Signatures of trans-Planckian dispersion in inflationary spectra

Abstract

The primordial spectra are calculated using dispersion relations that deviate from the relativistic one above a certain energy scale $\ensuremath{\Lambda}$. We determine the properties of the leading modifications with respect to the standard spectra when $\ensuremath{\Lambda}\ensuremath{\gg}H$, where $H$ is the Hubble scale during inflation. To be generic, we parameterize the lowest order deviation from the relativistic law by $\ensuremath{\alpha}$, the power of $P/\ensuremath{\Lambda}$, where $P$ is the proper momentum. When working in the asymptotic vacuum, the leading modification scales as $(H/\ensuremath{\Lambda}{)}^{\ensuremath{\alpha}}$ for all $\ensuremath{\alpha}$, except for a discrete set where the power is higher. Moreover, this modification is robust against introducing higher order terms in the dispersion relation. We then algebraically deduce the modifications of scalar and tensor power spectra in slow-roll inflation from modifications calculated in de Sitter space. The modifications do not exhibit oscillations unless the dispersion relation induces some nonadiabaticity near a given scale. Finally, we explore the much less studied regime where $H$ and $\ensuremath{\Lambda}$ are comparable. Our results indicate that the project of reconstructing the inflaton potential cannot be pursued without making some hypothesis about the dispersion relation of the fluctuation modes.