Observing Trans-Planckian Signatures in the Cosmic Microwave Background

TL;DR

The paper investigates how trans-Planckian physics could imprint a modulated component on the primordial perturbation spectrum and how this would appear in the CMB. It studies a concrete model where the modulation amplitude scales as $H/M$ and introduces a sinusoidal dependence tied to slow-roll parameters, framing the problem in terms of ten cosmological parameters including the trans-Planckian pair $(H/M,\phi)$. Using three forecasting methods—grid searches, Fisher matrix, and Markov Chain Monte Carlo—the authors find broadly consistent constraints, with detectability strongly aided by a large tensor signal $r$, and degeneracies manifesting as islands in likelihood space. They conclude that measuring the inflationary Hubble scale $H$ and the tensor-to-scalar ratio $r$ is critical to constraining any trans-Planckian corrections, and provide quantitative expectations such as a potential $H/M\sim 0.004$ detectability for $r\sim 0.15$ in an ideal measurement. The work emphasizes that, even if the specific model is not correct, the approach and qualitative insights should generalize to other modulated spectra and guide future CMB-oriented probes of Planck-scale physics.

Abstract

We examine the constraints cosmological observations can place on any trans-Planckian corrections to the primordial spectrum of perturbations underlying the anisotropies in the Cosmic Microwave Background. We focus on models of trans-Planckian physics which lead to a modulated primordial spectrum. Rather than looking at a generic modulated spectrum, our calculations are based on a specific model, and are intended as a case study for the sort of constraints one could hope to apply on a well-motivated model of trans-Planckian physics. We present results for three different approaches -- a grid search in a subset of the overall parameter space, a Fisher matrix estimate of the likely error ellipses, and a Monte Carlo Markov Chain fit to a simulated CMB sky. As was seen in previous analyses, the likelihood space has multiple peaks, and we show that their distribution can be reproduced via a simple semi-analytic argument. All three methods lead to broadly similar results. We vary 10 cosmological parameters (including two related to the trans-Planckian terms) and show that the amplitude of the tensor perturbations is directly correlated with the detectability of any trans-Planckian modulation. We argue that this is likely to be true for any trans-Planckian modulation in the paradigm of slow-roll inflation. For the specific case we consider, we conclude that if the tensor to scalar ratio, $r \sim 0.15$, the ratio between the inflationary Hubble scale $H$, and the scale of new physics $M$ has to be on the order of 0.004 if the modulation is detectable at the 2$σ$ level. For a lower value of $r$, the bound on $H/M$ becomes looser.

Figures (13)

• Figure 1: A coarse grid (100 points in the range $H/M = [0,0.05]$ by 75 points in the range $\phi = [0,3 \pi /2]$) covering a broad region of the trans-Planckian parameter space. The inner (colored) contours are drawn at the $1\sigma$, $2\sigma$, and $3\sigma$ levels, and the outer (black) contours are at a $\Delta\chi^2 = 50$ relative to the best fit to better show the shape of the likelihood function. The fiducial model has $H/M= 0.022$ and $\phi = 0.021$.
• Figure 2: A fine grid (100 points in the range $H/M = [0.020,0.024]$ by 100 points in the range $\phi = [0,0.4]$) showing detail in a small region of the parameter space from Fig. \ref{['fig:coarsegrid']}. The fiducial model is in the center of the plot, and contours are drawn at $1\sigma$ (inner, red), $2\sigma$ (middle, blue), and $3\sigma$ (outer, green) levels.
• Figure 3: Grid results for a fiducial model with $H/M = 0.01$ and $\phi = 2.0$, showing the broadening of the degeneracy for smaller amplitude. (Note that the fiducial model is missed due to finite grid size effects.)
• Figure 4: Grid results for the null hypothesis, $H/M = 0$, estimating an upper limit on $H/M$ if no trans-Planckian signal is detected.
• Figure 5: (Left) The value of $I$ plotted on the $(\lambda, \phi)$ plane, where the purple (gray) lines, white areas and black areas represent $I=1$, $I>1$ and $I<1$ regions respectively. (Right) The 3--d representation of $I$. The underlying "data" has $\phi_0=0$ and $\epsilon_0=0.15/16=\epsilon$, and the underlying value of $\lambda_0$ ranges from (top to bottom) 0.005, 0.01, 0.04 respectively.