MODERN COSMOLOGY

Acoustic oscillations 235

Figure 7.2. The G polarization power spectrum (full curve) and the cross-power TG
between temperature and polarization (dashed curve), for the same model as in figure 7.1.

density perturbations will set up standing acoustic waves in the plasma. Under
certain conditions, these waves leave a distinctive imprint on the power spectrum
of the microwave background, which in turn provides the basis for precision
constraints on cosmological parameters. This section reviews the basics of the
acoustic oscillations.

7.4.1 An oscillator equation

In their classic 1996 paper, Hu and Sugiyama transformed the basic equations
describing the evolution of perturbations into an oscillator equation. Combining
the zeroth moment of the photon Boltzmann equation with the baryon Euler
equation for a givenk-mode in the tight-coupling approximation (mean baryon
velocity equals mean radiation velocity) gives

' ̈ 0 +H R

1 +R

' ̇ 0 +k^2 c^2 s' 0 =− ̈−H R

1 +R

 ̇−^1

3

k^2 +, (7.27)

where ' 0 is the zeroth moment of the temperature distribution function
(proportional to the photon density perturbation),R= 3 ρb/ 4 ργis proportional
to the scale factora, H = ̇a/ais the conformal Hubble parameter, and the
sound speed is given byc^2 s = 1 /( 3 + 3 R). (All overdots are derivatives with