Statistical Description of the Anisotropies

The temperature pattern on the celestial sphere can be expanded in spherical harmonics

$${\Delta T \over T} = \sum_{\ell,m} a_\ell^m Y_{\ell}^m(\theta, \phi),$$ (1)

and the power spectrum of the temperature fluctuations, $C_\ell$, is defined by the mean square value of the coefficients $a_\ell^m$

$$C_{\ell} = \langle \vert a_{\ell}^m \vert^2 \rangle.$$ (2)

If the fluctuations in the early universe obey Gaussian statistics, as expected in most theories (see Section Testing Theories of Inflation ), each of the coefficients $a_\ell^m$ is independent and so the power spectrum $C_\ell$ provides a complete statistical description of the temperature anisotropies. The temperature power spectrum, $C_\ell$, is thus of fundamental importance in studies of the microwave background anisotropies. The temperature power spectrum can be estimated directly from observations by performing a spherical harmonic analysis, as has been done with the COBE data (Gorski et al.1994, Bond 1995, Tegmark 1996). However there are other ways of analyzing observations which are closely related to the temperature power spectrum, e.g. the temperature autocorrelation function (see e.g. Banday et al.1994, for an application to the COBE maps).

Figure 1.3 shows a calculation of the temperature power spectrum for a cold dark matter dominated universe with $\Omega_0=1$. These curves assume a scale-invariant initial fluctuation spectrum, as expected in the simplest models of inflation, a baryon density $\Omega_b = 0.05$and a Hubble constant h=0.5. The curve labeled `density' shows the power spectrum from density perturbations; these are the small irregularities in the early universe that grow under the action of gravity to form the structure in the Universe that we see today. The curve labeled `grav. waves' shows the power spectrum arising from gravitational waves generated during inflation (e.g. Starobinsky 1985, Davis et al.1992, Crittenden et al.1993). Notice the large differences in shape between the two curves which can be utilized to test models of inflation (see Section Testing Theories of Inflation).

The multipole $\ell$ tells us about anisotropies on an angular scale $\theta \sim 1/\ell$, as indicated by the scale at the top of the Figure. Thus COBE, which has an angular resolution of $\theta_{FWHM} \approx 7^\circ$ samples only low multipoles $\ell \lower.5ex\hbox{$\; \buildrel < \over \sim \;$}20$ (shown by the shaded bar in the upper panel). In contrast, the high angular resolution of PLANCK will allow measurements of multipoles up to $\ell \lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}1000$, thus sampling almost the full multipole range of the theoretical predictions. The shapes of these curves, and the physics that determine them, are described in the next section.

Figure 1.3: The power spectrum C of the microwave background anisotropies plotted against multipole for an inflationary cold dark matter cosmology. The angular scale corresponding to a given multipole is indicated by the scale at the top of the figure. The curve labeled density shows the contribution to the temperature power spectrum from small density fluctuations in the early universe. The curve labeled grav. waves shows the contribution to the temperature power spectrum from gravitational waves generated during inflation. The relative amplitude of these two contributions depends on the specific details of the model of inflation, as described in Section Testing Theories of Inflation. The bars show the range of multipoles (angular scales) probed by COBE and by PLANCK.

[last update: 1 August 1999 by P. Fosalba]