2ND-ORDER RECONSTRUCTION OF THE INFLATIONARY POTENTIAL

To first order in the deviation from scale invariance the inflationary potential and its rst two derivatives can be expressed in terms of th e spectral indices of the scalar and tensor perturbations, n and n(T), and their contributions to the variance of the quadrupole CBR tempera ture anisotropy, S and T. In addition, there is a ''consistency relati on'' between these quantities: n(T) = -1/7 (T/S). We discuss the overa ll strategy of perturbative reconstruction and derive the second-order expressions for the inflationary potential and its first two derivati ves and the first-order expression for its third derivative, all in te rms of n, n(T), S, T, and dn/d ln k. We also obtain the second-order c onsistency relation, n(T) = -1/7(T/S)[1 + 0.11(T/S) + 0.15(n - 1)]. As an example we consider the exponential potential, the only known case where exact analytic solutions for the perturbation spectra exist. We reconstruct the potential via Taylor expansion (with coefficients cal culated at both first and second order), and introduce the Pade approx imant as a greatly improved alternative.