The precision of slow-roll predictions for the CMBR anisotropies

TL;DR

This study quantifies how precisely slow-roll inflation can predict CMBR temperature anisotropies, focusing on the multipole spectrum $C_l$ and comparing slow-roll predictions (leading and next-to-leading order) to exact power-law inflation results. It shows that errors in $C_l$ from slow-roll generally exceed those in the primordial power spectrum, with leading-order errors around a few percent for amplitudes and tens of percent for the quadrupole, while next-to-leading order reduces errors only to a few percent unless the slow-roll parameters are very small. The authors demonstrate that achieving ~1% accuracy typically requires $| _S-1| aise.5exox0pt<0.02$ and/or $oldsymbol{ heta} o 0.01$ or smaller, and that optimizing the pivot scale toward acoustic oscillations ($k_0 r_{ m lss} oughly 100 e$) can help near the first peak but cannot bypass the fundamental limit of slow-roll, especially at high $l$. Consequently, for many realistic inflation models, slow-roll predictions remain larger than cosmic variance, indicating that full numerical mode-by-mode calculations are necessary for robust CMB forecasts and that higher-order slow-roll corrections beyond next-to-leading order are not meaningful.

Abstract

Inflationary predictions for the anisotropy of the cosmic microwave background radiation (CMBR) 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 $n_S=0.9$ the error from the leading order slow-roll approximation is $\approx 5%$ for the amplitudes and $\approx 20%$ for the quadrupoles. For the next-to-leading order the errors are within a few percent. The errors increase with $|n_S - 1|$. 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 $|n_S - 1| < 0.02$ and for the quadrupoles if $|n_S - 1| < 0.15$ only. The errors in the higher multipoles are even larger than those for the quadrupole, e.g. $\approx 15%$ for l=100, with $n_S = 0.9$ 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.

Figures (14)

• Figure 1: Error due to the long wavelength approximation in the transfer function for the scalar multipoles with a flat primordial spectrum. The exact multipoles are calculated by means of the CMBFAST code and are normalized to the quadrupole.
• Figure 2: The amplitudes of scalar and tensor perturbations. In the de Sitter limit $\gamma \to 0$ the scalar amplitude diverges. For larger values of $\gamma$ the perturbations are dominated by the tensor mode.
• Figure 3: The quadrupole moments of scalar and tensor perturbations.
• Figure 4: The tensor to scalar ratio of the quadrupole moments.
• Figure 5: Sketch of the effective potential for density perturbations and/or gravitational waves during inflation and radiation.