[Paper Review] The Cosmic Linear Anisotropy Solving System (CLASS) III: Comparision with CAMB for LambdaCDM

Julien Lesgourgues|arXiv (Cornell University)|Apr 14, 2011

Cosmology and Gravitation Theories6 references58 citations

TL;DR

This paper compares the CLASS and CAMB Boltzmann codes for minimal ΛCDM models, demonstrating sub-0.01% agreement in lensed CMB and matter power spectra when using identical recombination histories. It establishes that CLASS computes these spectra approximately 2.5 times faster than CAMB at the same precision level, confirming both codes' theoretical accuracy and positioning CLASS as a faster, reliable alternative for cosmological parameter estimation.

ABSTRACT

By confronting the two independent Boltzmann codes CLASS and CAMB, we establish that for concordance cosmology and for a given recombination history, lensed CMB and matter power spectra can be computed by current codes with an accuracy of 0.01%. We list a few tiny changes in CAMB which are necessary in order to reach such a level. Using the common limit of the two codes as a set of reference spectra, we derive precision settings corresponding to fixed levels of error in the computation of a CMB likelihood. We find that for a given precision level, CLASS is about 2.5 times faster than CAMB for computing the lensed CMB spectra of a LambdaCDM model. The nature of the main improvements in CLASS (which may each contribute to these performances) is discussed in companion papers.

Motivation & Objective

• To assess the theoretical accuracy of two independent Boltzmann codes, CLASS and CAMB, for minimal ΛCDM cosmology.
• To identify and eliminate systematic discrepancies between the codes by comparing their outputs at high precision.
• To calibrate precision settings in both codes that yield theoretical errors smaller than observational uncertainties.
• To evaluate computational performance, focusing on speed differences at fixed accuracy levels.
• To establish reference spectra from the common agreement of both codes as a benchmark for future cosmological likelihood analyses.

Proposed method

• Comparing lensed CMB and matter power spectra from CLASS and CAMB using identical cosmological parameters and recombination history (via RECFAST v1.5).
• Using the mutual agreement between CLASS and CAMB as a reference to define precision settings with guaranteed theoretical error levels.
• Measuring code performance by computing Δχ² between output spectra and the reference spectra, with execution time as a performance metric.
• Employing high-precision settings in both codes and measuring execution time on a single CPU to compare raw speed.
• Using the same compiler flags (-O4) and hardware to ensure fair comparison, minimizing implementation bias.
• Focusing on minimal ΛCDM with adiabatic initial conditions and scalar modes, excluding tensors, curvature, or massive neutrinos for this initial comparison.

Experimental results

Research questions

• RQ1 To what extent do CLASS and CAMB agree in computing lensed CMB and matter power spectra for ΛCDM models?
• RQ2 What level of precision can be achieved in both codes such that theoretical errors are smaller than observational errors in Planck and post-Planck data?
• RQ3 How do the computational speeds of CLASS and CAMB compare at fixed precision levels for ΛCDM models?
• RQ4 What are the key numerical and algorithmic differences that contribute to CLASS’s performance advantage over CAMB?
• RQ5 Can the mutual agreement between two independent codes be used as a reliable reference for cosmological likelihood computations?

Key findings

• CLASS and CAMB agree on lensed CMB and matter power spectra to within 0.01% for concordance ΛCDM, confirming their theoretical accuracy at a previously untested level.
• The mutual discrepancy between CLASS and CAMB is bounded by Δχ² = 0.027, which sets the intrinsic limit of theoretical error for current Boltzmann codes.
• CLASS computes lensed CMB spectra approximately 2.5 times faster than CAMB at the same precision level, with no significant performance penalty for matter power spectrum computation.
• The performance advantage of CLASS is attributed to its advanced numerical methods, including a stiff integrator, optimized loops, and approximation schemes for matter and dark energy domination.
• The comparison confirms that theoretical errors from Boltzmann codes are now fully under control and negligible compared to other systematics like recombination history or foreground modeling.
• Well-calibrated precision settings were derived for both codes, ensuring theoretical errors remain below 0.01% for likelihood analyses.