Volume 202, number 2 PHYSICS LETTERS B 3 March 1988 THE STABILITY OF THE DE SITTER SPACE-TIME IN FOURTH ORDER GRAVITY V. MLILLER, H.-J. SCHMIDT Central Institute for Astrophysics, Academy of Sciences of the GDR, Rosa-Luxemburg-StraJ3e 17a, DDR-15 91 Potsdam, GDR and A.A. STAROBINSKY Landau Institute for TheoreticalPhysics, USSR Academy of Sciences, ul. Kosygina 2, SU-117940 Moscow, USSR Received 5 November 1987 For a general non-linear lagrangian L = L (R) we derive results on the classical stability of the de Sitter solution. 1. Introduction The interest in fourth order gravity stems from the fact that one obtains curvature squared terms in the gravitational lagrangian at the one-loop approxima- tion of quantum gravity. The theory is of particular importance for black hole physics and for the model of the early Universe. Indeed, starting from the orig- inal version of the cosmological model with the ini- tial de Sitter stage [ 1 ], the combination of linear and quadratic curvature invariants in the lagrangian pro- vides, in its improved version [2,3], a viable reali- zation of the inflationary scenario. The question of the existence and stability of the Sitter solution in theories which stem from a general non-linear lagrangian L=L(R) has been discussed within the class of homogeneous and isotropic cos- mological models [4]. At the same time it became of much interest that the de Sitter stage is a quite com- mon phenomenon, realized not only in different the- oretical models, but also for a general class of inhomogeneous and anisotropic space-times [ 5 ]. We extend the analysis of Starobinsky and Schmidt [ 6 ] of the de Sitter space-time as an attractor for L=R 2 to a general non-linear function L=L(R) of differ- entiability class C 3. The results should be of a wider interest because of the conformal equivalence be- tween theories with non-linear lagrangians L(R) and the ordinary general relativity with a massive self-in- teracting scalar field [ 7,8 ]. 2. The main formulae The variation of the lagrangian L=L(R) for the metric leads to the field equation 1 8x/-gL =L,Rik_½Lgik L,,R;ik X/~ 8 gik -L'"R;~R;k+gik(L"I'qR+L"R;,,,R;")=O. (1) The function L(R) enters explicitly up to its third derivative, that is the cause for presuming L to be at least a C3-function. We do not require analyticity of L(R) as in ref. [4] for we want to cover also lagran- gians of the type R n ln(R/Ro) met in discussions of gravitational vacuum polarisation effects [9]. Restricted to a spatially fiat Friedmann model ds 2 =dt z -a2(t)(dx 2 +dy2Wdz 2) , (2) eq. (1) is equivalent to its 00-component, HL"(H+4H[t) + -~L'(/2/+ H 2) + ~6L=O. (3) Here, H=a/a= (d/dt) In a. The trace ofeq. (1) reads L'R-2L + 3L" []R + 3L"R kR "k =0. (4) 198 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)