The stress-energy tensor for trans-Planckian cosmology

TL;DR

The paper develops a covariant framework to derive the stress-energy tensor for a free scalar field with a general analytic dispersion relation in curved spacetime, enabling a consistent treatment of trans-Planckian physics in cosmology. By specializing to FLRW, it computes the energy density and pressure, enabling analysis of the equation of state of trans-Planckian modes and their backreaction on inflation. The authors demonstrate that gravitons with super-Planck momenta do not naturally reproduce the observed vacuum energy and that backreaction is generically significant for dispersion relations that alter the metric perturbation power spectrum, while pure de Sitter inflation leaves the spectrum’s shape unchanged (up to amplitude). Overall, the work generalizes previous Corley-Jacobson results and clarifies when trans-Planckian physics can imprint observable features without destabilizing the cosmological background.

Abstract

This article presents the derivation of the stress-energy tensor of a free scalar field with a general non-linear dispersion relation in curved spacetime. This dispersion relation is used as a phenomelogical description of the short distance structure of spacetime following the conventional approach of trans-Planckian modes in black hole physics and in cosmology. This stress-energy tensor is then used to discuss both the equation of state of trans-Planckian modes in cosmology and the magnitude of their backreaction during inflation. It is shown that gravitational waves of trans-Planckian momenta but subhorizon frequencies cannot account for the form of cosmic vacuum energy density observed at present, contrary to a recent claim. The backreaction effects during inflation are confirmed to be important and generic for those dispersion relations that are liable to induce changes in the power spectrum of metric fluctuations. Finally, it is shown that in pure de Sitter inflation there is no modification of the power spectrum except for a possible magnification of its overall amplitude independently of the dispersion relation.

Figures (2)

• Figure 1: Dispersion relation with an exponential decreasing tail, as envisaged in Ref. mbk. The dot-dashed line shows the linear dispersion relation, and the thick solid curve shows the modified dispersion relation. The dotted lines delimit three regions of different evolutions for the mode function: in region I, corresponding to the "tail", a mode has trans-Planckian momentum $k>ak_{\rm c}$ and sub-Hubble comoving pulsation $\omega(k)<aH$; in region II, a mode has $\omega(k)>aH$, and in region III, modes are super-horizon sized $k<aH$. The dashed area represents the "tail" modes defined by $k>K_+$ (equivalently $k_{\rm phys}>K_+/a$ in the figure).
• Figure 2: Example of a modified dispersion relation which breaks the WKB approximation in the trans-Planckian regime when $\omega_{\rm phys}<H$, as indicated.