On the Signature of Short Distance Scale in the Cosmic Microwave Background

We discuss the signature of the scale of short distance physics in the Cosmic Microwave Background. In addition to effects which depend on the ratio of Hubble scale H during inflation to the energy scale M of the short distance physics, there can be effects which depend on $dot{phi}^2/M^4$ where $phi$ is the {it classical background} of the inflaton field. Therefore, the imprints of short distance physics on the spectrum of Cosmic Microwave Background anisotropies generically involve a {it double expansion}. We present some examples of a single scalar field with higher order kinetic terms coupled to Einstein gravity, and illustrate that the effects of short distance physics on the Cosmic Microwave Background can be substantial even for H << M, and generically involve corrections that are not simply powers of H/M. The size of such effects can depend on the short distance scale non-analytically even though the action is local.