This package allows for the calculation of the physical properties of dark-matter halos.

Halo instances are created with the function halo.New and then halo properties are calculated through that halo's methods. The primary concern of this library is correcting for biases in the x-ray mass estimates of mass in large galaxy clusters, but it will work in other contexts as well.

Documentation can be found at http://godoc.org/bitbucket.org/phil-mansfield/halo. Documentation for a conveient ASCII table interface can be found at http://godoc.org/bitbucket.org/phil-mansfield/table.

The halo/cosmo subpackage defines a number of functions which calculate global cosmological parameters.

The halo/num subpackage is a small numerical computation library. It currently supports differentiation, several types of one-dimensional integration, and various searching functions.

The halo/table-scipts subpackage is a collection of quick analysis scripts. These can serve as usage examples. A real programmer wouldn't include these in a public repository, but I'm a physicist and not a programmer.

The following section is a work in progress.

The central purpose of this library is to correct cluster mass measurements derived from x-ray measurments. X-ray mass measurements are made by assuming that the clusters are in hydrostatic equilibrium, so the equation

dP_th/dr + rho_gas * GM/r^2 = 0

is valid. Since dP_th/dr and rho_gas can be measured directly cluster mass can be found at an arbitrary radius. However, from numerical simulations we know that there is a non-insignificant bulk flow in most clusters, resulting in an effective pressure gradient which is larger than the empirically measured thermal pressure gradient.

By dividing M_true (using dP_eff/dr) by M_bias (using fP_thermal/dr) we arrive at the following relation:

M_true(r)/M_bias(r) = b(r) = 1/f_th(r) (1 - beta(r)/alpha(r))

where

beta(r) = d ln(f_th)/dr

and

alpha(r) = d ln(P_th)/dr.

Here f_th = P_th / P_eff. f_th cannot be measured directly and must come from average relations in hydrodynamic N-body simulations (e.g. Battaglia, et al, 2012 and Nelson, et al, 2012 ).

It is straight-forward to insert fitted formulae into beta and alpha to arrive at a closed-form expression for b(r), but special care must be taken. Any reasonable fit for f_th will depend on at least the mass or the radius of the halo (for the purposes of this overview, a halo's radius and mass correspond to its 500c overdensity sphere, but this library is fully capable of working with alternate mass definitions). Since f_th is measured from simulations the fit will be made with respect to the true mass of the halo, but any any empirical fits for dP_th/dr (e.g. the Planck Pressure Profile, or Arnaud et al's Universal Pressure Profile ) will be made with respect to the halo's biased mass. For this reason it is strongly suggested that you use pressure profiles derived from simulations. For maximal internal consistency the pressure profile used in the calculate of f_th should be used.

It is still possible to compute a halo's bias from empirical pressure profiles, but if one starts with the true mass of the halo and wishes to compute the biased mass, one must solve

 0 = R_bias - R_overdenisty(rho_500, R_bias)

numerically. R_overdensity cannot be found trivially because biased halos do not follow an NFW profile (see below), so the dP_th/dr fit and its dependance on R_bias must be used.

Regardless of what type of pressure profile is assumed, calculating the bias of a halo based off of its biased mass requires numerically solving

0 = b - 1/fth(R_bias) (1 - alpha(R_Bias, M_true = b(R_bias)*M_bias) / beta(R_bias, M = b(R_bias)*M_bias*)

for b. This is neccesary because although haloes are traditionally assumed to follow NFW profiles, these profiles are measured using the true mass profiles found in simulations. There is no a priori garuantee that M_true(r) / b(r) can also be fit to the same functional form (in fact, for the profiles used here it most certainly cannot be). Thus, in order to calculate M_bias, onne must compute M_true(r)/b(r) and to calculate rho_bias(r) one must calculate (d/dr (M_true(r)/b(r)))/(4 pi r^2).

There are three quantities which are very easy to confuse, but are quantitatively fairly different. These are b(R_bias), b(R_true), and M_true/M_bias. Since the radius of a halo is related to its mass by the relation r = (m/(rho pi 4/3))^(1/3), a change in halo mass corresponds to a a change of radius meaning that b(R_bias) and b(R_true) are significantly different quantities. Furthermore, b(R_true) must not be confued with M_true/M_bias. M_bias(R_true) is neccesarily larger than M_bias(R_bias), meaning that to correct the measured x-ray masses it is insufficient to simply compute b(r). Instead equivelent haloes must actually be constructed.