.. cal-sky documentation master file, created by
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.. toctree::
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**Astropnysical Models**
------------------

Cosmic infrared background
^^^^^^^^^

Kinetic Sunyaev-Zel'dovich and Optical Depth
^^^^^^^^^

Patchy Reionization
''''''''''

Late time
'''''''''' 

21-cm intensity maps
^^^^^^^^^

Patchy Reionization
''''''''''''''

The 21-cm signal from patchy reionization is obtained, in the :math:`T_s >> T_{cmb}` limit, by using Lagrangian perturbation 
theory to displace mass cells to their Eulerian positions:

.. math::

   {\mathbf x}_c = {\mathbf q}_c + D(|{\mathbf q}|){\mathbf s}^{(1)}_c({\mathbf q}_c) + D^2(|{\mathbf q}|){\mathbf s}^{(2)}_c({\mathbf q}_c),

where :math:`D(z)` is the linear growth factor, :math:`{\mathbf x}` is the final, Eulerian position, :math:`{\mathbf q}` is the initial, Lagrangian, position, and :math:`{\mathbf s}^{(1)}` and :math:`{\mathbf s}^{(2)}` are the first and second order LPT displacement coefficients, respectively. All spatial coordinates are in comoving units and distance and redshift are used interchangeably as functional arguments. All mass elements correspond to the same comoving volume of :math:`a_{\mathrm latt}^3`, where :math:`a_{\mathrm latt}` is the comoving size of a grid cell, but are each located at a different redshift along the line of sight, :math:`z_c=z(|{\mathbf q}_c|)`.  For a sufficiently small frequency bin labeled with index :math:`i` of size :math:`\delta{\nu}_i`
in a pixel subtending a solid angle :math:`\Delta\Omega`, the mean differential 
brightness temperature in the corresponding frequency voxel is given by summing 
all the values of the comoving voxels that overlap the frequency voxel, weighted by the 
ratio of comoving voxel volume to frequency voxel, 

.. math::

   \delta{T}_b^{i}=\sum_c \overline{\delta{T}}_b(z_c) \frac{a_{\mathrm latt}^3}{V_{i}},

where the frequency voxel volume is 

.. math::

   V_{i}=\Delta\Omega\chi_i^2\frac{d\chi}{d\nu}_i\delta{\nu}_i.

   
The mean over a finite frequency bin, :math:`\Delta{\nu}`, corresponds to a sum over infinitesimal ones, 

.. math::

   \delta{T}_b=\frac{1}{\Delta{\nu}}\sum_i \delta{\nu}_i\delta{T}_b^{i}.

Combining all these expressions together, we obtain the weighting scheme to sum over comoving voxels 
when binning into finite size frequency voxels,

.. math::

   \delta{T}_b=\frac{a_{\mathrm latt}^3}{\Delta\Omega\Delta{\nu}}\sum_c \overline{\delta{T}}_b(z_c)\frac{1}{\chi_c^2}\left(\frac{d\nu}{d\chi}\right)_c\equiv \frac{a_{\mathrm latt}^3}{\Delta\Omega\Delta{\nu}}\sum_c \frac{W_{\mathrm dtb}(z_c)}{\chi_c^2}.

Note that the quantity outside the sum is a constant -- all pixels and comoving voxels have the same size -- while the numerator of the quantity inside the sum depends only on redshift and hence can be precomputed in a table for efficiency.  This is the origin of the line::

  float wdtb = Tbmean * nu0 / r0 * sqrt(Omegam*pow((1+z),3)+Omegal)*
      pow((1+z),-2);

in the function ``SetRedshift2WdtbTable`` in the file ``tables.C``. Note that the variable ``wdtb`` has units of mK.Hz/Mpc. The constant factor outside of the sum is multiplied in the line::

  dtbfac = Wdtb * xHI * 
	  pow(CellSize,3) / pow(r,2) * mapsize / 4. / 3.14159 / dnu;

in the function ``MakeMaps`` in the file ``makemaps.C``.  Note that this last quantity has units of mK and is what is added to the frequency voxel for each comoving voxel to generate the final map. 

HI in galaxies
''''''''''''''''

Inhomogeneous heating
'''''''''''''''''''''''

Lyman-alpha pumping
'''''''''''''''''''''

Lensing
^^^^^^^