( h + h + h ) t = 0( − δ ¨ ξ − δ ¨ ξ + δ ˙ ξ (cid:16) h + 2 h h + h h + h h − h h − h − h − hh − h h + h (cid:17) (27)+ δ ˙ ξ (cid:16) h + 2 h h + h h − h h − h − h − h h + h (cid:17) + δξ ( − h − h h − h h − h h − h h − h h h + 3 h − h h − h h h + 4 h h − h h h +6 h h + 3 h + 3 h + 6 h h )+ δξ ( − h − h h − h h − h h − h h − h h h − h h − h h h + 6 h h − h h h +4 h h + 3 h + 3 h + 6 h h )+ δξ ( − h h − h h − h h h + 2 h − h h − hh h − h h h + 6 h h + 6 h h − h h +2 h − h h + 3 h − h h + 6 h h ))( abc ) e ( h + h + h ) t = 012 ciPost Physics Proceedings Submission These equations can be redeuced to a single equation:( − h − h h − h h − h h − h h h + 2 h − h h + 2 h h − h h h (28)+6 h h + h (cid:16) h + 4 (cid:17) h + h + h h + h h + 2 h − h h + 8 h h ) ξ +( − h − h h − h h − h h − h h h + 2 h − h h + 2 h h − h h h + 6 h h + h (cid:16) h + 4 (cid:17) h + h + h h + h h + 2 h − h h + 8 h h ) ξ +( − h − h h − h h − h h − h h + 2 h h − h h h +4 h − h h + 2 h h + 9 hh h + h h h + 6 h h − h h h +4 h h − h + 3 h h + 4 h − h h + 2 h − h h − h h ) ξ +( − h + 3 hh − h h − h h − h h − h h + 2 h h + h + h + 2 h h + h h − h ) ˙ ξ +( − h − h h − h h h − h h − h h + 2 h h + h h h + h + h + 2 h h + h + h h − h ) ˙ ξ +( − h − h h − h h + 2 h h + h − h + 2 h − h + 3 h h − h − h ) ˙ ξ = 0The above equations 25-27 are long ordinary differential equations in terms of δξ ’s andtheir first and second derivatives. After solving them we get: δξ = c exp (cid:18) At (cid:19) + c exp (cid:18) Bt (cid:19) (29) A = − A − q A − A (30) B = − A + q A − A (31) A = 12 (2 h + 3 h h + 3 h h (32) − h − h − h − hh − h h + h − h h − h h + h ) A = 2( h + h h + 4 h h + h h + 6 h h + 5 h h h (33) − h − h h + 4 h h + 5 h h h − h h − h h + h + h h − h + h h − h − h h − h h ) δξ = c exp (cid:18) Ct (cid:19) + c exp (cid:18) Dt (cid:19) (34) C = − C − q C − C (35) D = − C + q C − C (36)13 ciPost Physics Proceedings Submission C = 12 ( − h − h h − h h h − h h (37)+ 3 h h − h h + h h h + h + 2 h − h − h + 3 h h + h ) C = 2( h + h h + 4 h h + h h + 6 h h + 5 h h h − h − h h + 4 h h (38)+ 5 h h h − h h − h h + h + h h − h + h h + 2 h − h h − h h ) δξ = c exp (cid:18) Et (cid:19) + c exp (cid:18) F t (cid:19) (39) E = − E − q E − E (40) F = − E + q E − E (41) E = 12 (cid:16) h + 3 h h + 2 h h − h h − h + h − h + h − h h + 2 h + 2 h (cid:17) (42) E = 2( − h − h h − h h − h h (43) − h h + 2 h h − h h h + 4 h − h h + 2 h h + 9 hh h + h h h + 6 h h − h h h + 4 h h − h + 3 h h + 4 h − h h + 2 h − h h − h h )In the above equations, provided that A , , C , , E , >
0, if we substitute very largenumbers for t ; (i.e t → ∞ ) we find out that δξ i → h , h or h since thepower of the exponent is negative. Consequently, we can deduce that the de Sitter universeis 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 largein the context of Nash gravity. As an alternative theory for gravity Einstein’s theory of gravity was developed and modifiedby John Nash. The theory is divergence free and considered to be of interest in constructingtheories of quantum gravity. In this paper, we examine the Einstein static and the de SitterUniverse solutions and quantify their stabilities for the Bianchi type I model using Nashmodified theory of gravity in the context of Bianchi universe geometry. We searched for thestability of Einstein and de Sitter Universes using Nash gravity Model. We used the metricof Bianch type I model in the Nash field equation. Then we substituted the appropriate scalefactors in the field equations depending on the type of universe under study. We have seen thatthe 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 , , C , , E , > ciPost Physics Proceedings Submission References [1] Komatsu, E., Smith, K. M., Dunkley, J., Bennett, C. L., Gold, B., Hinshaw, G., Jarosik,N., Larson, D., Nolta, M. R., Page, L., Spergel, D. N., Halpern, M., Hill, R. S., Kognut,A., Limon, M., Meyer, S. S., Odegard, N., Tucker, G. S., Weiland, J. L., Wollack, E.,Wright, E. L.: Astrophys. J. Suppl. 192 (2011) 18.[2] Hinshaw, G., Larson, D., Komatsu, E., Spergel, D. N., Bennett, C. L., Dunkley, J., Nolta,M. R., Halpern, M., Hill, R. 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