I'm studying from these lecture notes (Author: Wayne Hu, title: Lecture Notes on CMB Theory: From Nucleosynthesis to Recombination). At the very end of page 10 and beginning of page 11 he says:

Let us begin with the simple approximation that the temperature field at recombination is isotropic but inhomogeneous and the anisotropy viewed at the present is due to the observer seeing different portions of the recombination surface in different directions (see Fig. 8).

Now I'm not really understanding this argument. If, before last scattering the fluid is only inhomogeneous but isotropic I still expect CMB to be isotropic.

For example, let's suppose that with respect to an observer, before last scattering, at a distance $d$ in a direction $\hat{n}$ there is a positive fluctuation of temperature. I expect that, due to isotropy, there will be the same positive fluctuation in a direction $\hat{n}'$ at a distance $d$. After last scattering, let's say approximately after a time $t=d/c$ (not considering expansion of the universe here) the photons from that surface of radius $d$ arrive to the observer and it can measure the positive fluctuation in temperature in both the directions. Apply this to all the directions and we don't have any kind of anisotropy.

Am I missing something and there's a different way to understand it?


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