Conditional Mass Function

Class providing empirical models of the conditional stellar mass function (CSMF)—the mean number of galaxies \(\langle N(M_\star|M_\mathrm{halo})\rangle\) of stellar mass \(M_\star\) hosted by a dark matter halo of mass \(M_\mathrm{halo}\), separately for central and satellite galaxies. The CSMF is the key ingredient of halo occupation distribution (HOD) and stellar-to-halo mass relation (SHMR) models used to populate \(N\)-body simulations with galaxies and to interpret galaxy clustering observations without running a full semi-analytic model.

Default implementation: conditionalMassFunctionBehroozi2010

Methods

massFunction → double precision

Return the cumulative conditional mass function, \(\langle N(M_\star|M_\mathrm{halo}) \rangle \equiv \phi(M_\star|M_\mathrm{halo})\).

• double precision , intent(in ) :: massHalo , mass

• type (enumerationHaloModelGalaxyTypeType), intent(in ), optional :: galaxyType

massFunctionVariance → double precision

Return the variance in the cumulative conditional mass function, \(\langle N(M_\star|M_\mathrm{halo}) \rangle \equiv \phi(M_\star|M_\mathrm{halo})\).

• double precision, intent(in ) :: massHalo,massLow,massHigh

conditionalMassFunctionBehroozi2010

A conditional mass function class which implements the fitting functions of Behroozi et al. (2010):

\[\langle N_\mathrm{c}(M_\star|M)\rangle \equiv \int_{M_\star}^\infty \phi_\mathrm{c}(M_\star^\prime) \d \ln M_\star^\prime = {1 \over 2} \left[ 1 - \hbox{erf}\left( {\log_{10}M_\star - \log_{10} f_\mathrm{SHMR}(M) \over \sqrt{2}\sigma_{\log M_\star}} \right) \right].\]

Here, the function \(f_\mathrm{SHMR}(M)\) is the solution of

\[\log_{10}M = \log_{10}M_1 + \beta \log_{10}\left({M_\star \over M_{\star,0}}\right) + {(M_\star/M_{\star,0})^\delta \over 1 + (M_\star/M_{\star,0})^{-\gamma}} - {1/2}.\]

For satellites,

\[\langle N_\mathrm{s}(M_\star|M)\rangle \equiv \int_{M_\star}^\infty \phi_\mathrm{s}(M_\star^\prime) \d \ln M_\star^\prime = \langle N_\mathrm{c}(M_\star|M)\rangle \left({f^{-1}_\mathrm{SHMR}(M_\star) \over M_\mathrm{sat}}\right)^{\alpha_\mathrm{sat}} \exp\left(- {M_\mathrm{cut} \over f^{-1}_\mathrm{SHMR}(M_\star)} \right),\]

where

\[{M_\mathrm{sat} \over 10^{12} \mathrm{M}_\odot} = B_\mathrm{sat} \left({f^{-1}_\mathrm{SHMR}(M_\star) \over 10^{12} \mathrm{M}_\odot}\right)^{\beta_\mathrm{sat}},\]

and

\[{M_\mathrm{cut} \over 10^{12} \mathrm{M}_\odot} = B_\mathrm{cut} \left({f^{-1}_\mathrm{SHMR}(M_\star) \over 10^{12} \mathrm{M}_\odot}\right)^{\beta_\mathrm{cut}}.\]

By default, parameter values are taken from the fit of Leauthaud et al. (2012), specifically their SIG_MOD1 method for their \(z_1\) sample. These default values, and the Galacticus input parameters which can be used to adjust them are shown in Table 1. This method assumes that \(P_\mathrm{s}(N|M_\star,M;\delta \ln M_\star)\) is a Poisson distribution while \(P_\mathrm{c}(N|M_\star,M;\delta \ln M_\star)\) has a Bernoulli distribution, with each distribution’s free parameter fixed by requiring

\[\phi(M_\star;M) \delta \ln M_\star = \sum_{N=0}^\infty N P(N|M_\star,M;\delta \ln M_\star)\]

Table 1 Parameters of the Behroozi et al. (2010) conditional stellar mass function model, along with their default values and the corresponding Galacticus input parameters.

Parameter	Default	Galacticus name
\(\alpha_\mathrm{sat}\)	1.0	[conditionalStellarMassFunctionBehrooziAlphaSatellite]
\(\log_{10} M_1\)	12.520	[conditionalStellarMassFunctionBehrooziLog10M1]
\(\log_{10} M_{\star,0}\)	10.916	[conditionalStellarMassFunctionBehrooziLog10Mstar0]
\(\beta\)	0.457	[conditionalStellarMassFunctionBehrooziBeta]
\(\delta\)	0.5666	[conditionalStellarMassFunctionBehrooziDelta]
\(\gamma\)	1.53	[conditionalStellarMassFunctionBehrooziGamma]
\(\sigma_{\log M_\star}\)	0.206	[conditionalStellarMassFunctionBehrooziSigmaLogMstar]
\(B_\mathrm{cut}\)	1.47	[conditionalStellarMassFunctionBehrooziBCut]
\(B_\mathrm{sat}\)	10.62	[conditionalStellarMassFunctionBehrooziBSatellite]
\(\beta_\mathrm{cut}\)	\(-\)0.13	[conditionalStellarMassFunctionBehrooziBetaCut]
\(\beta_\mathrm{sat}\)	0.859	[conditionalStellarMassFunctionBehrooziBetaSatellite]

(Default implementation)

Methods

• compute — Compute the cumulative conditional mass function, \(\langle N(M_\star|M_\mathrm{halo}) \rangle \equiv \phi(M_\star|M_\mathrm{halo})\).

Parameters

• [alphaSatellite] (default 1.0d0) — The parameter \(\alpha_\mathrm{sat}\) from the fitting functions of Behroozi et al. (2010).

• [log10M1] (default 12.520d0) — The parameter \(\log_{10}M_1\) from the fitting functions of Behroozi et al. (2010).

• [log10Mstar0] (default 10.916d0) — The parameter \(\log_{10}M_{\star,0}\) from the fitting functions of Behroozi et al. (2010).

• [beta] (default 0.457d0) — The parameter \(\beta\) from the fitting functions of Behroozi et al. (2010).

• [delta] (default 0.5666d0) — The parameter \(\delta\) from the fitting functions of Behroozi et al. (2010).

• [gamma] (default 1.53d0) — The parameter \(\gamma\) from the fitting functions of Behroozi et al. (2010).

• [sigmaLogMstar] (default 0.206d0) — The parameter \(\sigma_{\log M_\star}\) from the fitting functions of Behroozi et al. (2010).

• [BCut] (default 1.47d0) — The parameter \(B_\mathrm{cut}\) from the fitting functions of Behroozi et al. (2010).

• [BSatellite] (default 10.62d0) — The parameter \(B_\mathrm{sat}\) from the fitting functions of Behroozi et al. (2010).

• [betaCut] (default -0.13d0) — The parameter \(\beta_\mathrm{cut}\) from the fitting functions of Behroozi et al. (2010).

• [betaSatellite] (default 0.859d0) — The parameter \(\beta_\mathrm{sat}\) from the fitting functions of Behroozi et al. (2010).