Optimising Boltzmann codes for the Planck era

TL;DR

The paper addresses the risk that numerical inaccuracies in Boltzmann codes like CAMB could bias Planck-era CMB parameter inference. It constructs reference spectra and an effective $\chi^2$ measure to quantify accuracy against Planck-like data, then uses a modified CosmoMC search to optimize 19 CAMB accuracy parameters under a time constraint. The results show that default CAMB settings can bias vanilla $\Lambda$CDM parameters by up to a few tenths of a standard deviation, while optimized configurations reduce potential biases to well below $0.1$ standard deviations. The study concludes that numerical errors are controllable with appropriate settings, and the main challenges for future CMB predictions will be astrophysical rather than numerical in nature.

Abstract

High precision measurements of the Cosmic Microwave Background (CMB) anisotropies, as can be expected from the Planck satellite, will require high-accuracy theoretical predictions as well. One possible source of theoretical uncertainty is the numerical error in the output of the Boltzmann codes used to calculate angular power spectra. In this work, we carry out an extensive study of the numerical accuracy of the public Boltzmann code CAMB, and identify a set of parameters which determine the error of its output. We show that at the current default settings, the cosmological parameters extracted from data of future experiments like Planck can be biased by several tenths of a standard deviation for the six parameters of the standard Lambda-CDM model, and potentially more seriously for extended models. We perform an optimisation procedure that leads the code to achieve sufficient precision while at the same time keeping the computation time within reasonable limits. Our conclusion is that the contribution of numerical errors to the theoretical uncertainty of model predictions is well under control -- the main challenges for more accurate calculations of CMB spectra will be of an astrophysical nature instead.

Figures (4)

• Figure 1: These diagrams illustrate the dependence of accuracy on the settings of individual parameters. We plot $\chi^2$ as a function of parameter value. The fiducial reference data sets were generated with all parameters set to a value of 2, except for the one scanned, which is set to an extremely high value (100 for dec_start, 5 for tc_largek, and 10 for all other parameters). When setting the accuracy parameters to their fiducial values, we obtain a $\chi^2$ of order $10^{-8}$ instead of 0. This effect stems from a numerical rounding error when outputting the fiducial data set, and is small enough not to be of relevance to any of our conclusions.
• Figure 2: These diagrams show the computation time $T$ (in seconds), varying the settings of all parameters individually while keeping all others fixed at a value of 2. The calculations were performed on a single core of a $2.40\rm{\ GHz}$ Intel T7700 CPU. The spikes for new_l_sample_boost, as well as the "bumps" at low settings of various other parameters are due to the fact that the Bessel functions were (re-)calculated at these points, adding a few seconds to the total time.
• Figure 3: Computation time versus $\chi^2$ for the $\sim 20000$ samples of our Monte Carlo run.
• Figure 4: Marginalised posterior probability densities of the cosmological parameters of the vanilla model. Thick black lines correspond to the exact results (using the same accuracy settings for the MCMC and for the fiducial data set), green lines are results obtained with the "Settings 2" column of Table \ref{['table:settings']} and the reference data set, and the red dotted lines represent the posteriors from the default accuracy settings of CosmoMC/CAMB and the reference data set.