.. _physics-darkMatterProfileShape:

Dark Matter Profile Shapes
==========================

Class providing the shape parameter :math:`\alpha` of dark matter halo density profiles such as the Einasto profile, :math:`\rho(r) \propto \exp\{-\frac{2}{\alpha}[(r/r_\mathrm{s})^\alpha - 1]\}`. The shape parameter controls the curvature of the profile near the center and is typically fit from :math:`N`-body simulations as a function of halo mass and redshift.

**Default implementation:** ``darkMatterProfileShapeGao2008``

Methods
-------

``shape`` → ``double precision``
   Returns the dimensionless shape parameter :math:`\alpha` of the Einasto dark matter density profile for the halo in ``node``, controlling the curvature of the inner profile with typical values in the range :math:`0.1`--:math:`0.3` from N-body simulations.

   * ``type(treeNode), intent(inout) :: node``

.. _physics-darkMatterProfileShapeBrown2021:

``darkMatterProfileShapeBrown2021``
-----------------------------------

A dark matter profile shape class in which the shape parameter for Einasto density profiles is computed using a fitting function from :cite:t:`brown_towards_2022`:

.. math::

   \alpha = \left\{ \begin{array}{ll} 8.52 \times 10^{-4} \nu_\alpha^4 + 0.166 & \hbox{ if } \nu_\alpha < 3.541 \\ 0.3 & \hbox{ if } \nu_\alpha \ge 3.541, \end{array} \right.

where :math:`\nu_\alpha=\delta_\mathrm{c}(t)/\sigma_\alpha(M)` is the peak height of the halo. The truncation at :math:`\alpha = 0.3` is included since :cite:t:`brown_towards_2022`'s fits do not probe this region and extremely large values of :math:`\alpha` are numerically troublesome.

This implementation accepts any :galacticus-class:`cosmologicalMassVarianceClass` object for use in computing for computing :math:`\sigma_\mathrm{c}(M)`. *However*, :cite:t:`brown_towards_2022` recommend using :math:`\sigma_\mathrm{c}(M)` computed using a generalized top-hat window function (:galacticus-class:`powerSpectrumWindowFunctionTopHatGeneralized`) with :math:`\mu_\mathrm{g}=0.01`.

.. _physics-darkMatterProfileShapeGao2008:

``darkMatterProfileShapeGao2008``
---------------------------------

A dark matter profile shape class in which the shape parameter for Einasto density profiles is computed using a fitting function from :cite:t:`gao_redshift_2008`:

.. math::

   \alpha = \left\{ \begin{array}{ll} 0.155 + 0.0095\nu^2 & \hbox{ if } \nu < 3.907 \\ 0.3 & \hbox{ if } \nu \ge 3.907,
   \end{array} \right.

where :math:`\nu=\delta_\mathrm{c}(t)/\sigma(M)` is the peak height of the halo. The truncation at :math:`\alpha = 0.3` is included since :cite:t:`gao_redshift_2008`'s fits do not probe this region and extremely large values of :math:`\alpha` are numerically troublesome.

**(Default implementation)**

**Parameters**

* ``[scatter]`` (default ``0.0d0``) — The scatter (in dex) to assume in the halo concentration distribution at fixed mass.

.. _physics-darkMatterProfileShapeKlypin2015:

``darkMatterProfileShapeKlypin2015``
------------------------------------

Dark matter halo shape parameters are computed using the algorithm of :cite:t:`klypin_multidark_2014`.

**Parameters**

* ``[sample]`` (default ``var_str('all')``) — The sample to use for the halo shape parameter algorithm of :cite:t:`klypin_multidark_2014`.
* ``[sample]`` (default ``var_str('planck200CritRelaxedMass')``) — The sample to use for the halo concentration algorithm of :cite:t:`klypin_multidark_2014`.