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Using the Planck Low Frequency Instrument (LFI) and WMAP data within the global Bayesian BEYONDPLANCK framework, we constrained the polarized foreground emission between 30 and 70 GHz. We combined, for the first time, full-resolution Planck LFI time-ordered data with low-resolution WMAP sky maps at 33, 40, and 61 GHz. The spectral parameters were fit with a likelihood defined at the native resolution of each frequency channel. This analysis represents the first implementation of true multi-resolution component separation applied to CMB observations for both amplitude and spectral energy distribution (SED) parameters. For the synchrotron emission, we approximated the SED as a power-law in frequency and we find that the low signal-to-noise ratio of the current data strongly limits the number of free parameters that can be robustly constrained. We partitioned the sky into four large disjoint regions (High Latitude; Galactic Spur; Galactic Plane; and Galactic Center), each associated with its own power-law index. We find that the High Latitude region is prior-dominated, while the Galactic Center region is contaminated by residual instrumental systematics. The two remaining regions appear to be signal-dominated, and for these we derive spectral indices of βsSpur=3.17±0.06 and βsPlane=3.03±0.07, which is in good agreement with previous results. For the thermal dust emission, we assumed a modified blackbody model and we fit a single power-law index across the full sky. We find βda =â 1.64±0.03, which is slightly steeper than the value reported in Planck HFI data, but still statistically consistent at the 2Ï confidence level.

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 constrained the level of polarized anomalous microwave emission (AME) on large angular scales using Planck Low-Frequency Instrument (LFI) and WMAP polarization data within a Bayesian cosmic microwave background (CMB) analysis framework. We modeled synchrotron emission with a power-law spectral energy distribution, as well as the sum of AME and thermal dust emission through linear regression with the Planck High-Frequency Instrument (HFI) 353 GHz data. This template-based dust emission model allowed us to constrain the level of polarized AME while making minimal assumptions on its frequency dependence. We neglected CMB fluctuations, but show through simulations that these fluctuations have a minor impact on the results. We find that the resulting AME polarization fraction confidence limit is sensitive to the polarized synchrotron spectral index prior. In addition, for prior means βsâ

We present a Bayesian calibration algorithm for cosmic microwave background (CMB) observations as implemented within the global end-to-end BEYONDPLANCK framework and applied to the Planck Low Frequency Instrument (LFI) data. Following the most recent Planck analysis, we decomposed the full time-dependent gain into a sum of three nearly orthogonal components: one absolute calibration term, common to all detectors, one time-independent term that can vary between detectors, and one time-dependent component that was allowed to vary between one-hour pointing periods. Each term was then sampled conditionally on all other parameters in the global signal model through Gibbs sampling. The absolute calibration is sampled using only the orbital dipole as a reference source, while the two relative gain components were sampled using the full sky signal, including the orbital and Solar CMB dipoles, CMB fluctuations, and foreground contributions. We discuss various aspects of the data that influence gain estimation, including the dipole-polarization quadrupole degeneracy and processing masks. Comparing our solution to previous pipelines, we find good agreement in general, with relative deviations of 0.67% (0.84%) for 30 GHz, 0.12% (0.04%) for 44 GHz and 0.03% (0.64%) for 70 GHz, compared to Planck PR4 and Planck 2018, respectively. We note that the BEYONDPLANCK calibration was performed globally, which results in better inter-frequency consistency than previous estimates. Additionally, WMAP observations were used actively in the BEYONDPLANCK analysis, which both breaks internal degeneracies in the Planck data set and results in an overall better agreement with WMAP. Finally, we used a Wiener filtering approach to smoothing the gain estimates. We show that this method avoids artifacts in the correlated noise maps as a result of oversmoothing the gain solution, which is difficult to avoid with methods like boxcar smoothing, as Wiener filtering by construction maintains a balance between data fidelity and prior knowledge. Although our presentation and algorithm are currently oriented toward LFI processing, the general procedure is fully generalizable to other experiments, as long as the Solar dipole signal is available to be used for calibration.

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 describe the correction procedure for Analog-to-Digital Converter (ADC) differential non-linearities (DNL) adopted in the Bayesian end-to-end BEYONDPLANCK analysis framework. This method is nearly identical to that developed for the official Planck Low Frequency Instrument (LFI) Data Processing Center (DPC) analysis, and relies on the binned rms noise profile of each detector data stream. However, rather than building the correction profile directly from the raw rms profile, we first fit a Gaussian to each significant ADC-induced rms decrement, and then derive the corresponding correction model from this smooth model. The main advantage of this approach is that only samples which are significantly affected by ADC DNLs are corrected, as opposed to the DPC approach in which the correction is applied to all samples, filtering out signals not associated with ADC DNLs. The new corrections are only applied to data for which there is a clear detection of the non-linearities, and for which they perform at least comparably with the DPC corrections. Out of a total of 88 LFI data streams (sky and reference load for each of the 44 detectors) we apply the new minimal ADC corrections in 25 cases, and maintain the DPC corrections in 8 cases. All these corrections are applied to 44 or 70 GHz channels, while, as in previous analyses, none of the 30 GHz ADCs show significant evidence of non-linearity. By comparing the BEYONDPLANCK and DPC ADC correction methods, we estimate that the residual ADC uncertainty is about two orders of magnitude below the total noise of both the 44 and 70 GHz channels, and their impact on current cosmological parameter estimation is small. However, we also show that non-idealities in the ADC corrections can generate sharp stripes in the final frequency maps, and these could be important for future joint analyses with the Planck High Frequency Instrument (HFI), Wilkinson Microwave Anisotropy Probe (WMAP), or other datasets. We therefore conclude that, although the existing corrections are adequate for LFI-based cosmological parameter analysis, further work on LFI ADC corrections is still warranted.

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 the intensity foreground algorithms and model employed within the BEYONDPLANCK analysis framework. The BEYONDPLANCK analysis is aimed at integrating component separation and instrumental parameter sampling within a global framework, leading to complete end-to-end error propagation in the Planck Low Frequency Instrument (LFI) data analysis. Given the scope of the BEYONDPLANCK analysis, a limited set of data is included in the component separation process, leading to foreground parameter degeneracies. In order to properly constrain the Galactic foreground parameters, we improve upon the previous Commander component separation implementation by adding a suite of algorithmic techniques. These algorithms are designed to improve the stability and computational efficiency for weakly constrained posterior distributions. These are: (1) joint foreground spectral parameter and amplitude sampling, building on ideas from MIRAMARE; (2) component-based monopole determination; (3) joint spectral parameter and monopole sampling; and (4) application of informative spatial priors for component amplitude maps. We find that the only spectral parameter with a significant signal-to-noise ratio using the current BEYONDPLANCK data set is the peak frequency of the anomalous microwave emission component, for which we find νpa =a 25.3±0.5a GHz; all others must be constrained through external priors. Future works will be aimed at integrating many more data sets into this analysis, both map and time-ordered based, thereby gradually eliminating the currently observed degeneracies in a controlled manner with respect to both instrumental systematic effects and astrophysical degeneracies. When this happens, the simple LFI-oriented data model employed in the current work will need to be generalized to account for both a richer astrophysical model and additional instrumental effects. This work will be organized within the Open Science-based COSMOGLOBE community effort.

We discuss the treatment of bandpass and beam leakage corrections in the Bayesian BEYONDPLANCK cosmic microwave background (CMB) analysis pipeline as applied to the Planck LFI measurements. As a preparatory step, we first applied three corrections to the nominal LFI bandpass profiles, including the removal of a known systematic effect in the ground measuring equipment at 61 GHz, along with a smoothing of standing wave ripples and edge regularization. The main net impact of these modifications is an overall shift in the 70 GHz bandpass of +0.6 GHz. We argue that any analysis of LFI data products, either from Planck or BEYONDPLANCK, should use these new bandpasses. In addition, we fit a single free bandpass parameter for each radiometer of the form δiâ =â δ0+δi, where δ0 represents an absolute frequency shift per frequency band and δi is a relative shift per detector. The absolute correction is only fitted at 30 GHz, with a full Ï 2-based likelihood, resulting in a correction of δ30â =â 0.24±0.03â GHz. The relative corrections were fitted using a spurious map approach that is fundamentally similar to the method pioneered by the WMAP team, but excluding the introduction of many additional degrees of freedom. All the bandpass parameters were sampled using a standard Metropolis sampler within the main BEYONDPLANCK Gibbs chain and the bandpass uncertainties were thus propagated to all other data products in the analysis. In summary, we find that our bandpass model significantly reduces leakage effects. For beam leakage corrections, we adopted the official Planck LFI beam estimates without any additional degrees of freedom and we only marginalized over the underlying sky model. We note that this is the first time that leakage from beam mismatch has been included for Planck LFI maps.

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.

We introduce a new formulation of the Conviqt convolution algorithm in terms of spin harmonics, and apply this to the problem of sidelobe correction for BEYONDPLANCK, the first end-to-end Bayesian Gibbs sampling framework for CMB analysis. We compare our implementation to the previous Planck LevelS implementation, and find good agreement between the two codes in terms of accuracy, but with a speed-up reaching a factor of 3-10, depending on the frequency bandlimits, lmax and mmax. The new algorithm is significantly simpler to implement and maintain, since all low-level calculations are handled through an external spherical harmonic transform library. We find that our mean sidelobe estimates for Planck LFI are in good agreement with previous efforts. Additionally, we present novel sidelobe rms maps that quantify the uncertainty in the sidelobe corrections due to variations in the sky model.

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.