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 \(T_s >> T_{cmb}\) limit, by using Lagrangian perturbation theory to displace mass cells to their Eulerian positions:

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

where \(D(z)\) is the linear growth factor, \({\mathbf x}\) is the final, Eulerian position, \({\mathbf q}\) is the initial, Lagrangian, position, and \({\mathbf s}^{(1)}\) and \({\mathbf s}^{(2)}\) 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 \(a_{\mathrm latt}^3\), where \(a_{\mathrm latt}\) is the comoving size of a grid cell, but are each located at a different redshift along the line of sight, \(z_c=z(|{\mathbf q}_c|)\). For a sufficiently small frequency bin labeled with index \(i\) of size \(\delta{\nu}_i\) in a pixel subtending a solid angle \(\Delta\Omega\), 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,

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

where the frequency voxel volume is

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

The mean over a finite frequency bin, \(\Delta{\nu}\), corresponds to a sum over infinitesimal ones,

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

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

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

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