Virial Density Contrasts¶

Class providing dark matter halo virial mean density contrasts, \(\Delta_\mathrm{vir}\). The virial radius of a halo is defined as the radius within which the mean enclosed density equals \(\Delta_\mathrm{vir}\) times the background matter density (or critical density, depending on convention). Different cosmologies and spherical-collapse approximations lead to different values of \(\Delta_\mathrm{vir}(z)\).

Default implementation: virialDensityContrastSphericalCollapseClsnlssMttrCsmlgclCnstnt

Methods¶

densityContrast → double precision

Returns the virial density contrast at the given epoch.

double precision mass [in]
double precision time (optional) [in]
double precision expansionFactor (optional) [in]
logical collapsing (optional) [in]

densityContrastRateOfChange → double precision

Returns the rate of change of virial density contrast at the given epoch.

double precision mass [in]
double precision time (optional) [in]
double precision expansionFactor (optional) [in]
logical collapsing (optional) [in]

turnAroundOverVirialRadii → double precision

Returns the ratio of turnaround and virial radii at the given epoch.

double precision mass [in]
double precision time (optional) [in]
double precision expansionFactor (optional) [in]
logical collapsing (optional) [in]

isMassDependent → logical

Returns true if the virial density contrast is mass-dependent.

`virialDensityContrastBryanNorman1998`¶

A dark matter halo virial density contrast class using the fitting functions given by Bryan and Norman (1998). As such, it is valid only for \(\Omega_\Lambda=0\) or \(\Omega_\mathrm{M}+\Omega_\Lambda=1\) cosmologies and will either abort on other cosmologies (if [allowUnsupportedCosmology]=false), or revert to a numerical solution from top-hat collapse (if [allowUnsupportedCosmology]=true).

Parameters

[allowUnsupportedCosmology] (boolean; default .false.) — If true, unsupported cosmologies revert to using a numerical solution from the top-hat collapse model. Otherwise, unsupported cosmologies result in an error.

`virialDensityContrastFixed`¶

A dark matter halo virial density contrast class which uses a fixed virial density contrast of [densityContrastValue], defined relative to critical or mean density as specified by [densityType].

Parameters

[densityContrastValue] (real; default 200.0d0) — The virial density contrast to use in the fixed value model.
[densityType] (string; one of critical, mean; default critical) — The reference density to use in the fixed value virial density contrast model. Either of critical and mean are allowed.
[turnAroundOverVirialRadius] (real; default 2.0d0) — The ratio of the turnaround to virial radii in the fixed value model.

`virialDensityContrastFriendsOfFriends`¶

Dark matter halo virial density contrasts based on the friends-of-friends algorithm linking length. The virial density contrast is computed consistent with a given friends-of-friends algorithm linking length. According to Lacey and Cole (1994), the friends-of-friends algorithm selects objects bounded by an isodensity contour of density contrast

\[\Delta_\mathrm{iso} = {3 \over 2 \pi b^3},\]

where \(b=\)[virialDensityContrastFoFLinkingLength] is the dimensionless linking length of the algorithm (i.e. the linking length in units of the mean interparticle spacing). The virial density contrast is then given by:

\[\Delta_\mathrm{vir} = {\bar{\rho}_\mathrm{vir} \over \rho(r_\mathrm{vir})} \Delta_\mathrm{iso},\]

where \(\bar{\rho}_\mathrm{vir}\) is the mean density inside the virial radius and \(\rho(r_\mathrm{vir})\) is the density at the virial radius. The ratio \(\bar{\rho}_\mathrm{vir} / \rho(r_\mathrm{vir})\) is specified via the parameter [virialDensityContrastFoFDensityRatio]. Its default value of \(4.688\) is appropriate for an NFW halo of concentration \(c=6.88\) which is the concentration found by Prada et al. (2012) for halos with \(\sigma=1.686\) which is the approximate critical overdensity for collapse).

Parameters

[linkingLength] (real; default 0.2d0) — The friends-of-friends linking length algorithm to use in computing virial density contrast.
[densityRatio] (real; default 4.688d0) — The ratio of mean virial density to density at the virial radius to assume when setting virial density contrasts in the friends-of-friends model. Default from: Value appropriate for an NFW profile with concentration \(c=6.88\) which is the concentration found by Prada et al. (2012) for halos with \(\sigma=1.686\) which is the approximate critical overdensity for collapse.

`virialDensityContrastKitayamaSuto1996`¶

A virial density contrast class using the fitting formula of Kitayama and Suto (1996), and so is valid only in flat cosmological models (an error will be reported in non-flat models). Specifically,

\[\Delta_\mathrm{virial}(t) = 18 \pi^2 [1+0.4093 \left\{{1\over \Omega_\mathrm{matter}(t)}-1\right\}(t)^{0.9052}].\]

`virialDensityContrastPercolation`¶

A dark matter halo virial density contrast class based on the percolation analysis of More et al. (2011). The virial density contrast is computed to be consistent with a given friends-of-friends algorithm linking length using the percolation-theory-motivated calibration of More et al. (2011). Specifically, the friends-of-friends algorithm is assumed to link together particles forming an isodensity surface of density \(\rho = \bar{\rho} n_\mathrm{c}/b^3\), where \(\bar{\rho}\) is the mean density of the universe, \(n_\mathrm{c}=0.652960\) is a critical density for percolation as given by More et al. (2011), and \(b\) is the linking length. Given this bounding density, the virial density contrast is found by requiring that the halo contain the required mass within such a bounding density, given the halo density profile.

Methods

tabulate

Tabulate the virial density contrast as a function of mass and time.

double precision mass [in]
double precision time [in]

Parameters

[linkingLength] (real; default 0.2d0) — The friends-of-friends linking length to use in computing virial density contrasts with the percolation analysis of More et al. (2011).