Planck 2013 results. XV. CMB power spectra and likelihood (1303.5075v2)

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)

• The paper establishes a robust likelihood framework for analyzing CMB power spectra, leveraging Bayesian methods to manage cosmic variance and foreground uncertainties.
• The analysis confirms the ΛCDM model up to multipole ℓ≈1500 while revealing a significant 5–10% power deficit at low multipoles.
• Extensive cross-verification with Planck’s full frequency dataset and rigorous consistency tests enhance the reliability of the derived cosmological parameters.

Analysis of Planck 2013 Results on CMB Power Spectra and Likelihood

This paper presents a comprehensive statistical analysis of the Cosmic Microwave Background (CMB) power spectra using data collected by the Planck satellite. The analysis hinges on a likelihood function, specifically constructed to capture the two-point correlation function of CMB temperature fluctuations. The objective is to assess the CMB angular power spectrum based on Planck’s observations across a multipole moment range from 2 to 2500.

Key Methodologies and Framework

1. Cosmic Variance and Error Management: The paper identifies cosmic variance as the primary source of error at multipole moments $\ell \leq 1500$. Beyond this, uncertainties due to foreground modeling and instrumental noise become predominant. The authors utilize a Bayesian component separation technique to extract cosmological signals at low $\ell$ from foreground emissions using frequency channels from 30 to 353 GHz.
2. Likelihood Approach: The analysis employs a correlated Gaussian likelihood approximation for $\ell \geq 50$, leveraging angular cross-spectra from 100, 143, and 217 GHz channels. Foreground templates are marginalized, ensuring a robust estimation of cosmological parameters.
3. Validation and Consistency: Extensive validation through consistency tests ensures the reliability of the likelihood. Internal agreement is noted within high-$\ell$ cross-spectra, with residuals remaining below a few $\mu\textrm{K}^2$.
4. Comparison and Cross-verification: The power spectra derived using this method were cross-verified against the foreground-cleaned CMB maps from Planck's full frequency dataset, including cross-spectra from the 70 GHz Planck map, showcasing broad agreement in residuals and cosmological metrics.

Results and Implications

• The analysis corroborates the $\Lambda$CDM model, showing a strong fit between Planck’s data up to $\ell \lesssim 1500$ and the predicted power spectra. The spectral index of scalar perturbations demonstrates a significant deviation from scale invariance ($n_s \neq 1$) with a deviation quantified at $5.4\,\sigma$, thereby reinforcing the non-scale-invariant nature of the primordial power spectrum.
• Extending the multipole range beyond 1500 did not notably enhance the precision of $\Lambda$CDM parameters but instead opened pathways to probe potential model extensions. The paper, however, finds no substantial evidence for deviations beyond the standard $\Lambda$CDM model.
• A tension arises between the best-fit $\Lambda$CDM model and the low-$\ell$ spectrum, characterized by a power deficit of 5–10% at $\ell \lesssim 40$, with a statistical significance of 2.5–3$\sigma$. This anomaly remains unexplained within the framework presented, marking it as an outlier in an otherwise consistent dataset.

Theoretical and Practical Contributions

The findings contribute significantly to the theoretical understanding of CMB fluctuations and the large-scale structure of the universe, encapsulated within the $\Lambda$CDM paradigm. Practically, the refined statistical methodologies for CMB analysis enhance the precision of cosmological parameter estimation, laying groundwork for future exploration of cosmic phenomena. The reported low-$\ell$ anomalies could potentially spur further investigation into new physics or calibration techniques in observational cosmology.

The work sets a benchmark for subsequent analyses within cosmological studies, given its comprehensive treatment of uncertainties and consistent results across multiple validation pipelines. Continued analyses could focus on exploring the unresolved power deficit and testing alternative hypotheses that might account for such deviations in the low-$\ell$ spectral domain.