Primordial Power Spectrum's Calculation and the Upper Bound on the Number of E-foldings of Inflation

In numerical calculations of the primordial power spectrum of perturbations produced during inflation, we find that for very little wavenumber $k$s, to implement the initial conditions required of the perturbation mode functions consistently, we must let the universe experience a bigger number of e-foldings of inflation than that required of it to solve the horizon, flatness and other pre-inflation problems. However, if the number of e-foldings of inflation has an upper bound, the initial conditions required of some little $k$ mode functions can not be implemented physically. If this upper bound takes the value estimated by T. Banks and W. Fischler, then for perturbations at scales about today's Hubble horizon, the initial conditions required of the mode functions cannot be implemented physically. We then go a step further, assume that perturbations at scales greater than the comoving horizon of the inflation beginning point do not contribute to the CMBA, under which the exceptional lowness of the observed little $l$ muli-poles of CMBA are reproduced numerically