onepower.pk.Spectra.compute_1h_term#

Spectra.compute_1h_term(profile_u, profile_v, mass, dndlnm)[source]#

Compute the 1-halo term for two fields u and v, e.g. matter, galaxy, intrinsic alignment

\(P^{\rm 1h}_{uv}(k)= \int W_{u}(k,z,M) W_{v}(k,z,M) n(M) {\rm d}M\)

If the fields are the same and they correspond to discrete tracers (e.g. satellite galaxies):

\(P^{\rm 1h}_{uv}(k)= 1/n_{x}^2 \int \langle N_{x}(M)[N_{x}(M)-1]\rangle U_{x}(k,z,M)^{2} n(M) {\rm d}M + 1/n_{x}\)

\(n_{x} = \int N_{x}(M) n(M) {\rm d}M\)

The shot noise term is removed as we do our measurements in real space where it only shows up at zero lag which is not measured. See eq 22 of Asgari, Mead, Heymans 2023 review paper.

But for satellite galaxis we use:

\(\langle N_{\rm sat}(N_{\rm sat}-1)\rangle = \mathcal{P} \langle N_{\rm sat}\rangle ^ {2}\):

\(P^{\rm 1h}_{\rm ss}(k)= 1/n_{\rm s}^{2} \int \mathcal{P} \langle N_{\rm sat}\rangle ^{2} U_{\rm s}(k,z,M)^{2} n(M) {\rm d}M\)

and write \(W_u = W_v = \langle N_{\rm sat}\rangle U_{\rm s}(k,z,M) \sqrt{\mathcal{P}}/n_{\rm s}\)

for matter halo profile is: \(W_{\rm m} = (M/\rho_{\rm m}) U_{\rm m}(z,k,M)\)

for galaxies: \(W_{\rm g} = (N_{\rm g}(M)/n_{\rm g}) U_{\rm g}(z,k,M)\)