# Matter power spectrum and matter transfer function variables¶

The various matter power spectrum functions, e.g. get_matter_power_interpolator(), can calculate power spectra for various quantities. Each variable used to form the power spectrum has a name as follows:

name number description
k/h 1 $$k/h$$
delta_cdm 2 $$\Delta_c$$, CDM density
delta_baryon 3 $$\Delta_b$$, baryon density
delta_photon 4 $$\Delta_\gamma$$, photon density
delta_neutrino 5 $$\Delta_r$$, for massless neutrinos
delta_nu 6 $$\Delta_\nu$$ for massive neutrinos
delta_tot 7 $$\frac{\rho_c\Delta_c+\rho_b\Delta_b+\rho_\nu\Delta_\nu}{\rho_c+\rho_b+\rho_\nu}$$, CDM+baryons+massive neutrino density
delta_nonu 8 $$\frac{\rho_c\Delta_c+\rho_b\Delta_b}{\rho_b+\rho_c}$$, CDM+baryon density
delta_tot_de 9 $$\frac{\rho_c\Delta_c+\rho_b\Delta_b+\rho_\nu\Delta_\nu +\rho_{ de}\Delta_{de}}{\rho_c+\rho_b+\rho_\nu}$$, CDM+baryons+massive neutrinos+ dark energy (numerator only) density
Weyl 10 $$k^2\Psi\equiv k^2(\phi+\psi)/2$$, the Weyl potential scaled by $$k^2$$ to scale in $$k$$ like a density.
v_newtonian_cdm 11 $$-v_{N,c}\, k/{\cal H}$$ (where $$v_{N,c}$$ is the Newtonian-gauge CDM velocity)
v_newtonian_baryon 12 $$-v_{N,b}\,k/{\cal H}$$ (Newtonian-gauge baryon velocity $$v_{N,b}$$)
v_baryon_cdm 13 $$v_b-v_c$$, relative baryon-CDM velocity

The number here corresponds to a corresponding numerical index, in Fortran these are the same as model.name, where name are the Transfer_xxx variable names: Transfer_kh=1,Transfer_cdm=2, Transfer_b=3, Transfer_g=4, Transfer_r=5, Transfer_nu=6, Transfer_tot=7, Transfer_nonu=8, Transfer_tot_de=9, Transfer_Weyl=10, Transfer_Newt_vel_cdm=11, Transfer_Newt_vel_baryon=12, Transfer_vel_baryon_cdm = 13.

So for example, requesting var1=’delta_b’, var2=’Weyl’ or alternatively var1=model.Transfer_b, var2=model.Transfer_Weyl would get the power spectrum for the cross-correlation of the baryon density with the Weyl potential. All density variables $$\Delta_i$$ here are synchronous gauge.

For transfer function variables (rather than matter power spectra), the variables are normalized corresponding to unit primordial curvature perturbation on super-horizon scales. The get_matter_transfer_data() function returns the above quantities divided by $$k^2$$ (so they are roughly constant at low $$k$$ on super-horizon scales).

The example notebook has various examples of getting the matter power spectrum, relating the Weyl-potential spectrum to lensing, and calculating the baryon-dark matter relative velocity spectra. There is also an explicit example of how to calculate the matter power spectrum manually from the matter transfer functions.

When generating dark-age 21cm power spectra (do21cm is set) the transfer functions are instead the model.name variables (see equations 20 and 25 of astro-ph/0702600)

name number description
Transfer_kh 1 $$k/h$$
Transfer_cdm 2 $$\Delta_c$$, CDM density
Transfer_b 3 $$\Delta_b$$, baryon density
Transfer_monopole 4 $$\Delta_s+(r_\tau-1)(\Delta_{b}-\Delta_{T_s})$$, 21cm monopole source
Transfer_vnewt 5 $$r_\tau kv_{N,b}/\mathcal{H}$$, 21cm Newtonian-gauge velocity source
Transfer_Tmat 6 $$\Delta_{T_m}$$, matter temperature perturbation
Transfer_tot 7 $$\frac{\rho_c\Delta_c+\rho_b\Delta_b+\rho_\nu\Delta_\nu}{\rho_c+\rho_b+\rho_\nu}$$, CDM+baryons+massive neutrino density
Transfer_nonu 8 $$\frac{\rho_c\Delta_c+\rho_b\Delta_b}{\rho_b+\rho_c}$$, CDM+baryon density
Transfer_tot_de 9 $$\frac{\rho_c\Delta_c+\rho_b\Delta_b+\rho_\nu\Delta_\nu +\rho_{ de}\Delta_{de}}{\rho_c+\rho_b+\rho_\nu}$$, CDM+baryons+massive neutrinos+ dark energy (numerator only) density
Transfer_Weyl 10 $$k^2\Psi\equiv k^2(\phi+\psi)/2$$, the Weyl potential scaled by $$k^2$$ to scale in $$k$$ like a density.
Transfer_Newt_vel_cdm 11 $$-v_{N,c}\, k/{\cal H}$$ (where $$v_{N,c}$$ is the Newtonian-gauge CDM velocity)
Transfer_Newt_vel_baryon 12 $$-v_{N,b}\,k/{\cal H}$$ (Newtonian-gauge baryon velocity $$v_{N,b}$$)
Transfer_vel_baryon_cdm 13 $$v_b-v_c$$, relative baryon-CDM velocity

If use_21cm_mK is set the 21cm results are multiplied by $$T_b$$ to give results in mK units.