.. _benchmark_more2015: Benchmark: More+2015 — BOSS CMASS mass-threshold samples ========================================================= **Model class**: ``MoreHODModel`` — More et al. 2015, ApJ 806, 2 (`arXiv:1407.1856 `_), BOSS CMASS z\ :sub:`eff` = 0.52. All variants fit :math:`w_p(r_p)` and :math:`\Delta\Sigma(R)` jointly using the beyond-linear halo bias correction (:class:`~hod_mod.core.beyond_linear_bias.BeyondLinearBiasMead21`). See the benchmark suite for the summary table. ---- .. _benchmark_more2015_logM11_12: Variant: more2015\_logM11\_12 — Joint wp+ΔΣ, logM*>11.1 --------------------------------------------------------- **MoreHODModel** fit to BOSS CMASS logM*>11.1, :math:`w_p + \Delta\Sigma` jointly. Beyond-linear halo bias (BNL) enabled. MAP: :math:`\chi^2/\text{dof} = 71.06 / 36 = 1.974`. **Status: PASSED**. Published :math:`\chi^2/\text{dof} \approx 0.8`. MCMC: 32 walkers × 2000 steps, 500 burn-in → 48 000 samples. .. list-table:: :header-rows: 1 :widths: 24 16 22 28 20 * - Parameter - MAP - MCMC median - MCMC :math:`1\sigma` interval - Published * - ``log10mmin`` - 13.169 - 13.164 - :math:`^{+0.302}_{-0.185}` - 13.13 ± 0.13 * - ``sigma_logm`` - 0.495 - 0.491 - :math:`^{+0.254}_{-0.263}` - 0.469 ± 0.13 * - ``log10m1`` - 14.192 - 14.220 - :math:`^{+0.079}_{-0.071}` - 14.21 ± 0.13 * - ``alpha`` - 1.870 - 1.928 - :math:`^{+0.175}_{-0.194}` - 1.13 ± 0.15 * - ``kappa`` - 2.619 - 1.862 - :math:`^{+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): .. list-table:: :header-rows: 1 :widths: 40 30 30 * - Quantity - Value - Unit * - :math:`\bar{n}_g` - :math:`(3.68^{+0.42}_{-0.78}) \times 10^{-4}` - :math:`h^3\,\mathrm{Mpc}^{-3}` * - Satellite fraction :math:`f_\mathrm{sat}` - 0.036 - — * - Mean halo mass :math:`\log_{10}\langle M_h\rangle` - 13.54 - :math:`\log_{10}(M_\odot/h)` .. figure:: _images/benchmarks__more2015_logM11_12__benchmark_more2015_logM11_12_combined.png :width: 90% :alt: more2015_logM11_12 combined MAP :math:`w_p(r_p)` (top) and :math:`\Delta\Sigma(R)` (middle) vs BOSS CMASS logM*>11.1 data, with residuals. .. figure:: _images/benchmarks__more2015_logM11_12__benchmark_more2015_logM11_12_hod.png :width: 70% :alt: more2015_logM11_12 HOD HOD occupation functions :math:`\langle N_c(M)\rangle`, :math:`\langle N_s(M)\rangle`, and :math:`\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. .. figure:: _images/benchmarks__more2015_logM11_12__benchmark_more2015_logM11_12_corner.png :width: 90% :alt: more2015_logM11_12 corner MCMC posterior corner plot (32 walkers × 2000 steps, 500 burn-in). Contours: 68% and 95% credible regions. Orange lines: published More+2015 values. ---- .. _benchmark_more2015_logM11p3_12: Variant: more2015\_logM11p3\_12 — Joint wp+ΔΣ, logM*>11.3 ---------------------------------------------------------- **MoreHODModel** fit to BOSS CMASS logM*>11.3, :math:`w_p + \Delta\Sigma` jointly. Beyond-linear halo bias (BNL) enabled. MAP: :math:`\chi^2/\text{dof} = 57.60 / 35 = 1.646`. **Status: PASSED**. Published :math:`\chi^2/\text{dof} \approx 1.3`. MCMC: 32 walkers × 3000 steps, 500 burn-in → 80 000 samples. .. list-table:: :header-rows: 1 :widths: 24 16 22 28 20 * - Parameter - MAP - MCMC median - MCMC :math:`1\sigma` interval - Published * - ``log10mmin`` - 13.549 - 13.347 - :math:`^{+0.323}_{-0.178}` - 13.45 ± 0.15 * - ``sigma_logm`` - 0.616 - 0.452 - :math:`^{+0.246}_{-0.238}` - 0.671 ± 0.19 * - ``log10m1`` - 14.548 - 14.395 - :math:`^{+0.093}_{-0.094}` - 14.51 ± 0.17 * - ``alpha`` - 2.361 - 1.982 - :math:`^{+0.295}_{-0.318}` - 1.14 ± 0.49 * - ``kappa`` - 0.148 - 1.553 - :math:`^{+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): .. list-table:: :header-rows: 1 :widths: 40 30 30 * - Quantity - Value - Unit * - :math:`\bar{n}_g` - :math:`(2.52^{+0.18}_{-0.24}) \times 10^{-4}` - :math:`h^3\,\mathrm{Mpc}^{-3}` * - Satellite fraction :math:`f_\mathrm{sat}` - 0.033 - — * - Mean halo mass :math:`\log_{10}\langle M_h\rangle` - 13.69 - :math:`\log_{10}(M_\odot/h)` .. figure:: _images/benchmarks__more2015_logM11p3_12__benchmark_more2015_logM11p3_12_combined.png :width: 90% :alt: more2015_logM11p3_12 combined MAP :math:`w_p(r_p)` and :math:`\Delta\Sigma(R)` vs BOSS CMASS logM*>11.3 data. .. figure:: _images/benchmarks__more2015_logM11p3_12__benchmark_more2015_logM11p3_12_hod.png :width: 70% :alt: more2015_logM11p3_12 HOD HOD occupation functions vs halo mass. Solid: MAP. Dashed + shaded: MCMC median and 16th–84th percentile. Orange: published values. .. figure:: _images/benchmarks__more2015_logM11p3_12__benchmark_more2015_logM11p3_12_corner.png :width: 90% :alt: more2015_logM11p3_12 corner MCMC posterior corner plot (32 walkers × 3000 steps, 500 burn-in). Contours: 68% and 95% credible regions. Orange lines: published More+2015 values. ---- .. _benchmark_more2015_logM11p4_12: Variant: more2015\_logM11p4\_12 — Joint wp+ΔΣ, logM*>11.4 ---------------------------------------------------------- **MoreHODModel** fit to BOSS CMASS logM*>11.4, :math:`w_p + \Delta\Sigma` jointly. Beyond-linear halo bias (BNL) enabled. MAP: :math:`\chi^2/\text{dof} = 63.30 / 35 = 1.809`. **Status: PASSED**. Published :math:`\chi^2/\text{dof} \approx 1.5`. MCMC: 32 walkers × 2000 steps, 500 burn-in → 48 000 samples. .. list-table:: :header-rows: 1 :widths: 24 16 22 28 20 * - Parameter - MAP - MCMC median - MCMC :math:`1\sigma` interval - Published * - ``log10mmin`` - 14.166 - 14.014 - :math:`^{+0.457}_{-0.444}` - 13.68 ± 0.16 * - ``sigma_logm`` - 0.875 - 0.804 - :math:`^{+0.196}_{-0.257}` - 0.889 ± 0.22 * - ``log10m1`` - 14.390 - 14.408 - :math:`^{+0.144}_{-0.303}` - 14.56 ± 0.25 * - ``alpha`` - 1.602 - 1.756 - :math:`^{+0.520}_{-0.747}` - 1.00 ± 0.44 * - ``kappa`` - 1.675 - 1.730 - :math:`^{+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): .. list-table:: :header-rows: 1 :widths: 40 30 30 * - Quantity - Value - Unit * - :math:`\bar{n}_g` - :math:`(0.92^{+0.96}_{-0.36}) \times 10^{-4}` - :math:`h^3\,\mathrm{Mpc}^{-3}` * - Satellite fraction :math:`f_\mathrm{sat}` - 0.016 - — * - Mean halo mass :math:`\log_{10}\langle M_h\rangle` - 13.75 - :math:`\log_{10}(M_\odot/h)` .. figure:: _images/benchmarks__more2015_logM11p4_12__benchmark_more2015_logM11p4_12_combined.png :width: 90% :alt: more2015_logM11p4_12 combined MAP :math:`w_p(r_p)` and :math:`\Delta\Sigma(R)` vs BOSS CMASS logM*>11.4 data. .. figure:: _images/benchmarks__more2015_logM11p4_12__benchmark_more2015_logM11p4_12_hod.png :width: 70% :alt: more2015_logM11p4_12 HOD HOD occupation functions vs halo mass. Solid: MAP. Dashed + shaded: MCMC median and 16th–84th percentile. Orange: published values. .. figure:: _images/benchmarks__more2015_logM11p4_12__benchmark_more2015_logM11p4_12_corner.png :width: 90% :alt: more2015_logM11p4_12 corner MCMC posterior corner plot (32 walkers × 2000 steps, 500 burn-in). Contours: 68% and 95% credible regions. Orange lines: published More+2015 values. ---- .. _benchmark_more2015_logM11_12_freecosmo: Variant: more2015\_logM11\_12\_freecosmo — Free cosmology ---------------------------------------------------------- Joint wp+ΔΣ fit with free :math:`\Omega_m` and :math:`S_8 = \sigma_8\sqrt{\Omega_m/0.3}`, using Planck 2018 Gaussian priors. Beyond-linear halo bias enabled (BNL). MAP: :math:`\chi^2/\text{dof} = 56.53 / 33 = 1.713`. **Status: PASSED**. Published :math:`\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. .. list-table:: :header-rows: 1 :widths: 28 22 22 28 * - Parameter - MAP - Published / prior - :math:`|\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 :math:`S_8 = 0.813` (0.34σ from Planck 2018) and :math:`\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. .. figure:: _images/benchmarks__more2015_logM11_12_freecosmo__benchmark_more2015_logM11_12_freecosmo_combined.png :width: 90% :alt: more2015_logM11_12_freecosmo combined MAP :math:`w_p(r_p)` and :math:`\Delta\Sigma(R)` vs BOSS CMASS logM*>11.1 data (free :math:`\Omega_m`, :math:`S_8` cosmology). .. figure:: _images/benchmarks__more2015_logM11_12_freecosmo__benchmark_more2015_logM11_12_freecosmo_hod.png :width: 70% :alt: more2015_logM11_12_freecosmo HOD HOD occupation curves for the MAP free-cosmology solution.