Preinflationary and inflationary fast-roll eras and their signatures in the low CMB multipoles

We study the entire coupled evolution of the inflaton φ(t) and the scale factor a(t) for general initial conditions φ(t0) and dφ(t0)/dt at a given initial time t0. The generic early Universe evolution has three stages: decelerated fast roll followed by inflationary fast roll and then inflationary slow roll (an attractor always reached for generic initial conditions). This evolution is valid for all regular inflaton potentials v(φ). In addition, we find a special (extreme) slow-roll solution starting at t=-∞ in which the fast-roll stages are absent. At some time t=t*, the evolution backwards in time from t0 reaches generically a mathematical singularity where a(t) vanishes and the Hubble parameter becomes singular. We determine the general behavior near the singularity. The classical homogeneous inflaton description turns to be valid for t-t*>10tPlanck well before the beginning of inflation, quantum loop effects are negligible there. The singularity is never reached in the validity region of the classical treatment and therefore it is not a real physical phenomenon here. Fast-roll and slow-roll regimes are analyzed in detail including the equation of state evolution, both analytically and numerically. The characteristic time scale of the fast-roll era turns to be t1=(1/m)√V(0)/[3M4]∼104tPlanck, where V is the double-well inflaton potential, m is the inflaton mass, and M the energy scale of inflation. The whole evolution of the fluctuations along the decelerated and inflationary fast-roll and slow-roll eras is computed. The Bunch-Davies initial conditions are generalized for the present case in which the potential felt by the fluctuations can never be neglected. The fluctuations feel a singular attractive potential near the t=t* singularity (as in the case of a particle in a central singular potential) with exactly the critical strength (-1/4) allowing the fall to the center. Precisely, the fluctuations exhibit logarithmic behavior describing the fall to t=t*. The power spectrum gets dynamically modified by the effect of the fast-roll eras and the choice of Bunch-Davies initial conditions at a finite time through the transfer function D(k) of initial conditions. The power spectrum vanishes at k=0.D(k) presents a first peak for k∼2/η0 (η0 being the conformal initial time), then oscillates with decreasing amplitude and vanishes asymptotically for k→∞. The transfer function D(k) affects the low cosmic microwave background multipoles Cℓ: the change ΔCℓ/Cℓ for 1≤ℓ≤5 is computed as a function of the starting instant of the fluctuations t0. Cosmic microwave background quadrupole observations indicate large suppressions, which are well reproduced for the range t0-t* 0.05/m10 100t Planck. © 2010 The American Physical Society.