Context

Trying to Model Completeness

Forward Model Approach

• I've tried one pass at estimating the completeness of the catalog using a "forward model" approach. The general approach I took is the following:

1. Define a "reference" sample of galaxies (all galaxies with z<0.06). We will "move" these to higher redshift and see if they end up getting observed via a detection model. I do this "moving" by converting them to an absolute magntiude, and then an "inverse" distance modulus to go to an observed magntiude at different z's.

2. Get the faint-end limits from the observed distributions of W1 and W2 (as the 99.9 percentile). These are the "limiting magntiudes" for the survey.

3. Approximate the SNR for the observations as 510^(0.4(m_lim - m)), motivated from the 5-sigma detection limit being associated with the limitng magntiudes above. Convert the SNR to magntiudes with 1.085/SNR. Set a minimum and max error allowed to be 0.03 and 0.33

4. Detection probability in each band is then given by: p_det(m) = 0.5 * [1 + erf((m_lim - m)/(sqrt(2)*sigma_m(m)))]

5. And the joint detection probability is the product of the two.

Pozzetti Approach

• Pozzetti+10 also has an approach to determine the completeness, which I've tried to implement as a comparison here. The idea there is:

1. Bin up redshifts into k bins.

2. Inside of each bin, convert galaxy masses to the "limiting mass" -- the mass they would have at that redshift if they had W1 equal to the limiting magntiude.

3. Select the faintest N% of targets inside of this redshift bin (I'm using 0.2).

4. Select the 95th percentile of galaxy masses from this faintest fraction. This is the limiting mass for that redshift bin.

Results

• Admittedly these tests are both still not the most well-understood (I'm still testing out sensitivity to parameter choices, and there's a chance I have some bugs or other revisions to make), but here are some diagnostics for the mass completness:

Injected W1 vs. Recoverd	Injected W2 vs. Recoverd	Injected Color vs. Recoverd	Histograms	Completeness Curve
[images/260421/hist_true_vs_test_vs_recovered_W1_largeN.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260421/hist_true_vs_test_vs_recovered_W2_largeN.png)	[images/260421/hist_true_vs_test_vs_recovered_W1_minus_W2_largeN.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260421/logM_histograms_by_zbin_with_masslimit_lines.png)	images/260421/mass_redshift_hist_with_95pct_limit.png

GSMF Follow Ups

• I've also tried to audit some of the GSMF work I've been doing (for the 1/vmax) with the help of some AI tooling. I still am getting under the hood of this but wanted to put this up for now as a work in progress. As a quick reminder, the general idea of the 1/vmax method is to:

1. For each galaxy in the catalog, find z_max, the max redshfit this galaxy would be observed given the limiting magntiude of the survey.
2. Turn the z_max into a comoving volume; each galaxy contributes 1/Vmax to its mass bin
3. Sum 1/vmax in mass bins and divided by bin width

• Note that I have two results here:
  • When splitting the GSMF into shells and computing 1/vmax, I wasn't sure what to do when the zmax fell outside of the upper resshift slice limit, so I have one case where I count that galaxy to its own z shell, and another where I don't count it. The "no hi-z clipping" appears to agree better with the LM24 result, but I need to look into this a bit more on what's typically done.

GSMFs	GSMFs (no hi-z clipping)	Histograms
[images/260421/gsmf_redshift_bins_with_masslimit_lines.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260421/gsmf_redshift_bins_with_masslimit_lines.png)	images/260421/logM_histograms_by_zbin_with_masslimit_lines.png

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