# How to translate from the CMB Power spectrum to a spectrum of temperature variations

The angular power spectrum of the cosmic microwave is the most frequently plotted quantity when talking about structure in the CMB.

But how is this quantity actually related to the rms temperature fluctuations in the CMB as a function of angular scale? I have seen one or two plots on the web which show $\langle(\Delta T)^2\rangle$ as a function of angular scale, and one or two vague statements about $l(l+1)C_l$ being proportional to the temperature variance.

Can anybody collect together a definitive statement about how the usual power spectrum and the temperature variations are connected?

• Afaik, there is no short path. The whole calculation is presented in "Physical Foundations of Cosmology" by V. Mukhanov, Chapter 9. It's a long computation and, to be honest, I never had the courage to read through it but after a quick look it seems to present all the necessary steps to establish the link between the temperature and the density/potential fluctuations, and also deals with the power spectrum of the temperature anisotropy. – Photon Apr 29 '17 at 16:33

Consider a direction $(\theta, \phi) = \hat{n}$, by fitting Planck's law to the radiation density you get the temperature $T(\hat{n})$. Define the quantity

$$\delta(\hat{n}) = \frac{T(\hat{n}) - \bar{T}}{\bar{T}} = \frac{\Delta T(\hat{n})}{\bar{T}} \tag{1}\label{1}$$

It is alway possible to expand Eq. $\ref{1}$ in spherical harmonics

$$\delta(\hat{n}) = \sum_{lm}a_{lm}Y_{lm}(\theta,\phi) \tag{2}\label{2}$$

If you think of $\delta$ as a random process, you can calculate things like the autocorrelation function of the temperature fluctuations

$$C(\theta) = \langle \delta(\hat{n}_1) \delta(\hat{n}_2)\rangle, ~~~\mbox{with}~~~ \cos\theta = \langle \hat{n}_1| \hat{n}_2\rangle \tag{3}\label{3}$$

It is not difficult to show that

$$C(\theta) = \frac{1}{4\pi} \sum_l(2l + 1) C_l P_l(\cos \theta), ~~~\mbox{where}~~~ C_l = \langle |a_{lm}|^2\rangle \tag{4}\label{4}$$

and $P_l$ the Legendre polynomial of degree $l$. Note Eq. $\ref{4}$ is just a link between $C_l$ and the autocorrelation of the random process $\delta$, as matter of fact the orthogonality of $P_l$ can be used to express $C_l$ as a function of $C(\theta)$.

The advantage of going this route is that in the linear regime $\delta$ is related with the mass perturbations $\Delta$, indeed

$$\delta = \frac{1}{4}\Delta$$

And there's a whole formalism to calculate $\Delta$ (See for instance Ch. 3 of this reference), for instance it is possible to show that for angular scales the density perturbations that give raise to the temperature fluctuations have a wavenumber $k \gg 2\pi a /ct_{\rm dec}$ ($t_{\rm dec}$ = time of decoupling), and

$$C_l \sim \frac{1}{l(l + 1)}$$

The way I find most useful to think about it is that if I drew circles of diameter $l$ on the CMB, what would be the average power (squared $\Delta T$) inside those circles? The CMB power spectrum tells us that most of the power is on degree scales (the biggest bump), but that there are contributions at specific scales (the smaller bumps).