Planck 2013 results. XV. CMB power spectra and likelihood

TL;DR

Planck 2013 results XV delivers a comprehensive likelihood for the CMB temperature two-point statistics, deriving a power spectrum over $2 \le \ell \le 2500$ by combining a Gibbs-based low-$\ell$ treatment with high-$\ell$ cross-spectra analyses. The work introduces two independent high-$\ell$ likelihoods, CamSpec and Plik, and develops a physically motivated foreground model (dust, CIB, Poisson, SZ) with carefully apodized masks, validated by extensive simulations. The six-parameter $\Lambda$CDM model fits the Planck data remarkably well up to $\ell \lesssim 1500$, with Planck alone detecting $n_s < 1$ at $>4\sigma$ and Planck+WP increasing the significance to $\sim5.4\sigma$, while highlighting a persistent low-$\ell$ power deficit at $\ell \lesssim 40$ at the level of 5–10% (2.5–3$\sigma$). The analysis emphasizes internal consistency across frequencies and maps, and demonstrates robustness of cosmological inferences to foreground modelling and data selections, although some degeneracies persist in the high-$\ell$ regime without external data. Overall, Planck achieves a near-cosmic-variance-limited temperature power spectrum for $\ell \gtrsim 0.1^{\circ}$, with polarization results broadly concordant with the LCDM expectation and foreground modelling refined through cross-experiment comparisons.

Abstract

We present the Planck likelihood, a complete statistical description of the two-point correlation function of the CMB temperature fluctuations. We use this likelihood to derive the Planck CMB power spectrum over three decades in l, covering 2 <= l <= 2500. The main source of error at l <= 1500 is cosmic variance. Uncertainties in small-scale foreground modelling and instrumental noise dominate the error budget at higher l's. For l < 50, our likelihood exploits all Planck frequency channels from 30 to 353 GHz through a physically motivated Bayesian component separation technique. At l >= 50, we employ a correlated Gaussian likelihood approximation based on angular cross-spectra derived from the 100, 143 and 217 GHz channels. We validate our likelihood through an extensive suite of consistency tests, and assess the impact of residual foreground and instrumental uncertainties on cosmological parameters. We find good internal agreement among the high-l cross-spectra with residuals of a few uK^2 at l <= 1000. We compare our results with foreground-cleaned CMB maps, and with cross-spectra derived from the 70 GHz Planck map, and find broad agreement in terms of spectrum residuals and cosmological parameters. The best-fit LCDM cosmology is in excellent agreement with preliminary Planck polarisation spectra. The standard LCDM cosmology is well constrained by Planck by l <= 1500. For example, we report a 5.4 sigma deviation from n_s /= 1. Considering various extensions beyond the standard model, we find no indication of significant departures from the LCDM framework. Finally, we report a tension between the best-fit LCDM model and the low-l spectrum in the form of a power deficit of 5-10% at l <~ 40, significant at 2.5-3 sigma. We do not elaborate further on its cosmological implications, but note that this is our most puzzling finding in an otherwise remarkably consistent dataset. (Abridged)

Figures (10)

• Figure 1: The covariance matrix blocks used in the likelihood, accounting for the correlations between cross-spectra estimated from the 100, 143, and 217 GHz channels.
• Figure 19: Comparison of cosmological parameters estimated from the CamSpec (blue) and Plik (red for mask C48; purple for mask C58) likelihoods. All parameters agree to better than $0.2\sigma$ when C58 is used for Plik, matching the sky area used by CamSpec at 100 GHz.
• Figure 20: Comparison of foreground parameters estimated with the CamSpec and Plik likelihoods. The red (purple) lines show the CamSpec (Plik) distributions using only Planck data, and the green (blue) lines show the CamSpec (Plik) results when additionally including ACT and SPT.
• Figure 21: Correlation matrix between all the cosmological (top block), foreground (middle block), and derived (bottom block) parameters, estimated using the Plik likelihood.
• Figure 22: Distribution of the difference between parameter mean (as estimated by Plik assuming foreground spectra are known) and the input values in units of the posterior standard deviation. This demonstrates the absence of methodological bias at the level of the intrinsic dispersion between realizations.