There are many differences between the peaks in Fig. 44 and the blue peaks in Fig. 45. There is also a marked difference between the observed power spectrum in Fig. 43 and the primordial power
spectrum proportional to a power of k. These differences are due to the presence of matter and gravity, as well as expansion and other smaller effects. It is from such differences that the amount of ordinary and dark matter as well as other cosmological parameters is determined. Once the matter content is known, the amount of dark energy present can be determined from the flatness of the universe.
Let us now discuss some of these corrections to the photon fluid approximation.
First of all, the minima in Fig. 45 between the peaks always reach down to zero, but not in Fig. 44. This is because realistically the initial oscillation does not start from rest. The resulting Doppler shift makes an out-of-phase contribution which fills in the zeros, and it also shifts the location of the peaks by a small amount.
Secondly, the peaks in Fig. 45 are of equal height but those in Fig. 44 decrease for higher l. Moreover, the peak at k = 0 in Fig. 45 is absent in Fig. 44. These differences are due mainly to the gravitational and matter effects explained below.
Without gravity, the compression phase and the stretching phase of the oscillation are symmetrical. In the presence of an attractive gravity, there would be more compression and less stretching, and hence the amplitude square of the compression (odd) peaks are higher and the stretching (even) peaks are lower than those shown in Fig. 45 (the peak at k = 0 in Fig. 45 is counted as the zeroth peak). This partly explains why the second peak in Fig. 44 is lower, but it does not explain why the third peak is lower still. That effect comes from the direct presence of ordinary and dark matter.
Note that there is a difference between ordinary matter and dark matter because the former interacts with photons and is a part of the oscillating plasma, but the latter is not coupled to the
THEZENINMODERNCOSMOLOGY
photon so it contributes only to the building up of gravitational potential.
The presence of ordinary matter provides an added inertia to the plasma oscillation. It also contributes to the energy density but not the pressure, so it slows down the velocity of sound cs
and shortens the sound horizon csη∗. These effects become more
and more important as we get closer and closer to the matter- dominated era.
In contrast, dark matter does not contribute directly to oscillation, but it contributes to gravity which compresses matter. In the matter-dominated era, this tendency to increase density is almost exactly balanced by the expansion of the universe which tends to decrease it. As a result, density, sound amplitude, and gravitational potential remain fairly constant throughout the matter era. Since disturbances with a low k re-enter the horizon directly into the matter era (see Fig. 42), the observed sound amplitudes remain constant so they are the same as their primordial initial amplitudes. In other words, the intervening universe is actually transparent to low frequency sound waves. This explains why at small k the observed power spectrum in Fig. 43 is the same as the primordial power spectrum, proportional to a power of k.
It also explains the absence of a peak at zero l in Fig. 44, although it is present in the simplified model of Fig. 45 at k = 0. When the disturbances re-enter directly into the matter era, pressure is absent so there is no oscillation, and the observed amplitude remains approximately constant. This is why an oscillation peak is absent for small k when gravitational and matter effects are included.
For disturbances with higher k which re-enter into the radiation-dominated era, the presence of photon pressure prevents a gravitational collapse, but the expansion of the universe is still
present to dilute the local density and gravity, and to reduce the observed amplitude, thus causing the peaks in Fig. 44 to decrease with increasing k, or l.
This also causes the power spectrum in Fig. 43 to drop away from the primordial power spectrum proportional to a power of k.
Indeed, the linear dependence in Fig. 43 stops close to the peak, at k about 0.01 h/Mpc. Here, h = 0.72 is the Hubble constant
H0 measured in units of 100 km/s/Mpc. This agrees with the
explanation above because it is equal to the aH/c value at the boundary between the radiation and the matter eras.[12] For values
of k larger than that, the disturbance re-enters in the radiation era, thus causing the observed power spectrum to drop.
Finally, there is another effect which causes the magnitude of the observed amplitude to decrease at large k or l. This effect is important for large l so it should definitely be taken into account for the Planck satellite data. I am referring to the leakage of photons. Up to this point we have assumed the photons to be completely trapped within the plasma and oscillate with it in unison. However, when k is large and the oscillation fast, photons cannot be bounced fast enough between the charged particles in the plasma to keep in sync, and a leakage will occur. This weakens the pressure and hence the oscillation amplitude.