CMB quadrupole suppression: II. The early fast roll stage

TL;DR

This work explains the anomalously low CMB quadrupole as a consequence of an early fast-roll stage preceding slow-roll inflation, arising from an initial inflaton equipartition between kinetic and potential energy within the effective field theory of inflation at the GUT scale. The fast-roll phase induces an attractive, localized potential $\mathcal{V}(\eta)$ in the wave equations for curvature and tensor perturbations, which, through a scattering-theory transfer function $D(k)$, suppresses the quadrupole without violating renormalization or backreaction constraints. A detailed Born-approximation analysis connects $\mathcal{V}(\eta)$ to changes in the primordial power and hence to $\Delta C_l/C_l$, showing a robust $10-20\%$ suppression of $C_2$ and a modest $2-4\%$ suppression of the tensor quadrupole for $N_{tot} \sim 59$, with higher multipoles diminishing as $1/l^2$. The results are model-agnostic within the EFT framework, predicting an upper limit $N_{tot} \lesssim 59$ to maintain observable quadrupole suppression, and they further connect to an inverse problem via the Gelfand-Levitan formalism for potential reconstruction from primordial power data.

Abstract

Within the effective field theory of inflation, an initialization of the classical dynamics of the inflaton with approximate equipartition between the kinetic and potential energy of the inflaton leads to a brief fast roll stage that precedes the slow roll regime. The fast roll stage leads to an attractive potential in the wave equations for the mode functions of curvature and tensor perturbations. The evolution of the inflationary perturbations is equivalent to the scattering by this potential and a useful dictionary between the scattering data and observables is established.Implementing methods from scattering theory we prove that this attractive potential leads to a suppression of the quadrupole moment for CMB and B-mode angular power spectra. The scale of the potential is determined by the Hubble parameter during slow roll. Within the effective field theory of inflation at the grand unification (GUT) energy scale we find that if inflation lasts a total number of efolds N_{tot} ~ 59, there is a 10-20% suppression of the CMB quadrupole and about 2-4% suppression of the tensor quadrupole. The suppression of higher multipoles is smaller, falling off as 1/l^2. The suppression is much smaller for N_{tot} > 59, therefore if the observable suppression originates in the fast roll stage, there is the upper bound N_{tot} ~ 59.

Figures (5)

• Figure 1: $y^2(\eta)$ vs. $\eta$ (left) and $y^2(N_e)$ vs. $N_e$ (right) for initial conditions with kinetic and potential inflaton energy of the same order.
• Figure 2: The potentials ${\mathcal{V}}_\mathcal{R}(\eta)/H^2_{i}$ (left panel) and ${\mathcal{V}}_T(\eta)/H^2_{i}$ (right panel) felt by curvature and tensor perturbations respectively vs $H_i\,\eta$, $H_i$ being the Hubble parameter during the slow roll stage (see fig.\ref{['fig:hubble']}).
• Figure 3: $H(t)/H_i$ vs number of e-folds
• Figure 4: $\Delta C_2/C_2$ vs. $\kappa/H_i$ (left panel) and $\Delta C_l/C_l$ vs. $l$ for $\kappa/H_i=0.8, \; 1, \; 2$ respectively for curvature perturbations. $\kappa = a_0 \, H_0/3.3$
• Figure 5: The odd function $\Psi(x)$ vs. $x$ for negative $x$ [see eq.(\ref{['defPsi']})]. This function convoluted with the potential $\mathcal{V}(\eta)$ yields the change on the quadrupole $\frac{\Delta C_2}{C_2}$ [see eq.(\ref{['cuaB']})].