A comparison of cosmological Boltzmann codes: are we ready for high precision cosmology?

TL;DR

The paper addresses the numerical precision of cosmological Boltzmann codes used for linear perturbation theory in high-precision cosmology. It compares three independent codes—NS, MW, and CMBFAST—against a common flat $\Lambda$CDM-like model with shared recombination outputs to isolate numerical sources of discrepancy. The study finds that, after eliminating major numerical instabilities, the relative error in the dark matter power spectrum is below $0.1\%$, and CMB spectra $C_l^{TT}$, $C_l^{EE}$, and $C_l^{TE}$ agree with sampling variance up to $l=3000$, with approximately 0.1% accuracy across most scales. This establishes that current Boltzmann codes are prepared for present and upcoming sub-percent cosmological tests, while highlighting recombination physics and certain physical extensions as areas for further verification.

Abstract

We compare three independent, cosmological linear perturbation theory codes to asses 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 0.1% for the dark matter power spectrum. For the cosmic microwave background anisotropies the agreement is below the sampling variance up to l=3000, with close to 0.1% 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.

Figures (4)

• Figure 1: Dark matter power spectrum for the 3 codes (top) and ratios between them (bottom). Also shown are $1\pm 0.1$% horizontal lines. The relative errors are below 0.1%.
• Figure 2: $C_l^{TT}$ for the 3 codes (top) and ratios between them (bottom). Also shown is the sampling variance limit $1\pm 3/l$ and $1\pm 0.1$% horizontal lines.
• Figure 3: Same as figure \ref{['fig2']} for $C_l^{EE}$.
• Figure 4: Same as figure \ref{['fig2']} for $C_l^{TE}$. At zero crossings of $C_l^{TE}$ the relative error is ill-defined, so we compare to a smoothed version, where the smoothing is $\Delta l =50$. The plotted sampling variance limit $1\pm 3/l$ is a lower limit to the actual sampling variance, as discussed in the text.