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We describe the computational infrastructure for end-to-end Bayesian cosmic microwave background (CMB) analysis implemented by the BeyondPlanck Collaboration. The code is called Commander3. It provides a statistically consistent framework for global analysis of CMB and microwave observations and may be useful for a wide range of legacy, current, and future experiments. The paper has three main goals. Firstly, we provide a high-level overview of the existing code base, aiming to guide readers who wish to extend and adapt the code according to their own needs or re-implement it from scratch in a different programming language. Secondly, we discuss some critical computational challenges that arise within any global CMB analysis framework, for instance in-memory compression of time-ordered data, fast Fourier transform optimization, and parallelization and load-balancing. Thirdly, we quantify the CPU and RAM requirements for the current BEYONDPLANCK analysis, finding that a total of 1.5 TB of RAM is required for efficient analysis and that the total cost of a full Gibbs sample for LFI is 170 CPU-hrs, including both low-level processing and high-level component separation, which is well within the capabilities of current low-cost computing facilities. The existing code base is made publicly available under a GNU General Public Library (GPL) license.

We present a Gibbs sampling solution to the mapmaking problem for cosmic microwave background (CMB) measurements that builds on existing destriping methodology. Gibbs sampling breaks the computationally heavy destriping problem into two separate steps: noise filtering and map binning. Considered as two separate steps, both are computationally much cheaper than solving the combined problem. This provides a huge performance benefit as compared to traditional methods and it allows us, for the first time, to bring the destriping baseline length to a single sample. Here, we applied the Gibbs procedure to simulated Planck 30 GHz data. We find that gaps in the time-ordered data are handled efficiently by filling them in with simulated noise as part of the Gibbs process. The Gibbs procedure yields a chain of map samples, from which we are able to compute the posterior mean as a best-estimate map. The variation in the chain provides information on the correlated residual noise, without the need to construct a full noise covariance matrix. However, if only a single maximum-likelihood frequency map estimate is required, we find that traditional conjugate gradient solvers converge much faster than a Gibbs sampler in terms of the total number of iterations. The conceptual advantages of the Gibbs sampling approach lies in statistically well-defined error propagation and systematic error correction. This methodology thus forms the conceptual basis for the mapmaking algorithm employed in the BEYONDPLANCK framework, which implements the first end-to-end Bayesian analysis pipeline for CMB observations.

We present Planck Low Frequency Instrument (LFI) frequency sky maps derived within the BEYONDPLANCK framework. This framework draws samples from a global posterior distribution that includes instrumental, astrophysical, and cosmological parameters, and the main product is an entire ensemble of frequency sky map samples, each of which corresponds to one possible realization of the various modeled instrumental systematic corrections, including correlated noise, time-variable gain, as well as far sidelobe and bandpass corrections. This ensemble allows for computationally convenient end-to-end propagation of low-level instrumental uncertainties into higher-level science products, including astrophysical component maps, angular power spectra, and cosmological parameters. We show that the two dominant sources of LFI instrumental systematic uncertainties are correlated noise and gain fluctuations, and the products presented here support -for the first time -full Bayesian error propagation for these effects at full angular resolution. We compared our posterior mean maps with traditional frequency maps delivered by the Planck Collaboration, and find generally good agreement. The most important quality improvement is due to significantly lower calibration uncertainties in the new processing, as we find a fractional absolute calibration uncertainty at 70 GHz of δg0/g0â =â 5Ã - 105, which is nominally 40 times smaller than that reported by Planck 2018. However, we also note that the original Planck 2018 estimate has a nontrivial statistical interpretation, and this further illustrates the advantage of the new framework in terms of producing self-consistent and well-defined error estimates of all involved quantities without the need of ad hoc uncertainty contributions. We describe how low-resolution data products, including dense pixel-pixel covariance matrices, may be produced from the posterior samples directly, without the need for computationally expensive analytic calculations or simulations. We conclude that posterior-based frequency map sampling provides unique capabilities in terms of low-level systematics modeling and error propagation, and may play an important role for future Cosmic Microwave Background (CMB) B-mode experiments aiming at nanokelvin precision.

We describe the BEYONDPLANCK project in terms of our motivation, methodology, and main products, and provide a guide to a set of companion papers that describe each result in more detail. Building directly on experience from ESA's Planck mission, we implemented a complete end-to-end Bayesian analysis framework for the Planck Low Frequency Instrument (LFI) observations. The primary product is a full joint posterior distribution P(Ï â £d), where Ï represents the set of all free instrumental (gain, correlated noise, bandpass, etc.), astrophysical (synchrotron, free-free, thermal dust emission, etc.), and cosmological (cosmic microwave background -CMB -map, power spectrum, etc.) parameters. Some notable advantages of this approach compared to a traditional pipeline procedure are seamless end-to-end propagation of uncertainties; accurate modeling of both astrophysical and instrumental effects in the most natural basis for each uncertain quantity; optimized computational costs with little or no need for intermediate human interaction between various analysis steps; and a complete overview of the entire analysis process within one single framework. As a practical demonstration of this framework, we focus in particular on low-â CMB polarization reconstruction with Planck LFI. In this process, we identify several important new effects that have not been accounted for in previous pipelines, including gain over-smoothing and time-variable and non-1/f correlated noise in the 30 and 44 GHz channels. Modeling and mitigating both previously known and newly discovered systematic effects, we find that all results are consistent with the Î CDM model, and we constrained the reionization optical depth to Ï â =â 0.066±0.013, with a low-resolution CMB-based Ï 2 probability to exceed of 32%. This uncertainty is about 30% larger than the official pipelines, arising from taking a more complete instrumental model into account. The marginal CMB solar dipole amplitude is 3362.7±1.4â μK, where the error bar was derived directly from the posterior distribution without the need of any ad hoc instrumental corrections. We are currently not aware of any significant unmodeled systematic effects remaining in the Planck LFI data, and, for the first time, the 44 GHz channel is fully exploited in the current analysis. We argue that this framework can play a central role in the analysis of many current and future high-sensitivity CMB experiments, including LiteBIRD, and it will serve as the computational foundation of the emerging community-wide COSMOGLOBE effort, which aims to combine state-of-the-art radio, microwave, and submillimeter data sets into one global astrophysical model.

We present the first application of the COSMOGLOBE analysis framework by analyzing nine-year WMAP time-ordered observations that uses similar machinery to that of BEYONDPLANCK for the Planck Low Frequency Instrument (LFI). We analyzed only the Q-band (41 GHz) data and report on the low-level analysis process based on uncalibrated time-ordered data to calibrated maps. Most of the existing BEYONDPLANCK pipeline may be reused for WMAP analysis with minimal changes to the existing codebase. The main modification is the implementation of the same preconditioned biconjugate gradient mapmaker used by the WMAP team. Producing a single WMAP Q1-band sample requires 22 CPU-hrs, which is slightly more than the cost of a Planck 44 GHz sample of 17 CPU-hrs; this demonstrates that a full end-to-end Bayesian processing of the WMAP data is computationally feasible. In general, our recovered maps are very similar to the maps released by the WMAP team, although with two notable differences. In terms of temperature, we find a ∼2â μK quadrupole difference that most likely is caused by different gain modeling, while in polarization we find a distinct 2.5â μK signal that has been previously referred to as poorly measured modes by the WMAP team. In the COSMOGLOBE processing, this pattern arises from temperature-to-polarization leakage from the coupling between the CMB Solar dipole, transmission imbalance, and sidelobes. No traces of this pattern are found in either the frequency map or TOD residual map, suggesting that the current processing has succeeded in modeling these poorly measured modes within the assumed parametric model by using Planck information to break the sky-synchronous degeneracies inherent in the WMAP scanning strategy.

We present posterior sample-based cosmic microwave background (CMB) constraints from Planck LFI and WMAP observations as derived through global end-to-end Bayesian processing within the BeyondPlanck framework. We first used these samples to study correlations between CMB, foreground, and instrumental parameters. We identified a particularly strong degeneracy between CMB temperature fluctuations and free-free emission on intermediate angular scales (400 ≤ ∫ ≤ 600), mitigated through model reduction, masking, and resampling. We compared our posterior-based CMB results with previous Planck products and found a generally good agreement, however, with notably higher noise due to our exclusion of Planck HFI data.We found a best-fit CMB dipole amplitude of 3362:7 ± 1:4 μK, which is in excellent agreement with previous Planck results. The quoted dipole uncertainty is derived directly from the sampled posterior distribution and does not involve any ad hoc contributions for Planck instrumental systematic effects. Similarly, we find a temperature quadrupole amplitude of φTT 2 = 229 ± 97 μK2, which is in good agreement with previous results in terms of the amplitude, but the uncertainty is one order of magnitude greater than the naive diagonal Fisher uncertainty. Concurrently, we find less evidence of a possible alignment between the quadrupole and octopole than previously reported, due to a much larger scatter in the individual quadrupole coeffcients that is caused both by marginalizing over a more complete set of systematic effects – as well as by requiring a more conservative analysis mask to mitigate the free-free degeneracy. For higher multipoles, we find that the angular temperature power spectrum is generally in good agreement with both Planck and WMAP. At the same time, we note that this is the first time that the sample-based, asymptotically exact Blackwell-Rao estimator has been successfully established for multipoles up to ∫ ≤ 600. It now accounts for the majority of the cosmologically important information. Overall, this analysis demonstrates the unique capabilities of the Bayesian approach with respect to end-to-end systematic uncertainty propagation and we believe it can and should play an important role in the analysis of future CMB experiments. Cosmological parameter constraints are presented in a companion paper.

End-to-end simulations play a key role in the analysis of any high-sensitivity cosmic microwave background (CMB) experiment, providing high-fidelity systematic error propagation capabilities that are unmatched by any other means. In this paper, we address an important issue regarding such simulations, namely, how to define the inputs in terms of sky model and instrument parameters. These may either be taken as a constrained realization derived from the data or as a random realization independent from the data. We refer to these as posterior and prior simulations, respectively. We show that the two options lead to significantly different correlation structures, as prior simulations (contrary to posterior simulations) effectively include cosmic variance, but they exclude realization-specific correlations from non-linear degeneracies. Consequently, they quantify fundamentally different types of uncertainties. We argue that as a result, they also have different and complementary scientific uses, even if this dichotomy is not absolute. In particular, posterior simulations are in general more convenient for parameter estimation studies, while prior simulations are generally more convenient for model testing. Before BEYONDPLANCK, most pipelines used a mix of constrained and random inputs and applied the same hybrid simulations for all applications, even though the statistical justification for this is not always evident. BEYONDPLANCK represents the first end-to-end CMB simulation framework that is able to generate both types of simulations and these new capabilities have brought this topic to the forefront. The BEYONDPLANCK posterior simulations and their uses are described extensively in a suite of companion papers. In this work, we consider one important applications of the corresponding prior simulations, namely, code validation. Specifically, we generated a set of one-year LFI 30 GHz prior simulations with known inputs and we used these to validate the core low-level BEYONDPLANCK algorithms dealing with gain estimation, correlated noise estimation, and mapmaking.

We present cosmological parameter constraints estimated using the Bayesian BeyondPlanck analysis framework. This method supports seamless end-to-end error propagation from raw time-ordered data onto final cosmological parameters. As a first demonstration of the method, we analyzed time-ordered Planck LFI observations, combined with selected external data (WMAP 33–61 GHz, Planck HFI DR4 353 and 857 GHz, and Haslam 408 MHz) in the form of pixelized maps that are used to break critical astrophysical degeneracies. Overall, all the results are generally in good agreement with previously reported values from Planck 2018 and WMAP, with the largest relative difference for any parameter amounting about 1φ when considering only temperature multipoles between 30 ≤ ∫ ≤ 600. In cases where there are differences, we note that the BeyondPlanck results are generally slightly closer to the high- ∫ HFI-dominated Planck 2018 results than previous analyses, suggesting slightly less tension between low and high multipoles. Using low- ∫ polarization information from LFI and WMAP, we find a best-fit value of φ = 0:066±0:013, which is higher than the low value of φ = 0:052 ± 0:008 derived from Planck 2018 and slightly lower than the value of 0:069 ± 0:011 derived from the joint analysis of offcial LFI and WMAP products. Most importantly, however, we find that the uncertainty derived in the BeyondPlanck processing is about 30% greater than when analyzing the offcial products, after taking into account the different sky coverage. We argue that this uncertainty is due to a marginalization over a more complete model of instrumental and astrophysical parameters, which results in more reliable and more rigorously defined uncertainties. We find that about 2000 Monte Carlo samples are required to achieve a robust convergence for a low-resolution cosmic microwave background (CMB) covariance matrix with 225 independent modes, and producing these samples takes about eight weeks on a modest computing cluster with 256 cores.

We present a Bayesian method for estimating instrumental noise parameters and propagating noise uncertainties within the global BEYONDPLANCK Gibbs sampling framework, which we applied to Planck Low Frequency Instrument (LFI) time-ordered data. Following previous works in the literature, we initially adopted a 1/f model for the noise power spectral density (PSD), but we found the need for an additional lognormal component in the noise model in the 30 and 44 GHz bands. We implemented an optimal Wiener-filter (or constrained realization) gap-filling procedure to account for masked data. We then used this procedure to both estimate the gapless correlated noise in the time-domain, ncorr, and to sample the noise PSD parameters, ξnâ =â {Ï0,â fknee,â α,â Ap}. In contrast to previous Planck analyses, we assumed piecewise stationary noise only within each pointing period (PID), and not throughout the full mission, but we adopted the LFI Data Processing Center results as priors on α and fknee. We generally found best-fit correlated noise parameters that are mostly consistent with previous results, with a few notable exceptions. However, a detailed inspection of the time-dependent results has revealed many important findings. First and foremost, we find strong evidence for statistically significant temporal variations in all noise PSD parameters, many of which are directly correlated with satellite housekeeping data. Second, while the simple 1/f model appears to be an excellent fit for the LFI 70 GHz channel, there is evidence for additional correlated noise that is not described by a 1/f model in the 30 and 44 GHz channels, including within the primary science frequency range of 0.1-1 Hz. In general, most 30 and 44 GHz channels exhibit deviations from 1/f at the 2-3Ï level in each one-hour pointing period, motivating the addition of the lognormal noise component for these bands. For certain periods of time, we also find evidence of strong common mode noise fluctuations across the entire focal plane. Overall, we conclude that a simple 1/f profile is not adequate for obtaining a full characterization of the Planck LFI noise, even when fitted hour-by-hour, and a more general model is required. These findings have important implications for large-scale CMB polarization reconstruction with the Planck LFI data and the current work is a first attempt at understanding and mitigating these issues.

The polarization of the cosmic microwave background (CMB) can be used to search for parity-violating processes like that predicted by a Chern-Simons coupling to a light pseudoscalar field. Such an interaction rotates E modes into E modes in the observed CMB signal through an effect known as cosmic birefringence. Even though isotropic birefringence can be confused with the rotation produced by a miscalibration of the detectors' polarization angles, the degeneracy between both effects is broken when Galactic foreground emission is used as a calibrator. In this work, we use realistic simulations of the High-Frequency Instrument of the Planck mission to test the impact that Galactic foreground emission and instrumental systematics have on the recent birefringence measurements obtained through this technique. Our results demonstrate the robustness of the methodology against the miscalibration of polarization angles and other systematic effects, like intensity-to-polarization leakage, beam leakage, or cross-polarization effects. However, our estimator is sensitive to the EB correlation of polarized foreground emission. Here we propose to correct the bias induced by dust EB by modeling the foreground signal with templates produced in Bayesian component-separation analyses that fit parametric models to CMB data. Acknowledging the limitations of currently available dust templates like that of the Commander sky model, high-precision CMB data and a characterization of dust beyond the modified blackbody paradigm are needed to obtain a definitive measurement of cosmic birefringence in the future.

We search for the signature of parity-violating physics in the cosmic microwave background, called cosmic birefringence, using the Planck data release 4. We initially find a birefringence angle of β=0.30°±0.11° (68% C.L.) for nearly full-sky data. The values of β decrease as we enlarge the Galactic mask, which can be interpreted as the effect of polarized foreground emission. Two independent ways to model this effect are used to mitigate the systematic impact on β for different sky fractions. We choose not to assign cosmological significance to the measured value of β until we improve our knowledge of the foreground polarization.

We describe the BeyondPlanck project in terms of motivation, methodology andmain products, and provide a guide to a set of companion papers that describeeach result in fuller detail. We implement a complete end-to-end Bayesiananalysis framework for the Planck LFI observations. The primary product is afull joint posterior distribution $P(\omega|d)$, where $\omega$ represents theset of all free instrumental, astrophysical, and cosmological parameters.Notable advantages of this approach are seamless end-to-end propagation ofuncertainties; accurate modeling of both astrophysical and instrumental effectsin the most natural basis for each uncertain quantity; optimized computationalcosts with little or no need for intermediate human interaction between variousanalysis steps; and a complete overview of the entire analysis process withinone single framework. We focus in particular on low-$\ell$ CMB polarizationreconstruction with Planck LFI. We identify several important new effects thathave not been accounted for in previous pipelines, including gainover-smoothing and time-variable and non-$1/f$ correlated noise in the 30 and44 GHz channels. We find that all results are consistent with the $\Lambda$CDMmodel, and we constrain the reionization optical depth to $\tau=0.066\pm0.013$,with a low-resolution $\chi^2$ probability-to-exceed of 32%. This uncertaintyis about 30% larger than the official pipelines, arising from taking intoaccount a more complete instrumental model. The marginal CMB Solar dipoleamplitude is $3362.7\pm1.4\mu\mathrm{K}$, where the error bar is deriveddirectly from the posterior distribution without the need of any ad-hocinstrumental corrections. We are currently not aware of any significantunmodelled systematic effects remaining in the Planck LFI data, and, for thefirst time, the 44 GHz channel is fully exploited. (Abridged.)

The shortest distance around the Universe through us is unlikely to be much larger than the horizon diameter if microwave background anomalies are due to cosmic topology. We show that observational constraints from the lack of matched temperature circles in the microwave background leave many possibilities for such topologies. We evaluate the detectability of microwave background multipole correlations for sample cases. Searches for topology signatures in observational data over the large space of possible topologies pose a formidable computational challenge.

Polarization of the cosmic microwave background (CMB) is sensitive to new physics violating parity symmetry, such as the presence of a pseudoscalar "axionlike" field. Such a field may be responsible for early dark energy (EDE), which is active prior to recombination and provides a solution to the so-called Hubble tension. The EDE field coupled to photons in a parity-violating manner would rotate the plane of linear polarization of the CMB and produce a cross-correlation power spectrum of E- and B-mode polarization fields with opposite parities. In this Letter, we fit the EB power spectrum predicted by the photon-axion coupling of the EDE model with a potential V(ϕ)∝[1-cos(ϕ/f)]^{3} to polarization data from Planck. We find that the unique shape of the predicted EB power spectrum is not favored by the data and obtain a first constraint on the photon-axion coupling constant, g=(0.04±0.16)M_{Pl}^{-1} (68% C.L.), for the EDE model that best fits the CMB and galaxy clustering data. This constraint is independent of the miscalibration of polarization angles of the instrument or the polarized Galactic foreground emission. Our limit on g may have important implications for embedding EDE in fundamental physics, such as string theory.