Annu. Rev. Astron. Astrophys. 1994. 32: 319-70
Copyright © 1994 by . All rights reserved

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If the underlying fluctuation spectrum is assumed to be stochastic in nature, then one is faced with the problem of trying to compare a theory which defines probability distributions as functions of the underlying parameters with just one sample drawn from these (our Universe). Only observations over an ensemble of universes would allow one to determine the parameters of the underlying theory unambiguously, even in principle. Since we can observe only our universe, there is an irremovable uncertainty in our ability to relate certain CMB measurements, no matter how precise, to parameters of the theory. The uncertainty introduced in determinations of theoretical parameters has been called "cosmic variance" or "theoretical uncertainty" (e.g. Abbott & Wise 1984a, b; Scaramella & Vittorio 1990, 1991, 1993b; Cayón et al 1991; White et al 1993).

In terms of the multipole moments the cosmic variance may be thought of as an uncertainty in relating

Equation 36 (36)

In a theory with Gaussian fluctuations, the aellm are independent Gaussian random variables and the aell2 are thus chi22ell+1 distributed (Abbott & Wise 1984a, b; Abbott & Schaefer 1986; Vittorio et al 1988). The uncertainty in relating aell2 and <aell2>ens is given by the width of the distribution, which scales as (ell + 1/2)-1/2. As one goes to smaller scales and probes larger ell, one thus becomes less sensitive to cosmic variance. On large scales, however, cosmic variance forms the limiting uncertainty in fixing (e.g.) the normalization of the power spectrum. For the quadrupole (with only 5 degrees of freedom), the actual amplitude in our Universe is likely to differ substantially from its expectation value (Gould 1993, Stark 1993).

While it is true that on smaller scales, which probe higher ell, the cosmic variance becomes negligible, this applies only to a full sky measurement. The variance from sampling only a fraction of the sky is larger than the cosmic variance one gets when sampling the whole sky. Since current small-scale experiments cover only small patches of the sky, this can be an important effect. In general for an experiment that samples a solid angle Omega, the cosmic variance is enhanced by a factor of 4pi / Omega (Scott et al 1994, see also Bunn et al 1994a). We should emphasize that this "sample variance" is completely independent of the experimental precision, and simply reflects the fact that the experiment has not covered enough of the sky to provide a good estimate of the rms value of DeltaT / T.

The spectrum of fluctuations depends not only on the primordial spectrum but on the evolution of perturbations as waves enter the horizon. This evolution introduces dependencies on Omega0, H0, and the dark matter. In addition, a knowledge of the ionization history of the universe is important since reionization would lead to both a reprocessing of primary fluctuations and generation of (small-scale) secondary fluctuations. (See Section 5.1.) A non-negligible ionized fraction during much of the history of the universe would also contribute a radiation drag which would affect the growth of baryon perturbations and alter the observed CMB anisotropy. In general, the uncertainties in the cosmological parameters and history produce very correlated shifts in the radiation power spectrum (Bond et al 1994, Hu & Sugiyama 1994b, c).

In addition, there are possible sources of "foreground" contamination such as synchrotron and free-free emission (at low frequency) and dust (at high frequency), These non-cosmological sources have been discussed in detail in the review of Readhead & Lawrence (1992) and we will not mention them further except to say that to discriminate between these sources it is necessary to have good frequency coverage and a wide range of angular resolutions. Note that, in particular, the galactic signal also makes it difficult to measure the quadrupole anisotropy (de Bernardis et al 1991, Bennett et al 1994).

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