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2.2 The Power Spectrum Deduced From the Cosmic Background Radiation

The microwave background is isotropic to about one part in 103. If one removes the anisotropy caused by our motion with respect to the cosmic background radiation (CBR) rest frame, then it is isotropic to about 30 parts-per-million. But as first discovered by the Cosmic Background Explorer (COBE), there are intrinsic fluctuation in the temperature of the CBR.

Figure 2. The angular power spectrum of CBR fluctuations (courtesy of Dick Bond and Llyod Knox).

Just as perturbations in the density field were expanded in terms of Fourier components, a similar expansion is useful for temperature fluctuations. Because the surface of observation about us can be described in terms of spherical angles theta and phi, the correct expansion basis is spherical harmonics, Ylm (theta, phi). If the average temperature is < T >, then one can expand

Equation 5 (5)

Of course < alm > = 0, but with proper averaging,

Equation 6 (6)

Cl as a function of l is called the angular power spectrum. In the six years since the first measurement of CBR fluctuations by COBE, a number of experiments have detected fluctuations. The present situation is illustrated in Fig. 2.

Associated with a multipole number l is a characteristic angle theta, and a length scale we can define as the distance subtended by theta on the surface of last scattering. Since the distance to the last scattering surface of the microwave background is so large, the temperature fluctuations represent the largest structures ever seen in the universe.

Contributing to the temperature anisotropies are fluctuations in the gravitational potential on the surface of last scattering. Photons escaping from regions of high density will suffer a larger than average gravitational redshift, hence will appear to originate from a cold region. In similar fashion, photons coming to us from a low-density region will appear hot. In this manner, temperature fluctuations can probe the density field on the surface of last scattering and provide information about the power spectrum on scales much larger than can be probed by conventional large-scale structure observations.

The region of wavenumber and amplitude of the power spectrum probed by COBE is illustrated in Fig. 3. There are now measurements of CBR fluctuations on smaller angular scale, corresponding to larger k.

Figure 3. The power spectrum deduced by measurements of large angular scale CBR temperature fluctuations.

Finally, Fig. 4 combines information from both large-scale structure surveys and CBR temperature fluctuations. The trend is obvious: on small distance scales the power spectrum is ``large,'' which implies a lot of structure. Matter is clustered on small scales. But on ``large'' scales the power spectrum decreases. As one examines the universe on larger scales, homogeneity and isotropy becomes a better and better approximation.

Figure 4. The ``grand unified'' power spectrum, including determinations from large-scale structure surveys (the points), and deduced from CBR temperature fluctuations (the box).

The data shown is only illustrative of many data sets. Although combining different data sets is uncertain and risky (problems with normalization, etc.) the qualitative features are the same. Figure 4 is best regarded as an impressionist representation of the situation.

Another thing to keep in mind is that the power spectrum may not be the entire story. The power spectrum contains all statistical information about the perturbations only if the fluctuations are Gaussian. This should be cause for concern, because even if the initial perturbations are Gaussian, eventually they will become non-Gaussian once the perturbations become nonlinear. Also, the power spectrum is not a useful discriminant for prominent features such as walls, voids, filaments, etc. In spite of its drawbacks, the power spectrum is remarkably useful - if we can't get the power spectrum right, then we are not on the right track.

Now we turn to an early-universe theory that can account for the power spectrum: inflation

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