Stability of the Einstein static universe in modified theories of gravity
Christian G. Boehmer, Lukas Hollenstein, Francisco S. N. Lobo, Sanjeev S. Seahra
aa r X i v : . [ g r- q c ] J a n August 19, 2018 23:33 WSPC - Proceedings Trim Size: 9.75in x 6.5in einsteinstatic Stability of the Einstein static universe in modified theories of gravity
C. G. B¨ohmer
Department of Mathematics and Institute of Origins, University College London,Gower Street, London, WC1E 6BT, United [email protected]
L. Hollenstein
D´epartement de Physique Th´eorique, Universit´e de Gen`eve24, Quai Ernest Ansermet, 1211 Gen`eve 4, [email protected]
F. S. N. Lobo
Centro de F´ısica Te´orica e Computacional, Faculdade de Ciˆencias da Universidadede Lisboa, Avenida Professor Gama Pinto 2, P-1649-003 Lisboa, Portugalfl[email protected]
S. S. Seahra
Department of Mathematics and Statistics, University of New Brunswick,Fredericton, New Brunswick, E3B 5A3, [email protected]
During the last decade, various modified theories of gravity have become verypopular, f ( R ) gravity theory probably being the most studied one. Although mostmodels that are in agreement with observations are very close to general relativity,we have developed a much deeper understanding of the theory we wanted to mod-ify. During the year 2009 it was Hoˇrava-Lifshitz gravity that excited the scientificcommunity with a new paper appearing on the subject area every other day. Hoˇravaproposed a power counting renormalizable theory for (3+1)-dimensional quantumgravity, which reduces to Einstein gravity with a non-vanishing cosmological con-stant in IR, but possesses improved UV behaviors.In this work, we explore the stability of the Einstein static universe in suchmodified theories of gravity, for this study in general relativity see. This can bemotivated from various points of view. From a cosmological viewpoint it is the pos-sibility that the universe might have started out in an asymptotically Einstein staticstate, in the inflationary universe context. On the other hand, the Einstein cosmoshas always been of great interest in various gravitational theories. In general rela-tivity for instance, generalizations with non-constant pressure have been analyzedin.
In the context of brane world models the Einstein static universe was investi-gated in, while its generalization within Einstein-Cartan theory can be found in. Finally, in the context of loop quantum cosmology, we refer the reader to.
Forthe Einstein static universe in modified Gauss-Bonnet gravity, see. Finally, stabil-ity of the Einstein static universe in Hoˇrava-Lifshitz gravity was analyzed in.
By analyzing a simple background model and its perturbations one can studymany properties of the modified theory in a rather explicit fashion. Let us give ugust 19, 2018 23:33 WSPC - Proceedings Trim Size: 9.75in x 6.5in einsteinstatic one example to elucidate this point. When f ( R ) gravity became popular, it wasbelieved that modifications of general relativity cannot stabilize solutions. However,as we showed explicitly in, this is not true and one can construct situations wherethe Einstein static universe, for instance, is stable with respect to a homogeneousperturbation. As we further showed, this result in fact holds for all non-degenerate f ( R ) ( f ′′ ( R ) = 0 etc) theories, see also. Generic stability results in f ( R ) gravityhave been know since 1983. The mathematics required to perform these investigations is straightforward andwell understood, namely linear perturbation theory. The principal idea is to expressall quantities u i ( x ) in the form u i ( x ) = u i bg ( x ) + ε u i ( x ) + ε u i ( x ) + . . . , (1)where the u i bg ( x ) describe a known exact solution of the field equations. It shouldbe noted that there are situations where knowledge of u i bg ( x ) is not even necessaryto solve to perturbed equations. In case of the Einstein static universe x would bethe cosmological time t , u i ( x ) = { a ( t ) , ρ ( t ) , p ( t ) } , with a ( t ), ρ ( t ) and p ( t ) beingthe scale factor, the energy density and the pressure respectively. The backgroundsolution would be a static solution of the field equations which effectively reduceto algebraic equations that determine the background values. In general, we canassume the field equations to take the form F i ( u i , ∇ α u i , . . . , ∇ α ∇ β ∇ γ ∇ δ u i ) = 0 . (2)Here we restrict ourselves to theories which contain up to four derivatives. Wealso assume that the number of field equations matches the numbers of unknownfunctions, we exclude over-determined and under-determined systems. Notice thatthe field equations sourced by a perfect fluid are under-determined as long as noequation of state is assumed. Therefore, we consider a perfect fluid (no anisotropicstresses) with a linear barotropic equation of state, p bg ( ρ bg ) = wρ bg . Now, if p ( ρ ) = wρ is also assumed to hold at perturbative level then the sound speed of adiabaticpressure perturbations is given as c s = w . Note that it would be interesting toinvestigate the situation when this assumption is dropped, c s = w .The approach described covers usual General Relativity and thus the entire fieldof cosmological perturbation theory, f ( R ) gravity, modified Gauss-Bonnet gravity,or f ( G ) gravity, and also Hoˇrava-Lifshitz gravity. Note that the above treatmentis sufficiently general to also include brane world models which pose additionaltechnical challenges. In order to investigate the behavior of the perturbation, one now inserts Eq. (1)into the field equations (2) and linearizes the equations with respect to ε (Taylorexpansion about ε = 0). In full generality, the field equations then become F i ( u i bg , ∇ α u i bg , . . . , ∇ α ∇ β ∇ γ ∇ δ u i bg )+ ε (cid:18) ∂F i ∂u i (cid:19) bg u i + . . . + ε (cid:18) ∂F i ∂ ∇ α ∇ β ∇ γ ∇ δ u i (cid:19) bg ∇ α ∇ β ∇ γ ∇ δ u i + O ( ε ) = 0 . (3) ugust 19, 2018 23:33 WSPC - Proceedings Trim Size: 9.75in x 6.5in einsteinstatic Those equations are now linear in the perturbed variables. In Gauss-Bonnet gravity,for example, this equation takes the simple form24 κ ρ (1 + w ) f ′′ (0) a ′′′′ ( t ) + 2 a ′′ ( t ) − κρ bg (1 + w )(1 + 3 w ) a ( t ) = 0 . (4)In this equation a is the perturbed scale factor and f is the function which deter-mines the modifications of the Gauss-Bonnet term.In general one finds a set of linear, coupled differential equations. Equationsof this type can always be solved analytically. We can therefore conclude that theadditional degrees of freedom in any modified gravity model lead to enhanced regionsof stability in the parameter space. Acknowledgments
We would like to thank Peter Dunsby, Naureen Goheer, Roy Maartens and LucaParisi for useful discussions. LH is supported by the Swiss National Science Foun-dation.
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