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

We study the entire coupled evolution of the inflaton and the scale factor for general initial conditions at a given initial time. The generic early universe evolution has three stages: decelerated fast-roll followed by inflationary fast roll and then inflationary slow-roll. This evolution is valid for all regular inflaton potentials. In addition, we find a special (extreme) slow-roll solution starting at t = -infty in which the fast-roll stages are absent. At some time t = t_*, the generic evolution backwards in time reaches a mathematical singu- larity where a(t) vanishes and Hubble becomes singular. We find the general behaviour near the singularity. The classical inflaton description is valid for t-t_* > 10 t_{Planck} 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. The whole evolution of the fluctuations is computed. The Bunch-Davies initial conditions (BDic) are generalized for the present case. The power spectrum gets dynamically modified by the effect of the fast-roll eras and the BDic choice at a finite time through the transfer function D(k) of initial conditions. D(0) = 0. D(k) presents a first peak for k ~ 2/eta_0 (eta_0 being the conformal initial time), then oscillates with decreasing amplitude and vanishes asymptotically for k -> infty. The transfer function D(k) affects the low CMB multipoles C_l: the change Delta C_l/C_l for l=1-5 is computed as a function of the starting instant of the fluctuations t_0. CMB quadrupole observations give large suppressions which are well reproduced here(Abridged)