The exact shape and position of the CMB peaks happen to depend on a large number of parameters. Since the precise shape of the peaks can very accurately be measured by high-precision CMB experiments such as WMAP or Planck, a wealth of informations can be extracted from them.

Sound speed

First, the frequency of the modes depend on the baryon density. For a pure radiation fluid, the sound speed is given by , where is the speed of light. When mixing radiation with non relativistic matter with negligible pressure, the sound speed squared becomes

,

where is the pressure plus density ratio between baryons and photons,

.

The temporal phase of the fluctuations therefore goes as

,

where is the conformal time. (It would reduce to if the sound speed was constant in time, which is not the case here.)

Gravity

As we said earlier, the smaller the pressure, the more unstable the propoagation of density perturbation is. Here, the presence of radiation insures that perturbations are stable, however, the presence of baryons modifies the zero point of the oscillations. One can show that the relevant quantity to study is not the photons density or pressure fluctuations, but a combination of the former with the gravitational potential

.

Thus, we see that the ratio between the even and odd peaks is directly proportional to the baryon density. Moreover, there exists a relation between the temperature fluctuations and the gravitational potential. This relation depends on the inflationary model, but for the simplest ones, it is

.

In this case, the ratio of amplitude between the initial time and half an oscillation is given by . By taking the square of this quantity, one obtains the ratio between the height of even and odd peaks in the CMB spectrum, which thus directly give an estimate of the baryon density. Maybe the most astounding success of CMB anisotropies (and the inflationary paradigm) is that the baryon density estimated that way is in good agreement with that coming from nucleosynthesis, whereas the two methods use completely different physics (the former relies on evolution of density fluctuations in an expanding universe at low temperature, whereas the latter comes from nuclear physics considerations).

Damping

The above results hold for large scales. At smaller scales, a number of microphysics effects have to be taken into account. In particular, the coupling between photons and baryons holds only at scales larger than the diffusion length,

,

where represents the differential opacity, that is the inverse of the photon free mean path. Below the diffusion scale, dissipation effect damp the fluctuations exponentially. In practice, this diffusion scale selects the relevant scales from an observational point of view since far below this scale no significant fluctuations are expected.

A second but less important damping effect comes from the fact that the CMB photons we observe were not all emitted exactly at the same epoch. Recombination is not instantaneous and lasts for a redshift interval around , so that the "exposure time" of the picture of the CMB you observe is not negligible. This effect is usually referred to as the finite width of the last scattering surface for obvious reasons. For scales which do not evolve significantly during recombination, this finite width is of no importance, but conversely for modes which experience several oscillations during this interval, the net effect is to average the fluctuations during this this epoch, which also translates into an exponential suppression of the observed fluctuations below the last scattering surface width. Note that it is only the observed anisotropies which are suppressed by this effect, the physical fluctuations are not suppressed here. The reason why this effect is negligible as compared to the former is that the width of the last scattering surface is usually significantly smaller than the diffusion length (at least for the expected values of the baryon density).

Summary

The effect of baryon density on CMB fluctuations is threefold:

 The amount of baryons modifies the sound speed, and hence the oscillation frequency of the modes. The wavelength of the mode which has experience one oscillation at recombination (say) is therefore dependant of the baryon density, so that the angular scale at which one sees the peaks.

 The constrast between even and odd peak reflect the unstable nature of propagation of sound in non relativistic media in the presence of gravity. Increasing the baryon density makes this contrast sharper.

 Dissipation damps fluctuations at small scales. The dffusion length decreases when baryon density increases as it reduces the photon free mean path.

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