  We describe how a survey of the polarization of the cosmic microwave
background induced by Compton scattering in galaxy clusters can be used to make
a full spatial reconstruction of the primordial (z~1089) matter distribution
inside our surface of last scattering. This "polarization tomography" can yield
a spatial map of the initial state of the Universe just as gravitational
collapse was beginning to drive structure formation. We present a transparent
method and simple formulas from which one can compute the 3D primordial map in
real and in Fourier space, given a 3D map of the polarization due to galaxy
clusters. The advantage of the real space reconstruction is that it is free
from the statistical uncertainties which are inherent in the Fourier space
reconstruction. We discuss how noise, partial sky covering and depth of the
survey can affect the results.
