text:
\section{Introduction}

The hyperfine transition of neutral hydrogen at 21 cm
provides a unique source of cosmological information from the
epoch of reionization (EOR) and the dark ages
\cite{Furlanetto,Pritchard:2011xb}.  It is the target of
several ongoing and near-future ground-based experiments
\cite{LOFARSKA}, as well as more distant prospects such as a
Lunar-based observatory \cite{Lunar}. During the two
relevant cosmological epochs, the dark ages and the EOR, the
hyperfine transition is observed in absorption or emission,
respectively, against the cosmic microwave background (CMB).  By
measuring this signal at different frequencies, the large
redshift volume of these two epochs can be used to generate
independent images of the spatial distribution of neutral
hydrogen at different redshifts
\cite{Zaldarriaga,LoebZald,CAMB_sources}.

The image of the 21-cm signal from the dark ages and/or EOR
should appear on the sky as a random field described by some
statistically isotropic two-point correlation function.
If, however, that image is distorted by weak gravitational
lensing from foreground matter---either the large-scale
inhomogeneous distribution of mass in the Universe or by
discrete objects, like galaxy clusters---then there may be local
departures from statistical isotropy induced
\cite{ZahnZald,Metcalf,MetcalfWhite,Hilbert,LuPen,Sigurdson:2005cp,Book:2011dz}.
Measurement of these local departures from statistical isotropy
may thus allow a measurement of the distribution of this
intervening matter.

The effects of gravitational lensing of 21-cm anisotropies are the
same as the analogous effects on CMB fluctuations and can
therefore be analyzed with the tools developed for lensing of
the CMB \cite{Seljak,Okamoto,HuDeDeo}.  However, the extension
of 21-cm fluctuations to far smaller angular scales (limited in
principle only by the baryonic Jeans mass \cite{LoebZald}) than
CMB fluctuations (which are suppressed on small scales by Silk
damping \cite{Silk:1967kq}), and the possibility to see
images of the 21-cm background at multiple redshifts, make
21-cm lensing far more promising, ultimately, for weak-lensing
studies.
In particular, constraining the parameters of galaxy cluster mass profiles using weak lensing reconstruction can be a useful tool in probing the evolution of dark energy \cite{Albrecht} and studying the properties of dark matter (e.g. through the characterization of substructure or the mapping of its distribution in merging clusters).

High-resolution imaging of the mass distribution in
galaxy clusters using 21-cm lensing was investigated in
Ref.~\cite{MetcalfWhite} (and with simulations in
Ref.~\cite{Hilbert}).  This work concluded that while
forthcoming experiments may have the potential to provide some
initial detections of galaxy clusters, the full promise of the
technique will likely have to await subsequent generations.

The goal of this paper is to revisit lensing of the 21-cm
background by galaxy clusters  with an analytic treatment
aimed primarily to help understand the dependence of the
detectability of the signal on experimental parameters.  The
aim will be to clarify the experimental requirements for such
detections and to assist in the design of experiments to make
such measurements.  More specifically, we use as a figure of
merit the smallest galaxy-cluster mass detectable by a given
experimental configuration and then investigate the dependence
of this threshold mass on the experimental configuration.

The plan of the paper is as follows:  In
Section II, we review the 21-cm signal and examine its dependence on the observation frequency
and on the bandwidth over which the signal is observed.  In Section III we
discuss the noise power spectrum of radio interferometers and
study its dependence on frequency, bandwidth, angular resolution,
observation time, and collecting area.  In Section IV we review
how lensing of the 21-cm background by galaxy-cluster masses is
accomplished.  We present
a quadratic estimator for the weak-lensing convergence and
derive the noise with which it can be measured. In Section V we
construct a minimum-variance estimator for the galaxy-cluster
mass obtained from the lensing convergence and derive a formula for the smallest detectable galaxy-cluster mass as a function of the various experimental parameters, assuming an Navarro-Frenk-White (NFW) mass profile.
In Section VI we show the results for galaxy-cluster mass detectability with next generation interferometers as well as futuristic ideal experiments and discuss different tradeoffs between the experimental parameters.
In Section VII we compare the prospects for
mass measurements from lensing of 21-cm fluctuations with other
mass measurements, mention possible improvements to our estimator and
discuss some relevant subtleties. We conclude in Section VIII.

\section{21-cm emission/absorption signal}

We begin by reviewing the physics responsible for producing the
21-cm signal from the dark ages and the EOR, whose relative comoving volumes are illustrated in Fig. \ref{fig:Volume}.
\begin{figure}[b!]
\includegraphics[width=0.8\linewidth]{volumeSmall.pdf}
\caption{An illustration of the comoving volume of the universe demonstrating the huge potential for extracting valuable information from 21-cm radiation in the dark ages and the reionization epochs.}
\label{fig:Volume}
\end{figure}
Defining the spin temperature $T_s$ as the excitation
temperature of the hyperfine transition (characterizing the
ratio between the number densities of hydrogen atoms in the
excited and ground-state levels), the rest-frame brightness
temperature of a patch of the sky is given by $T_b=T_{\rm
CMB}e^{-\tau}+T_S(1-e^{-\tau})$, where the optical depth for the
hyperfine transition is
\cite{field59}
\begin{eqnarray}
\tau & = & \frac{ 3 c^3 \hbar A_{10} \, n_{\rm HI}}{16
k \nu_0^2 \, T_S \, H(z) },
\label{tau}
\end{eqnarray}
$A_{10} = 2.85 \times 10^{-15}\, {\rm s}^{-1}$ is the Einstein
coefficient for the transition, $\nu_0=1420 \, {\rm MHz}$ its
rest-frame frequency and $n_{\rm HI}$ is the local neutral
hydrogen density.  The brightness temperature in this patch, at an
observed frequency $\nu$ corresponding to a redshift $1+z=\nu_0/\nu$,
and the CMB is given by the difference
\begin{eqnarray}
\delta T_b(\nu) & \approx & \frac{T_S - T_{\rm CMB}}{1+z} \, \tau .
\label{Tbright}
\end{eqnarray}
Hence if the excitation temperature $T_s$ in a region differs from
that of the CMB, the region will appear in emission (if $T_s>T_{\rm CMB}$)
or absorption (if $T_s<T_{\rm CMB}$) against the CMB.

During the dark ages, there is a redshift period $30<z<200$
where neutral hydrogen should be visible in absorption against
the CMB, as the spin temperature is coupled to the gas
temperature [and cools adiabatically as $(1+z)^2$] while
collisions are efficient and drops below the CMB temperature
[which only cools as $(1+z)$]. This process peaks at
$z\sim70$ and lasts until the Hubble expansion renders
collisions inefficient and $T_s\sim T_{\rm CMB}$ again at
$z\sim30$. The signal from this early epoch is not affected by
nonlinear density structures nor contaminated by astrophysical
sources, which have yet to form.

A 21-cm signal is also accessible during the EOR, at redshifts $7 \lesssim z \lesssim 13$, when
newly formed structure heated up neutral hydrogen but
before the hydrogen became fully ionized.  Predicting the
brightness-temperature signal in this epoch is much harder, as
astrophysical noise sources are substantial and our
uncertainties as to the beginning and duration of this period
are significant. Unlike the dark ages signal, in order to plot the signal during
reionization we need to trace the redshift behavior of the
neutral gas fraction which determines the optical depth,
Eq.~(\ref{tau}), for the hyperfine transition.

In the Appendix, we review the power spectrum of intensity fluctuations of 21-cm radiation. Eqs.~(\ref{eq:21PowerSpectrum})-(\ref{eq:21PowerSpectrumSmallScales}) describe the angular power spectrum and its approximated form in the limit of large and small scales. In observations of these fluctuations, an important factor is the damping that results from line-of-sight averaging in a width $\delta r$ (corresponding to an observed bandwidth $\Delta\nu$) around the distance $r$ to the desired frequency $\nu$, whose scale is determined for a given radio interferometer through the relation
\cite{Zaldarriaga,LoebZald}
\begin{equation}
\delta r/r\simeq 0.5(\Delta\nu/\nu)(1 + z)^{-1/2}.
\label{eq:damping}
\end{equation}
In Fig.~\ref{fig:CLDarkAges} we plot the 21-cm dark ages signal power
spectrum, Eq.~(\ref{eq:21PowerSpectrum}), in several redshifts using
CAMB \cite{CAMB_sources} in the limit of sharp frequency bandwidth
and with a bandwidth of $\Delta\nu=0.1\,{\rm MHz}$ (resulting in damping above $l\sim10^3$) in the linear
regime.
\begin{figure}[htbp]
\includegraphics[width=0.8\linewidth]{DarkAgesPSSmall.pdf}
\caption{21-cm power spectrum for different redshifts during the
dark ages, calculated using CAMB Sources
\cite{CAMB_sources}. Solid lines are calculated in the
narrow-bandwidth limit and the signal is damped by the effect of
baryon pressure at $l\gtrsim10^6$. Dashed lines are for
$\Delta\nu=0.1\,{\rm MHz}$, where the signal is damped due to
line-of-sight averaging over the bandwidth above $l\gtrsim10^3$
as predicted by Eq.~(\ref{eq:damping}) for these redshift ranges.}
\label{fig:CLDarkAges}
\end{figure}
The signal peaks at redshift $z\sim60$ where the deviation between the spin temperature and that of the CMB is maximal \cite{LoebZald}. We see that for a given bandwidth, the signal is roughly within the same order of magnitude for most redshifts up to very small angular scales, corresponding to arcsecond resolutions.

In Fig.~\ref{fig:CLEoR} we plot the 21-cm EOR signal power spectrum, assuming reionization is instantaneous and complete at $z\!=\!7$, using the approximations in Eqs.~(\ref{eq:21PowerSpectrumLargeScales}) and (\ref{eq:21PowerSpectrumSmallScales}) for a bandwidth of $1\,{\rm MHz}$. The damping scale is again approximately $l\sim10^3$ (as the redshift of reionization is an order of magnitude smaller), and the signal is roughly constant for small scales up to $\gtrsim10^4$, corresponding to slightly better than arcminute resolutions.
\begin{figure}[htbp]
\includegraphics[width=0.8\linewidth]{EoRPSSmall.pdf}
\caption{21-cm EOR power spectrum at redshift $z\!=\!7$, assuming complete and instantaneous reionization just after $z\!=\!7$, using Eqs.~(\ref{eq:21PowerSpectrumLargeScales}) and (\ref{eq:21PowerSpectrumSmallScales}) (with an amplitude $\bar{\delta T}_b(\nu)\sim20\,{\rm mK}$ \cite{Zaldarriaga}). The solid line is the signal power spectrum for a bandwidth of $\Delta\nu=1~{\rm MHz}$, which is damped around $l\lesssim10^3$ for this redshift according to Eq.~(\ref{eq:damping}). Dashed lines are noise power spectra for SKA (light gray), SKA with $10\times t_0$ (medium gray) and SKA with $10\times t_0$ and $4\times f_{\rm cover}$ (dark gray). We see that with the final setup we can reach the maximum $l$ observable with SKA's baseline of $6~{\rm km}$ which is roughly $l_{\rm cover}\sim2\times10^4$.}
\label{fig:CLEoR}
\end{figure}

In the next section we compare the 21-cm signal with the noise power
spectrum estimation of radio interferometers, before heading on
to discuss the reconstruction of galaxy clusters from the weak
lensing signal of the 21-cm radiation.

\section{Experimental Prospects}

Following Ref.~\cite{Zaldarriaga}, we examine the noise power
spectrum of a radio interferometer array.
We denote by $l_{\rm cover}(\nu)=2\pi D/\lambda$ the maximum
mode at frequency $\nu$ (corresponding to wavelength $\lambda$)
that can be measured with an array of dishes with  maximum
baseline $D$ covering a total area $A_{\rm total}$ with a covering
fraction $f_{\rm cover}\equiv N_{\rm dish} A_{\rm dish} /A_{\rm
total}$.
For uniform Fourier area coverage, a system temperature $T_{\rm
sys}$, a frequency window $\Delta \nu$, and an observing time
$t_o$, the noise power spectrum is
\begin{equation}
l^2 C_l^n= {(2\pi)^3 T_{\rm sys}^2(\nu) \over \Delta \nu t_o  f_{\rm cover}^2} \ \left({ l\over l_{\rm cover}(\nu)}\right)^2.
\end{equation}
Thus, for uniform Fourier coverage in an experiment targeting a
frequency $\nu$ with bandwidth $\Delta\nu$, we conclude that
$C^n_l \sim {\rm const}\equiv 2\pi\beta(\nu)(f_{\rm cover}^2\,
t_0 \Delta\nu)^{-1}$, where $\beta(\nu)= (2\pi)^2 T_{\rm
sys}^2(\nu) [l_{\rm cover}(\nu)^2]^{-1}$. As we can see from
Figs.~\ref{fig:CLDarkAges} and \ref{fig:CLEoR}, on the relevant (small) scales for cluster reconstruction, the 21-cm radiation
power spectrum at a given frequency and bandwidth behaves roughly (up to an order of magnitude) as $l^2C_l \sim {\rm const}
\equiv 2\pi\alpha(\nu,\Delta\nu)$. To find the value $l_{\rm
max}$ beyond which the noise $C^n_l$ is no longer negligible, we
compare $C_{l_{\rm max}}=C^n_{l_{\rm max}}$ to find
\begin{equation}
l^2_{\rm max}=\frac{\alpha(\nu,\Delta\nu)}{\beta(\nu)} (f_{\rm cover}^2\, t_0 \Delta\nu).
\label{l_max}
\end{equation}
This quantity depends trivially on the fractional coverage and observation time (they only affect the noise). The dependence on frequency and bandwidth is more elaborate, as we discuss later on below.

Observing the 21-cm anisotropies during the dark ages will be very challenging.
The redshift range $30<z<200$ of absorption of the CMB at 21-cm
($\nu = 1420\, {\rm MHz}$) in the rest frame of the
neutral-hydrogen gas corresponds to very low frequencies $7\,
{\rm MHz} < \nu < 46 \,{\rm MHz}$, where the sky temperature (dominated by
foreground sources of synchrotron emission from the Galaxy and
from extragalactic sources) in regions of minimum emission at
high Galactic latitudes, approximately given by $T_{\rm sys}
\sim 180\left(\nu/180\,{\rm MHz}\right)^{-2.6}$~K, reaches
$\sim 10^4-10^6 \,{\rm K}$, many orders of magnitude above the
signal $\sim 1\, {\rm mK}$ (as seen in Figs.~\ref{fig:CLDarkAges} and \ref{fig:CLEoR}).

While both Galactic and extragalactic emissions vary smoothly
and could be possibly subtracted by taking observations at two
closely spaced frequencies, the additional sources of
interference are harder to overcome. Terrestrial radio frequency
interference is abundant in this range and is not spectrally
smooth. Another source of contamination is the ionosphere which
causes phase distortions in the cosmic signal and turns opaque
at frequencies $\nu \lesssim 20\, {\rm MHz}$ (corresponding to
$z > 70$).
A futuristic experiment, based on placing a dark ages
observatory on the far side of the Moon (which has no permanent
ionosphere and its far side is shielded from terrestrial radio
interference) was suggested in Ref.~\cite{Lunar}. With a
baseline on the order of $\gtrsim10-100\,{\rm km}$, angular
resolutions corresponding to $l_{\rm max}\sim10^4\!-\!10^5$ (for
source redshifts $z=30-300$) might be reached in such an
experiment, compensating for covering fraction and system
temperature with prolonged observation time. With sufficient
frequency coverage, the huge volume of CMB absorption in neutral
hydrogen during the dark ages, $30 \lesssim z \lesssim 200$, can
be used to beat down the noise (by combining redshift slices, as we
describe in the next Section) and measure the mass profile
parameters of galaxy clusters to high accuracy.

To reach a measurement within the decade, the best candidate is
the currently planned next generation experiment to measure
21-cm emission from neutral hydrogen during the EOR, the
\emph{Square Kilometer
Array}\footnote{http://www.skatelescope.org} (SKA). With a
design based on an extended region of $D\! \sim\! 6$ km where
$l_{\rm cover}(\nu)\!\sim\! 10^4$ for the relevant frequencies in
the epoch of reionization and $f_{\rm cover}\! \sim\! 0.02$,
SKA can reach an observing time of order $t_0\sim\!1000\, {\rm
hrs}$ in one season covering up to an area of $\sim2\pi\, {\rm
sr}$ in the sky.

In Fig.~\ref{fig:CLEoR} we plot several noise power spectra matching the current plans for a three-season run of SKA measuring at a frequency corresponding to redshift $z\!=\!7$ with a bandwidth of $1\,{\rm MHz}$, as well as scenarios with an order of magnitude larger observation time, and with four times the coverage fraction. With the current plan, the maximum scale observable with a signal to noise greater than one is only $O(l_{\rm max}\!\sim\!10^3)$. To reach $l_{\rm max}\!\gtrsim\!10^4$ will require a considerable increase in coverage fraction and/or observation time.

\section{Lensing of 21-cm Fluctuations by a Galaxy Cluster}

We now review the distortion to the 21-cm fluctuations induced
by weak gravitational lensing.  We assume that the lensing
distortion takes place over a relatively small region of the
sky, as should occur for lensing by a galaxy cluster, so that we
can work in the flat-sky limit, where the analytic expressions
are simpler.

Let $I_0(\vec\theta)$ be the 21-cm intensity at position
$\vec\theta$ on the sky.  Lensing will deflect photons from
$\vec\theta$ by an amount $\delta\vec\theta(\vec\theta) = \nabla
\phi(\vec\theta)$.  Here, $\phi(\vec\theta)$ is the projected
potential, related to the convergence $\kappa(\vec\theta) =
\Sigma(\vec\theta)/\Sigma_{\rm cr}$, by $\nabla^2\phi = 2 \kappa$,
where $\Sigma(\vec\theta)$ is the surface mass density of the
intervening cluster, and $\Sigma_{\rm cr}^{-1}=4 \pi G D_d
D_{ds}/(c^2 D_{s})$ is the critical surface mass density in
terms of the observer-lens, lens-source, and observer-source
angular-diameter distances $D_d$, $D_{ds}$, and $D_{s}$,
respectively.  The observed intensity is thus $I(\vec\theta) =
I_0(\vec\theta+\delta\vec\theta) \simeq I_0(\vec\theta)+
\delta\vec\theta \cdot \nabla I_0(\vec\theta)$.

We suppose that intensity is measured over some square
patch of sky of solid angle $\Omega$, decomposed into $N_{\rm
pix}$ pixels at positions $\theta_i$, for $i=1,2,\ldots,{N_{\rm pix}}$,
surrounding the cluster.  The intensity can then
be written in terms of Fourier coefficients,
\begin{equation}
     I_{\vec l} = \frac{\Omega}{{N_{\rm pix}}} \sum_{\vec\theta_i} e^{i
     \vec l \cdot \vec\theta_i} I(\vec\theta_i),
\end{equation}
as
\begin{equation}
     I(\vec\theta_i) = \frac{1}{\Omega} \sum_{\vec l} e^{-i \vec l
     \cdot \vec\theta}I_{\vec l} ,
\end{equation}
where the sum is over the ${N_{\rm pix}}$ Fourier modes $\vec l$.  The
two-point intensity correlations are described by a power
spectrum $C_l$ defined by
\begin{equation}
     \VEV{ I_{\vec l} I_{\vec l'}^*} = \Omega C_l \delta_{\vec
     l,\vec l'}
\end{equation}
The Fourier coefficients for the observed intensity are related
to those of the unlensed intensity and deflection field by
\begin{equation}
     I_{\vec l} = I_{0 \vec l} - \frac{1}{\Omega} \sum_{\vec l'}
     \vec l' \cdot (\vec l- \vec l') \phi_{\vec l'} I_{0\, \vec
     l-\vec l'},
\end{equation}
from which it follows that for a fixed deflection field, the observed intensity satisfies
\begin{equation}
     \VEV{ I_{\vec l_1} I_{\vec l_2}} = \Omega C_{l_1}\delta_{\vec
     l_1,-\vec l_2} + \phi_{\vec L}
     \vec L \cdot (\vec l_1
     C_{l_1} + \vec l_2 C_{l_2} ),
\end{equation}
where $\vec L = \vec l_1 +\vec l_2$ and the ensemble average is for a fixed deflection field.

Since $\kappa_{\vec L} = -L^2 \phi_{\vec L}/2$,
each $\vec l_1$-$\vec l_2$ pair of
measured intensity Fourier modes with wavevectors $l_1+l_2 =
\vec L$ provides an estimator,
\begin{equation}
     \widehat{ \kappa_{\vec L}^{\vec l_1, \vec l_2}} =
     -\frac{L^2}{2} \frac{
     I_{\vec l_1} I_{\vec l_2}}{      \vec L \cdot (\vec l_1
     C_{l_1} + \vec l_2 C_{l_2} )},
\end{equation}
and the variance of this estimator is
\begin{equation}
     \VEV{ \left( \widehat{ \kappa_{\vec L}^{\vec l_1, \vec
     l_2}} \right)^2} =  \frac{L^4}{4} \frac{C_{l_1}^{\rm
     map} C_{l_2}^{\rm
     map} \Omega^2}{ \left[ \vec L \cdot (\vec l_1
     C_{l_1} + \vec l_2 C_{l_2} ) \right]^2},
\end{equation}
where $C_l^{\rm map}=C_l+ C_l^{\rm n}$ is the power spectrum of
the intensity map, including the noise $C_l^{\rm n}$.

We can then sum the estimators $\widehat {\kappa_{\vec L}^{\vec
l_1,\vec l_2}}$ over all $\vec l_1+\vec l_2 = \vec L$ pairs (correcting for double counting of triangles with $\vec l \leftrightarrow \vec L-\vec l$) with
inverse-variance weighting to obtain the minimum-variance
estimator,
\begin{eqnarray}
     \widehat{\kappa_{\vec L}}
     & = & -\frac{\Omega N_{\vec{L}}}{L^2 }
      \sum_{\vec l} \frac{I_{\vec l} I_{\vec L
     -\vec l} }{ \vec L \cdot(\vec l C_l + (\vec L-\vec l)
     C_{|\vec L-\vec l|})} \nonumber \\
     & & \times
     \left[ \frac{C_l^{\rm map} C_{|\vec
     L -\vec l|}^{\rm map} \Omega^2}{ [\vec L \cdot(\vec l C_l + (\vec L-\vec l)
     C_{|\vec L-\vec l|})]^2}\right]^{-1},
\end{eqnarray}
and
\begin{equation}
     N_{\vec{L}}^{-1} = \frac{2\Omega}{L^4}      \sum_{\vec l}
     \frac {
     [\vec L \cdot(\vec l C_l + (\vec L-\vec l)   C_{|\vec
     L-\vec l|})]^2}
     {C_l^{\rm map} C_{|\vec L -\vec l|}^{\rm map}
     \Omega^2}
\label{eq:huestimator}
\end{equation}
is the noise power spectrum for $\kappa_{\vec L}$.
Equivalently, $\langle| \widehat{\kappa_{\vec L}} |^2\rangle=(2\pi)^2\delta(0)N_{\vec{L}}=\Omega N_{\vec{L}}$ is the variance with which
$\kappa_{\vec L}$ can be measured.

We now use the continuum limit $\Omega^{-1} \sum_{\vec l} \to
\int d^2l/(2\pi)^2$ and employ the simplifying assumption,
Eq.~(\ref{l_max}), that the noise is small,
$C_l^{\rm n} \ll C_l$, so that $C_l^{\rm map}\simeq C_l$, up to
some scale $l_{\rm max}$.
Then, in the limit $L \ll l$ we approximate $|\vec L-\vec l|
\simeq l-L\cos\phi$, where $\cos\phi\equiv \hat L\cdot \hat l$,
and then to first order, $C_{|\vec L-\vec l|} \simeq C_l - L
(\cos\phi) (\partial C_l/\partial l)$, which
yields \cite{ZahnZald}
\begin{eqnarray}
     N_{\vec{L}}^{-1} \!&\simeq&\!  \frac{4}{L^4}\int {\frac{d^2l}{(2\pi)^2}}
     \frac{\left[\vec L\cdot \vec l C_{l} +\vec L\cdot (\vec L-\vec l) C_{|\vec L-\vec l|}\right]^2}{2 \,
     C_l \, C_{|\vec L-\vec l|}} \nonumber \\
                &\simeq&\!
                \frac{l_{\rm max}^2}{2\pi}\left[1+
                \frac{\partial\ln C_l}{\partial \ln l}
                  + \frac{3}{8} \left(\frac{\partial\ln
                  C_l}{\partial \ln l}\right)^2\right].
\label{eqn:vareqn}
\end{eqnarray}
Using the approximation leading to Eq.~(\ref{l_max}), that the the noise for a
single slice in an experiment at frequency $\nu$ with bandwidth
$\Delta\nu$, a maximum baseline corresponding to $l_{\rm
cover}$, a coverage fraction $f_{\rm cover}$, and observation
time $t_0$,  is approximately,
\begin{equation}
N_{\vec{L}} \sim \frac{4\pi}{ l_{\rm max}^2}=\frac{4\pi\beta(\nu)}{\alpha(\nu,\Delta\nu)(f_{\rm cover}^2\, t_0\Delta\nu)}.
\label{VarLimit}
\end{equation}

We can increase our signal considerably by changing the
frequency at which the 21-cm map is made and thereby focus on
spherical shells of neutral hydrogen at different redshifts. The
above results for a single redshift slice can be extended to
make use of the full redshift volume \cite{ZahnZald} by
discretizing the $z$ direction into radial components so that
$C_{l,k}$ is the power in a mode with angular component $l$ and
radial components $k=2\pi j/ R$, where $R$ is the
total radial length of the volume. Under the assumption that
modes with different $k$ are independent, a total-volume
estimator is built by summing the individual estimators, and the
corresponding noise variance is
\begin{eqnarray}
\frac{L^4/4}{N_{\vec{L}}}=\sum\limits_k \int {\frac{d^2l}{(2\pi)^2}} \frac{[\vec L\cdot \vec l C_{l,k}
    +  \vec L\cdot (\vec L-\vec l) C_{|\vec L-\vec l|,k}]^2}{2 C^{\rm
    map}_{l,k}C^{\rm map}_{|\vec L-\vec l|,k}} \nonumber \\
  \label{volumenoise}
\end{eqnarray}

In general, to estimate the number of slices
available, we can think of 21-cm maps at two
different frequencies that correspond to spherical shells
separated along the line of sight by a comoving distance $\Delta
r$. These will be statistically independent at the highest $l$
provided that $(\Delta r/r_{\nu}) \gtrsim l^{-1}$.  An
experiment that covers a spatial range $\delta r$ or a frequency
range $\nu_1-\nu_2$ around a frequency $\nu$ will yield a total
number of $N_z\sim(\delta r/\Delta r) \simeq l(\delta
r/r_{\nu})\simeq 0.5l((\nu_1-\nu_2)/\nu)(1 + z)^{-1/2}$
statistically independent maps, which is roughly $N_z\sim1500$
for the frequencies corresponding to the EOR at the maximum resolution of SKA and up to $N_z\sim5000$ for a
dark ages observatory with a baseline of $100\,{\rm km}$ (here we
neglect the fact that different frequency bins are also somewhat
correlated by contaminations). In practice, however, the
tradeoff is a complicated one. While the signal is larger for
narrow bandwidths, we saw in the last Section that the noise
power spectrum around a given frequency increases as the
frequency bandwidth is decreased, which limits the actual number
of slices that can be combined in Eq.~(\ref{volumenoise}).

\section{Estimating Cluster Properties}

Applying the estimator in Eq.~(\ref{eq:huestimator}) for the
convergence $\widehat{\kappa}_L$ to a patch of sky around a galaxy
cluster (assuming we know the location of its center to enough
accuracy, say from other sources such as Sunyaev-Zeldovich surveys), we can
retrieve a 2D image of the weak-lensing convergence of the
cluster which we can study versus theory or simulation.
The total number of pixels in Fourier space is the same as in real space.  We then denote the $N$ Fourier wavenumbers as $\vec{L_i}$ for $i=1,2,\ldots,N$. If $\widehat{\kappa_{\vec{L}_i}}$ is the
measured value for the pixel $i$ with variance
$\langle|\widehat{\kappa_{\vec{L}_i}}|^2\rangle=\Omega N_{\vec{L}_i}$
and the corresponding theoretical value is $\kappa^{\rm
th}_{\vec{L}_i}(M_{\rm fid})$ (calculated for some fiducial value
$M_{\rm fid}$), then an estimator for the halo mass per pixel is
given by
\begin{equation}
\widehat{M}_i =  \frac{\widehat{\kappa_{\vec{L}_i}}}{\kappa^{\rm
th}_{\vec{L}_i}(M_{\rm fid})}M_{\rm fid},
\label{MassPerPixel}
\end{equation}
with variance
\begin{equation}
     \langle|\widehat{M}_{i}|^2\rangle =  \frac{\Omega
     N_{\vec{L}_i}}{|\kappa^{\rm th}_{\vec{L}_i}(M_{\rm fid})|^2}M^2_{\rm
     fid}.
\label{MassVarPerPixel}
\end{equation}
From these we can build a minimum-variance estimator,
\begin{equation}\label{minvarestimator}
\widehat{M} =  \frac{\sum\limits_i
\widehat{M}_i/\langle|\widehat{M}_{i}|^2\rangle}{\sum\limits_i
1/\langle|\widehat{M}_{i}|^2\rangle},
\end{equation}
for the mass as a weighted sum over the pixels.  The variance of
this estimator is, under the null hypothesis,
\begin{eqnarray}
\sigma^{-2}_{M} &=&  \sum\limits_i
1/\langle|\widehat{M}_{i}|^2\rangle =  \sum\limits_{\vec L}
\frac{|\kappa^{\rm th}_{\vec L}(M_{\rm fid})|^2}{M^2_{\rm
fid}\Omega N_{\vec L}} \nonumber \\
&\simeq&  \frac{N_z l_{\rm max}^2}{M^2_{\rm
fid}\Omega4\pi}\sum\limits_{\vec L}  |\kappa^{\rm th}_{\vec
L}(M_{\rm fid})|^2|W_{\vec L}|^2 \nonumber \\
&\simeq&  \frac{N_z l_{\rm max}^2}{M^2_{\rm fid}4\pi}\int \frac{d^2\vec{L}}{(2\pi)^2}  |\kappa^{\rm th}_{\vec L}(M_{\rm fid})W_{\vec L}|^2 \nonumber \\
&=& \frac{N_z l_{\rm max}^2 }{M^2_{\rm fid}4\pi} \int d^2 \theta
\left[ \int d^2 \varphi \, W(\vec \varphi) \kappa(|\vec \theta -
\vec \varphi|) \right]^2, \nonumber \\
&&
\label{MassVarTotal}
\end{eqnarray}
where in the second line we substituted
\begin{equation}
N_{\vec{L}}=\frac{4\pi}{N_z l_{\rm max}^2}e^{L^2/2L_{\rm
max}^2}\equiv\frac{4\pi}{N_z l_{\rm
max}^2}W_{\vec{L}}^{-1},
\end{equation}
with the exponential included to describe roughly the
transition between those $L$ modes that can be measured and
those that cannot.  In the third line in
Eq.~(\ref{MassVarTotal}) we took the continuum limit $\sum_{\vec
L} \Leftrightarrow \Omega \int d^2\vec L/(2\pi)^2$, and in the
fourth we used Parseval's theorem to switch to real space,
resulting in a convolution of the convergence with a
two-dimensional Gaussian filter,
\begin{equation}
W(\varphi)=\frac{1}{2\pi \theta_s^2}e^{-\frac{\varphi^2}{2\theta_s^2}},
\end{equation}
with smoothing scale $\theta_s$ given by $\theta_s=\pi/L_{\rm max}$ (we assume here that we can push $L_{\rm max}$ close to $l_{\rm max}$ which is verified numerically). Assuming a spherically symmetric profile and cutting off the integral at $\Lambda$ when the signal becomes negligible, we get
\begin{eqnarray}
     \sigma^{-2}_{M} &=&  \frac{N_z l_{\rm max}^2}{2M^2_{\rm
     fid}}\int\limits^\Lambda \, \theta\,
     d\theta \nonumber \\
     &  \times &  \left[
     \int\limits^\Lambda \, \varphi\, d\varphi\, W(\varphi)
     \int\limits_0^{2\pi} \, d\phi\,
     \kappa(\sqrt{\theta^2+\varphi^2+2\theta\varphi\cos\phi})
     \right]^2. \nonumber \\
\label{signaltonoise}
\end{eqnarray}
This equation, together with Eq.~(\ref{l_max}), allows a
straightforward examination of the capabilities of different
telescopes to reconstruct a given lensing source. The focus of
this work is detection of a galaxy cluster, but a similar
formula can be used to estimate the signal-to-noise for
reconstruction of model parameters for other lensing sources, as
we discuss in the Conclusion.

The remaining task is to calculate the convergence profile of a
given cluster.  We model the mass profile of the galaxy cluster
by an NFW profile \cite{NFW},
\begin{equation}
     \rho(r)=\rho_s{1\over r/r_s(1+r/r_s)^2},
\end{equation}
where the scale radius $r_s$ and normalization $\rho_s$ are
often described by the concentration parameter $c\equiv r_{\rm
vir}/r_s$ and the cluster mass $M\equiv4\pi
r_s^3\rho_s[\ln(1+c)-c/(1+c)]$.  Here, $r_{\rm vir}$ is the
radius within which the enclosed mass $M$ is 200 times the
average mass of the same volume in a critical density universe.

The convergence $\kappa(\theta)$ of the NFW profile is given by
\begin{equation}
  \kappa(\theta = r/D_L) = \frac{2\rho_{\rm s}r_{\rm
  s}}{\Sigma_{\rm cr}}\,{f(\theta/(r_s/D_L))},
\end{equation}
where
the functional form of the projected mass density is
\begin{equation}
  f(x) = \begin{cases}
  {1\over x^2 - 1} \left[1 - {2\over\sqrt{x^2 - 1}}\arctan\sqrt{x-1\over x+1}\;\right], & x>1, \cr
  \frac{1}{3}, & x=1, \cr
 {1 \over x^2 - 1} \left[1 - {2\over\sqrt{1 - x^2}}\mathrel{\rm arctanh}\sqrt{1-x\over 1+x}\;\right], & x<1. \cr
  \end{cases}
\end{equation}

In Fig.~\ref{fig:ConvergencePerC} we plot the convergence of a
cluster at redshift $z=1$ with mass of
$M_1=5\times10^{14}h^{-1}M_{\odot}$ and concentration parameter
$c=3$ for a source at redshift $z=7$. Plugging this in the
integral of Eq.~(\ref{signaltonoise}), we find that the
signal-to-noise is saturated at the virial radius, so that
effectively $\Lambda=r_{\rm vir}$, an order of magnitude above
the corresponding Einstein radius. We also find that the
dependence on the concentration parameter is small.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{ConvIntegralThetaSmall.pdf}
\caption{The radial convergence of NFW clusters with mass of $M=5\times10^{14}h^{-1}M_{\odot}$ and concentration $c=3$ located at redshift $z=1$, where the 21-cm source is at redshift $z=7$. The vertical line marks the virial radii of the clusters where the signal becomes negligible. The two shaded regions show the amount of signal integrated with the ideal resolutions of SKA ({\it light shade}) at $1\,{\rm arcmin}$ and a dark ages observatory ({\it dark shade}) with a $100\,{\rm km}$ baseline at $\sim6\,{\rm arcsec}$.}
\label{fig:ConvergencePerC}
\end{figure}

\section{Results}
To estimate the signal-to-noise with which this cluster mass can
be reconstructed using the weak-lensing signal of 21-cm
radiation measured by a radio interferometer, we consider the
expected signal from the EOR from SKA (as
described in the previous Section), an upgraded SKA with four
times the collecting area, both with a bandwidth of $1\,{\rm MHz}$, and an ideal experiment (capable of
sustaining $C_l^s>C_l^n$ up to $l_{\rm max}=l_{\rm cover}$) with
the same resolution as SKA ($l_{\rm cover}\gtrsim10^4)$ with a bandwidth small enough to reach the maximum number of independent redshift slices. We also include extremely optimistic limits for an ideal dark ages
observatory with a baseline of $100\,{\rm km}$ (for which $l_{\rm
cover}\sim10^5)$.

In Fig.~\ref{fig:SNPerZLC} we plot the smallest detectable
mass as a function of redshift, for different experimental capabilities. We
see that with the current plans for SKA, the weak-lensing
reconstruction of clusters as considered in
Fig.~\ref{fig:ConvergencePerC} is completely beyond reach, unless about an order of magnitude more observation time is dedicated to the patch containing the target galaxy-cluster. Future experiments will narrow this gap and enable the mass measurement of significantly smaller mass halos using 21-cm weak lensing.

These detection prospects involve intricate tradeoffs between the different experimental parameters.  For example, if we increase observation time by 2, we can reach the same $l_{\rm max}$ with half the frequency bandwidth, which will also allow us to use twice the number of z-slices, increasing the S/N by $\sqrt{2}$. Alternatively, we could use this to reach a larger $l_{\rm max}$ (paying a small price for the damping in the signal power spectrum), which also increases the S/N, but this will still be limited by the maximum $l$ set by the baseline of the interferometer. As mentioned in Section IV, one does not always do better by splitting up into smaller bandwidth bins because this increases the noise, yielding a smaller $l_{\rm max}$, which reduces the S/N and might also leave galaxy-cluster scales beyond reach. In addition, as we also discussed in Section IV, there is a lower limit to the frequency bandwidth (which determines the width of the redshift slices) below which the slices become correlated.

Another experimental issue is that as the observed frequency is increased in order to use multiple redshift slices, a larger baseline is needed to cover the same scales. This means that for a given experimental baseline, the contribution of additional slices degrades with their redshift and the $S/N$ grows slower than $1/\sqrt{N_z}$. On the other hand, for high redshift clusters (which approach the sources of the EOR) the increase in lensing signal due to the larger line-of-sight distance traveled is larger than the loss in resolution, and so the $S/N$ grows faster than $1/\sqrt{N_z}$.

To demonstrate this last point quantitatively\footnote{A full treatment must also take into account the redshift dependence of the 21-cm signal amplitude, which is not trivial, as can be seen for the dark ages in Fig. \ref{fig:CLDarkAges}, but we neglect this for the purpose of this demonstration.}, we plot in Fig.~(\ref{fig:SNPerNz}) the $1\sigma$ detection limits for two cluster redshifts as the number of redshift slices accumulated beyond $z\!=\!7$ is increased. We assume a bandwidth of $\Delta\nu=0.1\,{\rm MHz}$ (yielding roughly $\sim800$ slices in the range $z\!=\!7-13$) and unlimited observation time so that the maximum resolution of SKA is reached. We see that the low redshift cluster gains less improvement in $S/N$ as the number of redshifts is increased, because the smallest observable scale with a given baseline decreases with redshift. We also see that for a high redshift cluster, which is closer to the redshifts of the EOR, the benefit from the inclusion of additional redshift slices more than compensates for this degradation in resolution.

Finally, our estimator, Eq.~\ref{minvarestimator}, can be generalized to include the free parameters of the cluster location, which in this work we assumed were already known from other surveys.

\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{SNperZSmall.pdf}
\caption{The $1\sigma$ limit for weak lensing reconstruction for clusters in different redshifts up to the end of reionization ($z\sim7$) using SKA ({\it \!thick solid line}), SKA with four times the coverage fraction or sixteen times the observing time ({\it \!thin solid line}), both with a bandwidth of $1\,{\rm MHz}$, and for an ideal experiment with SKA resolution ({\it \!dashed line}). We also plot the expected limits for an ideal dark ages observatory with a $100\,{\rm km}$ baseline ({\it \!dotted line}).}
\label{fig:SNPerZLC}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{SNNzSmall.pdf}
\caption{The $1\sigma$ limit for weak lensing reconstruction for clusters at two redshifts, $z\!=\!0.5$ ({\it dashed black}) and $z\!=\!5$ ({\it solid black}), as we include more and more redshift slices beyond the end of reionization ($z\sim7$), for SKA with frequency bandwidth $\Delta\nu=0.1{\rm MHz}$) and unlimited observation time. This takes into account the non-trivial source redshift dependence of the different slices. The gray lines are the results under the approximation of a trivial $1/\sqrt{N_z}$ dependence.}
\label{fig:SNPerNz}
\end{figure}

\section{Discussion}

We have constructed a minimum-variance estimator for the mass of
a galaxy cluster using the weak-lensing convergence
reconstruction of 21-cm fluctuations from the dark ages and the EOR.
As has been suggested for CMB lensing reconstruction, possible improvements to the results shown here can stem from
using modified weak-lensing estimators, enhancing sensitivity to small scales as in Ref.~\cite{Maturi} or iteratively approaching a maximum-likelihood estimator as in Ref.~\cite{YooZald}.
Another option which has been discussed for CMB measurements
(e.g. Refs.~\cite{Maturi,YooZald,HuDeDeo}) is to stack reconstructed
images of $N$ different clusters to yield a $\sqrt{N}$
improvement in the signal-to-noise for fitting an overall mass
profile (much like the use of different 21-cm redshift slices of
the same cluster in Eq.~(\ref{volumenoise})).

In comparison to other sources for weak-lensing measurements,
such as galaxy shapes \cite{Blandford,Bacon}, or
CMB fluctuations \cite{Seljak} (detected recently \cite{Smith}),
the potential in 21-cm measurements is far greater. Even deep
galaxy surveys will be limited to lower redshifts with small sky
coverage and will not exceed arcminute resolutions. CMB lensing
reconstruction at small scales is difficult because of the absence of power due to Silk
damping at these scales and due to the ambiguity caused by the presence of a similar signature from the kinetic
Sunyaev-Zeldovich effect on the CMB photons passing through the
moving clusters \cite{YooZald,HuDeDeo}.
Even with ideal experiments at arcminute resolutions, the corresponding minimum galaxy-cluster mass detectable according to Eq.~(\ref{signaltonoise}) with a CMB experiment is $M\sim10^{15}h^{-1}M_{\odot}$. An average mass profile of $M\sim10^{14}h^{-1}M_{\odot}$ can be measured with reasonable signal to noise by stacking thousands of galaxies, but the detailed reconstruction of an individual cluster is beyond the reach of these
alternative methods.

Finally, an issue which we have neglected in this work is the influence
of non-gaussianities which would have to be taken into account
to yield accurate predictions. Particularly important is the
effect of nonlinear structure \cite{LuPen,LuPenDore}, which is
relevant during the epoch of reionization, when certain patches
of the intergalactic medium become substantially ionized well before its
end. In order to properly account for these effects, the details of the reionization process will have to be uncovered.
However, under the assumption that these features appear in
higher resolutions than that of our considered interferometers
\cite{ZahnZald,MetcalfWhite}, there will be no connected
four-point function contribution to the variance of the quadratic lensing estimator used here, and this treatment remains valid.

\section{Conclusion}

When 21-cm fluctuations become accessible to observations at
small angular scales, the application of weak-lensing
reconstruction methods will open the door to unprecedented
precision measurements of local structure. We have found here
that galaxy clusters can be detected by lensing reconstruction
with futuristic experiments measuring CMB absorption by neutral
hydrogen during the dark ages $30 \lesssim z \lesssim 200$ down
to halo masses of order $M\gtrsim10^{12} h^{-1}\,M_\odot$. Next-generation
interferometers measuring emission from
hot neutral-hydrogen gas during the epoch of reionization
$7\lesssim z \lesssim 13$ will be limited to $M\gtrsim10^{15}\,
h^{-1}\,M_\odot$ with the currently planned specifications.  If,
however,  the collecting area or the observation time can be
increased, they may be able to push the limit down to
$M\gtrsim10^{13}\, h^{-1}\, M_\odot$ (in particular, this can be reached by increasing the observation time dedicated to the target patch alone by two orders of magnitude).
While these are challenging goals that remain unattainable in the near future,
the potential achievements discussed here provide more
motivation to invest in alleviating the experimental
limitations.

Meanwhile, we can use the prescription described here to
construct estimators for model parameters of other structures,
from standard isothermal spheres and voids to more exotic
structures such as cosmic textures \cite{Turok,TurokSpergel} or
overdensities created by pre-inflationary particles
\cite{Fialitzkov,Rathaus}. In future work we shall address the
weak-lensing detectability, with 21-cm weak lensing, of voids or
textures that might be responsible for the cold spot in WMAP CMB
data \cite{DasSpergel,BeNaSunny}.

\acknowledgments
We thank the anonymous referee for excellent comments and suggestions which helped improve the quality of this paper. EDK was supported by the National Science Foundation under Grant Number PHY-0969020 and by the Texas Cosmology Center. This work was supported at Johns Hopkins by DoE SC-0008108 and NASA NNX12AE86G. EDK would like to thank the hospitality of the Department of Physics and Astronomy at Johns Hopkins, where this work was partially carried out.

redpajama_set_name: RedPajamaArXiv


VANCOUVER, British Columbia, July 26, 2018 (GLOBE NEWSWIRE) -- NOVAGOLD RESOURCES INC. (TSX:NG) (NYSE American:NG) ("NOVAGOLD" or "the Company") today announced that it has entered into a binding agreement (the "Agreement") with Newmont Mining Corporation (NYSE:NEM) ("Newmont"), pursuant to which NOVAGOLD has agreed to sell its 50% interest in the Galore Creek project ("Project"), owned by the Galore Creek Partnership (GCP). The Project is located in northern British Columbia, Canada. All dollar amounts referred to in this release are stated in U.S. Dollars.
$75 million payable upon approval of Project construction.
Teck has waived its right of first refusal to acquire NOVAGOLD's interest in GCP. The transaction is expected to close on or about July 27, 2018.
As part of the transaction, Newmont will also acquire the Company's 40% interest in the adjacent Copper Canyon mineral property (the "Copper Canyon property"). The remaining 60% of the Copper Canyon property is owned by GCP. The sale will be accomplished by the transfer of 100% of the shares of a new company to be formed by the amalgamation of NovaGold Canada Inc. and Copper Canyon Resources Limited, two wholly-owned subsidiaries of the Company, to a subsidiary of Newmont. The Company expects to record an after-tax loss of approximately $92 million on completion of the sale of its interests in the Project, assuming no value is assigned to the final contingent payment.
RBC Capital Markets served as financial advisor to the Company's Board of Directors and B. Riley FBR, Inc. provided a fairness opinion to the Special Committee.
NOVAGOLD is a well-financed precious metals company focused on the permitting and development of its 50%-owned Donlin Gold project in Alaska, one of the safest mining jurisdictions in the world. With approximately 39 million ounces of gold in the measured and indicated resource categories, inclusive of proven and probable reserves (541 million tonnes at an average grade of approximately 2.2 grams per tonne), Donlin Gold is regarded to be one of the largest, highest grade, and most prospective known gold deposits in the world. According to the Second Updated Feasibility Study (as defined below), once in production, Donlin Gold is expected to produce an average of more than one million ounces per year over a 27-year mine life on a 100% basis. The Donlin Gold project has substantial exploration potential beyond the designed footprint which currently covers only three kilometers of an approximately eight-kilometer long gold-bearing trend. Current activities at Donlin Gold are focused on permitting, optimization work, community outreach and workforce development in preparation for the construction and operation of this top tier asset. NOVAGOLD anticipates applying the proceeds from the sale of its interest in the Galore Creek project and Copper Canyon property toward the development of Donlin Gold. With a strong balance sheet, NOVAGOLD is well positioned to stay the course by completing permitting and advancing optimization efforts at the Donlin Gold project.
This press release includes certain "forward-looking information" and "forward-looking statements" (collectively "forward-looking statements") within the meaning of applicable securities legislation, including the United States Private Securities Litigation Reform Act of 1995. All statements, other than statements of historical fact, included herein, without limitation, with respect to timing and completion of NOVAGOLD's sale of its interest in Galore Creek and the Copper Canyon mineral property, including the application of any proceeds from such sale, as well as statements relating to the recordation of an after-tax book loss. Statements regarding the permitting, potential development, exploration, construction and operation of Donlin Gold, statements relating to NOVAGOLD's future operating and financial performance, production estimates and outlook are forward-looking statements. Forward-looking statements are frequently, but not always, identified by words such as "expects", "anticipates", "believes", "intends", "estimates", "potential", "possible", and similar expressions, or statements that events, conditions, or results "will", "may", "could", "would" or "should" occur or be achieved. These forward-looking statements may include statements regarding; exploration potential of Donlin Gold; mine life and production estimates at Donlin Gold; perceived merit of properties; anticipated permitting timeframes; exploration results and budgets; mineral reserve and resource estimates; work programs; capital expenditures; timelines; strategic plans; benefits of the project; completion of transactions; market prices for precious and base metals; or other statements that are not statements of fact. Forward-looking statements involve various risks and uncertainties. There can be no assurance that such statements will prove to be accurate, and actual results and future events could differ materially from those anticipated in such statements. Important factors that could cause actual results to differ materially from NOVAGOLD's expectations include the timely satisfaction of closing conditions with respect to the sale of Galore Creek and the Copper Canyon mineral property, the uncertainties involving the interpretation of the drill results, the need to obtain permits and governmental approvals; the need for additional financing to explore and develop properties and availability of financing in the debt and capital markets; uncertainties involved in the interpretation of drilling results and geological tests and the estimation of reserves and resources; the need for continued cooperation with Barrick Gold Corporation for the continued exploration and development of the Donlin Gold property; the need for cooperation of government agencies and native groups in the development and operation of properties; risks of construction and mining projects such as accidents, equipment breakdowns, bad weather, non-compliance with environmental and permit requirements, unanticipated variation in geological structures, ore grades or recovery rates; unexpected cost increases, which could include significant increases in estimated capital and operating costs; fluctuations in metal prices and currency exchange rates; and other risk and uncertainties disclosed in NOVAGOLD's Annual Report filed on Form 10-K for the year-ended November 30, 2017 with the United States Securities and Exchange Commission, Canadian securities regulators, and in other NOVAGOLD reports and documents filed with applicable securities regulatory authorities from time to time. NOVAGOLD's forward-looking statements reflect the beliefs, opinions and projections on the date the statements are made. NOVAGOLD assumes no obligation to update the forward-looking statements of beliefs, opinions, projections, or other factors, should they change, except as required by law.
This press release has been prepared in accordance with the requirements of the securities laws in effect in Canada, which differ from the requirements of U.S. securities laws. Unless otherwise indicated, all resource and reserve estimates included in this press release have been prepared in accordance with Canadian National Instrument 43-101 Standards of Disclosure for Mineral Projects ("NI 43-101") and the Canadian Institute of Mining, Metallurgy and Petroleum (CIM)—CIM Definition Standards on Mineral Resources and Mineral Reserves, adopted by the CIM Council, as amended ("CIM Definition Standards"). NI 43-101 is a rule developed by the Canadian Securities Administrators which establishes standards for all public disclosure an issuer makes of scientific and technical information concerning mineral projects. Canadian standards, including NI 43-101, differ significantly from the requirements of the United States Securities and Exchange Commission (SEC), and resource and reserve information contained herein may not be comparable to similar information disclosed by U.S. companies. In particular, and without limiting the generality of the foregoing, the term "resource" does not equate to the term "reserves". Under U.S. standards, mineralization may not be classified as a "reserve" unless the determination has been made that the mineralization could be economically and legally produced or extracted at the time the reserve determination is made. The SEC\'s disclosure standards normally do not permit the inclusion of information concerning "measured mineral resources", "indicated mineral resources" or "inferred mineral resources" or other descriptions of the amount of mineralization in mineral deposits that do not constitute "reserves" by U.S. standards in documents filed with the SEC. Investors are cautioned not to assume that all or any part of "measured" or "indicated resources" will ever be converted into "reserves". Investors should also understand that "inferred mineral resources" have a great amount of uncertainty as to their existence and great uncertainty as to their economic and legal feasibility. Under Canadian rules, estimated "inferred mineral resources" may not form the basis of feasibility or pre-feasibility studies except in rare cases. Disclosure of "contained ounces" in a resource is permitted disclosure under Canadian regulations; however, the SEC normally only permits issuers to report mineralization that does not constitute "reserves" by SEC standards as in-place tonnage and grade without reference to unit measures. The requirements of NI 43-101 for identification of "reserves" are also not the same as those of the SEC, and reserves reported by NOVAGOLD in compliance with NI 43-101 may not qualify as "reserves" under SEC standards. Donlin Gold does not have known reserves, as defined under SEC Industry Guide 7. Accordingly, information concerning mineral deposits set forth herein may not be comparable with information made public by companies that report in accordance with U.S. standards.
1 Donlin Gold data as per the Updated Feasibility Study (as defined below). Donlin Gold measured resources of 8 Mt grading 2.52 g/t and indicated resources of 534 Mt grading 2.24 g/t, inclusive of proven reserves of 8 Mt grading 2.32 g/t and probable reserves of 497 Mt grading 2.08 g/t.


text:
£5 million to support water rescue
on Wednesday, 19 November 2014. Posted in Petrospot News
New support for charities keeping waterways safe
Transport Minister John Hayes today announced £5 million funding to support the work of the life-saving charities that run rescue boats on rivers, lakes and other inland waterways.
Charities around the UK are set to benefit from the money which will go towards enhancing the resources available to improve safety on the water.
John Hayes said:
"Common wisdom may have it that worse things happen at sea, but it is imperative that our rivers, lakes and estuaries are as safe as possible for those who take to the water.
"Britain benefits from countless volunteer and charitable organisations working round the clock to help during emergencies on inland waters, as well as during flooding and other extreme weather.
"The funding I am announcing today will ensure their tireless efforts are supported by the best equipment and save more lives."
Chief Secretary to the Treasury Danny Alexander said:
"Our volunteers are the backbone of our society. At Budget, this government announced that we would support the tremendous work of the charities keeping our waterways safe. And so today I can confirm that these charities can now apply for financial support from the £5m that has been set aside to make sure they have the best equipment necessary to provide emergency support.
This will be a huge boost to both those who bravely volunteer their services and to those who they assist, particularly when faced with incredibly challenging conditions such as extreme weather and flooding. Supporting our volunteers remains an integral part of this government's ongoing drive for a fairer society."
The Inshore and Inland Lifeboat Grant Scheme will run over the next five years, with £1 million available in each financial year.
Organisations will be able to bid for funding to support their work and have until 31 December to submit their bids for 2014-15. Bids will only be accepted by UK charities as defined in the Charities Act 2006.
Please click here to visit the official London International Shipping Week website.

redpajama_set_name: RedPajamaCommonCrawl


text:
Police Nabbed Serial Killer Suspect by Stumbling on His Son's DNA
80beatsBy Andrew MosemanJuly 10, 2010 12:46 AM
Los Angeles police say that Lonnie Franklin Jr. may be the "grim sleeper" serial killer they have sought for more than 20 years. And if indeed they do have their man, they have his son to thank—for getting arrested himself. Franklin is one of the first major suspects nabbed by police using familial DNA. With this controversial method, investigators look for partial matches between DNA left at a crime scene and DNA profiles that are stored in police databases; a partial match may indicate that the person is related to the target individual sought by the cops.
The trail began to heat up when the DNA of Franklin's son was entered in a state database after he was convicted in a weapons case, authorities said. The son's DNA was similar to genetic material found on the victims, and authorities soon began following around Franklin to get his DNA and see if he was the suspected killer [AP].
The cops posed as waiters at a restaurant where the elder Franklin ate, which is how they obtained a complete DNA sample from him--they grabbed a plate and napkin he tossed after eating a slice of pizza. The investigators say that when they found the match to the samples in their evidence, it eased 25 years of frustration at not being able to track him down.
In 2009, LAPD Det. Dennis Kilcoyne, who led a special unit assigned to find the Grim Sleeper, expressed frustration with knowing who the killer was, but only in a language of numbers and dashes. "We've got this beautiful DNA profile -- all these dashes and dots, and this and that, but there's no name to go with it," Kilcoyne said [CNN].
It was just two years ago that California Attorney General Jerry Brown gave the OK to testing the state's DNA data bank for these familial matches. Colorado is the only other state that currently allows the practice. Brown promises that only convicted felons are in that database for testing, but that's just the thing that raised the hackles of opponents.
Familial DNA database searches have come under fire from privacy and civil liberty advocates, who argue, among other things, that they put more minorities, who are disproportionally represented in the database, in an at-risk group [ABC News].
However, as DISCOVER blogger Razib Khan writes at Gene Expression
, this testing is coming whether we like it or not (and besides, it's probably more reliable than eyewitness testimony or fingerprinting). So, Khan says, the issue we should worry about is not privacy but accuracy. Related Content: Gene Expression: To Catch a Predator: Familial DNA
Gene Expression: The State May Have Your Genome Sooner Than You Think
80beats: Could Forensic Scientists ID You Based on Your "Bacterial Fingerprint?"
80beats: Think DNA Evidence Can't Be Faked? Think Again
Image: iStockphoto
How Playing Science Games is Advancing Genetic Research
How Do Climate Models Predict Global Warming?
How Tech Firms Have Tried to Stop Disinformation and Voter Intimidation — and Come up Short
HealthScientists Devise a Method to Edit Mitochondrial DNA. Here's How It Works and Why It Matters
TechnologyHow Playing Science Games is Advancing Genetic Research
HealthWhat Blood Type Has to Do With COVID-19 Risk
HealthCRISPR Treatment Is Injected Directly Into a Patient's Body — a First for Gene Editing
HealthHuman Evolution Isn't Over Yet
HealthThe Quest for an 'Asian Flush' Remedy


text:
Guardian Ad Litems Are A Better Choice To Help The Mentally Ill Than Public Defenders - Honolulu Civil Beat
Community Voice Guardian Ad Litems Are A Better Choice To Help The Mentally Ill Than Public Defenders
Guardian Ad Litems Are A Better Choice To Help The Mentally Ill Than Public Defenders
My uncle's experience illustrated the benefits of choosing caregivers over the courts.
By Caroline Kunitake
April 26, 2021 · 5 min read
Caroline Kunitake
Caroline Kunitake is a member of the American Association of University Women's Honolulu branch.
I believe that the intent of House Bill 345, Relating to Assisted Community Treatment, is to meet the basic mental health needs of people who suffer from severe mental illness.
This bill will mandate the appointment of a guardian ad litem to represent the best interest of a mentally ill individual in assisted community treatment proceedings and eliminate the requirement for the Office of the Public Defender to participate in the proceedings.
The law needs to be written and interpreted in such a way that the mentally incompetent who refuse necessary medical attention are not allowed to misuse their legal rights to further jeopardize the safety and well-being of themselves and others. Human beings who are misusing their legal rights in this way need immediate medical attention, not a lengthy legal process in court.
Imagine if you lost your mind and all connection to reality. What if you burned all of your bridges with those who cared and loved you? What if you refused all medical care when you were obviously very sick and needed to be admitted to a hospital? If that were you, would you rather have a guardian ad litem appointed to represent you or would you want to meet directly with a public defender to discuss your legal case?
I believe it is unrealistic to expect a public defender to successfully represent a mentally ill person who is incapable of understanding reality. The public defenders do not have the time, training or resources to invest in a mentally incompetent client.
People struggling with severe mental illness may be incapable of even brief, coherent conversations with anyone, let alone an attorney or judge in a court of law. My own personal experiences with a family member struggling with severe mental illness prompted me to submit testimony in support of HB 345 this legislative session.
It was a long, expensive, arduous and painful process to have the Caregiver Foundation (a local nonprofit) assigned to my 80-plus-year-old uncle as a legal guardian. My uncle was a Vietnam veteran who struggled with post-traumatic stress disorder. After he had a mental breakdown, he was uncooperative and very adamant about refusing any medical care to address his paranoia, hallucinations and the "voices" in his head.
My uncle stopped talking to all family members and soon forgot the names of his friends and neighbors that he had known for almost three decades. I continued to visit him at his apartment building, but he refused the groceries that I brought. I pleaded with him numerous times to get medical help, but he would always say, "Caroline, you're not a doctor. Go away."
My uncle would not have qualified for the MH-1, Involuntary Application for Mental Health Evaluation, because he didn't attempt suicide or commit a violent crime. My uncle would also not have requested the MH-2, Emergency Psychiatric Examination, to voluntarily commit himself and have the police transport him to a medical facility.
A current bill in the Legislature would change the way in which Hawaii approaches care for the mentally ill. Cory Lum/Civil Beat
Even if my uncle had qualified for the MH-1, the doctors would not have been able to get him to open up and discuss his broken state of mind. I know this because every time he went to the emergency room, he refused to speak to the attending physicians because he didn't want to be evaluated and committed to a psychiatric ward where he feared that communists from Vietnam would torture or kill him.
Soon my uncle stopped paying his rent and his bills. He could have easily been evicted to aimlessly roam the streets even though he had a full pension from the Air Force, Social Security benefits and owned a parcel of land on a neighbor island.
His sister took over his rent and bills so that he would have a place to live. This went on for a few years until my uncle's landlord died and the surviving relatives wanted to sell the apartment. At that point, my uncle needed a psychiatric evaluation to determine his competency level since he refused to move out.
Fortunately, it was determined that my uncle was no longer competent. Then the Caregiver Foundation became his legal guardian and tried to relocate him to a new residence. However my uncle repeatedly trespassed to get back to his old apartment and the police were called numerous times to remove him from his old high-rise building.
Eventually the courts ordered monthly injections with a sedative so that his 24-hour caregivers (paid for by his own retirement and assets) could care for him in his new apartment.
If my uncle had met with a public defender, he would have told the public defender to go to hell. Then the public defender would have been required to legally and ethically pursue my uncle's interests, which would be to refuse all medical care. Legal processes need to change to realistically work and benefit those who suffer from severe mental illness.
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Latest Comments (0)
Very sorry. Good insight. The PD can do a disservice to one in need who has no insight to his own needs. They need as much training as the police. But in present day reality , even a GAL may not have changed the course of events.Unfortunately, the course of Uncle's illness and what occurred may not have changed one iota with a GAL in todays mental health use of the ACT law and other laws. In other words, a GAL may not have helped much....uncle didn't listen to relatives or other officials due to his "illness" and having the name "GAL" next to someone's name may not have helped him voluntarily to listen as well. Uncle needed to get to a court to order the administer of medication a lot quicker.This is one of many major flaws with the ACT law as administered. Its the mental health system itself. It took months for an application to the court to order the administration of the medication needed. Check in with or join Mental Health Hawaii and NAMI who deal with this issue on a daily basis. Your voice is needed and the very best to uncle in the future. It's a long road.
Anson · 8 months ago
Any effort by gov't to grow itself even larger I do not support.
Ranger_MC · 8 months ago
HB345 will benefit not only individuals with severe mental illness , it will help the families whose hands are tied . It will help all facets of the community who actually have to actually deal with the individual . Hopefully there is enough support for this very , very important bill to help people deal with this part of our island community .
hanawaiman · 8 months ago
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River and Mercantile Solutions is an investment consultant and fiduciary manager for pension schemes and other institutional investors. We focus, single-mindedly, on providing our clients with outcome-orientated solutions that will help them meet their investment needs.
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text:
SO Dividend History
SO Stock Split History
SO Preferred Stock
SO Options Chain
SO Message Board
Southern (SO) has 2 splits in our SO split history database. The first split for SO took place on March 01, 1994. This was a 2 for 1 split, meaning for each share of SO owned pre-split, the shareholder now owned 2 shares. For example, a 1000 share position pre-split, became a 2000 share position following the split. SO's second split took place on April 03, 2001. This was a 10000 for 6109 split, meaning for each 6109 shares of SO owned pre-split, the shareholder now owned 10000 shares. For example, a 2000 share position pre-split, became a 3273.85824193812 share position following the split.
When a company such as Southern splits its shares, the market capitalization before and after the split takes place remains stable, meaning the shareholder now owns more shares but each are valued at a lower price per share. Often, however, a lower priced stock on a per-share basis can attract a wider range of buyers. If that increased demand causes the share price to appreciate, then the total market capitalization rises post-split. This does not always happen, however, often depending on the underlying fundamentals of the business.
Looking at the SO split history from start to finish, an original position size of 1000 shares would have turned into 3273.85824193812 today. Below, we examine the compound annual growth rate — CAGR for short — of an investment into Southern shares, starting with a $10,000 purchase of SO, presented on a split-history-adjusted basis factoring in the complete SO split history.
Dividends reinvested/share: $21.23
Years: 10.00
Dividends collected/share: $21.23
About Southern
Southern is a holding company. Through its subsidiaries, Alabama Power Company, Georgia Power Company, and Mississippi Power Company, Co. is engaged in the generation, transmission, distribution, and purchase of electricity and the sale of electric service in Alabama, Georgia, and Mississippi. Co.'s Southern Power Company subsidiary develops, constructs, acquires, owns, and manages power generation assets, including renewable energy projects, and sells electricity at market-based rates in the wholesale market. Co.'s Southern Company Gas subsidiary is an energy services holding company whose primary business is the distribution of natural gas in Illinois, Georgia, Virginia, and Tennessee. According to our SO split history records, Southern has had 2 splits.
SO Split History Table
04/03/2001 10000 for 6109
SO is categorized under the Utilities sector; below are some other companies in the same sector that also have a history of stock splits:
SPKE Split History
SR Split History
STR Split History
Also explore: SO shares outstanding history
SO Insider Buying
SO Split History | www.SplitHistory.com | Copyright © 2013 - 2020, All Rights Reserved


Re: Buying cores from merchants!
that would be a brutal combo abilities if you combine that with the vampire bats ability.
Hmmm, whatever bug had happened before must have been sorted out on the most recent update, cause it's worked without crash twice now. Woot!
You can't sacrifice cores obtained from power spells, Bynine, and certain event-specific creatures. You can only sacrifice cores from creatures that can be unlocked via creature tier upgrades.
If I buy a core from Bynine, like the Nightwing Gargoyle, can I extract more cores by using the first core to encounter more creatures in the next realm? Does this method work on cores from merchants, but not core from power spells or bynine's shop?
I had tried that a couple times and was actually about to try again to write down the bug. Because every time I sacrifice a core i bought from a merchant that I don't have unlocked, the game crashes on my first battle against the desired creature. I've only lost a few dear cores, but I haven't been able to mark down the error message yet, since it's on my phone and I didn't have paper and pen to jot down the error message.
You can sacrifice a core to have a high chance of encountering that same creature in the next realm you visit. That way you can get more cores to summon the creature you want.
I realized that since I'll most likely not be able to actually use the cores I buy from merchants (since you only get one and have a crazy low chance at having another two merchants with that same core) I've decided that I should at least show those that I find interesting or am excited to use in the future!
This one is the raven king! Talk about a sweet late-game ability! Can't wait to add him to my support team.
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