BI-TP 2010/01 Slow-roll inflation with a Gauss-Bonnet correction Zong-Kuan Guo ∗ and Dominik J. Schwarz † Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (Dated: October 10, 2018) We consider slow-roll inflation for a single scalar field with an arbitrary potential and an arbitrary nonminimal coupling to the Gauss-Bonnet term. By introducing a combined hierarchy of Hubble and Gauss-Bonnet flow functions, we analytically derive the power spectra of scalar and tensor perturbations. The standard consistency relation between the tensor-to-scalar ratio and the spectral index of tensor perturbations is broken. We apply this formalism to a specific model with a monomial potential and an inverse monomial Gauss-Bonnet coupling and constrain it by the 7-year Wilkinson Microwave Anisotropy Probe data. The Gauss-Bonnet term with a positive (or negative) coupling may lead to a reduction (or enhancement) of the tensor-to-scalar ratio and hence may revive the quartic potential ruled out by recent cosmological data. PACS numbers: 98.80.Cq, 98.80.Jk, 04.62.+v I. INTRODUCTION arXiv:1001.1897v2 [hep-th] 9 Jun 2010 Inflation in the early Universe has become the stan- dard model for the generation of cosmological perturba- tions in the Universe, the seeds for large-scale structure and temperature anisotropies of the cosmic microwave background. The simplest scenario of cosmological in- flation is based upon a single, minimally coupled scalar field with a flat potential. Quantum fluctuations of this inflaton field give rise to an almost scale-invariant power spectrum of isentropic perturbations (see Refs. [1, 2] for reviews). ative instabilities around a de Sitter background on small scales, in spite of the fact that scale-invariant scalar per- turbations can be achieved [10]. Besides the kinetic and GB terms, a scalar potential arises naturally from super- symmetry breaking or other nonperturbative effects. In a previous work, we investigated inflationary so- lutions and resulting cosmological perturbations for the special case of power-law inflation when both the GB cor- rection and the scalar potential are present [11]. Power- law inflation happens when both the potential and the GB coupling take an exponential form. In this model instabilities of either scalar or tensor perturbations show up on small scales for GB-dominated inflation. The GB correction with a positive (or negative) coupling may lead to a reduction (or enhancement) of the tensor-to-scalar ratio in the potential-dominated case. This effect leads to tight constraints on the magnitude of the GB cor- rection from the Wilkinson Microwave Anisotropy Probe (WMAP) 5-year analysis [12]. String theory is often regarded as the leading can- didate for unifying gravity with the other fundamental forces and for a quantum theory of gravity. It is known that the effective supergravity action from superstrings induces correction terms of higher order in the curvature, which may play a significant role in the early Universe. The simplest such correction is the Gauss-Bonnet (GB) term in the low-energy effective action of the heterotic string [3]. Such a term provides the possibility of avoiding the initial singularity of the Universe [4]. In the presence of an exponential potential for the modulus field, nonsin- gular cosmological solutions were found which begin in an asymptotically flat region, undergo superexponential inflation and end with a graceful exit to a phase with decreasing Hubble radius [5]. Here we generalize our previous work to the more gen- eral case of slow-roll inflation with an arbitrary potential and an arbitrary coupling. Making use of a combined hierarchy ( ϵ$_{i}$ , δ$_{i}$ ) of Hubble and GB flow functions (as defined below) with | ϵ$_{i}$ | ≪ 1 and | δ$_{i}$ | ≪ 1, analogous to the standard slow-roll approximation, we derive the power spectra of scalar and tensor perturbations. In this scenario the spectral index of scalar perturbations con- tains not only the Hubble flow parameters but also the GB flow parameters. Moreover, the standard consistency relation of single-field slow-roll inflation is modified. In order to impose observational constraints on such models, we focus on a specific model with a monomial potential and an inverse monomial GB coupling. We analyze the influence of the GB term on the scalar spectral index n$_{R}$ and the tensor-to-scalar ratio r . There are many works discussing accelerating cosmol- ogy with the GB correction in four and higher dimen- sions [6–9]. Recently it has been shown that the GB term might give rise to violent instabilities of tensor per- turbations [10]. A model in which inflation is driven by the GB term and a higher-order kinetic energy term was studied. When the GB term dominates the dynamics of the background, tensor perturbations exhibit violent neg- This paper is organized as follows. In Sec. II we define the Hubble and GB flow functions. Then by using the background equations of motion, we demonstrate that the slow-roll solution exists and is stable under asymp- totic conditions. In Sec. III we calculate the power spec- $^{∗}$Electronic address: guozk@physik.uni-bielefeld.de $^{†}$Electronic address: dschwarz@physik.uni-bielefeld.de