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The expected anisotropy in the microwave background radiation on both large and small angular scales has been calculated. Primordial adiabatic and isothermal fluctuations are considered for a variety of initial power-law fluctuation spectra |δm(k)|2 ∝ kn in a spatially flat universe. The calculated temperature anisotropy on small angular scales is below the current observational upper limits if Ω ≳ 1. An appreciable dipole and smaller quadrupole anisotropy are also produced. These are generated by fluctuations in the mean gravitational potential associated with small-amplitude large- wavelength fluctuations in the matter distribution on scales &≳10-100 Mpc. Most of the observed dipole anisotropy could be attributed to this effect, as could the possible detection of a quadrupole anisotropy. More generally, the large-scale radiation anisotropy measurements constrain the index of the initial fluctuation spectrum: n ≳ 0 for isothermal fluctuations and n ≳ 3 for adiabatic fluctuations. A further implication is that, in general, our peculiar velocity relative to distance shells of matter only coincides with that measured relative to the background radiation (inferred from the dipole moment) at a redshift z ≳ 0.1. At small redshifts there can be significant deviations in both direction and amplitude.
Wilson et al. (Thu,) studied this question.