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This paper presents a detailed description of the CamSpec likelihood which has been used to analyse Planck temperature and polarization maps of the cosmic microwave background since the first Planck data release. We have created a number of likelihoods using a range of Galactic sky masks and different methods of temperature foreground cleaning. Our most powerful likelihood uses 80 percent of the sky in temperature and polarization. Our results show that the six-parameter LCDM cosmology provides an excellent fit to the Planck data. There is no evidence for statistically significant internal tensions in the Planck TT, TE and EE spectra computed for different frequency combinations. We present evidence that the tendencies for the Planck temperature power spectra to favour a lensing amplitude A_L>1 and positive spatial curvature are caused by statistical fluctuations in the temperature power spectra. Using our statistically most powerful likelihood, we find that the A_L parameter differs from unity at no more than the 2.2 sigma level. We find no evidence for anomalous shifts in cosmological parameters with multipole range. In fact, we show that the combined TTTEEE likelihood over the restricted multipole range 2-800 gives cosmological parameters for the base LCDM cosmology that are very close to those derived from the full multipole range 2-2500. We present revised constraints on a few extensions of the base LCDM cosmology, focussing on the sum of neutrino masses, number of relativistic species and the tensor-scalar ratio. The results presented here show that the Planck data are remarkably consistent between detector-sets, frequencies and sky area. We find no evidence in our analysis that cosmological parameters determined from the CamSpec likelihood are affected to any significant degree by systematic errors in the Planck data (abridged).

We revisit the observational constraints on spatial curvature following recent claims that the Planck data favour a closed Universe. We use a new and statistically powerful Planck likelihood to show that the Planck temperature and polarization spectra are consistent with a spatially flat Universe, though because of a geometrical degeneracy cosmic microwave background spectra on their own do not lead to tight constraints on the curvature density parameter Omega_K. When combined with other astrophysical data, particularly geometrical measurements of baryon acoustic oscillations, the Universe is constrained to be spatially flat to extremely high precision, with Omega_ K = 0.0004 +/-0.0018 in agreement with the 2018 results of the Planck team. In the context of inflationary cosmology, the observations offer strong support for models of inflation with a large number of e-foldings and disfavour models of incomplete inflation.

We provide an analytical argument for understanding the likely nature of parameter shifts between those coming from an analysis of a dataset and from a subset of that dataset, assuming differences are down to noise and any intrinsic variance alone. This gives us a measure against which we can interpret changes seen in parameters and make judgements about the coherency of the data and the suitability of a model in describing those data.

We present angular power spectra and cosmological parameter constraints derived from the Planck PR4 (NPIPE) maps of the Cosmic Microwave Background. NPIPE, released by the Planck Collaboration in 2020, is a new processing pipeline for producing calibrated frequency maps from Planck data. We have created new versions of the CamSpec likelihood using these maps and applied them to constrain LCDM and single-parameter extensions. We find excellent consistency between NPIPE and the Planck 2018 maps at the parameter level, showing that the Planck cosmology is robust to substantial changes in the mapmaking. The lower noise of NPIPE leads to ~10% tighter constraints, and we see both smaller error bars and a shift toward the LCDM values for beyond-LCDM parameters including Omega_K and A_Lens.

In this paper I show how path integral techniques can be used to put measures on histories in \"susceptible-infectious-recovered\" (SIR)-type systems. The standard SIR solution emerges as the classical saddle point of the action describing the measure. One can then expand perturbatively around the background solution, and this paper goes on to work out the covariance of fluctuations around the background solution. Using a Green's function type approach, one simply needs to solve additional ordinary differential equations; an explicit matrix inversion is not required. The computed covariance matrix should be useful in the construction of fast likelihoods for fitting the parameters of SIR-type models to data. A comparison of the predictions of the approach to an ensemble of simulations is presented.

We present constraints on primordial B modes from large angular scale cosmic microwave background polarisation anisotropies measured with the Planck satellite. To remove Galactic polarised 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 polarisation-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 <= ell < 30) Planck polarisation 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.

We present a Bayesian parametric component separation method for polarised microwave sky maps. We solve jointly for the primary cosmic microwave background (CMB) signal and the main Galactic polarised 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 beta. We apply our method to low-resolution polarised 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 depolarisation of dust. Using the HFI SRoll2 maps, and applying wide priors on the spectral indices, we find a mean polarised synchrotron spectral index over the unmasked sky of beta-sync = -2.833 +- 0.620. For polarised dust emission, we obtain beta-dust = 1.429 +- 0.236. Our method returns correlated uncertainties for all components of the sky model. Using our recovered CMB maps and associated uncertainties, we constrain the optical depth to reionization, tau, using a cross-spectrum-based likelihood-approximation scheme (momento) to be tau = 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, albeit a fraction of a sigma higher than, that found by subtracting spatially uniform foreground templates. While the latter method is sufficient for current polarisation data from Planck, next-generation space-borne CMB experiments will need more powerful schemes such as the one presented here.

We investigate the applicability of the formalism of quantum mechanics to everyday life. It seems to be directly relevant for situations in which the very act of coming to a conclusion or decision on one issue affects one's confidence about conclusions or decisions on another issue. Lie algebra theory is argued to be a very useful tool in guiding the construction of quantum descriptions of such situations. Tests, extensions and speculative applications and implications, including for the encoding of thoughts in neural networks, are discussed. It is suggested that the recognition and incorporation of such mathematical structure into machine learning and artificial intelligence might lead to significant efficiency and generality gains in addition to ensuring probabilistic reasoning at a fundamental level.

Motivated by observed discrepancies between ACT DR4 and Planck 2018 cosmic microwave background (CMB) anisotropy power spectra, particularly in the cross-correlation of temperature and E-mode polarization, we investigate challenges that may be encountered in the comparison of satellite and ground-based CMB data. In particular, we focus on the effects of Fourier-space filtering and masking involving bright point sources. We show that the filtering operation generates bright cross-shaped artifacts in the map, which stretch far outside typical point-source masks. If not corrected, these artifacts can add bias or additional variance to cross-spectra, skewing results. However we find that the effect of this systematic is not large enough to explain the ACT-Planck differences presented with ACT DR4.

We present a technique for constructing suitable posterior probability distributions in situations for which the sampling distribution of the data is not known. This is very useful for modern scientific data analysis in the era of \"big data\", for which exact likelihoods are commonly either unknown, computationally prohibitively expensive or inapplicable because of systematic effects in the data. The scheme involves implicitly computing the changes in an approximate sampling distribution as model parameters are changed via explicitly-computed moments of statistics constructed from the data.