Concentration

The NFW profile needs two numbers, mass and concentration, and the halo mass function only supplies the first. \(c(M,z)\) is a structure-formation result — calibrated on N-body simulations at fixed cosmology, exactly like Halo Mass Function and Halo Bias — so it lives in clenspy.cosmology rather than clenspy.halo.

Concentration-mass relations at z=0.3 -- child18, child18_powerlaw, duffy08

\(c_{200c}(M)\) at \(z=0.3\): concentration falls with mass, since more massive haloes collapsed more recently and had less time to grow their inner density. child18 sits above duffy08 at cluster scales, tracking the trend into the Eq. 18 plateau.

Child et al. (2018): the \(c\)–\(M/M_\star\) form

Scaling mass by the nonlinear mass \(M_\star(z)\) collapses the whole redshift family onto one curve (Eq. 18):

\[ c_{200c} = A\left[ \left(\frac{M_{200c}/M_\star}{b}\right)^{m} \left(1 + \frac{M_{200c}/M_\star}{b}\right)^{-m} - 1 \right] + c_0, \]

a power law in \(M/M_\star\) below the threshold \(M_T = b\,M_\star\), flattening to a plateau \(c_0 \sim 3\)–4 above it. All of \(z\)’s dependence runs through \(M_\star(z)\), not through \(A\), \(b\), \(m\), or \(c_0\). Coefficients are CHILD18_TABLE1; the y3 production choice is the "individual_all" row.

\(M_\star(z)\) is defined by \(\sigma(R_\star, z) = \delta_c = 1.686\), \(M_\star = \tfrac{4\pi}{3}\bar\rho_m R_\star^3\), and Child et al. quote it at four anchors, \(\log_{10}(M_\star/h^{-1}M_\odot) = 12.5, 11, 9.5, 8\) at \(z = 0,1,2,3\) — exactly linear in \(z\), so m_star_hinv is that line, not a fit. m_star_from_sigma solves the definition exactly against any \(\sigma(R,z)\), for cosmologies away from Child et al.’s WMAP-7.

A power-law alternative, Eq. 19, needs no \(M_\star\) but is only valid for \(0 \le z \le 1\):

\[ c_{200c} = A\,(1+z)^{d}\,M^{m}. \]

Duffy et al. (2008): the older power law

A plain power law, kept because pyccl and much of the literature default to it:

\[ c = A\left(\frac{M}{2\times10^{12}\,h^{-1}M_\odot}\right)^{B}(1+z)^{C}. \]

Its WMAP-5 \(\sigma_8 = 0.796\) is low, so it sits below Child et al. at cluster scales. duffy08 takes a mass_def — "vir", "200m", or "200c" — since Duffy et al. tabulate all three as three different halo boundaries, not three fits to one; clenspy’s default is "200m", matching NfwProfile, not pyccl’s default of "200c".

Note

These relations were calibrated in \(h^{-1}M_\odot\) and on \(M_{200c}\) — the one place in clenspy where the h-free convention breaks, and a different halo boundary than NfwProfile/TinkerMassFunction’s \(M_{200m}\). Every mass argument here carries the unit and definition in its name (m200c_hinv), and clenspy does not vendor an \(M_{200m}\to M_{200c}\) conversion — the caller owns it.

\(\delta_c\) and scatter

The NFW characteristic overdensity, so that \(\rho_s = \delta_c\,\rho_{\rm ref}\):

\[ \delta_c(c) = \frac{200\,c^3/3}{\ln(1+c) - c/(1+c)}. \]

This is unrelated to \(\delta_c = 1.686\) above (spherical-collapse threshold) despite the shared name — one is \(O(1)\), the other \(O(10^4)\). Both Child et al. tables also carry a scatter note, \(\sigma_c = c_{200c}/3\) — a 33% halo-to-halo spread large enough that a stacked profile is not the profile at \(\langle c\rangle\); a stacked analysis wanting one number should use the "stacked_nfw" row instead.

Example

from clenspy.cosmology import child18, child18_powerlaw, delta_c, duffy08, m_star_hinv, scatter

# these relations were calibrated in h^-1 Msun and on M_200c, not M_200m --
# see the module docstring's two NOTEs before mixing them with the rest of
# clenspy, which is h-free and M_200m.
m200c_hinv = 1e14  # h^-1 Msun
z = 0.3

ms = m_star_hinv(z)  # Child et al.'s own anchor line, this cosmology's M*
print(f"M_star(z={z}) = {ms:.3e} h^-1 Msun,  M/M_star = {m200c_hinv / ms:.1f}")

c18 = child18(m200c_hinv, z, ms)
c19 = child18_powerlaw(m200c_hinv, z)
cd8 = duffy08(m200c_hinv, z, mass_def="200c")
print(f"c_200c: child18 = {c18:.3f}, child18_powerlaw = {c19:.3f}, "
      f"duffy08 = {cd8:.3f}")

# Duffy08's WMAP-5 sigma_8 sits below Child et al.'s, so it under-predicts
# concentration at cluster scales (Child et al. Fig. 12).
print(f"child18 / duffy08 = {c18 / cd8:.3f}  (> 1, as expected)")

print(f"NFW delta_c(c={c18:.3f}) = {delta_c(c18):.1f}, "
      f"scatter sigma_c = {scatter(c18):.3f}")


M_star(z=0.3) = 1.122e+12 h^-1 Msun,  M/M_star = 89.1
c_200c: child18 = 3.853, child18_powerlaw = 3.831, duffy08 = 3.634
child18 / duffy08 = 1.060  (> 1, as expected)
NFW delta_c(c=3.853) = 4854.4, scatter sigma_c = 1.284

See also: API Reference for the full clenspy.cosmology reference, Notation for the symbol table.