First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Parameter Estimation Methodology

TL;DR

This paper presents a rigorous, likelihood-based framework for estimating cosmological parameters from WMAP CMB data, emphasizing an accurate, Monte Carlo–calibrated treatment of the angular power spectrum likelihood and its covariance. It adopts Markov Chain Monte Carlo to efficiently explore multi-parameter space and to quantify uncertainties, while carefully incorporating external datasets (CBI, ACBAR, 2dFGRS, Ly-$\alpha$) to break degeneracies. The methodology includes reparameterizations to reduce degeneracies, a quartic fit to the likelihood surface for efficient step sizing, and a detailed model of large-scale structure effects (galaxy bias, redshift distortions, non-linear growth). The combined CMB+LSS analysis yields tighter cosmological constraints and demonstrates how robust uncertainty propagation across datasets can enhance parameter inferences in modern cosmology.

Abstract

We describe our methodology for comparing the WMAP measurements of the cosmic microwave background (CMB) and other complementary data sets to theoretical models. The unprecedented quality of the WMAP data, and the tight constraints on cosmological parameters that are derived, require a rigorous analysis so that the approximations made in the modeling do not lead to significant biases. We describe our use of the likelihood function to characterize the statistical properties of the microwave background sky. We outline the use of the Monte Carlo Markov Chains to explore the likelihood of the data given a model to determine the best fit cosmological parameters and their uncertainties. We add to the WMAP data the l>~700 CBI and ACBAR measurements of the CMB, the galaxy power spectrum at z~0 obtained from the 2dF galaxy redshift survey (2dFGRS), and the matter power spectrum at z~3 as measured with the Ly alpha forest. These last two data sets complement the CMB measurements by probing the matter power spectrum of the nearby universe. Combining CMB and 2dFGRS requires that we include in our analysis a model for galaxy bias, redshift distortions, and the non-linear growth of structure. We show how the statistical and systematic uncertainties in the model and the data are propagated through the full analysis.

Figures (8)

• Figure 1: Ratio of the effective sky coverage to the actual sky coverage. This correction factor calibrates the expression for the Fisher matrix to the value obtained from the Monte Carlo approach. Here we show the ratio obtained from 100,000 simulations (jagged line), the smooth curve shows the fit we use, equation (\ref{['eq:fitfskyeff']}). Note that, since we are switching between weighting schemes, the correction factors are not expected to smoothly interpolate between regimes.
• Figure 2: Correction factor for the noise. The lines are as in Figure \ref{['fig.calibint']}.Note that, since we are switching between weighting schemes, the correction factors are not expected to smoothly interpolate between regime
• Figure 3: Unconverged Markov chains. The left panel shown a trace plot of the likelihood values versus iteration number for one MCMC (these are the first 3000 steps from one of our $\Lambda$CDM model runs). Note the burn-in for the first $\sim 100$ steps. In the right panel, red dots are points of the chain in the ($n$, $A$) plane after discarding the burn-in. Green dots are from another MCMC for the same data-set and the same model. It is clear that, although the trace plot may appear to indicate that the chain has converged, it has not fully explored the likelihood surface. Using either of these two chains at this stage will give incorrect results for the best fit cosmological parameters and their errors.
• Figure 4: The CMB angular power spectrum (in $\mu$K$^2$) for our best fit $\Lambda$CDM model for $\ell >800$ and the Sunayev-Zel'dovich contribution for $\sigma_8=0.98$ for CBI wavelengths (dotted) and for ACBAR (dashed). The vertical line shows the adopted cutoff for CBI and ACBAR.
• Figure 5: The combined CMB and LSS data set. The CMB angular power spectrum in $\mu$K$^2$ (top panel) as a function of $k$ where $k$ is related to $\ell$ by $\ell=\eta_0 k$ (where $\eta_0 \sim 14400$ Mpc is the distance to the last scattering surface). Black points are the WMAP data, red points CBI, blue points ACBAR. The LSS data (bottom panel). Black points are the 2dFGRS measurements and green points are the Lyman $\alpha$ measurements. Both LSS power spectra are in units of (Mpc $h^{-1}$)$^3$ and have been rescaled to $z=0$. This plot only illustrates the scale coverage of all the data sets we consider. The various LSS power spectra as plotted here cannot be directly compared with the theory because of the effects outlined in § 5 (e.g., redshift-space distortions, non-linearities, bias and window function effect etc.).