CMB physics

In the framework of Big-Bang theory, the Universe started with a hot and dense phase about 15 billion years ago and cooled down while expanding. The first neutral atoms formed when the temperature was about 13.6 eV (160000 K), but due to the large number of photons compared to baryons (ratio $\simeq 10^9$), the Universe remained ionized until the temperature dropped below 0.3 eV (3000 K). At this moment, the mean free path of the photons increased drastically so that the photons that scattered at this time have not interacted with matter since then. This moment is known as matter-radiation decoupling or recombination. Those photons cooled down with the expansion of the Universe and are know observed at a temperature of 2.7 K. As the matter and radiation were at thermal equilibrium before decoupling, these photons have a pure blackbody spectrum and are homogeneously distributed on the celestial sphere. This radiation is known as the Cosmic Microwave Background (hereafter CMB).

The discovery of the CMB by Penzias and Wilson [1] and its interpretation in terms of a Big-Bang relic by Dicke and collaborators [2] was a major argument for the Big-Bang theory [3,4]. The CMB temperature was measured to be highly isotropic but tiny anisotropies were expected. These temperature fluctuations reflect the density fluctuations on the last scattering surface. These are necessary to explain the presence of structures in the Universe such as galaxies and clusters. The CMB anisotropies were discovered by the COBE satellite with a rms amplitude of about 30 $\mu\mathrm{K}$ [5] at scales larger than 7 degrees. COBE also measured its spectrum with high precision [6,7] proving its pure blackbody nature.

The CMB anisotropy typical physical size in the last scattering surface can be theoretically predicted while its angular size as seen from here and now depends on the geometry of the Universe along the path of the photons. Hence, mapping the CMB anisotropies is a powerful cosmological test.

The two competing paradigms for the origin of structures in the Universe, namely inflation and topological defects, predict significantly different distributions for the former density fluctuations. These distributions propagate to us in a cosmological parameters dependent way to describe the temperature anisotropies that we expect on the sky¹. It is therefore of deep interest to investigate their angular distribution and compare the measurements to cosmological models.

The temperature anisotropies on the sky are commonly described via their spherical harmonics expansion,

$$\frac{\delta T}{T}\left(\theta,\phi\right)=\sum_{\ell=0}^\in... ...m_{m=-\ell}^\ell a_{\ell m}Y_{\ell m}\left(\theta,\phi\right),$$ (1)

where $\ell$ is the multipole index, inversely proportional to the angular scale (1 degree roughly corresponds to $\ell=200$). The angular power spectrum of the temperature fluctuations of the CMB is defined as:

$$C_\ell=\frac{1}{2\ell+1}\sum_{m=-\ell}^\ell\left\vert a_{\ell m}\right\vert^2.$$ (2)

The evolution of the angular power spectrum of the CMB as a function of $\ell$ can be splitted into three major regions (see Figure ):

• On the low-$\ell$ part (large angular scales) no particular structure is expected as we are considering physical sizes on the last scattering surface larger than the horizon at the epoch of decoupling. No physical process is expected to have modified those fluctuations since the early Universe.
• Between $\ell\simeq 30$ and $\ell\simeq 1000$ (degree and sub-degree scales) we a re considering structures that had time to collapse and experience acoustic oscillations between the matter-radiation equality and the matter radiation decoupling. We therefore expect a series of acoustic peaks (the first one being located around $\ell=200$, corresponding to the size of the horizon at the epoch of decoupling) in the case of inflationary-like early Universe models where the oscillations are in phase. In the case of isocurvature fluctuations (such as topological defects), the oscillations are not in phase and a large bump is expected, but no multiple peaks.
• In the large $\ell$ part (arcminute scales and below), the power is expected to drop drastically due to the finite thickness of the last scattering surface and to the finite value of the mean free path of the photons before decoupling.

Figure: Expected CMB power spectrum $\left(\sqrt{\frac{\ell\left(\ell+1\right)C_\ell}{2\pi}}\right)$ for inflationary-like primordial density fluctuations (black curves) for three different cosmological models along with the latest measurements from BOOMERanG, MAXIMA and DASI and the earlier measurements from COBE.