CMB quadrupole depression produced by early fast-roll inflation: Monte Carlo Markov chains analysis of WMAP and SDSS data

Abstract

Generically, the classical evolution of the inflaton has a brief fast-roll stage that precedes the slow-roll regime. The fast-roll stage leads to a purely attractive potential in the wave equations of curvature and tensor perturbations (while the potential is purely repulsive in the slow-roll stage). This attractive potential leads to a depression of the CMB quadrupole moment for the curvature and $B$-mode angular power spectra. A single new parameter emerges in this way in the early universe model: the comoving wave number ${k}_{1}$ characteristic scale of this attractive potential. This mode ${k}_{1}$ happens to exit the horizon precisely at the transition from the fast-roll to the slow-roll stage. The fast-roll stage dynamically modifies the initial power spectrum by a transfer function $D(k)$. We compute $D(k)$ by solving the inflaton evolution equations. $D(k)$ effectively suppresses the primordial power for $k<{k}_{1}$ and possesses the scaling property $D(k)=\ensuremath{\Psi}(k/{k}_{1})$ where $\ensuremath{\Psi}(x)$ is a universal function. We perform a Monte Carlo Markov chain analysis of the WMAP and SDSS data including the fast-roll stage and find the value ${k}_{1}=0.266\text{ }\text{ }{\mathrm{Gpc}}^{\ensuremath{-}1}$. The quadrupole mode ${k}_{Q}=0.242\text{ }\text{ }{\mathrm{Gpc}}^{\ensuremath{-}1}$ exits the horizon earlier than ${k}_{1}$, about one-tenth of an $e$-fold before the end of fast roll. We compare the fast-roll fit with a fit without fast roll but including a sharp lower cutoff on the primordial power. Fast roll provides a slightly better fit than a sharp cutoff for the temperature--temperature, temperature--$E$ modes, and $E$ modes--$E$ modes. Moreover, our fits provide nonzero lower bounds for $r$, while the values of the other cosmological parameters are essentially those of the pure $\ensuremath{\Lambda}\mathrm{CDM}$ model. We display the real space two point ${C}^{\mathrm{TT}}(\ensuremath{\theta})$ correlator. The fact that ${k}_{Q}$ exits the horizon before the slow-roll stage implies an upper bound in the total number of $e$-folds ${N}_{\mathrm{tot}}$ during inflation. Combining this with estimates during the radiation dominated era we obtain ${N}_{\mathrm{tot}}\ensuremath{\sim}66$, with the bounds $62<{N}_{\mathrm{tot}}<82$. We repeated the same analysis with the WMAP-5, ACBAR-2007, and SDSS data confirming the overall picture.