Likelihood Analysis of CMB Temperature and Polarization Power Spectra

TL;DR

The paper tackles the problem of a reliable joint likelihood for the CMB temperature and polarization power spectra on partial sky, where cross-correlations (e.g., $C_l^{TE}$) and non-Gaussian estimator distributions bias simple options. It introduces a general, fast likelihood approximation for correlated Gaussian fields observed on part of the sky that is exact in the full-sky limit and precomputable from a fiducial covariance $\bm{M}_f$. The authors validate the method with full-sky and cut-sky tests, including anisotropic Planck-like noise, showing near-optimal parameter constraints and robustness to fiducial-model errors, outperforming standard Gaussian or per-$l$-based approaches in preserving the likelihood shape. This approach provides a practical route to robust cosmological parameter estimation from high-$l$ CMB data with realistic sky cuts and noise, without requiring expensive exact likelihood calculations.

Abstract

Microwave background temperature and polarization observations are a powerful way to constrain cosmological parameters if the likelihood function can be calculated accurately. The temperature and polarization fields are correlated, partial sky coverage correlates power spectrum estimators at different ell, and the likelihood function for a theory spectrum given a set of observed estimators is non-Gaussian. An accurate analysis must model all these properties. Most existing likelihood approximations are good enough for a temperature-only analysis, however they cannot reliably handle a temperature-polarization correlations. We give a new general approximation applicable for correlated Gaussian fields observed on part of the sky. The approximation models the non-Gaussian form exactly in the ideal full-sky limit and is fast to evaluate using a pre-computed covariance matrix and set of power spectrum estimators. We show with simulations that it is good enough to obtain correct results at ell >~ 30 where an exact calculation becomes impossible. We also show that some Gaussian approximations give reliable parameter constraints even though they do not capture the shape of the likelihood function at each ell accurately. Finally we test the approximations on simulations with realistically anisotropic noise and asymmetric foreground mask.

Figures (11)

• Figure 1: The plot compares various likelihood approximations on the full-sky for the case of a single field (temperature only) and no noise. The left-hand panel shows the difference between best-fit posterior amplitude of a $\Delta_l=10$ bin with the likelihood approximations and the exact likelihood over 10000 simulations where $A_{\text{Exact}}$ is the best fit amplitude of the exact likelihood and $A_{S}$, $A_{Q}$, $A_{D}$ and $A_{\text{WMAP}}$ are the best fit amplitude of the symmetric Gaussian, quadratic, Gaussian$_{D}$ and WMAP approximations respectively. The right-hand panel shows the root-mean-square difference. These two quantities are compared to the systematic error tolerance. Only the symmetric Gaussian and the quadratic approximations are clearly not good enough. The fiducial Gaussian and new likelihood approximations are not show as they are exactly unbiased in this simple test case with correct fiducial model.
• Figure 2: Single field likelihood approximation results for the likelihood as a function of bin amplitude, $A$. The plot compares the likelihood approximations to the exact likelihood for an azimuthal galactic cut with $f_{\text{sky}}=0.862$, $l_\text{max}=600$ and bin located at 200 $\leq$$l$$\leq 209$ for one realization.
• Figure 3: The likelihood as a function of bin amplitude, A, for the temperature and polarization fields in one realization. The black (solid) line is the exact likelihood, the red (dotted) line is the new likelihood and the blue (dashed) line is the fiducial Gaussian distribution. Unlike the fiducial Gaussian distribution which only agrees well around the peak, the new likelihood captures the shape of the exact one well. We used an azimuthal cut with $f_{\text{sky}}=0.862$, $l_\text{max}=500$ and bin at 150 $\leq$$l$$\leq 159$. Noise is isotropic and uncorrelated and the $E$ and $B$ modes noise is twice the $T$ noise.
• Figure 4: The difference between the average of the posterior amplitude and the true input model compared to the systematic error (red (solid) line). The green (long-dashed) and the black (dashed) lines represent the differences for the new likelihood and fiducial Gaussian, respectively. The curves clearly do not show any significant bias in the posterior amplitudes. The averages were taken over 5000 simulations (realizations) for $l_\text{max}=800$. Simulations were performed for spin-0 $T$ and $E$-mode only and for azimuthal cuts with $f_{\text{sky}}=0.862$ and a bin-size set to 10.
• Figure 5: Comparison between various binned and un-binned likelihood approximations. The left plot shows the average of the difference between the posterior amplitudes of these likelihoods and the right plot shows the variance, both compared to the systematic error (red (solid) line). The black (dashed), the green (long-dashed), the cyan (dashed long-dashed) and the blue (dotted dashed) lines represent the comparison between binned fiducial Gaussian and new likelihood, binned new likelihood and new likelihood, fiducial Gaussian and binned new likelihood and binned fiducial Gaussian and binned new likelihood, respectively. Averages were taken over 200 simulations (realizations) for $l_\text{max}=800$. Simulations were performed as previously mentioned. Results are all consistent to the required accuracy.