On the contribution of density perturbations and gravitational waves to the lower order multipoles of the cosmic microwave background radiation

The important studies of Peebles, and Bond and Efstathiou have led to the formula C-l = const./[l(l + 1)] aimed at describing the lower order multipoles of the CMBR temperature variations caused by density perturbations with the flat spectrum. Clearly, this formula requires amendments, as it predicts an infinitely large monopole C-0, and a dipole moment C-1 only 6/2 times larger than the quadrupole C-2, both predictions in conflict with observations. We restore the terms omitted in the course of the derivation of this formula, and arrive at a new expression. According to the corrected formula, the monopole moment is finite and small, while the dipole moment is sensitive to short-wavelength perturbations, and numerically much larger than the quadrupole, as one would expect on physical grounds. At the same time, the function l(l + 1)C-l deviates from a horizontal line and grows with l, for l greater than or equal to 2. We show that the inclusion of the modulating (transfer) function terminates the growth and forms the first peak, recently observed. We fit the theoretical curves to the position and height of the first peak, as well as to the observed dipole, varying three parameters: red-shift at decoupling, red-shift at matter-radiation equality, and slope of the primordial spectrum. It appears that there is always a deficit, as compared with the COBE observations, at small multipoles, l similar to 10. We demonstrate that a reasonable and theoretically expected amount of gravitational waves bridges this gap at small multipoles, leaving the other fits as good as before. We show that the observationally acceptable models permit somewhat "blue" primordial spectra. This allows one to avoid the infrared divergence of cosmological perturbations, which is otherwise present.