Scientific/Technical/Management Section

I. OBJECTIVES AND MOTIVATIONS

The aim of this proposal is to use NASA data and simulations to understand turbulence in the Interstellar Medium (ISM). NASA missions, in particular Planck, have measured the polarization of the Galactic radiation emitted at least in part by dust grains. The patterns of polarization encode information about the magnetic fields in which the grains reside, as the grains tend to align with the fields. The magnetic field itself is coupled to the gas in the ISM, so the Planck data contain a wealth of information about the complex multi-phase nature of our Galaxy and the turbulent fields within it.

What we already know from Planck and from magnetohydrodynamic (MHD) simulations is puzzling and suggests that there is much we do not understand about the turbulent ISM.

1. The E-mode polarization exceeds the B-mode by almost a factor of two [1] and that ratio is roughly constant across the sky. This is in contrast to the theoretical expectations that dust contributions to the E- and B- modes should be similar.

2. There is apparent alignment of the cold medium filamentary structure as traced by dust maps with the magnetic field in the plane of the sky, as traced by the polarization maps [2].

3. The intensity (written in terms of temperature T) and the E-mode of polarization are positively correlated (T E > 0). The hypothesis that the alignment between density and magnetic field structures is responsible for the EE/BB ratio cannot explain the T E cross correlation on angular scales ∼ 10^{∘}, larger than those typical of the interstellar filaments (although it remains plausible for smaller scales; see Ref. [3]).

4. The probability distributions of some key quantities such as the magnetic field and the E polarization are not gaussian. Indeed, Ref. [2] showed that there is an excess of strong filaments of aligned dust grains (see the shaded region of their Fig. 3) and MHD simulations have shown that the probability density function of the magnetic field strength is strongly non-gaussian [4], and that turbulence has a tendency to produce large tails of negative Epolarization [5].

It seems quite likely that these observations need to be explained by exploring the full complexity of magnetohydrodynamic (MHD) turbulence in a multi-phase medium. As a start, a fascinating question about the turbulence in the magnetized plasma of the ISM is which modes are excited: Alfvén, slow, or fast waves. Ref. [6] showed that any combination of these modes in weak MHD turbulence, even allowing for significant anisotropy, has a very difficult time reproducing these observations. Ref. [7] showed that it is a bit easier to explain the spectra if the Alfvén waves have low Mach numbers, while [8] showed that full MHD simulations have the potential to explain not only the spectra but also some of the other items listed above.

Our objective is to understand MHD turbulence in the ISM by (see Fig. 1) (i) running stateof-the-art simulations, (ii) expanding the suite of statistics used to analyze the data, and then (iii) applying these new techniques to both the simulations and the Planck data. More details are provided about the new techniques in §IIIB, but let us simply comment here that, while the EE, BB, and T E spectra, the two-point functions, are sufficient to characterize the isotropic, Gaussian

FIG. 1: Overview of the proposal. State-of-the-art MHD simulations (described in §IIIA) will generate synthetic data as close as possible to the Planck maps. New data analysis techniques (§IIIB) will be applied to both the simulations and the Planck data. The result will be the set of deliverables highlighted in the red box at right.

cosmic microwave background (CMB), they are obviously not sufficient to capture the highly anisotropic, non-gaussian Galactic radiation field. New statistics are clearly called for in order to extract all the information possible about the turbulent ISM.

II. IMPACT AND RELEVANCE TO NASA OBJECTIVES AND GOALS

While the structure of the Galaxy and the nature of the complex magnetic-fluid turbulence patterns within it are fascinating topics in of themselves, a further impact of this work will be to understand what has come to be known as foregrounds for the CMB radiation. The polarization of the CMB can be decomposed into E- and B-modes, and – at least on large scales (on small scales, B-modes are generated by gravitational lensing) – non-zero B-modes would provide strong evidence for primordial gravitational waves [9–12]. This is particularly interesting because many models of inflation predict that just such a background of stochastic waves should have been produced in the early universe [13]. Further, the amplitude of the gravitational wave (and therefore the B-mode) signal can be directly related to the energy scale of inflation, thereby offering the tantalizing possibility of learning about physics at energy scales ten orders of magnitude larger than those probed at the largest colliders ever constructed.

A robust detection of B-modes in the CMB will first require exquisite understanding of precisely the signal we aim to study in this proposal: that produced by dust grains residing in turbulent magnetic fields. Planck measured the polarization of radiation that is manifestly Galactic [14–16], so the amplitude of the Galactic signal is at least as large as the primordial signal over much of the sky and quite likely much larger [17]. Therefore, understanding this foreground is of paramount importance not just for its own sake, but also for the purposes of understanding the origins of the universe.

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”.

III. METHODOLOGY AND RESEARCH PROGRAM

Our approach involves three pieces:

• State-of-the-art MHD simulations that allow us to simulate many different possible types of turbulence and make maps of the ensuing polarization patterns (§IIIA)

• Introducing/developing new statistics that go beyond the simple 2-point functions that are appropriate for cosmology but that fail to capture all of the information needed to learn about the complex turbulent ISM (§IIIB)

• Apply these new statistics to both the simulations and Planck data (§IIIC)

A. Turbulence Simulations

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. In addition, the ISM is a highly compressible and magnetized medium consisting of neutral and ionized components with varying ionization fractions in its different phases (in the cold phase the ionization fraction is lower than in the hot and warm phases) [18]. In the cold phase, the low ionization degree results in different speeds for ionized and neutral components or ambipolar diffusion, which can result in the formation of sharp structures [19]. The complexity of the physical processes is enhanced by turbulent transport and mixing, turbulent heating in shocks, and various dissipation processes [20]. Correspondingly, turbulence in the ISM is a complex process [21] and cannot be described in terms of isotropic, homogeneous and incompressible Kolmogorov hydrodynamic turbulence [22]. Moreover, the ISM is an example of a non-equilibrium dissipative system that gains energy at large scales (around 100 pc) from gravitational instability, large-scale shear, gas accretion, stellar feedback, and loses it at much small scales through dissipation and radiative cooling [23]. To explore the complexity of hierarchical non-equilibrium dissipative system, Kritsuk et al. treated interstellar turbulence as an agent imposing “order in the form of coherent structures and correlations between flow fields emerging in a simple periodic box simulation when a statistically stationary state fully develops” [4]. The predicted magnetic field probability distribution function (PDF) is strongly non-Gaussian, and a relative alignment between magnetic field and density structures is observed [4]. These simulations were also used to reproduce the observed EE/BB ratio [8], although the physical understanding of what causes the excess remained open.

To make the medium turbulent the Reynolds number Re must be large enough for a large number of scales to be involved in the motion, as observed in turbulent flows. For the cold ISM Re ≃ 10^{5} − 10^{7}, and we expect the ISM to be turbulent if there are substantially strong forces to drive turbulence (driving mechanisms) [21]. Such driving mechanisms have different energy dissipation rates and they include supernova explosions (3 × 10^{−26}ergcm^{−3}s^{−1}), stellar winds (3 × 10^{−27}ergcm^{−3}s^{−1}), proto-stellar outflows (3 × 10^{−28}erg cm^{−3}s^{−1})), galactic spiral shocks and magneto-rotational instability (3 × 10^{−29}erg cm^{−3}s^{−1}), and HII regions 3 × 10^{−30}erg cm^{−3}s^{−1} [20]. Only 3–5% of the supernova energy supply is needed to drive turbulence in the ISM with the characteristic inertial scale around 50 − 100pc and with the characteristic speed around 10km s^{−1}, comparable with the sound speed in the ISM (or even can exceed it) [23]. As highlighted above turbulence in the ISM is magnetized (as well as turbulence in solar/stellar winds turbulence, radio galaxies and quasars) with the energy density of the magnetic field comparable to the kinetic energy of turbulence. The magnetic Reynolds number Re_{m} is high enough (Re_{m} ≫ Re ≫ 1). The magnetic field presence crucially changes the turbulent motions in ISM, see Ref. [24] for a recent study and references therein. In particular, the sensitivity of the ISM structure and outflow speed to magnetic field is viewed in simulations when dynamo action is allowed: in particular, apart from providing additional pressure support of the gas layer, magnetic fields reduce the density contrast between the warm and hot gas phases and quench galactic outflows [25].

In our proposed research, we plan to further explore turbulence in the ISM through a variety of numerical simulations, described below. Simulations will be performed in a semi-global geometry to capture the full vertical stratification, but are shearing-periodic in the radial direction. We then employ the shearing box approximation to incorporate the effects of galactic shear, which leads to a strong toroidal component of the magnetic field [26]. They incorporate density stratification and the different statistics of supernova I and II, which have different rates and scale heights. This was first done in our work of nearly 20 years ago [27]. In our new work, we will also incorporate the other aforementioned driving mechanisms that operate on different time scales and may therefore play an important role.

In a second type of simulations of the cold ISM in smaller domains, we can ignore shear and density stratification, and focus in highly supersonic turbulence that is just driven by volume forcing [26–29]. This is also the case in the recent work of [4]. To make contact with Planck 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^{1} community (for example see Refs. [30, 31]).

Given that astrophysical turbulence can take many different forms, we need to study the implications on the E- and B-modes. Turbulence can be helical or nonhelical, it can be magnetically dominated or subdominant, it can possess cross helicity^{2} with a systematic alignment or antialignment between magnetic field and velocity, etc. 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. These turbulence simulations provide a means of performing experiments in a variety of circumstances and environments. This allows us to isolate specific physical effects. We can also identify specific mathematical terms in the underlying equations with distinct features in the synthesized diagnostics which can be responsible for certain correlations in the E- and B-modes; see Fig. 2, of which we show the resulting Eand B-mode polarizations from a decaying magnetically dominant turbulence simulations. Here,

1 The Pencil Code is a high order finite difference code (sixth order in space and third order in time) that is publicly available through http://github.com/pencil-code, see also [32] for publications acknowledging the Pencil Code.

2 Helicity characterizes the swirl of a flow or a magnetic field.

FIG. 2: E-mode (left) and B-mode (right) polarization for isotropic fully helical magnetohydrodynamic turbulence using data from [33] (their Fig. 4d–f). Dark (light) shades indicate negative (positive) velocity. Adapted from Ref. [5].

the flow is mainly driven by the Lorentz force [34]. In Ref. [5], we found that for both dust and synchrotron emission, the PDF of the E polarization is strongly skewed, which contributes to an enhanced EE/BB ratio. This is comparable with the observed foreground polarization detected with Planck [1, 35], clearly more than what was theoretically expected [6] and was argued to be explicable with Alfvén waves [7].

There is the exciting possibility that the parity-odd EB and T B cross correlations can give indications about magnetic helicity [36]. This has been a subject of recent studies [5, 37], which demonstrated that this is only possible if turbulence in inhomogeneous. To explain the possibility of a nonvanishing T B cross correlations [35], Ref. [37] invoked the idea that the solar neighborhood is surrounded by a coherent helical tube-like structure. This possibility will need to be re-addressed.

Our recent work explained by even fully helical turbulence cannot produce a nonvanishing EB correlation. This is 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 circumstances, 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 Ref. [5] 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. In that case, there is a systematic EB correlation, with EB being positive in the north and negative in the south; see Refs. [5, 37] for our recent work on this. This is very promising and agrees with our intuition. We also found that in those simulations the BB correlation exceeds the EE correlation. This can be explained by a strong rotational influence, which leads to a strong parity-invariant signal and therefore to a large BB correlation. In addition, the magnetic field is here subdominant, in contrast to the situation

FIG. 3: Histograms of E and B polarizations for low (a) and high (b) ambipolar diffusion and their skewness versus the strength of ambipolar diffusion (c), as measured by an ambipolar Strouhal number, St_{AD}. Note the sign change of skew E. Adapted from Ref. [40].

shown in Fig. 2.

The Pencil Code has been successfully applied to a broad range of turbulent flows, including all the flows to be studied in this paper. In direct numerical simulations, where no shock viscosity or other large-eddy techniques are invoked, we can reach Mach numbers in the range 2–5 [38]. By comparison, using shock diffusivities, we can run high Mach numbers between ten [39] and 25 [28]. However, since those simulations are then no longer direct ones, it is in principle possible that the shock-capturing techniques lead to artifacts in the small-scales statistics of certain quantities. The E polarization is particularly sensitive [40], so we will first compare direct and large eddy simulations in equivalent situations, before studying different physical effects.

The most challenging simulations are those of supernova-driven turbulence, which have been described in details in Refs. [28, 29]. We use a parameterized cooling function, as done in [26]. We will simulate a domain of size 1 × 1 × 2 kpc^{3}, using typically 1152^{2} × 2304 meshpoints over about one Gyr, to accumulate sufficient statistics. For our project, it will also be important to solve for the differential motion of dust grains relative to the gas. Such studies have recently been done with the Pencil Code; see Ref. [41]. In a separate sets of simulations, we will also include partial ionization of the cold ISM, where ions can slip through the neutral component (ambipolar diffusion). This has recently been simulated using both single-fluid and two-fluid descriptions [40, 42]. It turned out that, similar to our earlier work on decaying turbulence [5], this can lead to a marked skewness in the E polarization; see Fig. 3. This suggests the possibility that by measuring the skewness of the E polarization in the dust foreground polarization, one can infer the degree of ionization.

Unlike dust polarization, where the polarized intensity is independent of the modulus of the magnetic field, for synchrotron radiation, it is either proportional to B_{⊥}^{1.9} [43] or to B^{2}B_{⊥}^{1.9}, if the cosmic ray electron density can be assumed to be proportional to the magnetic energy density [44]. The latter is justified by equipartition arguments, but this continues to be subject of debate [45] and therefore we will study both cases. In the presence of helical magnetic fields, its effect on the wavelength dependence of Faraday depolarization will be studied [46, 47] and compared with possible signatures in the EB cross correlation.

In summary, we will perform the following set of simulations:

• Semi-global simulations with rotation and shear (Ω = −S = 25 Gyr^{−1} for the solar neighborhood, but higher values will also be studied for comparison), supernova driving, covering a temperature range between 300 K and 10^{8} K, to determine EE/BB excess and to study the resulting synchrotron and dust emissions, and to study Faraday depolarization at different wavelengths.

• Local simulations of small isothermal (or nearly isothermal) domains reaching rms Mach numbers of around 10, to study the EE/BB excess.

• Study the skewness of E using idealized simulations with and without helicity, and with partial ionization.

• Study EB cross polarization in the ISM to determine and revisit the possibility of a lefthanded magnetic field in the solar neighborhood, as has recently been claimed [37].

All these simulations will also be analyzed using more advanced techniques that we describe next.

B. Beyond the Two-Point Function

The standard power spectrum C_{l} captures all of the information if the field is isotropic and Gaussian. Neither is true in this case, as the magnetic field and dust grain density depend sensitively on Galactic latitude and longitude, especially in and out of the Galactic arms. To extract more information about the ISM, we propose to introduce new statistics that go beyond the standard two-point function

C_{l} ≡ \frac{1}{2l + 1} ∑_{m} |a_{lm}|^{2}.  (1)

1. Searching for Anomalies

In this section we propose novel data driven machine learning (ML) methods for finding anomalous events in simulated and real data sets. These methods will enable us to learn what these anomalies could be due to and we will be able to propose follow-up observations or model these spots in such a way that they can be removed from maps that are used in standard 2-point analyses.

We propose methods for two different settings: i) detection of known structures and ii) anomalous event detection. In the “detection of known structures” task, we propose algorithms for the case in which some interesting events (e.g. Lagrangian coherent structures, vortices) have already been identified, and their goal is to find similar ones in the provided datasets. In the “anomalous event detection” problem the methods work in a completely unsupervised way; they require only real or simulated turbulence data and using this data they will try to identify the most anomalous parts automatically without requiring a labeled set of anomalous events. The methods proposed here will be applied to the 3D simulations. Once the anomalous regions have been detected in the 3D simulations, we can analyze the ensuing 2D maps to understand the characteristic signatures that are imprinted in the polarization data and then look for those signatures in the Planck data.

Detecting Known Structures in Turbulent Data. In many scientific problems, scientists have a collected set of labeled examples and their goal is to find similar examples in the rest of the data. For example, when our goal is to find vortices or other known structures in a turbulent dataset, we might already have a labeled collection of vortices and another labeled set of examples not containing vortices. Our goal is to develop a detection algorithm that can learn the similarities and dissimilarities between these classes and to use this acquired knowledge to detect more vortices (or other structures) in the rest of the dataset.

This problem belongs to the family of classification problems. There are numerous algorithms proposed for classification problems and they are very well-studied in ML. One crucial challenging issue, however, is that most ML algorithms developed for similar classification problems can operate only on simple finite dimensional feature vectors, and they cannot be directly applied to the simulations, where the inputs are not simple finite dimensional vectors. Advances in functional data analysis [48], nonparametric statistics, and ML, however, recently have started to study the question of how to develop learning systems that can process these complex data sets. This is a very rapidly developing field, with many open questions both in theory and practice.

(a)Positive  (b)Negative  (c)Negative  (d)Detected anomaly

FIG. 4: a,b,c: Training examples for the vortex classifier. d: Detected anomaly in the JHU[59] turbulence data. See Page 10 for further description.

As a specific example, suppose we are given a turbulence data set and we already identified a few interesting events in this data (e.g., vortices), and our goal is to find similar events. (See Fig. 4). In this case, we assume that these collected events are given as vector field patches, where each vector in the vector field patch has a start and an end position. We can represent any vector field patch as a set of vectors also known as a point cloud. Our goal can be rephrased as a supervised machine learning problem where given a set of positively and negatively labeled point clouds as training data (Fig. 4 (a,b,c)), we want to find other events (i.e. other point clouds) in our dataset that are most similar to the positively labeled examples (vortex events) and dissimilar to the negatively labeled examples (non-vortex). Co-I Poczos has recently proposed Reproducing Kernel Hilbert Space and Deep Neural Network based methods for this problem. We briefly summarize them below.

Reproducing Kernel Hilbert Space Based Estimators. Poczos introduced Reproducing Kernel Hilbert Space based methods (Support Vector Machines and Gaussian Processes) to the domain of vector fields, point cloud data, and distributions [49]. The main idea is that kernel based methods can operate on any input domains as long as we can define and evaluate a positive semi-definite kernel function k(p, q) on this domain. If the input instances are point clouds, then we can also think of each point cloud as a sample set sampled from an unknown distribution with unknown density p. Several kernel functions have been proposed that can operate on distributions and point clouds. They include the maximum mean discrepancy (MMD), polynomial kernel (k(p, q) = (1 + ∫ pq)^{d}, Gaussian kernel k(p, q) = c_{1} exp(− ∫ (p − q)^{2})/c_{2}), and many others. A crucial challenge is that in practice we do not know the density functions p and q of the point cloud data. All we have is some sample points sampled from p and q and we need to estimate k(p, q). Recently our group has demonstrated that under some conditions k(p, q) can be estimated efficiently from point cloud data. The estimators are completely non-parametric; we do not need to assume that p or q would have a Gaussian or other parametric distribution. One might also replace the L_{2} distance in the Gaussian kernel to other divergences D(p, q), such as Jensen-Shannon divergence, symmetric Renyi divergence or others. In this case the Gaussian kernel between two distributions becomes k(p, q) = c_{1} exp(−D(p, q)/c_{2}).

Recently we developed efficient nonparametric estimators for these divergences. They can be used to create a positive semi-definite kernel function and plug that kernel function into a support vector machine (or other kernel machines) that can be used for classification and regression. We call this method support distribution machine (SDM), and have demonstrated in many practical problems (cosmology [50, 51], computer vision [49], and civil engineering [52]) that these algorithms can often outperform the state-of-the art. We have also generalized other ML tools (such as Lasso [53], Ridge regression [54], and Deep Convolutional Neural networks [55] to distribution, set, function, and other complex inputs, and these algorithms can also be used to process turbulence data.

Nonparametric Divergence Estimation. Below we briefly summarize how we can use point cloud data with unknown densities p and q to estimate the kernel value k(p, q). A nice property of these estimators is that in many cases they are simple and can estimate these functionals of the density without estimating the density itself.

Let p, q be ℝ^{d} → ℝ density functions, and let α ∈ ℝ \ {1}. The Rényi-α divergence (D_{α}^{R}) is defined as the following functional of the densities p and q: D_{α}^{R}(p∥q) ≐ \frac{1}{α−1} log ∫ p^{α}(x)q^{1−α}(x)dx.

Let X_{1:n} ≐ (X_{1}, …, X_{n}) be an i.i.d. sample from a distribution with density p, and similarly let Y_{1:m} ≐ (Y_{1}, …, Y_{m}) be an i.i.d. sample from a distribution having density q. Let ρ_{k}(i) denote the Euclidean distance of the k^{th} nearest neighbor of X_{i} in the sample X_{1:n}, and similarly let ν_{k}(i) denote the distance of the k^{th} nearest neighbor of X_{i} in the sample Y_{1:m}. Let Γ denote the gamma function. Recently, Wang et al. [56] derived an estimator for the kl-divergence (D^{KL}(p, q) ≐ ∫ p log \frac{p}{q}), and we proposed a consistent estimator for the Rényi-α divergence (Póczos et al. [57]):

D̂^{KL}(X_{1:n}∥Y_{1:m}) ≐ \frac{d}{n} ∑_{i=1}^{n} log \frac{ν_{k}(i)}{ρ_{k}(i)} + log \frac{m}{n − 1}.

D̂_{α}^{R}(X_{1:n}∥Y_{1:m}) ≐ \frac{1}{α − 1} log \frac{1}{n} ∑_{i=1}^{n} (\frac{(n − 1)ρ_{k}^{d}(i)}{mν_{k}^{d}(i)})^{1−α} \frac{Γ(k)^{2}}{Γ(k − α + 1)Γ(k + α − 1)}.

In Figure 5, we illustrate how to calculate ρ_{k}(i) and ν_{k}(i) quantities that are needed for these estimators. They can be used to estimate the kernel function k(p, q).

FIG. 5: Calculating ρ_{k}(i) and ν_{k}(i). Blue dots with X signs and red dots show samples X_{1:n} and Y_{1:m}, respectively. We fixed k = 3 in this example, i.e. we need to calculate the 3rd nearest neighbors of each X_{i}.

Methods for anomalous event detection. In addition to finding instances of known patterns, our methods can also be generalized to find anomalous, rare events in a completely unsupervised way.

One possible approach is to flag those point cloud regions anomalous whose distributions—using the above listed MMD, KL, Renyi-α, or another divergence—is far from most of the distributions of the other regions of the point cloud data. Another possible approach is to use the so-called One-class SVM method [58] with an appropriate kernel that can operate on point cloud data. The One-class SVM method is an unsupervised method for detecting anomalous points in datasets.

Preliminary demonstrations on detection of known structures. To show a preliminary result on the applicability of these methods to detect vortices in turbulence datasets, we performed the following experiment. We trained an SDM classifier using turbulence data from the JHU Turbulence Data Cluster^{3} [59], (TDC). TDC simulates fluid flow through time on a 3-dimensional grid, calculating 3-dimensional velocities and pressures of the fluid at each step. We trained an SDM on a manually labeled training set of 11 positive (containing vortex) and 20 negative examples (not containing vortices). Training examples are shown in Fig. 4(a–c). This classifier was then used to evaluate groups along z-slices of the data. One slice of the resulting probability estimates is shown in Fig. 6(a); the arrows represent the mean velocity at each classification point. The highprobability region on the left is a canonical vortex, while the slightly-lower probability region in the upper-right deviates a little from the canonical form. Note that some other areas show somewhat complex velocity patterns but mostly have low probabilities. As we can see in the figure, our method can accurately detect vortices. Leave-one-out cross-validation yielded an accuracy of 97% (one mistake).

In this proposal we will extend these ideas and show that similar methods can also be used to detect other Lagrangian Coherent Structures in turbulent datasets. Using recent advances in ML on big data sets, we will also study how to scale these novel classification algorithms up to large datasets. In related function-to-function regression and distribution regression problems we were able to scale up regression methods to big data problems [60, 61].

Preliminary results on unsupervised anomalous event detection. In the above described TDC simulation we consider the vertices in a local cubic region as a group, and the goal is to find groups of vertices whose velocity distributions (i.e. moving patterns) are unusual and potentially interesting. To demonstrate anomaly detection with SDMs, we trained a “one-class SDM” on 100 typical distributions from the turbulence data, with centers chosen uniformly at random. This oneclass SDM method then was asked to process the data and find regions that are most untypical and look the most different from the examples in the training set. As we can see in Fig. 6(b), the two vortices in the data set are picked out, but the area with the highest score is a diamond-like velocity pattern; these may well be less common and therefore more anomalous in the dataset than vortices. In Fig. 4(d) we show another detected anomaly in the same TDC dataset. As we can see, this is indeed a rare and interesting interaction among three vortices positioned in a straight line.

In this proposal we will extend these ideas to be able to process large datasets and find different types of anomalies in the turbulence datasets. We believe that SDM classifiers and anomaly detectors can serve as simulation exploration tools that will allow scientists to iteratively look for anomalous phenomena and label some of them. Classifiers could then find more instances of those phenomena and compute statistics about their occurrence, while anomaly detection would be iteratively refined to highlight only what is truly new.

 3 http://turbulence.pha.jhu.edu

(a)Classification probabilities  (b)Anomaly scores

FIG. 6: Classification and anomaly scores along with velocities for one 54 × 48 slice of the turbulence data.

2. Anisotropic Correlation Functions

On of the puzzling features of the Planck observations [35] is that the ratio of EE/BB and the T E cross-correlations remain constant with angular scale and in different regions of the sky. Ref. [6] argued that this is difficult to account for in a simple MHD model because these spectra tend to be very sensitive to direction of the large scale magnetic field and the anisotropy of the MHD waves against this background (in their language, the spectra depend on the anisotropy parameter λ). More generally, this scale and region independence is an important clue to the underlying MHD turbulence in our Galaxy, and we propose to quantify it further in order to learn more about the underlying physics.

A first-pass way to do this is to compute the two-point functions (EE, BB, T E, T B) in small patches on the sky and then see how these vary as a function of Galactic latitude, longitude, largescale magnetic field strength, and column density. Some of this has been done already on the data (e.g., the spectral indices over six sky regions in [35]). Repeating and extending this treatment both on the data and on our simulations is a good first step into the project for a student.

We then propose to pursue more sophisticated approaches to the problem of variation of spectra as a function of background variation (be it the large scale magnetic field, Galactic latitude, etc.). There are several possible approaches, but all can be thought of as applications of marked correlation functions [62, 63]. Consider the Landy-Szalay [64] estimator for the correlation function

Ĉ(θ) = \frac{∑_{ij} Θ_{ij}(θ)T_{i}T_{j}}{∑_{ij} Θ_{ij}(θ)}  (2)

where Θ_{ij}(θ) is equal to zero unless the distance |θ⃗_{i} − θ⃗_{j}| falls within the bin θ, in which case it is equal to one and T_{i} is the temperature (which could be replaced by the E- or B- mode or some combination of these three) in a given pixel. This measures the correlation of the temperature across angular distances θ. The marked correlation function adds in a weight, so that the marked estimator is

Ĉ_{m}(θ) = \frac{∑_{ij} Θ_{ij}(θ)w_{i}w_{j}T_{i}T_{j}}{w̄^{2} ∑_{ij} Θ_{ij}(θ)} .  (3)

where w̄ is the mean value of the weight. For example, the weight could be Galactic latitude. In that case, in regions of high Galactic latitude, the correlation function may be larger than in medium latitude regions. This would be an indication that the temperature fluctuations are more highly correlated in the high Galactic latitude regions. Translating this to Fourier space, the marking enables us to quantify the extent to which the spectra depend on, e.g., Galactic latitude.

We will do this in simulations first, where we have a good understanding of the relation between the small and large scale magnetic fields and the density. From these simulation-based studies, we will aim to define the relevant anisotropic 2-point functions.

3. Quadratic Estimators

We will look at the information contained in the off-diagonal elements

∑_{l≠l′,m,m′} W_{ll′mm′}a_{lm}a_{l′m′}  (4)

with weights W chosen to optimize the information obtained from the data.

As a simple example, let us consider the model of Ref. [6]. In that linear Alfvén wave turbulence model the fluctuations in the intensity field are related to the velocities of the slow and fast wave modes via (Eq. 32 there)

T = C ∑_{s,f} F (ζ_{s,f})v_{s,f}  (5)

where T is the observed temperature; C is a constant; F is a function of magnetic field orientation as governed by the parameter ζ, which ultimately depends on the ratio of the pressure of the gas to that in the magnetic field (Equation 10 there); and v is the small-scale velocity field. The standard treatment is to take the 2-point function of the intensity field, either with itself (the auto-correlation) or with the polarization field (T E). However, since the large-scale variation in the magnetic field orientation modulates the velocity field, if T is decomposed into a_{lm}, then the expectation value of the off-diagonal elements in Eq. (4) will be non-zero. In fact, it is straightforward to show that in this simple model

⟨a_{lm}a_{l′m′}⟩_{l≠l′} ∝ P_{v}(l)F̃(L)  (6)

where P_{v} is the power spectrum of the small-scale velocity field, which can be suitably parameterized and measured, and L is the difference between the two angular momenta l, m and l′m′. Thus measuring the off-diagonal elements of the small-scale intensity field provides information about the large scale field modulating the signal. In this simple example, the result would be a map of the function F (after taking the inverse Fourier transform) or equivalently, the orientation angle of the magnetic field. To optimize the signal, one typically combines the information from many of these quadratic terms, summing as in Eq. (4) to obtain an estimator that is unbiased in F̃ in this example (so, the weights W would have P_{v} in the denominator) and has the minimum variance. An important part of this proposal is to develop these quadratic estimators for the large scale fields.

This technique of measuring the off-diagonal correlations in Fourier space on small scales to infer the properties of a large scale field modulating the signal has been successfully applied in the case of gravitational lensing of the CMB (e.g., from the proposers [65–69]. So, we do not expect any problems in principle. However, there are two important differences in practice that make this problem challenging. The first difference between this case of Galactic polarization and CMB lensing is quantitative: Planck has accurate polarization measurements down to about 10′ (l ∼ 600), while CMB lensing makes use of data that has ten times the resolving power. A more important difference is that in gravitational lensing the relation between the observed temperature field and the large scale field that modulates the fluctuations is known precisely [70]. Here, the situation is more complex, as we require theoretical models for how the large scale field modulate the observables. Part of this proposal then is to bring our expertise in MHD turbulence, both analytic and numerical, to bear on this question of how best to extract information from the data.

This is potentially the most exciting part of this proposal: estimates of, e.g., the magnetic field, via the quadratic estimator can be cross-correlated with other magnetic field maps (e.g., [71]). If the fields are correlated, then this indicates the success of the underlying model that was assumed in deriving the estimator. This method then of applying quadratic estimators to the data can be used in conjunction with more direct measurements to test our models of MHD turbulence in the Galaxy.

C. Applying the Statistics

Although this subsection is the shortest, it is here in the area of applying new statistics to the data and to the simulations where we will spend the most time. One advantage of this proposal (Fig. 1) is that the maps from the simulations will be made public, so the broader community can use them for similar and other analyzes to learn about turbulence.

The simulations will produce maps of polarization as produced by dust grains aligned with the magnetic field (although we have the ability to relax that constraint simply by adding in random rotation angles if the simulations that impose that assumption are unable to replicate Planck results). Therefore, the Planck map that is most relevant is the D353 map [72], designed to extract the contribution from dust.

One part of this proposal is to match the estimated features of the magnetic field derived via quadratic estimators assuming decomposition into Alfvén, fast, and slow modes [6, 7] against other measurements of the magnetic field. For this purpose, we will use published models of the large scale magnetic field (e.g., [71]) and also interface with the open IMAGINE Consortium [73].

IV. PROPOSAL TEAM

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 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, that will also be used in the present work. Drawing from his experience with supernova-driven turbulence in the interstellar medium [27], he will perform new simulations of interstellar turbulence. Having previously analyzed synchrotron emission from helical magnetic fields [46], he will use this expertise to compute the dust polarization signal from those simulations. He has analyzed data from NASA’s Solar Dynamics Observatory [74] and the SOLIS instrument [75] 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 [65, 66, 76– 80]. 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 [81], 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 imprints on CMB anisotropies, including the off-diagonal correlations [82, 83] and the parity-odd cross correlations T B and EB [36, 84–86]. These works were considered by the Planck collaboration as methodology when constraining primordial magnetic fields [87]. 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 data driven machine learning methods that can operate on distributions and point cloud data [49]. He successfully used these SDM based approaches to estimate the dynamical mass of galaxy clusters [50, 51]. SDMs were trained and tested on the distribution of line-of-sight velocities of galaxies. Our approach is a factor of two improvement over conventional methods. Recently we have also introduced deep learning based methods for a similar problem [88]. We also successfully applied SDM based approaches to find anomalous galaxy clusters [89, 90]. A grand challenge of the 21st century cosmology is to accurately estimate the cosmological parameters of our universe. In [55] we showed that it is possible to estimate these cosmological parameters directly from the point cloud distribution of matter in the universe using deep 3D convolutional networks in a distribution regression framework [54, 91].

• Wilson has developed a scalable Bayesian source separation algorithm that follows the Linear Mixture Model approach [92, 93] 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 [94]). 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.

In addition, the proposers have extensive collaboration experience; see Refs. [5, 33, 34, 86, 95– 106].

V. PROPOSAL TIME-LINE

• Year 1: Run several large simulations with a variety of initial assumptions, (anisotropic turbulence, including anisotropy in multiple directions, turbulence with and without helicity; wide range of temperature, 300–10^{8} K, varying the rotation and shear parameters, different fractions of ionization, two fluids approach, etc.) and physical conditions addressed (ambipolar diffusion, different driving mechanisms; with allowed and precluded dynamo action, with the imposed magnetic field) and produce intensity and polarization maps, and to determine under what circumstances the observed excess of E mode is produced, see for more details §3A. In addition to simulations of dust polarization, we will study the resulting synchrotron emission and Faraday depolarization at different wavenumbers. Under the direct supervision of the PI and co-I’s, one student will be introduced to the Pencil Code and the ray tracing code and another will begin analyzing both Planck data and the simulations. As a first exercise, the student will reproduce previous results on the spectra. Then, s/he will begin implementing the new statistics. The computing resources will be provided by the Swedish National Allocations Committee at the Center for Parallel Computers at the Royal Institute of Technology in Stockholm.

• Year 2: The students and (P,co-)I’s will work intensively during this year to implement all possible statistics discussed in §IIIB on both simulated and Planck maps. More simulations will be run (and all ensuing maps will be released to the public in HEALPix [107] format.). A particular interest is to study the parity odd cross correlations of polarization (T B and EB) and explore an option of the preferred handedness of the Galactic magnetic field and our Solar neighborhood.

• Year 3: Tweak the statistics and run additional simulations based on the findings of the first two years. Prepare a summary paper that highlights the statistics that will be most useful to the community. Compare Planck-based estimates of the magnetic field developed using the quadratic estimator with other magnetic field maps to see if the adopted MHD turbulence is a plausible explanation for the Galaxy.

VI. 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 [108]. The papers included: the proposal that sterile neutrinos could comprise the dark matter [109]; early analyzes of the Sloan Survey [110–116]; a well-cited review of CMB anisotropies [117]; the first predictions for CMB anisotropies for cosmic strings [118]; the signatures of massive neutrinos on the CMB [119]; early models of dark energy [120– 122]; an early analysis of CMB anisotropy data that pointed to a flat universe [123]; and proposals for discriminating models of inflation [124].

• 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. [125].

• NASA ATP Grant NNX10AC85G: Kahniashvili served as a PI for the Probing Cosmological Symmetry Breaking with the Cosmic Microwave Background (2010-2012). Within this project a number of papers have been published [36, 83, 95–99, 126–133]. One of major results including the following important precedents for the current proposal, is calculation of the microwave background temperature distortion due to aberration induced by our proper motion with respect to the microwave background rest frame [129]. 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 [134], 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 [135].

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