Monroy et al. (2022) (a.k.a. 2x2pt), and the $n(z)$ from this paper. As in Porredon et al. (2022), we also include in our analysis an additional likelihood constructed with the Shear Ratio (SR) measurements (Sánchez et al. 2022). This exploits galaxy–galaxy lensing signal at small scales ($<$6 Mpc h$^{-1}$) to provide further constraint to the redshift distributions and intrinsic alignment parameters. The ratio of a galaxy–galaxy lensing signal of each lens sample redshift bin computed with respect to two source sample bins results in a primarily geometric measurement, which has been proven to be a powerful method for constraining systematics and nuisance parameters. This adds independent information from SOMPZ and WZ to the source redshift calibration. The posterior distribution obtained follows the Bayes theorem:

$$P(p|D, M) \propto \mathcal{L}(D, p, M)\Pi(p|M), \quad (19)$$

where $\Pi(p|M)$ is the prior distribution for all the parameters of the model $M$. For the cosmological inference we use the CosmoSIS pipeline (Zuntz et al. 2015), and we sample the parameter posteriors using the PolyChord sampler (Handley, Hobson & Lasenby 2015a,b).

Our data vector $D = \{w(\theta), \gamma_i(\theta)\}$ is compared to theoretical predictions $T(p) = \{w(\theta, p), \gamma_i(\theta, p)\}$ in a Bayesian fashion, and the posterior of the parameters conditional on the data is evaluated assuming a Gaussian likelihood for the data:

$$\log \mathcal{L} \propto -\frac{1}{2}(D - T(p))^T C^{-1} (D - T(p)), \quad (20)$$

where $C$ is the measurement covariance. In our analysis, we vary five (or six) cosmological parameters assuming a $\Lambda$CDM (or wCDM) cosmology: $\Omega_m, \sigma_8, n_s, \Omega_b, h_{100}$, and $w$ for the wCDM case. Moreover, we also marginalize over ‘astrophysical’ nuisance parameters (describing intrinsic alignment effects and the galaxy-matter bias of the lens sample), and calibration parameters (redshift uncertainties, shear measurement uncertainties). In short, our set-up (covariance, parameters varied, prior ranges, etc.) is the same as the one adopted in Porredon et al. (2022), except for the redshift $n(z)$ and uncertainties priors of the lens sample, where the ones obtained in this work have been assumed, and other minor changes that we describe below. All modelling and analysis choices, together with the calculations of the theoretical two-point functions, are described in detail in Krause et al. (2021).

Our analyses were not ‘blinded’, since this work occurred after the ‘unblinding’ of the DES Y3 3x2pt results. We did not perform any cosmological analysis until the redshift distributions were frozen; no changes to the redshift distributions (and uncertainties prior) have been performed after looking at the cosmological constraints. To ensure the robustness of our final estimates, we adopted a $p$-value criteria on the best-fitting models to our data vector. Following Porredon et al. (2022), we required the goodness-of-fit $p$-value on unblinded data vectors was larger than 1 per cent. The goodness-of-fit has been computed using the predictive posterior distribution (PPD, Doux et al. 2021) and adopted in the main DES Y3 3x2pt analysis. The PPD methodology derives a calibrated probability-to-exceed $p$; in the case of goodness-of-fit tests, this is achieved by drawing realizations of the data vector for parameters drawn from the posterior under study which are then compared to actual observations. The distance metric ($\chi^2$) is computed in data space, which is then used to compute the $p$-value.

Concerning the redshift uncertainties, as it is the primary goal of this work, we proceeded using the fiducial DES Y3 methodology: we parametrize the redshift uncertainties with two parameters for each tomographic bin, which modify a fiducial $n(z)$ distribution with

a shift on the mean and a stretch on the width. The fiducial $n(z)$ is estimated by averaging the SOMPZ+WZ $n(z)$ realizations. The Gaussian priors on the mean and stretch parameters are centred at the mean and width of the fiducial $n(z)$, while the Gaussian priors width are measured from the variance in the mean and width of the $n(z)$ ensemble. This parametrization can be compared directly to the fiducial DES Y3 2x2pt analysis (Porredon et al. 2022). In Appendix D we describe an alternative marginalization of the redshift uncertainties, by marginalizing over the full sets of $n(z)$ realizations provided by the SOMPZ+WZ method. In principle, this latter method describes better the redshift uncertainties of our method. However, we find that the currently available techniques that marginalize over the full ensemble of realizations during cosmology inference are prohibitively computationally expensive. Therefore, we defer its application to future work.

Besides the different $n(z)$, we also ran a few analyses where we marginalized over magnification parameters of the lens samples over wide priors. This is different from Porredon et al. (2022), where magnification parameters have been fixed.

For the fiducial DES Y3 2x2pt analysis, the $p$-value from the data-model $\chi^2$ using all six bins of MAGLIM was not sufficient to pass the 1 per cent criteria. After a series of tests the consensus was that the two highest redshift tomographic bins were responsible for worsening the fit. Therefore, the analysis in Porredon et al. (2022) included only the first four MAGLIM bins. Here, we perform the analyses using all the six bins of the MAGLIM sample, but also using only the first four bins, to verify if the same applies also to this work using different redshift distributions.

In particular, we consider the following scenarios:

(i) $\Lambda$CDM ($w$CDM); four and six lens bins, fixed magnification. This is the fiducial analysis that mirrors the one presented in Porredon et al. (2022). Five (six) cosmological parameters are varied, including $\Omega_m, \sigma_8, n_s, \Omega_b, h_{100}$ (and $w$ for the $w$CDM case). Intrinsic alignment, shear measurement, and redshift uncertainties parameters (of both lenses and sources) and galaxy-matter linear biases of the lenses also are marginalized over. The magnification coefficients of the lens sample, however, are fixed to the values estimated from Balrog (Everett et al. 2022). Uncertainties in the redshift distributions of the lens sample are modelled as a shift and stretch in the distributions.

(ii) $\Lambda$CDM ($w$CDM); four and six lens bins, free magnification. Same parameters as the ones above, but magnification parameters are marginalized over using Gaussian priors. This is an additional set-up considered only after analysing the results from the aforementioned fixed magnification set-up.

In what follows, we will also quote results in terms of the $S_8$ parameter, defined as $S_8 = \sigma_8(\Omega_m/0.3)^{0.5}$. In Table 7 we summarize the best-fitting values of $S_8, \Omega_m, \sigma_8, w$, and the computed PPD goodness-of-fit $p$-value for all the different analyses.

## 6.1 $\Lambda$CDM results

### 6.1.1 Fiducial results: four bins, fixed magnification, and comparison with DNF results

The first cosmological constraints we analyse are the ones obtained assuming a $\Lambda$CDM cosmology, using four lens bins and fixed magnification parameters. The decision on which set of results will be quoted as ‘fiducial’ for this work had to be made before conducting any cosmological analysis on data. We initially planned to only run the fiducial analyses with fixed magnification, as in Porredon et al. (2021). The choice between four or six lens bins would depend on the
<table><caption>**Table 7.** Constraints on the cosmological parameters $\Omega_m$, $S_8$, and $\sigma_8$. For each parameter we report the mean of the posterior and the 68 per cent confidence interval. We also report the PPD goodness-of-fit *p*-value and the probability of the parameter difference (computed over the full parameter space) between the analyses considered in this work and Planck TTTEEEO01 lowl lowE (Aghanim et al. 2020). The fiducial results from this work is reported in bold in the first row, while the official, fiducial results of DES Y3 are reported in bold in the second to last row.</caption><thead><tr><th>n(z)</th><th>Model</th><th>bins</th><th>Magnif.</th><th>Ω<sub>m</sub></th><th>S<sub>8</sub></th><th>σ<sub>8</sub></th><th>w</th><th>p-value</th><th>Planck</th></tr></thead><tbody><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>4</td><td>Fixed</td><td><b>0.30 ± 0.04</b></td><td><b>0.81 ± 0.04</b></td><td><b>0.81 ± 0.07</b></td><td>—</td><td><b>0.029</b></td><td><b>1.15σ</b></td></tr><tr><td>SOMPZ (broad prior)</td><td>ΛCDM</td><td>4</td><td>Fixed</td><td>0.31 ± 0.04</td><td>0.76 ± 0.06</td><td>0.76 ± 0.09</td><td>—</td><td>0.037</td><td>—</td></tr><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>4</td><td>Gauss.</td><td>0.29 ± 0.04</td><td>0.81 ± 0.04</td><td>0.83 ± 0.08</td><td>—</td><td>0.035</td><td>1.11σ</td></tr><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>6</td><td>Fixed</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0.008</td><td>—</td></tr><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>6</td><td>Gauss.</td><td>0.28 ± 0.03</td><td>0.79 ± 0.03</td><td>0.82 ± 0.06</td><td>—</td><td>0.065</td><td>2.41σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>4</td><td>Fixed</td><td>0.29 ± 0.04</td><td>0.79 ± 0.06</td><td>0.81 ± 0.08</td><td>-1.2 ± 0.3</td><td>0.032</td><td>0.46σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>4</td><td>Gauss.</td><td>0.29 ± 0.04</td><td>0.79 ± 0.06</td><td>0.81 ± 0.07</td><td>-1.0 ± 0.3</td><td>0.035</td><td>0.46σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>6</td><td>Fixed</td><td>0.30 ± 0.04</td><td>0.78 ± 0.04</td><td>0.78 ± 0.06</td><td>-0.9 ± 0.3</td><td>0.012</td><td>2.29σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>6</td><td>Gauss.</td><td>0.31 ± 0.03</td><td>0.83 ± 0.04</td><td>0.82 ± 0.05</td><td>-0.7 ± 0.2</td><td>0.059</td><td>2.21σ</td></tr><tr><td>DNF+WZ</td><td>ΛCDM</td><td>4</td><td>Fixed</td><td>0.32 ± 0.04</td><td>0.78 ± 0.04</td><td>0.76 ± 0.07</td><td>—</td><td>0.019</td><td>1.0σ</td></tr><tr><td>DNF+WZ</td><td>wCDM</td><td>4</td><td>Fixed</td><td>0.32 ± 0.05</td><td>0.78 ± 0.05</td><td>0.76 ± 0.07</td><td>-1.0 ± 0.3</td><td>0.024</td><td>—</td></tr></tbody></table>

*p*-value criteria: if the ΛCDM, six bins, fixed magnification scenario were to yield a *p*-value above the specified threshold, then we would favour that configuration. This analysis though did not fulfil our *p*-value criteria (*p*-value = 0.008, see Table 7), similarly as for the analysis ran with the same settings but using the fiducial redshift distributions from DNF; hence, we do not show those results here. We then chose as fiducial the ΛCDM, four bins, fixed magnification analysis, which is equivalent to the ‘fiducial’ set-up assumed in Porredon et al. (2021), which also allows us to compare our results directly to the ones obtained using the DNF+WZ $n(z)$. The posterior on the cosmological parameters $\Omega_m$ and $S_8$ is shown in the left panel of Fig. 8; the marginalized mean values of $S_8$, $\Omega_m$, and $\sigma_8$, along with the 68 per cent confidence intervals, are

$$ \Omega_m = 0.30 \pm 0.04, \qquad (21) $$

$$ \sigma_8 = 0.81 \pm 0.07, \qquad (22) $$

$$ S_8 = 0.81 \pm 0.04. \qquad (23) $$

The PPD goodness-of-fit test for this analysis results into *p*-value = 0.029, well above our threshold (see also Table 7). In the left panel of Fig. 8 we also compare our results with the constraints obtained using the fiducial DNF+WZ $n(z)$. The size of the posteriors is similar for the two cases, but the two posteriors are slightly shifted; the distance between the posteriors’ peaks in the 2D $\Omega_m - S_8$ plane is $d \sim 0.4\sigma$. In DES Y3 we impose a $0.3\sigma$ threshold for differences in the $\Omega_m - S_8$ plane induced by different analysis choices, as larger statistical distances would indicate the presence of systematic uncertainties unaccounted for; these results would apparently violate this criteria. We note, however, that the (arbitrary) $0.3\sigma$ threshold adopted by DES refers to differences in the $\Omega_m - S_8$ plane when noiseless theory data vectors are assumed. In the presence of noisy data vectors these differences can become larger, without invalidating our criteria. Having said this, a $d \sim 0.4\sigma$ difference none the less show the large impact a different redshift calibration of the lens sample can have on the cosmological constraints. This is somewhat different from the results obtained for the source sample $n(z)$ (Amon et al. 2022), where uncertainties in the redshift calibration had a negligible impact on the cosmological constraints.

In Section 4.2 we explained how for the combination of the two methods we marginalize over possible functional forms of the unknown galaxy-matter bias of the MAGLIM sample, by means of the systematic function Sys(s) in our clustering model. The prior on

the parameters s is inferred from the clustering autocorrelation. We tested the impact on the redshift distributions of using a broader prior (the same used in Myles et al. 2021) in Section 5. We have tested the impact of using these $n(z)$ for the cosmological inference, and found that there is no change in constraining power and no shift for $\Omega_m$, but there is a shift on $S_8$ such to overlap with the fiducial results from DNF+WZ. Therefore, it is clear that the information carried by the autocorrelation is crucial in our cosmological analysis.

### 6.1.2 Four and six bins, free magnification

As supplementary analyses, we then proceed to relax the fixed priors on the magnification parameters for the lens sample. Instead of fixing them to the values estimated from Elvin-Poole et al. (2023) (as done in the previous section), we leave them as free parameters, using Gaussian priors. In short, Elvin-Poole et al. (2023) estimate the magnification parameters using Balrog, by injecting fake galaxies into the wide field with and without applying a small magnification; the difference between the number of galaxies passing the selection in the two cases is then used to estimate the magnification parameters of the sample. These parameters come with a small uncertainty, which is however ignored in the fiducial analysis, as the magnification parameters are assumed to be fixed to the mean Balrog value. The central values and the uncertainties are reported in Table C1 in Appendix C. One of the main reasons the DES Y3 fiducial analysis did not vary the magnification parameters was merely computational, as four (or six) additional parameters lengthen the parameter inference process. In principle there is no reason to doubt these estimates. Differences might be caused by the fact that the Balrog injections do not completely sample the full DES Y3 footprint, or in case our injections were not fully representative of the DES sample we are analysing.

When varying these parameters in our analyses, we find that the *p*-value computed using PPD indicates a good fit of the model to the data not only for the four bins case, but also for six bins case (see Table 7). Adding the last two lens bins significantly improves the constraining power on $\Omega_m$ by 30 per cent compared to the four bins case, whereas the constraints on $S_8$ are 20 per cent tighter.

## 6.2 wCDM results

We then proceed to analyse the results obtained with wCDM, for all four cases: four and six bins, fixed and free magnification, as described in the previous section. Parameter posteriors are shown in
<table><caption>**Table 7.** Constraints on the cosmological parameters $\Omega_m$, $S_8$, and $\sigma_8$. For each parameter we report the mean of the posterior and the 68 per cent confidence interval. We also report the PPD goodness-of-fit *p*-value and the probability of the parameter difference (computed over the full parameter space) between the analyses considered in this work and Planck TTTEE09 lowl lowE (Aghanim et al. 2020). The fiducial results from this work is reported in bold in the first row, while the official, fiducial results of DES Y3 are reported in bold in the second to last row.</caption><thead><tr><th>n(z)</th><th>Model</th><th>bins</th><th>Magnif.</th><th>Ω<sub>m</sub></th><th>S<sub>8</sub></th><th>σ<sub>8</sub></th><th>w</th><th>p-value</th><th>Planck</th></tr></thead><tbody><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>4</td><td>Fixed</td><td><b>0.30 ± 0.04</b></td><td><b>0.81 ± 0.04</b></td><td><b>0.81 ± 0.07</b></td><td>—</td><td><b>0.029</b></td><td><b>1.15σ</b></td></tr><tr><td>SOMPZ (broad prior)</td><td>ΛCDM</td><td>4</td><td>Fixed</td><td>0.31 ± 0.04</td><td>0.76 ± 0.06</td><td>0.76 ± 0.09</td><td>—</td><td>0.037</td><td>—</td></tr><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>4</td><td>Gauss.</td><td>0.29 ± 0.04</td><td>0.81 ± 0.04</td><td>0.83 ± 0.08</td><td>—</td><td>0.035</td><td>1.11σ</td></tr><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>6</td><td>Fixed</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0.008</td><td>—</td></tr><tr><td>SOMPZ+WZ</td><td>ΛCDM</td><td>6</td><td>Gauss.</td><td>0.28 ± 0.03</td><td>0.79 ± 0.03</td><td>0.82 ± 0.06</td><td>—</td><td>0.065</td><td>2.41σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>4</td><td>Fixed</td><td>0.29 ± 0.04</td><td>0.79 ± 0.06</td><td>0.81 ± 0.08</td><td>-1.2 ± 0.3</td><td>0.032</td><td>0.46σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>4</td><td>Gauss.</td><td>0.29 ± 0.04</td><td>0.79 ± 0.06</td><td>0.81 ± 0.07</td><td>-1.0 ± 0.3</td><td>0.035</td><td>0.46σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>6</td><td>Fixed</td><td>0.30 ± 0.04</td><td>0.78 ± 0.04</td><td>0.78 ± 0.06</td><td>-0.9 ± 0.3</td><td>0.012</td><td>2.29σ</td></tr><tr><td>SOMPZ+WZ</td><td>wCDM</td><td>6</td><td>Gauss.</td><td>0.31 ± 0.03</td><td>0.83 ± 0.04</td><td>0.82 ± 0.05</td><td>-0.7 ± 0.2</td><td>0.059</td><td>2.21σ</td></tr><tr><td>DNF+WZ</td><td>ΛCDM</td><td>4</td><td>Fixed</td><td>0.32 ± 0.04</td><td>0.78 ± 0.04</td><td>0.76 ± 0.07</td><td>—</td><td>0.019</td><td>1.0σ</td></tr><tr><td>DNF+WZ</td><td>wCDM</td><td>4</td><td>Fixed</td><td>0.32 ± 0.05</td><td>0.78 ± 0.05</td><td>0.76 ± 0.07</td><td>-1.0 ± 0.3</td><td>0.024</td><td>—</td></tr></tbody></table>

*p*-value criteria: if the ΛCDM, six bins, fixed magnification scenario were to yield a *p*-value above the specified threshold, then we would favour that configuration. This analysis though did not fulfil our *p*-value criteria (*p*-value = 0.008, see Table 7), similarly as for the analysis ran with the same settings but using the fiducial redshift distributions from DNF; hence, we do not show those results here. We then chose as fiducial the ΛCDM, four bins, fixed magnification analysis, which is equivalent to the ‘fiducial’ set-up assumed in Porredon et al. (2021), which also allows us to compare our results directly to the ones obtained using the DNF+WZ $n(z)$. The posterior on the cosmological parameters $\Omega_m$ and $S_8$ is shown in the left panel of Fig. 8; the marginalized mean values of $S_8$, $\Omega_m$, and $\sigma_8$, along with the 68 per cent confidence intervals, are

$$ \Omega_m = 0.30 \pm 0.04, \qquad (21) $$

$$ \sigma_8 = 0.81 \pm 0.07, \qquad (22) $$

$$ S_8 = 0.81 \pm 0.04. \qquad (23) $$

The PPD goodness-of-fit test for this analysis results into *p*-value = 0.029, well above our threshold (see also Table 7). In the left panel of Fig. 8 we also compare our results with the constraints obtained using the fiducial DNF+WZ $n(z)$. The size of the posteriors is similar for the two cases, but the two posteriors are slightly shifted; the distance between the posteriors’ peaks in the 2D $\Omega_m - S_8$ plane is $d \sim 0.4\sigma$. In DES Y3 we impose a $0.3\sigma$ threshold for differences in the $\Omega_m - S_8$ plane induced by different analysis choices, as larger statistical distances would indicate the presence of systematic uncertainties unaccounted for; these results would apparently violate this criteria. We note, however, that the (arbitrary) $0.3\sigma$ threshold adopted by DES refers to differences in the $\Omega_m - S_8$ plane when noiseless theory data vectors are assumed. In the presence of noisy data vectors these differences can become larger, without invalidating our criteria. Having said this, a $d \sim 0.4\sigma$ difference none the less show the large impact a different redshift calibration of the lens sample can have on the cosmological constraints. This is somewhat different from the results obtained for the source sample $n(z)$ (Amon et al. 2022), where uncertainties in the redshift calibration had a negligible impact on the cosmological constraints.

In Section 4.2 we explained how for the combination of the two methods we marginalize over possible functional forms of the unknown galaxy-matter bias of the MAGLIM sample, by means of the systematic function Sys(s) in our clustering model. The prior on

the parameters s is inferred from the clustering autocorrelation. We tested the impact on the redshift distributions of using a broader prior (the same used in Myles et al. 2021) in Section 5. We have tested the impact of using these $n(z)$ for the cosmological inference, and found that there is no change in constraining power and no shift for $\Omega_m$, but there is a shift on $S_8$ such to overlap with the fiducial results from DNF+WZ. Therefore, it is clear that the information carried by the autocorrelation is crucial in our cosmological analysis.

### 6.1.2 Four and six bins, free magnification

As supplementary analyses, we then proceed to relax the fixed priors on the magnification parameters for the lens sample. Instead of fixing them to the values estimated from Elvin-Poole et al. (2023) (as done in the previous section), we leave them as free parameters, using Gaussian priors. In short, Elvin-Poole et al. (2023) estimate the magnification parameters using Balrog, by injecting fake galaxies into the wide field with and without applying a small magnification; the difference between the number of galaxies passing the selection in the two cases is then used to estimate the magnification parameters of the sample. These parameters come with a small uncertainty, which is however ignored in the fiducial analysis, as the magnification parameters are assumed to be fixed to the mean Balrog value. The central values and the uncertainties are reported in Table C1 in Appendix C. One of the main reasons the DES Y3 fiducial analysis did not vary the magnification parameters was merely computational, as four (or six) additional parameters lengthen the parameter inference process. In principle there is no reason to doubt these estimates. Differences might be caused by the fact that the Balrog injections do not completely sample the full DES Y3 footprint, or in case our injections were not fully representative of the DES sample we are analysing.

When varying these parameters in our analyses, we find that the *p*-value computed using PPD indicates a good fit of the model to the data not only for the four bins case, but also for six bins case (see Table 7). Adding the last two lens bins significantly improves the constraining power on $\Omega_m$ by 30 per cent compared to the four bins case, whereas the constraints on $S_8$ are 20 per cent tighter.

## 6.2 wCDM results

We then proceed to analyse the results obtained with wCDM, for all four cases: four and six bins, fixed and free magnification, as described in the previous section. Parameter posteriors are shown in
<figure><img src="image_8.png" /><figcaption>**Figure 8.** Left panel: posterior distributions of the cosmological parameters $\Omega_m$ and $S_8$ for the $\Lambda$CDM analysis involving four bins and fixed magnification parameters. The ‘fiducial’ posteriors have been obtained using the DNF+WZ redshift distributions, and they are compared to the ones obtained using the SOMPZ+WZ redshift distributions. Right panel: posterior distributions of the cosmological parameters $\Omega_m$ and $S_8$ for the $\Lambda$CDM analysis for three different cases: (1) four bins and fixed magnification parameters (the blue contours in the two plots share the same analysis choices); (2) four bins and marginalized over magnification parameters (in solid green); (3) six bins and marginalizing over magnification parameters (in solid red). The 2D marginalized contours in both of these figures show the 68 per cent and 95 per cent confidence levels.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 9.** Posterior distributions of the cosmological parameters $\Omega_m$ and $S_8$ and $w$ for four different cases: (1) wCDM, four bins and fixed magnification parameters; (2) wCDM, six bins and fixed magnification parameters, (3) wCDM, four bins and free magnification parameters; (4) wCDM, six bins and free magnification parameters. The 2D marginalized contours in these figures show the 68 per cent and 95 per cent confidence levels. We note that the posteriors of $w$ for the six bins cases are partially affected by the prior edge ($w \in [-2, -0.33]$, Table C1); see the text for more details.</figcaption></figure>

Fig. 9, whereas $p$-values and parameters constraints are reported in Table 7. All the reported $p$-values are above our $p = 0.01$ threshold.

In general, the 2x2pt constraints on $w$ are loose and affected by the prior ($-2 < w < -0.3$), but compatible with a $\Lambda$CDM scenario. With respect to $\Lambda$CDM four bins case, freeing $w$ loosens the constraint on $S_8$ (both with fixed and with free magnification) by $\sim 30$ per cent, while leaves it unvaried for $\Omega_m$. For the six bins, we are unable to directly compare to the fixed magnification case, but for the free magnification the constraint on $S_8$ is $\sim 25$ per cent looser, while, similarly to the four bins case, it is unvaried for $\Omega_m$.

Passing from the four bins to the six bins configuration, besides increasing the constraints on $S_8$, also the constraints on $w$ improves (by $\sim 20$ per cent), although part of the improvement is due to the posterior partially hitting the prior edge.

Freeing the magnification parameters slightly shifts $w$ towards the upper edge of the prior ($w = -0.3$), and $S_8$ slightly towards higher values, due to a degeneracy between $w$, $S_8$, and the magnification parameters of the two highest lens bins, which are now fairly broad (see Table C1). Such a shift is not present in the case of four bins, as the Gaussian priors used for the first four magnification parameters are much tighter.

## 6.3 Statistical distance to Planck

We compute here the statistical distances between our cosmological constraints and the early Universe ones from the Planck satellite (Aghanim et al. 2020). To this aim, we used the algorithm presented in Raveri & Doux (2021), which estimates the probability of tension between parameters via Monte Carlo approximation. In particular, the probability of tension between parameters can be expressed as follows:

$$ P(\Delta\theta) = \int_{V_p} P_A(\theta)P_B(\theta - \Delta\theta)d\theta, \qquad (24) $$
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Left panel: posterior distributions of the cosmological parameters $\Omega_m$ and $S_8$ for the $\Lambda$CDM analysis involving four bins and fixed magnification parameters. The ‘fiducial’ posteriors have been obtained using the DNF+WZ redshift distributions, and they are compared to the ones obtained using the SOMPZ+WZ redshift distributions. Right panel: posterior distributions of the cosmological parameters $\Omega_m$ and $S_8$ for the $\Lambda$CDM analysis for three different cases: (1) four bins and fixed magnification parameters (the blue contours in the two plots share the same analysis choices); (2) four bins and marginalized over magnification parameters (in solid green); (3) six bins and marginalizing over magnification parameters (in solid red). The 2D marginalized contours in both of these figures show the 68 per cent and 95 per cent confidence levels.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Posterior distributions of the cosmological parameters $\Omega_m$ and $S_8$ and $w$ for four different cases: (1) wCDM, four bins and fixed magnification parameters; (2) wCDM, six bins and fixed magnification parameters, (3) wCDM, four bins and free magnification parameters; (4) wCDM, six bins and free magnification parameters. The 2D marginalized contours in these figures show the 68 per cent and 95 per cent confidence levels. We note that the posteriors of $w$ for the six bins cases are partially affected by the prior edge ($w \in [-2, -0.33]$, Table C1); see the text for more details.</figcaption></figure>

Fig. 9, whereas $p$-values and parameters constraints are reported in Table 7. All the reported $p$-values are above our $p = 0.01$ threshold.

In general, the 2x2pt constraints on $w$ are loose and affected by the prior ($-2 < w < -0.3$), but compatible with a $\Lambda$CDM scenario. With respect to $\Lambda$CDM four bins case, freeing $w$ loosens the constraint on $S_8$ (both with fixed and with free magnification) by $\sim 30$ per cent, while leaves it unvaried for $\Omega_m$. For the six bins, we are unable to directly compare to the fixed magnification case, but for the free magnification the constraint on $S_8$ is $\sim 25$ per cent looser, while, similarly to the four bins case, it is unvaried for $\Omega_m$.

Passing from the four bins to the six bins configuration, besides increasing the constraints on $S_8$, also the constraints on $w$ improves (by $\sim 20$ per cent), although part of the improvement is due to the posterior partially hitting the prior edge.

Freeing the magnification parameters slightly shifts $w$ towards the upper edge of the prior ($w = -0.3$), and $S_8$ slightly towards higher values, due to a degeneracy between $w$, $S_8$, and the magnification parameters of the two highest lens bins, which are now fairly broad (see Table C1). Such a shift is not present in the case of four bins, as the Gaussian priors used for the first four magnification parameters are much tighter.

## 6.3 Statistical distance to Planck

We compute here the statistical distances between our cosmological constraints and the early Universe ones from the Planck satellite (Aghanim et al. 2020). To this aim, we used the algorithm presented in Raveri & Doux (2021), which estimates the probability of tension between parameters via Monte Carlo approximation. In particular, the probability of tension between parameters can be expressed as follows:

$$ P(\Delta\theta) = \int_{V_p} P_A(\theta)P_B(\theta - \Delta\theta)d\theta, \qquad (24) $$
where $V_p$ represents the prior volume, while $P_A$ and $P_B$ represent two posterior parameter distributions under study. The probability of having a shift in the parameter space is described by the parameter shifts density:

$$ \Delta = \int_{P(\Delta\theta) > P(0)} P(\Delta\theta) d\Delta\theta. \qquad (25) $$

This refers to the posterior portion beyond the constant probability contour for no shift, $\Delta\theta = 0$. The integration in equation (25) is performed via Monte Carlo techniques.

The comparison between the results has been performed considering all the parameters shared by our analyses and Planck. The values are reported in the last column of Table 7; we find no sign of significant tension ($<3\sigma$) in any of the analysis set-ups considered. In particular, we find that for the four bins case for $\Lambda$CDM (both fixed and free magnification) there is good agreement (1.15$\sigma$, 1.11$\sigma$), similarly for wCDM with four bins we have 0.46$\sigma$ for both fixed and free magnification. For the six bins cases the values are larger (2.2–2.4$\sigma$), but still below the 3$\sigma$ threshold.

## 7 CONCLUSIONS

In this paper, we presented an alternative calibration of the MAGLIM lens sample redshift distributions from the DES first 3 yr of data (Y3). This new method, which has already been applied to the DES Y3 weak lensing sample (Myles et al. 2021), is based on a combination of an SOMPZ and clustering redshifts (WZ) to estimate redshift distributions and inherent uncertainties. The original redshift calibration of the MAGLIM sample (and cosmological results obtained adopting that calibration) have been originally presented in Porredon et al. (2022), and has been based on the photo-z code DNF (De Vicente et al. 2016) and WZ constraints (Cawthon et al. 2022). The methodology presented in this paper is meant to be more accurate than the original one. First, the SOMPZ method allows a better control over all the potential sources of uncertainties affecting the estimates compared to DNF; second, the clustering constraints (WZ) are incorporated through a rigorous joint likelihood framework which allows to draw $n(z)$ samples conditioned on both clustering and photometric measurements, improving the $n(z)$ estimates (e.g. the final ‘SOMPZ+WZ’ $n(z)$ have a smaller scatter, or uncertainty, compared to the SOMPZ ones, see Fig. 5).

We described in detail the methodology followed to produce the alternative MAGLIM $n(z)$ based on the SOMPZ+WZ approach, together with a detailed report on the main systematics dominating our calibration error budget. Our redshift uncertainties, in particular, are dominated by the impact of sample variance on the SOMPZ estimate (due to the limited area spanned by the deep field sample used in the calibration) and by the effect of the redshift evolution of the galaxy-matter bias of the MAGLIM sample on the WZ constraints. We then compared our SOMPZ+WZ $n(z)$ with the fiducial DNF+WZ $n(z)$ estimates; the means and widths of the 6 MAGLIM tomographic bins show moderate statistical distances, with the largest deviation of 2.7$\sigma$ in bin 2 (see Table 6). We also found the uncertainties on mean of the redshift distributions of the SOMPZ+WZ method to be slightly larger than the ones of the DNF+WZ method, due to a more conservative calibration of the tails of the redshift distributions. On the other hand, we found the two methods to have a similar constraining power on the widths of the distributions.

We then proceeded investigating the impact on the cosmological constraints of our new redshift calibration. In particular, we used the DES Y3 galaxy–galaxy lensing and galaxy clustering measurements (Prat et al. 2022; Rodríguez-Monroy et al. 2022) (a.k.a. 2x2pt), and

the $n(z)$ from this work, and compared to the results from Porredon et al. (2022). In the ‘fiducial’ configuration, which involves using the first four lens bins and assuming a $\Lambda$CDM cosmology, we obtained as marginalized mean values $\Omega_m = 0.30 \pm 0.04$, $\sigma_8 = 0.81 \pm 0.07$, and $S_8 = 0.81 \pm 0.04$. We noted a $\sim 0.4\sigma$ shift in the $\Omega - S_8$ plane compared to the Porredon et al. (2022) results, but no change in terms of constraining power. The shift indicates that the redshift calibration of the lens sample plays a key role on cosmological constraints from the 2x2pt analysis, contrary to the redshift calibration of the source sample (Amon et al. 2022). Subsequently, we explored different analysis set-ups; we tested the case where all the six MAGLIM redshift bins were included, a scenario where the magnification coefficients of the lens sample were marginalized during the inference, and last, we assumed a wCDM cosmology. We found that the inclusion of the last two redshift bins of the MAGLIM sample help improving the constraints on $\Omega_m$ by $\sim 25$ per cent, and on $S_8$ by $\sim 20$ per cent.

We also compared our results to the cosmological constraints from Planck (Aghanim et al. 2020), finding a no-tension of 1.15$\sigma$ between the results when four lens bins were considered. We did find a statistical distance of 2.41$\sigma$ in $\Lambda$CDM with free magnification coefficients when including in the analysis the two high-redshift bins ($z > 0.85$), which have not been included in the fiducial DES Y3 analysis (Porredon et al. 2022).

As a final comment, despite the SOMPZ+WZ method’s ability to produce $n(z)$ samples capturing the redshift uncertainties of our estimates, we could not efficiently marginalize over these realization during the cosmological inference, due to computational constraints. Our marginalization strategy followed the one adopted in Porredon et al. (2022): we adopted the mean of the SOMPZ+WZ samples as our fiducial $n(z)$, and marginalized over a shift in the mean and a stretch of the width of the distribution, using as priors the variances in the mean and widths of the SOMPZ+WZ $n(z)$ samples. While this strategy was deemed sufficient for this work, we plan to implement the full marginalization scheme for subsequent analyses of the lens samples with DES Y6 data.

## ACKNOWLEDGEMENTS

The project leading to these results has received funding from ‘la Caixa’ Foundation (ID 100010434), under the fellowship LCF/BQ/DI17/11620053 and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 713673.

Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey.

The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College
where $V_p$ represents the prior volume, while $P_A$ and $P_B$ represent two posterior parameter distributions under study. The probability of having a shift in the parameter space is described by the parameter shifts density:

$$ \Delta = \int_{P(\Delta\theta) > P(0)} P(\Delta\theta) d\Delta\theta. \qquad (25) $$

This refers to the posterior portion beyond the constant probability contour for no shift, $\Delta\theta = 0$. The integration in equation (25) is performed via Monte Carlo techniques.

The comparison between the results has been performed considering all the parameters shared by our analyses and Planck. The values are reported in the last column of Table 7; we find no sign of significant tension ($<3\sigma$) in any of the analysis set-ups considered. In particular, we find that for the four bins case for $\Lambda$CDM (both fixed and free magnification) there is good agreement ($1.15\sigma$, $1.11\sigma$), similarly for wCDM with four bins we have $0.46\sigma$ for both fixed and free magnification. For the six bins cases the values are larger ($2.2-2.4\sigma$), but still below the $3\sigma$ threshold.

## 7 CONCLUSIONS

In this paper, we presented an alternative calibration of the MAGLIM lens sample redshift distributions from the DES first 3 yr of data (Y3). This new method, which has already been applied to the DES Y3 weak lensing sample (Myles et al. 2021), is based on a combination of an SOMPZ and clustering redshifts (WZ) to estimate redshift distributions and inherent uncertainties. The original redshift calibration of the MAGLIM sample (and cosmological results obtained adopting that calibration) have been originally presented in Porredon et al. (2022), and has been based on the photo-z code DNF (De Vicente et al. 2016) and WZ constraints (Cawthon et al. 2022). The methodology presented in this paper is meant to be more accurate than the original one. First, the SOMPZ method allows a better control over all the potential sources of uncertainties affecting the estimates compared to DNF; second, the clustering constraints (WZ) are incorporated through a rigorous joint likelihood framework which allows to draw $n(z)$ samples conditioned on both clustering and photometric measurements, improving the $n(z)$ estimates (e.g. the final 'SOMPZ+WZ' $n(z)$ have a smaller scatter, or uncertainty, compared to the SOMPZ ones, see Fig. 5).

We described in detail the methodology followed to produce the alternative MAGLIM $n(z)$ based on the SOMPZ+WZ approach, together with a detailed report on the main systematics dominating our calibration error budget. Our redshift uncertainties, in particular, are dominated by the impact of sample variance on the SOMPZ estimate (due to the limited area spanned by the deep field sample used in the calibration) and by the effect of the redshift evolution of the galaxy-matter bias of the MAGLIM sample on the WZ constraints. We then compared our SOMPZ+WZ $n(z)$ with the fiducial DNF+WZ $n(z)$ estimates; the means and widths of the 6 MAGLIM tomographic bins show moderate statistical distances, with the largest deviation of $2.7\sigma$ in bin 2 (see Table 6). We also found the uncertainties on mean of the redshift distributions of the SOMPZ+WZ method to be slightly larger than the ones of the DNF+WZ method, due to a more conservative calibration of the tails of the redshift distributions. On the other hand, we found the two methods to have a similar constraining power on the widths of the distributions.

We then proceeded investigating the impact on the cosmological constraints of our new redshift calibration. In particular, we used the DES Y3 galaxy-galaxy lensing and galaxy clustering measurements (Prat et al. 2022; Rodríguez-Monroy et al. 2022) (a.k.a. 2x2pt), and

the $n(z)$ from this work, and compared to the results from Porredon et al. (2022). In the 'fiducial' configuration, which involves using the first four lens bins and assuming a $\Lambda$CDM cosmology, we obtained as marginalized mean values $\Omega_m = 0.30 \pm 0.04$, $\sigma_8 = 0.81 \pm 0.07$, and $S_8 = 0.81 \pm 0.04$. We noted a $\sim 0.4\sigma$ shift in the $\Omega - S_8$ plane compared to the Porredon et al. (2022) results, but no change in terms of constraining power. The shift indicates that the redshift calibration of the lens sample plays a key role on cosmological constraints from the 2x2pt analysis, contrary to the redshift calibration of the source sample (Amon et al. 2022). Subsequently, we explored different analysis set-ups; we tested the case where all the six MAGLIM redshift bins were included, a scenario where the magnification coefficients of the lens sample were marginalized during the inference, and last, we assumed a wCDM cosmology. We found that the inclusion of the last two redshift bins of the MAGLIM sample help improving the constraints on $\Omega_m$ by $\sim 25$ per cent, and on $S_8$ by $\sim 20$ per cent.

We also compared our results to the cosmological constraints from Planck (Aghanim et al. 2020), finding a no-tension of $1.15\sigma$ between the results when four lens bins were considered. We did find a statistical distance of $2.41\sigma$ in $\Lambda$CDM with free magnification coefficients when including in the analysis the two high-redshift bins ($z > 0.85$), which have not been included in the fiducial DES Y3 analysis (Porredon et al. 2022).

As a final comment, despite the SOMPZ+WZ method's ability to produce $n(z)$ samples capturing the redshift uncertainties of our estimates, we could not efficiently marginalize over these realization during the cosmological inference, due to computational constraints. Our marginalization strategy followed the one adopted in Porredon et al. (2022): we adopted the mean of the SOMPZ+WZ samples as our fiducial $n(z)$, and marginalized over a shift in the mean and a stretch of the width of the distribution, using as priors the variances in the mean and widths of the SOMPZ+WZ $n(z)$ samples. While this strategy was deemed sufficient for this work, we plan to implement the full marginalization scheme for subsequent analyses of the lens samples with DES Y6 data.

## ACKNOWLEDGEMENTS

The project leading to these results has received funding from 'la Caixa' Foundation (ID 100010434), under the fellowship LCF/BQ/DI17/11620053 and has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 713673.

Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey.

The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College
London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium.

Based in part on observations at Cerro Tololo Inter-American Observatory at NSF's NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.

The DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MINECO under grants AYA2015-71825, ESP2015-66861, FPA2015-68048, SEV-2016-0588, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We acknowledge support from the Brazilian Instituto Nacional de Ciência e Tecnologia (INCT) e-Universe (CNPq grant 465376/2014-2).

This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics| Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatorio Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

## 8 DATA AVAILABILITY

The DES Y3 data products used in this work, as well as the full ensemble of DES Y3 MAGLIM sample redshift distributions described by this work, are publicly available at https://des.ncsa.illinois.edu/releases. As cosmology likelihood sampling software we use cosmosis, available at https://github.com/joezuntz/cosmosis.

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London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium.

Based in part on observations at Cerro Tololo Inter-American Observatory at NSF's NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.

The DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MINECO under grants AYA2015-71825, ESP2015-66861, FPA2015-68048, SEV-2016-0588, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We acknowledge support from the Brazilian Instituto Nacional de Ciência e Tecnologia (INCT) e-Universe (CNPq grant 465376/2014-2).

This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics| Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatorio Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

## 8 DATA AVAILABILITY

The DES Y3 data products used in this work, as well as the full ensemble of DES Y3 MAGLIM sample redshift distributions described by this work, are publicly available at https://des.ncsa.illinois.edu/releases. As cosmology likelihood sampling software we use cosmosis, available at https://github.com/joezuntz/cosmosis.

## REFERENCES

* Abbott T. M. C. et al., 2018, ApJS, 239, 18

* Abolfathi B. et al., 2018, ApJS, 235, 42

* Aghanim N. et al., 2020, A&A, 641, A6

* Ahumada R. et al., 2020, ApJS, 249, 3

* Aihara H. et al., 2018, PASJ, 70, S4

* Alam S. et al., 2021, MNRAS, 504, 4667

* Alarcon A. et al., 2021, MNRAS, 501, 6103

* Amon A. et al., 2022, Phys. Rev. D, 105, 023514

* Becker M. R., 2013, MNRAS, 435, 115

* Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013, ApJ, 762, 109

* Benjamin J. et al., 2013, MNRAS, 431, 1547

* Blanton M. R. et al., 2017, AJ, 154, 28

* Bonnett C. et al., 2016, Phys. Rev. D, 94, 042005

* Buchs R. et al., 2019, MNRAS, 489, 820

* Cawthon R. et al., 2022, MNRAS, 513, 5517

* Conroy C., Wechsler R. H., Kravtsov A. V., 2006, ApJ, 647, 201

* Cordero J. P. et al., 2022, MNRAS, 511, 2170

* Cunha C. E., Huterer D., Busha M. T., Wechsler R. H., 2012, MNRAS, 423, 909

* Davis M., Peebles P. J. E., 1983, ApJ, 267, 465

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## APPENDIX A: MAGLIM SAMPLE IN SIMULATIONS

Due to the small but existing differences in magnitude/colour space between the Buzzard simulation and the DES data (DeRose et al. 2019), we expect the simulated sample to not be a perfect copy of the data sample, although we do not expect this to have a sensible impact on any of the conclusions drawn in this work. The direct application of the fiducial MAGLIM selection (equation 1) to the Buzzard catalogue leads to slightly different number densities and colour distributions with respect to data. We therefore redefine a more adequate MAGLIM selection for Buzzard, with the goal of achieving the same number density as the data sample. The new Buzzard MAGLIM selection is a piece-wise linear selection in redshift and magnitude, similar to equation (1) but with coefficients redefined by minimizing the quadratic sum of the difference in number density with the values in data, for each tomographic bin, in order to avoid discontinuities in the selection. Such a redefined selection guarantees similar number densities as the data sample. We then ensure similar colour distributions by an additional reweighting procedure of the mock catalogue, so as to resemble the colour distributions of the data sample. In particular, we iteratively reweight based on *i*, *r* magnitudes and *i*-*r* colours, with the final distributions matching closely the data ones, as shown in Fig. A1.

The new MAGLIM selection in Buzzard for each tomographic bin is then the following:

(i) Bin 1: $i < 2.017 *z_{\text{mean}} + 18.882$

(ii) Bin 2: $i < 2.687 *z_{\text{mean}} + 18.614$

(iii) Bin 3: $i < 5.705 *z_{\text{mean}} + 16.954$

(iv) Bin 4: $i < 2.399 *z_{\text{mean}} + 19.268$

(v) Bin 5: $i < 9.455 *z_{\text{mean}} + 13.271$

(vi) Bin 6: $i < -0.960 *z_{\text{mean}} + 23.165$

We list in Table A1 the number densities of MAGLIM in Buzzard, obtained with the fiducial selection and with the adapted in simulations.
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Lehmann B. V., Mao Y.-Y., Becker M. R., Skillman S. W., Wechsler R. H., 2017, ApJ, 834, 37

Lidman C. et al., 2020, MNRAS, 496, 19

Lilly S. J. et al., 2009, ApJS, 184, 218

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Lupton R. H., Gunn J. E., Szalay A. S., 1999, AJ, 118, 1406

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Masters D. C. et al., 2019, ApJ, 877, 81

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Ménard B., Scranton R., Schmidt S., Morrison C., Jeong D., Budavari T., Rahman M., 2013, preprint (arXiv:1303.4722)

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Morrison C. B., Hildebrandt H., Schmidt S. J., Baldry I. K., Bilicki M., Choi A., Erben T., Schneider P., 2017, MNRAS, 467, 3576

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Scodeggio M. et al., 2018, A&A, 609, A84

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Zuntz J. et al., 2015, Astron. Comput., 12, 45

## APPENDIX A: MAGLIM SAMPLE IN SIMULATIONS

Due to the small but existing differences in magnitude/colour space between the Buzzard simulation and the DES data (DeRose et al. 2019), we expect the simulated sample to not be a perfect copy of the data sample, although we do not expect this to have a sensible impact on any of the conclusions drawn in this work. The direct application of the fiducial MAGLIM selection (equation 1) to the Buzzard catalogue leads to slightly different number densities and colour distributions with respect to data. We therefore redefine a more adequate MAGLIM selection for Buzzard, with the goal of achieving the same number density as the data sample. The new Buzzard MAGLIM selection is a piece-wise linear selection in redshift and magnitude, similar to equation (1) but with coefficients redefined by minimizing the quadratic sum of the difference in number density with the values in data, for each tomographic bin, in order to avoid discontinuities in the selection. Such a redefined selection guarantees similar number densities as the data sample. We then ensure similar colour distributions by an additional reweighting procedure of the mock catalogue, so as to resemble the colour distributions of the data sample. In particular, we iteratively reweight based on *i*, *r* magnitudes and *i*-*r* colours, with the final distributions matching closely the data ones, as shown in Fig. A1.

The new MAGLIM selection in Buzzard for each tomographic bin is then the following:

(i) Bin 1: $i < 2.017 *z_{\text{mean}} + 18.882$

(ii) Bin 2: $i < 2.687 *z_{\text{mean}} + 18.614$

(iii) Bin 3: $i < 5.705 *z_{\text{mean}} + 16.954$

(iv) Bin 4: $i < 2.399 *z_{\text{mean}} + 19.268$

(v) Bin 5: $i < 9.455 *z_{\text{mean}} + 13.271$

(vi) Bin 6: $i < -0.960 *z_{\text{mean}} + 23.165$

We list in Table A1 the number densities of MAGLIM in Buzzard, obtained with the fiducial selection and with the adapted in simulations.
<figure><img src="image_10.png" /><figcaption>**Figure A1.** Comparison of riz-band magnitudes and r-i, z-i colours of the six bins of the MAGLIM sample, between data (blue) and simulations, before (green) and after reweighting (red). The re-weighting process has proven successful in yielding magnitude distributions that closely resemble those observed in the actual data.</figcaption></figure>

<table><caption>**Table A1.** Number densities of the MAGLIM sample in Buzzard as obtained with the fiducial MAGLIM selection, and with the one adapted for Buzzard.</caption><thead><tr><th>Number density</th><th>Bin 1</th><th>Bin 2</th><th>Bin 3</th><th>Bin 4</th><th>Bin 5</th><th>Bin 6</th></tr></thead><tbody><tr><td>Before (Fiducial MAGLIM selection)</td><td>1.10</td><td>0.90</td><td>1.12</td><td>0.97</td><td>0.69</td><td>0.76</td></tr><tr><td>After (Buzzard MAGLIM selection)</td><td>0.98</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.98</td></tr></tbody></table>
<figure><img src="image_12.png" /><figcaption>**Figure A1.** Comparison of riz-band magnitudes and *r* − *i*, *z* − *i* colours of the six bins of the MAGLIM sample, between data (blue) and simulations, before (green) and after reweighting (red). The re-weighting process has proven successful in yielding magnitude distributions that closely resemble those observed in the actual data.</figcaption></figure>

<table><caption>**Table A1.** Number densities of the MAGLIM sample in Buzzard as obtained with the fiducial MAGLIM selection, and with the one adapted for Buzzard.</caption><thead><tr><th>Number density</th><th>Bin 1</th><th>Bin 2</th><th>Bin 3</th><th>Bin 4</th><th>Bin 5</th><th>Bin 6</th></tr></thead><tbody><tr><td>Before (Fiducial MAGLIM selection)</td><td>1.10</td><td>0.90</td><td>1.12</td><td>0.97</td><td>0.69</td><td>0.76</td></tr><tr><td>After (Buzzard MAGLIM selection)</td><td>0.98</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.98</td></tr></tbody></table>
## APPENDIX B: VALIDATION IN SIMULATIONS

The validity of our methodology and pipeline has been tested in the Buzzard N-body simulation, introduced in Section 2.8. The measurements of redshift distributions using both phenotypes and clustering were validated in simulations to ensure unbiased estimates with respect to the true redshift distributions. The MAGLIM sample has been recreated in the Buzzard simulations, as described in Section 2. The sample selection has been altered to reproduce as faithfully as possible the number density and colour distributions of the data.

<figure><img src="image_11.png" /><figcaption>**Figure B1.** Estimated $n(z)$ in four tomographic bins using a 12 × 12 cell deep SOM and 32 × 32 cell wide SOM trained on Buzzard simulations. In the top row we have bins 1 and 4, in the middle row bins 2 and 5, and in the bottom row bins 3 and 6. The redshift sample used here has 100 000 galaxies drawn from 1.38 deg$^2$, such that after the MAGLIM selection it yields ~15 000 unique galaxies, which is the same order of magnitude as the redshift samples in data, see Table 2. The deep sample is drawn from three fields of size 3.32, 3.29, and 1.94 deg$^2$, respectively, from the Buzzard simulated sky catalogue. The black dashed line marks the true value, the transparent bands are the 3sDir set of $n(z)$, and the solid bands are the realizations once combined with clustering redshifts. We can appreciate the effect of the combined likelihood, resulting in distributions more constrained in terms of shape, and still consistent with the truth.</figcaption></figure>

As described in Section 4.1.1, we generated 300 simulated deep field realizations that we used to estimate the SOMPZ method uncertainty, which we report in Table 4, and add into our overall error budget. Here, we illustrate that the uncertainty predicted by the 3sDir and the 3sDir + WZ models is consistent with the true $n(z)$ in one of these simulated realizations. We start by selecting one of these simulated realizations, which includes the four deep fields and their corresponding Balrog and redshift samples. We then proceeded to perform the 3sDir analytical sample variance estimation for that one specific realization. The geometry and resolution of the SOM used in simulations are the same as the ones used in data. There are two differences between our simulated scenario and real data: (1) we use the true redshifts from the Buzzard simulations; (2) we assume all redshift information comes from one of our four deep fields. This latter point matches the modelling assumption of 3sDir, which also assumes that the redshift information only comes from one out of four fields. This is a conservative choice that inflates the modelled error due to sample variance in real data for the term $p(z|c)$, and it avoids modelling the highly non-trivial selection function of spectroscopic samples coming from fields other than the COSMOS field. We note that the sample variance contribution to the colour distribution $p(c)$ is modelled correctly as coming from all four fields. The SOMPZ redshift distributions, and their uncertainties estimated through the 3sDir method, are in agreement with the true distribution, as shown in Fig. B1. In Table B1 we summarize the mean and width of the simulated $n(z)$ of the SOMPZ and SOMPZ+WZ methods in each tomographic bin, and of the true $n(z)$, together with the respective statistical distances from the truth.

We also repeated in simulations the same procedure as for data also for the WZ estimates. We created a mock BOSS/eBOSS catalogue to use as a reference sample. As in data, also in simulations the BOSS/eBOSS sample is divided into 50 bins spanning the 0.1 < z < 1.1 range of the catalogue (width $\Delta z = 0.02$). Before proceeding with combining the SOMPZ and WZ information through the combined likelihood, the compatibility between SOMPZ and WZ was checked. This was tested by inferring the windowed means and widths of the WZ and SOMPZ redshift estimates, following Gatti et al. (2022). The window has been determined such that magnification effects related to the WZ measurements can be neglected. As for WZ, we used a ‘simple’ estimator for the redshift distribution, inverting equation (17) and ignoring magnification effects (this is possible as we are considering only windowed quantities). The means and widths computed in this way for the two methods were compatible

<table><caption>**Table B1.** SIMULATIONS: Summary of values for centre values for mean (top panel) and width (bottom panel) for the $n(z)$ distributions as measured in the Buzzard simulations. The values related to SOMPZ and SOMPZ+WZ refer to Fig. B1. Note that the uncertainties quoted here only include sample variance and shot noise.</caption><thead><tr><th></th><th></th><th>Bin 1<br>z &isin; [0.2, 0.4]</th><th>Bin 2<br>z &isin; [0.4, 0.55]</th><th>Bin 3<br>z &isin; [0.55, 0.7]</th><th>Bin 4<br>z &isin; [0.7, 0.85]</th><th>Bin 5<br>z &isin; [0.85, 0.95]</th><th>Bin 6<br>z &isin; [0.95, 1.05]</th></tr></thead><tbody><tr><td rowspan="2">&lt;z&gt;</td><td>SOMPZ</td><td>0.319 &plusmn; 0.009</td><td>0.484 &plusmn; 0.007</td><td>0.623 &plusmn; 0.006</td><td>0.784 &plusmn; 0.006</td><td>0.891 &plusmn; 0.007</td><td>0.993 &plusmn; 0.010</td></tr><tr><td>SOMPZ+WZ</td><td>0.313 &plusmn; 0.008</td><td>0.466 &plusmn; 0.006</td><td>0.613 &plusmn; 0.005</td><td>0.774 &plusmn; 0.007</td><td>0.876 &plusmn; 0.007</td><td>0.988 &plusmn; 0.007</td></tr><tr><td rowspan="2">&Delta;&lt;z&gt;</td><td>SOMPZ</td><td>1.46</td><td>2.55</td><td>0.20</td><td>0.28</td><td>0.55</td><td>1.08</td></tr><tr><td>SOMPZ+WZ</td><td>0.83</td><td>0.42</td><td>1.81</td><td>1.28</td><td>2.49</td><td>2.10</td></tr><tr><td rowspan="2">&sigma;<sub>z</sub></td><td>SOMPZ</td><td>0.075 &plusmn; 0.010</td><td>0.064 &plusmn; 0.007</td><td>0.062 &plusmn; 0.006</td><td>0.056 &plusmn; 0.005</td><td>0.060 &plusmn; 0.005</td><td>0.068 &plusmn; 0.007</td></tr><tr><td>SOMPZ+WZ</td><td>0.077 &plusmn; 0.005</td><td>0.057 &plusmn; 0.005</td><td>0.064 &plusmn; 0.004</td><td>0.068 &plusmn; 0.005</td><td>0.064 &plusmn; 0.005</td><td>0.060 &plusmn; 0.003</td></tr><tr><td rowspan="2">&Delta;&sigma;<sub>z</sub></td><td>SOMPZ</td><td>0.53</td><td>0.59</td><td>1.17</td><td>1.42</td><td>0.79</td><td>0.10</td></tr><tr><td>SOMPZ+WZ</td><td>0.46</td><td>2.08</td><td>2.13</td><td>0.93</td><td>1.50</td><td>2.09</td></tr></tbody></table>
## APPENDIX B: VALIDATION IN SIMULATIONS

The validity of our methodology and pipeline has been tested in the Buzzard N-body simulation, introduced in Section 2.8. The measurements of redshift distributions using both phenotypes and clustering were validated in simulations to ensure unbiased estimates with respect to the true redshift distributions. The MAGLIM sample has been recreated in the Buzzard simulations, as described in Section 2. The sample selection has been altered to reproduce as faithfully as possible the number density and colour distributions of the data.

<figure><img src="image_13.png" /><figcaption>**Figure B1.** Estimated $n(z)$ in four tomographic bins using a 12 × 12 cell deep SOM and 32 × 32 cell wide SOM trained on Buzzard simulations. In the top row we have bins 1 and 4, in the middle row bins 2 and 5, and in the bottom row bins 3 and 6. The redshift sample used here has 100 000 galaxies drawn from 1.38 deg$^2$, such that after the MAGLIM selection it yields ~15 000 unique galaxies, which is the same order of magnitude as the redshift samples in data, see Table 2. The deep sample is drawn from three fields of size 3.32, 3.29, and 1.94 deg$^2$, respectively, from the Buzzard simulated sky catalogue. The black dashed line marks the true value, the transparent bands are the 3sDir set of $n(z)$, and the solid bands are the realizations once combined with clustering redshifts. We can appreciate the effect of the combined likelihood, resulting in distributions more constrained in terms of shape, and still consistent with the truth.</figcaption></figure>

As described in Section 4.1.1, we generated 300 simulated deep field realizations that we used to estimate the SOMPZ method uncertainty, which we report in Table 4, and add into our overall error budget. Here, we illustrate that the uncertainty predicted by the 3sDir and the 3sDir + WZ models is consistent with the true $n(z)$ in one of these simulated realizations. We start by selecting one of these simulated realizations, which includes the four deep fields and their corresponding Balrog and redshift samples. We then proceeded to perform the 3sDir analytical sample variance estimation for that one specific realization. The geometry and resolution of the SOM used in simulations are the same as the ones used in data. There are two differences between our simulated scenario and real data: (1) we use the true redshifts from the Buzzard simulations; (2) we assume all redshift information comes from one of our four deep fields. This latter point matches the modelling assumption of 3sDir, which also assumes that the redshift information only comes from one out of four fields. This is a conservative choice that inflates the modelled error due to sample variance in real data for the term $p(z|c)$, and it avoids modelling the highly non-trivial selection function of spectroscopic samples coming from fields other than the COSMOS field. We note that the sample variance contribution to the colour distribution $p(c)$ is modelled correctly as coming from all four fields. The SOMPZ redshift distributions, and their uncertainties estimated through the 3sDir method, are in agreement with the true distribution, as shown in Fig. B1. In Table B1 we summarize the mean and width of the simulated $n(z)$ of the SOMPZ and SOMPZ+WZ methods in each tomographic bin, and of the true $n(z)$, together with the respective statistical distances from the truth.

We also repeated in simulations the same procedure as for data also for the WZ estimates. We created a mock BOSS/eBOSS catalogue to use as a reference sample. As in data, also in simulations the BOSS/eBOSS sample is divided into 50 bins spanning the 0.1 < z < 1.1 range of the catalogue (width $\Delta z = 0.02$). Before proceeding with combining the SOMPZ and WZ information through the combined likelihood, the compatibility between SOMPZ and WZ was checked. This was tested by inferring the windowed means and widths of the WZ and SOMPZ redshift estimates, following Gatti et al. (2022). The window has been determined such that magnification effects related to the WZ measurements can be neglected. As for WZ, we used a ‘simple’ estimator for the redshift distribution, inverting equation (17) and ignoring magnification effects (this is possible as we are considering only windowed quantities). The means and widths computed in this way for the two methods were compatible

<table><caption>**Table B1.** SIMULATIONS: Summary of values for centre values for mean (top panel) and width (bottom panel) for the $n(z)$ distributions as measured in the Buzzard simulations. The values related to SOMPZ and SOMPZ+WZ refer to Fig. B1. Note that the uncertainties quoted here only include sample variance and shot noise.</caption><thead><tr><th></th><th></th><th>Bin 1<br>z &isin; [0.2, 0.4]</th><th>Bin 2<br>z &isin; [0.4, 0.55]</th><th>Bin 3<br>z &isin; [0.55, 0.7]</th><th>Bin 4<br>z &isin; [0.7, 0.85]</th><th>Bin 5<br>z &isin; [0.85, 0.95]</th><th>Bin 6<br>z &isin; [0.95, 1.05]</th></tr></thead><tbody><tr><td rowspan="2">&lt;z&gt;</td><td>SOMPZ</td><td>0.319 &plusmn; 0.009</td><td>0.484 &plusmn; 0.007</td><td>0.623 &plusmn; 0.006</td><td>0.784 &plusmn; 0.006</td><td>0.891 &plusmn; 0.007</td><td>0.993 &plusmn; 0.010</td></tr><tr><td>SOMPZ+WZ</td><td>0.313 &plusmn; 0.008</td><td>0.466 &plusmn; 0.006</td><td>0.613 &plusmn; 0.005</td><td>0.774 &plusmn; 0.007</td><td>0.876 &plusmn; 0.007</td><td>0.988 &plusmn; 0.007</td></tr><tr><td rowspan="2">&Delta;&lt;z&gt;</td><td>SOMPZ</td><td>1.46</td><td>2.55</td><td>0.20</td><td>0.28</td><td>0.55</td><td>1.08</td></tr><tr><td>SOMPZ+WZ</td><td>0.83</td><td>0.42</td><td>1.81</td><td>1.28</td><td>2.49</td><td>2.10</td></tr><tr><td rowspan="2">&sigma;<sub>z</sub></td><td>SOMPZ</td><td>0.075 &plusmn; 0.010</td><td>0.064 &plusmn; 0.007</td><td>0.062 &plusmn; 0.006</td><td>0.056 &plusmn; 0.005</td><td>0.060 &plusmn; 0.005</td><td>0.068 &plusmn; 0.007</td></tr><tr><td>SOMPZ+WZ</td><td>0.077 &plusmn; 0.005</td><td>0.057 &plusmn; 0.005</td><td>0.064 &plusmn; 0.004</td><td>0.068 &plusmn; 0.005</td><td>0.064 &plusmn; 0.005</td><td>0.060 &plusmn; 0.003</td></tr><tr><td rowspan="2">&Delta;&sigma;<sub>z</sub></td><td>SOMPZ</td><td>0.53</td><td>0.59</td><td>1.17</td><td>1.42</td><td>0.79</td><td>0.10</td></tr><tr><td>SOMPZ+WZ</td><td>0.46</td><td>2.08</td><td>2.13</td><td>0.93</td><td>1.50</td><td>2.09</td></tr></tbody></table>
<figure><img src="image_12.png" /><figcaption>**Figure B2.** Posterior distributions of the cosmological parameters $\Omega_m$, $S_8$, and $w$ for the $\Lambda$CDM and $w$CDM analyses. These have been run with six bins and fixed magnification parameters.</figcaption></figure>

within statistical (and systematic) errors, hence the SOMPZ and WZ could be combined together.

The posterior obtained in simulations from multiplying the two likelihoods is shown in Fig. B1, in which the effect of the combination immediately stands out: the additional information from clustering redshifts places a tight constraint on the shape of the $n(z)$, while still being in agreement with the true distribution. This larger constraining power derives from the fact that in clustering the number density for each redshift bin correlates across neighbouring bins, which restrains the joint likelihood to prefer smoother realizations and reject the ones with more uncorrelated values of clustering.

As the second phase of the validation process, a full 2x2pt cosmological analysis was performed. We utilized the data vector consisting of the two-point measurements from the Buzzard simulations and the redshift distributions obtained from the SOMPZ+WZ method, obtained as described in the previous paragraph. We considered both $\Lambda$CDM and $w$CDM models, fixing magnification parameters and including all six MAGLIM tomographic bins. Additionally, we fixed the source galaxies redshift distributions, to ensure any deviation from the true parameter values of the simulation would be caused by the lens $n(z)$ alone. The mean values of $S_8$, $\Omega_m$ (and $w$), with their respective 68 per cent confidence intervals, are

(i) $\Lambda$CDM: $S_8 = 0.73 \pm 0.18$, $\Omega_m = 0.31 \pm 0.07$;

(ii) $w$CDM: $S_8 = 0.71 \pm 0.18$, $\Omega_m = 0.30 \pm 0.08$, $w = -1.3 \pm 0.4$.

For both analyses, the posterior distributions successfully recovered the input parameters (see Section 2), as displayed in Fig. B2.

## APPENDIX C: COSMOLOGICAL PARAMETERS

In Table C1 are listed all the cosmological parameters included in our fiducial analysis.

<table><caption>**Table C1.** The parameters and their priors used in the fiducial MAGLIM $\Lambda$CDM, and $w$CDM analyses. The parameter $w$ is fixed to $-1$ in $\Lambda$CDM. The square brackets denote a flat prior, while the parentheses denote a Gaussian prior of the form $N(\mu, \sigma)$.</caption><thead><tr><th>Parameter</th><th>Fiducial</th><th>Prior</th></tr></thead><tbody><tr><td colspan="3"><strong>Cosmology</strong></td></tr><tr><td>Ω<sub>m</sub></td><td>0.3</td><td>[0.1, 0.9]</td></tr><tr><td>A<sub>s</sub> 10<sup>9</sup></td><td>2.19</td><td>[0.5, 5.0]</td></tr><tr><td>n<sub>s</sub></td><td>0.97</td><td>[0.87, 1.07]</td></tr><tr><td>w</td><td>-1.0</td><td>[-2, -0.33]</td></tr><tr><td>Ω<sub>b</sub></td><td>0.048</td><td>[0.03, 0.07]</td></tr><tr><td>h<sub>0</sub></td><td>0.69</td><td>[0.55, 0.91]</td></tr><tr><td>Ω<sub>v</sub>h<sup>2</sup> 10<sup>3</sup></td><td>0.83</td><td>[0.6, 6.44]</td></tr><tr><td colspan="3"><strong>Linear galaxy bias</strong></td></tr><tr><td>b<sub>i</sub></td><td>1.5, 1.8, 1.8, 1.9, 2.3, 2.3</td><td>[0.8,3.0]</td></tr><tr><td colspan="3"><strong>Lensmagnification</strong></td></tr><tr><td>C<sub>1</sub></td><td>0.43</td><td>(0.43, 0.51)</td></tr><tr><td>C<sub>2</sub></td><td>0.30</td><td>(0.30, 0.48)</td></tr><tr><td>C<sub>3</sub></td><td>1.75</td><td>(1.75, 0.39)</td></tr><tr><td>C<sub>4</sub></td><td>1.94</td><td>(1.94, 0.35)</td></tr><tr><td>C<sub>5</sub></td><td>1.56</td><td>(1.56, 0.71)</td></tr><tr><td>C<sub>6</sub></td><td>2.96</td><td>(2.96, 0.95)</td></tr><tr><td colspan="3"><strong>Lens photo-z</strong></td></tr><tr><td>Δz<sub>1</sub><sup>1</sup></td><td>0.0</td><td>(0.0, 0.0164)</td></tr><tr><td>Δz<sub>1</sub><sup>2</sup></td><td>0.0</td><td>(0.0, 0.0100)</td></tr><tr><td>Δz<sub>1</sub><sup>3</sup></td><td>0.0</td><td>(0.0, 0.0085)</td></tr><tr><td>Δz<sub>1</sub><sup>4</sup></td><td>0.0</td><td>(0.0, 0.0084)</td></tr><tr><td>Δz<sub>1</sub><sup>5</sup></td><td>0.0</td><td>(0.0, 0.0094)</td></tr><tr><td>Δz<sub>1</sub><sup>6</sup></td><td>0.0</td><td>(0.0, 0.0116)</td></tr><tr><td>σz<sub>1</sub><sup>1</sup></td><td>1.0</td><td>(1.0, 0.0639)</td></tr><tr><td>σz<sub>1</sub><sup>2</sup></td><td>1.0</td><td>(1.0, 0.0624)</td></tr><tr><td>σz<sub>1</sub><sup>3</sup></td><td>1.0</td><td>(1.0, 0.0315)</td></tr><tr><td>σz<sub>1</sub><sup>4</sup></td><td>1.0</td><td>(1.0, 0.0409)</td></tr><tr><td>σz<sub>1</sub><sup>5</sup></td><td>1.0</td><td>(1.0, 0.0515)</td></tr><tr><td>σz<sub>1</sub><sup>6</sup></td><td>1.0</td><td>(1.0, 0.0650)</td></tr><tr><td colspan="3"><strong>Intrinsic alignment</strong></td></tr><tr><td>a<sub>i</sub> (i ∈ [1, 2])</td><td>0.7, -1.36</td><td>[-5, 5]</td></tr><tr><td>η<sub>i</sub> (i ∈ [1, 2])</td><td>-1.7, -2.5</td><td>[-5, 5]</td></tr><tr><td>b<sub>TA</sub></td><td>1.0</td><td>[0, 2]</td></tr><tr><td>z<sub>0</sub></td><td>0.62</td><td>Fixed</td></tr><tr><td colspan="3"><strong>Source photo-z</strong></td></tr><tr><td>Δz<sub>s</sub><sup>1</sup></td><td>0.0</td><td>(0.0, 0.018)</td></tr><tr><td>Δz<sub>s</sub><sup>2</sup></td><td>0.0</td><td>(0.0, 0.013)</td></tr><tr><td>Δz<sub>s</sub><sup>3</sup></td><td>0.0</td><td>(0.0, 0.006)</td></tr><tr><td>Δz<sub>s</sub><sup>4</sup></td><td>0.0</td><td>(0.0, 0.013)</td></tr><tr><td colspan="3"><strong>Shear calibration</strong></td></tr><tr><td>m<sup>1</sup></td><td>-0.006</td><td>(-0.006, 0.008)</td></tr><tr><td>m<sup>2</sup></td><td>-0.010</td><td>(-0.010, 0.013)</td></tr><tr><td>m<sup>3</sup></td><td>-0.026</td><td>(-0.026, 0.009)</td></tr><tr><td>m<sup>4</sup></td><td>-0.032</td><td>(-0.032, 0.012)</td></tr></tbody></table>

## APPENDIX D: REDSHIFT UNCERTAINTIES
### SAMPLING STRATEGY

How redshift uncertainties are propagated in the cosmological analysis can have an impact on the final result. In this section we discuss different strategies to marginalize over the redshift uncertainties of our sample during the cosmological inference. Because we have can rely on a full ensemble of $n(z)$ shapes capturing our redshift uncertainties, we can compare three different sampling methods:
<figure><img src="image_14.png" /><figcaption>**Figure B2.** Posterior distributions of the cosmological parameters $\Omega_m$, $S_8$, and $w$ for the $\Lambda$CDM and $w$CDM analyses. These have been run with six bins and fixed magnification parameters.</figcaption></figure>

within statistical (and systematic) errors, hence the SOMPZ and WZ could be combined together.

The posterior obtained in simulations from multiplying the two likelihoods is shown in Fig. B1, in which the effect of the combination immediately stands out: the additional information from clustering redshifts places a tight constraint on the shape of the $n(z)$, while still being in agreement with the true distribution. This larger constraining power derives from the fact that in clustering the number density for each redshift bin correlates across neighbouring bins, which restrains the joint likelihood to prefer smoother realizations and reject the ones with more uncorrelated values of clustering.

As the second phase of the validation process, a full 2x2pt cosmological analysis was performed. We utilized the data vector consisting of the two-point measurements from the Buzzard simulations and the redshift distributions obtained from the SOMPZ+WZ method, obtained as described in the previous paragraph. We considered both $\Lambda$CDM and $w$CDM models, fixing magnification parameters and including all six MAGLIM tomographic bins. Additionally, we fixed the source galaxies redshift distributions, to ensure any deviation from the true parameter values of the simulation would be caused by the lens $n(z)$ alone. The mean values of $S_8$, $\Omega_m$ (and $w$), with their respective 68 per cent confidence intervals, are

$$ (i) \quad \Lambda \text{CDM: } S_8 = 0.73 \pm 0.18, \Omega_m = 0.31 \pm 0.07; $$

$$ (ii) \quad w \text{CDM: } S_8 = 0.71 \pm 0.18, \Omega_m = 0.30 \pm 0.08, w = -1.3 \pm 0.4. $$

For both analyses, the posterior distributions successfully recovered the input parameters (see Section 2), as displayed in Fig. B2.

## APPENDIX C: COSMOLOGICAL PARAMETERS

In Table C1 are listed all the cosmological parameters included in our fiducial analysis.

<table><caption>**Table C1.** The parameters and their priors used in the fiducial MAGLIM $\Lambda$CDM, and $w$CDM analyses. The parameter $w$ is fixed to $-1$ in $\Lambda$CDM. The square brackets denote a flat prior, while the parentheses denote a Gaussian prior of the form $N(\mu, \sigma)$.</caption><thead><tr><th>Parameter</th><th>Fiducial</th><th>Prior</th></tr></thead><tbody><tr><td colspan="3"><strong>Cosmology</strong></td></tr><tr><td>Ω<sub>m</sub></td><td>0.3</td><td>[0.1, 0.9]</td></tr><tr><td>A<sub>s</sub> 10<sup>9</sup></td><td>2.19</td><td>[0.5, 5.0]</td></tr><tr><td>n<sub>s</sub></td><td>0.97</td><td>[0.87, 1.07]</td></tr><tr><td>w</td><td>-1.0</td><td>[-2, -0.33]</td></tr><tr><td>Ω<sub>b</sub></td><td>0.048</td><td>[0.03, 0.07]</td></tr><tr><td>h<sub>0</sub></td><td>0.69</td><td>[0.55, 0.91]</td></tr><tr><td>Ω<sub>v</sub>h<sup>2</sup>10<sup>3</sup></td><td>0.83</td><td>[0.6, 6.44]</td></tr><tr><td colspan="3"><strong>Linear galaxy bias</strong></td></tr><tr><td>b<sub>i</sub></td><td>1.5, 1.8, 1.8, 1.9, 2.3, 2.3</td><td>[0.8,3.0]</td></tr><tr><td colspan="3"><strong>Lensmagnification</strong></td></tr><tr><td>C<sub>1</sub></td><td>0.43</td><td>(0.43, 0.51)</td></tr><tr><td>C<sub>2</sub></td><td>0.30</td><td>(0.30, 0.48)</td></tr><tr><td>C<sub>3</sub></td><td>1.75</td><td>(1.75, 0.39)</td></tr><tr><td>C<sub>4</sub></td><td>1.94</td><td>(1.94, 0.35)</td></tr><tr><td>C<sub>5</sub></td><td>1.56</td><td>(1.56, 0.71)</td></tr><tr><td>C<sub>6</sub></td><td>2.96</td><td>(2.96, 0.95)</td></tr><tr><td colspan="3"><strong>Lens photo-z</strong></td></tr><tr><td>Δz<sub>1</sub><sup>1</sup></td><td>0.0</td><td>(0.0, 0.0164)</td></tr><tr><td>Δz<sub>1</sub><sup>2</sup></td><td>0.0</td><td>(0.0, 0.0100)</td></tr><tr><td>Δz<sub>1</sub><sup>3</sup></td><td>0.0</td><td>(0.0, 0.0085)</td></tr><tr><td>Δz<sub>1</sub><sup>4</sup></td><td>0.0</td><td>(0.0, 0.0084)</td></tr><tr><td>Δz<sub>1</sub><sup>5</sup></td><td>0.0</td><td>(0.0, 0.0094)</td></tr><tr><td>Δz<sub>1</sub><sup>6</sup></td><td>0.0</td><td>(0.0, 0.0116)</td></tr><tr><td>σz<sub>1</sub><sup>1</sup></td><td>1.0</td><td>(1.0, 0.0639)</td></tr><tr><td>σz<sub>1</sub><sup>2</sup></td><td>1.0</td><td>(1.0, 0.0624)</td></tr><tr><td>σz<sub>1</sub><sup>3</sup></td><td>1.0</td><td>(1.0, 0.0315)</td></tr><tr><td>σz<sub>1</sub><sup>4</sup></td><td>1.0</td><td>(1.0, 0.0409)</td></tr><tr><td>σz<sub>1</sub><sup>5</sup></td><td>1.0</td><td>(1.0, 0.0515)</td></tr><tr><td>σz<sub>1</sub><sup>6</sup></td><td>1.0</td><td>(1.0, 0.0650)</td></tr><tr><td colspan="3"><strong>Intrinsic alignment</strong></td></tr><tr><td>a<sub>i</sub> (i ∈ [1, 2])</td><td>0.7, -1.36</td><td>[-5, 5]</td></tr><tr><td>η<sub>i</sub> (i ∈ [1, 2])</td><td>-1.7, -2.5</td><td>[-5, 5]</td></tr><tr><td>b<sub>TA</sub></td><td>1.0</td><td>[0, 2]</td></tr><tr><td>z<sub>0</sub></td><td>0.62</td><td>Fixed</td></tr><tr><td colspan="3"><strong>Source photo-z</strong></td></tr><tr><td>Δz<sub>s</sub><sup>1</sup></td><td>0.0</td><td>(0.0, 0.018)</td></tr><tr><td>Δz<sub>s</sub><sup>2</sup></td><td>0.0</td><td>(0.0, 0.013)</td></tr><tr><td>Δz<sub>s</sub><sup>3</sup></td><td>0.0</td><td>(0.0, 0.006)</td></tr><tr><td>Δz<sub>s</sub><sup>4</sup></td><td>0.0</td><td>(0.0, 0.013)</td></tr><tr><td colspan="3"><strong>Shear calibration</strong></td></tr><tr><td>m<sup>1</sup></td><td>-0.006</td><td>(-0.006, 0.008)</td></tr><tr><td>m<sup>2</sup></td><td>-0.010</td><td>(-0.010, 0.013)</td></tr><tr><td>m<sup>3</sup></td><td>-0.026</td><td>(-0.026, 0.009)</td></tr><tr><td>m<sup>4</sup></td><td>-0.032</td><td>(-0.032, 0.012)</td></tr></tbody></table>

## APPENDIX D: REDSHIFT UNCERTAINTIES
### SAMPLING STRATEGY

How redshift uncertainties are propagated in the cosmological analysis can have an impact on the final result. In this section we discuss different strategies to marginalize over the redshift uncertainties of our sample during the cosmological inference. Because we have can rely on a full ensemble of $n(z)$ shapes capturing our redshift uncertainties, we can compare three different sampling methods:
(i) **Shift**: we compress the realizations by computing their average, and marginalize over a shift on the mean;

(ii) **Shift and stretch**: we compress the realizations by computing their average, and marginalize over both a shift on the mean and on a stretch on the width;

(iii) **Full shape**: we provide as input all the produced realizations and we rank them by one of their properties using the *Hyperrank* method (Cordero et al. 2022), marginalizing over the full shape of the distributions.

Using only *shifts* is the methodology usually adopted to model redshift uncertainties in weak lensing sample, as the weak lensing kernel is mostly sensitive to the mean of the redshift distributions. On the other hand, clustering and galaxy-galaxy lensing measurements are also very sensitive to the width of the lens redshift distributions; therefore, the *shift and stretch* approach is preferred. The full shape marginalization, in theory, is more accurate, because it accounts for the uncertainties in the higher order moments of the distribution; however, depending on the science case, it might not make a huge impact on the final constraints. The full shape marginalization is implemented via hyperrank (Cordero et al. 2022), which is an algorithm that orders realizations of the ensemble according to a parameter, which facilitates the sampling and marginalization over the $n(z)$ ensemble within the cosmological likelihood Markov chains. Hyperrank was also implemented for the WL sources, although it had a negligible impact on the results. The quantity chosen for the ranking in that case was the mean. We decided for this case it would be more appropriate to perform the optimized ranking of the realization by the 68 per cent sigma rather than the mean, and we tested it indeed improved the performance of the sampling. To test the different sampling strategies, we built a synthetic noiseless data vector based on theory predictions at fixed cosmology and we used as $n(z)$ the realizations average of the SOMPZ+WZ estimates in data. We then marginalized over redshift uncertainties using the three approaches aforementioned. We performed this test both using four or six lens bins, although here we are just going to show the posteriors obtained with four bins as they are not qualitatively different from the ones with six bins. The results of this test are shown in Fig. D1, where we show the posterior of $\sigma_8$, $\Omega_m$ and for sake of simplicity, two out of the four galaxy-matter linear biases.

Focusing on the shift and shift+stretch contours, one can notice that the width of the contour in the direction perpendicular to the degeneration axis is larger for the shift + stretch. This is related to impact of the additional marginalization over the width of the distributions. One caveat is that in our marginalization scheme (as adopted in the main DES Y3 2x2pt analysis), we are implicitly neglecting correlations between the uncertainties in the mean and widths of the distributions, which usually show a certain degree of correlation (from ~10 per cent to ~30 per cent, depending from the tomographic bin). These are neglected, which might translate in a slight overestimation of our constraints. When marginalizing over the uncertainties using the hyperrank framework, on the other hand, such correlations are implicitly accounted for. Indeed, one can notice that the hyperrank posteriors are slightly tighter than the shift or shift-stretch posteriors.

Unfortunately, we did not manage to successfully apply hyperrank to the data. When performing the cosmological analysis on data using hyperrank, we found significantly less smooth posteriors compared to our tests on simulations. A similar behaviour has also been found when applying hyperrank to the DES Y3 source sample (Amon et al. 2022), and it has been interpreted as a consequence of a possible larger degree of complexity of the redshift distributions of

<figure><img src="image_13.png" /><figcaption>**Figure D1.** Posterior distributions of the cosmological parameters $\Omega_m$, $S_8$, and two out of four of the galaxy-matter biases ($b_2$, $b_4$) for the $\Lambda$CDM analysis involving four bins and fixed magnification parameters. These analyses have been obtained assuming a theoretical data vector and adopting different marginalization schemes on the redshift distribution of the lens sample.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure D2.** Same as the left panel of Fig. 8, but with two additional posteriors overplotted representing the constraints obtained using the redshift distributions with ‘clipped’ tails.</figcaption></figure>

our data compared to simulations. We attempted both to artificially smooth our $n(z)$ and to increase the number of samples from the SOMPZ+WZ method, without reaching a satisfactory level. Due to the very high computational cost of running a cosmological chain using hyperrank, we could only test a few different levels of smoothing before deciding to abandon hyperrank for this work, and choose the shift + stretch as photo-z uncertainty marginalization
(i) **Shift**: we compress the realizations by computing their average, and marginalize over a shift on the mean;

(ii) **Shift and stretch**: we compress the realizations by computing their average, and marginalize over both a shift on the mean and on a stretch on the width;

(iii) **Full shape**: we provide as input all the produced realizations and we rank them by one of their properties using the *Hyperrank* method (Cordero et al. 2022), marginalizing over the full shape of the distributions.

Using only *shifts* is the methodology usually adopted to model redshift uncertainties in weak lensing sample, as the weak lensing kernel is mostly sensitive to the mean of the redshift distributions. On the other hand, clustering and galaxy–galaxy lensing measurements are also very sensitive to the width of the lens redshift distributions; therefore, the *shift and stretch* approach is preferred. The full shape marginalization, in theory, is more accurate, because it accounts for the uncertainties in the higher order moments of the distribution; however, depending on the science case, it might not make a huge impact on the final constraints. The full shape marginalization is implemented via hyperrank (Cordero et al. 2022), which is an algorithm that orders realizations of the ensemble according to a parameter, which facilitates the sampling and marginalization over the $n(z)$ ensemble within the cosmological likelihood Markov chains. Hyperrank was also implemented for the WL sources, although it had a negligible impact on the results. The quantity chosen for the ranking in that case was the mean. We decided for this case it would be more appropriate to perform the optimized ranking of the realization by the 68 per cent sigma rather than the mean, and we tested it indeed improved the performance of the sampling. To test the different sampling strategies, we built a synthetic noiseless data vector based on theory predictions at fixed cosmology and we used as $n(z)$ the realizations average of the SOMPZ+WZ estimates in data. We then marginalized over redshift uncertainties using the three approaches aforementioned. We performed this test both using four or six lens bins, although here we are just going to show the posteriors obtained with four bins as they are not qualitatively different from the ones with six bins. The results of this test are shown in Fig. D1, where we show the posterior of $\sigma_8$, $\Omega_m$ and for sake of simplicity, two out of the four galaxy-matter linear biases.

Focusing on the shift and shift+stretch contours, one can notice that the width of the contour in the direction perpendicular to the degeneration axis is larger for the shift + stretch. This is related to impact of the additional marginalization over the width of the distributions. One caveat is that in our marginalization scheme (as adopted in the main DES Y3 2x2pt analysis), we are implicitly neglecting correlations between the uncertainties in the mean and widths of the distributions, which usually show a certain degree of correlation (from ~ 10 per cent to ~ 30 per cent, depending from the tomographic bin). These are neglected, which might translate in a slight overestimation of our constraints. When marginalizing over the uncertainties using the hyperrank framework, on the other hand, such correlations are implicitly accounted for. Indeed, one can notice that the hyperrank posteriors are slightly tighter than the shift or shift-stretch posteriors.

Unfortunately, we did not manage to successfully apply hyperrank to the data. When performing the cosmological analysis on data using hyperrank, we found significantly less smooth posteriors compared to our tests on simulations. A similar behaviour has also been found when applying hyperrank to the DES Y3 source sample (Amon et al. 2022), and it has been interpreted as a consequence of a possible larger degree of complexity of the redshift distributions of

<figure><img src="image_15.png" /><figcaption>**Figure D1.** Posterior distributions of the cosmological parameters $\Omega_m$, $S_8$, and two out of four of the galaxy-matter biases ($b_2$, $b_4$) for the $\Lambda$CDM analysis involving four bins and fixed magnification parameters. These analyses have been obtained assuming a theoretical data vector and adopting different marginalization schemes on the redshift distribution of the lens sample.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure D2.** Same as the left panel of Fig. 8, but with two additional posteriors overplotted representing the constraints obtained using the redshift distributions with ‘clipped’ tails.</figcaption></figure>

our data compared to simulations. We attempted both to artificially smooth our $n(z)$ and to increase the number of samples from the SOMPZ+WZ method, without reaching a satisfactory level. Due to the very high computational cost of running a cosmological chain using hyperrank, we could only test a few different levels of smoothing before deciding to abandon hyperrank for this work, and choose the shift + stretch as photo-z uncertainty marginalization
methodology. For DES Y6, we plan to apply several tools that will speed up our cosmological inference, enabling more tests on hyperrank, which has great potential and whose implementation is a goal for the DES Y6 analysis.

## D1 Cosmological constraints with clipped $n(z)$ tails

Here, we test whether the difference between DNF+WZ and SOMPZ+WZ constraints (Fig. 8) were only due to the different treatment of redshift outliers and of the tails of the redshift distributions. We artificially removed the tails from the DNF+WZ and SOMPZ+WZ $n(z)$ (i.e. we set the distributions to zero), and repeated our cosmological analysis. We used as definition of the tails the same interval used to calibrate the DNF distribution with the WZ constraints adopted in Porredon et al. (2022). Results for the $\Lambda$CDM case, four bins, and fixed magnification are shown in Fig. D2. By removing the tails, both posteriors are shifted, which means that the calibration of the tails of the redshift distribution is important for our cosmological analysis. Since the two posteriors are shifted but they still do not overlap, we can assume that the differences in the bulk of the redshift distributions inferred by two methods is also crucially driving the differences at the constraints level seen in Fig. 8.

<footnote n="1">¹Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote n="2">²Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote n="3">³Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, E-08193 Bellaterra (Barcelona) Spain</footnote>

<footnote n="4">⁴Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA</footnote>

<footnote n="5">⁵Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA</footnote>

<footnote n="6">⁶Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA</footnote>

<footnote n="7">⁷Department of Physics, The Ohio State University, Columbus, OH 43210, USA</footnote>

<footnote n="8">⁸Institut d'Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain</footnote>

<footnote n="9">⁹Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, E-08193 Barcelona, Spain</footnote>

<footnote n="10">¹⁰Physics Department, William Jewell College, Liberty, MO 64068, USA</footnote>

<footnote n="11">¹¹Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA</footnote>

<footnote n="12">¹²Kavli Institute for Particle Astrophysics & Cosmology, PO Box 2450, Stanford University, Stanford, CA 94305, USA</footnote>

<footnote n="13">¹³SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA</footnote>

<footnote n="14">¹⁴Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA</footnote>

<footnote n="15">¹⁵Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="16">¹⁶Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="17">¹⁷Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="18">¹⁸Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil</footnote>

<footnote n="19">¹⁹Institute for Astronomy, University of Hawaii'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA</footnote>

<footnote n="20">²⁰Physics Department, University of Wisconsin-Madison, 2320 Chamberlin Hall, 1150 University Avenue Madison, WI 53706-1390, USA</footnote>

<footnote n="21">²¹Department of Physics, Northeastern University, Boston, MA 02115, USA</footnote>

<footnote n="22">²²Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ 20921-400, Brazil</footnote>

<footnote n="23">²³Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15312, USA</footnote>

<footnote n="24">²⁴Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="25">²⁵Departamento Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="26">²⁶Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St, Urbana, IL 61801, USA</footnote>

<footnote n="27">²⁷Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="28">²⁸Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK</footnote>

<footnote n="29">²⁹Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain</footnote>

<footnote n="30">³⁰Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA</footnote>

<footnote n="31">³¹NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA</footnote>

<footnote n="32">³²Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA</footnote>

<footnote n="33">³³Department of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA</footnote>

<footnote n="34">³⁴Departments of Statistics and Data Science, University of Texas at Austin, Austin, TX 78757, USA</footnote>

<footnote n="35">³⁵Faculty of Physics, University Observatory, Ludwig-Maximilians-Universität, Scheinerstr 1, D-81679 Munich, Germany</footnote>

<footnote n="36">³⁶Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ 20921-400, Brazil</footnote>

<footnote n="37">³⁷School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, UK</footnote>

<footnote n="38">³⁸Department of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290 Versoix, Switzerland</footnote>

<footnote n="39">³⁹Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, UK</footnote>

<footnote n="40">⁴⁰Department of Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA</footnote>

<footnote n="41">⁴¹Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton BN1 9QH, UK</footnote>

<footnote n="42">⁴²Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK</footnote>

<footnote n="43">⁴³Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA</footnote>

<footnote n="44">⁴⁴Department of Physics, Duke University, Durham NC 27708, USA</footnote>

<footnote n="45">⁴⁵Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile</footnote>

<footnote n="46">⁴⁶Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK</footnote>

<footnote n="47">⁴⁷Institut d'Astrophysique de Paris, CNRS, Sorbonne Université, UMR 7095, F-75014, Paris, France</footnote>

<footnote n="48">⁴⁸Institut d'Astrophysique de Paris, Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, F-75014 Paris, France</footnote>

<footnote n="49">⁴⁹Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy</footnote>

<footnote n="50">⁵⁰INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy</footnote>

<footnote n="51">⁵¹Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34014 Trieste, Italy</footnote>

<footnote n="52">⁵²Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany</footnote>

<footnote n="53">⁵³Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India</footnote>

<footnote n="54">⁵⁴Institute of Theoretical Astrophysics, University of Oslo. PO Box 1029 Blindern, NO-0315 Oslo, Norway</footnote>

<footnote n="55">⁵⁵Instituto de Física Teórica UAM/CSIC, Universidad Autonoma de Madrid, E-28049 Madrid, Spain</footnote>

<footnote n="56">⁵⁶Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="57">⁵⁷School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia</footnote>

<footnote n="58">⁵⁸Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA</footnote>

<footnote n="59">⁵⁹Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="60">⁶⁰Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia</footnote>
methodology. For DES Y6, we plan to apply several tools that will speed up our cosmological inference, enabling more tests on hyperrank, which has great potential and whose implementation is a goal for the DES Y6 analysis.

## D1 Cosmological constraints with clipped $n(z)$ tails

Here, we test whether the difference between DNF+WZ and SOMPZ+WZ constraints (Fig. 8) were only due to the different treatment of redshift outliers and of the tails of the redshift distributions. We artificially removed the tails from the DNF+WZ and SOMPZ+WZ $n(z)$ (i.e. we set the distributions to zero), and repeated our cosmological analysis. We used as definition of the tails the same interval used to calibrate the DNF distribution with the WZ constraints adopted in Porredon et al. (2022). Results for the $\Lambda$CDM case, four bins, and fixed magnification are shown in Fig. D2. By removing the tails, both posteriors are shifted, which means that the calibration of the tails of the redshift distribution is important for our cosmological analysis. Since the two posteriors are shifted but they still do not overlap, we can assume that the differences in the bulk of the redshift distributions inferred by two methods is also crucially driving the differences at the constraints level seen in Fig. 8.

<footnote n="1">¹Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote n="2">²Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote n="3">³Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, E-08193 Bellaterra (Barcelona) Spain</footnote>

<footnote n="4">⁴Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA</footnote>

<footnote n="5">⁵Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA</footnote>

<footnote n="6">⁶Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA</footnote>

<footnote n="7">⁷Department of Physics, The Ohio State University, Columbus, OH 43210, USA</footnote>

<footnote n="8">⁸Institut d'Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain</footnote>

<footnote n="9">⁹Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, E-08193 Barcelona, Spain</footnote>

<footnote n="10">¹⁰Physics Department, William Jewell College, Liberty, MO 64068, USA</footnote>

<footnote n="11">¹¹Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA</footnote>

<footnote n="12">¹²Kavli Institute for Particle Astrophysics & Cosmology, PO Box 2450, Stanford University, Stanford, CA 94305, USA</footnote>

<footnote n="13">¹³SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA</footnote>

<footnote n="14">¹⁴Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA</footnote>

<footnote n="15">¹⁵Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="16">¹⁶Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="17">¹⁷Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="18">¹⁸Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil</footnote>

<footnote n="19">¹⁹Institute for Astronomy, University of Hawaii'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA</footnote>

<footnote n="20">²⁰Physics Department, University of Wisconsin-Madison, 2320 Chamberlin Hall, 1150 University Avenue Madison, WI 53706-1390, USA</footnote>

<footnote n="21">²¹Department of Physics, Northeastern University, Boston, MA 02115, USA</footnote>

<footnote n="22">²²Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ 20921-400, Brazil</footnote>

<footnote n="23">²³Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15312, USA</footnote>

<footnote n="24">²⁴Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="25">²⁵Departamento Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="26">²⁶Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St, Urbana, IL 61801, USA</footnote>

<footnote n="27">²⁷Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="28">²⁸Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK</footnote>

<footnote n="29">²⁹Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain</footnote>

<footnote n="30">³⁰Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA</footnote>

<footnote n="31">³¹NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA</footnote>

<footnote n="32">³²Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA</footnote>

<footnote n="33">³³Department of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA</footnote>

<footnote n="34">³⁴Departments of Statistics and Data Science, University of Texas at Austin, Austin, TX 78757, USA</footnote>

<footnote n="35">³⁵Faculty of Physics, University Observatory, Ludwig-Maximilians-Universität, Scheinerstr 1, D-81679 Munich, Germany</footnote>

<footnote n="36">³⁶Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ 20921-400, Brazil</footnote>

<footnote n="37">³⁷School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, UK</footnote>

<footnote n="38">³⁸Department of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290 Versoix, Switzerland</footnote>

<footnote n="39">³⁹Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, UK</footnote>

<footnote n="40">⁴⁰Department of Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA</footnote>

<footnote n="41">⁴¹Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton BN1 9QH, UK</footnote>

<footnote n="42">⁴²Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK</footnote>

<footnote n="43">⁴³Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA</footnote>

<footnote n="44">⁴⁴Department of Physics, Duke University, Durham NC 27708, USA</footnote>

<footnote n="45">⁴⁵Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile</footnote>

<footnote n="46">⁴⁶Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK</footnote>

<footnote n="47">⁴⁷Institut d'Astrophysique de Paris, CNRS, Sorbonne Université, UMR 7095, F-75014, Paris, France</footnote>

<footnote n="48">⁴⁸Institut d'Astrophysique de Paris, Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, F-75014 Paris, France</footnote>

<footnote n="49">⁴⁹Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy</footnote>

<footnote n="50">⁵⁰INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy</footnote>

<footnote n="51">⁵¹Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34014 Trieste, Italy</footnote>

<footnote n="52">⁵²Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany</footnote>

<footnote n="53">⁵³Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India</footnote>

<footnote n="54">⁵⁴Institute of Theoretical Astrophysics, University of Oslo. PO Box 1029 Blindern, NO-0315 Oslo, Norway</footnote>

<footnote n="55">⁵⁵Instituto de Física Teórica UAM/CSIC, Universidad Autonoma de Madrid, E-28049 Madrid, Spain</footnote>

<footnote n="56">⁵⁶Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="57">⁵⁷School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia</footnote>

<footnote n="58">⁵⁸Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA</footnote>

<footnote n="59">⁵⁹Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="60">⁶⁰Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia</footnote>
⁶¹Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA

⁶²Centre for Gravitational Astrophysics, College of Science, The Australian National University, ACT 2601, Australia

⁶³The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia

⁶⁴Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, SP 05314-970, Brazil

⁶⁵Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA

⁶⁶Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain

⁶⁷Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

⁶⁸School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK

⁶⁹Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

This paper has been typeset from a `TEX/LATEX` file prepared by the author.
⁶¹Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA

⁶²Centre for Gravitational Astrophysics, College of Science, The Australian National University, ACT 2601, Australia

⁶³The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia

⁶⁴Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, SP 05314-970, Brazil

⁶⁵Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA

⁶⁶Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain

⁶⁷Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

⁶⁸School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK

⁶⁹Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

This paper has been typeset from a `TEX/LATEX` file prepared by the author.