On Inflation with the Generalized Uncertainty Principle

In this paper we revisit the formulation of scalar field theories on de Sitter backgrounds subject to the generalized uncertainty principle (GUP). The GUP arises in several contexts in string theory, but is most readily thought of as resulting from using strings as effective probes of geometry, which suggests an uncertainty relation incorporating the string scale ls. After reviewing the string theoretic case for the GUP, which implies a minimum length scale ls, we follow in the footsteps of Kempf and concern ourselves as to how one might write down field theories which respect the GUP. We uncover a new representation of the GUP, which unlike previous studies, readily permits analytical solutions for the mode functions of a scalar field on de Sitter backgrounds. We find that scalar fields cannot be quantized on inflationary backgrounds with a Hubble radius H−1 smaller than the string scale, implying a potential sub-planckian cutoff on the scale of inflation resulting from the GUP. We also compute order (Hls) 2 (as opposed to (Hlpl) 2) corrections to the two point correlation function and comment on the future prospects of observing such corrections in the event our universe is described by a weakly coupled string theory.