Cosmic inversion: Reconstructing the primordial spectrum from CMB anisotropy

We investigate the possibility of reconstructing the initial spectrum of density fluctuations from the cosmic microwave background (CMB) anisotropy. As a first step toward this program, we consider a spatially flat, CDM dominated universe. In this case, it is shown that, with a good accuracy, the initial spectrum satisfies a first order differential equation with the source determined by the CMB angular correlation function. The equation is found to contain singularities arising from zeros of the acoustic oscillations in the transfer functions. Nevertheless, we find these singularities are not fatal, and the equation can be solved nicely. We test our method by considering simple analytic forms for the transfer functions. We find the initial spectrum is reproduced within $5\%$ accuracy even for a spectrum that has a sharp spike.