Power Spectra

The final step in calculating the anisotropy spectra is to integrate over the k-modes. The power spectra of temperature and polarization anisotropies today are defined as, e.g. $C_\ell^{\Theta\Theta}\equiv\left\langle \vert a_{\ell m} \vert^2\right\rangle$for $\Theta = \sum a_{\ell m} Y_\ell^m$ with the average being over the ($2\ell+1$) m-values. In terms of the moments of the previous section  

$$(2\ell+1)^2 C_\ell^{X\widetilde X} = {2 \over \pi} \int {dq... ... \sum_{m=-2}^2 q^3 \ X_\ell^{(m)*} \widetilde X_\ell^{(m)}\, ,$$ (35)

where X takes on the values $\Theta$, E and B for the temperature, electric polarization and magnetic polarization evaluated at the present. For a closed geometry, the integral is replaced by a sum over $q/\vert K\vert=3,4,5\ldots$Note that there is no cross correlation $C_\ell^{\Theta B}$ or $C_\ell^{E B}$due to parity.

We caution the reader that power spectra for the metric fluctuation sources $P_h(q) = \left\langle h^*(q) h(q) \right\rangle$ must be defined in a similar fashion for consistency and choices between various authors differ by factors related to the curvature (see [19] for further discussion). To clarify this point, the initial power spectra of the metric fluctuations for a scale-invariant spectrum of scalar modes and minimal inflationary gravity wave modes [3] are 

$$\begin{array} {rcl}P_{\Phi}(q) &\propto& {\displaystyle{1 \o... ...laystyle{(q^2+4)\over q^3(q^2+1)}} \tanh(\pi q/2)\,,\end{array}$$

where the normalization of the power spectrum comes from the underlying theory for the generation of the perturbations. This proportionality constant is related to the amplitude of the matter power spectrum on large scales or the energy density in long-wavelength gravitational waves [19]. The vector perturbations have only decaying modes and so are only present in seeded models. The other initial conditions follow from detailed balance of the evolution equations and gauge transformations (see Appendix A).

Our conventions for the moments also differ from those in [13,14]. They are related to those of [13] by

$$\begin{array} {rcl}(2\ell+1)\Delta_{T\ell}^{(S)} &=& \Theta_... ...^{(T)} &=& \sqrt{2}\Theta_\ell^{(2)}/(2\pi)^{3/2}\,,\end{array}$$

where the factor of $\sqrt{2}$ in the latter comes from the quadrature sum over equal m=2 and -2 contributions. Similar relations for $\Delta_{(E,B)\ell}^{(S,T)}$ occur but with an extra minus sign so that $C_{C,\ell}=-C_\ell^{\Theta E}$ with the other power spectra unchanged. The output of CMBFAST continues to be $C_{C,\ell}$ with the sign convention of [13]. In the notation of [14], the temperature power spectra agree but for polarization $C^{EE,BB}_\ell=C^{G,C}_\ell/2$ and $C^{\Theta E}_\ell=-C^{TG}_\ell/\sqrt{2}$.

  

Figure 1: The scalar (left) and tensor (right) angular power spectra for anisotropies in a critical density model (thick lines) and an open model (thin lines) with $\Omega_0=0.4$.Solid lines are $C_\ell^{\Theta\Theta}$, dashed $C_\ell^{EE}$ and dotted $C_\ell^{BB}$.