Probing cosmological parameters with the CMB: Forecasts from full Monte Carlo simulations

TL;DR

This study demonstrates that Fisher-matrix forecasts can substantially misestimate parameter errors for Planck-like CMB data, especially in multi-parameter cosmologies where likelihoods are non-Gaussian due to degeneracies. By contrast, full Monte Carlo analyses with CosmoMC on synthetic data reveal the true posterior shapes, showing significant differences for parameters such as the neutrino mass fraction $f_{\nu}$ and $\Omega_{dm} h^2$ in minimal models, and persistently notable discrepancies in extended models. Incorporating CMB lensing information mitigates some non-Gaussian effects and improves agreement, though residual differences remain in the eleven-parameter case, particularly for $N_{\rm eff}$ and $m_ν$. The authors advocate using Monte Carlo-based forecasts for robust error predictions and argue that this approach is readily extendable to complementary probes beyond Planck.

Abstract

The Fisher matrix formalism has in recent times become the standard method for predicting the precision with which various cosmological parameters can be extracted from future data. This approach is fast, and generally returns accurate estimates for the parameter errors when the individual parameter likelihoods approximate a Gaussian distribution. However, where Gaussianity is not respected (due, for instance, to strong parameter degeneracies), the Fisher matrix formalism loses its reliability. In this paper, we compare the results of the Fisher matrix approach with those from Monte Carlo simulations. The latter method is based on the publicly available CosmoMC code, but uses synthetic realisations of data sets anticipated for future experiments. We focus on prospective cosmic microwave background (CMB) data from the Planck satellite, with or without CMB lensing information, and its implications for a minimal cosmological scenario with eight parameters and an extended model with eleven parameters. We show that in many cases, the projected sensitivities from the Fisher matrix and the Monte Carlo methods differ significantly, particularly in models with many parameters. Sensitivities to the neutrino mass and the dark matter fraction are especially susceptible to change.

Figures (6)

• Figure 1: Projected 1$\sigma$ errors for $\{\Omega_bh^2, \tau, Y_{He}, n_s, A_s, \theta \}$ in the eight-parameter model of section \ref{['sec:minimal']}. The first five points and error bars (red) in each plot are the Monte Carlo estimates from independent mock datasets. The sixth one (dark blue) is obtained by replacing the mock data spectra by the fiducial spectra. The last error bar (light blue) corresponds to the estimate from the Fisher matrix method, centred on the fiducial value of the parameter of interest (horizontal lines).
• Figure 2: Same as Figure \ref{['fig2']} but for $\{f_{\nu}, \Omega_{dm} h^2\}$. In addition to the mean values (diamonds), we show also the best-fit values (green crosses) obtained from the Markov Chains.
• Figure 3: Projected $68\%$ and $95\%$ confidence levels from the Monte Carlo (colored/shaded) and the Fisher matrix (black lines) methods, for Planck without lensing extraction and the minimal, eight-parameter $\Lambda$MDM model of section \ref{['sec:minimal']}. The diagonal plots show the corresponding marginalised probabilities for each cosmological parameter.
• Figure 4: Projected $68\%$ and $95\%$ confidence levels from the Monte Carlo (colored/shaded) and the Fisher matrix (black lines) methods, for Planck with lensing extraction and the minimal, eight-parameter $\Lambda$MDM model of section \ref{['sec:minimal']}. The Monte Carlo results of the Figure \ref{['fig3']} (for Planck without lensing extraction) are shown again for comparison (blue dashed lines). The diagonal plots show the corresponding the marginalised probabilities for each cosmological parameter, with (red) lensing extraction (red) and without (blue dashed).
• Figure 5: Projected $68\%$ and $95\%$ confidence levels from the Monte Carlo (colored/shaded) and the Fisher matrix (black lines) methods, for Planck without lensing extraction and the extended, eleven-parameter $\Lambda$MDM model of section \ref{['sec:non-minimal']}. The diagonal plots show the corresponding marginalised probabilities for each cosmological parameter.