variance on the sphere $\sigma^2$ is represented as:

$$\sigma^2 = \frac{1}{4 \pi} \sum_l \sum_m (a_l^m)^2 = \sum_l \Delta
T_l^2 \,$$ where $\Delta T_l^2$ is a $l$–th spherical harmonic power.

In any real experiment the analyzed signal $a^m_{l,signal}$ is affected by a transfer function $W_l$ (thereafter we use the COBE antenna transfer function (Wright et al., 1994a) as $W_l$) and some noise $a^m_{l,noise}$ is added:

$$\Delta T (\theta, \varphi) = \sum (W_l a^m_{l,signal}+a^m_{l,noise})
Y_{l,m}(\theta, \varphi) \,$$

The cosmological signal on the sphere is a sample of a random process. In a standard cosmological scenario the inflation creates adiabatic scalar and tensor Gauss fields. So the spectrum is assumed to be a sample of a normal random process with independent harmonics, $\langle a_l^m a_k^n \rangle
\sim \delta
(l,k) \delta (m,n)$.

If initial density perturbation spectrum is $(\delta \rho / \rho)^2 = A k^n$, one may find for the mean lowest multipoles in expansion $\Delta T(\theta, \varphi) $ (Bond, Efstathiou, 1987):

$$\langle (a_l^m)^2 \rangle \sim \frac{\Gamma(l+(n-1)/2)
\Gamma((9-n)/2)}
            {\Gamma(l+(5-n)/2) \Gamma((3+n)/2)}
\label{approx_DTT}$$

A consideration of other physical phenomena ( evolution effects, acoustic oscillation, Silk effect etc.) causes a more complicated spectrum (White et al., 1994 and references therein). Hereafter we approximate a real spectrum with 