CMB power spectrum parameter degeneracies in the era of precision cosmology

TL;DR

This study probes how precise CMB power spectra constrain cosmological parameters in the presence of geometrical degeneracies, by systematically separating numerical artefacts from true physical breaking effects. Using CAMB with high-accuracy settings, the authors map degeneracies in non-flat and flat ΛCDM-like models, including massive neutrinos and general dark energy, and forecast Planck-like constraints via χ_eff^2 and MCMC approaches. They show that lensing and ISW effects are the primary physical channels breaking degeneracies, while numerical artefacts can be controlled with boosted accuracy and improved interpolation. A key methodological advance is a January 2012 interpolation scheme that dramatically reduces interpolation-induced biases, enabling reliable, fast parameter inference at Planck precision. Overall, the paper demonstrates that with current numerical tools and physical modelling, CMB data alone can constrain complex cosmological scenarios, though degeneracies will persist and require high-precision data and analysis to fully resolve.

Abstract

Cosmological parameter constraints from the CMB power spectra alone suffer several well-known degeneracies. These degeneracies can be broken by numerical artefacts and also a variety of physical effects that become quantitatively important with high-accuracy data e.g. from the Planck satellite. We study degeneracies in models with flat and non-flat spatial sections, non-trivial dark energy and massive neutrinos, and investigate the importance of various physical degeneracy-breaking effects. We test the CAMB power spectrum code for numerical accuracy, and demonstrate that the numerical calculations are accurate enough for degeneracies to be broken mainly by true physical effects (the integrated Sachs-Wolfe effect, CMB lensing and geometrical and other effects through recombination) rather than numerical artefacts. We quantify the impact of CMB lensing on the power spectra, which inevitably provides degeneracy-breaking information even without using information in the non-Gaussianity. Finally we check the numerical accuracy of sample-based parameter constraints using CAMB and CosmoMC. In an appendix we document recent changes to CAMB's numerical treatment of massive neutrino perturbations, which are tested along with other recent improvements by our degeneracy exploration results.

Figures (28)

• Figure 1: CMB power spectrum obtained using camb for nearly degenerate geometries in non-flat $\Lambda$CDM models with no lensing (left) and the fractional differences from the fiducial-model spectrum (right). Both $\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ were fixed to their fiducial values in all cases to preserve the pre-recombination physics. Low accuracy and values of 1 for lSampleBoost, AccuracyBoost and lAccuracyBoost were used for the calculations.
• Figure 2: Difference between the unlensed power spectra of a range of non-flat degenerate models and the fiducial model. Models are computed at low accuracy with default accuracy parameters (left), high accuracy with default accuracy parameters (middle) and high accuracy with parameters boosted to 2 (right). The right-hand figure shows the physical differences in the spectra, with very little residual numerical error.
• Figure 3: Numerical errors in the computation of a range of non-flat degenerate models. For a given model, the numerical errors are estimated by subtracting the spectrum from one calculated at high accuracy with accuracy parameters boosted to 2. Errors are plotted for a low (left) and high (right) accuracy calculation. In all cases, the other accuracy parameters are at their default values.
• Figure 4: Fractional differences between nearly degenerate geometries and the fiducial model, comparing boosts of 2 in all the accuracy parameters (solid red) to boosts of 2 in only lSampleBoost (dashed blue). In all cases, high accuracy is used for the calculations.
• Figure 5: Minimum $\chi_{\text{eff}}^{2}$ for a range of degenerate geometries close to the fiducial model. Solid (black) is high accuracy and each of lSampleBoost, lAccuracyBoost and AccuracyBoost boosted to 2; dashed (green) is high accuracy with default accuracy parameter values; dot-dashed (blue) is high accuracy with only lSampleBoost boosted to 2; and triple-dot-dashed (red) is low accuracy with no accuracy boosts. In all cases we use only $l \ge 100$ in the calculation of $\chi_{\text{eff}}^{2}$.