Natural Covariant Planck Scale Cutoffs and the Cosmic Microwave Background Spectrum

We calculate the impact of quantum gravity–motivated ultraviolet cutoffs on inflationary predictions for the cosmic microwave background spectrum. We model the ultraviolet cutoffs fully covariantly to avoid possible artifacts of covariance breaking. Imposing these covariant cutoffs results in the production of small, characteristically $k$-dependent oscillations in the spectrum. The size of the effect scales linearly with the ratio of the Planck to Hubble lengths during inflation. Consequently, the relative size of the effect could be as large as one part in $10^5$; i.e., eventual observability may not be ruled out.

Figure 1

Relative change in $\delta {\varphi }_k$ as a function of comoving mode number with $|\eta |=1$ fixed. Horizon crossing ($k|\eta |=1$) is marked with a solid line. The ratio of Planck to Hubble scales is set to $H/\mathrm{\Omega }={10}^{-2}$ so that the numerical computation remains tractable. The oscillations are due to an oscillatory integrand that appears in $G_F^{\mathrm{\Omega }}$, which dominates at small $k$. Roughly, the oscillations may be understood as an interference effect, with dips occurring when an integer number of wavelengths fit between the Planck and Hubble scales.

Figure 2

Relative change in $\delta {\varphi }_k$ as a function of the ratio of Planck to Hubble scales. The upper series of points is at horizon crossing ($k|\eta |=1$), and the lower series is well past horizon crossing ($k|\eta |=1/20$). The inset shows a sliver of the lower series at higher resolution, where the apparent scatter in the data is resolved as rapid oscillations.

Figure 3

Relative change in the fluctuation spectrum for a slowly varying Hubble parameter. For each $k$, $|\eta |$ is set to $1/k$ and the Hubble parameter is matched to that of a power law spacetime: $H(\eta )=p[C(p-1)|\eta |{]}^{1/(p-1)}$, where $a(t)=C{t}^p$ in cosmic time. Here, we set $p=10$ and we fixed $C$ such that $H(-1)=1$, so as to coincide with horizon crossing in Fig. 1. The value of $H$ ranges from 1 to 7.74 as $k$ goes from 1 to ${10}^{-8}$; this change is tracked by the linear trend in $\mathrm{\Delta }(\delta {\varphi }_k)/\delta {\varphi }_k$. The inset shows a detail of the oscillations on top of the trend.