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C. Franceschet

Publications and source records attributed to C. Franceschet.

At least 37 records · Page 2Linked to original sources
arXiv2024

LSPE-STRIP on-sky calibration strategy using bright celestial sources

In this paper we describe the global on-sky calibration strategy of the LSPE-Strip instrument. Strip is a microwave telescope operating in the Q- and W-bands (central frequencies of 43 and 95 GHz respectively) from the Observatorio del Teide in Tenerife, with the goal to observe and characterise the polarised Galactic foreground emission, and complement the observations of the polarisation of the cosmic microwave background to be performed by the LSPE-SWIPE instrument and other similar experiments operating at higher frequencies to target the detection of the B-mode signal from the inflationary epoch of the Universe. Starting from basic assumptions on some of the instrument parameters (NET, 1/f noise knee frequency, beam properties, observing efficiency) we perform realistic simulations to study the level of accuracy that can be achieved through observations of bright celestial calibrators in the Strip footprint (sky fraction of 30 %) on the determination and characterisation of the main instrument parameters: global and relative gain factors (in intensity and in polarisation), polarisation direction, polarisation efficiency, leakage from intensity to polarisation, beams, window functions and pointing model.

astro-ph.CO↗
arXiv2023

Impact of beam far side-lobe knowledge in the presence of foregrounds for LiteBIRD

We present a study of the impact of an uncertainty in the beam far side-lobe knowledge on the measurement of the Cosmic Microwave Background $B$-mode signal at large scale. It is expected to be one of the main source of systematic effects in future CMB observations. Because it is crucial for all-sky survey missions to take into account the interplays between beam systematic effects and all the data analysis steps, the primary goal of this paper is to provide the methodology to carry out the end-to-end study of their effect for a space-borne CMB polarization experiment, up to the cosmological results in the form of a bias $δr$ on the tensor-to-scalar ratio $r$. LiteBIRD is dedicated to target the measurement of CMB primordial $B$ modes by reaching a sensitivity of $σ\left( r \right) \leq 10^{-3}$ assuming $r=0$. As a demonstration of our framework, we derive the relationship between the knowledge of the beam far side-lobes and the tentatively allocated error budget under given assumptions on design, simulation and component separation method. We assume no mitigation of the far side-lobes effect at any stage of the analysis pipeline. We show that $δr$ is mostly due to the integrated fractional power difference between the estimated beams and the true beams in the far side-lobes region, with little dependence on the actual shape of the beams, for low enough $δr$. Under our set of assumptions, in particular considering the specific foreground cleaning method we used, we find that the integrated fractional power in the far side-lobes should be known at a level as tight as $\sim 10^{-4}$, to achieve the required limit on the bias $δr < 1.9 \times 10^{-5}$. The framework and tools developed for this study can be easily adapted to provide requirements under different design, data analysis frameworks and for other future space-borne experiments beyond LiteBIRD.

astro-ph.CO↗
arXiv2023

LiteBIRD Science Goals and Forecasts: A full-sky measurement of gravitational lensing of the CMB

We explore the capability of measuring lensing signals in $LiteBIRD$ full-sky polarization maps. With a $30$ arcmin beam width and an impressively low polarization noise of $2.16\,μ$K-arcmin, $LiteBIRD$ will be able to measure the full-sky polarization of the cosmic microwave background (CMB) very precisely. This unique sensitivity also enables the reconstruction of a nearly full-sky lensing map using only polarization data, even considering its limited capability to capture small-scale CMB anisotropies. In this paper, we investigate the ability to construct a full-sky lensing measurement in the presence of Galactic foregrounds, finding that several possible biases from Galactic foregrounds should be negligible after component separation by harmonic-space internal linear combination. We find that the signal-to-noise ratio of the lensing is approximately $40$ using only polarization data measured over $90\%$ of the sky. This achievement is comparable to $Planck$'s recent lensing measurement with both temperature and polarization and represents a four-fold improvement over $Planck$'s polarization-only lensing measurement. The $LiteBIRD$ lensing map will complement the $Planck$ lensing map and provide several opportunities for cross-correlation science, especially in the northern hemisphere.

astro-ph.CO↗
arXiv2023

LiteBIRD Science Goals and Forecasts: Improving Sensitivity to Inflationary Gravitational Waves with Multitracer Delensing

We estimate the efficiency of mitigating the lensing $B$-mode polarization, the so-called delensing, for the $LiteBIRD$ experiment with multiple external data sets of lensing-mass tracers. The current best bound on the tensor-to-scalar ratio, $r$, is limited by lensing rather than Galactic foregrounds. Delensing will be a critical step to improve sensitivity to $r$ as measurements of $r$ become more and more limited by lensing. In this paper, we extend the analysis of the recent $LiteBIRD$ forecast paper to include multiple mass tracers, i.e., the CMB lensing maps from $LiteBIRD$ and CMB-S4-like experiment, cosmic infrared background, and galaxy number density from $Euclid$- and LSST-like survey. We find that multi-tracer delensing will further improve the constraint on $r$ by about $20\%$. In $LiteBIRD$, the residual Galactic foregrounds also significantly contribute to uncertainties of the $B$-modes, and delensing becomes more important if the residual foregrounds are further reduced by an improved component separation method.

astro-ph.CO↗
arXiv2023

Measuring the CMB primordial B-modes with Bolometric Interferometry

The Q&U Bolometric Interferometer for Cosmology (QUBIC) is the first bolometric interferometer designed to measure the primordial B-mode polarization of the Cosmic Microwave Background (CMB). Bolometric interferometry is a novel technique that combines the sensitivity of bolometric detectors with the control of systematic effects that is typical of interferometry, both key features in the quest for the faint signal of the primordial B-modes. A unique feature is the so-called "spectral imaging", i.e., the ability to recover the sky signal in several sub-bands within the physical band during data analysis. This feature provides an in-band spectral resolution of Δν/ν \sim 0.04 that is unattainable by a traditional imager. This is a key tool for controlling the Galactic foregrounds contamination. In this paper, we describe the principles of bolometric interferometry, the current status of the QUBIC experiment and future prospects.

astro-ph.CO↗
arXiv2023

Tensor-to-scalar ratio forecasts for extended LiteBIRD frequency configurations

LiteBIRD is a planned JAXA-led CMB B-mode satellite experiment aiming for launch in the late 2020s, with a primary goal of detecting the imprint of primordial inflationary gravitational waves. Its current baseline focal-plane configuration includes 15 frequency bands between 40 and 402 GHz, fulfilling the mission requirements to detect the amplitude of gravitational waves with the total uncertainty on the tensor-to-scalar ratio, $δr$, down to $δr<0.001$. A key aspect of this performance is accurate astrophysical component separation, and the ability to remove polarized thermal dust emission is particularly important. In this paper we note that the CMB frequency spectrum falls off nearly exponentially above 300 GHz relative to the thermal dust SED, and a relatively minor high frequency extension can therefore result in even lower uncertainties and better model reconstructions. Specifically, we compare the baseline design with five extended configurations, while varying the underlying dust modeling, in each of which the HFT (High-Frequency Telescope) frequency range is shifted logarithmically towards higher frequencies, with an upper cutoff ranging between 400 and 600 GHz. In each case, we measure the tensor-to-scalar ratio $r$ uncertainty and bias using both parametric and minimum-variance component-separation algorithms. When the thermal dust sky model includes a spatially varying spectral index and temperature, we find that the statistical uncertainty on $r$ after foreground cleaning may be reduced by as much as 30--50 % by extending the upper limit of the frequency range from 400 to 600 GHz, with most of the improvement already gained at 500 GHz. We also note that a broader frequency range leads to better ability to discriminate between models through higher $χ^2$ sensitivity. (abridged)

astro-ph.CO↗
arXiv2023

Cosmoglobe DR1 results. II. Constraints on isotropic cosmic birefringence from reprocessed WMAP and Planck LFI data

Cosmic birefringence is a parity-violating effect that might have rotated the plane of linearly polarized light of the cosmic microwave background (CMB) by an angle $β$ since its emission. This has recently been measured to be non-zero at a statistical significance of $3.6σ$ in the official Planck PR4 and 9-year WMAP data. In this work, we constrain $β$ using the reprocessed BeyondPlanck LFI and Cosmoglobe DR1 WMAP polarization maps. These novel maps have both lower systematic residuals and a more complete error description than the corresponding official products. Foreground $EB$ correlations could bias measurements of $β$, and while thermal dust $EB$ emission has been argued to be statistically non-zero, no evidence for synchrotron $EB$ power has been reported. Unlike the dust-dominated Planck HFI maps, the majority of the LFI and WMAP polarization maps are instead dominated by synchrotron emission. Simultaneously constraining $β$ and the polarization miscalibration angle, $α$, of each channel, we find a best-fit value of $β=0.35^{\circ}\pm0.70^{\circ}$ with LFI and WMAP data only. When including the Planck HFI PR4 maps, but fitting $β$ separately for dust-dominated, $β_{>70\,\mathrm{GHz}}$, and synchrotron-dominated channels, $β_{\leq 70\,\mathrm{GHz}}$, we find $β_{\leq 70\,\mathrm{GHz}}=0.53^{\circ}\pm0.28^\circ$. This differs from zero with a statistical significance of $1.9σ$, and the main contribution to this value comes from the LFI 70 GHz channel. While the statistical significances of these results are low on their own, the measurement derived from the LFI and WMAP synchrotron-dominated maps agrees with the previously reported HFI-dominated constraints, despite the very different astrophysical and instrumental systematics involved in all these experiments.

astro-ph.CO↗
arXiv2023

Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey

LiteBIRD, the Lite (Light) satellite for the study of B-mode polarization and Inflation from cosmic background Radiation Detection, is a space mission for primordial cosmology and fundamental physics. The Japan Aerospace Exploration Agency (JAXA) selected LiteBIRD in May 2019 as a strategic large-class (L-class) mission, with an expected launch in the late 2020s using JAXA's H3 rocket. LiteBIRD is planned to orbit the Sun-Earth Lagrangian point L2, where it will map the cosmic microwave background (CMB) polarization over the entire sky for three years, with three telescopes in 15 frequency bands between 34 and 448 GHz, to achieve an unprecedented total sensitivity of 2.2$μ$K-arcmin, with a typical angular resolution of 0.5$^\circ$ at 100 GHz. The primary scientific objective of LiteBIRD is to search for the signal from cosmic inflation, either making a discovery or ruling out well-motivated inflationary models. The measurements of LiteBIRD will also provide us with insight into the quantum nature of gravity and other new physics beyond the standard models of particle physics and cosmology. We provide an overview of the LiteBIRD project, including scientific objectives, mission and system requirements, operation concept, spacecraft and payload module design, expected scientific outcomes, potential design extensions and synergies with other projects.

astro-ph.IM↗
arXiv2023

From BeyondPlanck to Cosmoglobe: Open Science, Reproducibility, and Data Longevity

The BeyondPlanck and Cosmoglobe collaborations have implemented the first integrated Bayesian end-to-end analysis pipeline for CMB experiments. The primary long-term motivation for this work is to develop a common analysis platform that supports efficient global joint analysis of complementary radio, microwave, and sub-millimeter experiments. A strict prerequisite for this to succeed is broad participation from the CMB community, and two foundational aspects of the program are therefore reproducibility and Open Science. In this paper, we discuss our efforts toward this aim. We also discuss measures toward facilitating easy code and data distribution, community-based code documentation, user-friendly compilation procedures, etc. This work represents the first publicly released end-to-end CMB analysis pipeline that includes raw data, source code, parameter files, and documentation. We argue that such a complete pipeline release should be a requirement for all major future and publicly-funded CMB experiments, noting that a full public release significantly increases data longevity by ensuring that the data quality can be improved whenever better processing techniques, complementary datasets, or more computing power become available, and thereby also taxpayers' value for money; providing only raw data and final products is not sufficient to guarantee full reproducibility in the future.

astro-ph.IM↗
arXiv2023

Cosmoglobe DR1 results. I. Improved Wilkinson Microwave Anisotropy Probe maps through Bayesian end-to-end analysis

We present Cosmoglobe Data Release 1, which implements the first joint analysis of WMAP and Planck LFI time-ordered data, processed within a single Bayesian end-to-end framework. This framework builds directly on a similar analysis of the LFI measurements by the BeyondPlanck collaboration, and approaches the CMB analysis challenge through Gibbs sampling of a global posterior distribution, simultaneously accounting for calibration, mapmaking, and component separation. The computational cost of producing one complete WMAP+LFI Gibbs sample is 812 CPU-hr, of which 603 CPU-hrs are spent on WMAP low-level processing; this demonstrates that end-to-end Bayesian analysis of the WMAP data is computationally feasible. We find that our WMAP posterior mean temperature sky maps and CMB temperature power spectrum are largely consistent with the official WMAP9 results. Perhaps the most notable difference is that our CMB dipole amplitude is $3366.2 \pm 1.4\ \mathrm{μK}$, which is $11\ \mathrm{μK}$ higher than the WMAP9 estimate and $2.5\ σ$ higher than BeyondPlanck; however, it is in perfect agreement with the HFI-dominated Planck PR4 result. In contrast, our WMAP polarization maps differ more notably from the WMAP9 results, and in general exhibit significantly lower large-scale residuals. We attribute this to a better constrained gain and transmission imbalance model. It is particularly noteworthy that the W-band polarization sky map, which was excluded from the official WMAP cosmological analysis, for the first time appears visually consistent with the V-band sky map. Similarly, the long standing discrepancy between the WMAP K-band and LFI 30 GHz maps is finally resolved, and the difference between the two maps appears consistent with instrumental noise at high Galactic latitudes. All maps and the associated code are made publicly available through the Cosmoglobe web page.

astro-ph.CO↗
arXiv2022

Status of QUBIC, the Q&U Bolometer for Cosmology

The Q&U Bolometric Interferometer for Cosmology (QUBIC) is a novel kind of polarimeter optimized for the measurement of the B-mode polarization of the Cosmic Microwave Back-ground (CMB), which is one of the major challenges of observational cosmology. The signal is expected to be of the order of a few tens of nK, prone to instrumental systematic effects and polluted by various astrophysical foregrounds which can only be controlled through multichroic observations. QUBIC is designed to address these observational issues with a novel approach that combines the advantages of interferometry in terms of control of instrumental systematics with those of bolometric detectors in terms of wide-band, background-limited sensitivity.

astro-ph.IM↗
arXiv2022

BeyondPlanck XIII. Intensity foreground sampling, degeneracies, and priors

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 $\texttt{Commander}$ 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 $ν_{\mathrm{p}}=25.3\pm0.5$ 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.

astro-ph.CO↗
arXiv2022

BeyondPlanck I. Global Bayesian analysis of the Planck Low Frequency Instrument data

We describe the BeyondPlanck project in terms of motivation, methodology and main products, and provide a guide to a set of companion papers that describe each result in fuller detail. We implement a complete end-to-end Bayesian analysis framework for the Planck LFI observations. The primary product is a full joint posterior distribution $P(ω|d)$, where $ω$ represents the set of all free instrumental, astrophysical, and cosmological parameters. Notable advantages of this approach 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. We focus in particular on low-$\ell$ CMB polarization reconstruction with Planck LFI. 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. We find that all results are consistent with the $Λ$CDM model, and we constrain the reionization optical depth to $τ=0.066\pm0.013$, with a low-resolution $χ^2$ probability-to-exceed of 32%. This uncertainty is about 30% larger than the official pipelines, arising from taking into account a more complete instrumental model. The marginal CMB Solar dipole amplitude is $3362.7\pm1.4μ\mathrm{K}$, where the error bar is derived directly from the posterior distribution without the need of any ad-hoc instrumental corrections. We are currently not aware of any significant unmodelled systematic effects remaining in the Planck LFI data, and, for the first time, the 44 GHz channel is fully exploited. (Abridged.)

astro-ph.CO↗
arXiv2022

BeyondPlanck IV. On end-to-end simulations in CMB analysis -- Bayesian versus frequentist statistics

End-to-end simulations play a key role in the analysis of any high-sensitivity CMB experiment, providing high-fidelity systematic error propagation capabilities 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 Bayesian and frequentist simulations, respectively. We show that the two options lead to significantly different correlation structures, as frequentist simulations, contrary to Bayesian simulations, effectively include cosmic variance, but exclude realization-specific correlations from non-linear degeneracies. Consequently, they quantify fundamentally different types of uncertainties, and we argue that they therefore also have different and complementary scientific uses, even if this dichotomy is not absolute. Before BeyondPlanck, most pipelines have used a mix of constrained and random inputs, and used 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 Bayesian BeyondPlanck simulations and their uses are described extensively in a suite of companion papers. In this paper we consider one important applications of the corresponding frequentist simulations, namely code validation. That is, we generate a set of 1-year LFI 30 GHz frequentist simulations with known inputs, and use these to validate the core low-level BeyondPlanck algorithms; gain estimation, correlated noise estimation, and mapmaking.

astro-ph.CO↗
arXiv2022

BeyondPlanck XI. Bayesian CMB analysis with sample-based end-to-end error propagation

We present posterior sample-based cosmic microwave background (CMB) constraints from Planck LFI and WMAP observations derived through global end-to-end Bayesian processing. We use these samples to study correlations between CMB, foreground, and instrumental parameters, and we identify a particularly strong degeneracy between CMB temperature fluctuations and free-free emission on intermediate angular scales, which is mitigated through model reduction, masking, and resampling. We compare our posterior-based CMB results with previous Planck products, and find generally good agreement, but with higher noise due to exclusion of HFI data. We find a best-fit CMB dipole amplitude of $3362.7\pm1.4μK$, in excellent agreement with previous Planck results. The quoted uncertainty is derived directly from the sampled posterior distribution, and does not involve any ad hoc contribution for systematic effects. Similarly, we find a temperature quadrupole amplitude of $σ^{TT}_2=229\pm97μK^2$, in good agreement with previous results in terms of the amplitude, but the uncertainty is an order of magnitude larger than the diagonal Fisher uncertainty. Relatedly, we find lower evidence for a possible alignment between $\ell = 2$ and $\ell = 3$ than previously reported due to a much larger scatter in the individual quadrupole coefficients, caused both by marginalizing over a more complete set of systematic effects, and by our more conservative analysis mask. For higher multipoles, we find that the angular temperature power spectrum is generally in good agreement with both Planck and WMAP. This is the first time the sample-based asymptotically exact Blackwell-Rao estimator has been successfully established for multipoles up to $\ell\le600$, and it now accounts for the majority of the cosmologically important information. Cosmological parameter constraints are presented in a companion paper. (Abriged)

astro-ph.CO↗
arXiv2022

BeyondPlanck X. Planck LFI frequency maps with sample-based error propagation

We present Planck 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. This ensemble allows for computationally convenient end-to-end propagation of low-level instrumental uncertainties into higher-level science products. 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 compare 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 $δg_{0}/g_{0} =5 \cdot 10^{-5}$, which is nominally 40 times smaller than that reported by Planck 2018. However, the original Planck 2018 estimate has a non-trivial 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 directly from the posterior samples 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 modelling and error propagation, and may play an important role for future CMB B-mode experiments. (Abridged.)

astro-ph.CO↗
arXiv2022

BeyondPlanck VI. Noise characterization and modelling

We present a Bayesian method for estimating instrumental noise parameters and propagating noise uncertainties within the global BeyondPlanck Gibbs sampling framework, and apply this to Planck Low Frequency Instrument (LFI) time-ordered data. Following previous literature, we initially adopt a $1/f$ model for the noise power spectral density (PSD), but find the need for an additional lognormal component in the noise model for the 30 and 44\,GHz bands. We implement an optimal Wiener-filter (or constrained realization) gap-filling procedure to account for masked data. We then use this procedure to both estimate the gapless correlated noise in the time-domain, $n_\mathrm{corr}$, and to sample the noise PSD parameters, $ξ^n = \{σ_0, f_\mathrm{knee}, α, A_\mathrm{p}\}$. In contrast to previous \textit{Planck} analyses, we assume piecewise stationary noise only within each pointing period (PID), not throughout the full mission, but we adopt the LFI Data Processing Center (DPC) results as priors on $α$ and $f_\mathrm{knee}$. On average, we find 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 reveals 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 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. (Abridged)

astro-ph.CO↗
arXiv2022

Total power horn-coupled 150 GHz LEKID array for space applications

We have developed two arrays of lumped element kinetic inductance detectors working in the D-band, and optimised for the low radiative background conditions of a satellite mission aiming at precision measurements of the Cosmic Microwave Background (CMB). The first detector array is sensitive to the total power of the incoming radiation to which is coupled via single-mode waveguides and corrugated feed-horns, while the second is sensitive to the polarisation of the radiation thanks to orthomode transducers. Here, we focus on the total power detector array, which is suitable, for instance, for precision measurements of unpolarised spectral distortions of the CMB, where detecting both polarisations provides a sensitivity advantage. We describe the optimisation of the array design, fabrication and packaging, the dark and optical characterisation, and the performance of the black-body calibrator used for the optical tests. We show that almost all the detectors of the array are photon-noise limited under the radiative background of a 3.6 K black-body. This result, combined with the weak sensitivity to cosmic rays hits demonstrated with the OLIMPO flight, validates the idea of using lumped elements kinetic inductance detectors for precision, space-based CMB missions.

astro-ph.IM↗