Comparison of cosmological Boltzmann codes: Are we ready for high precision cosmology?

We compare three independent, cosmological linear perturbation theory codes to assess the level of agreement between them and to improve upon it by investigating the sources of discrepancy. By eliminating the major sources of numerical instability the final level of agreement between the codes was improved by an order of magnitude. The relative error is now below ${10}^{-3}$ for the dark matter power spectrum. For the cosmic microwave background anisotropies and using identical ionization histories the agreement is below the sampling variance up to $l=3000,$ with close to ${10}^{-3}$ accuracy reached over most of this range of scales. The same level of agreement is also achieved for the polarization spectrum and the temperature-polarization cross-spectrum. Linear perturbation theory codes are thus well prepared for the present and upcoming high precision cosmological observations. The remaining systematic errors are likely to be dominated by the uncertainties in the recombination history.