Inflationary potentials yielding constant scalar perturbation spectral indices

Inflation, a cornerstone of the modern framework for understanding the early universe @1,2#, predicts the initial conditions for the formation of structure and the cosmic microwave background ~CMB! anisotropies. During inflation, the primordial scalar ~density! and tensor ~gravitational wave! perturbations generated by quantum fluctuations are redshifted beyond the Hubble radius, becoming frozen as perturbations in the background metric @3–7#. However, even when there is only one scalar field—the inflaton—the number of inflation models proposed in the literature is large @2#. Determination of the properties of the scalar perturbations and tensor perturbations from CMB and large-scale structure observations allows one to constrain the space of possible inflation models @8–14#. It is often adequate to characterize inflationary perturbations in terms of four quantities: the scalar and tensor power spectra, PR and Pg , and the scalar and tensor spectral indices n and nT . In this paper we focus on the scalar spectral index which, unless explicitly indicated otherwise, we refer to simply as the ‘‘spectral index.’’ Successful inflation models predict n close to 1 ~the so-called Harrison-Zel’dovich spectrum!, and n typically has a small scale dependence. The best data available to date, combining the Wilkinson Microwave Anisotropy Probe @15# and Sloan Digital Sky Survey @16# data sets, indicate that the evidence for anything other than a scale-invariant spectra is marginal at best, with no evidence for significant running of the scalar spectral index @17#. Moreover, one of us has recently argued that when information criteria are used to carry out cosmological model selection based on the current data sets available, then the