Radical Compression of Cosmic Microwave Background Data

TL;DR

The paper tackles biases arising from Gaussian likelihood assumptions when compressing CMB data by introducing radical compression via bandpowers and non-Gaussian likelihoods. It develops two practical approximations—the offset lognormal form and the equal-variance form—for the bandpower likelihood, each incorporating a noise-offset parameter $x$ to capture non-Gaussian tails. The methods are tested on COBE/DMR, Saskatoon, OVRO, SP, and SuZIE datasets, yielding robust maximum-likelihood power spectra and cosmological constraints that closely match full likelihoods and reduce biases. This approach enables simple, scalable, and joint analyses of diverse CMB datasets, with a clear pathway for reporting results that preserve non-Gaussian information and facilitate parameter estimation.

Abstract

Powerful constraints on theories can already be inferred from existing CMB anisotropy data. But performing an exact analysis of available data is a complicated task and may become prohibitively so for upcoming experiments with \gtrsim10^4 pixels. We present a method for approximating the likelihood that takes power spectrum constraints, e.g., ``band-powers'', as inputs. We identify a bias which results if one approximates the probability distribution of the band-power errors as Gaussian---as is the usual practice. This bias can be eliminated by using specific approximations to the non-Gaussian form for the distribution specified by three parameters (the maximum likelihood or mode, curvature or variance, and a third quantity). We advocate the calculation of this third quantity by experimenters, to be presented along with the maximum-likelihood band-power and variance. We use this non-Gaussian form to estimate the power spectrum of the CMB in eleven bands from multipole moment ell = 2 (the quadrupole) to ell=3000 from all published band-power data. We investigate the robustness of our power spectrum estimate to changes in these approximations as well as to selective editing of the data.

Figures (11)

• Figure 1: DMR Likelihoods $P(\Delta|{\cal C}_\ell$) for various values of $\ell$, as marked. The horizontal axis is ${\cal C}_\ell=\ell(\ell+1)C_\ell/(2\pi)$. The upper right panel gives the cumulative probability. The solid (black) line is the full likelihood calculated exactly. The dashed (red) line is the Gaussian approximation about the peak.
• Figure 2: Full and approximate COBE/DMR likelihoods $P(\Delta|{\cal C}_\ell)$ for various values of $\ell$, as marked. The horizontal axis is ${\cal C}_\ell=\ell(\ell+1)C_\ell/(2\pi)$. The upper right panel gives the cumulative probability. The solid (black) line is the full likelihood calculated exactly. The short-dashed (red) line is the Gaussian approximation about the peak. The dotted (cyan) line is a Gaussian in $\ln{{\cal C}_\ell}$; the dashed (magenta) line is a Gaussian in $\ln{({\cal C}_\ell+x_\ell)}$, as discussed in the text. The dot-dashed (green) line is the equal-variance approximation.
• Figure 3: Exact and approximate likelihood contours for COBE/DMR, for the cosmological parameters $n_s$ and $\sigma_8$ (with otherwise standard CDM values). Contours are for ratios of the likelihood to its maximum equal to $\exp{-\nu^2/2}$ with $\nu=1,2,3$. Upper panel is for the full likelihood (dashed) and its offset lognormal approximation as a Gaussian in $\ln{({\cal C}_\ell+x_\ell)}$ (solid; see text); lower panel shows the full likelihood and its approximation as a Gaussian in ${\cal C}_\ell$.
• Figure 4: Full and approximate Saskatoon likelihoods. As in Fig. \ref{['fig:dmrlike']}. The solid (black) line is the full likelihood calculated exactly. The short-dashed (red) line is the Gaussian approximation about the peak. The dotted (cyan) line is a Gaussian in $\ln{{\cal C}_\ell}$; the dashed (magenta) line is a Gaussian in $\ln{({\cal C}_\ell+x_\ell)}$, as discussed in the text. The dot-dashed (green) line is the equal-variance approximation.
• Figure 5: Likelihood contours for the Saskatoon experiment alone, as in Figure \ref{['fig:dmrlikecontours']}, but using the "orthogonalized bands" of Sec. \ref{['sec:ortho']}.