.. _physics-powerSpectrumPrimordial:

Primordial Power Spectrum
=========================

Class providing the primordial power spectrum. The late-time power spectrum is typically computed from the specified primordial power spectrum and the :galacticus-class:`transferFunction` and normalized to a value of :math:`\sigma_8`.

**Default implementation:** ``powerSpectrumPrimordialPowerLaw``

Methods
-------

``power`` → ``double precision``
   Return the (unnormalized) power in the primordial power spectrum at the given ``wavenumber`` (specified in units of Mpc\ :math:`^{-1}`).

   * ``double precision, intent(in ) :: wavenumber``

``logarithmicDerivative`` → ``double precision``
   Return the logarithmic derivative with respect to wavenumber of the primordial power spectrum at the given ``wavenumber`` (specified in units of Mpc\ :math:`^{-1}`).

   * ``double precision, intent(in ) :: wavenumber``

.. _physics-powerSpectrumPrimordialCosmologicalCube:

``powerSpectrumPrimordialCosmologicalCube``
-------------------------------------------

A primordial power spectrum class which truncates a primordial power spectrum into a cosmological simulation cube. Specifically, it assumes that wave-vectors with all components (:math:`k_x,k_y,k_z`) smaller than :math:`\Delta k = 2 \pi f / L`, where :math:`L=`\ ``[lengthCube]`` is the simulation cube length, and :math:`f=`\ ``wavenumberMinimumFactor`` specifies the minimum wavenumber (in units of :math:`2 \pi / L`) to which the power spectrum is integrated, are missing from the power spectrum. For wavenumbers of magnitude :math:`k` the fraction of power missed due to these missing wavenumbers is computed. The total power at that wavenumber is then reduced by that amount.

The fractional suppression in power, :math:`f(x)`, as a function of :math:`x=k/\Delta k` is given by:

.. math::

   f(x) = \left\{ \begin{array}{ll} 0 & \hbox{ for } x \le 1, \\ 3(1-x^{-1}) & \hbox{ for } 1 < x \le \sqrt{2}, \\ 3 x{-1} [ 1 + 2 \pi^{-1} x \sin^{-1}(\{x^2-1\}^{-1}) - 4 \pi^{-1} x \sin^{-1}(\{1-x^2\}^{-1/2})  ] & \hbox{ for } \sqrt{2} < x \le \sqrt{3}, \\ 1 & \hbox{ for } \sqrt{3} < x. \end{array} \right.

For the :math:`1 < x \le \sqrt{2}` the solution is found by considering the 6 spherical caps of the sphere which protrude from the faces of the cube. The solution for :math:`\sqrt{2} < x \le \sqrt{3}` is more complicated and follows the solution given by :cite:t:`achille2013`.

**Parameters**

* ``[lengthCube]`` — The length of the cosmological cube.
* ``[wavenumberMinimumFactor]`` (default ``0.5d0``) — The minimum wavenumber (in units of :math:`2 \pi / L`) to which the power spectrum is integrated.

.. _physics-powerSpectrumPrimordialPiecewisePowerLaw:

``powerSpectrumPrimordialPiecewisePowerLaw``
--------------------------------------------

Implements a piecewise power-law primordial power spectrum, possibly with a running index. The primordial power spectrum has the form:

.. math::

   P(k) \propto A_\mathrm{i} k^{n_\mathrm{eff,i}(k)},

where

.. math::

   n_\mathrm{eff,i}(k) = n_\mathrm{s,i} + {1\over 2}{\d n \over \d \ln k}_i \ln \left( {k \over k_\mathrm{ref,i}} \right) + {1\over 6}{\d^2 n \over \d \ln k^2}_i \left[ \ln \left( {k \over k_\mathrm{ref}} \right) \right]^2,

where :math:`n_\mathrm{s,i}=`\ ``[index]`` is the power spectrum index at wavenumber :math:`k_\mathrm{ref,i}=`\ ``[wavenumberReference]``, :math:`\d n / \d \ln k_i=`\ ``[running]``, and :math:`\d^2 n / \d \ln k^2_i=`\ ``[runningRunning]`` describes the running of this index with wavenumber. The subscript "i", which runs from :math:`1` to :math:`N` refers to each interval of the piecewise power-law. Note that :math:`k_\mathrm{ref,i}` is defined only for :math:`i\ge 2`. For the first (:math:`i=1`) interval, the wavenumber ranges from :math:`0` to :math:`k_\mathrm{ref,2}`.

The amplitudes, :math:`A_i`, are chosen to make the power spectrum continuous.

**Methods**

* ``indexEffective`` — Compute the local effective index of the power spectrum.
* ``indices`` — Compute the indices corresponding to the given wavenumber.

**Parameters**

* ``[index]`` (default ``[0.9649d0,0.9649d0]``) — The index of the power-law primordial power spectrum.
* ``[running]`` (default ``[0.0d0,0.0d0]``) — The running, :math:`\d n_\mathrm{s} / \d \ln k`, of the power spectrum index.
* ``[runningRunning]`` (default ``[0.0d0,0.0d0]``) — The running-of-the-running, :math:`\d^2 n_\mathrm{s} / \d \ln k^2`, of the power spectrum index.
* ``[wavenumberReference]`` (default ``[1.0d0]``) — When a running power spectrum index is used, this is the wavenumber, :math:`k_\mathrm{ref}`, at which the index is equal to ``[index]``.

.. _physics-powerSpectrumPrimordialPowerLaw:

``powerSpectrumPrimordialPowerLaw``
-----------------------------------

Implements a power-law primordial power spectrum, possibly with a running index. The primordial power spectrum has the form:

.. math::

   P(k) \propto k^{n_\mathrm{eff}(k)},

where

.. math::

   n_\mathrm{eff}(k) = n_\mathrm{s} + {1\over 2}{\d n \over \d \ln k} \ln \left( {k \over k_\mathrm{ref}} \right) + {1\over 6}{\d^2 n \over \d \ln k^2} \left[ \ln \left( {k \over k_\mathrm{ref}} \right) \right]^2,

where :math:`n_\mathrm{s}=`\ ``[index]`` is the power spectrum index at wavenumber :math:`k_\mathrm{ref}=`\ ``[wavenumberReference]``, :math:`\d n / \d \ln k=`\ ``[running]``, and :math:`\d^2 n / \d \ln k^2=`\ ``[runningRunning]`` describes the running of this index with wavenumber.

**(Default implementation)**

**Parameters**

* ``[exponent]`` (default ``1.73d0``) — Exponent of the differential luminosity function.
* ``[rateHydrogenIonizingPhotonsMinimum]`` (default ``1.0d48``) — The minimum ionizing photon production rate (:math:`Q_\mathrm{H,min}`, in photons/s) below which the power-law HII region luminosity function is truncated to zero.
* ``[rateHydrogenIonizingPhotonsMaximum]`` (default ``huge(0.0d0)``) — The maximum ionizing photon production rate (:math:`Q_\mathrm{H,max}`, in photons/s) above which the power-law HII region luminosity function is truncated to zero.
* ``[exponent]`` (default ``1.0d0``) — Halo masses will be (pseudo-)uniformly distributed in :math:`[\log(M)]^{1/(1+\alpha)}` where :math:`\alpha=`\ ``exponent``.
* ``[wavelengthMinimum]`` — The minimum wavelength (in units of \AA) for the power-law spectrum.
* ``[wavelengthMaximum]`` — The maximum wavelength (in units of \AA) for the power-law spectrum.
* ``[exponent]`` — The exponent of the power-law spectrum.
* ``[normalization]`` — The normalization (in units of :math:`L_\odot / \AA`) of the power-law spectrum.
* ``[normalization]`` — Parameter :math:`\sigma_{12}` appearing in model for random errors in the halo mass function.
* ``[fractionalErrorHighMass]`` — Parameter :math:`\sigma_\infty` appearing in model for random errors in the halo mass function.
* ``[exponent]`` — Parameter :math:`\gamma` appearing in model for random errors in the halo mass function. Specifically, the fractional error is given by :math:`\sigma(M) = \left[ \sigma^2_{12} \left({M_\mathrm{halo} \over 10^{12}\mathrm{M}_\odot}\right)^{2\gamma} + \sigma^2_\infty \right]^{1/2}`, where :math:`\sigma_{12}=`\ ``[normalization]`` and :math:`\gamma=`\ ``[exponent]``.
* ``[correlationModelTrivial]`` (default ``.true.``) — If true, the correlation between mass errors of pairs of halos is unity for halos with identical mass and time, and zero otherwise. If false, a power-law correlation model in mass ratio and expansion factor ratio is used instead.
* ``[correlationNormalization]`` (default ``0.0d0``) — Variable :math:`C_0` in the model for the correlation between halo mass errors: :math:`C_{12} = C_0 [M_2/M_1]^\alpha [(1+z_2)/(1+z_1)]^\beta`.
* ``[correlationMassExponent]`` (default ``0.0d0``) — Variable :math:`\alpha` in the model for the correlation between halo mass errors: :math:`C_{12} = C_0 [M_2/M_1]^\alpha [(1+z_2)/(1+z_1)]^\beta`.
* ``[correlationRedshiftExponent]`` (default ``0.0d0``) — Variable :math:`\beta` in the model for the correlation between halo mass errors: :math:`C_{12} = C_0 [M_2/M_1]^\alpha [(1+z_2)/(1+z_1)]^\beta`.
* ``[radiusLow]`` (default ``+0.0154d0``) — The low-mass limit of the characteristic scale radius :math:`r_0` (in Mpc) in the power-law scale radius model, giving the scale radius normalization for low-mass halos as a function of peak height and expansion factor.
* ``[radiusHigh]`` (default ``+0.0962d0``) — The high-mass limit of the characteristic scale radius :math:`r_1` (in Mpc) in the power-law scale radius model, giving the scale radius normalization for high-mass halos.
* ``[radiusTransition]`` (default ``+1.2137d0``) — The peak height :math:`\nu` at which the characteristic scale radius transitions between its low-mass and high-mass limiting values in the power-law scale radius model.
* ``[radiusWidth]`` (default ``+0.5482d0``) — The parameter :math:`\Delta r` in the power-law scale radius model.
* ``[massLow]`` (default ``+0.3895d0``) — The parameter :math:`\alpha_0` in the power-law scale radius model.
* ``[massHigh]`` (default ``+0.2984d0``) — The parameter :math:`\alpha_1` in the power-law scale radius model.
* ``[massTransition]`` (default ``-0.2583d0``) — The parameter :math:`\alpha_\nu` in the power-law scale radius model.
* ``[massWidth]`` (default ``+16.6050d0``) — The parameter :math:`\Delta \alpha` in the power-law scale radius model.
* ``[expansionFactorLow]`` (default ``-0.6977d0``) — The parameter :math:`\beta_0` in the power-law scale radius model.
* ``[expansionFactorHigh]`` (default ``+0.7972d0``) — The parameter :math:`\beta_1` in the power-law scale radius model.
* ``[expansionFactorTransition]`` (default ``+0.5395d0``) — The parameter :math:`\beta_\nu` in the power-law scale radius model.
* ``[expansionFactorWidth]`` (default ``+0.4282d0``) — The parameter :math:`\Delta \beta` in the power-law scale radius model.
* ``[scatter]`` (default ``+0.1513d0``) — The scatter (in dex) in the scale radius at fixed halo mass and redshift in the power-law scale radius model, representing the intrinsic halo-to-halo variation in concentration.
* ``[index]`` (default ``0.9649d0``) — The index of the power-law primordial power spectrum.
* ``[running]`` (default ``0.0d0``) — The running, :math:`\d n_\mathrm{s} / \d \ln k`, of the power spectrum index.
* ``[runningRunning]`` (default ``0.0d0``) — The running-of-the-running, :math:`\d^2 n_\mathrm{s} / \d \ln k^2`, of the power spectrum index.
* ``[wavenumberReference]`` (default ``1.0d0``) — When a running power spectrum index is used, this is the wavenumber, :math:`k_\mathrm{ref}`, at which the index is equal to ``[index]``.
* ``[runningSmallScalesOnly]`` (default ``.false.``) — If ``true`` then the index runs only for :math:`k > k_\mathrm{ref}`, for smaller :math:`k` the index is constant.