Benchmark: More+2015 — BOSS CMASS mass-threshold samples

Model class: MoreHODModel — More et al. 2015, ApJ 806, 2 (arXiv:1407.1856), BOSS CMASS z_eff = 0.52.

All variants fit \(w_p(r_p)\) and \(\Delta\Sigma(R)\) jointly using the beyond-linear halo bias correction (BeyondLinearBiasMead21). See the benchmark suite for the summary table.

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Variant: more2015_logM11_12 — Joint wp+ΔΣ, logM*>11.1

MoreHODModel fit to BOSS CMASS logM*>11.1, \(w_p + \Delta\Sigma\) jointly. Beyond-linear halo bias (BNL) enabled.

MAP: \(\chi^2/\text{dof} = 71.06 / 36 = 1.974\). Status: PASSED. Published \(\chi^2/\text{dof} \approx 0.8\). MCMC: 32 walkers × 2000 steps, 500 burn-in → 48 000 samples.

Parameter	MAP	MCMC median	MCMC \(1\sigma\) interval	Published
log10mmin	13.169	13.164	\(^{+0.302}_{-0.185}\)	13.13 ± 0.13
sigma_logm	0.495	0.491	\(^{+0.254}_{-0.263}\)	0.469 ± 0.13
log10m1	14.192	14.220	\(^{+0.079}_{-0.071}\)	14.21 ± 0.13
alpha	1.870	1.928	\(^{+0.175}_{-0.194}\)	1.13 ± 0.15
kappa	2.619	1.862	\(^{+0.793}_{-1.025}\)	1.25 ± 0.45

The mass-scale parameters (log10mmin, sigma_logm, log10m1) agree with the published values within 0.3σ. alpha is 5.2σ from the published value; kappa is within 1.4σ at the MCMC median.

Derived quantities (at MCMC median):

Quantity	Value	Unit
\(\bar{n}_g\)	\((3.68^{+0.42}_{-0.78}) \times 10^{-4}\)	\(h^3\,\mathrm{Mpc}^{-3}\)
Satellite fraction \(f_\mathrm{sat}\)	0.036	—
Mean halo mass \(\log_{10}\langle M_h\rangle\)	13.54	\(\log_{10}(M_\odot/h)\)

MAP \(w_p(r_p)\) (top) and \(\Delta\Sigma(R)\) (middle) vs BOSS CMASS logM*>11.1 data, with residuals.

HOD occupation functions \(\langle N_c(M)\rangle\), \(\langle N_s(M)\rangle\), and \(\langle N(M)\rangle\) vs halo mass. Solid lines: MAP. Dashed lines + shaded bands: MCMC median and 16th–84th percentile posterior. Orange: published More+2015 parameters.

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

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Variant: more2015_logM11p3_12 — Joint wp+ΔΣ, logM*>11.3

MoreHODModel fit to BOSS CMASS logM*>11.3, \(w_p + \Delta\Sigma\) jointly. Beyond-linear halo bias (BNL) enabled.

MAP: \(\chi^2/\text{dof} = 57.60 / 35 = 1.646\). Status: PASSED. Published \(\chi^2/\text{dof} \approx 1.3\). MCMC: 32 walkers × 3000 steps, 500 burn-in → 80 000 samples.

Parameter	MAP	MCMC median	MCMC \(1\sigma\) interval	Published
log10mmin	13.549	13.347	\(^{+0.323}_{-0.178}\)	13.45 ± 0.15
sigma_logm	0.616	0.452	\(^{+0.246}_{-0.238}\)	0.671 ± 0.19
log10m1	14.548	14.395	\(^{+0.093}_{-0.094}\)	14.51 ± 0.17
alpha	2.361	1.982	\(^{+0.295}_{-0.318}\)	1.14 ± 0.49
kappa	0.148	1.553	\(^{+0.966}_{-0.952}\)	—

The halo mass parameters log10mmin, sigma_logm, log10m1 all agree within 0.7σ of the published values at the MCMC median.

Derived quantities (at MCMC median):

Quantity	Value	Unit
\(\bar{n}_g\)	\((2.52^{+0.18}_{-0.24}) \times 10^{-4}\)	\(h^3\,\mathrm{Mpc}^{-3}\)
Satellite fraction \(f_\mathrm{sat}\)	0.033	—
Mean halo mass \(\log_{10}\langle M_h\rangle\)	13.69	\(\log_{10}(M_\odot/h)\)

MAP \(w_p(r_p)\) and \(\Delta\Sigma(R)\) vs BOSS CMASS logM*>11.3 data.

HOD occupation functions vs halo mass. Solid: MAP. Dashed + shaded: MCMC median and 16th–84th percentile. Orange: published values.

MCMC posterior corner plot (32 walkers × 3000 steps, 500 burn-in). Contours: 68% and 95% credible regions. Orange lines: published More+2015 values.

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Variant: more2015_logM11p4_12 — Joint wp+ΔΣ, logM*>11.4

MoreHODModel fit to BOSS CMASS logM*>11.4, \(w_p + \Delta\Sigma\) jointly. Beyond-linear halo bias (BNL) enabled.

MAP: \(\chi^2/\text{dof} = 63.30 / 35 = 1.809\). Status: PASSED. Published \(\chi^2/\text{dof} \approx 1.5\). MCMC: 32 walkers × 2000 steps, 500 burn-in → 48 000 samples.

Parameter	MAP	MCMC median	MCMC \(1\sigma\) interval	Published
log10mmin	14.166	14.014	\(^{+0.457}_{-0.444}\)	13.68 ± 0.16
sigma_logm	0.875	0.804	\(^{+0.196}_{-0.257}\)	0.889 ± 0.22
log10m1	14.390	14.408	\(^{+0.144}_{-0.303}\)	14.56 ± 0.25
alpha	1.602	1.756	\(^{+0.520}_{-0.747}\)	1.00 ± 0.44
kappa	1.675	1.730	\(^{+0.873}_{-1.033}\)	—

At the MCMC median, sigma_logm and log10m1 are within 0.4σ and 0.6σ of the published values. log10mmin remains 2.1σ away, reflecting a genuine tension for the most massive threshold sample.

Derived quantities (at MCMC median):

Quantity	Value	Unit
\(\bar{n}_g\)	\((0.92^{+0.96}_{-0.36}) \times 10^{-4}\)	\(h^3\,\mathrm{Mpc}^{-3}\)
Satellite fraction \(f_\mathrm{sat}\)	0.016	—
Mean halo mass \(\log_{10}\langle M_h\rangle\)	13.75	\(\log_{10}(M_\odot/h)\)

MAP \(w_p(r_p)\) and \(\Delta\Sigma(R)\) vs BOSS CMASS logM*>11.4 data.

HOD occupation functions vs halo mass. Solid: MAP. Dashed + shaded: MCMC median and 16th–84th percentile. Orange: published values.

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

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Variant: more2015_logM11_12_freecosmo — Free cosmology

Joint wp+ΔΣ fit with free \(\Omega_m\) and \(S_8 = \sigma_8\sqrt{\Omega_m/0.3}\), using Planck 2018 Gaussian priors. Beyond-linear halo bias enabled (BNL).

MAP: \(\chi^2/\text{dof} = 56.53 / 33 = 1.713\). Status: PASSED. Published \(\chi^2/\text{dof} \approx 0.8\).

Note

MCMC not yet run for this variant. The table below shows MAP values only; run --mcmc --force-mcmc to generate the posterior.

Parameter	MAP	Published / prior	\(|\Delta|/\sigma\)
Omega_m	0.297	0.31 ± 0.02	0.64σ
S8	0.813	0.798 ± 0.044	0.34σ
log10mmin	13.088	13.13 ± 0.13	0.32σ
sigma_logm	0.401	0.469 ± 0.13	0.52σ
log10m1	14.324	14.21 ± 0.13	0.87σ
alpha	2.492	1.13 ± 0.15	9.08σ ⚠
kappa	2.339	1.25 ± 0.45	2.42σ ⚠

The free-cosmology MAP recovers \(S_8 = 0.813\) (0.34σ from Planck 2018) and \(\Omega_m = 0.297\) (0.64σ from prior center). HOD mass parameters agree within 0.9σ. The alpha offset is the same Nelder-Mead satellite-slope degeneracy seen in the fixed-cosmology fits.

MAP \(w_p(r_p)\) and \(\Delta\Sigma(R)\) vs BOSS CMASS logM*>11.1 data (free \(\Omega_m\), \(S_8\) cosmology).

HOD occupation curves for the MAP free-cosmology solution.