Efficient Cosmological Parameter Estimation from Microwave Background Anisotropies

TL;DR

This work tackles the problem of efficiently estimating cosmological parameters from high-precision CMB power spectra. It introduces a physically motivated parameter set that isolates independent physical effects, yielding near-linear power-spectrum dependence on the parameters and enabling Monte Carlo exploration with up to $10^5$ model evaluations per second. The authors demonstrate that this approach produces accurate MAP-like error regions and clean degeneracy structures, with results consistent with Fisher-matrix estimates in the physical parameter space, and they show substantial speedups over traditional grid or direct Boltzmann-code approaches. They also emphasize the importance of controlling systematics and numerical precision, arguing for improved codes and robust likelihood pipelines to fully exploit upcoming CMB data, while noting the method naturally extends to polarization and tensor components.

Abstract

We revisit the issue of cosmological parameter estimation in light of current and upcoming high-precision measurements of the cosmic microwave background power spectrum. Physical quantities which determine the power spectrum are reviewed, and their connection to familiar cosmological parameters is explicated. We present a set of physical parameters, analytic functions of the usual cosmological parameters, upon which the microwave background power spectrum depends linearly (or with some other simple dependence) over a wide range of parameter values. With such a set of parameters, microwave background power spectra can be estimated with high accuracy and negligible computational effort, vastly increasing the efficiency of cosmological parameter error determination. The techniques presented here allow calculation of microwave background power spectra $10^5$ times faster than comparably accurate direct codes (after precomputing a handful of power spectra). We discuss various issues of parameter estimation, including parameter degeneracies, numerical precision, mapping between physical and cosmological parameters, and systematic errors, and illustrate these considerations with an idealized model of the MAP experiment.

Figures (12)

• Figure 1: Temperature power spectra for the fiducial cosmological model (top panel, heavy line), plus models varying the parameter ${\cal A}$ upwards by 10% (curve shifts to lower $l$ values) and downwards by 10% (curve shifts to higher $l$ values) while keeping ${\cal B}$, ${\cal M}$, ${\cal V}$, ${\cal R}$, ${\cal S}$ and $n$ fixed. The bottom panel displays the fractional error between the fiducial model and the other two models with $l$-axes rescaled by the factor ${\cal A}/{\cal A_0}$.
• Figure 2: Temperature power spectra for the fiducial cosmological model (top panel, heavy line), plus models varying the parameter ${\cal B}$ upwards by 25% (higher first peak) and downwards by 25% (lower first peak) while keeping ${\cal A}$, ${\cal M}$, ${\cal V}$, ${\cal R}$, ${\cal S}$, and $n$ fixed. The bottom panel displays $C_l$ as a function of ${\cal B}$, for $l=50$, 100, 200, 500, and 1000. The horizontal axis shows the fractional change in $\cal B$, while the vertical axis gives the change in $C_l$ as a fraction of the cosmic variance at that multipole.
• Figure 3: Same as Fig. \ref{['Cl_B']}, except varying the parameter ${\cal R}$ while keeping the others fixed. Larger ${\cal R}$ increases the peak heights. The substantial glitch in the bottom panel is due to systematic errors in CMBFAST.
• Figure 4: Same as Fig. \ref{['Cl_B']}, except varying the parameter ${\cal M}$ while keeping the others fixed. Larger $\cal M$ increases the first peak height. Again, the glitch in the bottom panel is due to systematic errors in CMBFAST.
• Figure 5: Varying the the parameter ${\cal V}$, keeping the others fixed, for low $l$ (top) and higher $l$ (center) values; note the scales of the vertical axes. Only the lowest multipoles have any significant variation, arising from the Integrated Sachs-Wolfe effect at late times; the bottom panel shows the dependence of the $l=5$, 10, and 20 multipoles on ${\cal V}$. A linear approximation is rough but reasonable; higher-order approximations will model this dependence better. At higher $l$, the variation between the curves is a measure of the numerical accuracy of the code generating the $C_l$ curves.