Modifiers for merger tree branching probabilities

Class providing multiplicative modifiers to the merger tree branching probability rate, applied at each potential branching event during tree construction. Modifiers allow the branching probability to be re-weighted based on halo properties (mass, overdensity, redshift) beyond what the base branching probability class provides. This enables fitting to N-body simulation merger rates or implementing environmental or assembly-bias corrections.

Default implementation: mergerTreeBranchingProbabilityModifierIdentity

Methods

rateModifier → double precision

Return the multiplicative modifier to the tree branch probability rate.

• type (treeNode), intent(inout) :: nodeParent

• double precision , intent(in ) :: massParent, sigmaParent, sigmaChild, timeParent

mergerTreeBranchingProbabilityModifierIdentity

A merger tree branching probability modifier class which always applies the identity modifier.

(Default implementation)

Parameters

• [redshift] (default 0.0d0) — The redshift at which the transfer function is defined.

mergerTreeBranchingProbabilityModifierMulti

Chains multiple other merger tree branch probability modifiers by taking their product.

Methods

• columnDescriptions — Return a description of the columns.

• elementCount — Return the number of properties in the tuple.

• extractDouble — Extract the double properties from the given node.

• extractInteger — Extract the integer properties from the given node.

• names — Return the names of the properties extracted.

• descriptions — Return descriptions of the properties extracted.

• unitsInSI — Return the units of the properties extracted in the SI system.

• units — Return an object containing units metadata for the properties.

• ranks — Return the ranks of the properties extracted.

• metaData — Populate a hash with meta-data for the property.

mergerTreeBranchingProbabilityModifierParkinson2008

A merger tree branching probability modifier class which uses the model of Parkinson et al. (2008). Specifically, the branching probability is multiplied by:

\[G(\delta_\mathrm{p},\sigma_\mathrm{c},\sigma_\mathrm{p}) = G_0 \left({\sigma_\mathrm{c}\over\sigma_\mathrm{p}}\right)^{\gamma_1} \left({\delta_\mathrm{p}\over\sigma_\mathrm{p}}\right)^{\gamma_2}\]

where \(\delta_\mathrm{p}\) is the current critical overdensity for collapse for the parent halo, and \(\sigma_\mathrm{c}\) and \(\sigma_\mathrm{p}\) are the root-variance of the smooth mass-density field on scales corresponding to the masses of child and parent halos respectively. The parameters of the fit can be adjusted via input parameters: \(G_0=\)[G0], \(\gamma_1=\)[gamma1], and \(\gamma_2=\)[gamma2].

Parameters

• [G0] (default 0.57d0) — The parameter \(G_0\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma1] (default 0.38d0) — The parameter \(\gamma_1\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma2] (default -0.01d0) — The parameter \(\gamma_2\) appearing in the modified merger rate expression of Parkinson et al. (2008).

mergerTreeBranchingProbabilityModifierPCHPlus

Provides a merger tree branching probability rate modifier which uses the model of Parkinson et al. (2008) plus an additional term. Specifically, the modifier becomes

\[G\left( {\sigma_1 \over \sigma_2} , {\delta_2 \over \sigma_2} \right) = G_0 \left({\sigma_1\over\sigma_2}\right)^{\gamma_1} \left({\delta_2\over\sigma_2}\right)^{\gamma_2} \left(1 - {\sigma_2^2 \over \sigma_1^2}\right)^{\gamma_3},\]

where \(\sigma_i=\sigma(M_i)\), \(\sigma(M)\) is the usual present-day, linear-theory mass-variance in spheres enclosing an average mass \(M\), \(M_2\) is the mass of the parent halo, \(M_1\) is the mass of the child halo, and \(\delta_2\) is the critical overdensity for collapse at the epoch of the parent.

Parameters

• [G0] (default 0.57d0) — The parameter \(G_0\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma1] (default 0.38d0) — The parameter \(\gamma_1\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma2] (default -0.01d0) — The parameter \(\gamma_2\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma3] (default 0.0d0) — The parameter \(\gamma_3\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma4] (default 0.0d0) — The parameter \(\gamma_4\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma5] (default 0.0d0) — The parameter \(\gamma_5\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [accuracyFirstOrder] (default 0.1d0) — Limits the step in \(\delta_\mathrm{crit}\) when constructing merger trees using the Parkinson et al. (2008) algorithm, so that it never exceeds accuracyFirstOrder\(\sqrt{2[\sigma^2(M_2/2)-\sigma^2(M_2)]}\).

• [precisionHypergeometric] (default 1.0d-6) — The fractional precision required in evaluates of hypergeometric functions in the modified Press-Schechter tree branching calculations.

• [hypergeometricTabulate] (default .true.) — Specifies whether hypergeometric factors should be precomputed and tabulated in modified Press-Schechter tree branching functions.

• [cdmAssumptions] (default .false.) — If true, assume that \(\alpha(=-\mathrm{d}\log \sigma/\mathrm{d}\log M)>0\) and \(\mathrm{d}\alpha/\mathrm{d}M>0\) (as is true in the case of CDM) when constructing merger trees using the Parkinson et al. (2008).

• [tolerateRoundOffErrors] (default .false.) — If true, round-off errors in integrations of branching probability will be tolerated. This may degrade the accuracy of solutions, but can be unavoidable in models with cut-offs in their power spectra.

• [G0] (default 0.57d0) — The parameter \(G_0\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma1] (default 0.38d0) — The parameter \(\gamma_1\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma2] (default -0.01d0) — The parameter \(\gamma_2\) appearing in the modified merger rate expression of Parkinson et al. (2008).

• [gamma3] (default 0.0d0) — The parameter \(\gamma_3\) appearing in the extension of the modified merger rate expression of Parkinson et al. (2008) as defined in this class.

mergerTreeBranchingProbabilityModifierZhang2014

Provides a merger tree branching probability rate modifier which uses the model of Zhang et al. (2014) to account for environmental-dependence of the merger rate. Specifically, the modifier is:

\[1 + \kappa \beta \alpha - \sqrt{2\pi} \kappa \nu (1-\alpha)^{3/2} + \pi \kappa \left(\nu^2\frac{\delta_\mathrm{d}}{\delta_\mathrm{c}}-1\right) (1-\alpha)^{3/2} \exp\left[\frac{\nu^2}{2}\right] \hbox{erfc}\left[\frac{\nu}{\sqrt{2}}\right]\]

where \(\kappa\) characterizes the non-Markovian behavior in the excursion set random walks, and whose value depends on the shape of the window function (Zhang et al., 2014), \(\alpha = S_\mathrm{d}/S_\mathrm{p}\) with \(S_\mathrm{d}\) being the variance on the scale of the descendant (i.e., parent) halo, and \(S_\mathrm{p}\) being the variance on the scale of the progenitor (i.e., child) halo,

\[\beta = -2 + \frac{(1-\alpha)^{3/2}}{2\alpha}\ln\left(\frac{1+\sqrt{1-\alpha}}{1-\sqrt{1-\alpha}}\right)+\frac{1}{\alpha}+2\alpha,\]

\(\nu = \delta_\mathrm{c}/\sqrt{S_\mathrm{d}}\), \(\delta_\mathrm{e}\) is the overdensity on the scale of the environment (evaluated at \(z=0\)), and \(\delta_\mathrm{c}\) is the critical overdensity at the time of the descendant halo.

Note that this implementation uses the solutions for the limit of large environment scale (\(S_\mathrm{e} \rightarrow 0\); equation 30 of Zhang et al. 2014). A warning is emitted if \(S_\mathrm{e}\) is sufficiently large to break this assumption.

Parameters

• [kappa] (default 0.44d0) — The parameter characterizing non-Markovian behavior in excursion set random walks (Zhang et al., 2014).