Trans-Planckian footprints in inflationary cosmology

TL;DR

This work addresses trans-Planckian effects in inflation by imposing a minimum-uncertainty vacuum at a fixed scale $\Lambda$ and applying it to power-law inflation. The authors derive a sub-Planckian effective theory for a massless scalar, solve the mode equation exactly for $a(\eta)=(\bar{\eta}/\eta)^q$, and compute the resulting power spectrum $P_\phi$, finding a leading correction of order $H_k/\Lambda$ with an oscillatory phase. The corrections depend on $k$ through the crossing condition and are of order a few percent in physically relevant ranges, aligning with de-Sitter results in the appropriate limit. The results suggest that trans-Planckian physics can leave observable imprints on the CMBR, highlighting the importance of high-energy initial conditions in cosmological perturbation theory.

Abstract

We consider a minimum uncertainty vacuum choice at a fixed energy scale Lambda as an effective description of trans-Planckian physics, and discuss its implications for the linear perturbations of a massless scalar field in power-law inflationary models. We find possible effects with a magnitude of order H/Λin the power spectrum, in analogy with previous results for de-Sitter space-time.

Figures (1)

• Figure 1: The ratio $R_q$ for $P_\phi$ in Eq. (\ref{['Pexact']}) and $q=2$ (solid line), $q=3/2$ (dotted line) and $q=4/3$ (dashed line). The momentum index $k$ is in units with $\Lambda=\bar{\eta}=1$ and the regions of physical interest are those for $k\gtrsim 10^2$ ($q=2$), $k\gtrsim 10^3$ ($q=3/2$) and $k\gtrsim 10^4$ ($q=4/3$).