Real space tomography of the primordial Universe with cluster polarization

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 ($z\sim 1089$) 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.

Figure 1

Last scattering surfaces seen by an observer at the origin at $\eta ={\eta }_0$ (solid black circle) and by two clusters at times ${\eta }_1$ and ${\eta }_2$ (dashed gray circles—dashed red and blue circles in the online version.) The variable $x$ is physical distance in units of the distance to our LSS, $x=(\eta -{\eta }_{\mathrm{rec}})/({\eta }_0-{\eta }_{\mathrm{rec}})$. The distance from the origin to the LSS of a given cluster at $x$ spans the interval $|1-2x|\le x^{\prime }\le 1$.

Figure 2

Noisy estimator $K_{\ell }^{+}(\mathbb{1}+N_{\ell })K_{\ell }$ for $\ell =10$ (top) and $\ell =30$ (bottom), assuming zero errors (black stars) and assuming Gaussian errors with variances ${\sigma }_0=0.05$, 0.2, 0.5 (boxes of dark, medium and light gray.) We used 500 realizations of $N_{\ell }$ for each ${\sigma }_0$, and we use $f_{\mathrm{sky}}=2/3$. In this example we have assumed a hypothetical survey with 100 redshift bins up to $z=10$, and we recover information on 50 redshift bins.