# 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.