Context

Page Overview

• This page contains three result groups:
  • baseline/raw GSMF and completeness diagnostics using Mstar_cluver14_clipped,
  • ODR calibration diagnostics (global and local z<0.05 fits to the joint K-band mass),
  • ODR-corrected GSMF/completeness diagnostics (global and local corrections).
• For each mass treatment, the same completeness families are compared:
  • scalar survey completeness,
  • Pozzetti-style mass completeness,
  • local forward-model completeness,
  • 20x20 fine-grid completeness diagnostics.
• Tables are organized so each row is a diagnostic type and each column is a redshift slice or correction case.

How the GSMF is Computed

1. Start from the merged catalog and use zPhoto_Corr, W1, and a chosen stellar-mass column.
2. Split the catalog into redshift bins: 0.0-0.1, 0.1-0.2, 0.2-0.3, and 0.3-0.4.
3. Bin galaxies in stellar mass using 0.1 dex bins over log(M*/Msun)=7-13.
4. For the raw GSMF, compute number density as N / (V_shell * dlogM) using a flat LCDM cosmology with H0=70 and Om0=0.3.
5. For 1/Vmax, estimate the maximum redshift at which each galaxy would still pass the W1<17 selection using luminosity-distance scaling:
   • D_L,max = D_L(z) * 10 ** (0.2 * (W1_limit - W1_obs)).
6. Clip each galaxy's accessible volume to the current redshift shell and sum 1/Vmax_eff in each mass bin.
7. All GSMF plots include the LM24 curve as a visual reference.

Completeness Methods

1. Scalar Survey Completeness

• This is the simplest comparison case.
• The WISExSuperCOSMOS paper quotes an approximate high-latitude catalog completeness of 90%.
• The correction is just phi = phi_1/Vmax / 0.90.
• This is useful as a reference but is not a true mass/redshift-dependent selection function.

2. Pozzetti-Style Mass Completeness

• For every galaxy, estimate the stellar mass it would have if observed at the W1 flux limit:
  • logM_lim = logM + 0.4 * (W1_obs - W1_limit).
• In each redshift bin, use the faintest 20% of galaxies in W1.
• The main threshold is the 95th percentile of these logM_lim values.
• For cell-by-cell completeness, compute the fraction of faint-end logM_lim values below each mass-bin center.
• The GSMF correction is phi = phi_1/Vmax / C_pozzetti(M,z), with a floor of 5% to avoid infinite corrections.

3. Local Forward-Model Completeness

• Select a local reference set with z<0.05.
• Infer each local galaxy's absolute W1 from its observed W1 and luminosity distance.
• Reobserve each local galaxy at the midpoint of each redshift bin.
• A reobserved galaxy is recovered if its shifted apparent W1 remains brighter than W1=17.
• In each mass/redshift cell, compute the recovered fraction.
• The GSMF correction is phi = phi_1/Vmax / C_forward(M,z), again with a 5% floor.
• This is intentionally simple and diagnostic. It does not yet model B/R cuts, angular mask structure, K-correction evolution, or population evolution.

4. Fine-Grid Completeness Diagnostics

• In addition to the broad GSMF redshift bins, I made 20 x 20 completeness heatmaps over the same M-z range.
• These are intended to show where the completeness boundary moves continuously in the M-z plane.

Raw / Original Cluver14 Masses

• This section uses Mstar_cluver14_clipped directly, with no ODR mass correction.

Completeness Diagnostics (Raw Masses)

How to read the line-based completeness plots (applies to overlay/line plots below)

• Red/purple curves in completeness_limits_overlay_mz:
  • red = Pozzetti cell-completeness limits,
  • purple = forward-model recovered-fraction limits,
  • solid/dashed distinguish different completeness levels (e.g. 90% vs 95%).
• Black horizontal segments in completeness_limits_overlay_mz:
  • these are coarse-bin Pozzetti 95% thresholds from the 4 broad GSMF redshift bins,
  • each black segment is drawn at a fixed stellar-mass threshold for one broad bin, from that bin's z_min to z_max,
  • they are therefore a coarse/binned version of the finer red Pozzetti curve.
• Fine-grid cell values (20x20):
  • each cell's fraction is computed directly in that (M,z) cell,
  • Pozzetti cells use faint-end logM_lim = logM + 0.4*(W1-W1_limit) statistics,
  • forward-model cells use local (z<0.05) templates reobserved at the cell/bin redshift and count recovered fraction.
• 1/Vmax accessible volume fraction heatmap:
  • for each galaxy, compute Vmax_eff/V_shell in its redshift shell,
  • the heatmap cell value is the mean of that ratio for galaxies in the cell,
  • values near 1 mean objects are visible through most of the shell, values near 0 mean strong flux-limit truncation and large 1/Vmax weighting.

Diagnostic	Broad Grid / Main View	Fine Grid / Secondary View	Additional Check
Completeness limits	[images/260424/original_completeness_limits_overlay_mz.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_fine_grid_completeness_heatmap_mass_selection_completeness.png)	images/260424/original_fine_grid_completeness_heatmap_forward_model_local_reobserve.png
Pozzetti / forward heatmaps	[images/260424/original_completeness_heatmap_mass_selection_completeness.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_completeness_heatmap_forward_model_local_reobserve.png)	images/260424/original_completeness_heatmap_vmax_volume_access_fraction.png

Combined GSMFs Across Redshift Bins (Raw Masses)

• The plots below show GSMFs computed from the raw masses in this section (no K-band mass correction).
• The columns compare no correction, 1/Vmax only, and 1/Vmax plus different completeness assumptions.

Raw	1/Vmax	1/Vmax + scalar	1/Vmax + Pozzetti	1/Vmax + forward model
[images/260424/original_gsmf_all_bins_raw_volume_uncorrected.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_gsmf_all_bins_vmax_only.png)	[images/260424/original_gsmf_all_bins_vmax_plus_survey_completeness.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_gsmf_all_bins_vmax_plus_pozzetti_cell.png)	images/260424/original_gsmf_all_bins_vmax_plus_forward_model.png

Per-Redshift Results (Raw Masses)

• The plots below show one redshift bin per column for the same raw-mass setup.
• "Method comparison" overlays all correction choices within each redshift bin; "Mass histogram" shows the underlying mass distributions used for those GSMFs.

Diagnostic	z=0.0-0.1	z=0.1-0.2	z=0.2-0.3	z=0.3-0.4
Method comparison GSMF	[images/260424/original_gsmf_methods_comparison_z0.00_0.10.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_gsmf_methods_comparison_z0.10_0.20.png)	[images/260424/original_gsmf_methods_comparison_z0.20_0.30.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_gsmf_methods_comparison_z0.30_0.40.png)
Mass histogram	[images/260424/original_mass_hist_z0.00_0.10.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_mass_hist_z0.10_0.20.png)	[images/260424/original_mass_hist_z0.20_0.30.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/original_mass_hist_z0.30_0.40.png)

ODR-Corrected Masses

• The fitted relation is joint_K_mass = slope * Mstar_cluver14_clipped + intercept.
• The corrected mass is defined by applying this fitted line to the WISE/Cluver14 mass.
• The global fit uses all galaxies with finite joint K-band masses.
• The local fit uses only the z<0.05 subset with finite joint K-band masses.
• Note I have a "local" and "global" ODR -- the only difference is which set of galaxies are used for the fit. In the "local" case, as mentioned above, I limit galaxies to z <0.05 for the ODR fit; in the global case, I use everything. It probably makes most sense physically to use the local fit.

ODR Calibration Plots

Fit Input / No Correction	Global ODR-Corrected	Local ODR-Corrected
[images/260424/mass_calibration_wise_vs_joint_odr_fits.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/mass_calibration_corrected_vs_joint_global_odr.png)	images/260424/mass_calibration_corrected_vs_joint_local_odr.png

Completeness Diagnostics (Corrected Masses)

Mass Case	Completeness Limits	Pozzetti Heatmap	Forward Heatmap	Fine Pozzetti Grid	Fine Forward Grid	Accessible Volume Fraction
Global ODR	[images/260424/global_odr_completeness_limits_overlay_mz.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_completeness_heatmap_mass_selection_completeness.png)	[images/260424/global_odr_completeness_heatmap_forward_model_local_reobserve.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_fine_grid_completeness_heatmap_mass_selection_completeness.png)	[images/260424/global_odr_fine_grid_completeness_heatmap_forward_model_local_reobserve.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_completeness_heatmap_vmax_volume_access_fraction.png)
Local ODR	[images/260424/local_odr_completeness_limits_overlay_mz.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_completeness_heatmap_mass_selection_completeness.png)	[images/260424/local_odr_completeness_heatmap_forward_model_local_reobserve.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_fine_grid_completeness_heatmap_mass_selection_completeness.png)	[images/260424/local_odr_fine_grid_completeness_heatmap_forward_model_local_reobserve.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_completeness_heatmap_vmax_volume_access_fraction.png)

Combined GSMFs Across Redshift Bins (Corrected Masses)

• The plots below show GSMFs after applying the ODR mass correction for each row (global ODR or local ODR).
• The columns compare no correction, 1/Vmax only, and 1/Vmax plus different completeness assumptions.

Mass Case	Raw	1/Vmax	1/Vmax + scalar	1/Vmax + Pozzetti	1/Vmax + forward model
Global ODR	[images/260424/global_odr_gsmf_all_bins_raw_volume_uncorrected.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_gsmf_all_bins_vmax_only.png)	[images/260424/global_odr_gsmf_all_bins_vmax_plus_survey_completeness.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_gsmf_all_bins_vmax_plus_pozzetti_cell.png)	images/260424/global_odr_gsmf_all_bins_vmax_plus_forward_model.png
Local ODR	[images/260424/local_odr_gsmf_all_bins_raw_volume_uncorrected.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_gsmf_all_bins_vmax_only.png)	[images/260424/local_odr_gsmf_all_bins_vmax_plus_survey_completeness.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_gsmf_all_bins_vmax_plus_pozzetti_cell.png)	images/260424/local_odr_gsmf_all_bins_vmax_plus_forward_model.png

Per-Redshift GSMF Method Comparisons (Corrected Masses)

• The plots below show one redshift bin per column for each ODR correction row.
• Each panel overlays raw, 1/Vmax, and completeness-corrected GSMFs so the impact of each correction can be compared within the same redshift bin.

Mass Case	z=0.0-0.1	z=0.1-0.2	z=0.2-0.3	z=0.3-0.4
Global ODR	[images/260424/global_odr_gsmf_methods_comparison_z0.00_0.10.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_gsmf_methods_comparison_z0.10_0.20.png)	[images/260424/global_odr_gsmf_methods_comparison_z0.20_0.30.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_gsmf_methods_comparison_z0.30_0.40.png)
Local ODR	[images/260424/local_odr_gsmf_methods_comparison_z0.00_0.10.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_gsmf_methods_comparison_z0.10_0.20.png)	[images/260424/local_odr_gsmf_methods_comparison_z0.20_0.30.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_gsmf_methods_comparison_z0.30_0.40.png)

Mass Histograms (Corrected Masses)

Mass Case	z=0.0-0.1	z=0.1-0.2	z=0.2-0.3	z=0.3-0.4
Global ODR	[images/260424/global_odr_mass_hist_z0.00_0.10.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_mass_hist_z0.10_0.20.png)	[images/260424/global_odr_mass_hist_z0.20_0.30.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/global_odr_mass_hist_z0.30_0.40.png)
Local ODR	[images/260424/local_odr_mass_hist_z0.00_0.10.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_mass_hist_z0.10_0.20.png)	[images/260424/local_odr_mass_hist_z0.20_0.30.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260424/local_odr_mass_hist_z0.30_0.40.png)