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DOE OSTI2023

BEYONDPLANCK III. Commander3

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.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

BeyondPlanck: VII. Bayesian estimation of gain and absolute calibration for cosmic microwave background experiments

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.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

$\mu$-distortion around stupendously large primordial black holes

In a variety of mechanisms generating primordial black holes, each black hole is expected to form along with a surrounding underdense region that roughly compensates the black hole mass. This region will propagate outwards and expand as a shell at the speed of sound in the homogeneous background. Dissipation of the shell due to Silk damping could lead to detectable μ-distortion in the CMB spectrum: if black holes are rare on the last scattering surface, the signal(s) would be pointlike; whereas if there are a sufficient number of them, we could have a uniform distortion in the CMB sky. While the current bound on the average μ-distortion is |$\bar{μ}$| ≲ 10>- 4 , the standard ΛCDM model predicts |$\bar{μ}$| ~10 -8 , which could possibly be detected in future missions. It is shown in this work that the non-observation of $\bar{μ}$ beyond ΛCDM can place a new upper bound on the density of supermassive primordial black holes within the mass range 10 6 M ⊙ ≲ M ≲ 10 15 M ⊙ . Finally, black holes with initial mass M ≳ 10 12 M ⊙ could leave a pointlike distortion with μ ≳10-8> at an angular scale ~ 1° in CMB, and its non-observation would impose an even more stringent bound on the population of these stupendously large primordial black holes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Cosmology at high redshift — a probe of fundamental physics

An observational program focused on the high redshift (2<z<6) Universe has the opportunity to dramatically improve over upcoming LSS and CMB surveys on measurements of both the standard cosmological model and its extensions. Using a Fisher matrix formalism that builds upon recent advances in Lagrangian perturbation theory, we forecast constraints for future spectroscopic and 21-cm surveys on the standard cosmological model, curvature, neutrino mass, relativistic species, primordial features, primordial non-Gaussianity, dynamical dark energy, and gravitational slip. We compare these constraints with those achievable by current or near-future surveys such as DESI and Euclid, all under the same forecasting formalism, and compare our formalism with traditional linear methods. Here, our Python code FishLSS — used to calculate the Fisher information of the full shape power spectrum, CMB lensing, the cross-correlation of CMB lensing with galaxies, and combinations thereof — is publicly available.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Early dark energy constraints with late-time expansion marginalization

Abstract Early dark energy (EDE) is an extension to the ΛCDM model that includes an additional energy density contribution near recombination. The model was proposed to reduce the tension between the measurements of the Hubble constantH 0 from the cosmic microwave background (CMB) and from the local cosmic distance ladder. Some analyses in the recent literature have shown intriguing hints for EDE. However, this model increases the tension in the derived clustering of galaxies (as measured by the so-calledS 8 parameter) between CMB and large scale structure (LSS) measurements. This new tension limits the contribution of EDE during recombination, and thus its effect on the Hubble tension. In this work, we investigate whether the inclusion of a general, smooth late-time dark energy modification can increase back the EDE contribution when LSS data is included in the analysis. In order to generalize the late expansion with respect to the ΛCDM model, we substitute the cosmological constant by a late dark energy fluid model with a piecewise constant equation of state w(z) in redshift bins. We show that, when analysing this generalized model with combinations of CMB, LSS and type Ia supernovae data from several experiments no significant changes onS 8 and EDE parameter constraints is found. The contribution to the EDE fraction constraint with late-time expansion marginalization isf EDE = 0.067 +0.019 -0.027 using 3 redshift bins, with similar results for 5 and 10 redshift bins. This work shows that in order to solve simultaneously the Hubble andS 8 tensions, one needs a mechanism for increasing the clustering of matter at late times different from a simple change in the background evolution of late dark energy.

Astronomy & Astrophysics↗
DOE OSTI2024

Dark radiation isocurvature from cosmological phase transitions

Abstract Cosmological first order phase transitions are typically associated with physics beyond the Standard Model, and thus of great theoretical and observational interest. Models of phase transitions where the energy is mostly converted to dark radiation can be constrained through limits on the dark radiation energy density (parameterized by ΔN eff ). However, the current constraint (ΔN eff < 0.3) assumes the perturbations are adiabatic. We point out that a broad class of non-thermal first order phase transitions that start during inflation but do not complete until after reheating leave a distinct imprint in the scalar field from bubble nucleation. Dark radiation inherits the perturbation from the scalar field when the phase transition completes, leading to large-scale isocurvature that would be observable in the CMB. We perform a detailed calculation of the isocurvature power spectrum and derive constraints on ΔN eff based on CMB+BAO data. For a reheating temperature ofT rh and a nucleation temperatureT * , the constraint is approximately ΔN eff ≲ 10 -5 (T * /T rh ) -4 , which can be much stronger than the adiabatic result. We also point out that since perturbations of dark radiation have a non-Gaussian origin, searches for non-Gaussianity in the CMB could place a stringent bound on ΔN eff as well.

Astronomy & Astrophysics↗
DOE OSTI2024

Cosmic neutrino decoupling and its observable imprints: insights from entropic-dual transport

Abstract Very different processes characterize the decoupling of neutrinos to form the cosmic neutrino background (CνB) and the much later decoupling of photons from thermal equilibrium to form the cosmic microwave background (CMB). The CνB emerges from the fuzzy, energy-dependent neutrinosphere and encodes the physics operating in the early universe in the temperature rangeT∼ 10 MeV toT∼ 10 keV. This is the epoch where beyond Standard Model (BSM) physics, especially in the neutrino sector, may be influential in setting the light element abundances, the necessarily distorted fossil neutrino energy spectra, and other light particle energy density contributions. Here we use techniques honed in extensive CMB studies to analyze the CνB as calculated in detailed neutrino energy transport and nuclear reaction simulations of the protracted weak decoupling and primordial nucleosynthesis epochs. Our moment method, relative entropy, and differential visibility approach can leverage future high precision CMB and light element primordial abundance measurements to provide new insights into the CνB and any BSM physics it encodes. We demonstrate that the evolution of the energy spectrum of the CνB throughout the weak decoupling epoch is accurately captured in the Standard Model by only three parameters per species, a non-trivial conclusion given the deviation from thermal equilibrium and the impact of the decrease of electron-positron pairs. Furthermore, we can interpret each of the three parameters as physical characteristics of a non-equilibrium system. Though the treatment presented here makes some simplifying assumptions including ignoring neutrino flavor oscillations, the success of our compact description within the Standard Model motivates its use also in BSM scenarios. We further demonstrate how observations of primordial light element abundances can be used to place constraints on the CνB energy spectrum, deriving response functions that can be applied for general deviations from a thermal spectrum. Combined with the description of those deviations that we develop here, our methods provide a convenient and powerful framework to constrain the impact of BSM physics on the CνB.

Astronomy & Astrophysics↗
DOE OSTI2025

The Atacama Cosmology Telescope: Large-scale velocity reconstruction with the kinematic Sunyaev-Zel'dovich effect and DESI LRGs

The kinematic Sunyaev-Zel'dovich (kSZ) effect induces a non-zero density-density-temperature bispectrum, which we can use to reconstruct the large-scale velocity field from a combination of cosmic microwave background (CMB) and galaxy density measurements, in a procedure known as “kSZ velocity reconstruction”. This method has been forecast to constrain large-scale modes with future galaxy and CMB surveys, improving their measurement beyond what is possible with the galaxy surveys alone. Such measurements will enable tighter constraints on large-scale signals such as primordial non-Gaussianity, deviations from homogeneity, and modified gravity. In this work, we demonstrate a statistically significant measurement of kSZ velocity reconstruction for the first time, by applying quadratic estimators to the combination of the ACT DR6 CMB+kSZ map and the DESI LRG galaxies (with photometric redshifts) in order to reconstruct the velocity field. We do so using a formalism appropriate for the 2-dimensional projected galaxy fields that we use, which naturally incorporates the curved-sky effects important on the largest scales. We find evidence for the signal by cross-correlating with an external estimate of the velocity field from the spectroscopic BOSS survey and rejecting the null (no-kSZ) hypothesis at 3.8σ. Our work presents a first step towards the use of this observable for cosmological analyses.

Sunyaev-Zeldovich effect↗
DOE OSTI2025

Abundance and properties of dark radiation from the cosmic microwave background

We study the cosmological signatures of new light relics that are collisionless like standard neutrinos or are strongly interacting. We provide a simple and succinct rephrasing of their physical effects in the cosmic microwave background, as well as the resulting parameter degeneracies with other cosmological parameters, in terms of the total radiation abundance and the fraction thereof that freely streams. In these more general terms, interacting and noninteracting light relics are differentiated by their respective decrease and increase of the free-streaming fraction, and, moreover, the scale-dependent interplay thereof with a common, correlated reduction of the fraction of matter in baryons. We then derive updated constraints on various dark-radiation scenarios with the latest cosmological observations, employing this language to identify the physical origin of the impact of each dataset. The “PR4” reanalyses of Planck CMB data prefer a larger primordial helium yield and therefore also slightly more radiation than the 2018 analysis; we investigate the differences between the two releases that drives these shifts. Smaller free-streaming fractions are disfavored by the excess lensing of the CMB measured in lensing reconstruction data from Planck and the Atacama Cosmology Telescope. On the other hand, baryon acoustic oscillation measurements from the Dark Energy Spectroscopic Instrument drive marginal detections of new, strongly interacting light relics due to that data's preference for lower matter fractions. Finally, we forecast measurements from the CMB-S4 experiment.

cosmological parameters from CMBR↗
DOE OSTI2026

H 0 without the sound horizon (or supernovae): A 2% measurement in DESI DR1

The sound horizon scale rs is a key source of information for measurements of H 0 from early-time data, and is therefore a common target of new physics proposed to solve the Hubble tension. We present a sub-2% measurement of the Hubble constant that is independent of this scale, using data from the first data release of the Dark Energy Spectroscopic Instrument (DESI DR1). Building on previous work, we remove dependency on the sound horizon size using a heuristic rescaling procedure at the power spectrum level. A key innovation is the inclusion of uncalibrated (agnostic to r s ) post-reconstruction BAO measurements from DESI DR1, as well as using the CMB acoustic scale θ* as a high-redshift anchor. Uncalibrated type-Ia supernovae are often included as an independent source of Ω m information; here we demonstrate the robustness of our results by additionally considering two supernova-independent alternative datasets. We find somewhat higher values of H 0 relative to our previous work: 69.2 +1.3 -1.4 , 70.3 +1.4 -1.2 , and 69.6 +1.3 -1.8 km s -1 Mpc -1 respectively when including measurements from i) Planck/ACT CMB lensing × unWISE galaxies, ii) the DES Year 3 6×2pt analysis, and iii) Planck/ACT CMB lensing + the DES Year 5 supernova analysis. These remarkably consistent constraints achieve better than 2% precision; they are among the most stringent sound horizon-independent measurements from LSS to date, and provide a powerful avenue for probing the origin of the Hubble tension.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Including massive neutrinos in thermal Sunyaev Zeldovich power spectrum and cluster counts analyses

We consistently include the effect of massive neutrinos in the thermal Sunyaev Zeldovich (SZ) power spectrum and cluster counts analyses, highlighting subtle dependencies on the total neutrino mass and data combination. In particular, we find that using the transfer functions for cold dark matter (CDM) + baryons in the computation of the halo mass function, instead of the transfer functions including neutrino perturbations, as prescribed in recent work, yields an ≈0.25 percent downward shift of the σ 8 constraint from tSZ power spectrum data, with a fiducial neutrino mass Σm ν = 0.06 eV. In ΛCDM, with an X-ray mass bias corresponding to the expected hydrostatic mass bias, i.e. (1 - b) ≃ 0.8, our constraints from Planck SZ data are consistent with the latest results from SPT, DES-Y1, and KiDS+VIKING-450. In νΛCDM, our joint analyses of Planck SZ with Planck 2015 primary CMB yield a small improvement on the total neutrino mass bound compared to the Planck 2015 primary CMB constraint, as well as (1 - b) = 0.64 ± 0.04 (68 percent CL). For forecasts, we find that competitive neutrino mass measurements using cosmic variance limited SZ power spectrum require masking the heaviest clusters and probing the small-scale SZ power spectrum up to ℓ max ≈ 10 4 . Although this is challenging, we find that SZ power spectrum can realistically be used to tightly constrain intracluster medium properties: we forecast a 2 percent determination of the X-ray mass bias by combining CMB-S4 and our mock SZ power spectrum with ℓ max = 10 3 .

Astronomy & Astrophysics↗
DOE OSTI2020

Hemispherical variance anomaly and reionization optical depth

ABSTRACT Cosmic microwave background (CMB) full-sky temperature data show a hemispherical asymmetry in power nearly aligned with the Ecliptic, with the Northern hemisphere displaying an anomalously low variance, while the Southern hemisphere appears consistent with expectations from the best-fitting theory, Lambda Cold Dark Matter (ΛCDM). The low signal-to-noise ratio in current polarization data prevents a similar comparison. Polarization realizations constrained by temperature data show that in ΛCDM the lack of variance is not expected to be present in polarization data. Therefore, a natural way of testing whether the temperature result is a fluke is to measure the variance of CMB polarization components. In anticipation of future CMB experiments that will allow for high-precision large-scale polarization measurements, we study how the variance of polarization depends on ΛCDM-parameter uncertainties by forecasting polarization maps with Planck’s Markov chain Monte Carlo chains. We show that polarization variance is sensitive to present uncertainties in cosmological parameters, mainly due to current poor constraints on the reionization optical depth τ, which drives variance at low multipoles. We demonstrate how the improvement in the τ measurement seen between Planck’s two latest data releases results in a tighter constraint on polarization variance expectations. Finally, we consider even smaller uncertainties on τ and how more precise measurements of τ can drive the expectation for polarization variance in a hemisphere close to that of the cosmic-variance-limited distribution.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Dark Energy Survey Year 1 results: the lensing imprint of cosmic voids on the cosmic microwave background

ABSTRACT Cosmic voids gravitationally lens the cosmic microwave background (CMB) radiation, resulting in a distinct imprint on degree scales. We use the simulated CMB lensing convergence map from the Marenostrum Institut de Ciencias de l’Espai (MICE) N-body simulation to calibrate our detection strategy for a given void definition and galaxy tracer density. We then identify cosmic voids in Dark Energy Survey (DES) Year 1 data and stack the Planck 2015 lensing convergence map on their locations, probing the consistency of simulated and observed void lensing signals. When fixing the shape of the stacked convergence profile to that calibrated from simulations, we find imprints at the 3σ significance level for various analysis choices. The best measurement strategies based on the MICE calibration process yield S/N ≈ 4 for DES Y1, and the best-fitting amplitude recovered from the data is consistent with expectations from MICE (A ≈ 1). Given these results as well as the agreement between them and N-body simulations, we conclude that the previously reported excess integrated Sachs–Wolfe (ISW) signal associated with cosmic voids in DES Y1 has no counterpart in the Planck CMB lensing map.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

The picasso map-making code: application to a simulation of the QUIJOTE northern sky survey

ABSTRACT Map-making is an important step for the data analysis of cosmic microwave background (CMB) experiments. It consists of converting the data, which are typically a long, complex, and noisy collection of measurements, into a map, which is an image of the observed sky. We present in this paper a new map-making code named picasso (Polarization and Intensity CArtographer for Scanned Sky Observations), which was implemented to construct intensity and polarization maps from the Multi Frequency Instrument (MFI) of the QUIJOTE (Q-U-I Joint TEnerife) CMB polarization experiment. picasso is based on the destriping algorithm, and is suited to address specific issues of ground-based microwave observations, with a technique that allows the fit of a template function in the time domain, during the map-making step. This paper describes the picasso code, validating it with simulations and assessing its performance. For this purpose, we produced realistic simulations of the QUIJOTE-MFI survey of the northern sky (approximately ∼20 000 deg2), and analysed the reconstructed maps with picasso, using real and harmonic space statistics. We show that, for this sky area, picasso is able to reconstruct, with high fidelity, the injected signal, recovering all the scales with ℓ > 10 in TT, EE, and BB. The signal error is better than 0.001 per cent at 20 < ℓ < 200. Finally, we validated some of the methods that will be applied to the real wide-survey data, like the detection of the CMB anisotropies via cross-correlation analyses. Despite that the implementation of picasso is specific for QUIJOTE-MFI data, it could be adapted to other experiments.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Improved galactic foreground removal for B-mode detection with clustering methods

ABSTRACT Characterizing the sub-mm Galactic emission has become increasingly critical especially in identifying and removing its polarized contribution from the one emitted by the cosmic microwave background (CMB). In this work, we present a parametric foreground removal performed on to sub-patches identified in the celestial sphere by means of spectral clustering. Our approach takes into account efficiently both the geometrical affinity and the similarity induced by the measurements and the accompanying errors. The optimal partition is then used to parametrically separate the Galactic emission encoding thermal dust and synchrotron from the CMB one applied on two nominal observations of forthcoming experiments from the ground and from the space. Moreover, the clustering is performed on tracers that are different from the data used for component separation, e.g. the spectral index maps of dust and synchrotron. Performing the parametric fit singularly on each of the clustering derived regions results in an overall improvement: both controlling the bias and the uncertainties in the CMB B-mode recovered maps. We finally apply this technique using the map of the number of clouds along the line of sight, $\mathcal {N}_c$, as estimated from H i emission data and perform parametric fitting on to patches derived by clustering on this map. We show that adopting the $\mathcal {N}_c$ map as a tracer for the patches related to the thermal dust emission, results in reducing the B-mode residuals post-component separation. The code is made publicly available https://github.com/giuspugl/fgcluster.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Signatures of Light Massive Relics on non-linear structure formation

ABSTRACT Cosmologies with Light Massive Relics (LiMRs) as a subdominant component of the dark sector are well-motivated from a particle physics perspective, and can also have implications for the σ8 tension between early and late time probes of clustering. The effects of LiMRs on the cosmic microwave background (CMB) and structure formation on large (linear) scales have been investigated extensively. In this paper, we initiate a systematic study of the effects of LiMRs on smaller, non-linear scales using cosmological N-body simulations; focusing on quantities relevant for photometric galaxy surveys. For most of our study, we use a particular model of non-thermal LiMRs but the methods developed generalizing to a large class of LiMR models – we explicitly demonstrate this by considering the Dodelson–Widrow velocity distribution. We find that, in general, the effects of LiMR on small scales are distinct from those of a ΛCDM universe, even when the value of σ8 is matched between the models. We show that weak lensing measurements around massive clusters, between ∼0.1 h−1Mpc and ∼10 h−1Mpc, should have sufficient signal-to-noise in future surveys to distinguish between ΛCDM and LiMR models that are tuned to fit both CMB data and linear scale clustering data at late times. Furthermore, we find that different LiMR cosmologies indistinguishable by conventional linear probes can be distinguished by non-linear probes if their velocity distributions are sufficiently different. LiMR models can, therefore, be best tested by jointly analyzing the CMB and late-time structure formation on both large and small scales.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Cosmological measurements from void-galaxy and galaxy-galaxy clustering in the Sloan Digital Sky Survey

We present the cosmological implications of measurements of void-galaxy and galaxy-galaxy clustering from the Sloan Digital Sky Survey (SDSS) Main Galaxy Sample (MGS), Baryon Oscillation Spectroscopic Survey (BOSS), and extended BOSS (eBOSS) luminous red galaxy catalogues from SDSS Data Release 7, 12, and 16, covering the redshift range 0.07 < $z$ < 1.0. We fit a standard ΛCDM cosmological model as well as various extensions, including a constant dark energy equation of state not equal to −1, a time-varying dark energy equation of state, and these same models allowing for spatial curvature. Results on key parameters of these models are reported for void-galaxy and galaxy-galaxy clustering alone, both of these combined, and all these combined with measurements from the cosmic microwave background (CMB) and supernovae (SN). For the combination of void-galaxy and galaxy-galaxy clustering, we find tight constraints of Ω m = 0.356 ± 0.024 for a base ΛCDM cosmology, Ω m = $0.391^{+0.028}_{-0.021}$, w = $-1.50^{+0.43}_{-0.28}$ additionally allowing the dark energy equation of state $w$ to vary, and Ω m = $0.331^{+0.067}_{-0.094}$, $w=-1.41^{+0.70}_{-0.31}$, and Ω k = $0.06^{+0.18}_{-0.13}$ further extending to non-flat models. The combined SDSS results from void-galaxy and galaxy-galaxy clustering in combination with CMB+SN provide a 30 per cent improvement in parameter Ω m over CMB+SN for ΛCDM, a 5 per cent improvement in parameter Ω m when $w$ is allowed to vary, and a 32 per cent and 68 per cent improvement in parameters Ω m and Ω k when allowing for spatial curvature.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Bias-hardened estimators of patchy screening profiles

Detecting anisotropic screening of the cosmic microwave background (CMB) holds the promise of revealing the distribution of gas in the Universe, characterizing the complex processes of galaxy formation and feedback, and studying the epoch of reionization. Estimators for inhomogeneous screening, including some recently proposed small-scale (stacked) estimators, are quadratic or higher order in the CMB temperature or polarization fields and are therefore subject to contamination from CMB lensing. We review the origin of this lensing bias and show that, when stacking on unWISE galaxies, the expected lensing bias dominates the signal if left unmitigated. Hardening techniques that null the lensing bias have been proposed for standard quadratic estimators, whereas only approximate methods have been proposed for stacked estimators. In conclusion, we review these techniques and apply the former to stacked estimators, presenting several strategies (including the optimal strategy) to null lensing contamination when stacking on any large-scale structure tracer.

Cosmic microwave background↗