Figure 2

A model-independent method for determining the spectrum of the x-ray sources that heated the Universe. We compare W k as measured from mock observations to the predicted W k lin from the radial distribution of x-ray flux as measured in our simulations. We compare our two cases of a hard x-ray SED (left panel) or a soft SED (right panel). We show W k at a range of redshifts, z = 12 (solid red), 13 (dashed red), and 14 (dotted red) for the hard SED, and z = 14 (solid red), 16 (dashed red), and 18 (dotted red) for the soft SED. Each case is compared to W k lin (which changes little with z ) at the central redshift (black curve); for contrast we also show W k lin from the other panel (gray, dot-dashed line). We only wish to compare shapes, so all the curves are normalized to unity at k = 0.1 Mpc − 1 (but to illustrate the overall amplitude at these redshifts, we note that the highest fluctuation at k = 0.2 is 3.3 mK for the hard SED at z = 14 , and 12.4 mK for the soft SED at z = 16 ). We note that the slopes of W k and W k lin agree only on scales on which the 21-cm fluctuations are dominated by heating fluctuations; also note that in this figure we are able to go to much larger scales than in Fig. 1, since the isotropically averaged power spectrum can be determined more accurately in a given simulation box than parameters that depend on the angular variation.