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\begin{document}

\centerline{\bf{\large{Scientific/Technical/Management Section }}}

%TK -modified title
%\section{Objectives}
\section{Objectives and Motivations}
%TK replaced by Scott's sentence from Summary at NSPIRES
The aim of this proposal is to use NASA data and simulations to understand turbulence in the Interstellar Medium (ISM).
NASA missions, in particular {\it 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  {\it 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 {\it Planck} and from magnetohydrodynamic (MHD) simulations is puzzling and suggests that there is much we do not understand about the turbulent ISM.
\begin{enumerate}
\item The $E$-mode polarization exceeds the
$B$-mode by almost a factor of two \cite{Adam:2014bub} 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.
\item 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
 \cite{Ade:2015mbc}.
%AB: it is not obvious what is surprising here. Or is there something that I missed?
%AB: I thought we knew for a long time and understand this at least in basic ways.
%AB: If there is something concrete here, we better reveal this here.
\item The intensity
(written in terms of temperature $T$)
and the $E$-mode of polarization are positively correlated ($TE>0$).
The hypothesis that the alignment between density and magnetic
field structures is responsible for the
$EE/BB$ ratio cannot explain the $TE$ cross correlation
on angular scales $\sim10^{\circ}$, larger than those typical
of the interstellar filaments (although it remains plausible for
smaller scales; see Ref.~\cite{clark2014}).
%\item The probability distributions of some key quantities are
%AB: I think it is better to be concrete here. For example the
%AB: B polarization is gaussian, and it is also a key quantity.
\item The probability distributions of some key quantities such
as the magnetic field and the $E$ polarization are
%AB.
%not gaussian. As examples, Ref.~\cite{Ade:2015mbc} showed that the
%AB: no "the". Also, since I now added the examples above, we should
%AB: not say "As examples", but could say "Indeed". But for a bullet
%AB: point this is perhaps too much detail already.
not gaussian. Indeed, Ref.~\cite{Ade:2015mbc} showed that
there is an excess of strong filaments of aligned dust grains
%(left shaded region of Fig.~3 there) and MHD simulations have shown
%AB: modified
(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~\cite{Kritsuk:2017},
and that turbulence has a tendency to produce large tails of negative
$E$-polarization~\cite{Brandenburg:2018khp}.
\end{enumerate}

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: Alfv\'en, slow, or fast waves. Ref.~\cite{Caldwell:2016xkd} 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.~\cite{Kandel:2017xjx} showed that it is a bit easier to explain the spectra if the Alfv\'en waves have low Mach numbers, while \cite{Kritsuk:2017aab} showed that full MHD simulations have the potential to explain not only the spectra but also some of the other items listed above.

\begin{figure}[htb]
  \centering
    \includegraphics[width=15cm]{fig/overview}
   % \vspace*{-2cm}
    \caption{Overview of the proposal. State-of-the-art MHD simulations (described in \S IIIA) will generate synthetic data as close as possible to the Planck maps. New data analysis techniques (\S 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.}
  \label{fig:overview}
\end{figure}



Our objective is to understand MHD turbulence in the ISM by (see Fig.~1) (i) running state-of-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 {\it Planck} data. More details are provided about the new techniques in \S IIIB, but let us simply comment here that, while the $EE$, $BB$, and $TE$ spectra, the two-point functions, are sufficient to characterize the isotropic, Gaussian cosmic microwave background (CMB), they are obviously {\bf 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.

\section{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 {\it 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 \cite{Kamionkowski:1996zd,Kamionkowski:1996ks,
Seljak:1996gy,Zaldarriaga:1996xe}.
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~\cite{Starobinsky:1979ty}. 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. {\it Planck} measured the polarization of radiation that is manifestly Galactic
\cite{Mortonson:2014kja,Flauger:2014gra,Tassis:2014bea}, 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
\cite{Ade:2015tva}. Therefore, understanding this {\it 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''.
%The science questions addressed in this proposal are central questions of the Physics of the Cosmos program.

\section{Methodology and Research Program}

Our approach involves three pieces:
\begin{itemize}
\item 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 (\S IIIA)
%\item Machine learning techniques that will be applied to both {\it Planck} data and the simulations to carry out the E- and B- decompositions
\item 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 (\S IIIB)
\item Apply these new statistics to both the simulations and {\it Planck} data (\S IIIC)
\end{itemize}


\subsection{Turbulence Simulations}

{\it Overview of 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.
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) \cite{Kaplan:1970}. 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
\cite{Brandenburg:1994}. The complexity of the physical processes
is enhanced by turbulent transport and mixing,
turbulent heating in shocks, and various dissipation processes
\cite{Scalo:2004wk}.
Correspondingly, turbulence in the ISM
%is a complex process \cite{Elmegreen:2004wj} and cannot be fully
%AB: better omit "fully"
is a complex process \cite{Elmegreen:2004wj} and cannot be
described in terms of isotropic, homogeneous and incompressible
Kolmogorov hydrodynamic turbulence \cite{Monin:1975}. 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 \cite{MacLow:2003mz}. 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" \cite{Kritsuk:2017}.
The predicted magnetic field probability distribution function (PDF) is
strongly non-Gaussian, and a relative alignment between magnetic
field and density structures is observed \cite{Kritsuk:2017}.
These simulations were also used to reproduce the observed $EE/BB$ ratio
\cite{Kritsuk:2017aab}, 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 $Re \simeq 10^5-10^7$, and we expect the ISM to be turbulent if there are substantially strong forces to drive turbulence ({\it driving mechanisms}) \cite{Elmegreen:2004wj}. Such driving mechanisms have different energy dissipation rates and they include supernova explosions ($3\times10^{-26}$erg${\rm cm}^{-3}$s$^{-1}$), stellar winds ($3\times10^{-27}$erg${\rm cm}^{-3}$s$^{-1}$), proto-stellar outflows ($3\times10^{-28}$erg~${\rm cm}^{-3}$s$^{-1}$), galactic spiral shocks and magneto-rotational instability ($3\times10^{-29}$erg~${\rm cm}^{-3}$s$^{-1}$), and HII regions $3\times10^{-30}$erg~${\rm cm}^{-3}$s$^{-1}$ \cite{Scalo:2004wk}.
Only $3$--$5$\% of the supernova energy supply is needed
to drive turbulence in the ISM with the characteristic {\it inertial} scale around $50-100$pc and with the characteristic speed around $10$km~${\rm s}^{-1}$, comparable to or even exceeding the sound speed in the ISM~\cite{MacLow:2003mz}. 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. When the magnetic Reynolds number $Re_m$ is high enough ($Re_m \gg Re \gg 1$), the field presence crucially changes the turbulent motions in ISM~\cite{Evirgen:2019}. Indeed, the sensitivity of the ISM structure and outflow speeds to the magnetic field is observed in simulations when the 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 \cite{Shukurov:2018}.

Turbulence comes in many flavors: it can be helical or nonhelical; it can be magnetically dominated or
subdominant; and it can possess cross helicity\footnote{%TK added
Helicity characterizes the swirl of a flow or a magnetic field.
} with a systematic alignment
or anti-alignment between magnetic field and velocity.
Of particular interest for the present proposal is our ability to
consider turbulence that can be strongly anisotropic, with multiple
preferred directions, so lacking mirror symmetry statistically, i.e.,
being helical.
There is the exciting possibility that {\it Planck} data can give indications about magnetic helicity
\cite{Kahniashvili:2014dfa}.
This has been a subject of recent studies
\cite{Brandenburg:2018khp,Bracco:2018vyi}, which demonstrated
that parity-odd $EB$ and
$TB$ cross correlations are possible only if turbulence in inhomogeneous.
%To explain the possibility of a nonvanishing $TB$ cross correlations
%\cite{Akrami:2018wkt}, Ref.~\cite{Bracco:2018vyi} invoked the idea
%that the solar neighborhood is surrounded by a coherent helical
%tube-like structure.
%This possibility will need to be re-addressed.


The turbulence simulations described below provide a means of performing numerical experiments in
a variety of circumstances and environments that will allow us to isolate specific physical effects that may be responsible for the {\it Planck} observations.
We can even isolate
specific mathematical terms in the underlying equations that can be responsible
for certain correlations in the $E$- and $B$-modes. For example, \Fig{peb4} shows
the $E$- and $B$-mode polarizations from a
decaying magnetically dominant turbulence simulation.
Here, the flow is mainly driven by the Lorentz force \cite{Brandenburg:2017rnt}.
In Ref.~\cite{Brandenburg:2018khp}, and we found that for both
dust and synchrotron emission, the PDF of the
$E$ polarization is strongly skewed, thereby leading to an enhanced
${EE}/{BB}$ ratio.
%This is comparable with the observed foreground polarization detected
%with {\em Planck} \cite{Adam:2014bub,Akrami:2018wkt}, clearly more
%than what was theoretically expected \cite{Caldwell:2016xkd} and was
%argued to be explicable with Alfv\'en waves \citep{Kandel:2017xjx}.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.96\textwidth]{peb4}
\end{center}\caption[]{\small{
$E$-mode (left) and $B$-mode (right) polarization for isotropic
fully helical magnetohydrodynamic turbulence using data from
\cite{Brandenburg:2016odr} (their Fig.~4d--f).
Dark (light) shades indicate negative (positive) velocity.
Adapted from Ref.~\cite{Brandenburg:2018khp}.}
}\label{peb4}\end{figure*}



\newcommand\pencil{{\sc Pencil Code}}

{\it Details of the simulations.} Co-I Brandenburg is the main developer of the \pencil, a high order finite difference code
(sixth order in space and third order in time) that is publicly
available through
\url{http://github.com/pencil-code}.
It is currently being developed by 72 members of the scientific community. The total number of publications acknowledging the {\sc Pencil Code} is around 500~\cite{PCcits}. The {\sc Pencil Code} has been successfully applied to a broad range of
turbulent flows, including all the flows to be studied in this paper.


In our proposed research, we plan to further explore turbulence in the ISM
through a variety of numerical simulations using the \pencil.
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 \cite{BKM07}.
The simulations will incorporate density stratification and the different statistics of
Supernova Types I and II, which have different rates and scale heights.
This was first done in our work of nearly 20 years ago \cite{KBSTN99}.
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, we will focus on the cold ISM in smaller domains, where we can ignore shear and density
stratification, and focus on highly supersonic turbulence that is just
driven by volume forcing~\cite{KBSTN99,BKM07,Gent13a,Gent13b,Kritsuk:2017}.


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 \cite{HBM04}. By comparison, using shock diffusivities, we can run high Mach
numbers between ten \cite{Gustafsson:2006} and 25 \cite{Gent13a}.
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 \cite{Brandenburg:2019},
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 detail in Refs.~\cite{Gent13a,Gent13b}.
We use a parameterized cooling function, as done in \cite{BKM07}.
We will simulate a domain of size $1\times1\times2$ kpc$^3$, using typically
$1152^2\times2304$ 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 {\sc Pencil Code}~\cite{Mattsson19}.
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 \cite{Xu_etal19,Brandenburg:2019}.
It turned out that, similar to our earlier work on decaying turbulence
\cite{Brandenburg:2018khp}, this can lead to a marked skewness in the
$E$ polarization; see \Fig{phisto_eb_comp_selected}.
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.


{\it Maps.} In all the simulations, to make contact with {\it Planck} observations, we will integrate the intrinsic
polarization along the line of sight assuming that dust grains are aligned with the magnetic field. This feature of the {\sc Pencil Code}\ has already been implemented widely (for example see Refs.~\cite{Bhat2013,Brandenburg:2014zia}). The end product will be maps of intensity and polarization of the radiation emitted by dust as well as synchrotron emission from the gas circulating in the magnetic fields, all in HEALPix~\cite{Gorski:2004by} format.
Unlike dust polarization, where the polarized intensity is independent
of the modulus of the magnetic field, for synchrotron radiation, it is
either proportional to $\BB_\perp^{1.9}$ \cite{GS65}
or to $\BB^2\BB_\perp^{1.9}$, if the cosmic ray
electron density can be assumed to be proportional to the
magnetic energy density \cite{BB96}.
The latter is justified by equipartition arguments, but this
continues to be subject of debate \cite{SBMS06} and therefore we will
study both cases.
Faraday rotation measure maps will also be produced and in particular, we will study the impact of helical magnetic fields on the wavelength
dependence of Faraday depolarization~\cite{BS14,HF14}.
%and compared with possible signatures in the $EB$ cross correlation.



%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.~\cite{Brandenburg:2018khp} 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.~\cite{Brandenburg:2018khp,Bracco:2018vyi} 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 shown in \Fig{peb4}.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.98\textwidth]{phisto_eb_comp_selected}
\end{center}\caption[]{\small{
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,
${\rm St}_{\rm AD}$.
Note the sign change of skew~$E$.
Adapted from Ref.~\cite{Brandenburg:2019}.}
}\label{phisto_eb_comp_selected}\end{figure*}



In summary, we will perform the following set of simulations:
\vspace{-1mm}
\begin{itemize}\itemsep-1pt
\item
Semi-global simulations with rotation and shear ($\Omega=-S=25\Gyr^{-1}$
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.
\item
Local simulations of small isothermal (or nearly isothermal) domains
reaching rms Mach numbers of around 10 to study the $EE/BB$ excess.
\item
Idealized simulations with and without
helicity and with partial ionization to understand to PDF of the $E$-mode of polarization.
%\item
%Study $EB$ cross polarization in the ISM to determine and revisit the
%possibility of a left-handed magnetic field in the solar neighborhood,
%as has recently been claimed \cite{Bracco:2018vyi}.
\end{itemize}
All these simulations will also be analyzed using more advanced
techniques that we describe next.

\subsection{Beyond the Two-Point Function}

The standard
%TK C_l is the power spectrum
power spectrum
%two-point function
%AB: ok, that's better
$C_l$ 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
\begin{equation}
C_l\equiv \frac{1}{2l+1}\,\sum_m \vert a_{lm} \vert^2.\eql{cldiag}
\end{equation}

\newcommand\be{\begin{equation}}
\newcommand\ee{\end{equation}}

\subsubsection{Searching for Anomalies}

\input{AnomalyDetection}

\subsubsection{Marked Correlation Functions}

On of the puzzling features of the Planck observations~\cite{Akrami:2018wkt} is that the ratio of $EE/BB$ and the $TE$ cross-correlations remain constant with angular scale and in different regions of the sky. Ref.~\cite{Caldwell:2016xkd} argued that this is difficult to account for in a simple MHD model. In these models, the spectra tend to be very sensitive to direction of the large scale magnetic field and the anisotropy of the MHD waves propagating in the background field (in their language, the spectra depend on the anisotropy parameter $\lambda$). Since the field and the ensuing anisotropy vary across the sky, the fact that the spectra remain stable is quite difficult to understand.
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
($EE,BB,TE,TB$) in small patches on the sky and then see how these vary as a function of Galactic latitude, longitude, large-scale 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 ~\cite{Akrami:2018wkt}). 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~\cite{Sheth:2005aj,Satpathy:2019nvo}. Consider the Landy-Szalay~\cite{Landy:1993yu}  estimator for the correlation function
\be
\hat C(\theta) = \frac{\sum_{ij} \Theta_{ij}(\theta) T_iT_j}{\sum_{ij} \Theta_{ij}(\theta) }
\ee
where $\Theta_{ij}(\theta)$ is equal to zero unless the distance $\vert\vec \theta_i-\vec \theta_j\vert$ falls within the bin $\theta$, 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 $\theta$. The marked correlation function adds in a weight, so that the marked estimator is
\be
\hat C_m(\theta) = \frac{\sum_{ij} \Theta_{ij}(\theta) w_iw_jT_iT_j}{\bar w^2\sum_{ij} \Theta_{ij}(\theta) }.
\ee
where $\bar w$ 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.
%This has been done for the magnetic field in ~\cite{2016ApJ...818..178L}, but we propose to extend the work to


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 marked 2-point functions.

\subsubsection{Quadratic Estimators}

The basic statistics used to analyze maps such as those produced by the {\it Planck} team are the two-point functions, such as the $C_l$ defined in \ec{cldiag}.
The estimators for these decompose the maps into spherical harmonics with coefficients $a_{lm}$ and then take the product $\vert a_{lm}\vert^2$. This captures all the information if the underlying field is gaussian and isotropic. Since neither is true for the radiation produced by the turbulent ISM, we will look at the information contained in the off-diagonal elements
\begin{equation}
\sum_{l\ne l',m,m'} W_{ll'mm'}a_{lm} a_{l'm'}\eql{offd}
\end{equation}
with weights $W$ chosen to optimize the information obtained from the data.

As a simple example, let us consider the model of Ref.~\cite{Caldwell:2016xkd}. In that weak
MHD turbulence model,
the fluctuations in the temperature and polarization fields are related to the velocities of the slow and fast wave modes. For example, we can schematically write Eq.~(32) there for the temperature field as
\be
T = C \sum_{s,f} F(\zeta_{s,f})  v_{s,f}
\ee
where $T$ is the observed temperature; $C$ is a constant; $F$ is a function of magnetic field orientation as governed by the parameter $\zeta$, 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 for each of the modes. 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 ($TE$). 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 \ec{offd} will be non-zero. In fact, it is straightforward to show that in this simple model
\be
\langle a_{lm} a_{l'm'} \rangle_{l\ne l'} \propto P_v(l) \tilde F(L)
\ee
where $P_v$ is the power spectrum of the small-scale velocity field, which can be suitably parameterized and measured; $L$ is the difference between the two angular momenta $l,m$ and $l'm'$; and $\tilde F$ is the Fourier transform of the function of the magnetic field orientation. So, $F$ or its Fourier transform $\tilde F$ can be thought of as a map of the direction of the Galactic magnetic field. 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 \ec{offd} to obtain an estimator that is unbiased in $\tilde F$ 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:
\begin{itemize}
\item Gravitational lensing of the CMB (e.g., from the PI~\cite{Baxter:2014frs,
Baxter:2016ziy,GL,Baxter:2018kap,Abbott:2018ydy})
\item Distortions in the CMB due to the aberration induced by our proper motion with respect to the microwave background rest frame (co-I Kahnishvilli~\cite{Kosowsky:2010jm}).
\end{itemize}
%We first pointed out that non-diagonal multipole correlations from aberration would be detectable in  {\it Planck} sky maps, and the effect was then detected at the predicted level by {\it Planck} team in the 2013 results Paper XXVII \cite{Aghanim:2013suk},
So, we do not expect any problems in principle. However, there are two differences between this case and, e.g., CMB lensing,  that make this problem challenging. The first difference is quantitative: Planck has accurate polarization measurements down to about $10'$ ($l\sim600$), 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~\cite{Hu:2001tn}. 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 one of 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.,~\cite{Jansson:2012pc}). 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.



\subsection{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 {\it Planck} results). Therefore, the {\it Planck} map that is most relevant is the D353 map~\cite{Abergel:2013fza}, 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 Alfv\'en, fast, and slow modes~\cite{Caldwell:2016xkd,Kandel:2017xjx} against other measurements of the magnetic field. For this purpose, we will use published models of the large scale magnetic field (e.g., ~\cite{Jansson:2012pc}) and also interface with the open IMAGINE Consortium \cite{Boulanger:2018zrk}.

\section{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,
\begin{itemize}
\item
{\bf Brandenburg} has led the development of the {\sc Pencil Code},
%a public domain tool box (\url{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 \citep{KBSTN99}, he will perform new simulations of
interstellar turbulence.
Having previously analyzed synchrotron emission from helical magnetic
fields \cite{BS14}, he will use this expertise to compute the dust
polarization signal from those simulations.
%He has analyzed data from NASA's {\em Solar Dynamics Observatory} \cite{BPS17} and the {\em SOLIS} instrument \cite{Singh_etal18} and will utilize this experience to analyze the $E$- and $B$-mode polarization of solar data.
%
\item
{\bf Dodelson}
has broad experience analyzing data from the South Pole Telescope~\cite{Baxter:2017ixz,Manzotti:2017net,Soergel:2016mce,
Baxter:2016ziy,Kirk:2015dpw,Giannantonio:2015ahz,Baxter:2014frs}. 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~\cite{Poh:2016rfa}, 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.
%
\item
{\bf Kahniashvili} has expertise in turbulence and in proposing new ways of analyzing microwave data~\cite{Durrer:1998ya,Kahniashvili:2008sh,Kahniashvili:2005xe,Kahniashvili:2014dfa,
Caprini:2003vc,Kahniashvili:2016bkp}. These works were considered by the {\it Planck} collaboration as methodology when constraining primordial magnetic fields \cite{Ade:2015cva}. In the proposed research,
Kahniashvili will work on further extensions of the ISM
module of the {\sc Pencil code} and she
will lead the effort to interpret the simulated MHD turbulence data.
%and the parity-odd cross correlations $TB$ and $EB$
%physics
%(primordial magnetic fields, Lorentz and parity symmetry violations, etc.),
%imprints on CMB anisotropies, including the

%
\item{\bf Poczos} has expertise in machine learning that he has applied to a wide variety of astrophysical problems by working with astronomers at Carnegie Mellon. He developed data driven machine learning methods that can operate on distributions and point cloud data \cite{poczos12CVPR}. He successfully used  these SDM based approaches to \emph{estimate the dynamical mass of galaxy clusters} \cite{Ntampaka16Dynamical,Ntampaka15Machine}. SDMs were trained and tested on the distribution of line-of-sight velocities of galaxies, leading to % Our approach is
a factor of two improvement over conventional methods. Recently he also introduced deep learning based methods for a similar problem \cite{Ho:2019zap}
and successfully applied SDM based approaches to find anomalous galaxy clusters \cite{xiong11hierarchical,xiong11group}.
    A grand challenge of the 21st century cosmology is to accurately estimate the cosmological parameters of our universe. In \cite{Ravanbakhsh16ICML} 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 \cite{poczos13DFDR,szabo16JMLR}.
%
\item
{\bf Wilson} has developed %a scalable Bayesian source separation algorithm that follows the Linear Mixture Model approach \cite{Wilson:2012quiros,Wilson:2008wilson} 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
algorithms for $E,B$-mode decomposition that scale so they can run on Planck-sized data (following work in \cite{Wilson:2004whiley}). On a modest cluster, an all-sky reconstruction can be completed in under an hour. He will work on scaling the codes that implement some of these algorithms so they can be realistically applied to both the simulations and the {\it Planck} data.
%TK: Wilson's stuff must be rewritten
%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.
\end{itemize}
In addition, the proposers have extensive collaboration experience; see
Refs.~\cite{
Kahniashvili:2010gp,
Kahniashvili:2012vt,
Tevzadze:2012kk,
Kahniashvili:2012uj,
Brandenburg:2014mwa,
Kahniashvili:2015msa,
Kahniashvili:2016bkp,
Brandenburg:2016odr,
Brandenburg:2017rcb,
Brandenburg:2017rnt,
Brandenburg:2017neh,
Brandenburg:2018ptt,
Pol:2018pao,
Brandenburg:2018khp,
Kahniashvili:2018mzl,
Pol:2019yex}.
%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.
%
\section{Proposal Timeline}
\begin{itemize}
\item{\bf 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 \S 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
    {\sc Pencil Code}
    and the ray tracing code and another will begin analyzing both {\it 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.
%\item{\bf Year 2:} The students and (P,co-)I's will work furiously during this year
%AB: intensively
\item{\bf Year 2:} The students and (P,co-)I's will work intensively during this year
to implement all possible statistics discussed in \S IIIB on both simulated and  {\it Planck} maps. More simulations will be run (and all ensuing maps will be released to the public in HEALPix~\cite{Gorski:2004by} format.).
    %TK added
    A particular interest is to study the parity odd cross correlations of polarization ($TB$ and $EB$) and explore an option of the preferred handedness of the Galactic magnetic field and our Solar neighborhood.
\item{\bf 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 {\it Planck}-based estimates of the magnetic field developed using the quadratic estimator with other magnetic field maps to see if the adopted
    %TK or should we say considered? we cannot say linear MHD turbulence
    MHD turbulence is a plausible explanation for the Galaxy.
\end{itemize}

\section{Results from Prior NASA Support}

%{\bf Dodelson} (PI) and {\bf Kahniashvili} (Co-I) have been supported through the following NASA grants:
\begin{itemize}
\item {\bf Dodelson}\\
\underline{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~\cite{Dodelson:2003ft}. %The papers included: the proposal that sterile neutrinos could comprise the dark matter~\cite{Dodelson:1993je}; early analyzes of the Sloan Survey~\cite{Dodelson:2001ux,Scranton:2001xa,Szalay:2001rw,
%Zehavi:2001nr,Stoughton:2002ae,Tegmark:2003ud,Abazajian:2003jy}; a well-cited review of CMB anisotropies~\cite{Hu:2001bc}; the first predictions for CMB anisotropies for cosmic strings~\cite{Allen:1997ag}; the signatures of massive neutrinos on the CMB~\cite{Dodelson:1995es}; early models of dark energy~\cite{Coble:1996te,Dodelson:2001fq,Ahmed:2002mj}; an early analysis of CMB anisotropy data that pointed to a flat universe~\cite{Dodelson:1999am}; and proposals for discriminating models of inflation~\cite{Dodelson:1997hr}.\\
\underline{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.~\cite{Dodelson:2009zz}.
%
\item {\bf Kahniashvili}\\
\underline{NASA ATP Grant NNX10AC85G:}
{ Kahniashvili} served as a PI for the  {\it Probing Cosmological Symmetry Breaking with the Cosmic
Microwave Background} (2010-2012). Within this project
a number of papers have been published
\cite{
Kahniashvili:2008sh,
Kahniashvili:2009mf,
Kahniashvili:2009qi,
Kahniashvili:2010us,
Kahniashvili:2010wm,
Kahniashvili:2010gp,
Kosowsky:2010jm,
Chitov:2009ph,
Campanelli:2011qd,
Kahniashvili:2012vt,
Tevzadze:2012kk,
Kahniashvili:2012dy,
Kahniashvili:2012uj,
Brandenburg:2014mwa,
Kahniashvili:2014dfa}.
One of major results, which has relevance to the current proposal, is a
calculation of the microwave background temperature distortion due to
aberration induced by our proper motion with respect to the microwave background rest frame \cite{Kosowsky:2010jm}.
She used a similar techniques to investigate whether a dipole modulation of the microwave background anisotropies could be responsible for the hemispherical power asymmetry seen in both WMAP and {\it Planck} sky maps \cite{Aiola:2015rqa}.
\end{itemize}

\newpage



\begin{thebibliography}{widest-label}
%




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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
%\bibliographystyle{apalike}
%\bibliography{refs}

%\vfill\bigskip\noindent\tiny\begin{verbatim}$Header: /var/cvs/brandenb/tex/tina/prop/ADAP19EB/NASA19.tex,v 1.63 2019/05/17 17:00:41 tina Exp $\end{verbatim}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%TK
%PLANCK









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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%






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%%
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