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, for very little wavenumber $k$s, to implement the initial conditions required of the perturbation mode functions consistently, we must let the universe experience more 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, then for perturbations at scales greater than some critical value, the initial conditions required of the mode functions cannot be implemented physically. Because at the inflation beginning point, these perturbation modes lie outside the horizon. We propose that such perturbations do not contribute to the Cosmological Microwave Background Anisotropy (CMBA). Under this proposition, the exceptional lowness of the observed little $l$ muli-poles of CMBA is reproduced numerically. In Linde's $\phi^2$ model, the upper bound on the number of e-foldings of inflation is determined to be 65 approximately. 
Ding-fang Zeng and Yi-hong Gao