The search for primordial gravitational waves in the Cosmic Microwave Background (CMB) will soon be limited by our ability to remove the lensing contamination to $B$-mode polarization. The often-used quadratic estimator for lensing is known to be suboptimal for surveys that are currently operating and will continue to become less and less efficient as instrumental noise decreases. While foregrounds can in principle be mitigated by observing in more frequency bands, progress in delensing hinges entirely on algorithmic advances. We demonstrate here a new inference method that solves this problem by sampling the exact Bayesian posterior of any desired cosmological parameters, of the gravitational lensing potential, and of the delensed CMB maps, given lensed temperature and polarization data. We validate the method using simulated CMB data with non-white noise and masking on up to 650\,deg$^2$ patches of sky. A unique strength of this approach is the ability to jointly estimate cosmological parameters which control both the primordial CMB and the lensing potential, which we demonstrate here for the first time by sampling both the tensor-to-scalar ratio, $r$, and the amplitude of the lensing potential, $A_ϕ$. The method allows us to perform the most precise check to-date of several important approximations underlying CMB-S4 $r$ forecasting, and we confirm these yield the correct expected uncertainty on $r$ to better than 10%.
We present a new method for delensing $B$ modes of the cosmic microwave background (CMB) using a lensing potential reconstructed from the same realization of the CMB polarization (CMB internal delensing). The $B$-mode delensing is required to improve sensitivity to primary $B$ modes generated by, e.g., the inflationary gravitational waves, axion-like particles, modified gravity, primordial magnetic fields, and topological defects such as cosmic strings. However, the CMB internal delensing suffers from substantial biases due to correlations between observed CMB maps to be delensed and that used…
One of the primary scientific targets of current and future CMB polarization experiments is the search for a stochastic background of gravity waves in the early universe. As instrumental sensitivity improves, the limiting factor will eventually be B-mode power generated by gravitational lensing, which can be removed through use of so-called delensing algorithms. We forecast prospects for delensing using lensing maps which are obtained externally to CMB polarization: either from large-scale structure observations, or from high-resolution maps of CMB temperature. We conclude that the forecasts i…
We study the impact of the nonlinear growth of the large-scale structure (LSS) on the removal of the gravitational lensing effect (delensing) in cosmic microwave background (CMB) $B$ modes. The importance of the nonlinear growth of the LSS in the gravitational lensing analysis of CMB has been recently recognized by several works, while its impact on delensing is not yet explored. The delensing using mass-tracers such as galaxies and cosmic infrared background (CIB) could be also affected by the nonlinear growth. We find that the nonlinear growth of the LSS leads to $\sim 0.3\%$ corrections to …
As confusion with lensing B-modes begins to limit experiments that search for primordial B-mode polarization, robust methods for delensing the CMB polarization sky are becoming increasingly important. We investigate in detail the possibility of delensing the CMB with the cosmic infrared background (CIB), emission from dusty star-forming galaxies that is an excellent tracer of the CMB lensing signal, in order to improve constraints on the tensor-to-scalar ratio $r$. We find that the maps of the CIB, such as current Planck satellite maps at 545 GHz, can be used to remove more than half of the le…
We present a first internal delensing of CMB maps, both in temperature and polarization, using the public foreground-cleaned (SMICA) Planck 2015 maps. After forming quadratic estimates of the lensing potential, we use the corresponding displacement field to undo the lensing on the same data. We build differences of the delensed spectra to the original data spectra specifically to look for delensing signatures. After taking into account reconstruction noise biases in the delensed spectra, we find an expected sharpening of the power spectrum acoustic peaks with a delensing efficiency of $29\,\%$…
Delensing, the removal of the limiting lensing B-mode background, is crucial for the success of future cosmic microwave background (CMB) surveys in constraining inflationary gravitational waves (IGWs). In recent work, delensing with large-scale structure tracers has emerged as a promising method both for improving constraints on IGWs and for testing delensing methods for future use. However, the delensing fractions (i.e., the fraction of the lensing-B mode power removed) achieved by recent efforts have been only $20-30\%$. In this work, we provide a detailed characterization of a full-sky, dus…
We present a method to delens the acoustic peaks of the CMB temperature and polarization power spectra internally, using lensing maps reconstructed from the CMB itself. We find that when delensing CMB acoustic peaks with a lensing potential map derived from the same CMB sky, a large bias arises in the delensed power spectrum. The cause of this bias is that the noise in the reconstructed potential map is derived from, and hence correlated with, the CMB map when delensing. This bias is more significant relative to the signal than an analogous bias found when delensing CMB B modes. We calculate t…
Next generation CMB experiments such as CMB-S4 aim at measuring the CMB lensing potential at sub-percent precision where most of the constraining power will come from CMB polarization. We investigate the prospects of achieving this goal in the presence of large-scale, diffuse galactic foreground emission by using non-Gaussian sky simulations and exploit multi-frequency information to clean those. We show that, while prior to foreground cleaning, cosmological parameter estimates from the contaminated lensing potential estimation can be significantly biased, these can be successfully mitigated b…
A detection of excess cosmic microwave background (CMB) B-mode polarization on large scales allows the possibility of measuring not only the amplitude of these fluctuations but also their scale dependence, which can be parametrized as the tensor tilt $n_T$. Measurements of this scale dependence will be hindered by the secondary B-mode polarization anisotropy induced by gravitational lensing. Fortunately, these contaminating B modes can be estimated and removed with a sufficiently good estimate of the intervening gravitational potential and a good map of CMB E-mode polarization. We present fore…
The next-generation CMB experiments are expected to constrain the tensor-to-scalar ratio $r$ with high precision. Delensing is an important process as the observed CMB $B$-mode polarization that contains the primordial tensor perturbation signal is dominated by a much larger contribution due to gravitational lensing. To do so successfully, it is useful to explore methods for lensing reconstruction beyond the traditional quadratic estimator (QE) (which will become suboptimal for the next-generation experiments), and the maximum a posterior estimator (which is still currently under development).…
We explore the potential use of the Radio Continuum (RC) survey conducted by the Square Kilometre Array (SKA) to remove (delens) the lensing-induced B-mode polarization and thus enhance future cosmic microwave background (CMB) searches for inflationary gravitational waves. Measurements of large-scale B-modes of the CMB are considered to be the best method for probing gravitational waves from the cosmic inflation. Future CMB experiments will, however, suffer from contamination by non-primordial B-modes, one source of which is the lensing B-modes. Delensing, therefore, will be required for furth…
Most of the CMB experiments proposed for the next generation aim to detect the Primordial Gravitational Wave Background (PGWB). The fulfillment of this objective depends on our capacity to separate Galactic foreground emissions and to \emph{delens} the secondary B-mode component induced by weak gravitational lensing. Focusing on the latter of these efforts, in this work we briefly review the basic aspects of lensing, and exhaustively compare the performance of current delensing methodologies and implementations within the Born approximation as a preparation for the analysis of the data to come…
The primordial B-modes component of the cosmic microwave background (CMB) polarization is a promising experimental dataset to probe the inflationary paradigm. B-modes are indeed a direct consequence of the presence of gravitational waves in the early universe. However, several secondary effects in the low redshift universe will produce \textit{non-primordial} B-modes. In particular, the gravitational interactions of CMB photons with large-scale structures will distort the primordial E-modes, adding a lensing B-mode component to the primordial signal. Removing the lensing component ("delensing"…
Efforts to detect a primordial $B$-mode of CMB polarization generated by inflationary gravitational waves ought to mitigate the large variance associated with the $B$-modes produced by gravitational lensing, a process known as delensing. A popular approach to delensing entails building a lensing $B$-mode template by mimicking the lensing operation, either at gradient order or non-perturbatively, using high-resolution $E$-mode observations and some proxy of the lensing potential. By explicitly calculating all contributions to two-loop order in lensing to the power spectrum of $B$-modes delensed…
Cosmic microwave background (CMB) experiments that constrain the tensor-to-scalar ratio $r$ are now approaching the sensitivity at which delensing---removing the $B$ modes induced by the gravitational lensing of large-scale structure---is necessary. We consider the improvement in delensing that maps of large-scale structure from tomographic line intensity mapping (IM) experiments targeting $2 < z < 10$ could provide. Compared to a nominal baseline of cosmic infrared background and internal delensing at CMB-S4 sensitivity, we find that the addition of high-redshift IM data could improve delensi…
We present a demonstration of delensing the observed cosmic microwave background (CMB) B-mode polarization anisotropy. This process of reducing the gravitational-lensing generated B-mode component will become increasingly important for improving searches for the B modes produced by primordial gravitational waves. In this work, we delens B-mode maps constructed from multi-frequency SPTpol observations of a 90 deg$^2$ patch of sky by subtracting a B-mode template constructed from two inputs: SPTpol E-mode maps and a lensing potential map estimated from the $\textit{Herschel}$ $500\,μm$ map of th…
Delensing is an increasingly important technique to reverse the gravitational lensing of the cosmic microwave background (CMB) and thus reveal primordial signals the lensing may obscure. We present a first demonstration of delensing on Planck temperature maps using the cosmic infrared background (CIB). Reversing the lensing deflections in Planck CMB temperature maps using a linear combination of the 545 and 857GHz maps as a lensing tracer, we find that the lensing effects in the temperature power spectrum are reduced in a manner consistent with theoretical expectations. In particular, the char…
We develop the first algorithm able to jointly compute the maximum {\it a posteriori} estimate of the Cosmic Microwave Background (CMB) temperature and polarization fields, the gravitational potential by which they are lensed, and cosmological parameters such as the tensor-to-scalar ratio, $r$. This is an important step towards sampling from the joint posterior probability function of these quantities, which, assuming Gaussianity of the CMB fields and lensing potential, contains all available cosmological information and would yield theoretically optimal constraints. Attaining such optimal con…
The effects of gravitational lensing of the cosmic microwave background (CMB) have been measured at high significance with existing data and will be measured even more precisely in future surveys. Reversing the effects of lensing on the observed CMB temperature and polarization maps provides a variety of benefits. Delensed CMB spectra have sharper acoustic peaks and more prominent damping tails, allowing for improved inferences of cosmological parameters that impact those features. Delensing reduces $B$-mode power, aiding the search for primordial gravitational waves and allowing for lower var…
The search for primordial $B$-mode polarization of the CMB is limited by the sample variance of $B$-modes produced at later times by gravitational lensing. Constraints can be improved by `delensing': using some proxy of the matter distribution to partially remove the lensing-induced $B$-modes. Current and soon-upcoming experiments will infer a matter map -- at least in part -- from the temperature anisotropies of the CMB. These reconstructions are contaminated by extragalactic foregrounds: radio-emitting galaxies, the cosmic infrared background, or the Sunyaev--Zel'dovich effects. Using the We…
The most promising avenue for detecting primordial gravitational waves from cosmic inflation is through measurements of degree-scale CMB $B$-mode polarisation. This approach must face the challenge posed by gravitational lensing of the CMB, which obscures the signal of interest. Fortunately, the lensing effects can be partially removed by combining high-resolution $E$-mode measurements with an estimate of the projected matter distribution. For near-future experiments, the best estimate of the latter will arise from co-adding internal reconstructions (derived from the CMB itself) with external …
Impacts of observational systematic errors on the lensing analysis of the cosmic microwave background (CMB) polarization are investigated by numerical simulations. We model errors of gain, angle, and pointing in observation of the CMB polarization and simulate polarization fields modulated by the errors. We discuss the response of systematics-induced $B$-modes to amplitude and spatial scale of the imposed errors and show that the results of the lensing reconstruction and delensing analysis behave according to it. It is observed that error levels expected in the near future lead to no significa…
We address a current caveat in the deconvolution of the Primordial Power Spectrum (PPS), from observed Cosmic Microwave Background (CMB) temperature anisotropy, in the presence of weak lensing of the CMB by the large scale structure (LSS) in the Universe. Richardson-Lucy (RL) deconvolution algorithm has been used in the context of reconstructing a free-form PPS, $P_R(k)$ from the observed lensed CMB temperature anisotropy power spectrum $\widetilde{C}_{\ell}^{TT}$. We propose and demonstrate that the RL algorithm works in the context of a non-linear convolution where the non-linear contributio…
Gravitational lensing by large-scale structure significantly impacts observations of the cosmic microwave background (CMB): it smooths the acoustic peaks in temperature and $E$-mode polarization power spectra, correlating previously uncorrelated modes; and it converts $E$-mode polarization into $B$-mode polarization. The act of measuring and removing the effect of lensing from CMB maps, or delensing, has been well studied in the context of $B$ modes, but little attention has been given to the delensing of the temperature and $E$ modes. In this paper, we model the expected delensed $T$ and $E$ …
Searches for the imprint of primordial gravitational waves in degree-scale CMB $B$-mode polarisation data must account for significant contamination from gravitational lensing. Fortunately, the lensing effects can be partially removed by combining high-resolution $E$-mode measurements with an estimate of the projected matter distribution. In the near future, experimental characteristics will be such that the latter can be reconstructed internally with high fidelity from the observed CMB, with the $EB$ quadratic estimator providing a large fraction of the signal-to-noise. It is a well-known phe…
A Fast Generative Framework for High-dimensional Posterior Sampling: Application to CMB Delensing
Department of Physics, Université de Montréal（蒙特利尔大学物理系）；Mila - Quebec AI Institute（魁北克人工智能研究院）；Ciela, Montreal Institute for Astrophysics and Machine Learning（蒙特利尔天体物理与机器学习研究所）；Center for Computational Astrophysics, Flatiron Institute（Flatiron研究所计算天文学中心）
We demonstrate how to obtain optimal constraints on a primordial gravitational wave component in lensed Cosmic Microwave Background (CMB) data under ideal conditions. We first derive an estimator of the tensor-to-scalar ratio, $r$, by using an error-controlled close approximation to the exact posterior, under the assumption of Gaussian primordial CMB and lensing deflection potential. This combines fast internal iterative lensing reconstruction with optimal recovery of the unlensed CMB. We evaluate its performance on simulated low-noise polarization data targeted at the recombination peak. We c…
One of the major targets for next-generation cosmic microwave background (CMB) experiments is the detection of the primordial B-mode signal. Planning is under way for Stage-IV experiments that are projected to have instrumental noise small enough to make lensing and foregrounds the dominant source of uncertainty for estimating the tensor-to-scalar ratio $r$ from polarization maps. This makes delensing a crucial part of future CMB polarization science. In this paper we present a likelihood method for estimating the tensor-to-scalar ratio $r$ from CMB polarization observations, which combines th…
The cosmic microwave background (CMB) represents a unique source for the study of gravitational lensing. It is extended across the entire sky, partially polarized, located at the extreme distance of z=1100, and is thought to have the simple, underlying statistics of a Gaussian random field. Here we review the weak lensing of the CMB, highlighting the aspects which differentiate it from the weak lensing of other sources, such as galaxies. We discuss the statistics of the lensing deflection field which remaps the CMB, and the corresponding effect on the power spectra. We then focus on methods fo…
We present a constraint on the tensor-to-scalar ratio, $r$, derived from measurements of cosmic microwave background (CMB) polarization $B$-modes with "delensing," whereby the uncertainty on $r$ contributed by the sample variance of the gravitational lensing $B$-modes is reduced by cross-correlating against a lensing $B$-mode template. This template is constructed by combining an estimate of the polarized CMB with a tracer of the projected large-scale structure. The large-scale-structure tracer used is a map of the cosmic infrared background derived from Planck satellite data, while the polari…
The primordial CMB $B$-mode search is on the spotlight of the scientific community due to the large amount of cosmological information that is encoded in the primeval signal. However, the detection of this signal is challenging from the data analysis point of view, due to the relative low amplitude compared to the foregrounds, the lensing contamination coming from the leakage of $E$-modes, and the instrumental noise. Here, we studied the viability of the detection of the primordial polarization $B$-mode with a ground-based telescope operating in the microwave low-frequency regime (i.e., from 1…
Gravitational lensing information from the two and higher point statistics of the CMB temperature and polarization fields are intrinsically correlated because they are lensed by the same realization of structure between last scattering and observation. Using an analytic model for lens sample covariance, we show that there is one mode, separately measurable in the lensed CMB power spectra and lensing reconstruction, that carries most of this correlation. Once these measurements become lens sample variance dominated, this mode should provide a useful consistency check between the observables tha…
Gravitational lensing of the CMB is a valuable cosmological signal that correlates to tracers of large-scale structure and acts as a important source of confusion for primordial $B$-mode polarization. State-of-the-art lensing reconstruction analyses use quadratic estimators, which are easily applicable to data. However, these estimators are known to be suboptimal, in particular for polarization, and large improvements are expected to be possible for high signal-to-noise polarization experiments. We develop a method and numerical code, $\rm{LensIt}$, that is able to find efficiently the most pr…
At linear order the only expected source of a curl-like, B mode, polarization pattern in the cosmic microwave background (CMB) is primordial gravitational waves. At second-order B modes are also produced from purely scalar, density, initial conditions. Unlike B modes from primordial gravitational waves, these B modes are expected to be non-Gaussian and not independent from the temperature and gradient-like polarization, E mode, CMB anisotropies. We find that the three point function between a second-order B mode and two first-order T/E modes is a powerful probe of second-order B modes and shou…
The next generation of ground-based CMB experiments aim to measure temperature and polarization fluctuations up to $\ell_{\rm max} \approx 5000$ over half of the sky. Combined with Planck data on large scales, this will provide improved constraints on primordial non-Gaussianity. However, the impressive resolution of these experiments will come at a price. Besides signal confusion from galactic foregrounds, extra-galactic foregrounds and late-time gravitational effects, gravitational lensing will introduce large non-Gaussianity that can become the leading contribution to the bispectrum covarian…
The Cosmic Microwave Background (CMB) radiation B mode polarization signal contains the unique signature of primordial metric perturbations produced during the inflation. The separation of the weak CMB B-mode signal from strong foreground contamination in observed maps is a complex task, and proposed new generation low noise satellite missions compete with the weak signal level of this gravitational background. In this article, for the first time, we employ a foreground model-independent internal linear combination (ILC) method to reconstruct the CMB B mode signal using simulated observations …
