The Power Spectrum of CMB Fluctuations

So far, we have discussed the evolution of perturbations in the dark matter, photons and baryons at any given point, for some particular set of initial conditions. However, our theories are not so specific. Rather, they give us only statistical information about the initial perturbations. 

It turns out that everything will be easier to understand in Fourier space, so we first define our conventions  

In particular, we will rely on the power spectrum, defined as 

Similarly, we can transform the temperature fluctuations on the sky using the spherical harmonics, V7 , (x) 

We can then define the power spectrum of temperature fluctuations, Ci, (Bond and Efstathiou, 1987) 

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