Cosmic Confusion: Degeneracies among Cosmological Parameters Derived from Measurements of Microwave Background Anisotropies

TL;DR

The paper analyzes degeneracies among cosmological parameters inferred from CMB anisotropies in a CDM framework with adiabatic fluctuations, using a Fisher-matrix approach and principal component analysis to reveal that a small number of components dominate parameter variance and that a nearly exact geometrical degeneracy between curvature and the cosmological constant cannot be broken by linear CMB data alone. It demonstrates that the Fisher-matrix can overestimate parameter precision and that correlated errors in the CMB power spectrum can bias inferences, while external priors from $H_0$, the age of the Universe, large-scale structure, and Type Ia supernovae, along with tensor constraints from polarization, can substantially improve constraints. The work also shows how Doppler-peak positions and heights encode degeneracies, and it emphasizes the need for joint analyses and careful treatment of systematics—especially for Planck—to robustly constrain cosmological parameters. Overall, the results highlight both the power of next-generation CMB data and the necessity of complementary observations to break fundamental degeneracies and obtain reliable parameter estimates.

Abstract

In the near future, observations of the cosmic microwave background (CMB) anisotropies will provide accurate determinations of many fundamental cosmological parameters. In this paper, we analyse degeneracies among cosmological parameters to illustrate some of the limitations inherent in CMB parameter estimation. For simplicity, throughout our analysis we assume a cold dark matter universe with power-law adiabatic scalar and tensor fluctuation spectra. We show that most of the variance in cosmological parameter estimates is contributed by a small number (two or three) principal components. An exact likelihood analysis shows that the usual Fisher matrix approach can significantly overestimate the errors on cosmological parameters. We show that small correlated errors in estimates of the CMB power spectrum at levels well below the cosmic variance limits, (caused, for example, by Galactic foregrounds or scanning errors) can lead to significant biases in cosmological parameters. Estimates of cosmological parameters can be improved very significantly by applying theoretical restrictions to the tensor component and external constraints derived from more conventional astronomical observations such as measurements of he Hubble constant, Type 1a supernovae distances and observations of galaxy clustering and peculiar velocities.

Figures (17)

• Figure 1: Figure \ref{['figure1']}a shows degeneracy lines of constant ${\cal R}$ in the $\Omega_\Lambda$-$\Omega_k$ plane. The value of ${\cal R}$ is given next to each line. In computing this figure, the matter density parameter is fixed by the constraint $\Omega_m = 1 - \Omega_k - \Omega_\Lambda$. Figure \ref{['figure1']}b shows lines of constant ${\cal R}$ in the $\omega_\Lambda$-$\omega_k$ plane for a universe with $\omega_m = 0.250$. The five dots in each figure show the locations of the five models with nearly degenerate $C_\ell$ spectra plotted in Figure \ref{['figure2']}. The target model with the parameters given in Section \ref{['sec:3.1']} is located at the origin in each panel.
• Figure 2: The upper panel shows the scalar power spectrum for the nearly degenerate models shown by the dots in Figures \ref{['figure1']}. The target model has $\omega_\Lambda = \omega_k = 0$, $\omega_b = 0.0125$ and $\omega_c = 0.2375$ and $n_s = 1$. The lower panel shows the residuals of these open universe models with respect to the spatially flat target model. The numbers next to each curve give the value of $\omega_\Lambda$ for each model. The two lines with long dashes show the standard deviation of the residuals from cosmic variance alone.
• Figure 3: Likelihood ratio contours in the $\omega_\Lambda$-$\omega_k$ plane for models containing only scalar modes. The models have fixed values of $\omega_b=0.0125$ and $\omega_c=0.2375$. The likelihood ratio for each model has been computed assuming the experimental parameters of OMAP (Table \ref{['tab1']}) and our standard spatially flat target model. The contours show where $-2{\rm ln}({\cal L}/{\cal L}_{max})$ has values of $2.3$, $6.2$ and $11.8$ corresponding approximately to $1\sigma$, $2 \sigma$ and $3\sigma$ contours assuming all other parameters are known. The full line shows the degeneracy curve with the same value of ${\cal R}$ as the target model ( cf Figure \ref{['figure1']}b) and the dashed line shows the Taylor series approximation to this curve, $\omega_\Lambda = 7 \omega_k$ (see Section \ref{['sec:3.3']}).
• Figure 4: Results of Monte-Carlo simulations of parameter estimation as described in the text assuming the parameters of OMAP as given in Table \ref{['tab1']}. The panels show correlations between various pairs of parameters. The lines in the $\omega_\Lambda$-$\omega_k$ and $\Omega_\Lambda$-$\Omega_k$ plane show the degeneracy lines given by equation (\ref{['eq:10b']}). The line in the $h$-$\Omega_\Lambda$ plane shows the degeneracy line given by equation (\ref{['eq:12']}).
• Figure 5: Derivatives of $C_\ell$ with respect to the seven parameters of the spatially flat target model defined in the text. The derivatives with respect to $\omega_b$, $n_t$ and $r$ have been multiplied by the factor indicated in the figure ( e.g. the derivative $\partial C_\ell/\partial \omega_b$ has been divided by a factor of ten).