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We measure the three-dimensional power spectrum (P3D) of the transmitted flux in the Lyman- |$\alpha$| (Ly  |$\alpha$|⁠) forest using the complete extended Baryon Oscillation Spectroscopic Survey data release 16 (eBOSS DR16). This sample consists of |$\sim$| 205 000 quasar spectra in the redshift range |$2\le z \le 4$| at an effective redshift |$z=2.334$|⁠. We propose a pair-count spectral estimator in configuration space, weighting each pair by |$\exp (i\mathbf {k}\cdot \mathbf {r})$|⁠ , for wave vector |$\mathbf {k}$| and pixel pair separation |$\mathbf {r}$|⁠ , effectively measuring the anisotropic power spectrum without the need for fast Fourier transforms. This accounts for the window matrix in a tractable way, avoiding artefacts found in Fourier-transform based power spectrum estimators due to the sparse sampling transverse to the line of sight of Ly  |$\alpha$| skewers. We extensively test our pipeline on two sets of mocks: (i) idealized Gaussian random fields with a sparse sampling of Ly  |$\alpha$| skewers, and (ii) log-normal LyaCoLoRe mocks including realistic noise levels, the eBOSS survey geometry and contaminants. On eBOSS DR16 data, the Kaiser formula with a non-linear correction term obtained from hydrodynamic simulations yields a good fit to the power spectrum data in the range |$(0.02 \le k \le 0.35)\, h\, {\rm Mpc}^{-1}\,$| at the 1–2σ level with a covariance matrix derived from LyaCoLoRe mocks. We demonstrate a promising new approach for full-shape cosmological analyses of Ly  |$\alpha$| forest data from cosmological surveys such as eBOSS, the currently observing Dark Energy Spectroscopic Instrument and future surveys such as the Prime Focus Spectrograph, WEAVE-QSO, and 4MOST. [ABSTRACT FROM AUTHOR]

We present constraints on primordial B modes from large-angular scale cosmic microwave background polarization anisotropies measured with the Planck satellite. To remove Galactic polarized foregrounds, we use a Bayesian parametric component separation method, modelling synchrotron radiation as a power law and thermal dust emission as a modified blackbody. This method propagates uncertainties from the foreground cleaning into the noise covariance matrices of the maps. We construct two likelihoods: (i) a semi-analytical cross-spectrum-based likelihood-approximation scheme (momento), and (ii) an exact polarization-only pixel-based likelihood (pixLike). Since momento is based on cross-spectra, it is statistically less powerful than pixLike , but is less sensitive to systematic errors correlated across frequencies. Both likelihoods give a tensor-to-scalar ratio, r , that is consistent with zero from low-multipole (2 ≤ ℓ < 30) Planck polarization data. From full-mission maps, we obtain r 0.05 < 0.274 at 95 per cent confidence at a pivot scale of k * ≡ 0.05 Mpc−1, using pixLike. momento gives a qualitatively similar but weaker 95 per cent confidence limit of r 0.05 < 0.408. [ABSTRACT FROM AUTHOR]

We present a Bayesian parametric component separation method for polarized microwave sky maps. We solve jointly for the primary cosmic microwave background (CMB) signal and the main Galactic polarized foreground components. For the latter, we consider electron-synchrotron radiation and thermal dust emission, modelled in frequency as a power law and a modified blackbody, respectively. We account for inter-pixel correlations in the noise covariance matrices of the input maps and introduce a spatial correlation length in the prior matrices for the spectral indices β. We apply our method to low-resolution polarized Planck 2018 Low and High Frequency Instrument (LFI/HFI) data, including the SRoll2 re-processing of HFI data. We find evidence for spatial variation of the synchrotron spectral index, and no evidence for depolarization of dust. Using the HFI SRoll2 maps, and applying wide priors on the spectral indices, we find a mean polarized synchrotron spectral index over the unmasked sky of |$\bar{\beta }_{\rm sync}=-2.83\pm 0.62$|⁠. For polarized thermal dust emission, we obtain |$\bar{\beta }_{\rm dust}=1.43\pm 0.24$|⁠. Using our recovered CMB maps and associated uncertainties, we constrain the optical depth to reionization, τ, using a cross-spectrum-based likelihood-approximation scheme (momento) to be τ = 0.0598 ± 0.0059. We confirm our findings using a pixel-based likelihood (pixLike). In both cases, we obtain a result that is consistent with that found by subtracting spatially uniform foreground templates. While the latter method is sufficient for current polarization data from Planck , next-generation space-borne CMB experiments will need more powerful schemes such as the one presented here. [ABSTRACT FROM AUTHOR]

This paper explores methods for constructing low multipole temperature and polarization likelihoods from maps of the cosmic microwave background anisotropies that have complex noise properties and partial sky coverage. We use Planck 2018 High Frequency Instrument (HFI) and updated SRoll2 temperature and polarization maps to test our methods. We present three likelihood approximations based on quadratic cross spectrum estimators: (i) a variant of the simulation-based likelihood (SimBaL) techniques used in the Planck legacy papers to produce a low multipole EE likelihood; (ii) a semi-analytical likelihood approximation (momento) based on the principle of maximum entropy; (iii) a density-estimation 'likelihood-free' scheme (delfi). Approaches (ii) and (iii) can be generalized to produce low multipole joint temperature-polarization (TTTEEE) likelihoods. We present extensive tests of these methods on simulations with realistic correlated noise. We then analyse the Planck data and confirm the robustness of our method and likelihoods on multiple inter- and intra-frequency detector set combinations of SRoll2 maps. The three likelihood techniques give consistent results and support a low value of the optical depth to reoinization, τ, from the HFI. Our best estimate of τ comes from combining the low multipole SRoll2 momento (TTTEEE) likelihood with the CamSpec high multipole likelihood and is |$\tau = 0.0627^{+0.0050}_{-0.0058}$|⁠. This is consistent with the SRoll2 team's determination of τ, though slightly higher by ∼0.5σ, mainly because of our joint treatment of temperature and polarization. [ABSTRACT FROM AUTHOR]