Scientific/Technical/Management Section

I. MOTIVATION, OBJECTIVES, AND INTELLECTUAL MERIT

Cosmic microwave background (CMB) polarization measurements offer a unique possibility to reconstruct the physical conditions in the very early universe, and to test the application of fundamental physics laws toward Planck energy scales (10^{19} GeV) that are far beyond the reach of any laboratory tests or particle colliders like the Large Hadron Collider (LHC) at CERN with targeted energy scales around 14 TeV. In particular, the detection of a primary B-mode (curl-type) polarization and the determination of its source are major challenging tasks for the CMB community; see Ref. [1] for a recent study.

Primary gravitational wave (GW) detection through B-mode polarization [2–5] would be a direct proof of the amplification of quantum-mechanical fluctuations [6] during a very early inflationary [7, 8] epoch and therefore open a window on physics at ultra-high energy scales and the very earliest moments of the universe. In 2014, the BICEP2 collaboration announced a possible detection of relic GWs through the Bpolarization signal [9], but further investigation showed that this signal can be fully explained by galactic dust rather than being of cosmological origin [10–13]. The presence of foreground dust emission does not exclude the presence of a cosmological B-polarization signal, but, given the presence of B-modes from dust and galactic synchrotron emission, an unambiguous detection of a primordial signal will require multifrequency measurements and sophisticated modeling of foreground polarization signals;^{1} see e.g. [15–23].

Even in the absence of foregrounds, extracting the B-mode signal from a cut sky presents formidable challenges, especially in the current case when both the intensity (Stokes I) and the E-modes are significantly larger than the B-modes. As such, this proposal is focused on two main objectives:

• Understanding the nature of the dust polarization signal and properly account for synchrotron emission by applying both state-of-the-art simulations and careful comparisons with data products.

• Creating new techniques for extracting the faint B-mode signal in the realistic case where it competes with larger signals and noise.

One of the most intriguing features of Planck’s dust-polarization measurements is that the E-mode polarization exceeds the B-mode by almost a factor of two [24]. This was in contrast to the theoretical expectations that dust contributions to the E- and B- modes should be similar. The Planck collaboration suggested that the spectral E- and B- mode asymmetry, as well as the positive value of the temperature (T) cross correlation with E-polarization (T E), might be due to the alignment of the cold medium filamentary structure with the magnetic field [25]. This possibility has been quantified theoretically in Ref. [26]. However, the relevant mechanism that might produce this alignment is unclear. Further, analyzes of the latest version of the Planck data suggest that the hypothesis of the alignment between density and magnetic field structures cannot explain the T E cross correlation at low multipoles ℓ ( < 40 ) which characterize the dust polarization on angular scales ∼ 10^{∘}, larger than those typical of the interstellar filaments (although it remains plausible for ℓ > 40; see Ref. [27]). Moreover, recent analysis of the Planck data also shows a non-vanishing cross correlation between temperature and B-polarization (T B) ([20]). This is an important, yet unexplored, result, which suggests the presence of parity-violating processes in the interstellar medium (ISM).

There are therefore a number of mysteries surrounding the Planck dust maps, and we propose to address these with a combination of enhanced analysis techniques and simulations. Efforts to explain these puzzles have relied on studying the interaction of magnetohydrodynamic (MHD) turbulence with dust grains.

1 The B-polarization signal from thermal dust and synchrotron emission dominate over the primordial signal at high and low frequencies, respectively [14]

The standard model for calculating the dust polarization foreground is based on the effect of interstellar turbulence. In this case, dust grains align perpendicular to the magnetic field lines (see [28] and references therein). However, a recent study has pointed out that the observed ratio between E- and B-mode polarization spectra together with a positive T E cross correlation cannot normally be achieved within simple MHD turbulence models [26]. Therefore, more complex MHD turbulence models are called for. Interstellar turbulence can be magnetically dominated and strongly anisotropic toward small scales [29–31]. In that case, compressible MHD turbulence can in principle lead to the observed ratio between E- and B-mode spectra (EE/BB) [32]. In addition, it has been shown that interstellar turbulence can be sub-Alfvénic at high galactic latitudes, and density and magnetic field perturbations are weakly correlated [14]. On the other hand, the EE/BB ratio exhibits an almost scale-invariant nature. Hence, a successful turbulent model of galactic dust polarization maps should account for weak scale-invariance of the turbulence anisotropy. Another major obstacle to detect primary B-polarization mode is the galactic synchrotron emission [33].

Previously we have performed numerical simulations of compressible hydrodynamic and MHD turbulence in the early universe (in post-inflationary epochs, during cosmological phases transitions, until recombination) using the Pencil Code, adopting various initial conditions [34–45]. We will utilize and develop the ”interstellar” module of the Pencil Code, building upon our earlier work [46, 47] and that of others in the Pencil Code community [48, 49]. Recently we studied nonhelical turbulence and suggested that a strong non-vanishing skewness of the E-polarization is responsible for an enhanced ratio EE/BB in both helical and nonhelical cases [50, 51].

Understanding the properties of dust polarization is interesting in its own right, but also can serve as a tool for better component separation, see [52] for a recent study and references therein. Even for all-sky data coverage, the extraction of E- and B-mode dust polarization maps is a complex task: handling the contribution to the spectra of point sources or very bright objects, such as the galactic disk, is not trivial. The difficulty for statistical and machine learning separation methods is that there is a trade-off between a statistical model that describes point sources and the disk, which is necessarily complex, and being able to implement the algorithm at the scale of Planck data. Hence scalability and modelling are key challenges for statistical and machine learning approaches to source separation. The usual solution is to simply apply a mask to these sources and so be able to use a practical model. Thus, masks are usually applied to the full-sky data and E-to-B leakage may occur. This is why mask apodization is a key step in computing polarization power spectra both in small sky patches and in full sky maps. In practice the apodized masks are tested with simulated sky maps in order to validate the standard procedure [53]. In the case of finite cut-sky CMB maps, the extraction of the pure E- and pure B-mode signals is even more complex [54, 55], and different methods have been proposed [56–60]. In our program we propose to use machine learning (ML) methods to extract the pure E and B modes being motivated to see if ML can improve component separation.

Understanding the detailed properties of foreground dust and synchrotron emission polarization is important to be able to separate dust from the CMB, which has different statistical properties. Assuming that the galactic foreground with its highly non-gaussian statistics is strongest at low galactic latitudes, we can use advanced ML tools to separate it from the CMB with its gaussian statistical properties. Thus, to facilitate this approach, a detailed understanding of the properties of dust polarization is essential.

As another example of the importance of understanding the properties of dust, usually parity-odd cross correlations T B and EB are a priori assumed to be zero and used to calibrate detectors [61, 62]. We aim to investigate if this calibration can be justified when accounting for dust polarization.

The proposed research is multidisciplinary, involving data analysis and ML, physics of the ISM and MHD turbulence, and the CMB. The obtained results will improve ISM turbulence modeling, CMB sky component separation techniques, and the results will advance our understanding of galactic microwave polarization and potentially will lead to critical developments in extracting a primary B-mode signal.

Below we summarize the major goals of our proposed research program.

II. BRIEF SUMMARY OF THE PROPOSED RESEARCH PROGRAM

We propose to study the following issues:

• Interstellar dust is a key component of the ISM to probe the interstellar gas mass, [63]. This is important both for astrophysical processes in the local Universe, namely for star formation activity in the Galaxy, and for accurately probing the cosmic times, as the ISM is an unavoidable filter for primordial lights coming to us. In the context of studying primordial GWs and their imprint on the CMB, dust polarized emission is today becoming a main focus. The amount of polarization coming from interstellar dust however is still poorly constrained. Dust polarization depends (i) on intrinsic dust properties, such as dust chemical composition, shape, and emissivity; (ii) on the statistical alignment of the dust grains with the local interstellar magnetic field; and (iii) on the morphology of the galactic magnetic field. Since the net polarization is the combination of all such effects, modeling dust polarization is a complex issue with a high degree of degeneration. Nevertheless, recent observational data will allow us to further explore dust properties in total and polarized emissions and to test the latest state-of-the-art models [64, 65]

• Because of the degenerate problem described above, we propose to tackle the dust polarization modeling from two different perspectives. First of all, we will focus on investigating the role of interstellar turbulence, and its impact on the density and magnetic field structures, and, in turn, their effect on the observed polarization. Assuming uniform intrinsic dust properties in the ISM [66], we propose to perform large eddy and direct numerical simulations of MHD turbulence using a variety of physical conditions, including supernova-driven turbulence in the interstellar medium as well as artificially forced turbulence and free decay cases. We consider isotropic and anisotropic turbulence, and compare theoretical polarization spectra with Planck dust polarization ones. Second, we will attempt to add dust properties using the Polaris code [67], to disentangle the effects related to the magnetic field structure from those of dust grains on the observed polarization.

• Given the knowledge acquired through the previous goals, we propose to elaborate a recipe to properly remove the dust and synchrotron foreground in order to study new physics related to the CMB B-mode polarization. We use advanced ML algorithms to separate contributions from CMB and dust. Our approach is based on their systematically different statistical properties. We also propose to use an ML approach to decompose CMB polarization maps in order to extract pure E- and pure B- mode signals. As explained further below, we will also apply these methods to other data sets where we know the answer. This will allow us to test the ML technique, and it enables us to see if the separation can be improved. As described below, our ML approach based on an Independent Component Analysis (ICA) method developed by our group is superior to previous ML approaches used in this context.

• An important application of our work is to determine the influence of magnetic and kinetic helicity on the polarization signature, and to find out if helicity can be detected in parity-odd cross correlations between E- and B-mode polarization. We will therefore use advanced tools to analyze the cross correlation between E- and B-modes (EB) and compare with models and other observational data such as from the solar surface [51], where we can measure magnetic helicity also directly from the inferred magnetic field [68–70]. To isolate different physical effects, we apply the tool of analyzing E- and B-mode polarizations in other circumstances such as scattering emission of polarized light from the magnetized solar surface. These applications to solar data have the advantage that we have independent means of determining the underlying magnetic field, allowing us to validate our approach. This argument also applies to simulation data discussed below.

• One of our approaches to source separation uses a Bayesian algorithm with an emphasis on providing reasonable and principled measures of uncertainty around the estimates of the polarization sources such as dust and CMB. This gives important information on how secure we can be as regards our theory for the causes of the polarization, based on the data that we currently have, and can provide input into what new missions should target. The algorithm that first appeared in [71] will be used, modified from its initial use for temperature map separation and taking account of other known properties of the sources.

III. THE RELEVANCE OF THE PROPOSED WORK TO NASA OBJECTIVES AND GOALS

Some of the research will use Planck (and WMAP) data in novel ways and help to understand better the galactic microwave polarization and to remove foreground effects when analyzing B-polarization data. As a result, our investigation aims to extract more information about and provides constraints on fundamental physics. The research proposed here will also help to guide the development of future CMB polarization missions.

The objectives and goals of our proposal are directly related to the NASA Science Goal “Discover how the universe works, explore how the universe began and evolved”, and objectives “Understand the origin and destiny of the universe, and understand the many phenomena and processes associated with galaxy, stellar, and planetary system formation and evolution from the earliest epochs to today”. The science questions addressed in this proposal are central questions of the Physics of the Cosmos program.

IV. METHODOLOGY AND APPROACH

In this section we review previous studies to model galactic microwave polarization (including ISM turbulence) and data analysis tools that we aim to use.

A. Dust: the Link between Magnetic Fields and Polarization

Observations of dust polarized emission are today one of the most efficient indirect probes of magnetic fields in the ISM. Here we provide a brief description of the main mechanisms that link dust polarization to the magnetized ISM. We begin by summarizing the standard view for how the magnetic field impacts the polarization of radiation. It is observed that starlight is polarized, and the more extinction along the line of sight, the more polarized it is, with maximum polarization in the visual roughly equal to p_{max} = 0.03A_{V}, where A_{V} is the extinction coefficient in V-band. The orientation of the polarization tends to follow the magnetic field of the Galaxy, so this suggests that dust is influenced by the magnetic field and in turn impacts the polarization of radiation. Interstellar dust permeates the ISM. Large dust grains (from μm to mm sizes) dominate the visible extinction and the far-infrared/sub-mm domains of the galactic electromagnetic spectrum [72]. Since these particles are typically not spherical, both starlight extinction and dust thermal emission consist of anisotropic radiation that is linearly polarized. Magnetic fields in the ISM are responsible for grain alignment [73]. Large grains in the ISM spin around their axis of maximal inertia (their shortest axis) at superthermal velocities due to torques produced by collisions with atoms and molecules, ejection of H_{2} molecules formed on their surfaces, or radiation field (radiative torques, see Ref. [74]). Since dust grains are generally charged particles, their rotation contributes to their total magnetic moment M. Thus, in the presence of an external magnetic field B, a magnetic torque M × B is exerted on the grains. If the angular momentum J of the grains is not aligned with the field, the torque forces the grains to precess around the field lines. Paramagnetic relaxation [75], or radiative torques [76], yield then a global alignment of the grains with the ambient magnetic field by dissipating the kinetic energy associated with the precession motion. The amount of observed polarization is therefore tightly linked to the degree of alignment of the grains with the field. Given a certain degree of grain alignment, which is however still poorly constrained by observations ([64] and references therein), one may consider dust grains as compasses in the ISM, so that, by observing dust polarized emission, it is possible to infer the orientation of the magnetic field in the ISM. In particular, the observed polarization from dust is maximal when the magnetic field is perpendicular to the line of sight, and minimal when the magnetic field is pointing toward the observer. Hence, dust polarization represents a suitable tracer for the orientation of the magnetic field projected on the plane of the sky.

B. Modeling ISM Turbulence

The ISM is multiphased, consisting of hot, warm, and cold phases, which are being segregated by heating through supernova driving and the temperature-dependence of the cooling function. Such simulations will be performed in the shearing box approximation to incorporate the effects of galactic shear, which leads to a strong toroidal component of the magnetic field, [46]. They incorporate density stratification and the different statistics of supernovae I and II, which have different rates and scale heights. This was first done in our work of nearly 20 years ago [47]. On smaller length scales, it is customary to ignore shear and density stratification, and focus in highly supersonic turbulence that is just driven by volume forcing [46–49]. This is also the case in the recent work of [77].

To make contact with observations, we integrate the intrinsic polarization along the line of sight. This has been done extensively also in other applications, including also a number of cases studied in the Pencil Code community (for example see Refs. [78, 79]).

To characterize dust polarization, two are the main observables: the polarization fraction p and the polarization angle ψ. The former carries information about the fraction of detected intensity that is polarized compared to the total intensity, and it depends on dust properties (composition and grain alignment) and on the geometry of the field with respect to the observer. The polarization angle, instead, allows us to infer the orientation of the magnetic field projected on the plane of the sky. In practice, polarization observations provide the three Stokes parameters I, Q, and U of linear polarization. While the Stokes I parameter represents the total intensity of the incoming radiation, and is defined as a positive quantity, Stokes Q and U include the geometry of the polarization plane and can assume both positive and negative values. The two aforementioned observables are ψ = \frac{1}{2} arctan (U/Q). Since sub-mm dust thermal emission is optically thin, these quantities are representative of the whole line of sight. For dust polarization one can write:

I = ∫ S_{ν} e^{−τ_{ν}} [1 − p_{dust}R (cos^{2}γ − \frac{2}{3})] dτ_{ν}, Q + iU = ∫ p_{dust}R S_{ν} e^{−τ_{ν}+2iψ} cos^{2}γ dτ_{ν}, (1)

where S_{ν} is the source function of dust emission (usually considered as a modified black-body spectrum), τ_{ν} is the dust optical depth, γ is the angle between the magnetic field and the plane of the sky, p_{dust} is the intrinsic polarization fraction of dust (corresponding to the ratio C_{pol}/C_{avg}, where C_{pol} and C_{avg} are the polarization and the average cross-sections of dust, as defined in Ref. [66]), and R is the Rayleigh reduction factor, which characterizes the degree of dust grain alignment with the local magnetic field [80, 81].

C. E and B-Modes Polarization Analysis

As detailed above, linear polarization is measured in terms of the two Stokes parameters Q and U in the plane of the sky. For our purpose, however, in order to statistically characterize the polarization sky and compare it with numerical simulations, it is useful to introduce two rotational invariants, which can be directly derived from the Stokes parameters. In the simulations, on the other hand, we extract the magnetic field components perpendicular to the plane of the sky, b_{x} and b_{y}, say. We then compute the linear polarization as complex polarization P = Q + iU = (b_{y} − ib_{x})^{2}, which assumes that the electric field is 90^{∘} rotated relative to the magnetic field. In both observations and simulations, we compute the E and B polarizations in Fourier space, denoted by tildes, as functions of the wavevector (k_{x}, k_{y}) as Ẽ + iB̃ = (k_{x} − ik_{y})^{2}P̃. This approach is valid for short enough separations in the sky. Next, we compute spectra, averaged over concentric shells in k space, for example C_{EB}(k) = Re⟨Ẽ(k)B̃^{∗}(k)⟩_{ϕ}. As said in the beginning, the ratio C_{EE}(k)/C_{BB}(k) is found to be around 2 in interstellar dust polarization and the correlation C_{EB}(k) might reflect helicity. We prefer to characterize these in terms of quantities that are between −1 and +1, namely

c_{S}(k) ≡ [C_{EE}(k) − C_{BB}(k)]/[C_{EE}(k) + C_{BB}(k)], c_{A}(k) ≡ 2C_{EB}(k)/[C_{EE}(k) + C_{BB}(k)]. (2)

Thus, C_{EE}(k)/C_{BB}(k) = 2 corresponds to c_{S} = 1/3, and c_{A}(k) characterizes parity breaking.

In the solar context, there is a large-scale modulation across the solar disk that affects the sign of the helicity between the northern and southern hemispheres [70], or, in the context of cross helicity, over the scale of sunspot pairs [82]. or over the scale of sunspot pairs. In both cases, we are able to account for this variation characterized by the wavevector K using the two-scale method [70, 82, 83] by computing C_{EB}(k; K) = −Im⟨Ẽ(k + K/2)B̃^{∗}(k − K/2)⟩_{ϕ}. We have already extended this method to measuring c_{A}(k) by taking the north/south antisymmetry into account; see below.

D. Machine Learning – Component Separation

The observations from the Planck dataset contain a convoluted mixture of various sources including CMB, galactic synchrotron, dust emissions, Sunyaev-Zeldovich effect in clusters, etc. The pixel intensity distribution of the foreground source objects are expected to be non-gaussian, while the CMB is expected to be close to gaussian. In this task our goal is to separate the CMB component from the other components. We are not the first trying to accomplish this goal; there have been numerous attempts proposed to this problems with varying success. Below we briefly summarize some of these methods, and propose an alternate method that has the potential to achieve better results.

a. Internal Linear Combination. One of the simplest component separation method is the Internal Linear Combination (ILC) [84]. In this setting, for any available directions ξ in the sky, M channels of the Planck dataset at frequencies ν_{1}, … ν_{M} provide an M-dimensional observational data vector Y (ξ) = [y_{1}(ξ), …, y_{M}(ξ)]^{T} ∈ ℝ^{M}. The ILC method assumes that the template of emission of the component of interest is the same at all frequencies of observation and for any available directions ξ in the sky, and for each frequency channel i = 1, …, M the following equation holds:

y_{i}(ξ) = s(ξ) + f_{i}(ξ) + n_{i}(ξ),

where f_{i}(ξ) and n_{i}(ξ) are foregrounds and noise contributions respectively in channel i. Our goal is to estimate the CMB component s(ξ) from the observations y_{i}(ξ). To accomplish this goal, ILC defines an estimator, ŝ(ξ) = ∑_{i=1}^{M} w_{i}y_{i}(ξ), and the weights w_{i} are chosen such that ∑_{i=1}^{M} w_{i} = 1 and the variance of the estimated component ŝ(ξ) is minimal.

b. Linear Mixture Model. The ILC approach makes very strong assumptions and hence the quality of the results it can provide is quite limited. A more general model for component separation is the Linear Mixture Model. This model assumes that for each frequency channel i the observations are the results of a linear mixture of several sources s_{j} [85]:

y_{i}(ξ) = ∑_{j=1}^{K} A_{ij}s_{j}(ξ) + n_{i}(ξ), (i = 1, …, M).  (3)

Here each entry s_{j}(ξ) contains a sky-pattern for the emission of a given component at sky location ξ. There have been numerous prior work on this problem [84–90]. Some methods assume that the mixing matrix A ∈ ℝ^{M×K} is known, and some try to estimate this matrix from the observations Y (ξ). These methods usually are based on Wiener filtering, wavelet analysis, or some kind of algorithms for solving the Independent Component Analysis (ICA) problem. The ICA based methods also assume that the components s_{1}, s_{2}, …, s_{K} are statistically independent. They either use simulated data or older released versions of the Planck data.

c. Bayesian ICA Model. Given the large uncertainties that most likely will exist in the estimation of the B-mode and its interactions with other modes, and the availability of a large amount of a priori information about it properties, a Bayesian learning approach has some attractions. From a model that originally appeared in [71], a Bayesian source separation algorithm will be modified so that it can be applied to polarization data. This makes use of the linear mixture model, but with a set of options for modelling prior properties of the sources such as their distribution (Gaussian vs. non-Gaussian) and degree of spatial smoothness. For example, in [91], dependencies between sources were modelled. Spatial smoothness can be modelled by an intrinsic Gaussian prior that constrains neighboring pixels in a source map to differ by a zero-mean Gaussian:

s_{j}(ξ_{k}) − s_{j}(ξ_{k′}) ∼ N(0, τ_{j}),

for ’neighboring’ directions in the pixel discretization ξ_{k} and ξ_{k′}. The Bayesian approach’s strength is in correctly propagating the uncertainties due to measurement error, and the uncertainty in other information that we have, to give a principled estimate of the uncertainty of the B-mode estimate given the information that we have, through the posterior distribution. Implementation of the inference at the scale of Planck data is certainly an issue and to this end a method based on the INLA approach [92] has been used.

V. PROPOSED RESEARCH PROGRAM

A. Dust Polarization and ISM Turbulence Modeling

Our aim is to statistically compare observational data with synthetic data produced with numerical simulations (see next sections). Thus, the data analysis will consist in applying the same methodology to the two datasets. With respect to the observational data, we will make use of the latest full-sky observations made with the Planck satellite. The Planck data consist of maps in intensity and polarization between 30 GHz and 353 GHz [93], allowing us to constrain the CMB polarization after accounting for galactic synchrotron and dust and synchrotron foreground emissions. We will mainly focus on the latter two, which is today the least characterized foreground signal [14, 33]. In order to compare observations and simulations, we will develop ML algorithms (see next section) and use standard approaches with statistical tools already in hands, such as power spectra both in intensity and in polarization (i.e., Ref. [24]). Since we have access to the whole sky, we will be able to perform both statistics on the sphere and on cut-out images of the full sky, allowing us to study angular scales larger than a few degrees, which are those interesting for the detected E-B modes signals in the Planck data (i.e., see Figs. 2–5 of Ref. [20]). Before proceeding with the comparison between observations and numerical simulations, both at the power spectrum level and at the map level, we will test our algorithms using toy models of the polarized sky [19], in order to validate our methodology without the complexity of the real data.

1. Turbulence Simulations

Astrophysical turbulence comes in a multitude of different forms: it can be helical or nonhelical, it can be magnetically dominated or subdominant, it can possess cross helicity, with a systematic alignment

FIG. 1: E-mode (left) and B-mode (right) polarization for isotropic fully helical magnetohydrodynamic turbulence using data from [40] (their Fig. 4d–f). Dark (light) shades indicate negative (positive) velocity. The right panel shows that c_{S}(k) exceeds 1/3, so the ratio of EE to BB exceeds the factor of 2 as k increases. Adapted from Ref. [51].

or anti-alignment between magnetic field and velocity. These turbulence simulations provide a means of performing experiments in a variety of circumstances and environments. This allows us to isolate specific physical effects or even just the mathematical terms in the underlying equations and associate them with distinct features in the synthesized diagnostics which, in this case, are correlations in the E- and B-modes; see Fig. 1. Of particular interest for the present proposal is our ability to consider turbulence that can be strongly anisotropic, with multiple preferred directions, and lacking statistically mirror symmetry, i.e., being helical.

In the context of early universe turbulence, we have studied decaying MHD turbulence which is magnetically dominated, i.e., the magnetic energy exceeds the kinetic energy by typically a factor of ten [43]. The turbulence is then mostly driven by the Lorentz force. The resulting E- and B-mode polarizations are shown in Fig. 1 for a particular example. We find that c_{S}(k) > 1/3 for large k, i.e., C_{EE}/C_{BB} > 2; see the right-most panel of Fig. 1. Our recent work [51] now suggests that for both dust and synchrotron emission, the probability density function of the E polarization is strongly skewed, which contributes to an enhanced C_{EE}/C_{BB} ratio. This is comparable with the observed foreground polarization detected with Planck [20, 24], clearly more than what is theoretically expect [26] and may be explicable with Alfvén waves [14].

It is interesting to note that, even though the turbulence is nearly fully helical with a fractional helicity of about 98%, the EB correlation, C_{EB}(k), is actually zero. In hindsight, this is not surprising because the parity-odd polarization cross correlation EB characterizes the shape of two-dimensional structures on a surface, e.g., it can distinguish between the letters p and q, which are mirror images of one another. In three dimensions, however we can flip any structure and view it from the backside.

There can be certain physical circumstance, however, that give preference of one over the other side of a plane. The CMB may be one such example. Rotating stratified convection is another quite intuitive example; see Fig. 2 for the results of such a simulation. There one sees cyclonic convection in the northern hemisphere viewed from the top, so all converging inflows attain a counterclockwise swirl, and so do all diverging outflows. It is only when inspecting this pattern from beneath that we will see a mirror image of the original pattern and therefore the opposite sign of the B polarization. This can be seen in Fig. 3, where we plot c_{S}(k), c_{A}(k), and c_{S} versus c_{A} for north (red) and south (blue) in a parametric representation.

FIG. 2: Magnetic field vectors and line-of-sight component (color-coded; left) as well as E-mode (middle) and Bmode polarization (red) in rotating convection viewed from the top onto the convecting layer. Adapted from Ref. [51].

FIG. 3: The functions c_{A}(k) and c_{S}(k) for rotating convection against k for two runs A and B. Red (blue) lines and symbols are for north (south), demonstrating opposite signs of c_{A}(k), but the same sign of c_{S}(k). Adapted from Ref. [51].

There is now a systematic EB correlation, so c_{A}(k) is positive in the north and negative in the south; see Refs. Brandenburg:2018khp,Bracco:2018vyi for our recent work on this. This is very promising and agrees with our intuition. Surprisingly, however, c_{S}(k) is now negative, so the BB correlation now exceeds the EE correlation. This can be explained by a strong rotational influence, which leads to a strong parityinvariant signal and therefore to a large BB correlation. In addition, the magnetic field is here subdominant, in contrast to the situation shown in Fig. 1.

2. Simulations of the Interstellar Medium

Using shock diffusivities in the Pencil Code, we can run high Mach numbers of around ten, as was done in [94], or around 25, as done in [48]. Turbulence will be driven by supernova explosions, as done in [48, 49]. We use a parameterized cooling function, as done in [46]. We will simulate a domain of size 1 × 1 × 2 kpc^{3} over about one Gyr, to accumulate sufficient statistics.

B. ML Algorithms for Component Separation

We will apply ICA to separate signals of different origins in sky maps. i) Instead of the previously used ICA methods, we propose to use more robust ICA algorithms that are more tolerant to noise and outliers. ii) The previously used ICA based methods assumed linear mixing of the hidden components. We will investigate the application of nonlinear ICA methods that are capable of dealing with nonlinear mixing instead of linear. iii) The previously used component separation methods used older versions of the Planck dataset. We will demonstrate the applicability and effectiveness of these methods on the new Planck data.

1. Robust ICA

Using vector and matrix notation, the above described Linear Mixture Model of Eq. (3) can be formulated as Y = AS + N, where Y = [y_{1}, …, y_{M}] ∈ ℝ^{M} is our observation, A ∈ ℝ^{M×K} is the mixing matrix, S ∈ ℝ^{K} denotes the hidden components we want to estimate, and N ∈ ℝ^{M} is a noise vector. We assume that the coordinates of S(ξ), that is s_{1}(ξ), …, s_{K}(ξ) are statistically independent, and therefore their mutual information is zero.

Most ICA algorithms seek for the solution in the form of Ŝ = W Y, where W ∈ ℝ^{K×M}. Since according to our assumptions, the coordinates of S(ξ) are statistically independent, a natural cost function is to minimize their mutual information (MI) of the coordinates of Ŝ: min_{W} I(ŝ_{1}, …, ŝ_{K}). The quality of the solution thus depends on how accurate is the corresponding MI estimation. The problem with using MI is that without a parametric assumption on the functional form of the joint distribution, MI cannot be evaluated exactly, and numerical estimation can be both inaccurate and computationally expensive. Estimating the mutual information is usually quite challenging, therefore many approximations have been proposed. The FastICA algorithm [95, 96] for example, uses kurtosis or other higher order moments to achieve this goal.

Higher order moment estimation, however, can be very sensitive to noise and outliers. As an extreme case, for example, one simple large outlier can completely ruin the statistics for mean, variance, or other higher-order moment estimation. Instead, we will use ICA methods developed by our team that are more robust to noise and outliers [97].

A key component our method is to use rank statistics and the copula of the multivariate distribution S(ξ) ∈ ℝ^{K} to estimate the statistical dependence of its coordinates. For simplicity, we explain the main ideas with two statistically dependent one-dimensional random variables X and Y, but the method can be naturally generalized to higher-dimensions. Let (X, Y) ∈ ℝ^{2} be distributed according to a bivariate probability distribution P. Assume we are given N samples of (X, Y), D = {(x_{1}, y_{1}), …, (x_{N}, y_{N})}. Let the rank r_{x}(x) be the number of x_{i}, i = 1, …, N such that x > x_{i}, and let r_{y}(y) be defined similarly.

Many nonlinear dependence measures are based on ranks. Among most commonly used are Kendall’s τ and Spearman’s ρ rank correlation coefficients. Both τ and ρ have a range of [−1, 1] and are equal to 0 if X and Y are independent. However, the converse is not true; both τ and ρ can be 0 even if X and Y are not independent. Therefore, while they are robust to outliers, neither ρ nor τ is a good ICA objective function. Rank correlations can be extended from samples to distributions with the help of copulas, distributions over continuous multivariate ranks. Below we will propose an effective robust contrast for ICA using a measure of dependence for copulas.

A bivariate copula C is a probability function (cdf) defined on a unit square, C : [0, 1] × [0, 1] → [0, 1] such that its univariate marginals are uniform, i.e., C(u, 1) = u, C(1, v) = v, ∀u, v, ∈ [0, 1]. Let U = P_{x}(X) and V = P_{y}(Y) denote the corresponding cdfs for the previously defined random variables X and Y. Then, for (u, v) ∈ [0, 1] × [0, 1], define C as C(u, v) = P (P_{x}^{−1}(u), P_{y}^{−1}(v)). It is easy to verify that C is a copula. Sklar’s theorem [98] states that such copula exists for any distribution P, and that it is unique on the range of values of the marginal distributions. Copulas can also be viewed as a canonical form of multivariate distributions as they preserve multivariate dependence properties of the corresponding families of distribution. Therefore, if we want to measure the dependence between two random variables X and Y, it is enough to measure the dependence between the marginal variables of their copula distributions.

As the true distribution P and its copula C are not known, rank statistics can be used to estimate the copula with the so-called empirical copula [99]. For a data set {(x_{1}, y_{1}), …, (x_{N}, y_{N})}, an empirical copula C_{N} is given by

C_{N} (\frac{i}{N} , \frac{j}{N} ) = \frac{# of (x_{k}, y_{k}) s.t. x_{k} ≤ x_{i} and y_{k} ≤ y_{j}}{N} .

As we can see from this expression, the copula distribution can be estimated with rank statistics only, and hence it is robust to outliers. Well-known sample versions of several nonlinear dependence measures, such as Kendall’s τ and Spearman’s ρ, can be obtained using the empirical copula (e.g., [100]). As we discussed, part of the problem with Kendall’s τ and Spearman’s ρ as a dependence measure for ICA is a property that their value may be 0 even though the corresponding variables X and Y are not independent. Instead, we suggest using Schweizer-Wolff σ, a measure of dependence between two continuous random variables [101]:

σ = 12 ∫_{[0,1]^{2}} |C(u, v) − uv|dudv.

Here σ can be viewed as an L_{1} norm between a copula for the distribution and a product of uniform independent variables. It has a range of [0, 1], with an important property that σ = 0 if and only if the corresponding variables are mutually independent. A sample version of σ can be derived as:

σ̂ = \frac{12}{N^{2} − 1} ∑_{i=1}^{N} ∑_{j=1}^{N} |C_{N} (\frac{i}{N} , \frac{j}{N} ) − \frac{i}{N} × \frac{j}{N} |  (4)

To use this dependence measure in an ICA algorithm, we just simply have to minimize the pairwise dependencies measured by the Schweizer-Wolff dependence criterion (4). We call this ICA method Schweizer-Wolff ICA (SWICA). We note that other measures of dependence can also be potentially used as an ICA contrast. We will investigate the L_{∞} version of σ too: 4 sup_{[0,1]^{2}} |C(u, v) − uv|, a dependence measure similar to the Kolmogorov-Smirnov univariate statistic ([101]).

As our preliminary results indicate (see below), these dependence measures can be used to derive new efficient ICA methods. During the dependence estimation, they only use rank statistics, and hence they are robust to outliers.

2. Post-nonlinear ICA

The previous methods in the Linear Mixture Model, Y (ξ) = AS(ξ) + N(ξ), assumed that the observations Y (ξ) are results of linear mixtures of statistically independent hidden components. Using nonlinear ICA methods developed by our group [102] and others, we propose to investigate the more general case, when this mixing can be still unknown but nonlinear. In particular, efficient algorithms exist for the postnonlinear mixing case when

y_{i}(ξ) = f ∑_{j=1}^{K}(A_{ij}s_{j}(ξ)) + n_{i}(ξ), (i = 1, …, M).

Here f is an unknown monotone function, and A is an unknown mixing matrix. Our goal is to estimate the unknown independent components (s_{j}(ξ)) from the observations (y_{i}(ξ)).

3. E- and B-Modes Separation with Convolutional Neural Networks

Recently, Isola et al. [103] proposed a new neural network architecture based on Conditional Generative Adversarial Neural Networks. This architecture is capable of learning a map from a wide range of input images to output images. Hence, the name of this network is Pix2Pix. As particular applications, it has been demonstrated that Pix2Pix can be used to colorize grayscale images, remove background from pictures, predict Aerial photos from their maps, or predict the original images from their sketched versions. The algorithm assumes that we have a set of input-output pair images and this training set can be used to learn the map between the input and output images.

Our goal in this task is to investigate how well these Pix2Pix neural networks (or their modifications) can be used to learn a map between an image patch of the Planck data and its E and B components. We assume that a dataset of simulated images with their ground truth E and B components are available to us for training the neural networks. The only difference between our setting and Pix2Pix is that in our task the output consists of two images, the E and B components, instead of one single component. We are planning to use both real and simulated datasets in this task. Neural networks usually require large amounts of data, but as it has been reported by [103] in some application Pix2Pix can provide satisfactory results with using only 400 images. We will have access to much larger real datasets and we can also generate large amounts of simulated data that can be also used for training these neural networks. Recently we have successfully used Generative Adversarial Neural Networks for generating realistic looking fake galaxy images [104, 105].

4. Preliminary Results: Robustness of Component Separation

FIG. 4: Box plot of Amari errors (multiplied by 100) for the mixed sounds with outliers. Plot was computed over 100 experiments. Here on a synthetic example we demonstrate the robustness of our component separation method to outliers. We mixed N = 1000 samples from 8 sound pieces of an ICA benchmark

[106] mixing matrix. Then we added 20 outliers to this mixture and demixed them using SWICA and 4 state-of-the-art ICA algorithms, [95, 107–109]. Figure 4 shows that SWICA outperforms its competitor methods on this task. The separation error is measured by the Amari-error [96]. The closer this number is to zero, the better the separation is.

As a second preliminary experiment to demonstrate the robustness of SWICA, we mixed 4 natural images of size 128 × 256. The pixel intensities we normalized in the [0, 255] interval. Each image was considered as a realization of a stochastic variable with 32768 sample points. We mixed these 4 images by a 4 × 4 random mixing matrix, resulting in a matrix of size 4 × 32768. Then we added large outliers to 3% randomly selected points of this mixture, and then estimated the original components. Figure 5 shows that SWICA outperforms its competitor methods on this task. In this project we will use SWICA for separating the hidden components of Planck data.

VI. GENERAL PLAN OF WORK

As highlighted above, our research program consists of two major parts: modeling of dust polarization based on MHD simulations and CMB component separation.

The proposed research requires a range of technical expertise including aspects of CMB and ISM physics, dust polarization and MHD turbulence modeling, data analysis, and ML. The proposers have complementary

FIG. 5: Separation of outlier-corrupted mixed images. (a) The original images. (b) the mixed images corrupted with outliers. (c)-(e) The separated images using SWICA, FastICA, and RADICAL algorithms, respectively. The Amari error of the SWICA, FastICA, Radical was 0.10, 0.30, 0.29 respectively. The quality of the KernelICA and JADE was similar to that of FastICA and RADICAL.

research expertise and have done fundamental work in all of the relevant areas of this proposal. In particular,

• Brandenburg has led the development of the Pencil Code, a public domain tool box (https: //github.com/pencil-code) that will also be used in the present work. Drawing from his experience with supernova-driven turbulence in the interstellar medium [47], he will perform new simulations of interstellar turbulence. Having previously analyzed synchrotron emission from helical magnetic fields [110], he will use this expertise to compute the dust polarization signal from those simulations. He has analyzed data from NASA’s Solar Dynamics Observatory [70] and the SOLIS instrument [82] and will utilize this experience to analyze the E- and B-mode polarization of solar data.

• Dodelson has broad experience analyzing data from the South Pole Telescope [111–117]. Accounting for foregrounds has been a major effort in much of that work. Further, he and a student considered the impact of multiple dust clouds along the line of sight [118], and this complexity is something we propose to add into our analysis. He will oversee all aspects of the proposal, coordinating the simulations with the map-making and spectral analysis, as well as implementing the ML algorithms for component separation.

• Kahniashvili has an expertise in studying non-standard physics (primordial magnetic fields, Lorentz and parity symmetry violations, etc.), imprints on CMB anisotropies, including the off-diagonal correlations [119, 120] and the parity-odd cross correlations T B and EB [41, 121–123]. These works were considered by the Planck collaboration as methodology when constraining primordial magnetic fields [124]. In the framework of the proposed research, Kahniashvili will work on further extensions of the ISM module of the Pencil code and she will lead the effort to interpret the simulated MHD turbulence data.

• Poczos has been working on developing robust methods for independent component analysis and its generalizations. He studied the cases when the components are noisy, corrupted by outliers [97], nonlinearly mixed [102], or mixed by convolutions [125]. He also introduced new methods for the setting when not all the components are independent from each others [126]. In multi-dimensional ICA and Independent Subspace Analysis it is assumed that the components can be organized into groups, the groups are independent from each other, but the components in the groups can be statistically dependent. These ICA based methods will be used to separate CMB from dust and other components as described in Sections IV D, and V B. Poczos also has a long-term expertise in developing convolutional and adversarial neural networks [127, 128] and he has successfully applied them in other cosmology applications [104, 105, 129, 130]. In this project conditional adversarial neural networks that will be used for E and B mode separation (Section V B 3).

• Wilson has developed a scalable Bayesian source separation algorithm that follows the Linear Mixture Model approach [71, 91] and also allows for (a) incorporation of prior information about the mixing matrix coefficients and properties of the sources such as their spatial smoothness (b) provides estimates of the uncertainty in the estimated values of CMB as well as the linear mixture model parameters and (c) makes use of an approximation that allows for parallelization of the algorithm so that it can scale to Planck-sized data (following work in [131]). On a modest cluster, an all-sky reconstruction can be completed in under an hour, allowing for multiple runs of the code to explore optimal parameter settings and prior sensitivity. The goal for work in this project is to make the code more accessible to users through a front-end written in Python and adapt it from its current use for temperature map separation to polarization map separation.

In addition, the proposers have extensive collaboration experience; see Refs. [34–45, 50, 51, 132, 133]. The earlier successful collaboration of the team members, as well as their experience in the fields of cosmology, MHD turbulence modeling, data analysis, and ML, provide an excellent base that should ensure that the project goals are successfully accomplished.

A. Proposal Time-line

The first and second years of the proposed research will be devoted to the analysis of MHD simulations with dust included and to develop the ML algorithms for the CMB polarization component separation. In addition, during the second year, we will test our formalism and compare numerical synthetic data with observations. While Planck data will be utilized during the entire research, we will devote the third year of the proposed research to the application of the next techniques developed so far.

In addition to the main participants identified by names above, two young researchers (graduate students) will be involved in the investigation. We request financial support for graduate students at CMU. We also anticipate undergraduate students participation in the proposed research (through different available funds). We also request partial salary support for Kahniashvili (research faculty at CMU supported by external research grants), and partial summer supports for Brandenburg, Dodelson, and Poczos. Collaborators Bracco and Wilson will participate on the proposal with no salary support.

VII. RESULTS FROM PRIOR NASA SUPPORT

Dodelson (PI) and Kahniashvili (Co-I) have been supported through the following NASA grants:

• NASA ATP Grant NAG5-10842: From 1991 until 2006, Dodelson was supported on a large NASA ATP group grant to the Fermilab Astrophysics Group. During the final three years of the grant, he was a PI (NASA Grant NAG5-10842, “Fundamental Physics from Space”). Various federal regulations made it more difficult for a DOE lab to obtain NASA funding, so that support ended in 2006.

Throughout those years, he wrote 88 papers, most of which were focused on the connection between cosmology and particle physics, and one textbook [134]. The papers included: the proposal that sterile neutrinos could comprise the dark matter [135]; early analyzes of the Sloan Survey [136– 142]; a well-cited review of CMB anisotropies [143]; the first predictions for CMB anisotropies for cosmic strings [144]; the signatures of massive neutrinos on the CMB [145]; early models of dark energy [146–148]; an early analysis of CMB anisotropy data that pointed to a flat universe [149]; and proposals for discriminating models of inflation [150].

• NASA Mission Concept Study Award NNX08AT71G S01: Dodelson also led the effort to build the science case for a CMB Polarization mission to the 2010 Decadal Survey. This work was funded by NASA Mission Concept Study Award NNX08AT71G S01 and culminated in 6 articles posted to the arXiv and then collected in Ref. [151].

• NASA ATP Grant NNX10AC85G: Kahniashvili has been awarded by the NASA ATP program grant Probing Cosmological Symmetry Breaking with the Cosmic Microwave Background (2010-2012), NNXlOAC85G. Within this project a number of papers have been published [34–38, 120, 121, 152–159]. Major results including the following important precedents for the current proposal:

(i) Calculation of the microwave background temperature distortion due to aberration induced by our proper motion with respect to the microwave background rest frame [155]. We first pointed out that non-diagonal multipole correlations from aberration would be detectable in Planck sky maps, and the effect was then detected at the predicted level by Planck team in the 2013 results Paper XXVII [160], the only new effect detected by Planck 2013. We have used the similar techniques to investigate a possible dipole modulation of the microwave background anisotropies to characterize the hemispherical power asymmetry seen in both WMAP and Planck sky maps [161].

(ii) Computation of the main characteristics of gravitational waves generated by the cosmological magnetic fields induced via bubble collisions during the phase transitions [152] using aero-acoustic approximation described above. In the follow-up study we have computed gravitational wave polarization degree assuming that parity violating sources were present during cosmological phase transitions [162].

(iii) Direct numerical simulations of forced and freely decaying 3D magnetohydrodynamic turbulence in order to model magnetic field evolution during cosmological phase transitions [34]. We have studied magnetic field decay laws during phase-transition turbulence and for the first time showed numerically that a magnetic field with a single initial scale is redistributed over a wide ranges of wavenumbers having a Batchelor spectrum at large length scales, and the scaling laws are almost independent on the Prandtl numbers [36, 37]. We also studied the evolution of an inflation-generated magnetic field during cosmological phase transitions [35].

(iv) Constraints on parity-violating magnetic helicity in the early universe [121]. In the follow-up study, [41], we have shown that if the mean magnetic helicity density of a helical field is measured, it can be used to determine a lower bound on the duration of inflation.

[1] D. Poletti, Giulio Fabbian, M. Le Jeune, J. Peloton, K. Arnold, C. Baccigalupi, D. Barron, S. Beckman, J. Borrill, S. Chapman, Y. Chinone, A. Cukierman, A. Ducout, T. Elleflot, J. Errard, S. Feeney, N. Goeckner-Wald, J. Groh, G. Hall, M. Hasegawa, M. Hazumi, C. Hill, Logan Howe, Yuki Inoue, Andrew H. Jaffe, O. Jeong, N. Katayama, B. Keating, R. Keskitalo, T. Kisner, A. Kusaka, A. T. Lee, D. Leon, E. Linder, L. Lowry, F. Matsuda, M. Navaroli, H. Paar, G. Puglisi, C. L. Reichardt, C. Ross, P. Siritanasak, N. Stebor, B. Steinbach, R. Stompor, A. Suzuki, O. Tajima, G. Teply, N. Whitehorn, “Making maps of Cosmic Microwave Background polarization for B-mode studies: the POLARBEAR example” Astron. Astrophys. 600, A60 (2017).

[2] M. Kamionkowski, A. Kosowsky and A. Stebbins, “A Probe of primordial gravity waves and vorticity,” Phys. Rev. Lett. 78, 2058 (1997).

[3] M. Kamionkowski, A. Kosowsky and A. Stebbins, “Statistics of cosmic microwave background polarization,” Phys. Rev. D 55, 7368 (1997).

[4] U. Seljak and M. Zaldarriaga, “Signature of gravity waves in polarization of the microwave background,” Phys. Rev. Lett. 78, 2054 (1997).

[5] M. Zaldarriaga and U. Seljak, “An all sky analysis of polarization in the microwave background,” Phys. Rev. D 55, 1830 (1997).

[6] L. P. Grishchuk, “Amplification of gravitational waves in an isotropic universe,” Sov. Phys. JETP 40, 409 (1975) [Zh. Eksp. Teor. Fiz. 67, 825 (1974)].

[7] A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23, 347 (1981).

[8] A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. 108B, 389 (1982).

[9] P. A. R. Ade, R. W. Aikin, D. Barkats, S. J. Benton, C. A. Bischoff, J. J. Bock, J. A. Brevik, I. Buder, E. Bullock, C. D. Dowell, L. Duband, J. P. Filippini, S. Fliescher, S. R. Golwala, M. Halpern, M. Hasselfield, S. R. Hildebrandt, G. C. Hilton, V. V. Hristov, K. D. Irwin, K. S. Karkare, J. P. Kaufman, B. G. Keating, S. A. Kernasovskiy, J. M. Kovac, C. L. Kuo, E. M. Leitch, M. Lueker, P. Mason, C. B. Netterfield, H. T. Nguyen, R. O’Brient, R. W. Ogburn IV, A. Orlando, C. Pryke, C. D. Reintsema, S. Richter, R. Schwarz, C. D. Sheehy, Z. K. Staniszewski, R. V. Sudiwala, G. P. Teply, J. E. Tolan, A. D. Turner, A. G. Vieregg, P. Wilson, C. L. Wong, K. W. Yoon, [BICEP2 Collaboration], “BICEP2 I: Detection Of B-mode Polarization at Degree Angular Scales,” Phys. Rev. Lett. 112, 241101 (2014).

[10] M. J. Mortonson and U. Seljak, “A joint analysis of Planck and BICEP2 B modes including dust polarization uncertainty,” JCAP 1410, 035 (2014).

[11] R. Flauger, J. C. Hill and D. N. Spergel, “Toward an Understanding of Foreground Emission in the BICEP2 Region,” JCAP 1408, 039 (2014).

[12] K. Tassis and V. Pavlidou, “Searching for Inflationary B-modes: Can dust emission properties be extrapolated from 350 GHz to 150 GHz?,” Mon. Not. Roy. Astron. Soc. 451, L90 (2015).

[13] P. A. R. Ade, N. Aghanim, Z. Ahmed, R. W. Aikin, K. D. Alexander, M. Arnaud, J. Aumont, C. Baccigalupi, A. J. Banday, D. Barkats, R. B. Barreiro, J. G. Bartlett, N. Bartolo, E. Battaner, K. Benabed, A. Benoit, A. Benoit-Levy, S. J. Benton, J.-P. Bernard, M. Bersanelli, P. Bielewicz, C. A. Bischoff, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, J. A. Brevik, M. Bucher, I. Buder, E. Bullock, C. Burigana, R. C. Butler, V. Buza, E. Calabrese, J.-F. Cardoso, A. Catalano, A. Challinor, R.-R. Chary, H. C. Chiang, P. R. Christensen, L. P. L. Colombo, C. Combet, J. Connors, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J.-M. Delouis, F.-X. Desert, C. Dickinson, J. M. Diego, H. Dole, S. Donzelli, O. Dore, M. Douspis, C. D. Dowell, L. Duband, A. Ducout, J. Dunkley, X. Dupac, C. Dvorkin, G. Efstathiou, F. Elsner, T. A. Ensslin, H. K. Eriksen, E. Falgarone, J. P. Filippini, F. Finelli, S. Fliescher, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli, K. Ganga, T. Ghosh, M. Giard, E. Gjerlow, S. R. Golwala, J. Gonzalez-Nuevo, K. M. Gorski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, M. Halpern, F. K. Hansen, D. Hanson, D. L. Harrison, M. Hasselfield, G. Helou, S. Henrot-Versille, D. Herranz, S. R. Hildebrandt, G. C. Hilton, E. Hivon, M. Hobson, W. A. Holmes, W. Hovest, V. V. Hristov, K. M. Huffenberger, H. Hui, G. Hurier, K. D. Irwin, A. H. Jaffe, T. R. Jaffe, J. Jewell, W. C. Jones, M. Juvela, A. Karakci, K. S. Karkare, J. P. Kaufman, B. G. Keating, S. Kefeli, E. Keihanen, S. A. Kernasovskiy, R. Keskitalo, T. S. Kisner, R. Kneissl, J. Knoche, L.

Knox, J. M. Kovac, N. Krachmalnicoff, M. Kunz, C. L. Kuo, H. Kurki-Suonio, G. Lagache, A. Lahteenmaki, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, E. M. Leitch, R. Leonardi, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Linden-Vornle, M. Lopez-Caniego, P. M. Lubin, M. Lueker, J. F. Macias-Perez, B. Maffei, D. Maino, N. Mandolesi, A. Mangilli, M. Maris, P. G. Martin, E. Martinez-Gonzalez, S. Masi, P. Mason, S. Matarrese, K. G. Megerian, P. R. Meinhold, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, S. Mitra, M.-A. Miville-Deschnes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, H. T. Nguyen, H. U. Nrgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, R. O’Brient, R. W. Ogburn IV, A. Orlando, L. Pagano, F. Pajot, R. Paladini, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, T. J. Pearson, O. Perdereau, L. Perotto, V. Pettorino, F. Piacentini, M. Piat, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, G. W. Pratt, S. Prunet, C. Pryke, J.-L. Puget, J. P. Rachen, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, S. Richter, I. Ristorcelli, G. Rocha, M. Rossetti, G. Roudier, M. Rowan-Robinson, J. A. Rubino-Martin, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, R. Schwarz, D. Scott, M. D. Seiffert, C. D. Sheehy, L. D. Spencer, Z. K. Staniszewski, V. Stolyarov, R. Sudiwala, R. Sunyaev, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, G. P. Teply, L. Terenzi, K. L. Thompson, L. Toffolatti, J. E. Tolan, M. Tomasi, M. Tristram, M. Tucci, A. D. Turner, L. Valenziano, J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, A. G. Vieregg, F. Villa, L. A. Wade, B. D. Wandelt, R. Watson, A. C. Weber, I. K. Wehus, M. White, S. D. M. White, J. Willmert, C. L. Wong, K. W. Yoon, D. Yvon, A. Zacchei, A. Zonca, [BICEP2 and Planck Collaborations], “Joint Analysis of BICEP2/Keck − Array and P lanck Data,” Phys. Rev. Lett. 114, 101301 (2015).

[14] D. Kandel, A. Lazarian and D. Pogosyan, “Statistical properties of galactic CMB foregrounds: dust and synchrotron,” Mon. Not. Roy. Astron. Soc. 478, no. 1, 530 (2018)

[15] J. Lizarraga, J. Urrestilla, D. Daverio, M. Hindmarsh, M. Kunz and A. R. Liddle, “Fitting BICEP2 with defects, primordial gravitational waves and dust,” J. Phys. Conf. Ser. 600, 012025 (2015).

[16] R. Stompor, J. Errard and D. Poletti, “Forecasting performance of CMB experiments in the presence of complex foreground contaminations,” Phys. Rev. D 94, 083526 (2016).

[17] N. Aghanim, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, K. Benabed, J.-P. Bernard, M. Bersanelli, P. Bielewicz, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, A. Bracco, C. Burigana, E. Calabrese, J.-F. Cardoso, H. C. Chiang, L. P. L. Colombo, C. Combet, B. Comis, B. P. Crill, A. Curto, F. Cuttaia, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J.-M. Delouis, E. Di Valentino, C. Dickinson, J. M. Diego, O. Dor, M. Douspis, A. Ducout, X. Dupac, S. Dusini, G. Efstathiou, F. Elsner, T. A. Enlin, H. K. Eriksen, E. Falgarone, Y. Fantaye, F. Finelli, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Gnova-Santos, M. Gerbino, T. Ghosh, M. Giard, J. Gonzlez-Nuevo, K. M. Grski, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, F. K. Hansen, G. Helou, D. Herranz, E. Hivon, Z. Huang, A. H. Jaffe, W. C. Jones, E. Keihnen, R. Keskitalo, T. S. Kisner, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lhteenmki, J.- M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Jeune, F. Levrier, M. Liguori, P. B. Lilje, M. Lpez-Caniego, P. M. Lubin, J. F. Macas-Prez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, M. Maris, P. G. Martin, E. Martnez-Gonzlez, S. Matarrese, N. Mauri, J. D. McEwen, A. Melchiorri, A. Mennella, M. Migliaccio, S. Mitra, M.-A. Miville-Deschnes, D. Molinari, A. Moneti, L. Montier, G. Morgante, A. Moss, P. Naselsky, H. U. Nrgaard-Nielsen, C. A. Oxborrow, L. Pagano, D. Paoletti, B. Partridge, L. Patrizii, O. Perdereau, L. Perotto, V. Pettorino, F. Piacentini, S. Plaszczynski, G. Polenta, J.-L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, M. Rossetti, G. Roudier, J. A. Rubio-Martn, B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, C. Sirignano, G. Sirri, L. Stanco, A.-S. Suur-Uski, J. A. Tauber, M. Tenti, L. Toffolatti, M. Tomasi, M. Tristram, T. Trombetti, J. Valiviita, J. Vansyngel, F. Van Tent, P. Vielva, B. D. Wandelt, I. K. Wehus, A. Zacchei, A. Zonca [Planck Collaboration], “Planck intermediate results – L. Evidence of spatial variation of the polarized thermal dust spectral energy distribution and implications for CMB B-mode analysis,” Astron. Astrophys. 599, A51 (2017).

[18] T. Ghosh, F. Boulanger, P. G. Martin, A. Bracco, F. Vansyngel, J. Aumont, J. Bock, O. Dor, U. Haud, P. M. W. Kalberla, P. Serra, “Modelling and simulation of large-scale polarized dust emission over the southern Galactic cap using the GASS HI data,” Astron. Astrophys. 601, A71 (2017).

[19] F. Vansyngel, F. Boulanger, T. Ghosh, B. D. Wandelt, J. Aumont, A. Bracco, F. Levrier, P. G. Martin, L. Montier et al., “Statistical simulations of the dust foreground to cosmic microwave background polarization,” Astron. Astrophys. 603, A62 (2017).

[20] Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, K. Benabed, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, A. Bracco, M. Bucher, C. Burigana, E. Calabrese, J.-F. Cardoso, J. Carron, H. C. Chiang, C. Combet, B. P. Crill, P. de Bernardis, G. de Zotti, J. Delabrouille, J.-M. Delouis, E. Di Valentino, C. Dickinson, J. M. Diego, A. Ducout, X. Dupac, F. Elsner, T. A. Enlin, E. Falgarone, Y. Fantaye, K. Ferrire, F. Finelli, F. Forastieri, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Gnova-Santos, T. Ghosh, J. Gonzlez-Nuevo, K. M. Grski, A. Gruppuso, J. E. Gudmundsson, V. Guillet, W. Handley, F. K. Hansen, D. Herranz, Z. Huang, A. H. Jaffe, W. C. Jones, E. Keihnen, R. Keskitalo, K. Kiiveri, J. Kim, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, J.-M. Lamarre, A. Lasenby, M. Le Jeune, F. Levrier, M. Liguori, P. B. Lilje, V. Lindholm, M. Lpez-Caniego, P. M. Lubin, Y.-Z. Ma, J. F. Macas-Prez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, P. G. Martin, E. Martnez-Gonzlez, S. Matarrese, J. D. McEwen, P. R. Meinhold, A. Melchiorri, M. Migliaccio, M.-A. Miville-Deschnes, D. Molinari, A. Moneti, L. Montier, G. Morgante, P. Natoli, L. Pagano, D. Paoletti, V. Pettorino, F. Piacentini, G. Polenta, J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubio-Martn, B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, J. D. Soler, L. D. Spencer, J. A. Tauber, D. Tavagnacco, L. Toffolatti, M. Tomasi, T. Trombetti, J. Valiviita, F. Vansyngel, F. Van Tent, P. Vielva, F. Villa, N. Vittorio, I. K. Wehus, A. Zacchei, A. Zonca, [Planck Collaboration], “Planck intermediate results. LIV. Polarized dust foregrounds,” arXiv:1801.04945 [astro-ph.GA].

[21] Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, K. Benabed, M. Bersanelli, P. Bielewicz, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, E. Calabrese, J.-F. Cardoso, J. Carron, B. Casaponsa, A. Challinor, L. P. L. Colombo, C. Combet, B. P. Crill, F. Cuttaia, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J.-M. Delouis, E. Di Valentino, C. Dickinson, J. M. Diego, S. Donzelli, O. Dor, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enlin, H. K. Eriksen, E. Falgarone, R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Gnova-Santos, M. Gerbino, T. Ghosh, J. Gonzlez-Nuevo, K. M. Grski, S. Gratton, A. Gruppuso, J. E. Gudmundsson, W. Handley, F. K. Hansen, G. Helou, D. Herranz, Z. Huang, A. H. Jaffe, A. Karakci, E. Keihnen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Jeune, F. Levrier, M. Liguori, P. B. Lilje, V. Lindholm, M. Lpez-Caniego, P. M. Lubin, Y.-Z. Ma, J. F. Macas-Prez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero, P. G. Martin, E. Martnez-Gonzlez, S. Matarrese, N. Mauri , J. D. McEwen, P. R. Meinhold, A. Melchiorri, A. Mennella, M. Migliaccio, M.-A. Miville-Deschnes, D. Molinari, A. Moneti, L. Montier, G. Morgante, P. Natoli, F. Oppizzi, L. Pagano, D. Paoletti, B. Partridge, M. Peel, V. Pettorino, F. Piacentini, G. Polenta, J.-L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, G. Roudier, J. A. Rubio-Martn, B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, D. S. Seljebotn, C. Sirignano, L. D. Spencer, A.-S. Suur-Uski, J. A. Tauber, D. Tavagnacco, M. Tenti, H. Thommesen, L. Toffolatti, M. Tomasi, T. Trombetti, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K. Wehus, A. Zacchei, A. Zonca, [Planck Collaboration], “Planck 2018 results. IV. Diffuse component separation,” arXiv:1807.06208 [astro-ph.CO].

[22] P. A. R. Ade, Z. Ahmed, R. W. Aikin, K. D. Alexander, D. Barkats, S. J. Benton, C. A. Bischoff, J. J. Bock, R. Bowens-Rubin, J. A. Brevik, I. Buder, E. Bullock, V. Buza, J. Connors, J. Cornelison, B. P. Crill, M. Crumrine, M. Dierickx, L. Duband, C. Dvorkin, J. P. Filippini, S. Fliescher, J. Grayson, G. Hall, M. Halpern, S. Harrison, S. R. Hildebrandt, G. C. Hilton, H. Hui, K. D. Irwin, J. Kang, K. S. Karkare, E. Karpel, J. P. Kaufman, B. G. Keating, S. Kefeli, S. A. Kernasovskiy, J. M. Kovac, C. L. Kuo, N. A. Larsen, K. Lau, E. M. Leitch, M. Lueker, K. G. Megerian, L. Moncelsi, T. Namikawa, C. B. Netterfield, H. T. Nguyen, R. O’Brient, R. W. Ogburn IV, S. Palladino, C. Pryke, B. Racine, S. Richter, A. Schillaci, R. Schwarz, C. D. Sheehy, A. Soliman, T. St. Germaine, Z. K. Staniszewski, B. Steinbach, R. V. Sudiwala, G. P. Teply, K. L. Thompson, J. E. Tolan, C. Tucker, A. D. Turner, C. Umilta, A. G. Vieregg, A. Wandui, A. C. Weber, D. V. Wiebe, J. Willmert, C. L. Wong, W. L. K. Wu, H. Yang, K. W. Yoon, C. Zhang, [BICEP2 and Keck Array Collaborations], “BICEP2 / Keck Array X: Constraints on Primordial Gravitational Waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season,” Submitted to: Phys.Rev.Lett. [arXiv:1810.05216 [astro-ph.CO]].

[23] S. von Hausegger, A. G. Ravnebjerg and H. Liu, “Statistical properties of polarized CMB foreground maps,” arXiv:1811.02470 [astro-ph.CO].

[24] R. Adam, P. A. R. Ade, N. Aghanim, M. Arnaud, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, N. Bartolo, E. Battaner, K. Benabed, A. Benoit-Livy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, A. Bracco, M. Bucher, C. Burigana, R.

C. Butler, E. Calabrese, J.-F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, R.-R. Chary, H. C. Chiang, P. R. Christensen, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, F. Couchot, A. Coulais, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, G. de Zotti, J. Delabrouille, J.-M. Delouis, F.-X. Desert, C. Dickinson, J. M. Diego, K. Dolag, H. Dole, S. Donzelli, O. Dore, M. Douspis, A. Ducout, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Ensslin, H. K. Eriksen, E. Falgarone, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli, K. Ganga, T. Ghosh, M. Giard, Y. Giraud-Heraud, E. Gjerlow, J. Gonzalez-Nuevo, K. M. Gorski, S. Gratton, A. Gregorio, A. Gruppuso, V. Guillet, F. K. Hansen, D. Hanson, D. L. Harrison, G. Helou, S. Henrot-Versille, C. Hernandez-Monteagudo, D. Herranz, E. Hivon, W. A. Holmes, K. M. Huffenberger, G. Hurier, A. H. Jaffe, T. R. Jaffe, J. Jewell, W. C. Jones, M. Juvela, E. Keihanen, R. Keskitalo, T. S. Kisner, R. Kneissl, J. Knoche, L. Knox, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, J. P. Leahy, R. Leonardi, J. Lesgourgues, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vornle, M. Lopez-Caniego, P. M. Lubin, J. F. Macias-Perez, B. Maffei, D. Maino, N. Mandolesi, A. Mangilli, M. Maris, P. G. Martin, E. Martinez-Gonzalez, S. Masi, S. Matarrese, P. Mazzotta, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, S. Mitra, M.-A. Miville-Deschnes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, H. U. Nörgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, L. Pagano, F. Pajot, R. Paladini, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, T. J. Pearson, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, G. W. Pratt, S. Prunet, J.-L. Puget, J. P. Rachen, W. T. Reach, R. Rebolo, M. Remazeilles, C. Renault, A. Renzi, S. Ricciardi, I. Ristorcelli, G. Rocha, C. Rosset, M. Rossetti, G. Roudier, B. Rouill d’Orfeuil, J. A. Rubino-Martin, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, D. Scott, J. D. Soler, L. D. Spencer, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, L. Terenzi, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, L. Valenziano, J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, R. Watson, I. K. Wehus, M. White, S. D. M. White, D. Yvon, A. Zacchei, A. Zonca, [Planck Collaboration], “Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes,” Astron. Astrophys. 586, A133 (2016).

[25] P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, E. Battaner, K. Benabed, A. Benoit-Lvy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, A. Bracco, C. Burigana, E. Calabrese, J.-F. Cardoso, A. Catalano, A. Chamballu, R.-R. Chary, H. C. Chiang, P. R. Christensen, L. P. L. Colombo, C. Combet, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J.-M. Delouis, C. Dickinson, J. M. Diego, H. Dole, S. Donzelli, O. Dor, M. Douspis, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enlin, H. K. Eriksen, E. Falgarone, K. Ferrire, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, T. Ghosh, M. Giard, E. Gjerlw, J. Gonzlez-Nuevo, K. M. Grski, A. Gruppuso, V. Guillet, F. K. Hansen, D. L. Harrison, G. Helou, C. Hernndez-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, A. Hornstrup, W. Hovest, Z. Huang, K. M. Huffenberger, G. Hurier, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihnen, R. Keskitalo, T. S. Kisner, R. Kneissl, J. Knoche, M. Kunz, H. Kurki-Suonio, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, R. Leonardi, J. Len-Tavares, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vrnle, M. Lpez-Caniego, P. M. Lubin, J. F. Macas-Prez, B. Maffei, D. Maino, N. Mandolesi, M. Maris, P. G. Martin, E. Martnez-Gonzlez, S. Masi, S. Matarrese, P. McGehee, A. Melchiorri, A. Mennella, M. Migliaccio, M.-A. Miville-Deschnes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, D. Novikov, I. Novikov, N. Oppermann, C. A. Oxborrow, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, O. Perdereau, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, G. W. Pratt, S. Prunet, J.-L. Puget, J. P. Rachen, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha, C. Rosset, M. Rossetti, G. Roudier, J. A. Rubio-Martn, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, D. Scott, P. Serra, J. D. Soler, V. Stolyarov, R. Sudiwala, R. Sunyaev, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, I. K. Wehus, D. Yvon, A. Zacchei, A. Zonca, [Planck Collaboration], “Planck intermediate results. XXXVIII. E- and B-modes of dust polarization from the magnetized filamentary structure of the interstellar medium,” Astron. Astrophys. 586, A141 (2016).

[26] R. R. Caldwell, C. Hirata and M. Kamionkowski, “Dust polarization and ISM turbulence,” Astrophys. J. 839,

91 (2017).

[27] S. E. Clark, J. E. G. Peek, and M. E. Putman ”Magnetically Aligned HI Fibers and the Rolling Hough Transform” Astrophys. J. 789, 82 (2014).

[28] B.-G. Andersson, A. Lazarian, and J. E. Vaillancourt, “Interstellar dust Grain Alignment”, Ann. Rev. Astron. Astrophys. 53, 501 (2015).

[29] P. Goldreich and S. Sridhar, “Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence”, Astrophys J. 438, 763 (1995).

[30] J. Cho and E. T. Vishniac, “The Anisotropy of Magnetohydrodynamic Alfvénic Turbulence”, Astrophys J. 539, 273 (2000).

[31] J. Maron and P. Goldreich, “Simulations of Incompressible Magnetohydrodynamic Turbulence”, Astrophys. J. 554, 1175 (2001).

[32] D. Kandel, A. Lazarian and D. Pogosian, ”Can the observed E/B ratio for dust galactic foreground be explained by sub-Alfvénic turbulence?” MNRAS Lett. 472, L10 (2017).

[33] N. Krachmalnicoff, E. Carretti, C. Baccigalupi, G. Bernardi, S. Brown, B.M. Gaensler, M. Haverkorn, M. Kesteven, F. Perrotta, S. Poppi, L. Staveley-Smith, “The S-PASS view of polarized Galactic Synchrotron at 2.3 GHz as a contaminant to CMB observations,” Astron. Astrophys. 618, A166 (2018).

[34] T. Kahniashvili, A. Brandenburg, A. G. Tevzadze and B. Ratra, “Numerical simulations of the decay of primordial magnetic turbulence,” Phys. Rev. D 81, 123002 (2010).

[35] T. Kahniashvili, A. Brandenburg, L. Campanelli, B. Ratra and A. G. Tevzadze, “Evolution of inflation-generated magnetic field through phase transitions,” Phys. Rev. D 86, 103005 (2012).

[36] A. G. Tevzadze, L. Kisslinger, A. Brandenburg and T. Kahniashvili, “Magnetic Fields from QCD Phase Transitions,” Astrophys. J. 759, 54 (2012).

[37] T. Kahniashvili, A. G. Tevzadze, A. Brandenburg and A. Neronov, ”Evolution of Primordial Magnetic Fields from Phase Transitions,” Phys. Rev. D 87, 083007 (2013).

[38] A. Brandenburg, T. Kahniashvili and A. G. Tevzadze, “Nonhelical inverse transfer of a decaying turbulent magnetic field,” Phys. Rev. Lett. 114, 075001 (2015).

[39] T. Kahniashvili, A. Brandenburg and A. G. Tevzadze, “The evolution of primordial magnetic field since its generation,” Phys. Scripta 91, 104008 (2016).

[40] A. Brandenburg and T. Kahniashvili, “Classes of hydrodynamic and magnetohydrodynamic turbulent decay,” Phys. Rev. Lett. 118, 055102 (2017).

[41] T. Kahniashvili, A. Brandenburg, R. Durrer, A. G. Tevzadze and W. Yin, “Scale-invariant helical magnetic field evolution and the duration of inflation,” JCAP 1712, 002 (2017).

[42] A. Brandenburg, J. Schober, I. Rogachevskii, T. Kahniashvili, A. Boyarsky, J. Frohlich, O. Ruchayskiy and N. Kleeorin, “The turbulent chiral-magnetic cascade in the early universe,” Astrophys. J. 845, L21 (2017)

[43] A. Brandenburg, T. Kahniashvili, S. Mandal, A. R. Pol, A. G. Tevzadze and T. Vachaspati, “The dynamo effect in decaying helical turbulence,” arXiv:1710.01628 [physics.flu-dyn].

[44] A. Brandenburg, T. Kahniashvili, S. Mandal, A. R. Pol, A. G. Tevzadze and T. Vachaspati, “Evolution of hydromagnetic turbulence from the electroweak phase transition,” Phys. Rev. D 96, 123528 (2017).

[45] A. Brandenburg, R. Durrer, T. Kahniashvili, S. Mandal and W. W. Yin, “Statistical Properties of Scale-Invariant Helical Magnetic Fields and Applications to Cosmology,” arXiv:1804.01177 [astro-ph.CO].

[46] A. Brandenburg, M. J. Korpi and A. J. Mee, ”Thermal instability in shearing and periodic turbulence”, Astrophys. J. 654, 945 (2007).

[47] M. J. Korpi, A. Brandenburg, A. Shukurov, I. Tuominen, A. Nordlund, ”A Supernova-regulated Interstellar Medium: Simulations of the Turbulent Multiphase Medium”, Astrophys. J. Lett. 514, L99 (1999).

[48] F. A. Gent, A. Shukurov, A. Fletcher, G. R. Sarson and M. J. Mantere, ”The supernova-regulated ISM - I. The multiphase structure”, MNRAS 432, 1396 (2013).

[49] F. A. Gent, A. Shukurov, A. Fletcher, G. R. Sarson and M. J. Mantere, ”The supernova-regulated ISM - II. The mean magnetic field” MNRAS 430, L40 (2013).

[50] A. Bracco, S. Candelaresi, F. Del Sordo and A. Brandenburg, “Is there a left-handed magnetic field in the solar neighborhood? Exploring helical magnetic fields in the interstellar medium through dust polarization power spectra,” arXiv:1807.10188 [astro-ph.GA].

[51] A. Brandenburg, A. Bracco, T. Kahniashvili, S. Mandal, A. R. Pol, G. Petrie, J.D. and N. K. Singh, “E and B polarizations from inhomogeneous and solar surface turbulence,” arXiv:1807.11457 [astro-ph.SR].

[52] H. Liu, J. Creswell and P. Naselsky, “E and B families of the Stokes parameters in the polarized synchrotron

and thermal dust foregrounds,” JCAP 1805, 059 (2018).

[53] A. Benoit-Levy, T. Dechelette, K. Benabed, J. F. Cardoso, D. Hanson and S. Prunet, “Full-sky CMB lensing reconstruction in presence of sky-cuts,” Astron. Astrophys. 555, A37 (2013).

[54] E. F. Bunn, M. Zaldarriaga, M. Tegmark and A. de Oliveira-Costa, “E/B decomposition of finite pixelized CMB maps,” Phys. Rev. D 67, 023501 (2003)

[55] C. Baccigalupi, F. Perrotta, G. De Zotti, G. F. Smoot, C. Burigana, D. Maino, L. Bedini and E. Salerno, “Extracting cosmic microwave background polarisation from satellite astrophysical maps,” Mon. Not. Roy. Astron. Soc. 354, 55 (2004).

[56] W. Zhao and D. Baskaran, “Separating E and B types of polarization on an incomplete sky,” Phys. Rev. D 82, 023001 (2010).

[57] D. L. Larson, H. K. Eriksen, B. D. Wandelt, K. M. Gorski, G. Huey, J. B. Jewell and I. J. O’Dwyer, “Estimation of Polarized Power Spectra by Gibbs sampling,” Astrophys. J. 656, 653 (2007).

[58] E. F. Bunn and B. Wandelt, “Pure E and B polarization maps via Wiener filtering,” Phys. Rev. D 96, 043523 (2017).

[59] K. K. Rogers, H. V. Peiris, B. Leistedt, J. D. McEwen and A. Pontzen, “Spin-SILC: CMB polarization component separation with spin wavelets,” Mon. Not. Roy. Astron. Soc. 463, 2310 (2016).

[60] M. Remazeilles, C. Dickinson, H. K. Eriksen and I. K. Wehus, “Joint Bayesian estimation of tensor and lensing B-modes in the power spectrum of CMB polarization data,” Mon. Not. Roy. Astron. Soc. 474, 3889 (2018).

[61] J. P. Kaufman, N.J. Miller, M. Shimon, D. Barkats, C. Bischoff, I. Buder, B.G. Keating, J.M. Kovac, P.A.R. Ade, R. Aikin, J.O. Battle, E.M. Bierman, J. J. Bock, H.C. Chiang, C.D. Dowell, L. Duband, J. Filippini, E.F. Hivon, W.L. Holzapfel, V.V. Hristov, W.C. Jones, S.S. Kernasovskiy, C.L. Kuo, E.M. Leitch, P.V. Mason, T. Matsumura, H.T. Nguyen, N. Ponthieu, C. Pryke, S. Richter, G. Rocha, C. Sheehy, M. Su, Y.D. Takahashi, J.E. Tolan, K.W. Yoon [BICEP1 Collaboration], “Self-Calibration of BICEP1 Three-Year Data and Constraints on Astrophysical Polarization Rotation,” Phys. Rev. D 89, 062006 (2014).

[62] B. Koopman, J. Austermann, H.-M. Cho, K. P. Coughlin, S. M. Duff, P. A. Gallardo, M. Hasselfield, S. W. Henderson, S.-P. P. Ho, J. Hubmayr, K. D. Irwin, D. Li, J. McMahon, F. Nati, Michael D. Niemack, Laura Newburgh, Lyman A. Page, Maria Salatino, A. Schillaci, B. L. Schmitt, S. M. Simon, E. M. Vavagiakis, J. T. Ward, E. J. Wollack “Optical modeling and polarization calibration for CMB measurements with ACTPol and Advanced ACTPol,” Proc. SPIE Int. Soc. Opt. Eng. 9914, 99142T (2016).

[63] R. H. Hildebrand, “The Determination of Cloud Masses and Dust Characteristics from Submillimetre Thermal Emission,” Q. J. Roy. Astron. Soc. 24, 267 (1983).

[64] V. Guillet, L. Fanciullo, L. Verstraete, F. Boulanger, A. P. Jones, M. A. Miville-Deschnes, N. Ysard, F. Levrier, M. Alves, ”Dust models compatible with Planck intensity and polarization data in translucent lines of sight”, Astron. Astrophys. 610, A16 (2018).

[65] A.P. Jones, L. Fanciullo, M. Koehler, L. Verstraete, V. Guillet, M. Bocchio, N. Ysard, ”The evolution of amorphous hydrocarbons in the ISM: dust modelling from a new vantage point”, Astron. Astrophys. 558, A62 (2013).

[66] P. A. R. Ade, N. Aghanim, D. Alina, M. I. R. Alves, G. Aniano, C. Armitage-Caplan, M. Arnaud, D. Arzoumanian, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, E. Battaner, K. Benabed, A. Benoit-Lvy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, A. Bracco, C. Burigana, J.-F. Cardoso, A. Catalano, A. Chamballu, H. C. Chiang, P. R. Christensen, S. Colombi, L. P. L. Colombo, C. Combet, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, C. Dickinson, J. M. Diego, S. Donzelli, O. Dor, M. Douspis, X. Dupac, G. Efstathiou, T. A. Enlin, H. K. Eriksen, E. Falgarone, L. Fanciullo, K. Ferrire, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, S. Galeotta, K. Ganga, T. Ghosh, M. Giard, Y. Giraud-Hraud, J. Gonzlez-Nuevo, K. M. Grski, A. Gregorio, A. Gruppuso, V. Guillet, F. K. Hansen, D. L. Harrison, G. Helou, C. Hernndez-Monteagudo, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, K. M. Huffenberger, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihnen, R. Keskitalo, T. S. Kisner, R. Kneissl, J. Knoche, M. Kunz, H. Kurki-Suonio, G. Lagache, J.-M. Lamarre, A. Lasenby, C. R. Lawrence, R. Leonardi, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vrnle, M. Lpez-Caniego, P. M. Lubin, J. F. Macas-Prez, D. Maino, N. Mandolesi, M. Maris, D. J. Marshall, P. G. Martin, E. Martnez-Gonzlez, S. Masi, S. Matarrese, P. Mazzotta, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, M.-A. Miville-Deschnes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, F. Noviello, D. Novikov, I. Novikov, C. A. Oxborrow, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, V.-M. Pelkonen, O. Perdereau, L. Perotto, F. Perrotta, F. Piacentini, M. Piat, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, L. Popa, G. W. Pratt, S. Prunet, J.-L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, C. Renault, S. Ricciardi, T. Riller, I. Ristorcelli, G. Rocha, C. Rosset, G. Roudier, B. Rusholme, M. Sandri, D. Scott, J. D. Soler, L. D. Spencer, V. Stolyarov, R. Stompor, R. Sudiwala, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, A. Zonca, [Planck Collaboration], “Planck intermediate results. XX. Comparison of polarized thermal emission from Galactic dust with simulations of MHD turbulence,” Astron. Astrophys. 576, A105 (2015).

[67] http://www1.astrophysik.uni-kiel.de/ polaris/

[68] H. Zhang, A. Brandenburg, and D. D. Sokoloff, ”Magnetic helicity and energy spectra of a solar active region” Astrophys. J. Lett. 784, L45 (2014).

[69] H. Zhang, A. Brandenburg, and D. D. Sokoloff, ”Evolution of magnetic helicity and energy spectra of solar active regions”, Astrophys. J. 819, 146 (2016).

[70] A. Brandenburg, G. J. D. Petrie, and N. K. Singh, ”Two-scale analysis of solar magnetic helicity”, Astrophys. J. 836, 21 (2017).

[71] S.P. Wilson, E. E. Kuruoğlu and E. Salerno, “Fully Bayesian source separation of astrophysical images modelled by mixture of Gaussians,” IEEE Journal of Selected Topics in Signal Processing: Special Issue on Signal Processing for Astronomical and Space Research Applications, 2, 685 (2008).

[72] B. T. Draine and A. A. Fraisse, “Polarized Far-Infrared and Submillimeter Emission from Interstellar Dust,” Astrophys. J. 696, 1 (2009). Erratum: [Astrophys. J. 757, 106 (2012)].

[73] E. Fermi, ”Nucleon Polarization in Pion Proton Scattering”, Phys. Rev. 91, 947 (1953).

[74] A. Lazarian and T. Hoang, “Alignment of Dust by Radiative Torque: Recent Developments,” ASP Conf. Ser. 449, 116 (2011).

[75] L. Davis and J. L. Greenstein, ”The Polarization of Starlight by Aligned Dust Grains” Astrophys. J. 114, 206 (1951).

[76] A. Lazarian and T. Hoang, “Subsonic Mechanical Alignment of Irregular Grains,” Astrophys. J. 669, L77 (2007).

[77] A. G. Kritsuk, R. Flauger and S. D. Ustyugov, “Dust-polarization maps for local interstellar turbulence,” arXiv:1711.11108 [astro-ph.GA].

[78] P. Bhat and K. Subramanian, ”Fluctuation dynamos and their Faraday rotation signatures” MNRAS 429, 2469 (2013).

[79] A. Brandenburg and R. Stepanov, “Faraday signature of magnetic helicity from reduced depolarization,” Astrophys. J. 786, 91 (2014).

[80] H. M. Lee and B. T. Draine ”Infrared extinction and polarization due to partially aligned spheroidal grains - Models for the dust toward the BN object ” Astrophys. J. 290, 211 (1985).

[81] R. Hildebrand, ”Magnetic alignment of grains”, Astrophysical Letters and Communications 26, 263 (1988).

[82] N. K. Singh, M. J. Käpylä, A. Brandenburg, P. J. Käpylä, A. Lagg, I. Virtanen, ”Bihelical spectrum of solar magnetic helicity and its evolution”, Astrophys. J. submitted, 1804.04994 [astro-ph.SR] (2018).

[83] H. Zhang and A. Brandenburg, ”Solar kinetic energy and cross helicity spectra”, Astrophys. J. submitted, 1804.10321 [astro-ph.SR] (2018).

[84] J. Delabrouille and J.-F. Cardoso, ”Diffuse Source Separation in CMB Observations”, In: Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 159–205, (2009).

[85] J.-F. Cardoso, M. Le Jeune, J. Delabrouille, M. Betoule, and G. Patanchon, ”Component Separation With Flexible Models Application to Multichannel Astrophysical Observations”, IEEE Journal of Selected Topics in Signal Processing, 2, 735 (2008).

[86] J. Delabrouille, J.-F. Cardoso, and G. Patanchon, ”Multidetector multicomponent spectral matching and applications for cosmic microwave background data analysis”, MNRAS 346, 1089 (2003).

[87] S. M. Leach, J.-F. Cardoso, C. Baccigalupi, R. B. Barreiro, M. Betoule, J. Bobin, A. Bonaldi, J. Delabrouille, G. de Zotti, C. Dickinson, H. K. Eriksen, J. González-Nuevo, F. K. Hansen, D. Herranz, M. Le Jeune, M. López-Caniego, E. Martínez-González, M. Massardi, J.-B. Melin, M.-A. Miville-Deschênes, G. Patanchon, S. Prunet, S. Ricciardi, E. Salerno, J. L. Sanz, J.-L. Starck, F. Stivoli, V. Stolyarov, R. Stompor, and P. Vielva. ”Component separation methods for the PLANCK mission”, Astron. Astrophys. 491, 597, (2008).

[88] M. Betoule, E. Pierpaoli, J. Delabrouille, M. Le Jeune, and J.-F. Cardoso, ”Measuring the tensor to scalar ratio from CMB B-modes in the presence of foregrounds”, Astron. Astrophys. 503, 691 (2009).

[89] C. Baccigalupi, L Bedini, C Burigana, G. De Zotti, A Farusi, D. Maino, M Maris, F Perrotta, E. Salerno, L. Toffolatti, and A. Tonazzini, ”Neural networks and the separation of cosmic microwave background and astrophysical signals in sky maps”, MNRAS 318, 769 (2000).

[90] F. Vansyngel. ”The blind Bayesian approach to Cosmic Microwave Background data analysis”, Theses, Université Pierre et Marie Curie - Paris VI, (2014).

[91] A. Quiros Carraterro and S.P. Wilson, “Dependent Gaussian mixture models for source separation,” Adv. Sig. Proc. 239, (2012).

[92] H. Rue, S. Martino and N. Chopin, ”Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations,” Journal of the Royal Statistical Society Series B, 71, 319 (2009).

[93] P. A. R. Ade, N. Aghanim, M. I. R. Alves, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, H. Aussel, C. Baccigalupi, A. J. Banday, R. B. Barreiro, R. Barrena, M. Bartelmann, J. G. Bartlett, N. Bartolo, S. Basak, E. Battaner, R. Battye, K. Benabed, A. Benot, A. Benoit-Lvy, J.-P. Bernard, M. Bersanelli, B. Bertincourt, M. Bethermin, P. Bielewicz, I. Bikmaev, A. Blanchard, J. Bobin, J. J. Bock, H. Bhringer, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, H. Bourdin, J. W. Bowyer, M. Bridges, M. L. Brown, M. Bucher, R. Burenin, C. Burigana, R. C. Butler, E. Calabrese, B. Cappellini, J.-F. Cardoso, R. Carr, P. Carvalho, M. Casale, G. Castex, A. Catalano, A. Challinor, A. Chamballu, R.-R. Chary, X. Chen, H. C. Chiang, L.-Y Chiang, G. Chon, P. R. Christensen, E. Churazov, S. Church, M. Clemens, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, B. Comis, F. Couchot, A. Coulais, B. P. Crill, M. Cruz, A. Curto, F. Cuttaia, A. Da Silva, H. Dahle, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, T. Dchelette, J. Delabrouille, J.-M. Delouis, J. Dmocls, F.-X. Dsert, J. Dick, C. Dickinson, J. M. Diego, K. Dolag, H. Dole, S. Donzelli, O. Dor, M. Douspis, A. Ducout, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enlin, H. K. Eriksen, O. Fabre, E. Falgarone, M. C. Falvella, Y. Fantaye, J. Fergusson, C. Filliard, F. Finelli, I. Flores-Cacho, S. Foley, O. Forni, P. Fosalba, M. Frailis, A. A. Fraisse, E. Franceschi, M. Freschi, S. Fromenteau, M. Frommert, T. C. Gaier, S. Galeotta, J. Gallegos, S. Galli, B. Gandolfo, K. Ganga, C. Gauthier, R. T. Gnova-Santos, T. Ghosh, M. Giard, G. Giardino, M. Gilfanov, D. Girard, Y. Giraud-Hraud, E. Gjerlw, J. Gonzlez-Nuevo, K. M. Grski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, J. Haissinski, J. Hamann, F. K. Hansen, M. Hansen, D. Hanson, D. L. Harrison, A. Heavens, G. Helou, A. Hempel, S. Henrot-Versill, C. Hernndez-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, S. Ho, M. Hobson, W. A. Holmes, A. Hornstrup, Z. Hou, W. Hovest, G. Huey, K. M. Huffenberger, G. Hurier, S. Ilic, A. H. Jaffe, T. R. Jaffe, J. Jasche, J. Jewell, W. C. Jones, M. Juvela, P. Kalberla, P. Kangaslahti, E. Keihnen, J. Kerp, R. Keskitalo, I. Khamitov, K. Kiiveri, J. Kim, T. S. Kisner, R. Kneissl, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, F. Lacasa, G. Lagache, A. Lhteenmki, J.-M. Lamarre, M. Langer, A. Lasenby, M. Lattanzi, R. J. Laureijs, A. Lavabre, C. R. Lawrence, M. Le Jeune, S. Leach, J. P. Leahy, R. Leonardi, J. Len-Tavares, C. Leroy, J. Lesgourgues, A. Lewis, C. Li, A. Liddle, M. Liguori, P. B. Lilje, M. Linden-Vrnle, V. Lindholm, M. Lpez-Caniego, S. Lowe, P. M. Lubin, J. F. Macas-Prez, C. J. MacTavish, B. Maffei, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero, D. Marinucci, M. Maris, F. Marleau, D. J. Marshall, P. G. Martin, E. Martnez-Gonzlez, S. Masi, M. Massardi, S. Matarrese, T. Matsumura, F. Matthai, L. Maurin, P. Mazzotta, A. McDonald, J. D. McEwen, P. McGehee, S. Mei, P. R. Meinhold, A. Melchiorri, J.-B. Melin, L. Mendes, E. Menegoni, A. Mennella, M. Migliaccio, K. Mikkelsen, M. Millea, R. Miniscalco, S. Mitra, M.-A. Miville-Deschnes, D. Molinari, A. Moneti, L. Montier, G. Morgante, N. Morisset, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, M. Negrello, N. P. H. Nesvadba, C. B. Netterfield, H. U. Nrgaard-Nielsen, C. North, F. Noviello, D. Novikov, I. Novikov, I. J. O’Dwyer, F. Orieux, S. Osborne, C. O’Sullivan, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, R. Paladini, S. Pandolfi, D. Paoletti, B. Partridge, F. Pasian, G. Patanchon, P. Paykari, D. Pearson, T. J. Pearson, M. Peel, H. V. Peiris, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, P. Platania, D. Pogosyan, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, T. Poutanen, G. W. Pratt, G. Przeau, S. Prunet, J.-L. Puget, A. R. Pullen, J. P. Rachen, B. Racine, A. Rahlin, C. Rth, W. T. Reach, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, A. Riazuelo, S. Ricciardi, T. Riller, C. Ringeval, I. Ristorcelli, G. Robbers, G. Rocha, M. Roman, C. Rosset, M. Rossetti, G. Roudier, M. Rowan-Robinson, J. A. Rubio-Martn, B. Ruiz-Granados, B. Rusholme, E. Salerno, M. Sandri, L. Sanselme, D. Santos, M. Savelainen, G. Savini, B. M. Schaefer, F. Schiavon, D. Scott, M. D. Seiffert, P. Serra, E. P. S. Shellard, K. Smith, G. F. Smoot, T. Souradeep, L. D. Spencer, J.-L. Starck, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, F. Sureau, P. Sutter, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, D. Tavagnacco, D. Taylor, L. Terenzi, D. Texier, L. Toffolatti, M. Tomasi, J.-P. Torre, M. Tristram, M. Tucci, J. Tuovinen, M. Trler, M. Tuttlebee, G. Umana, L. Valenziano, J.

Valiviita, B. Van Tent, J. Varis, L. Vibert, M. Viel, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, C. Watson, R. Watson, I. K. Wehus, N. Welikala, J. Weller, M. White, S. D. M. White, A. Wilkinson, B. Winkel, J.-Q. Xia, D. Yvon, A. Zacchei, J. P. Zibin, A. Zonca, [Planck Collaboration], “Planck 2013 results. I. Overview of products and scientific results,” Astron. Astrophys. 571, A1 (2014).

[94] M. Gustafsson, A. Brandenburg, J. L. Lemaire, D. Field, “The nature of turbulence in OMC1 at the star forming scale: observations and simulations,” Astron. Astrophys. 454, 815 (2006).

[95] A. Hyvarinen, ”Fast and robust fixed-point algorithms for independent component analysis” IEEE Trans. on Neural Networks, 10, (1999).

[96] A. Hyvarinen, J. Karhunen, and E. Oja. Independent component analysis. New York: John Wiley, (2001).

[97] S. Kirshner and B. Póczos, ”ICA and ISA Using Schweizer-Wolff Measure of Dependence”, In: Proceedings of the 25th International Conference on Machine Learning, ICML ’08, pp. 464–471, New York, NY, USA, (2008).

[98] A. Sklar. ”Fonctions de repartition ‘a n dimensions et leures marges”, Publications de lInstitut de Statistique de LUniversite de Paris, 8, 229 (1959).

[99] P. Deheuvels, ”La fonction de dependance empirique et ses proprietes, un test non parametrique dindependance”, In: Bulletin de lAcademie Royale de Belgique, Classe des Sciences, pp. 274292, (1979).

[100] R. B Nelsen, An introduction to copulas, Springer Series in Statistics. Springer. 2nd edition (2006).

[101] B. Schweizer and E. F. Wolff, ”On nonparametric measures of dependence for random variables”, The Annals of Statistics, 9, 879 (1981).

[102] Z. Szabo, B. Poczos, G. Szirtes, and A. Lorincz, ”Post nonlinear independent subspace analysis”, In:International Conference on Artificial Neural Networks (ICANN), 4668, (2007).

[103] P. Isola, J. Zhu, T. Zhou, and A. Efros, ”Image-to-Image Translation with Conditional Adversarial Nets”, In:International Conference on Computer Vision (ICCV)International Conference on Computer Vision (ICCV), (2017).

[104] F. Lanusse, M. Ravanbakhsh, B. Poczos, J. Schneider, and R. Mandelbaum, ”Deep Generative Models of Galaxy Images for the Calibration of the Next Generation of Weak Lensing Surveys”, In:American Astronomical Society Annual Meeting, (Grapevine, TX), (2017).

[105] S. Ravanbakhsh, F. Lanusse, R. Mandelbaum, J. Schneider, and B. Poczos. “Enabling dark energy science with deep generative models of galaxy images” In in Proceedings of the Thirty First AAAI Conference on Artificial Intelligence, (2017).

[106] http://research.ics.aalto.fi/ica/demos.shtml

[107] E. Learned-Miller, J. Fisher III, ”ICA Using Spacings Estimates of Entropy”, Journal of Machine Learning Research, 4, 1271 (2007).

[108] J. Cardoso, ”High-order contrasts for independent component analysis”, Neural Computation, 11 (1), 157 (1999).

[109] F. Bach and M. Jordan, ”Kernel Independent Component Analysis”, Journal of Machine Learning Research, 3, 1 (2002).

[110] A. Brandenburg and R. Stepanov, “Faraday signature of magnetic helicity from reduced depolarization,” Astrophys. J. 786, 91 (2014).

[111] E. J. Baxter, R. Keisler, S. Dodelson, K. A. Aird, S. W. Allen, M. L. N. Ashby, M. Bautz, M. Bayliss, B. A. Benson, L. E. Bleem, S. Bocquet, M. Brodwin, J. E. Carlstrom, C. L. Chang, I. Chiu, H-M. Cho, A. Clocchiatti, T. M. Crawford, A. T. Crites, S. Desai, J. P. Dietrich, T. de Haan, M. A. Dobbs, R. J. Foley, W. R. Forman, E. M. George, M. D. Gladders, A. H. Gonzalez, N. W. Halverson, N. L. Harrington, C. Hennig, H. Hoekstra, G. P. Holder, W. L. Holzapfel, Z. Hou, J. D. Hrubes, C. Jones, L. Knox, A. T. Lee, E. M. Leitch, J. Liu, M. Lueker, D. Luong-Van, A. Mantz, D. P. Marrone, M. McDonald, J. J. McMahon, S. S. Meyer, M. Millea, L. M. Mocanu, S. S. Murray, S. Padin, C. Pryke, C. L. Reichardt, A. Rest, J. E. Ruhl, B. R. Saliwanchik, A. Saro, J. T. Sayre, K. K. Schaffer, E. Shirokoff, J. Song, H. G. Spieler, B. Stalder, S. A. Stanford, Z. Staniszewski, A. A. Stark, K. T. Story, A. van Engelen, K. Vanderlinde, J. D. Vieira, A. Vikhlinin, R. Williamson, O. Zahn, A. Zenteno, “A Measurement of Gravitational Lensing of the Cosmic Microwave Background by Galaxy Clusters Using Data from the South Pole Telescope,” Astrophys. J. 806 247 (2015).

[112] T. Giannantonio, P. Fosalba, R. Cawthon, Y. Omori, M. Crocce, F. Elsner, B. Leistedt, S. Dodelson, A. Benoit-Levy, E. Gaztanaga, G. Holder, H. V. Peiris, W. J. Percival, D. Kirk, A. H. Bauer, B. A. Benson, G. M. Bernstein, J. Carretero, T. M. Crawford, R. Crittenden, D. Huterer, B. Jain, E. Krause, C. L. Reichardt, A. J. Ross, G. Simard, B. Soergel, A. Stark, K. T. Story, J. D. Vieira, J. Weller, T. Abbott, F. B. Abdalla, S. Allam, R. Armstrong, M. Banerji, R. A. Bernstein, E. Bertin, D. Brooks, E. Buckley-Geer, D. L. Burke, D. Capozzi, J. E. Carlstrom, A. Carnero Rosell, M. Carrasco Kind, F. J. Castander, C. L. Chang, C. E. Cunha, L. N. da Costa, C. B. D’Andrea, D. L. DePoy, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, T. F. Eifler, A. E. Evrard, A. Fausti Neto, E. Fernandez, D. A. Finley, B. Flaugher, J. Frieman, D. Gerdes, D. Gruen, R. A. Gruendl, G. Gutierrez, W. L. Holzapfel, K. Honscheid, D. J. James, K. Kuehn, N. Kuropatkin, O. Lahav, T. S. Li, M. Lima, M. March, J. L. Marshall, P. Martini, P. Melchior, R. Miquel, J. J. Mohr, R. C. Nichol, B. Nord, R. Ogando, A. A. Plazas, A. K. Romer, A. Roodman, E. S. Rykoff, M. Sako, B. R. Saliwanchik, E. Sanchez, M. Schubnell, I. Sevilla-Noarbe, R. C. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, M. E. C. Swanson, G. Tarle, J. Thaler, D. Thomas, V. Vikram, A. R. Walker, R. H. Wechsler, J. Zuntz, [DES Collaboration], “CMB lensing tomography with the DES Science Verification galaxies,” Mon. Not. Roy. Astron. Soc. 456 3213 (2016).

[113] D. Kirk, Y. Omori, A. Benoit-Lvy, R. Cawthon, C. Chang, P. Larsen, A. Amara, D. Bacon, T. M. Crawford, S. Dodelson, P. Fosalba, T. Giannantonio, G. Holder, B. Jain, T. Kacprzak, O. Lahav, N. MacCrann, A. Nicola, A. Refregier, E. Sheldon, K. T. Story, M. A. Troxel, J. D. Vieira, V. Vikram, J. Zuntz, T. M. C. Abbott, F. B. Abdalla, M. R. Becker, B. A. Benson, G. M. Bernstein, R. A. Bernstein, L. E. Bleem, C. Bonnett, S. L. Bridle, D. Brooks, E. Buckley-Geer, D. L. Burke, D. Capozzi, J. E. Carlstrom, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, M. Crocce, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, T. F. Eifler, A. E. Evrard, B. Flaugher, J. Frieman, D. W. Gerdes, D. A. Goldstein, D. Gruen, R. A. Gruendl, K. Honscheid, D. J. James, M. Jarvis, S. Kent, K. Kuehn, N. Kuropatkin, M. Lima, M. March, P. Martini, P. Melchior, C. J. Miller, R. Miquel, R. C. Nichol, R. Ogando, A. A. Plazas, C. L. Reichardt, A. Roodman, E. Rozo, E. S. Rykoff, M. Sako, E. Sanchez, V. Scarpine, M. Schubnell, I. Sevilla-Noarbe, G. Simard, R. C. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas, R. H. Wechsler, J. Weller, [DES Collaboration], “Cross-correlation of gravitational lensing from DES Science Verification data with SPT and Planck lensing,” Mon. Not. Roy. Astron. Soc. 459 (2016) no.1, 21 doi:10.1093/mnras/stw570 [arXiv:1512.04535 [astro-ph.CO]].

[114] E. Baxter, J. Clampitt, T. Giannantonio, S. Dodelson, B. Jain, D. Huterer, L. E. Bleem, T. M. Crawford, G. Efstathiou, P. Fosalba, D. Kirk, J. Kwan, C. Snchez, K. T. Story, M. A. Troxel, T. M. C. Abbott, F. B. Abdalla, R. Armstrong, A. Benoit-Lvy, B. A. Benson, G. M. Bernstein, R. A. Bernstein, E. Bertin, D. Brooks, J. E. Carlstrom, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, R. Chown, M. Crocce, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, A. E. Evrard, A. Fausti Neto, B. Flaugher, J. Frieman, D. Gruen, R. A. Gruendl, G. Gutierrez, T. de Haan, G. P. Holder, K. Honscheid, Z. Hou, D. J. James, K. Kuehn, N. Kuropatkin, M. Lima, M. March, J. L. Marshall, P. Martini, P. Melchior, C. J. Miller, R. Miquel, J. J. Mohr, B. Nord, Y. Omori, A. A. Plazas, C. L. Reichardt, A. K. Romer, E. S. Rykoff, E. Sanchez, I. Sevilla-Noarbe, E. Sheldon, R. C. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, A. A. Stark, M. E. C. Swanson, G. Tarle, D. Thomas, A. R. Walker, R. H. Wechsler, “Joint measurement of lensing?galaxy correlations using SPT and DES SV data,” Mon. Not. Roy. Astron. Soc. 461, 4099 (2016).

[115] A. Manzotti, K. T. Story, W. L. K. Wu, J. E. Austermann, J. A. Beall, A. N. Bender, B. A. Benson, L. E. Bleem, J. J. Bock, J. E. Carlstrom, C. L. Chang, H. C. Chiang, H-M. Cho, R. Citron, A. Conley, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, S. Dodelson, W. Everett, J. Gallicchio, E. M. George, A. Gilbert, N. W. Halverson, N. Harrington, J. W. Henning, G. C. Hilton, G. P. Holder, W. L. Holzapfel, S. Hoover, Z. Hou, J. D. Hrubes, N. Huang, J. Hubmayr, K. D. Irwin, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, D. Li, J. J. McMahon, S. S. Meyer, L. M. Mocanu, T. Natoli, J. P. Nibarger, V. Novosad, S. Padin, C. Pryke, C. L. Reichardt, J. E. Ruhl, B. R. Saliwanchik, J.T. Sayre, K. K. Schaffer, G. Smecher, A. A. Stark, K. Vanderlinde, J. D. Vieira, M. P. Viero, G. Wang, N. Whitehorn, V. Yefremenko, M. Zemcov, [SPT and Herschel Collaborations], “CMB Polarization B-mode Delensing with SPTpol and Herschel,” Astrophys. J. 846 45 (2017).

[116] B. Soergel, S. Flender, K. T. Story, L. Bleem, T. Giannantonio, G. Efstathiou, E. Rykoff, B. A. Benson, T. Crawford, S. Dodelson, S. Habib, K. Heitmann, G. Holder, B. Jain, E. Rozo, A. Saro, J. Weller, F. B. Abdalla, S. Allam, J. Annis, R. Armstrong, A. Benoit-Lvy, G. M. Bernstein, J. E. Carlstrom, A. Carnero Rosell, M. Carrasco Kind, F. J. Castander, I. Chiu, R. Chown, M. Crocce, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, T. de Haan, S. Desai, H. T. Diehl, J. P. Dietrich, P. Doel, J. Estrada, A. E. Evrard, B. Flaugher, P. Fosalba, J. Frieman, E. Gaztanaga, D. Gruen, R. A. Gruendl, W. L. Holzapfel, K. Honscheid, D. J. James, R. Keisler, K. Kuehn, N. Kuropatkin, O. Lahav, M. Lima, J. L. Marshall, M. McDonald, P. Melchior, C. J. Miller, R. Miquel, B. Nord, R. Ogando, Y. Omori, A. A. Plazas, D. Rapetti, C. L. Reichardt, A. K. Romer, A. Roodman, B. R. Saliwanchik, E. Sanchez, M. Schubnell, I. Sevilla-Noarbe, E. Sheldon, R. C. Smith, M. Soares-Santos, F. Sobreira, A. Stark, E. Suchyta, M. E. C. Swanson, G. Tarle, D. Thomas, J. D. Vieira, A. R. Walker, N.

Whitehorn, [DES and SPT Collaborations], “Detection of the kinematic Sunyaev?Zel’dovich effect with DES Year 1 and SPT,” Mon. Not. Roy. Astron. Soc. 461, 3172 (2016).

[117] E. J. Baxter, S. Raghunathan, T. M. Crawford, P. Fosalba, Z. Hou, G. P. Holder, Y. Omori, S. Patil, E. Rozo, T. M. C. Abbott, J. Annis, K. Aylor, A. Benoit-Lvy, B. A. Benson, E. Bertin, L. Bleem, E. Buckley-Geer, D. L. Burke, J. Carlstrom, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, C. L. Chang, H-M. Cho, A. T. Crites, M. Crocce, C. E. Cunha, L. N. da Costa, C. B. D’Andrea, C. Davis, T. de Haan, S. Desai, M. A. Dobbs, S. Dodelson, J. P. Dietrich, P. Doel, A. Drlica-Wagner, J. Estrada, W. B. Everett, A. Fausti Neto, B. Flaugher, J. Frieman, J. Garca-Bellido, E. M. George, E. Gaztanaga, T. Giannantonio, D. Gruen, R. A. Gruendl, J. Gschwend, G. Gutierrez, N. W. Halverson, N. L. Harrington, W. G. Hartley, W. L. Holzapfel, K. Honscheid, J. D. Hrubes, B. Jain, D. J. James, M. Jarvis, T. Jeltema, L. Knox, E. Krause, K. Kuehn, S. Kuhlmann, N. Kuropatkin, O. Lahav, A. T. Lee, E. M. Leitch, T. S. Li, M. Lima, D. Luong-Van, A. Manzotti, M. March, D. P. Marrone, J. L. Marshall, P. Martini, J. J. McMahon, P. Melchior, F. Menanteau, S. S. Meyer, C. J. Miller, R. Miquel, L. M. Mocanu, J. J. Mohr, T. Natoli, B. Nord, R. L. C. Ogando, S. Padin, A. A. Plazas, C. Pryke, D. Rapetti, C. L. Reichardt, A. K. Romer, A. Roodman, J. E. Ruhl, E. Rykoff, M. Sako, E. Sanchez, J. T. Sayre, V. Scarpine, K. K. Schaffer, R. Schindler, M. Schubnell, I. Sevilla-Noarbe, E. Shirokoff, M. Smith, R. C. Smith, M. Soares-Santos, F. Z. Staniszewski, A. Stark, K. Story, E. Suchyta, G. Tarle, D. Thomas, M. A. Troxel, K. Vanderlinde, J. D. Vieira, A. R. Walker, R. Williamson, Y. Zhang, J. Zuntz, [DES and SPT Collaborations], “A Measurement of CMB Cluster Lensing with SPT and DES Year 1 Data,” arXiv:1708.01360 [astro-ph.CO].

[118] J. Poh and S. Dodelson, “Line-of-Sight Extrapolation Noise in Dust Polarization,” Phys. Rev. D 95, 103511 (2017).

[119] R. Durrer, T. Kahniashvili and A. Yates, “Microwave background anisotropies from Alfven waves,” Phys. Rev. D 58, 123004 (1998).

[120] T. Kahniashvili, G. Lavrelashvili and B. Ratra, “CMB Temperature Anisotropy from Broken Spatial Isotropy due to an Homogeneous Cosmological Magnetic Field,” Phys. Rev. D 78, 063012 (2008).

[121] T. Kahniashvili, Y. Maravin, G. Lavrelashvili and A. Kosowsky, “Primordial Magnetic Helicity Constraints from WMAP Nine-Year Data,” Phys. Rev. D 90, 083004 (2014).

[122] C. Caprini, R. Durrer and T. Kahniashvili, “The Cosmic microwave background and helical magnetic fields: The Tensor mode,” Phys. Rev. D 69, 063006 (2004).

[123] T. Kahniashvili and B. Ratra, “Effects of Cosmological Magnetic Helicity on the Cosmic Microwave Background,” Phys. Rev. D 71, 103006 (2005).

[124] P. A. R. Ade, N. Aghanim, M. Arnaud, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, E. Battaner, K. Benabed, A. Benot, A. Benoit-Lvy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J.-F. Cardoso, A. Catalano, A. Chamballu, H. C. Chiang, J. Chluba, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, F.-X. Dsert, J. M. Diego, K. Dolag, H. Dole, S. Donzelli, O. Dor, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enlin, H. K. Eriksen, J. Fergusson, F. Finelli, E. Florido, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, S. Galeotta, S. Galli, K. Ganga, M. Giard, Y. Giraud-Hraud, E. Gjerlw, J. Gonzlez-Nuevo, K. M. Grski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, F. K. Hansen, D. Hanson, D. L. Harrison, G. Helou, S. Henrot-Versill, C. Hernndez-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, K. M. Huffenberger, G. Hurier, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihnen, R. Keskitalo, J. Kim, T. S. Kisner, J. Knoche, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lhteenmki, J.-M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, J. P. Leahy, R. Leonardi, J. Lesgourgues, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vrnle, M. Lpez-Caniego, P. M. Lubin, J. F. Macas-Prez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, M. Maris, P. G. Martin, E. Martnez-Gonzlez, S. Masi, S. Matarrese, P. McGehee, P. R. Meinhold, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, S. Mitra, M.-A. Miville-Deschnes, D. Molinari, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, H. U. Nrgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, N. Oppermann, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, G. Patanchon, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, L. Popa, G. W. Pratt, G. Przeau, S. Prunet, J.-L. Puget, J. P. Rachen, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha, C. Rosset, M. Rossetti, G. Roudier, J. A. Rubio-Martn, B. Ruiz-Granados, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, D. Scott, M. D. Seiffert, E. P. S. Shellard, M. Shiraishi, L. D. Spencer, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, D. Sutton, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, G. Umana, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, I. K. Wehus, D. Yvon, A. Zacchei, A. Zonca, [Planck Collaboration], “Planck 2015 results. XIX. Constraints on primordial magnetic fields,” Astron. Astrophys. 594, A19 (2016).

[125] Z. Szabo, B. Poczos, and A. Lorincz, ”Undercomplete Blind Subspace Deconvolution”, Journal of Machine Learning Research, 8, 1063, (2007).

[126] B. Poczos and A. Lorincz, ”Independent Subspace Analysis Using Geodesic Spanning Trees”, In:International Conference on Machine Learning (ICML), pp. 673–680, (2005).

[127] C. Li, W. Chang, Y. Cheng, Y. Yang, and B. Poczos, ”MMD GAN: Towards Deeper Understanding of Moment Matching Network” In:Proceedings of the Neural Information Processing Systems (NIPS), (2017).

[128] J. Chang, C. Li, B. Poczos, B. Kumar, and A. Sankaranarayanan, ”One Network to Solve Them All — Solving Linear Inverse Problems using Deep Projection Models”, In:International Conference on Computer Vision (ICCV)International Conference on Computer Vision (ICCV), (2017).

[129] M. Ravanbakhsh, J. Oliva, S. Fromenteau, L. Price, S. Ho, J. Schneider, and B. Poczos, ”Estimating Cosmological Parameters from the Dark Matter Distribution”, In:International Conference on Machine Learning (ICML), (2016).

[130] F. Lanusse, Q. Ma, N. Li, T. Collett, C. Li, M. Ravanbakhsh, R. Mandelbaum, and B. Poczos, ”CMU DeepLens: Deep Learning For Automatic Image-based Galaxy-Galaxy Strong Lens Finding”, Monthly Notices of the Royal Astronomical Society Main Journal (MNRAS) (2017).

[131] M. Whiley and S.P. Wilson, “Parallel algorithms for Markov chain Monte Carlo methods in latent spatial Gaussian models,” Stat. Comput. 14, 171 (2004).

[132] A. Roper Pol, A. Brandenburg, T. Kahniashvili, A. Kosowsky and S. Mandal, “The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence,” arXiv:1807.05479 [physics.fludyn].

[133] T. Kahniashvili, A. Brandenburg, A. Kosowsky, S. Mandal and A. R. Pol, “Magnetism in the Early Universe,” arXiv:1810.11876 [astro-ph.CO].

[134] S. Dodelson, “Modern Cosmology,” Amsterdam, Netherlands: Academic Pr. (2003).

[135] S. Dodelson and L. M. Widrow, “Sterile-neutrinos as dark matter,” Phys. Rev. Lett. 72, 17 (1994).

[136] S. Dodelson et al. [SDSS Collaboration], “The three-dimensional power spectrum from angular clustering of galaxies in early SDSS data,” Astrophys. J. 572, 140 (2001).

[137] R. Scranton et al. [SDSS Collaboration], “Analysis of systematic effects and statistical uncertainties in angular clustering of galaxies from Early SDSS Data,” Astrophys. J. 579, 48 (2002).

[138] A. S. Szalay et al. [SDSS Collaboration], “KL estimation of the power spectrum parameters from the angular distribution of galaxies in early SDSS data,” Astrophys. J. 591, 1 (2003).

[139] I. Zehavi, M. R. Blanton, J. A. Frieman, D. H. Weinberg, H. J. Mo, M. A. Strauss, S. F. Anderson, J. Annis, N. A. Bahcall, M. Bernardi, J. W. Briggs, J. Brinkmann, S. Burles, L. Carey, F. J. Castander, A. J. Connolly, Istvan Csabai11, Julianne J. Dalcanton, S. Dodelson1, M. Doi, D. Eisenstein, M. L. Evans, D. P. Finkbeiner, S. Friedman, M. Fukugita, J. E. Gunn, G. S. Hennessy, R. B. Hindsley, Z. Ivezic, S. Kent, G. R. Knapp, R. Kron, P. Kunszt, D. Q. Lamb, R. French Leger, D. C. Long, J. Loveday, Robert H. Lupton, T. McKay, A. Meiksin, A. Merrelli, J. A. Munn, V. Narayanan, M. Newcomb, R. C. Nichol, R. Owen, J. Peoples, A. Pope, C. M. Rockosi, D. Schlegel, D. P. Schneider, R. Scoccimarro, R. K. Sheth, W. Siegmund, Smee, Y. Snir, A. Stebbins, S. Stoughton, M. SubbaRao, A. S. Szalay, I. Szapudi, M. Tegmark, D. L. Tucker, A. Uomoto, D. Vanden Berk, M. S. Vogeley29, P. Waddell, B. Yanny, and D. G. York, [SDSS Collaboration], “Galaxy clustering in early SDSS redshift data,” Astrophys. J. 571, 172 (2002).

[140] C. Stoughton, R. H. Lupton, M. Bernardi, M. R. Blanton, S. Burles, F. J. Castander, A. J. Connolly, D. J. Eisenstein, Joshua A. Frieman, G. S. Hennessy, R. B. Hindsley, Z. Ivezic, Stephen Kent1,3, Peter Z. Kunszt, B. C. Lee1, A. Meiksin, J. A. Munn, H. J. Newberg, R. C. Nichol1, Tom Nicinski1, Jeffrey R. Pier, G. T. Richards, M. W. Richmond, D. J. Schlegel, J. A. Smith, M. A. Strauss, M. SubbaRao, A. S. Szalay, A. R. Thakar, D. L. Tucker1, D. E. Vanden Berk, B. Yanny, J. K. Adelman, J. E. Anderson, Jr., S. F. Anderson, J. Annis, N. A. Bahcall, J. A. Bakken1, M. Bartelmann, Steven Bastian, A. Bauer, E. Berman, H. Bohringer, W. N. Boroski, S. Bracker, C. Briegel, J. W. Briggs, J. Brinkmann, R. Brunner, L. Carey, M. A. Carr, B. Chen, D. Christian, P. L. Colestock, J. H. Crocker, I. Csabai, P. C. Czarapata, J. Dalcanton, A. F. Davidsen, J. E. Davis, W. Dehnen, S. Dodelson, M. Doi, T. Dombeck, M. Donahue, N. Ellman, B. R. Elms, M. L. Evans, L. Eyer, Xiaohui Fan, G. R. Federwitz, S. Friedman, M. Fukugita, R. Gal, B. Gillespie, Karl Glazebrook, J. Gray, E. K. Grebel, B. Greenawalt, Gretchen Greene, J. E. Gunn2, E. de Haas, Z. Haiman, M. Haldeman1, P. B. Hall, M. Hamabe, B. Hansen, F. H. Harris, H. Harris, M. Harvanek, S. L. Hawley, J. J. E. Hayes, T. M. Heckman, A. Helmi, A. Henden, C. J. Hogan, D. W. Hogg, D. J. Holmgren1, J. Holtzman, C.-H. Huang, C. Hull, S. Ichikawa, T. Ichikawa, D. E. Johnston, G. Kauffmann, R. S. J. Kim, Tim Kimball, E. Kinney, M. Klaene, S. J. Kleinman, A. Klypin, G. R. Knapp, J. Korienek1, J Krolik, Richard G. Kron, J. Krzesinski, D. Q. Lamb, R. F. Leger1, S. Limmongkol, C. Lindenmeyer1, D. C. Long, C. Loomis, J. Loveday, B. MacKinnon, E. J. Mannery, P. M. Mantsch1, B. Margon, P. McGehee, T. A. McKay, B. McLean, K. Menou, A. Merelli18, H. J. Mo, D. G. Monet, O. Nakamura, V. K. Narayanan, T. Nash1, E. H. Neilsen, Jr., P. R. Newman, A. Nitta, M. Odenkirchen, N. Okada, Sadanori Okamura, J. P. Ostriker, R. Owen, A. G. Pauls, J. Peoples, R. S. Peterson1, D. Petravick, A. Pope, R. Pordes, M. Postman, A. Prosapio1, T. R. Quinn, R. Rechenmacher1, C. H. Rivetta1, H.-W. Rix, C. M. Rockosi, R. Rosner, K. Ruthmansdorfer, D. Sandford, D. P. Schneider, R. Scranton, M. Sekiguchi, G. Sergey, R. Sheth, K. Shimasaku, S. Smee, S. A. Snedden, A. Stebbins, C. Stubbs, I. Szapudi, P. Szkody, G. P. Szokoly, S. Tabachnik, Z. Tsvetanov, A. Uomoto, M. S. Vogeley, W. Voges, P. Waddell, R. Walterbos, S. Wang, M. Watanabe, D. H. Weinberg, R. L. White, S. D. M. White, B. Wilhite, D. Wolfe, N. Yasuda, D. G. York, I. Zehavi1, and W. Zheng, [SDSS Collaboration], “The Sloan Digital Sky Survey: Early Data Release,” Astron. J. 123, 485 (2002).

[141] M. Tegmark, M. Strauss, M. Blanton, K. Abazajian, S. Dodelson, H. Sandvik, X. Wang, D. Weinberg, I. Zehavi, N. Bahcall, F. Hoyle, D. Schlegel, R. Scoccimarro, M. Vogeley, A. Berlind, T. Budavari, A. Connolly, D. Eisenstein, D. Finkbeiner, J. Frieman, J. Gunn, L. Hui, B. Jain, D. Johnston, S. Kent, H. Lin, R. Nakajima, R. Nichol, J. Ostriker, A. Pope, R. Scranton, U. Seljak, R. Sheth, A. Stebbins, A. Szalay, I. Szapudi, Y. Xu, J. Annis, J. Brinkmann1, S. Burles, F. J. Castander, I. Csabai, J. Loveday, M. Doi, M. Fukugita, B. Gillespie, G. Hennessy, D. W. Hogg, Z. Ivezic, G. R. Knapp, D. Q. Lamb, B. C. Lee, R. H. Lupton, T. A. McKay, P. Kunszt, J. A. Munn, L. OConnell, J. Peoples, J. R. Pier, M. Richmond, C. Rockosi, Donald P. Schneider, C. Stoughton, D. L. Tucker, D. E. Vanden Berk, B. Yanny, D. G. York [SDSS Collaboration], “Cosmological parameters from SDSS and WMAP,” Phys. Rev. D 69, 103501 (2004).

[142] K. Abazajian, J. K. Adelman-McCarthy, M. A. Agüeros, S. S. Allam, S. F. Anderson, J. Annis, N. A. Bahcall, I. K. Baldry, S. Bastian, A. Berlind, M. Bernardi, M. R. Blanton, N. Blythe, J. J. Bochanski Jr., W. N. Boroski, H. Brewington, J. W. Briggs, J. Brinkmann, R. J. Brunner, T. Budavari, L. N. Carey, M. A. Carr, Francisco J. Castander, Kuenley Chiu, M. J. Collinge, A. J. Connolly, Kevin R. Covey, István Csabai, J. J. Dalcanton, S. Dodelson, M. Doi, F. Dong, D. J. Eisenstein, M. L. Evans, X. Fan, P. D. Feldman, D. P. Finkbeiner, S. D. Friedman, J. A. Frieman, M. Fukugita, R. R. Gal, B. Gillespie, K. Glazebrook, C. F. Gonzalez, J. Gray, E. K. Grebel, L. Grodnicki, J. E. Gunn, V. K. Gurbani, P. B. Hall, L. Hao, D. Harbeck, F. H. Harris, H. C. Harris, M. Harvanek, S. L. Hawley, T. M. Heckman, J. F. Helmboldt, J. S. Hendry, G. S. Hennessy, R. B. Hindsley, D. W. Hogg, D. J. Holmgren, J. A. Holtzman, L. Homer, L. Hui, S. Ichikawa, T. Ichikawa, J. P. Inkmann, Ž. Ivezić, S. Jester, D. E. Johnston, B. Jordan, W. P. Jordan, A. M. Jorgensen, M. Jurić, G. Kauffmann, S. M. Kent, S. J. Kleinman, G. R. Knapp, A. Yu. Kniazev, R. G. Kron, J. Krzesinski, P. Z. Kunszt, N. Kuropatkin, D. Q. Lamb, H. Lampeitl, B. E. Laubscher, B. C. Lee, R. F. Leger, N. Li, A. Lidz, H. Lin, Y. Loh, D. C. Long, J. Loveday, R. H. Lupton, T. Malik, B. Margon, P. M. McGehee, T. A. McKay, A. Meiksin, G. A. Miknaitis, B. K. Moorthy, J. A. Munn, T. Murphy, R. Nakajima, V. K. Narayanan, T. Nash, E. H. Neilsen, Jr., H. J. Newberg, P. R. Newman, R. C. Nichol, T. Nicinski, M. Nieto-Santisteban, A. Nitta, M. Odenkirchen, S. Okamura, J. P. Ostriker, R. Owen, N. Padmanabhan, J. Peoples, J. R. Pier, B. Pindor, A. C. Pope, T. R. Quinn, R. R. Rafikov, S. N. Raymond, G. T. Richards, M. W. Richmond, H.-W. Rix, C. M. Rockosi, J. Schaye, D. J. Schlegel, D. P. Schneider, J. Schroeder, R. Scranton, M. Sekiguchi, U. Seljak, G. Sergey, B. Sesar, E. Sheldon, K. Shimasaku, W. A. Siegmund, N. M. Silvestri, A. J. Sinisgalli, E. Sirko, J. Smith, V. Smolčić, S. A. Snedden, A. Stebbins, C. Steinhardt, G. Stinson, C. Stoughton, I. V. Strateva, M. A. Strauss, M. SubbaRao, A. S. Szalay, I. Szapudi, P. Szkody, L. Tasca, M. Tegmark, A. R. Thakar, C. Tremonti, D. L. Tucker, A. Uomoto, D. E. Vanden Berk, J. Vandenberg, M. S. Vogeley, W. Voges, N. P. Vogt, L. M. Walkowicz, D. H. Weinberg, A. A. West, S. D.M. White, B. C. Wilhite, B. Willman, Y. Xu, B. Yanny, J. Yarger, N. Yasuda, C.-W. Yip, D. R. Yocum, D. G. York, N. L. Zakamska, I. Zehavi, W. Zheng, S. Zibetti, D. B. Zucker, [SDSS Collaboration], “The First data release of the Sloan Digital Sky Survey,” Astron. J. 126, 2081 (2003).

[143] W. Hu and S. Dodelson, “Cosmic microwave background anisotropies,” Ann. Rev. Astron. Astrophys. 40, 171 (2002).

[144] B. Allen, R. R. Caldwell, S. Dodelson, L. Knox, E. P. S. Shellard and A. Stebbins, “CMB anisotropy induced by cosmic strings on angular scales ¿ approximately 15-minutes,” Phys. Rev. Lett. 79, 2624 (1997).

[145] S. Dodelson, E. Gates and A. Stebbins, “Cold + hot dark matter and the cosmic microwave background,” Astrophys. J. 467, 10 (1996).

[146] K. Coble, S. Dodelson and J. A. Frieman, “Dynamical Lambda models of structure formation,” Phys. Rev. D 55, 1851 (1997).

[147] S. Dodelson, M. Kaplinghat and E. Stewart, “Solving the coincidence problem : Tracking oscillating energy,” Phys. Rev. Lett. 85, 5276 (2000).

[148] M. Ahmed, S. Dodelson, P. B. Greene and R. Sorkin, “Everpresent lambda,” Phys. Rev. D 69, 103523 (2004).

[149] S. Dodelson and L. Knox, “Dark energy and the CMB,” Phys. Rev. Lett. 84, 3523 (2000).

[150] S. Dodelson, W. H. Kinney and E. W. Kolb, “Cosmic microwave background measurements can discriminate among inflation models,” Phys. Rev. D 56, 3207 (1997).

[151] S. Dodelson et al., “Theory and foregrounds: CMBPol mission concept study. Proceedings, CMB Polarization Workshop, Batavia, USA, June 23-26, 2008,” AIP Conf. Proc. 1141, pp.1 (2009).

[152] T. Kahniashvili, L. Kisslinger and T. Stevens, “Gravitational Radiation Generated by Magnetic Fields in Cosmological Phase Transitions,” Phys. Rev. D 81, 023004 (2010).

[153] T. Kahniashvili and G. Lavrelashvili, “CMB two- and three-point correlation functions from Alfvén waves,” arXiv:1010.4543 [astro-ph.CO].

[154] T. Kahniashvili, A. G. Tevzadze, S. K. Sethi, K. Pandey and B. Ratra, “Primordial magnetic field limits from cosmological data,” Phys. Rev. D 82, 083005 (2010).

[155] A. Kosowsky and T. Kahniashvili, “The Signature of Proper Motion in the Microwave Sky,” Phys. Rev. Lett. 106, 191301 (2011).

[156] L. Campanelli, G. L. Fogli, T. Kahniashvili, A. Marrone and B. Ratra, “Galaxy cluster number count data constraints on cosmological parameters,” Eur. Phys. J. C 72, 2218 (2012).

[157] G. Y. Chitov, T. August, A. Natarajan and T. Kahniashvili, “Mass Varying Neutrinos, Quintessence, and the Accelerating Expansion of the Universe,” Phys. Rev. D 83, 045033 (2011).

[158] T. Kahniashvili, Y. Maravin, A. Natarajan, N. Battaglia and A. G. Tevzadze, “Constraining primordial magnetic fields through large scale structure,” Astrophys. J. 770, 47 (2013).

[159] T. Kahniashvili, A. G. Tevzadze, B. Ratra, “Phase Transition Generated Cosmological Magnetic Field at Large Scales,” Astrophys. J. 726, 78 (2011).

[160] N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, A. J. Banday, R. B. Barreiro, J. G. Bartlett, K. Benabed, A. Benoit-Lvy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bridges, C. Burigana, R. C. Butler, J.-F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, L.-Y Chiang, H. C. Chiang, P. R. Christensen, D. L. Clements, L. P. L. Colombo, F. Couchot, B. P. Crill, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J. M. Diego, S. Donzelli, O. Dor, X. Dupac, G. Efstathiou, T. A. Enlin, H. K. Eriksen, F. Finelli, O. Forni, M. Frailis, E. Franceschi, S. Galeotta, K. Ganga, M. Giard, G. Giardino, J. Gonzlez-Nuevo, K. M. Grski, S. Gratton, A. Gregorio, A. Gruppuso, F. K. Hansen, D. Hanson, D. Harrison, G. Helou, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, W. Hovest, K. M. Huffenberger, W. C. Jones, M. Juvela, E. Keihnen, R. Keskitalo, T. S. Kisner, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, A. Lhteenmki, J.-M. Lamarre, A. Lasenby, C. R. Lawrence, R. Leonardi, A. Lewis, M. Liguori, P. B. Lilje, M. Linden-Vrnle, M. Lpez-Caniego, P. M. Lubin, J. F. Macas-Prez, M. Maris, D. J. Marshall, P. G. Martin, E. Martnez-Gonzlez, S. Masi, S. Matarrese, P. Mazzotta, P. R. Meinhold , A. Melchiorri, L. Mendes, M. Migliaccio, S. Mitra, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, P. Naselsky, F. Nati, P. Natoli, H. U. Nrgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, S. Osborne, C. A. Oxborrow, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, G. Patanchon, O. Perdereau, F. Perrotta, F. Piacentini, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, G. W. Pratt, G. Przeau, J.-L. Puget, J. P. Rachen, W. T. Reach, M. Reinecke, S. Ricciardi, T. Riller, I. Ristorcelli, G. Rocha, C. Rosset, J. A. Rubio-Martn, B. Rusholme, D. Santos, G. Savini, D. Scott, M. D. Seiffert, E. P. S. Shellard, L. D. Spencer, R. Sunyaev, F. Sureau, A.-S. Suur-Uski, J.-F. Sygnet, J. A. Tauber, D. Tavagnacco, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, M. Trler, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, M. White, D. Yvon, A. Zacchei, J. P. Zibin, A. Zonca, [Planck Collaboration], “Planck 2013 results. XXVII. Doppler boosting of the CMB: Eppur si muove,” Astron. Astrophys. 571, A27 (2014)

[161] S. Aiola, B. Wang, A. Kosowsky, T. Kahniashvili and H. Firouzjahi, “Microwave Background Correlations

from Dipole Anisotropy Modulation,” Phys. Rev. D 92, 063008 (2015).

[162] L. Kisslinger and T. Kahniashvili, “Polarized Gravitational Waves from Cosmological Phase Transitions,” Phys. Rev. D 92, 043006 (2015)

[163] [ M. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. Salakhutdinov, and A. Smola. “Deep sets” preprint arXiv:1703.06114, (2017). In Advances in Neural Information Processing Systems

[164] S. Ravanbakhsh, J. Schneider, and B. Poczos “Equivariance through parameter-sharing” arXiv preprint arXiv:1702.08389 (2017).

[165] F. Lanusse, S. Ravanbakhsh, R. Mandelbaum, J. Schneider, and B. Poczos “Deep Generative Models of Galaxy Images for the Calibration of the Next Generation of Weak Lensing Surveys” American Astronomical Society Meeting Abstracts, volume 229 of American Astronomical Society Meeting Abstracts, page 342.05, January 2017.

[166] Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, Eric P Xing “Stochastic Variational Deep Kernel Learning” Neural Information Processing System, (2016)

[167] K. Kandasamy, J. Schneider, and B. Póczos, “Query Efficient Posterior Estimation in Scientific Experiments via Bayesian Active Learning” Artificial Intelligence Journal, (2016)

$Header: /var/cvs/brandenb/tex/tina/prop/ADAP19EB/DustEB.tex,v 1.1 2019/04/24 14:17:24 brandenb Exp $