Oscillations in the Primordial Plasma

Armed with an understanding of the evolution of the ionization fraction and of perturbations, we can actually understand most of the qualitative features of the CMB. 

But any complete description of the evolution of perturbations in the universe will link all of these terms initial velocity and density perturbations to the various components (baryons, dark matter, photons) evolve prior to last scattering as discussed above, and so photon overdensities occur in potential wells, and velocity perturbations occur in response to gravitational and pressure forces. Indeed, to solve this problem in its most general form, we must resort to the Boltzmann equation. The Boltzmann equation gives the evolution of the distribution function, fi(xp,Pp) for a particle of species i with position Xp, and momentum p/(. In its most general form, the Boltzmann equation is formally 

If we subtract this zeroth order solution, fourier transform the x coordinates, convert the time coordinate to conformal time, r), defined by dr) = dt/a, and ignore vector and tensor perturbations (discussed in the lectures by J. Bartlett on polarization at this school), the Liouville operator becomes a first-order partial differential operator for /( (k, p, rj), depending also on the general-relativistic potentials, (I and T. We further define the temperature fluctuation at a point, 0(jfc, p) = f( lj i lodf 0 1 / 9To) 1 where To is the average temperature and )i = cos 6 in the polar coordinates for wavevector k. 

The collision term, C[f] is the rate for interactions to change /(k. p. rj). For photons, this is from Thomson scattering off of electrons, with differential cross section 

we can only investigate some of the simplest qualitative features of the solution. First consider the effect of free-streaming since recombination, 

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CJoherent oscillations in the primordial plasma [...] leave an imprint in the linear power spectrum of the dark matter",... 

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