We have hitherto focused on the statistical properties of the cosmological perturbations as probed by a variety of observations. However, we on Earth occupy a unique location in a specific realization of wave modes that we have argued are drawn from a specific set of waves with particular amplitudes and phases, despite these supposedly being drawn from a statistical distribution (in much the same way that different pieces of music are distinct despite having similar power spectra). It should not be too long before we can produce a 3-dimensional “map” of our universe out to recombination based on a suite of observations, presuming that our standard cosmology and theory for the evolution of perturbations is correct.

(a) Calculate the comoving radii of recombination, reionization, and the most distant galaxies and quasars.

(b) Suppose that we have noise-free CMB fluctuation maps in temperature and polarization up to spherical harmonic quantum number l = 100. How many numbers would be contributed by these measurements?

(c) How many numbers would we need to measure to describe the potential and the associated density perturbations out to the radius of recombination with comparable resolution?

(d) Would these modes still be in the linear regime today?
(e) Challenge. Explore some of the practical challenges of carrying out this program, paying attention to the investigations we have described in Sec. 28.6 and assuming them to be carried out over the whole sky.