Volume 245, number 3, 4 PHYSICS LETTERS B 16 August 1990 Effects of R 3 and RE]R terms on R 2 inflation Andrew L. Berkin 1 and Kei-ichi Maeda 2 Department of Physics, Waseda University, Shinjuka-ku, Tokyo 169, Japan Received 2 March 1990 To study the generality of inflation from lagrangians containing higher orders of curvature, we consider the effects of R 3 and R[]R terms on the R 2 inflationary scenario. Both direct analysis and conformal techniques are used. We find that both terms restrict the initial conditions which will lead to a satisfactory inflationary cosmology. The inflationary universe scenario has been one of the major cosmological developments of the past decade [1]. Several mechanisms have been used to generate the rapid expansion needed to solve some long standing cosmological problems. One particular mechanism, dubbed "R: inflation", adds terms quadratic in the curvature tensors to the standard Einstein-Hilbert action [2,3]. For a certain range of initial parameters, these quadratic terms will generate sufficient expansion without the need for the special matter terms used in most inflationary mechanisms. The justification for addition of quadratic terms is usually twofold. First, renormalization of quantum field theories in curved spacetime usually mandates such terms [4], although with coefficients much smaller than required for a fully satisfactory inflationary picture. Second, as there is no de facto reason why the action should be only linear, perhaps nature really does contain these terms, which would be observationally undetectable at our present epoch [3]. With both reasons, the question of the influence of even higher order terms in the action arises. The renormalization of higher loop processes introduces higher order corrections to the action, while if one believes that nature is not merely linear in the gravitational action then certainly terms of all orders are possible. Because each higher order is suppressed by the inverse Planck mass squared, their effects will only be significant near the Planck era, in which case not just quadratic but all terms will contribute. In this paper we consider the effects of third order terms on the early universe, where RDR is regarded as third order since two derivatives are dimensionally equivalent to one curvature tensor. If these cubic curvature contributions further increase the amount of expansion, then the feasibility of inflation from higher order terms seems strengthened. Indeed, these terms have been claimed to enhance the viability of such models by producing effects such as double inflation [5]. However, we will show that these cubic terms restrict the range of parameters which lead to a successful inflationary scenario. Because such terms arise naturally in any theory which admits quadratic contributions, the viability of higher order inflation is weakened. We use a Robertson-Walker universe, so to quadratic order only the R 2 term contributes. For simplicity and to facilitate comparison with previous work we consider third order terms which contain only the Ricci scalar and its derivatives ~1. First the effects of an R 3 term and an RI3R term are analyzed separately, then both effects are taken into account. A term of the form V,RV~'R also exists, but upon integration by parts it may be 1 E-mail address: berkin@jpnwas00.bitnet 2 E-mail address: maeda@jpnwas00.bitnet ~1 Second order terms are discussed in ref [4]. All possible third order terms, with coefficients due to quantum corrections, are found in ref. [6]. For terms through fourth order, see ref. [7]. 348 0370-2693/90/$ 03.50 O 1990- Elsevier Science Publishers B.V. (North-Holland)