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... Coordinators: Andrey Kravtsov, Dan Marrone, and Peng Oh
Scientific Advisors: Maxim Markevitch…

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Coordinators: Andrey Kravtsov, Dan Marrone, and Peng Oh
Scientific Advisors: Maxim Markevitch, Megan Donahue, John Carlstrom, Richard Bond, Gus Evrard, and Mark Voit

SBCS 4.2 Spherical-Template Mass
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... Make these brief bullet-like points.
Discussion: Expand upon the strengths and weaknesses her…

...

Make these brief bullet-like points.
Discussion: Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here) 4.2.1.4.2.2. NFW-Profile Lensing
Notation: {cue_mathinline.gif}
M_{\Delta, {\rm lens,NFW}}

...

\Delta .
Algorithmic Description: Azimuthally-averaged tangential shear must first be measured as described in section 4.2.0. This procedure requires an algorithm for determining the cluster center, which must be specified explicitly and can introduce uncertainty and bias into the mass measurement. Once the shear has been measured, an NFW mass model can be fit to the shear signal. If the lensing data are of very high quality, it may be possible for the fitting procedure to constrain the scale radius {cue_mathinline.gif}
r_s ,
or equivalently the concentration {cue_mathinline.gif}

...

/ r_s
(e.g. Okabe et al. 2010). Otherwise, one must assume a relationship between halo mass and concentration (see Hoekstra 2007 for an example). The derived lensing masses will then depend on the mass-concentration relation adopted in the fitting procedure.
[Algorithmic description currently lacks an explanation of how to average over the distribution of source redshifts.]

SBCS 4.2 Spherical-Template Mass
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... \rho_{\rm M} (r) \propto r^{-1} (r+r_s)^{-2} \; ,
where {cue_mathinline.gif}
r_s
is a s…

...

\rho_{\rm M} (r) \propto r^{-1} (r+r_s)^{-2} \; ,
where {cue_mathinline.gif}
r_s
is a scale radius at which the power-law slope of the density profile is {cue_mathinline.gif}

...

r^{-2} .
Integrating over this density profile gives the mass profile \rho_{\rm M} (r)M( < r) \propto r^{-1} (r+r_s)^{-2} \;\ln \left( 1 + \frac {r} {r_s} \right) - \frac {r} {r + r_s} .
After fitting this mass profile to the lensing data, one can calculate the cluster's mass within a chosen overdensity threshold {cue_mathinline.gif}
\Delta .
Algorithmic Description: Azimuthally-averaged tangential shear must first be measured as described in section 4.2.0. This procedure requires an algorithm for determining the cluster center, which must be specified explicitly and can introduce uncertainty and bias into the mass measurement. Once the shear has been measured, an NFW mass model can be fit to the shear signal. If the lensing data are of very high quality, it may be possible for the fitting procedure to constrain the scale radius {cue_mathinline.gif}
r_s ,
or equivalently the concentration {cue_mathinline.gif}
c_\Delta \equiv r_\Delta / r_s
(e.g. Okabe et al. 2010). Otherwise, one must assume a relationship between halo mass and concentration (see Hoekstra 2007 for an example). The derived lensing masses will then depend on the mass-concentration relation adopted in the fitting procedure.
[Algorithmic description currently lacks an explanation of how to average over the distribution of source redshifts.]
Links to Software: Add links to any publicly available software. This will be a great way to get others to use it.
Strengths:

SBCS 4.2 Spherical-Template Mass
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... \sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} } ,
where {…

...

\sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} } ,
where {cue_mathinline.gif}
\theta
is the angular distance from the cluster center at which the average shear has been measured. A spherical-overdensity cluster mass defined with respect to the critical density is then
M_{\Delta{\rm c,lens,iso}} = \frac {4 \sigma} {G H(z) \Delta^{1/2}} ,

...

gives the mean density contrast adopted for the mass definition, along with an "m" or a "c" to specify the reference density, as described in Section 1.2 on spherical-overdensity masses.)
Introduction: Algorithmic Description: Azimuthally-averaged tangential shear must first be measured as described in section 4.2.0. This procedure requires an algorithm for determining the cluster center, which must be specified explicitly and can introduce uncertainty and bias intoOne of the most widely used mass measurement. Oncemodels for clusters is the shear has been measured, it can be related tospherically-symmetric NFW profile (Navarro et al. 1997). In this model the mass model through
\gamma_Tmatter density is
\rho_{\rm M} (r) = \frac {r_E} {2r} ,
where the Einstein radius (in radians) can be expressed in terms of the line-of-sight velocity dispersion {cue_mathinline.gif}
\sigma
as
r_E = 4 \pi \left( \frac {\sigma} {c} \right)^2 \frac {D_{ls}} {D_s} ,
assuming that halo orbits are isotropic. Alternatively, the velocity-dispersion parameter {cue_mathinline.gif}
\sigma
can be related to shear through
\sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} }\propto r^{-1} (r+r_s)^{-2} \; ,
where {cue_mathinline.gif} \theta
is the angular distance from the cluster centerr_s
is a scale radius at which the average shear has been measured. A spherical-overdensity cluster mass defined with respect topower-law slope of the critical density profile is then
M_{\Delta c,{\rm lens,iso}} = \frac {4 \sigma} {G H(z) \Delta^{1/2}} ,
and using the mean density as the reference{cue_mathinline.gif}
\rho_{\rm M} (r) \propto r^{-2} .
Integrating over this density profile gives
M_{\Delta m,{\rm lens,iso}} = \frac {4 \sigma} {G H(z) \Omega_{\rm the mass profile
\rho_{\rm M} \Delta^{1/2}}(r) \propto r^{-1} (r+r_s)^{-2} \; .
[Algorithmic description currently lacks an explanation of how
After fitting this mass profile to average over the distribution of source redshifts.]lensing data, one can calculate the cluster's mass within a chosen overdensity threshold {cue_mathinline.gif}
\Delta .
Algorithmic Description:
Links to Software: Add links to any publicly available software. This will be a great way to get others to use it.
Strengths:

SBCS 4.2 Spherical-Template Mass
edited
... can be related to shear through
\sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} …

...

can be related to shear through
\sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} } ,
where {cue_mathinline.gif}
\theta
is the angular distance from the cluster center at which the average shear has been measured. A spherical-overdensity cluster mass defined with respect to the critical density is then
M_{\Delta{\rm c,lens,iso}} = \frac {4 \sigma} {G H(z) \Delta^{1/2}} ,
and using the mean density as the reference density gives
M_{\Delta{\rm m,lens,iso}} = \frac {4 \sigma} {G H(z) \Omega_{\rm M} \Delta^{1/2}} .
[Algorithmic description currently lacks an explanation of how to average over the distribution of source redshifts.]
Links to Software: Add links to any publicly available software. This will be a great way to get others to use it.
Strengths:
Make these brief bullet-like points.
Weaknesses:
Make these brief bullet-like points.
Discussion: Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)
4.2.1. NFW-Profile Lensing Masses
Notation: {cue_mathinline.gif}
M_{\Delta, {\rm lens,NFW}}
(The density contrast factor {cue_mathinline.gif}
\Delta
gives the mean density contrast adopted for the mass definition, along with an "m" or a "c" to specify the reference density, as described in Section 1.2 on spherical-overdensity masses.)
Introduction:
Algorithmic Description: Azimuthally-averaged tangential shear must first be measured as described in section 4.2.0. This procedure requires an algorithm for determining the cluster center, which must be specified explicitly and can introduce uncertainty and bias into the mass measurement. Once the shear has been measured, it can be related to the mass model through
\gamma_T (r) = \frac {r_E} {2r} ,
where the Einstein radius (in radians) can be expressed in terms of the line-of-sight velocity dispersion {cue_mathinline.gif}
\sigma
as
r_E = 4 \pi \left( \frac {\sigma} {c} \right)^2 \frac {D_{ls}} {D_s} ,
assuming that halo orbits are isotropic. Alternatively, the velocity-dispersion parameter {cue_mathinline.gif}
\sigma
can be related to shear through
\sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} } ,
where {cue_mathinline.gif}
\theta
is the angular distance from the cluster center at which the average shear has been measured. A spherical-overdensity cluster mass defined with respect to the critical density is then
M_{\Delta c,{\rm lens,iso}} = \frac {4 \sigma} {G H(z) \Delta^{1/2}} ,
and using the mean density as the reference density gives
M_{\Delta m,{\rm lens,iso}} = \frac {4 \sigma} {G H(z) \Omega_{\rm M} \Delta^{1/2}} .
[Algorithmic description currently lacks an explanation of how to average over the distribution of source redshifts.]
Links to Software: Add links to any publicly available software. This will be a great way to get others to use it.
Strengths:

SBCS 4.2 Spherical-Template Mass
edited
... \Delta
can then be computed from the best fitting isothermal mass model.
Algorithmic Descrip…

...

\Delta
can then be computed from the best fitting isothermal mass model.
Algorithmic Description: Preparation of the lensing data for model fitting begins with the same basic steps for all spherical-template analyses. Here we outline those preparatory steps. Subsequent specific procedures unique to each spherical template areAzimuthally-averaged tangential shear must first be measured as described below.
Weak-lensing analysis of a galaxy cluster begins with the detection of stars and galaxies in the same region of the sky, followed by unbiased measurement of the shapes of faint background galaxies that will be used to quantify the lensing signal. That signal can be expressed in terms of weighted quadrupole moments,
I_{\rm ij} = \int d^2 {\bf x} \, \, x_i x_j W({\bf x}) f(x) \, ,
where {cue_mathinline.gif}
W({\bf x})
is a Gaussian with a dispersion {cue_mathinline.gif}
r_g ,
which is matched to the size of the object (see Kaiser et al. 1995; Hoekstra et al. 1998,section 4.2.0. This procedure requires an algorithm for more details). These quadrupole moments are then combined to formdetermining the two-component polarization
e_1 = \frac {I_{11} - I_{22}} {I_{11} + I_{22}} \; \; , \; \;
e_2 = \frac {I_{12}} {I_{11} + I_{22}} \; \; .
Various instrumental effects, including but not limited to a position-dependent PSF,cluster center, which must then be removed to obtain a corrected polarization for each faint galaxy.
The weak-lensing signal measured fromspecified explicitly and can introduce uncertainty and bias into the net galaxy polarization undermass measurement. Once the assumption that the galaxies are randomly orientedshear has been measured, it can be quantified by computing the azimuthally averaged tangential shear {cue_mathinline.gif}
\gamma_T .
Shear is related to the surface mass densitymodel through
\langle \gamma_T \rangle
\gamma_T (r) = \frac { \bar{\Sigma}( < r) - \bar{\Sigma}(r)} {\Sigma_{\rm crit}}{r_E} {2r} ,
where {cue_mathinline.gif}
\bar{\Sigma}( < r)
is the mean surface density withinEinstein radius {cue_mathinline.gif}
r , \, \, \bar{\Sigma}(r)
is(in radians) can be expressed in terms of the mean surface density at radius {cue_mathinline.gif}
r,
andline-of-sight velocity dispersion {cue_mathinline.gif}
\Sigma_{\rm crit}
\sigma
as
r_E = \frac {c^2} {44 \pi G}\left( \frac {\sigma} {c} \right)^2 \frac {D_{ls}} {D_s} {{D_l} {D_{ls}}}
is the critical surface density computed from the observer-lens angular diameter distance {cue_mathinline.gif}
D_l ,
the observer-source angular diameter distance {cue_mathinline.gif}
D_s ,
and
assuming that halo orbits are isotropic. Alternatively, the lens-source angular diameter distance {cue_mathinline.gif}
D_{ls} .
It can alsovelocity-dispersion parameter {cue_mathinline.gif}
\sigma
can be quantified in terms of the convergence {cue_mathinline.gif}
\kappa (r) ,
equalrelated to the surface density divided by the critical surface density. Note that the critical surface density, and by extension the convergence, both depend on the distance to the source and therefore on the redshift distribution of the lensed objects used to compute the shear. And in practice, the directly observed quantity is not {cue_mathinline.gif}
\gamma_T
but rather the reduced shear {cue_mathinline.gif}
g_T (r)through
\sigma = c \, \sqrt{ \frac {\gamma_T (r)} {1 - \kappa (r)} \,{\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} } ,
which must be corrected for mass-sheet degeneracy.
Azimuthally averaged quantities such as {cue_mathinline.gif}
\langle \gamma_T \rangle (r)
depend on the algorithm used to identify the cluster's center. An offset between the adopted center and the halo's ``true" center will lead to an underestimate of the cluster's mass. Substructure in the cluster core can therefore cause bias in mass estimates (see Hoekstra et al. 2002). Using wide-field lensing data to establish the halo's center can minimize these biases. When such data are not available, X-ray imaging or the location of the brightest cluster galaxy (BCG) can be used to estimate the halo's center, but the BCG in particular may be substantially offset from the halo's true center.
Once the azimuthally averaged shear has been measured, it can be fit with a specific mass model as detailed below. All such weak-lensing mass measurements are subject to uncertainty arising from cosmic variance in the amount of matter along the line of sight to the cluster but not belonging to the cluster itself (see Hoekstra 2001, Hoekstra 2003).
Links to Software: Add links to any publicly available software. This will be a great way to get others to use it.
Strengths:

Notation: {cue_mathinline.gif}
M_{\Delta, {\rm lens}} ).
Introduction: Gravitational-lensing measurements of cluster mass within a spherical region of radius {cue_mathinline.gif}
r

...

M_{\Delta, {\rm lens,iso}}
(The density contrast factor $\Delta$ gives the mean density contrast adopted for the mass definition, along with an "m" or a "c" to specify the reference density, as described in Section 1.2 on spherical-overdensity masses.)
Introduction: The simplest spherical mass model one can fit to weak-lensing data is a singular isothermal sphere, in which {cue_mathinline.gif}
M( < r) \propto r .
The mass and radius corresponding to a given overdensity {cue_mathinline.gif}
\Delta
can then be computed from the best fitting isothermal mass model.
Algorithmic Description: Preparation of the lensing data for model fitting begins with the same basic steps for all spherical-template analyses. Here we outline those preparatory steps. Subsequent specific procedures unique to each spherical template are described below.
Weak-lensing analysis of a galaxy cluster begins with the detection of stars and galaxies in the same region of the sky, followed by unbiased measurement of the shapes of faint background galaxies that will be used to quantify the lensing signal. That signal can be expressed in terms of weighted quadrupole moments,

SBCS 4.2 Spherical-Template Mass
edited
... 4.2 Spherical-Template Mass
4.2.0. Generic Spherical-Template Mass
Notation: A unique symbo…

...

4.2 Spherical-Template Mass
4.2.0. Generic Spherical-Template Mass
Notation: A unique symbolic identifier (e.g., {cue_mathinline.gif} {cue_mathinline.gif}
M_{\Delta, {\rm lens}}
).

...

Discussion: Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)
4.2.1. Isothermal-Sphere Lensing Masses
Notation: A unique symbolic identifier (e.g., {cue_mathinline.gif} {cue_mathinline.gif}
M_{\Delta, {\rm lens}}
).
Introduction: Gravitational-lensing measurements of cluster mass within a spherical region of radius {cue_mathinline.gif}
r
require some assumptions aboutlens,iso}}
(The density contrast factor $\Delta$ gives the mean density contrast adopted for the 3-D mass distribution in that region. The simplest way to proceed is by assuming a spherically-symmetric mass distribution describeddefinition, along with an "m" or a simple parametric model. Two of"c" to specify the most commonly used models arereference density, as described below. Once a mass model has been chosen, one can fit it to the lensing data and derive an approximatein Section 1.2 on spherical-overdensity mass for any desired scale radius.masses.)
Introduction:
Algorithmic Description: Preparation of the lensing data for model fitting begins with the same basic steps for all spherical-template analyses. Here we outline those preparatory steps. Subsequent specific procedures unique to each spherical template are described below.
Weak-lensing analysis of a galaxy cluster begins with the detection of stars and galaxies in the same region of the sky, followed by unbiased measurement of the shapes of faint background galaxies that will be used to quantify the lensing signal. That signal can be expressed in terms of weighted quadrupole moments,

SBCS 4.2 Spherical-Template Mass
edited
... \gamma_T .
Shear is related to the surface mass density through
... \frac { \bar{\Sigma}[…

...

\gamma_T .
Shear is related to the surface mass density through

...

\frac { \bar{\Sigma}[0,r]\bar{\Sigma}( < r) - \bar{\Sigma}(r)}
where {cue_mathinline.gif} \bar{\Sigma}[0,r]\bar{\Sigma}( < r)
is the mean surface density within radius {cue_mathinline.gif}
r , \, \, \bar{\Sigma}(r)

SBCS 4.2 Spherical-Template Mass
edited
... \gamma_T .
Shear is related to the surface mass density through
... \frac { \bar{\Sigma}[…

...

\gamma_T .
Shear is related to the surface mass density through

...

\frac { \bar{\Sigma}[0,r]\bar{\Sigma}( < r) - \bar{\Sigma}(r)}
where {cue_mathinline.gif} \bar{\Sigma}[0,r]\bar{\Sigma}( < r)
is the mean surface density within radius {cue_mathinline.gif}
r , \, \, \bar{\Sigma}(r)

...

Make these brief bullet-like points.
Discussion: Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here) and this is a less-than sign:
2 < 3
DId that work?
Shear is related to the surface mass density through
\langle \gamma_T \rangle (r) = \frac { \bar{\Sigma}( < r ) - \bar{\Sigma}(r)} {\Sigma_{\rm crit}}
where {cue_mathinline.gif}
\bar{\Sigma}( < r)
is the mean surface density within