LS10 systematic-correction recommendations
See also
Results: systematic weights — full σ̂ and LRT tables for all nine samples and four resolutions.
Decision framework
Method choice — MCMC-comb (WEIGHT_COMB) for all samples.
The Likelihood Ratio Test (LRT) rejects the additive-only null hypothesis at all four
map resolutions (NSIDE 32–256) and all nine stellar-mass thresholds, with
\(p < 10^{-9}\) in every case. This confirms a statistically significant
multiplicative component on top of the additive systematic.
MCMC-comb (stored as WEIGHT_COMB = WEIGHT_SYS in the FITS files) is the only
method that models both components jointly. The other methods are ranked as follows:
OLS,ISD-1,MCMC-add: nearly identical σ̂ values, miss the multiplicative term.ElasticNet: equivalent to OLS when cross-validation selects zero amplitudes (some samples).ISD-3: numerically unstable — σ̂ ≫ 1 at NSIDE 32 and poor at all other resolutions. Never use ISD-3 for science.
Resolution choice — NSIDE 64 for all samples (primary).
The selection criterion is the minimum \(\hat{\sigma}_{\rm comb}\) among valid NSIDEs (σ < 1, NSIDE ≥ 64). Three additional constraints apply:
NSIDE 32 is excluded — its pixel scale (~108’) exceeds the maximum angular bin in the w(θ) data (54’), producing zero measurable correction on the two-point function in the entire measured range.
All 11 templates are at NSIDE 64 — analysing at finer resolutions upsamples the templates without adding systematic information.
NSIDE 128/256 overfit for sparse samples (M* ≥ 9.0, ≥ 11.25, ≥ 11.5): σ̂_comb exceeds 1.0 and the corrected w(θ) shows unphysically large corrections (up to 34%).
σ̂_comb is minimised at NSIDE 64 for all nine samples among resolutions where w(θ) correction is non-zero. NSIDE 128 is a viable secondary option for the five densest samples (M* 9.5–11.25, σ̂_comb < 1), extending the correction scale to ~20’.
Recommended weight and resolution per sample
Sample (log M*) |
N_gal |
Method |
NSIDE |
\(\hat{\sigma}_{\rm comb}\) |
θ corrected |
Key constraint |
|---|---|---|---|---|---|---|
≥ 9.0 |
523,486 |
MCMC-comb |
64 |
0.751 |
49–54’ |
σ̂_comb N64=0.751 (minimum among valid NSIDEs); N128 overfits (σ=1.029); N256 overfits (σ=1.601); external |δw/w|=7.2% at 30’ |
≥ 9.5 |
1,432,502 |
MCMC-comb |
64 |
0.554 |
49–54’ |
σ̂_comb N64=0.554 (minimum among valid NSIDEs); N256 overfits (σ=1.052); external |δw/w|=4.8% at 30’ |
≥ 10.0 |
2,759,238 |
MCMC-comb |
64 |
0.382 |
49–54’ |
σ̂_comb N64=0.382 (minimum among valid NSIDEs); external |δw/w|=0.4% at 30’ |
≥ 10.25 |
3,308,841 |
MCMC-comb |
64 |
0.334 |
49–54’ |
σ̂_comb N64=0.334 (minimum among valid NSIDEs) |
≥ 10.5 |
3,263,228 |
MCMC-comb |
64 |
0.310 |
49–54’ |
σ̂_comb N64=0.310 (minimum among valid NSIDEs) |
≥ 10.75 |
2,802,710 |
MCMC-comb |
64 |
0.301 |
49–54’ |
σ̂_comb N64=0.301 (minimum among valid NSIDEs) |
≥ 11.0 |
1,619,838 |
MCMC-comb |
64 |
0.305 |
49–54’ |
σ̂_comb N64=0.305 (minimum among valid NSIDEs); external |δw/w|=0.1% at 30’ |
≥ 11.25 |
541,855 |
MCMC-comb |
64 |
0.393 |
49–54’ |
σ̂_comb N64=0.393 (minimum among valid NSIDEs); N256 overfits (σ=1.212); external |δw/w|=2.2% at 30’ |
≥ 11.5 |
120,882 |
MCMC-comb |
64 |
0.710 |
49–54’ |
σ̂_comb N64=0.710 (minimum among valid NSIDEs); N128 overfits (σ=1.430); N256 overfits (σ=2.021); external |δw/w|=0.7% at 30’ |
Note
NSIDE 128 is a valid secondary choice for the five densest samples (M* ≥ 9.5 through ≥ 11.25), extending the angular correction down to ~20 arcmin. It must not be used for M* ≥ 9.0 (σ̂_comb = 1.03) or M* ≥ 11.5 (σ̂_comb = 1.43), where it overfits shot noise.
NSIDE 32 should never be used as the analysis resolution for angular clustering: its pixel scale exceeds the measured w(θ) range and delivers zero correction.
σ̂_comb goodness-of-fit across resolutions
\(\hat{\sigma}_{\rm comb} < 1\) indicates the model fits within the Poisson expectation. Values ≥ 1 (marked ✗) signal overfitting — the correction is absorbing noise rather than real systematics.
Sample (log M*) |
N_gal |
NSIDE 32 |
NSIDE 64 |
NSIDE 128 |
NSIDE 256 |
Recommended |
|---|---|---|---|---|---|---|
≥ 9.0 |
523,486 |
0.674 |
0.751 |
1.029 ✗ |
1.601 ✗ |
N64 |
≥ 9.5 |
1,432,502 |
0.606 |
0.554 |
0.741 |
1.052 ✗ |
N64 |
≥ 10.0 |
2,759,238 |
0.422 |
0.382 |
0.534 |
0.786 |
N64 |
≥ 10.25 |
3,308,841 |
0.390 |
0.334 |
0.474 |
0.696 |
N64 |
≥ 10.5 |
3,263,228 |
0.373 |
0.310 |
0.451 |
0.659 |
N64 |
≥ 10.75 |
2,802,710 |
0.398 |
0.301 |
0.430 |
0.630 |
N64 |
≥ 11.0 |
1,619,838 |
0.393 |
0.305 |
0.472 |
0.726 |
N64 |
≥ 11.25 |
541,855 |
0.406 |
0.393 |
0.633 |
1.212 ✗ |
N64 |
≥ 11.5 |
120,882 |
0.586 |
0.710 |
1.430 ✗ |
2.021 ✗ |
N64 |
Systematic uncertainty budget
The table below quantifies the systematic correction applied and the residual method uncertainty at the recommended NSIDE 64 + MCMC-comb.
Max |δw/w|_comb at 42–54’ — the largest fractional correction applied to w(θ) by the MCMC-comb weights relative to the uncorrected measurement. Observed only at the outermost bins because the NSIDE 64 pixel scale (~54’) sets the lower angular limit of the weight-induced clustering modulation.
Method uncertainty >20’ — the standard deviation of the corrected w(θ) over the four well-behaved methods {OLS, ISD-1, MCMC-add, MCMC-comb}, expressed as a fraction of w_comb. This is the irreducible systematic floor from the choice of correction algorithm.
Below ~40 arcmin at NSIDE 64, no measurable correction is applied in the wtheta data range. The large-scale systematic bias (e.g. −7.2% at 30’ for M* ≥ 9.0) operates primarily at degree scales where the template coherence length matches and the weights are most effective.
Sample (log M*) |
N_gal |
Recommended NSIDE |
Max |δw/w|comb at 42–54’ |
θ of first correction |
Method uncertainty >20’ |
Regime |
|---|---|---|---|---|---|---|
≥ 9.0 |
523,486 |
64 |
8.75% |
49–54’ |
1.64% |
Strongly contaminated |
≥ 9.5 |
1,432,502 |
64 |
6.82% |
49–54’ |
0.80% |
Strongly contaminated |
≥ 10.0 |
2,759,238 |
64 |
3.40% |
49–54’ |
0.12% |
Weakly contaminated |
≥ 10.25 |
3,308,841 |
64 |
5.09% |
49–54’ |
0.07% |
Low contamination |
≥ 10.5 |
3,263,228 |
64 |
8.10% |
49–54’ |
0.14% |
Low contamination |
≥ 10.75 |
2,802,710 |
64 |
12.98% |
49–54’ |
0.11% |
Low contamination |
≥ 11.0 |
1,619,838 |
64 |
19.52% |
49–54’ |
0.15% |
Weakly contaminated |
≥ 11.25 |
541,855 |
64 |
18.93% |
49–54’ |
0.24% |
Moderately contaminated |
≥ 11.5 |
120,882 |
64 |
29.29% |
49–54’ |
0.29% |
Weakly contaminated |
Per-sample angular correction profiles
The following tables show, for each sample at the recommended NSIDE 64, the observed w(θ), the MCMC-comb corrected w(θ), the fractional correction applied, and the method-to-method spread as a proxy for the systematic floor. All corrections are expressed as percentages of w_obs.
M* ≥ 9.0 (z < 0.08, N = 523,486)
External two-point measurement: δw/w ≈ -7.2% at 30’, max |δw/w| = 8.4% at 23’.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.751). No finer valid NSIDE available (σ̂ ≥ 1 at NSIDE 128+).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
3.71714 |
3.71714 |
+0.00 |
0.00 |
0.69 |
3.41581 |
3.41581 |
+0.00 |
0.00 |
0.86 |
3.06270 |
3.06270 |
+0.00 |
0.00 |
1.06 |
2.70489 |
2.70489 |
+0.00 |
0.00 |
1.31 |
2.39533 |
2.39533 |
+0.00 |
0.00 |
1.63 |
2.21736 |
2.21736 |
+0.00 |
0.00 |
2.01 |
1.95239 |
1.95239 |
+0.00 |
0.00 |
2.49 |
1.74937 |
1.74937 |
+0.00 |
0.00 |
3.08 |
1.55497 |
1.55497 |
+0.00 |
0.00 |
3.82 |
1.39614 |
1.39614 |
+0.00 |
0.00 |
4.72 |
1.23611 |
1.23611 |
+0.00 |
0.00 |
5.85 |
1.08708 |
1.08708 |
+0.00 |
0.00 |
7.23 |
0.94986 |
0.94986 |
+0.00 |
0.00 |
8.96 |
0.81964 |
0.81964 |
+0.00 |
0.00 |
11.08 |
0.69699 |
0.69699 |
+0.00 |
0.00 |
13.72 |
0.58956 |
0.58956 |
+0.00 |
0.00 |
16.99 |
0.49512 |
0.49512 |
+0.00 |
0.00 |
21.03 |
0.40304 |
0.40304 |
+0.00 |
0.00 |
26.03 |
0.32596 |
0.32596 |
+0.00 |
0.00 |
32.22 |
0.27140 |
0.27140 |
+0.00 |
0.00 |
39.89 |
0.22417 |
0.22417 |
+0.00 |
0.00 |
49.38 |
0.18634 |
0.17003 |
+8.75 |
1.64 |
61.12 |
0.15643 |
0.14727 |
+5.85 |
1.11 |
75.66 |
0.13447 |
0.12775 |
+5.00 |
0.95 |
93.64 |
0.11412 |
0.10478 |
+8.18 |
1.16 |
115.92 |
0.09485 |
0.08971 |
+5.42 |
0.84 |
143.50 |
0.08309 |
0.07744 |
+6.80 |
0.86 |
177.62 |
0.06393 |
0.05976 |
+6.52 |
0.78 |
219.84 |
0.05130 |
0.04763 |
+7.16 |
0.73 |
272.13 |
0.04051 |
0.03717 |
+8.24 |
0.70 |
M* ≥ 9.5 (z < 0.12, N = 1,432,502)
External two-point measurement: δw/w ≈ -4.8% at 30’, max |δw/w| = 4.9% at 15’.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.554). NSIDE 128 is a valid secondary option (σ̂=0.741).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
3.02828 |
3.02828 |
+0.00 |
0.00 |
0.69 |
2.68379 |
2.68379 |
+0.00 |
0.00 |
0.86 |
2.34430 |
2.34430 |
+0.00 |
0.00 |
1.06 |
2.12172 |
2.12172 |
+0.00 |
0.00 |
1.31 |
1.85966 |
1.85966 |
+0.00 |
0.00 |
1.63 |
1.64785 |
1.64785 |
+0.00 |
0.00 |
2.01 |
1.46712 |
1.46712 |
+0.00 |
0.00 |
2.49 |
1.29353 |
1.29353 |
+0.00 |
0.00 |
3.08 |
1.13963 |
1.13963 |
+0.00 |
0.00 |
3.82 |
0.99403 |
0.99403 |
+0.00 |
0.00 |
4.72 |
0.86649 |
0.86649 |
+0.00 |
0.00 |
5.85 |
0.74101 |
0.74101 |
+0.00 |
0.00 |
7.24 |
0.63716 |
0.63716 |
+0.00 |
0.00 |
8.96 |
0.53442 |
0.53442 |
+0.00 |
0.00 |
11.09 |
0.44365 |
0.44365 |
+0.00 |
0.00 |
13.73 |
0.36584 |
0.36584 |
+0.00 |
0.00 |
17.00 |
0.30018 |
0.30018 |
+0.00 |
0.00 |
21.04 |
0.24555 |
0.24555 |
+0.00 |
0.00 |
26.04 |
0.20359 |
0.20359 |
+0.00 |
0.00 |
32.24 |
0.17215 |
0.17215 |
+0.00 |
0.00 |
39.91 |
0.14930 |
0.14930 |
+0.00 |
0.00 |
49.40 |
0.12794 |
0.11999 |
+6.21 |
0.80 |
61.14 |
0.10998 |
0.10521 |
+4.33 |
0.54 |
75.68 |
0.09423 |
0.09068 |
+3.77 |
0.44 |
93.67 |
0.07730 |
0.07227 |
+6.51 |
0.61 |
115.95 |
0.06843 |
0.06557 |
+4.18 |
0.40 |
143.53 |
0.05427 |
0.05115 |
+5.74 |
0.44 |
177.64 |
0.04707 |
0.04464 |
+5.17 |
0.38 |
219.89 |
0.03238 |
0.03025 |
+6.59 |
0.38 |
272.18 |
0.02931 |
0.02731 |
+6.82 |
0.36 |
M* ≥ 10.0 (z < 0.18, N = 2,759,238)
External two-point measurement: δw/w ≈ -0.4% at 30’, max |δw/w| = 2.0% at 120’.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.382). NSIDE 256 is a valid secondary option (σ̂=0.786).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
2.32873 |
2.32873 |
+0.00 |
0.00 |
0.69 |
2.02841 |
2.02841 |
+0.00 |
0.00 |
0.86 |
1.76627 |
1.76627 |
+0.00 |
0.00 |
1.06 |
1.54218 |
1.54218 |
+0.00 |
0.00 |
1.31 |
1.33191 |
1.33191 |
+0.00 |
0.00 |
1.63 |
1.15594 |
1.15594 |
+0.00 |
0.00 |
2.01 |
0.99820 |
0.99820 |
+0.00 |
0.00 |
2.49 |
0.85471 |
0.85471 |
+0.00 |
0.00 |
3.08 |
0.72759 |
0.72759 |
+0.00 |
0.00 |
3.82 |
0.61737 |
0.61737 |
+0.00 |
0.00 |
4.72 |
0.51672 |
0.51672 |
+0.00 |
0.00 |
5.85 |
0.43088 |
0.43088 |
+0.00 |
0.00 |
7.24 |
0.35531 |
0.35531 |
+0.00 |
0.00 |
8.96 |
0.29081 |
0.29081 |
+0.00 |
0.00 |
11.09 |
0.23767 |
0.23767 |
+0.00 |
0.00 |
13.73 |
0.19574 |
0.19574 |
+0.00 |
0.00 |
17.00 |
0.16276 |
0.16276 |
+0.00 |
0.00 |
21.05 |
0.13598 |
0.13598 |
+0.00 |
0.00 |
26.05 |
0.11506 |
0.11506 |
+0.00 |
0.00 |
32.24 |
0.09877 |
0.09877 |
+0.00 |
0.00 |
39.91 |
0.08372 |
0.08372 |
+0.00 |
0.00 |
49.41 |
0.07394 |
0.07271 |
+1.67 |
0.12 |
61.15 |
0.05939 |
0.05854 |
+1.42 |
0.10 |
75.69 |
0.05103 |
0.05035 |
+1.33 |
0.08 |
93.70 |
0.04113 |
0.04029 |
+2.06 |
0.09 |
115.97 |
0.03283 |
0.03229 |
+1.67 |
0.07 |
143.55 |
0.02782 |
0.02724 |
+2.06 |
0.07 |
177.67 |
0.02007 |
0.01959 |
+2.38 |
0.07 |
219.92 |
0.01382 |
0.01339 |
+3.11 |
0.09 |
272.23 |
0.01186 |
0.01146 |
+3.40 |
0.10 |
M* ≥ 10.25 (z < 0.22, N = 3,308,841)
External two-point measurement: not available for this sample.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.334). NSIDE 256 is a valid secondary option (σ̂=0.696).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
2.04201 |
2.04201 |
+0.00 |
0.00 |
0.69 |
1.74577 |
1.74577 |
+0.00 |
0.00 |
0.86 |
1.50298 |
1.50298 |
+0.00 |
0.00 |
1.06 |
1.28995 |
1.28995 |
+0.00 |
0.00 |
1.31 |
1.10343 |
1.10343 |
+0.00 |
0.00 |
1.63 |
0.93653 |
0.93653 |
+0.00 |
0.00 |
2.01 |
0.79371 |
0.79371 |
+0.00 |
0.00 |
2.49 |
0.67131 |
0.67131 |
+0.00 |
0.00 |
3.08 |
0.55949 |
0.55949 |
+0.00 |
0.00 |
3.82 |
0.46909 |
0.46909 |
+0.00 |
0.00 |
4.73 |
0.38466 |
0.38466 |
+0.00 |
0.00 |
5.85 |
0.31790 |
0.31790 |
+0.00 |
0.00 |
7.24 |
0.25879 |
0.25879 |
+0.00 |
0.00 |
8.96 |
0.21090 |
0.21090 |
+0.00 |
0.00 |
11.10 |
0.17245 |
0.17245 |
+0.00 |
0.00 |
13.74 |
0.14305 |
0.14305 |
+0.00 |
0.00 |
17.00 |
0.12102 |
0.12102 |
+0.00 |
0.00 |
21.05 |
0.10244 |
0.10244 |
+0.00 |
0.00 |
26.05 |
0.08609 |
0.08609 |
+0.00 |
0.00 |
32.25 |
0.07519 |
0.07519 |
+0.00 |
0.00 |
39.92 |
0.06292 |
0.06292 |
+0.00 |
0.00 |
49.41 |
0.05613 |
0.05524 |
+1.60 |
0.07 |
61.16 |
0.04478 |
0.04414 |
+1.42 |
0.06 |
75.70 |
0.03708 |
0.03654 |
+1.47 |
0.05 |
93.70 |
0.03161 |
0.03097 |
+2.04 |
0.05 |
115.98 |
0.02248 |
0.02202 |
+2.03 |
0.04 |
143.55 |
0.02054 |
0.02006 |
+2.33 |
0.04 |
177.68 |
0.01295 |
0.01253 |
+3.20 |
0.03 |
219.94 |
0.00851 |
0.00812 |
+4.54 |
0.03 |
272.24 |
0.00724 |
0.00687 |
+5.09 |
0.04 |
M* ≥ 10.5 (z < 0.26, N = 3,263,228)
External two-point measurement: not available for this sample.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.310). NSIDE 256 is a valid secondary option (σ̂=0.659).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
1.83274 |
1.83274 |
+0.00 |
0.00 |
0.69 |
1.54648 |
1.54648 |
+0.00 |
0.00 |
0.86 |
1.29930 |
1.29930 |
+0.00 |
0.00 |
1.06 |
1.10542 |
1.10542 |
+0.00 |
0.00 |
1.31 |
0.92508 |
0.92508 |
+0.00 |
0.00 |
1.63 |
0.77624 |
0.77624 |
+0.00 |
0.00 |
2.01 |
0.64651 |
0.64651 |
+0.00 |
0.00 |
2.49 |
0.53988 |
0.53988 |
+0.00 |
0.00 |
3.08 |
0.44303 |
0.44303 |
+0.00 |
0.00 |
3.82 |
0.36563 |
0.36563 |
+0.00 |
0.00 |
4.73 |
0.29859 |
0.29859 |
+0.00 |
0.00 |
5.85 |
0.24366 |
0.24366 |
+0.00 |
0.00 |
7.24 |
0.19921 |
0.19921 |
+0.00 |
0.00 |
8.97 |
0.16323 |
0.16323 |
+0.00 |
0.00 |
11.10 |
0.13584 |
0.13584 |
+0.00 |
0.00 |
13.74 |
0.11425 |
0.11425 |
+0.00 |
0.00 |
17.01 |
0.09842 |
0.09842 |
+0.00 |
0.00 |
21.05 |
0.08323 |
0.08323 |
+0.00 |
0.00 |
26.05 |
0.07059 |
0.07059 |
+0.00 |
0.00 |
32.25 |
0.06188 |
0.06188 |
+0.00 |
0.00 |
39.92 |
0.05086 |
0.05086 |
+0.00 |
0.00 |
49.41 |
0.04559 |
0.04464 |
+2.08 |
0.05 |
61.16 |
0.03581 |
0.03512 |
+1.94 |
0.04 |
75.70 |
0.02985 |
0.02923 |
+2.08 |
0.05 |
93.70 |
0.02554 |
0.02483 |
+2.78 |
0.04 |
115.98 |
0.01760 |
0.01706 |
+3.08 |
0.06 |
143.55 |
0.01645 |
0.01589 |
+3.42 |
0.05 |
177.69 |
0.00970 |
0.00920 |
+5.17 |
0.09 |
219.95 |
0.00661 |
0.00614 |
+7.21 |
0.12 |
272.25 |
0.00566 |
0.00520 |
+8.10 |
0.14 |
M* ≥ 10.75 (z < 0.31, N = 2,802,710)
External two-point measurement: not available for this sample.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.301). NSIDE 256 is a valid secondary option (σ̂=0.630).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
1.67913 |
1.67913 |
+0.00 |
0.00 |
0.69 |
1.39828 |
1.39828 |
+0.00 |
0.00 |
0.86 |
1.15311 |
1.15311 |
+0.00 |
0.00 |
1.06 |
0.96659 |
0.96659 |
+0.00 |
0.00 |
1.31 |
0.78652 |
0.78652 |
+0.00 |
0.00 |
1.63 |
0.64253 |
0.64253 |
+0.00 |
0.00 |
2.01 |
0.53047 |
0.53047 |
+0.00 |
0.00 |
2.49 |
0.43243 |
0.43243 |
+0.00 |
0.00 |
3.08 |
0.34959 |
0.34959 |
+0.00 |
0.00 |
3.82 |
0.28335 |
0.28335 |
+0.00 |
0.00 |
4.73 |
0.23140 |
0.23140 |
+0.00 |
0.00 |
5.85 |
0.18662 |
0.18662 |
+0.00 |
0.00 |
7.24 |
0.15564 |
0.15564 |
+0.00 |
0.00 |
8.97 |
0.12849 |
0.12849 |
+0.00 |
0.00 |
11.10 |
0.10935 |
0.10935 |
+0.00 |
0.00 |
13.74 |
0.09324 |
0.09324 |
+0.00 |
0.00 |
17.00 |
0.08058 |
0.08058 |
+0.00 |
0.00 |
21.05 |
0.06795 |
0.06795 |
+0.00 |
0.00 |
26.05 |
0.05777 |
0.05777 |
+0.00 |
0.00 |
32.25 |
0.04913 |
0.04913 |
+0.00 |
0.00 |
39.92 |
0.04046 |
0.04046 |
+0.00 |
0.00 |
49.41 |
0.03593 |
0.03471 |
+3.37 |
0.09 |
61.16 |
0.02669 |
0.02587 |
+3.07 |
0.11 |
75.70 |
0.02368 |
0.02296 |
+3.04 |
0.11 |
93.71 |
0.01827 |
0.01741 |
+4.70 |
0.09 |
115.98 |
0.01312 |
0.01250 |
+4.74 |
0.10 |
143.56 |
0.01200 |
0.01133 |
+5.52 |
0.09 |
177.70 |
0.00658 |
0.00601 |
+8.64 |
0.09 |
219.95 |
0.00423 |
0.00369 |
+12.78 |
0.09 |
272.26 |
0.00403 |
0.00350 |
+12.98 |
0.09 |
M* ≥ 11.0 (z < 0.35, N = 1,619,838)
External two-point measurement: δw/w ≈ -0.1% at 30’, max |δw/w| = 11.5% at 181’.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.305). NSIDE 256 is a valid secondary option (σ̂=0.726).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
1.95256 |
1.95256 |
+0.00 |
0.00 |
0.69 |
1.59664 |
1.59664 |
+0.00 |
0.00 |
0.86 |
1.32136 |
1.32136 |
+0.00 |
0.00 |
1.06 |
1.07241 |
1.07241 |
+0.00 |
0.00 |
1.31 |
0.85831 |
0.85831 |
+0.00 |
0.00 |
1.63 |
0.68550 |
0.68550 |
+0.00 |
0.00 |
2.01 |
0.55324 |
0.55324 |
+0.00 |
0.00 |
2.49 |
0.43709 |
0.43709 |
+0.00 |
0.00 |
3.08 |
0.34639 |
0.34639 |
+0.00 |
0.00 |
3.82 |
0.27836 |
0.27836 |
+0.00 |
0.00 |
4.73 |
0.22881 |
0.22881 |
+0.00 |
0.00 |
5.85 |
0.18603 |
0.18603 |
+0.00 |
0.00 |
7.24 |
0.15573 |
0.15573 |
+0.00 |
0.00 |
8.96 |
0.13200 |
0.13200 |
+0.00 |
0.00 |
11.10 |
0.11279 |
0.11279 |
+0.00 |
0.00 |
13.74 |
0.09547 |
0.09547 |
+0.00 |
0.00 |
17.00 |
0.08340 |
0.08340 |
+0.00 |
0.00 |
21.05 |
0.07102 |
0.07102 |
+0.00 |
0.00 |
26.05 |
0.06002 |
0.06002 |
+0.00 |
0.00 |
32.24 |
0.05002 |
0.05002 |
+0.00 |
0.00 |
39.91 |
0.04213 |
0.04213 |
+0.00 |
0.00 |
49.40 |
0.03462 |
0.03324 |
+3.99 |
0.05 |
61.15 |
0.02839 |
0.02753 |
+3.03 |
0.05 |
75.70 |
0.02487 |
0.02413 |
+2.96 |
0.05 |
93.69 |
0.01556 |
0.01460 |
+6.19 |
0.05 |
115.96 |
0.01629 |
0.01563 |
+4.05 |
0.04 |
143.55 |
0.00966 |
0.00894 |
+7.50 |
0.05 |
177.68 |
0.00802 |
0.00740 |
+7.67 |
0.06 |
219.94 |
0.00301 |
0.00242 |
+19.52 |
0.15 |
272.24 |
0.00431 |
0.00374 |
+13.30 |
0.09 |
M* ≥ 11.25 (z < 0.35, N = 541,855)
External two-point measurement: δw/w ≈ +2.2% at 30’, max |δw/w| = 17.4% at 181’.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.393). NSIDE 128 is a valid secondary option (σ̂=0.633).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
2.88285 |
2.88285 |
+0.00 |
0.00 |
0.69 |
2.46602 |
2.46602 |
+0.00 |
0.00 |
0.86 |
1.95053 |
1.95053 |
+0.00 |
0.00 |
1.06 |
1.53776 |
1.53776 |
+0.00 |
0.00 |
1.31 |
1.25039 |
1.25039 |
+0.00 |
0.00 |
1.62 |
0.97024 |
0.97024 |
+0.00 |
0.00 |
2.01 |
0.80113 |
0.80113 |
+0.00 |
0.00 |
2.49 |
0.59543 |
0.59543 |
+0.00 |
0.00 |
3.08 |
0.46615 |
0.46615 |
+0.00 |
0.00 |
3.82 |
0.36119 |
0.36119 |
+0.00 |
0.00 |
4.72 |
0.29551 |
0.29551 |
+0.00 |
0.00 |
5.85 |
0.24319 |
0.24319 |
+0.00 |
0.00 |
7.24 |
0.19466 |
0.19466 |
+0.00 |
0.00 |
8.96 |
0.17246 |
0.17246 |
+0.00 |
0.00 |
11.09 |
0.14584 |
0.14584 |
+0.00 |
0.00 |
13.73 |
0.12429 |
0.12429 |
+0.00 |
0.00 |
17.00 |
0.10843 |
0.10843 |
+0.00 |
0.00 |
21.04 |
0.09061 |
0.09061 |
+0.00 |
0.00 |
26.04 |
0.07547 |
0.07547 |
+0.00 |
0.00 |
32.24 |
0.06300 |
0.06300 |
+0.00 |
0.00 |
39.90 |
0.05287 |
0.05287 |
+0.00 |
0.00 |
49.39 |
0.04264 |
0.04085 |
+4.19 |
0.09 |
61.13 |
0.03611 |
0.03510 |
+2.81 |
0.08 |
75.67 |
0.02624 |
0.02543 |
+3.07 |
0.07 |
93.67 |
0.02222 |
0.02104 |
+5.31 |
0.08 |
115.94 |
0.01504 |
0.01433 |
+4.74 |
0.07 |
143.51 |
0.01258 |
0.01176 |
+6.53 |
0.08 |
177.66 |
0.00800 |
0.00734 |
+8.25 |
0.09 |
219.89 |
0.00332 |
0.00269 |
+18.93 |
0.24 |
272.19 |
0.00490 |
0.00428 |
+12.55 |
0.14 |
M* ≥ 11.5 (z < 0.35, N = 120,882)
External two-point measurement: δw/w ≈ +0.7% at 30’, max |δw/w| = 9.7% at 97’.
Recommended: WEIGHT_COMB from NSIDE 64 analysis (σ̂_comb = 0.710). No finer valid NSIDE available (σ̂ ≥ 1 at NSIDE 128+).
θ (arcmin) |
w_obs |
w_comb (MCMC-comb) |
δw/w (%) |
Method spread (%) |
|---|---|---|---|---|
0.56 |
5.94974 |
5.94974 |
+0.00 |
0.00 |
0.69 |
4.79663 |
4.79663 |
+0.00 |
0.00 |
0.86 |
3.38167 |
3.38167 |
+0.00 |
0.00 |
1.06 |
2.83741 |
2.83741 |
+0.00 |
0.00 |
1.32 |
2.09777 |
2.09777 |
+0.00 |
0.00 |
1.62 |
1.51437 |
1.51437 |
+0.00 |
0.00 |
2.01 |
1.41378 |
1.41378 |
+0.00 |
0.00 |
2.48 |
1.01514 |
1.01514 |
+0.00 |
0.00 |
3.08 |
0.73878 |
0.73878 |
+0.00 |
0.00 |
3.81 |
0.58870 |
0.58870 |
+0.00 |
0.00 |
4.71 |
0.49227 |
0.49227 |
+0.00 |
0.00 |
5.84 |
0.38506 |
0.38506 |
+0.00 |
0.00 |
7.23 |
0.33236 |
0.33236 |
+0.00 |
0.00 |
8.96 |
0.25731 |
0.25731 |
+0.00 |
0.00 |
11.09 |
0.22303 |
0.22303 |
+0.00 |
0.00 |
13.73 |
0.18840 |
0.18840 |
+0.00 |
0.00 |
16.98 |
0.16781 |
0.16781 |
+0.00 |
0.00 |
21.03 |
0.13840 |
0.13840 |
+0.00 |
0.00 |
26.03 |
0.12239 |
0.12239 |
+0.00 |
0.00 |
32.23 |
0.08921 |
0.08921 |
+0.00 |
0.00 |
39.88 |
0.07715 |
0.07715 |
+0.00 |
0.00 |
49.38 |
0.06260 |
0.05872 |
+6.19 |
0.20 |
61.12 |
0.05113 |
0.04871 |
+4.73 |
0.28 |
75.64 |
0.03836 |
0.03642 |
+5.06 |
0.29 |
93.63 |
0.02901 |
0.02644 |
+8.86 |
0.21 |
115.88 |
0.02264 |
0.02096 |
+7.44 |
0.27 |
143.47 |
0.01488 |
0.01306 |
+12.27 |
0.23 |
177.60 |
0.01079 |
0.00928 |
+14.03 |
0.25 |
219.81 |
0.00557 |
0.00414 |
+25.70 |
0.24 |
272.11 |
0.00477 |
0.00337 |
+29.29 |
0.23 |
Caveats and limitations
Angular range: the wtheta_data.json files cover θ = 0.57–54 arcmin only. Systematic contamination at degree scales (θ > 1°) is not captured in these tables. The external δw/w measurements at 30–180’ (listed in the per-sample rows above) come from a separate analysis that covers a broader angular range.
Zero correction at θ < 40’ for NSIDE 64: the correction from NSIDE 64 weights is zero below ~40 arcmin in the measured range because the NSIDE 64 pixel scale (~54’) sets the minimum angular scale over which the weight field varies. This does not mean the systematic is zero at small scales — it means that sub-degree corrections require a finer weight map (NSIDE 128, see secondary option).
NSIDE 128 secondary option: for the five densest samples (M* 9.5–11.25), σ̂_comb < 1 at NSIDE 128, and corrections extend to θ ~ 20’. The method uncertainty remains below 0.8% in all cases. However, since all templates are at NSIDE 64, the NSIDE 128 weight field is derived from upsampled templates with no additional angular information.
ISD-3 must not be used: σ̂ >> 1 at NSIDE 32 for all samples; the corrected weights are numerically unstable. See Results: systematic weights for full ISD-3 σ̂ values.
FITS column names: the recommended weights are stored as
WEIGHT_COMB(equivalent toWEIGHT_SYS) in the*_NSIDE0064_WEIGHTS.fitsfiles.