3.3. Acoustic oscillations and the horizon at last scattering
For fluctuation modes smaller than the horizon size, more complicated physics comes into play. Even a summary of the many effects that determine the precise shape of the CMB multipole spectrum is beyond the scope of these lectures, and the student is referred to Refs. [13] for a more detailed discussion. However, the dominant process that occurs on short wavelengths is important to us. These are acoustic oscillations in the baryon/photon plasma. The idea is simple: matter tends to collapse due to gravity onto regions where the density is higher than average, so the baryons "fall" into overdense regions. However, since the baryons and the photons are still strongly coupled, the photons tend to resist this collapse and push the baryons outward. The result is "ringing", or oscillatory modes of compression and rarefaction in the gas due to density fluctuations. The gas heats as it compresses and cools as it expands, and this creates fluctuations in the temperature of the CMB. This manifests itself in the C_{} spectrum as a series of bumps (Fig. 8). The specific shape and location of the bumps is created by complicated, although well-understood physics, involving a large number of cosmological parameters. The shape of the CMB multipole spectrum depends, for example, on the baryon density _{b}, the Hubble constant H_{0}, the densities of matter _{M} and cosmological constant _{}, and the amplitude of primordial gravitational waves (see Sec. 4.5). This makes interpretation of the spectrum something of a complex undertaking, but it also makes it a sensitive probe of cosmological models. In these lectures, I will primarily focus on the CMB as a probe of inflation, but there is much more to the story.
These oscillations are sound waves in the direct sense: compression waves in the gas. The position of the bumps in is determined by the oscillation frequency of the mode. The first bump is created by modes that have had time to go through half an oscillation in the age of the universe (compression), the second bump modes that have gone through one full oscillation (rarefaction), and so on. So what is the wavelength of a mode that goes through half an oscillation in a Hubble time? About the horizon size at the time of recombination, 300,000 light years or so! This is an immensely powerful tool: it in essence provides us with a ruler of known length (the wavelength of the oscillation mode, or the horizon size at recombination), situated at a known distance (the distance to the surface of last scattering at z = 1100). The angular size of this ruler when viewed at a fixed distance depends on the curvature of the space that lies between us and the surface of last scattering (Fig. 7).
Figure 7. The effect of geometry on angular size. Objects of a given angular size are smaller in a closed space than in a flat space. Conversely, objects of a given angular size are larger in an open space. (Figure courtesy of Wayne Hu [21].) |
If the space is negatively curved, the ruler will subtend a smaller angle than if the space is flat; ^{(5)} if the space is positively curved, the ruler will subtend a larger angle. We can measure the "angular size" of our "ruler" by looking at where the first acoustic peak shows up on the plot of the C_{} spectrum of CMB fluctuations. The positions of the peaks are determined by the curvature of the universe. ^{(6)} This is how we measure with the CMB. Fig. 8 shows an = 1 model and an = 0.3 model along with the current data. The data allow us to clearly distinguish between flat and open universes. Figure 9 shows limits from Type Ia supernovae and the CMB in the space of _{M} and _{}.
Figure 8. C_{} spectra for a universe with _{M} = 0.3 and _{} = 0.7 (blue line) and for _{M} = 0.3 and _{} = 0 (red line). The open universe is conclusively ruled out by the current data [20] (black crosses). |
Figure 9. Limits on _{M} and _{} from the CMB and from Type Ia supernovae. The two data sets together strongly favor a flat universe (CMB) with a cosmological constant (SNIa). [22] |
^{5} To paraphrase Gertrude Stein, "there's more there there." Back.
^{6} Not surprisingly, the real situation is a good deal more complicated than what I have described here. [13] Back.