Precision of slow-roll predictions for the cosmic microwave background radiation anisotropies

Abstract

Inflationary predictions for the anisotropy of the cosmic microwave background radiation are often based on the slow-roll approximation. We study the precision with which the multipole moments of the temperature two-point correlation function can be predicted by means of the slow-roll approximation. We ask whether this precision is good enough for the forthcoming high precision observations by means of the MAP and Planck satellites. The error in the multipole moments due to the slow-roll approximation is demonstrated to be bigger than the error in the power spectrum. For power-law inflation with ${n}_{\mathrm{S}}=0.9$ the error from the leading order slow-roll approximation is $\ensuremath{\approx}5%$ for the amplitudes and $\ensuremath{\approx}20%$ for the quadrupoles. For the next-to-leading order the errors are within a few percent. The errors increase with $|{n}_{\mathrm{S}}\ensuremath{-}1|.$ To obtain a precision of $1%$ it is necessary, but in general not sufficient, to use the next-to-leading order. In the case of power-law inflation this precision is obtained for the spectral indices if $|{n}_{\mathrm{S}}\ensuremath{-}1|l0.02$ and for the quadrupoles if $|{n}_{\mathrm{S}}\ensuremath{-}1|l0.15$ only. The errors in the higher multipoles are even larger than those for the quadrupole, e.g. $\ensuremath{\approx}15%$ for $l=100,$ with ${n}_{\mathrm{S}}=0.9$ at the next-to-leading order. We find that the accuracy of the slow-roll approximation may be improved by shifting the pivot scale of the primordial spectrum (the scale at which the slow-roll parameters are fixed) into the regime of acoustic oscillations. Nevertheless, the slow-roll approximation cannot be improved beyond the next-to-leading order in the slow-roll parameters.