Probing Trans-Planckian Signatures in the Early Universe: A Bayesian Analysis of the Generalized Sasaki-Mukhanov Equation

TL;DR

The paper investigates trans-Planckian corrections to inflationary perturbations by generalizing the Sasaki–Mukhanov equation with a time-dependent term ∝ $1/\eta$. It derives exact mode solutions in terms of Whittaker functions, enabling an analytic form for the modified primordial power spectrum and its scale-dependent oscillations. Implementing the model in Cobaya, the authors analyze Planck 2018 data together with ACT DR6, finding a stringent 95% C.L. bound of $|f| \le 1\times 10^{-4}$ and a mild preference for negative $f$ that slightly improves the low-$\ell$ CMB fit without compromising small-scale performance. The work highlights the potential to address low-$\ell$ anomalies within an EFT-consistent framework, while stressing the need for careful treatment of backreaction and non-Gaussian signatures, and points to future polarization and large-scale structure data as critical tests.

Abstract

We present a rigorous and comprehensive investigation of a generalized inflationary perturbation theory designed to address persistent large-scale anomalies in the Cosmic Microwave Background (CMB). Motivated by the Trans-Planckian problem and potential non-canonical dynamics in the early Universe, we introduce a generalized Sasaki-Mukhanov equation characterized by a time-dependent correction term, parameterized by a coupling constant f. Unlike the standard slow-roll approximation, we derive the exact analytical solutions for the mode functions in terms of Whittaker functions, ensuring a precise treatment of the mode evolution across the horizon. We compute the resulting primordial scalar power spectrum, which exhibits scale-dependent oscillatory modulations and a distinct suppression of power at low multipoles. We numerically implement this modified framework within the Cobaya Bayesian inference engine. Utilizing the latest Planck 2018 temperature and polarization likelihoods combined with high-resolution data from the Atacama Cosmology Telescope (ACT) DR6, we perform a robust Monte Carlo Markov Chain (MCMC) analysis. Our results place stringent constraints on the modification parameter, |f| <= 10^-4, at a 95% confidence level. However, we find intriguing hints that the generalized model provides a better fit to the low-l CMB spectrum compared to the standard LambdaCDM model, effectively alleviating the low-quadrupole anomaly without compromising the fit at smaller scales. We discuss the implications of these findings for the energy scale of inflation and the validity of the effective field theory description during the inflationary epoch.

Figures (5)

• Figure 1: 1D posterior distributions for $H_0$, $f$, $n_s$, and $\ln(10^{10}A_s)$. The shift in distributions when adding ACT data (orange) highlights the constraining power of small-scale measurements.
• Figure 2: 2D marginalized contours (68% and 95% C.L.) for key parameter pairs. The correlation between $f$ and $n_s$ is visible, indicating a partial degeneracy in how they affect the spectral shape.
• Figure 3: Full triangle plot (corner plot) showing the posterior distributions for all sampled parameters. The stability of the $\Lambda$CDM parameters against the introduction of $f$ is evident.
• Figure 4: Scatter plot of MCMC samples in the $n_s$ vs. $\ln(10^{10}A_s)$ plane, color-coded by the value of $f$. Darker blue points (negative $f$) cluster in specific regions, visualizing the multi-dimensional structure of the likelihood.
• Figure 5: Comparison of the CMB temperature power spectrum ($D_\ell^{TT}$). The generalized model (red dashed) follows the $\Lambda$CDM prediction (black solid) at high multipoles but exhibits a suppression at low $\ell$, providing a better fit to the Planck data (cyan points). The residuals panel explicitly shows the reduction in tension at $\ell=2$.