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The theory of the anisotropies in the CMB is rich with details about the contents and structure of the early universe. Consequently, this theory can become quite complicated. However, because of this same richness, this branch of cosmology holds the potential to provide meaningful constraints on a very large number of quantities of cosmological interest. Our focus here is to provide the reader with a conceptual understanding of why and how CMB anisotriopies can be used to determine cosmological parameters. We will place particular emphasis on the density parameters corresponding to the spatial curvature of the universe Omegak, and the baryon density Omegab. The reader seeking more detail should consult Ref. 17 and the references therein.

5.1. Anisotropies in the CMB

The "hot big bang" model is widely accepted as the standard model of the early universe. According to this idea, our universe started in a very hot, very dense state that suddenly began to expand, and the expansion is continuing today. All of space was contained in that dense point. It is not possible to observe the expansion from an outside vantagepoint and it is not correct to think of the big bang as happening at one point in space. The big bang happened everywhere at once.

During the first fraction of a second after the big bang, it is widely believed that the universe went through a brief phase of exponential expansion called inflation. [18] Baryonic matter formed in about the first second; and the nuclei of the light elements began to form (nucleosynthesis) when the universe was only several minutes old. Baryons are particles made up of three quarks; the most familiar baryons are the protons and neutrons in the nuclei of atoms. Since all of the matter that we normally encounter is made up of atoms, baryonic matter is considered to be the "ordinary" matter in the universe.

The very early universe was hot enough to keep matter ionized, so the universe was filled with nucleons and free electrons. The density of free electrons was so high that Thomson scattering effectively made the universe opaque to electromagnetic radiation. The universe remained a baryonic plasma until around 300,000 years after the big bang when the universe had expanded and cooled to approximately 3000 K. At this point, the universe was sufficiently cool that the free electrons could join with protons to form neutral hydrogen. This process is called recombination. With electrons being taken up by atoms, the density of free electrons became sufficiently low that the mean free path of the photons became much larger (on the order of the size of the universe); and light was free to propagate. The light that was freed during recombination has now cooled to a temperature of about T0 = 2.73. This light is what we observe today as the cosmic microwave background. We see the CMB as if it were coming from a spherical shell called the surface of last scattering (Fig. 3). This shell has a finite thickness because recombination occurred over a finite amount of time.

Figure 3

Figure 3. The surface of last scattering (SLS), the fundamental acoustic mode, and the sound horizon (rs, thetas). The photons of the CMB underwent Thomson scattering in the early universe and acoustic oscillation left their imprint at recombination.

Today, over very large scales, the universe is homogeneous. However, as evidenced by our own existence, and the existence of galaxies and groups of galaxies, etc., inhomogeneities exist up to scales on the order of 100 Mpc. Theories of structure formation require that the seeds of the structure we observe today must have been inhomogeneities in the matter density of the early universe. These inhomogeneities would have left their imprint in the CMB which we would observe today as temperature anisotropies. So, in order to explain the universe in which we live, there should be bumps in the CMB; and these bumps should occur over angular scales that correspond to the scale of observed structure. In 1992, the COBE satellite measured temperature fluctuations deltaT in the CMB, deltaT/T ~ 10-5 on a 7° angular scale [19], where T is the ambient temperature of the CMB. The anisotropies detected by COBE are considered to be large-scale variations caused by nonuniformities generated at the creation of the universe. However, recent observations [20] - [22] have found small-scale anisotropies that correspond to the physical scale of today's observed structure. It is believed that these latter anisotropies are the result of quantum fluctuations in density that existed prior to inflation which were greatly amplified during inflation. These amplified fluctuations became the intrinsic density perturbations which are the seeds of structure formation.

The small-scale anisotropies in the CMB can be separated into two categories: primary and secondary. Primary anisotropies are due to effects that occur at the time of recombination and are "imprinted" in the CMB as the photons leave the surface of last scattering. Secondary anisotropies arise through scattering along the line of sight between the surface of last scattering and the observer. In this paper, we will only be concerned with the primary anisotropies. There are three main sources for primary anisotropies in the microwave background. These are the Sachs-Wolfe effect, intrinsic (adiabatic) perturbations, and a Doppler effect.

For the largest of these primary anisotropies the dominant mechanism is the Sachs-Wolfe effect. At the surface of last scattering, matter density fluctuations will lead to perturbations in the gravitational potential, delta Phi. These perturbations cause a gravitational redshift of the photons coming from the surface of last scattering as they "climb out" of the potential wells. This effect is described by, deltaT/T = delta Phi / c2. These same perturbations in the gravitational potential also cause a time dilation at the surface of last scattering, so these photons appear to come from a younger, hotter universe. This effect is described by, deltaT/T = -2(delta Phi) / 3c2. Combining these two processes gives the Sachs-Wolfe effect [23],

Equation 41 (41)

On intermediate scales, the main effect is due to adiabatic perturbations. Recombination occurs later in regions of higher density, so photons emanating from overly dense regions experience a smaller redshift from the universal expansion and thus appear hotter. The observed temperature anisotropy resulting from this process is given by [23],

Equation 42 (42)

Finally, on smaller scales there is a Doppler effect that becomes important. This effect arises because the photons are last scattered in a moving plasma. The temperature anisotropy corresponding to this effect is described by [23],

Equation 43 (43)

where r hat denotes the direction along the line of sight and vector v is a characteristic velocity of the material in the scattering medium.

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