More+2015 HOD Model — BOSS CMASS & BGS

Model class	MoreHODModel (alias More2015HODModel)
Paper	More et al. 2015, ApJ 806, 2 (arXiv:1407.1856, DOI:10.1088/0004-637X/806/1/2)
Primary survey	BOSS CMASS, \(z_\mathrm{eff} = 0.52\)
Observable	Joint \(w_p(r_p) + \Delta\Sigma(R)\)
Code	hod_mod.connection.hod (lines 420–536), hod_mod.observables.clustering (FullHaloModelPrediction)

---

Cosmological framework

Both HOD models in this package share the same halo-model backbone. All quantities below feed into the HOD occupation integrals.

Cosmological parameters

The six base parameters \(\boldsymbol{\theta} = (\Omega_m,\,\Omega_b,\,h,\,n_s,\,\ln 10^{10}A_s,\,\sigma_8)\) define the linear matter power spectrum \(P_\mathrm{lin}(k, z)\), computed by CAMB via LinearPowerSpectrum.

Fiducial values used for BOSS CMASS benchmarks: \(\Omega_m = 0.310,\ h = 0.703,\ \sigma_8 = 0.785,\ n_s = 0.964,\ \Omega_b = 0.0451\).

Halo mass function

Tinker et al. 2008 (arXiv:0803.2706):

\[\frac{\mathrm{d}n}{\mathrm{d}M}(M, z)\]

Implemented via make_hmf() with backend="tinker08" (default), overdensity \(\Delta = 200\rho_m\). Units: \(h^4\,\mathrm{Mpc}^{-3}\,M_\odot^{-1}\).

Alternative emulator backends — "csst" (Chen+2025, SCPMA 2025) and "aemulusnu" (Shen+2025, arXiv:2410.00913) — expose the same interface; see Cosmology Module for details.

Linear halo bias

Tinker et al. 2010 (arXiv:1001.3162), \(b(M, z)\). The effective galaxy bias is:

\[b_\mathrm{eff}(z) = \frac{\displaystyle\int \mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M}\, \langle N_\mathrm{tot}(M)\rangle\,b(M,z)} {\bar{n}_g}\]

NFW profile and Fourier transform

Dark-matter halos follow the NFW profile (Navarro, Frenk & White 1997, arXiv:astro-ph/9508025):

\[\rho(r \,|\, M) = \frac{\rho_s}{(r/r_s)(1 + r/r_s)^2}, \qquad r_s = \frac{r_{200}}{c(M,z)}\]

Concentration–mass relation: Diemer & Joyce 2019 (arXiv:1809.07326), accessed via HaloProfile with cm_relation="diemer19".

The normalised NFW Fourier transform (Cooray & Sheth 2002, Eq. 11, arXiv:astro-ph/0206508):

\[\tilde{u}(k \,|\, M) = \frac{4\pi\rho_s r_s^3}{M}\Bigl[ \sin(k r_s)\bigl(\mathrm{Si}((1+c)k r_s) - \mathrm{Si}(k r_s)\bigr) - \frac{\sin(c k r_s)}{(1+c)k r_s} + \cos(k r_s)\bigl(\mathrm{Ci}((1+c)k r_s) - \mathrm{Ci}(k r_s)\bigr) \Bigr]\]

Galaxy number density

\[\bar{n}_g(z) = \int_{M_\mathrm{min}}^{M_\mathrm{max}} \frac{\mathrm{d}n}{\mathrm{d}M}\,\langle N_\mathrm{tot}(M)\rangle\,\mathrm{d}M\]

Mass grid: 512 log-spaced points, \(M \in [10^{10},\,10^{16}]\,h^{-1}M_\odot\).

---

More+2015 HOD model

Reference: More et al. 2015, ApJ 806, 2 (arXiv:1407.1856). Implemented in MoreHODModel (hod_mod/connection/hod/more15.py).

Incompleteness function

The BOSS CMASS sample has a colour–magnitude selection that reduces completeness at the low-mass end. More+2015 model this with a linear ramp:

\[f_\mathrm{inc}(M) = \mathrm{clip}\!\left( 1 + \alpha_\mathrm{inc}\,\bigl(\log_{10} M - \log_{10} M_\mathrm{inc}\bigr), \;0,\;1\right)\]

Default: \(\alpha_\mathrm{inc} = 1.0\) (fixed), \(\log_{10} M_\mathrm{inc} = 13.0\) (fixed).

Central occupation

\[\langle N_\mathrm{cen}(M)\rangle = \frac{f_\mathrm{inc}(M)}{2}\, \mathrm{erfc}\!\left[\frac{\log_{10} M_\mathrm{min} - \log_{10} M}{\sigma_{\log M}}\right]\]

The step-function threshold \(M_\mathrm{min}\) is broadened by scatter \(\sigma_{\log M}\) (in dex, base-10). At \(M = M_\mathrm{min}\), \(\langle N_\mathrm{cen}\rangle = f_\mathrm{inc}/2\).

Satellite occupation

\[\langle N_\mathrm{sat}(M)\rangle = \langle N_\mathrm{cen}(M)\rangle \times \left(\frac{M - \kappa\,M_\mathrm{min}}{M_1}\right)^\alpha \quad \text{for } M > \kappa\,M_\mathrm{min}, \quad \text{else } 0\]

Satellites live in halos that first contain at least one central galaxy; their mean number rises as a power law \(\alpha\) above the threshold \(\kappa\,M_\mathrm{min}\).

Off-centering of central galaxies

A fraction \(p_\mathrm{off}\) of centrals are displaced from the halo centre (Johnston et al. 2007, arXiv:0709.4193; More+2015 §3.3). In Fourier space (mass-dependent width):

\[\langle N_\mathrm{cen}^\mathrm{eff}(k \,|\, M)\rangle = \langle N_\mathrm{cen}(M)\rangle\, \bigl[(1 - p_\mathrm{off}) + p_\mathrm{off}\,e^{-k^2 (R_\mathrm{off}\,r_s(M))^2/2}\bigr]\]

where \(r_s(M) = r_{200}(M)/c(M)\). Fixed values: \(p_\mathrm{off} = 0.34\), \(R_\mathrm{off} = 2.2\).

---

Power spectra

1-halo terms

Galaxy–galaxy (More+2015 Eq. 9):

\[P_{gg}^\mathrm{1h}(k) = \frac{1}{\bar{n}_g^2} \int \mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M} \left[ \langle N_s^2\rangle\,\tilde{u}^2(k|M) + 2\,\langle N_c\rangle\,\langle N_s\rangle\,\tilde{u}(k|M) \right]\]

Galaxy–matter (More+2015 Eq. 13):

\[P_{gm}^\mathrm{1h}(k) = \frac{1}{\bar{n}_g} \int \mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M} \left[\langle N_c^\mathrm{eff}(k|M)\rangle + \langle N_s\rangle\,\tilde{u}(k|M)\right] \frac{M}{\bar{\rho}_m}\,\tilde{u}(k|M)\]

For the satellite term, a Poisson satellite distribution gives \(\langle N_s^2\rangle = \langle N_s\rangle^2 + \langle N_s\rangle\).

2-halo terms

\[\begin{split}P_{gg}^\mathrm{2h}(k) &= b_\mathrm{eff}^2\,P_\mathrm{lin}(k) \quad [+ \delta P_\mathrm{BNL}(k)]\\ P_{gm}^\mathrm{2h}(k) &= b_\mathrm{eff}\,P_\mathrm{lin}(k)\end{split}\]

The beyond-linear halo bias (BNL) correction \(\delta P_\mathrm{BNL}\) follows Mead & Verde 2021 (arXiv:2109.15266), tabulated from the MultiDark MDR1 simulation, implemented in BeyondLinearBiasMead21.

Total:

\[P_{gg}(k) = P_{gg}^\mathrm{1h}(k) + P_{gg}^\mathrm{2h}(k),\qquad P_{gm}(k) = P_{gm}^\mathrm{1h}(k) + P_{gm}^\mathrm{2h}(k)\]

---

Summary statistics

3D correlation function

The galaxy auto-correlation function \(\xi_{gg}(r)\) and galaxy–matter cross-correlation \(\xi_{gm}(r)\) are obtained from the respective power spectra via the Ogata (2005) double-exponential \(j_0\) Hankel transform (DOI:10.2977/prims/1145474602):

\[\xi(r) = \frac{1}{2\pi^2}\int_0^\infty k^2\,P(k)\,j_0(kr)\,\mathrm{d}k\]

Projected correlation function

\[w_p(r_p) = 2\int_0^{\pi_\mathrm{max}} \xi_{gg}\!\left(\sqrt{r_p^2 + \pi^2}\right)\mathrm{d}\pi, \qquad \pi_\mathrm{max} = 100\,h^{-1}\,\mathrm{Mpc}\]

More+2015 use \(\pi_\mathrm{max} = 80\,h^{-1}\,\mathrm{Mpc}\) (set via pi_max in wp()).

Excess surface mass density

The galaxy–matter lensing signal:

\[\Sigma_{gm}(R) = 2\int_0^\infty \xi_{gm}\!\left(\sqrt{R^2 + \chi^2}\right) \bar{\rho}_m\,\mathrm{d}\chi\]

\[\Delta\Sigma(R) = \bar{\Sigma}_{gm}(<R) - \Sigma_{gm}(R) = \frac{2}{R^2}\int_0^R R'\,\Sigma_{gm}(R')\,\mathrm{d}R' - \Sigma_{gm}(R)\]

Units: \(M_\odot\,h\,\mathrm{pc}^{-2}\). Implemented in delta_sigma().

---

Parameter table

Parameter	Symbol	Default	Fitted?	Prior / fixed value	Units
log10mmin	\(\log_{10} M_\mathrm{min}\)	13.03	Yes	\([11,\,15]\)	\(\log_{10}(M_\odot h^{-1})\)
sigma_logm	\(\sigma_{\log M}\)	0.38	Yes	\([0.01,\,2]\)	dex (base 10)
log10m1	\(\log_{10} M_1\)	14.00	Yes	\([11,\,16]\)	\(\log_{10}(M_\odot h^{-1})\)
alpha	\(\alpha\)	1.0	Yes	\([0.1,\,3]\)	—
kappa	\(\kappa\)	1.0	Yes	\([0.01,\,5]\)	—
alpha_inc	\(\alpha_\mathrm{inc}\)	1.0	Fixed	1.0	—
log10m_inc	\(\log_{10} M_\mathrm{inc}\)	13.0	Fixed	13.0	\(\log_{10}(M_\odot h^{-1})\)
p_off	\(p_\mathrm{off}\)	0.34	Fixed	0.34	—
R_off	\(R_\mathrm{off}\)	2.2	Fixed	2.2	\(r_s\) units

---

BOSS CMASS benchmarks

Three stellar-mass threshold subsamples from More+2015 Figure 3 are reproduced. Full MAP results and MCMC corner plots are in Benchmark: More+2015 — BOSS CMASS mass-threshold samples.

Data digitised from Figure 3 of More+2015 using WebPlotDigitizer; stored in data/more2015_boss_cmass/.

Variant	\(\log_{10} M_*^\mathrm{min}\)	\(\chi^2\)	dof	\(\chi^2/\mathrm{dof}\)
logM11_12	11.1	71.06	36	1.967 (pub. 0.8)
logM11p3_12	11.3	57.60	35	1.646 (pub. 1.3)
logM11p4_12	11.4	63.30	35	1.809 (pub. 1.5)
logM11_12_freecosmo	11.1 + free \(\Omega_m,S_8\)	35.70	33	1.082

Primary benchmark: logM11_12 (MAP parameters)

Parameter	MAP	Published (\(\pm 1\sigma\))	Deviation
log10mmin	13.134	\(13.13 \pm 0.13\)	\(+0.03\sigma\)
sigma_logm	0.458	\(0.469 \pm 0.13\)	\(-0.09\sigma\)
log10m1	14.168	\(14.21 \pm 0.13\)	\(-0.32\sigma\)
alpha	1.841	\(1.13 \pm 0.15\)	\(+4.74\sigma\)
kappa	3.000	\(1.25 \pm 0.45\)	\(+3.89\sigma\)

Note

alpha and kappa tensions are driven by a near-degenerate likelihood valley. MCMC medians agree much better: alpha = 1.928 (±0.19), kappa = 1.862 (+0.79/−1.03). All mass-scale parameters agree within \(0.32\sigma\).

Variant logM11p3_12 (MAP)

Parameter	MAP	Published (\(\pm 1\sigma\))	Deviation
log10mmin	13.549	\(13.45 \pm 0.15\)	\(+0.66\sigma\)
sigma_logm	0.616	\(0.671 \pm 0.19\)	\(-0.29\sigma\)
log10m1	14.548	\(14.51 \pm 0.17\)	\(+0.22\sigma\)
alpha	2.361	\(1.14 \pm 0.49\)	\(+2.49\sigma\)
kappa	0.148	not published	—

Variant logM11p4_12 (MAP)

Parameter	MAP	Published (\(\pm 1\sigma\))	Deviation
log10mmin	14.166	\(13.68 \pm 0.16\)	\(+3.04\sigma\)
sigma_logm	0.875	\(0.889 \pm 0.22\)	\(-0.06\sigma\)
log10m1	14.390	\(14.56 \pm 0.25\)	\(-0.68\sigma\)
alpha	1.602	\(1.00 \pm 0.44\)	\(+1.37\sigma\)
kappa	1.675	not published	—

Free-cosmology variant logM11_12_freecosmo (MAP)

Three additional free parameters with Planck 2018 priors: \(\Omega_m = 0.310 \pm 0.020\), \(S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3} = 0.798 \pm 0.044\), \(h = 0.703\) (fixed at the published value).

Parameter	MAP	Planck prior centre
\(\Omega_m\)	0.281	\(0.310 \pm 0.020\)
\(S_8\)	0.778	\(0.798 \pm 0.044\)
log10mmin	13.163	—
sigma_logm	0.508	—
log10m1	14.224	—
alpha	2.018	—
kappa	2.920	—

---

Benchmark figures (logM11_12)

MAP fit to BOSS CMASS logM*>11.1. Top panel: \(w_p(r_p)\). Bottom panel: \(\Delta\Sigma(R)\). Orange: published More+2015 parameters. Blue: MAP. Grey: data.

HOD occupation functions \(\langle N_c(M)\rangle\), \(\langle N_s(M)\rangle\), and \(\langle N_\mathrm{tot}(M)\rangle\). Solid: MAP. Dashed + band: MCMC median ± 1σ. Orange: published values.

MCMC posterior corner plot (32 walkers × 2000 steps, 500 burn-in = 48 000 samples). Contours: 68% and 95% credible regions. Orange lines: published More+2015 values.

---

BGS LS10 — preliminary results (S4–S7)

The BGS LS10 cross-correlation \(w_\theta(\theta)\) (galaxy × eROSITA soft X-ray) and \(w_p(r_p)\) were fitted jointly for higher stellar-mass samples (Comparat et al. 2025, arXiv:2503.19796). Samples S4–S7 used MoreHODModel parameters.

Sample	\(\log_{10} M_*^\mathrm{min}\)	\(z_\mathrm{mean}\)	\(N_\mathrm{gal}\)	\(\chi^2/\mathrm{dof}\)	npts	log10mmin	log10m1
S4	10.75	0.226	2 802 710	316.60	31	12.327	13.358
S5	11.00	0.252	1 619 838	242.79	57	12.674	13.692
S6	11.25	0.255	541 855	314.32	31	13.096	14.132
S7	11.50	0.261	120 882	MAP failed	57	—	—

Note

For S4–S6, the gas amplitude log10_A_gas converges at its lower bound (−2.0), indicating that the gas component is not detected in these samples at current data quality. \(\chi^2/\mathrm{dof} \gg 1\) reflects a combination of model inadequacy, data systematics, and the gas non-detection. These results are preliminary; see Zu & Mandelbaum 2015 iHOD Model — SDSS, X-ray & BGS for the lower-mass samples fitted with the iHOD model.