Galaxy cluster mass profiles from weak gravitational lensing assuming spherical symmetry

Package to infer galaxy cluster mass profiles from weak-lensing data assuming only spherical symmetry (and not assuming a specific mass profile such as an NFW profile). See arXiv:2408.07026 for details.

Installation

To use this package, first install it using the julia package manager. It's not yet uploaded to the official julia package repository. As a workaround it's possible to install like this

pkg> add https://github.com/tmistele/SphericalClusterMass.jl#master

Usage

First import the package. We also import Unitful and UnitfulAstro for later use of units and LinearAlgebra to conveniently construct diagonal matrices.

using SphericalClusterMass
using Unitful
using UnitfulAstro
using LinearAlgebra

Then you can get the deprojected mass from the azimuthally averaged tangential reduced shear $G_+ = \langle g_+ \Sigma_{\mathrm{crit}} \rangle$ and the azimuthally averaged inverse critical surface density $f_c = \langle \Sigma_{\mathrm{crit}}^{-1} \rangle$ (alternatively, if individual source redshifts are not available, you can use $G_+ = \langle g_+ \rangle / \langle \Sigma_{\mathrm{crit}}^{-1} \rangle$ and $f_c = \langle \Sigma_{\mathrm{crit}}^{-2} \rangle / \langle \Sigma_{\mathrm{crit}}^{-1} \rangle$). The following code calculates (a form of) the deprojected mass profile and the associated covariance matrix,

R=[.2, .5, .7] .* u"Mpc"
result = calculate_gobs_and_covariance_in_bins(
    # Observational data.
    R=R,
    G=1e3 .* [.3, .2, .1] .* u"Msun/pc^2",
    f=1e-3 .* [.9, .9, .9] ./ u"Msun/pc^2",
    # Covariance matrix.
    # This corresponds to no actual covariance, just 10% statistical uncertainties on G
    G_covariance=diagm((1e2 .* [.3, .2, .1] .* u"Msun/pc^2") .^ 2),
    # Extrapolate beyond last data point assuming G ~ 1/R.
    # Corresponds to a singular isothermal sphere.
    extrapolate=ExtrapolatePowerDecay(1),
    # Interpolate linearly between discrete data points in "R-space".
    # - Quadratic interpolation is also a good choice, but a bit slower.
    # - To interpolate in "ln(R)-space", use `InterpolateLnR(..)`.
    #   This may be the better choice for logarithmic bins.
    interpolate=InterpolateR(1),
)

This will run for a few seconds on the first run to compile the code. Subsequent runs will be fast, unless the number of data points changes, which requires recompilation. result is a named tuple with fields gobs, gobs_stat_cov and gobs_stat_err where gobs refers to the acceleration $G M(r) /r^2$. The deprojected mass $M$ and its statistical uncertainties and covariance matrix can be inferred from this. For example,

M = result.gobs .* R .^ 2 ./ u"G" .|> u"Msun"
M_stat_err = result.gobs_stat_err .* R .^ 2 ./ u"G" .|> u"Msun"

using Plots
plot(R, M, yerror=M_stat_err)

Faster calculation assuming constant $f_c$

In practice, $f_c$ is often reasonably constant as a function of radius. A constant $f_c$ makes the calculation simpler and faster. This faster calculation will automatically be done if $f_c$ is passed as a scalar instead of a vector,

result = calculate_gobs_and_covariance_in_bins(
    R=R,
    G=1e3 .* [.3, .2, .1] .* u"Msun/pc^2",
    f=1e-3 * .9 / u"Msun/pc^2", # <-- This is now a scalar instead of a vector
    G_covariance=diagm((1e2 .* [.3, .2, .1] .* u"Msun/pc^2") .^ 2),
    extrapolate=ExtrapolatePowerDecay(1),
    interpolate=InterpolateR(1),
)

Faster calculation without covariance matrix

If one is not interested in the statistical uncertainties and covariances, one can use calculate_gobs_fgeneral

result = calculate_gobs_fgeneral(
    R=R,
    G=1e3 .* [.3, .2, .1] .* u"Msun/pc^2",
    f=1e-3 .* [.9, .9, .9] ./ u"Msun/pc^2",
    extrapolate=ExtrapolatePowerDecay(1),
    interpolate=InterpolateR(1),
).(R ./ u"Mpc")

or, for the case assuming $f_c = \mathrm{const}$, one can use calculate_gobs_fconst,

result = calculate_gobs_fconst(
    R=R,
    G=1e3 .* [.3, .2, .1] .* u"Msun/pc^2",
    f=1e-3 * .9 / u"Msun/pc^2", # <-- This is now a scalar instead of a vector
    extrapolate=ExtrapolatePowerDecay(1),
    interpolate=InterpolateR(1),
).(R ./ u"Mpc")

Correct for miscentering

To correct for miscentering (to leading order in $R_{\mathrm{mc}}/R$ where $R_{\mathrm{mc}}$ is the miscentering radius), you can use the miscenter_correct argument:

result = calculate_gobs_and_covariance_in_bins(
    R=R,
    G=1e3 .* [.3, .2, .1] .* u"Msun/pc^2",
    f=1e-3 .* [.9, .9, .9] ./ u"Msun/pc^2",
    G_covariance=diagm((1e2 .* [.3, .2, .1] .* u"Msun/pc^2") .^ 2),
    extrapolate=ExtrapolatePowerDecay(1),
    interpolate=InterpolateR(1),
    # This will (approximately) correct for miscentering by `Rmc²`.
    # The uncertainty `σ_Rmc²` on `Rmc²` will be propagated into the covariance matrix.
    # The default is `MiscenterCorrectNone()`, which does not correct for miscentering.
    miscenter_correct=MiscenterCorrectSmallRmc(
        Rmc²=(.16u"Mpc")^2,
        σ_Rmc²=(.16u"Mpc")^2
    )
)

The functions calculate_gobs_fconst and calculate_gobs_fgeneral also support the miscenter_correct argument.