<figure><img src="image_10.png" /><figcaption>**Figure 10.** The difference in CDFs depending on whether or not we account for reduced shear effects, $\Delta P = P^{\text{RS}} - P^{\text{fid}}$. The high-redshift bins, especially when looking at the 2-field CDFs, see the largest impact given source planes at high redshift have larger values of $\kappa$ and thus the $1/(1-\kappa)$ term for the reduced shear is larger. The deviations are still within $\lesssim 0.1\sigma$ in all cases and are 2–3 orders of magnitude below the cosmological signal. The total signal-to-noise of reduced shear-induced residuals is $0.3\sigma$.</figcaption></figure>

Duffy et al. 2010), which consequently will impact the weak lensing signal (Chisari et al. 2018). These baryonic imprints have a strong mass/redshift dependence (Lovell et al. 2018; Beltz-Mohrmann & Berlind 2021; Anbajagane, Evrard & Farahi 2022a) and this mass/redshift-dependent impact on the halo potential can vary across simulation prescriptions (e.g. Shao, Anbajagane & Chang 2022; Anbajagane et al. 2022b).

Recently, Schneider et al. (2019) implemented a halo-based model that can alter N-body simulations – which are cheaper to run than full hydrodynamic simulations with galaxy formation – to then model the baryon imprints on the density/convergence field. This technique provides a higher-level, approximate galaxy formation model that depends only on ‘macro’ properties like the halo baryon fraction, the baryon density profiles, dark matter density profile etc. and the flexibility manifesting from the method’s approximate nature is particularly useful both for matching the range of halo property scaling relations found in the latest hydro simulations (e.g. Anbajagane et al. 2020, 2022b; Lim et al. 2021; Cui et al. 2022; Lee et al. 2022; Stiskalek et al. 2022; Anbajagane, Evrard & Farahi 2022a) and for handling differences between the evolution of gas in observations and simulations as found in different analyses (e.g. Hill et al. 2018; Amodeo et al. 2021; Pandey et al. 2022; Anbajagane et al. 2022c, 2023a).

In this section, we once again compute residuals between CDFs measured on maps from N-body simulations and maps that have been ‘baryonified’. Both sets of maps used in this section come from the COSMOGRID suite, and the baryonification was performed with the same model as Schneider et al. (2019). The parameters of the baryonification model were all given their default values, except for some of the gas model parameters which we give values of $M_c = 13.82$ and $v = 0$. These parameters are part of a reparametrization

done in Fluri et al. (2022) and control the gas density profiles’ slopes. We take the true convergence fields from COSMOGRID and post-process them using the same pipeline described in Section 3.3. We make 800 DES Y3 cutouts from each set of maps. All results are averages over these realizations.

Fig. 11 shows the residuals due to baryonic imprints on DES Y3-like mock maps. In all cases, the baryon impacts are below $1\sigma$. However, note that the maps from COSMOGRID have a resolution of NSIDE = 512, and thus the pixel resolution is $6.4'$ arcmin, instead of the $3.2'$ arcmin minimum scale used in this work. Since the baryons’ dominant contribution is on smaller scales, it is likely that the true residuals at $3' < \theta < 6.4'$ are actually larger than what is presented in Fig. 11 but are currently suppressed due to the pixel resolution of the COSMOGRID maps. Nevertheless, we can state that the baryon imprints for $\theta > 10'$ have a significance that is approximately 1–2 orders of magnitude below the cosmological signal.

The impact is also highest for the extreme thresholds in the CDF – the $k = -0.006$ and $k = -0.020$ thresholds – and this has been seen in previous, theoretical works. Osato, Liu & Haiman (2021) compared hydrodynamic simulations with a dark matter-only counterpart and showed the lensing PDF can be impacted by more than 10 per cent at the tails of the distribution (see their fig. 5). Sunseri, Li & Liu (2023) used the same set of simulations to show that the impact of baryons on haloes, filaments, and voids affects different parts of the matter PDF.

## 5.7 Scale cuts

In the above sections, we have determined the impact of different systematics and modelling approximations on the CDF data vector. Some systematics are negligible for the whole data vector, such
<figure><img src="image_12.png" /><figcaption>**Figure 10.** The difference in CDFs depending on whether or not we account for reduced shear effects, $\Delta P = P^{\text{RS}} - P^{\text{fid}}$. The high-redshift bins, especially when looking at the 2-field CDFs, see the largest impact given source planes at high redshift have larger values of $\kappa$ and thus the $1/(1-\kappa)$ term for the reduced shear is larger. The deviations are still within $\lesssim 0.1\sigma$ in all cases and are 2–3 orders of magnitude below the cosmological signal. The total signal-to-noise of reduced shear-induced residuals is $0.3\sigma$.</figcaption></figure>

Duffy et al. 2010), which consequently will impact the weak lensing signal (Chisari et al. 2018). These baryonic imprints have a strong mass/redshift dependence (Lovell et al. 2018; Beltz-Mohrmann & Berlind 2021; Anbajagane, Evrard & Farahi 2022a) and this mass/redshift-dependent impact on the halo potential can vary across simulation prescriptions (e.g. Shao, Anbajagane & Chang 2022; Anbajagane et al. 2022b).

Recently, Schneider et al. (2019) implemented a halo-based model that can alter N-body simulations – which are cheaper to run than full hydrodynamic simulations with galaxy formation – to then model the baryon imprints on the density/convergence field. This technique provides a higher-level, approximate galaxy formation model that depends only on ‘macro’ properties like the halo baryon fraction, the baryon density profiles, dark matter density profile etc. and the flexibility manifesting from the method’s approximate nature is particularly useful both for matching the range of halo property scaling relations found in the latest hydro simulations (e.g. Anbajagane et al. 2020, 2022b; Lim et al. 2021; Cui et al. 2022; Lee et al. 2022; Stiskalek et al. 2022; Anbajagane, Evrard & Farahi 2022a) and for handling differences between the evolution of gas in observations and simulations as found in different analyses (e.g. Hill et al. 2018; Amodeo et al. 2021; Pandey et al. 2022; Anbajagane et al. 2022c, 2023a).

In this section, we once again compute residuals between CDFs measured on maps from N-body simulations and maps that have been ‘baryonified’. Both sets of maps used in this section come from the COSMOGRID suite, and the baryonification was performed with the same model as Schneider et al. (2019). The parameters of the baryonification model were all given their default values, except for some of the gas model parameters which we given values of $M_c = 13.82$ and $v = 0$. These parameters are part of a reparametrization

done in Fluri et al. (2022) and control the gas density profiles’ slopes. We take the true convergence fields from COSMOGRID and post-process them using the same pipeline described in Section 3.3. We make 800 DES Y3 cutouts from each set of maps. All results are averages over these realizations.

Fig. 11 shows the residuals due to baryonic imprints on DES Y3-like mock maps. In all cases, the baryon impacts are below $1\sigma$. However, note that the maps from COSMOGRID have a resolution of NSIDE = 512, and thus the pixel resolution is $6.4' \text{ arcmin}$, instead of the $3.2' \text{ arcmin}$ minimum scale used in this work. Since the baryons’ dominant contribution is on smaller scales, it is likely that the true residuals at $3' < \theta < 6.4'$ are actually larger than what is presented in Fig. 11 but are currently suppressed due to the pixel resolution of the COSMOGRID maps. Nevertheless, we can state that the baryon imprints for $\theta > 10'$ have a significance that is approximately 1–2 orders of magnitude below the cosmological signal.

The impact is also highest for the extreme thresholds in the CDF – the $k = -0.006$ and $k = -0.020$ thresholds – and this has been seen in previous, theoretical works. Osato, Liu & Haiman (2021) compared hydrodynamic simulations with a dark matter-only counterpart and showed the lensing PDF can be impacted by more than 10 per cent at the tails of the distribution (see their fig. 5). Sunseri, Li & Liu (2023) used the same set of simulations to show that the impact of baryons on haloes, filaments, and voids affects different parts of the matter PDF.

## 5.7 Scale cuts

In the above sections, we have determined the impact of different systematics and modelling approximations on the CDF data vector. Some systematics are negligible for the whole data vector, such
<figure><img src="image_11.png" /><figcaption>**Figure 11.** The difference in CDFs measured on dark matter-only (DMO) simulations and ‘baryonified’ DMO simulations. As expected, baryon imprints are a significant effect on the data vector. The grey band shows the scales below $\theta < 6.4'$, which is the pixel resolution of the COSMOGRID DES Y3 maps, and is a factor of 2 larger than the other maps we consider in this work. Thus, the baryon effects we estimate below that scale are an underestimate of the true effect given the pixel resolution will suppress these effects. The total signal-to-noise of baryon imprints is $3.5\sigma$, though this is a lower bound given the suppression due to map resolution.</figcaption></figure>

as the PSFs (Section 5.3) and the reduced shear approximation (Section 5.5), while others are prominent at a subset of scales, such as baryon imprints (Section 5.6). Thus, using the CDFs to robustly infer cosmological constraints will require us to discard some parts of the fiducial data vector – namely the parts where the amplitude of the systematics is high – and obtain constraints using the remaining fraction of the data vector.

Amongst all the systematic effects considered in this work, the most significant are the baryon imprints (Fig. 11) and the source clustering effect (Fig. 8). These will determine how the data vector is truncated. Our scale cuts are determined by requiring that the parameter bias due to unmodelled systematic effects is below a certain threshold. We compute this bias using the extended Fisher formalism of Amara & Réfrégier (2008) and Asgari et al. (2021a),

$$ \Delta_p^{\text{bias}} = \sum_q (F^{-1})_{pq} \frac{d\tilde{X}_{\text{fid}}}{dp} C^{-1} (\tilde{X}_{\text{biased}} - \tilde{X}_{\text{fid}}), \quad (20) $$

where both p and q are indexes over the cosmological parameters of interest. The average bias in the data vector, $\tilde{X}_{\text{biased}} - \tilde{X}_{\text{fid}}$, is a quantity we have already computed and presented in the above subsections. We then summarize this bias-per-parameter, $\Delta_p^{\text{bias}}$, into a bias for the full N-D posterior as

$$ \delta = \sqrt{\sum_{p,q} \Delta_p^{\text{bias}} (C^{-1})_{pq} \Delta_q^{\text{bias}}}, \quad (21) $$

where C is the covariance of the parameters, and so $C^{-1}$ is just the Fisher matrix, F. Our procedure for scale cuts is simply removing data points until $\delta < X$, where X is some chosen threshold. We will use $X \in \{0.3, 0.2, 0.1\}$. The choice $X = 0.3$ matches the tests done in the main methodology pipeline for DES Y3 (e.g. Krause et al.

2021; Amon et al. 2022; Secco et al. 2022a) while the other values are chosen to explore more stringent cuts that could be reflective of Stage IV surveys. Note that this threshold, X, is somewhat arbitrary, but that is not a concern as our goal is to see how the scale-cuts for the CDFs compare to those for the moments; as long as the same choices are applied across both statistics, the arbitrariness of the choices is not relevant.

The other component we must decide is how to determine and discard data points to achieve the condition $\delta < X$, as there is significant freedom in doing so. We could throw away all data points for every bin/threshold corresponding to aperture scales below a certain chosen value. However, the choice of a fixed scale threshold is suboptimal as the impact of systematics at a chosen scale varies across bins and thresholds (as seen in any of the Figures above). Thus, our choice here is a scale cut done bin-by-bin (and threshold-by-threshold, in the case of CDFs) and follows the approach of Amon et al. (2022); Secco et al. (2022a). We compute the chi-squared of a given effect in a specific tomographic bin combination (and also specific threshold, in the case of CDFs), and remove the data points corresponding to the smallest scales until we satisfy the relation,

$$ (\tilde{X}_{\text{sub,biased}} - \tilde{X}_{\text{sub,fid}}) C_{\text{sub}}^{-1} (\tilde{X}_{\text{sub,biased}} - \tilde{X}_{\text{sub,fid}})^T < \Delta \chi_{\text{thresh}}^2, \quad (22) $$

where $\tilde{X}_{\text{sub,biased}}$ and $\tilde{X}_{\text{sub,fid}}$ are subsets of the data vectors used in equation (20), where the subsets correspond to specific tomographic bin combination (and threshold, when using CDFs), $C_{\text{sub}}$ is the covariance matrix of the subset, and $\Delta \chi_{\text{thresh}}^2$ is the maximum change in $\chi^2$ we allow for the full data vector. In practice, we vary $\Delta \chi_{\text{thresh}}^2$ until the parameter bias goes below our required threshold. The data points that have been removed to achieve this condition define the scale cuts.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** The difference in CDFs measured on dark matter-only (DMO) simulations and ‘baryonified’ DMO simulations. As expected, baryon imprints are a significant effect on the data vector. The grey band shows the scales below $\theta < 6.4'$, which is the pixel resolution of the COSMOGRID DES Y3 maps, and is a factor of 2 larger than the other maps we consider in this work. Thus, the baryon effects we estimate below that scale are an underestimate of the true effect given the pixel resolution will suppress these effects. The total signal-to-noise of baryon imprints is $3.5\sigma$, though this is a lower bound given the suppression due to map resolution.</figcaption></figure>

as the PSFs (Section 5.3) and the reduced shear approximation (Section 5.5), while others are prominent at a subset of scales, such as baryon imprints (Section 5.6). Thus, using the CDFs to robustly infer cosmological constraints will require us to discard some parts of the fiducial data vector – namely the parts where the amplitude of the systematics is high – and obtain constraints using the remaining fraction of the data vector.

Amongst all the systematic effects considered in this work, the most significant are the baryon imprints (Fig. 11) and the source clustering effect (Fig. 8). These will determine how the data vector is truncated. Our scale cuts are determined by requiring that the parameter bias due to unmodelled systematic effects is below a certain threshold. We compute this bias using the extended Fisher formalism of Amara & Réfrégier (2008) and Asgari et al. (2021a),

$$ \Delta_p^{\text{bias}} = \sum_q (F^{-1})_{pq} \frac{d\tilde{X}_{\text{fid}}}{dp} C^{-1} (\tilde{X}_{\text{biased}} - \tilde{X}_{\text{fid}}), \quad (20) $$

where both $p$ and $q$ are indexes over the cosmological parameters of interest. The average bias in the data vector, $\tilde{X}_{\text{biased}} - \tilde{X}_{\text{fid}}$, is a quantity we have already computed and presented in the above subsections. We then summarize this bias-per-parameter, $\Delta_p^{\text{bias}}$, into a bias for the full N-D posterior as

$$ \delta = \sqrt{\sum_{p,q} \Delta_p^{\text{bias}} (C^{-1})_{pq} \Delta_q^{\text{bias}}}, \quad (21) $$

where $C$ is the covariance of the parameters, and so $C^{-1}$ is just the Fisher matrix, $F$. Our procedure for scale cuts is simply removing data points until $\delta < X$, where $X$ is some chosen threshold. We will use $X \in \{0.3, 0.2, 0.1\}$. The choice $X = 0.3$ matches the tests done in the main methodology pipeline for DES Y3 (e.g. Krause et al.

2021; Amon et al. 2022; Secco et al. 2022a) while the other values are chosen to explore more stringent cuts that could be reflective of Stage IV surveys. Note that this threshold, $X$, is somewhat arbitrary, but that is not a concern as our goal is to see how the scale-cuts for the CDFs compare to those for the moments; as long as the same choices are applied across both statistics, the arbitrariness of the choices is not relevant.

The other component we must decide is how to determine and discard data points to achieve the condition $\delta < X$, as there is significant freedom in doing so. We could throw away all data points for every bin/threshold corresponding to aperture scales below a certain chosen value. However, the choice of a fixed scale threshold is suboptimal as the impact of systematics at a chosen scale varies across bins and thresholds (as seen in any of the Figures above). Thus, our choice here is a scale cut done bin-by-bin (and threshold-by-threshold, in the case of CDFs) and follows the approach of Amon et al. (2022); Secco et al. (2022a). We compute the chi-squared of a given effect in a specific tomographic bin combination (and also specific threshold, in the case of CDFs), and remove the data points corresponding to the smallest scales until we satisfy the relation,

$$ (\tilde{X}_{\text{sub,biased}} - \tilde{X}_{\text{sub,fid}}) C_{\text{sub}}^{-1} (\tilde{X}_{\text{sub,biased}} - \tilde{X}_{\text{sub,fid}})^T < \Delta \chi_{\text{thresh}}^2, \quad (22) $$

where $\tilde{X}_{\text{sub,biased}}$ and $\tilde{X}_{\text{sub,fid}}$ are subsets of the data vectors used in equation (20), where the subsets correspond to specific tomographic bin combination (and threshold, when using CDFs), $C_{\text{sub}}$ is the covariance matrix of the subset, and $\Delta \chi_{\text{thresh}}^2$ is the maximum change in $\chi^2$ we allow for the full data vector. In practice, we vary $\Delta \chi_{\text{thresh}}^2$ until the parameter bias goes below our required threshold. The data points that have been removed to achieve this condition define the scale cuts.
<table><caption>**Table 2.** The Fisher information constraints presented in this work for CDFs measured on simulations and for a joint analysis of $\Omega_m$, $\sigma_8$, and $w_0$, but after implementing various types of scale cuts. From top to bottom, we do (i) simple, fixed angular scale cuts, and then cuts based on (ii) baryonic imprints and (iii) source clustering. The cuts are made by removing data points until $\delta < X$, where $\delta$ – defined in equation (21) – is the total parameter bias in a full N-D parameter space. We show the size of the modified data vector in the rightmost column. The FoM is quoted relative to the FoM of the CDFs constraints with no scale cuts.</caption><thead><tr><th>Scale Cut</th><th>σ(Ω<sub>m</sub>)</th><th>σ(σ<sub>8</sub>)</th><th>σ(w<sub>0</sub>)</th><th>FoM</th><th>N<sub>dof</sub></th></tr></thead><tbody><tr><td colspan="6" style="text-align: center; font-style: italic;">Fixed angular cuts</td></tr><tr><td>θ > 3'</td><td><strong>0.018</strong></td><td><strong>0.025</strong></td><td><strong>0.15</strong></td><td><strong>1.0</strong></td><td><strong>460</strong></td></tr><tr><td>θ > 10'</td><td>0.022</td><td>0.032</td><td>0.18</td><td>0.85</td><td>410</td></tr><tr><td>θ > 20'</td><td>0.025</td><td>0.035</td><td>0.21</td><td>0.59</td><td>270</td></tr><tr><td colspan="6" style="text-align: center; font-style: italic;">Baryonic imprints cuts</td></tr><tr><td>δ < 0.3</td><td>0.033</td><td>0.053</td><td>0.33</td><td>0.14</td><td>129</td></tr><tr><td>δ < 0.2</td><td>0.035</td><td>0.061</td><td>0.39</td><td>0.10</td><td>100</td></tr><tr><td>δ < 0.1</td><td>0.036</td><td>0.062</td><td>0.42</td><td>0.08</td><td>92</td></tr><tr><td colspan="6" style="text-align: center; font-style: italic;">Source clustering cuts</td></tr><tr><td>δ < 0.3</td><td>0.038</td><td>0.063</td><td>0.44</td><td>0.07</td><td>84</td></tr><tr><td>δ < 0.2</td><td>0.038</td><td>0.078</td><td>0.57</td><td>0.05</td><td>71</td></tr><tr><td>δ < 0.1</td><td>0.042</td><td>0.109</td><td>0.82</td><td>0.03</td><td>34</td></tr></tbody></table>

Once the scale-cuts have been defined, we recompute the Fisher constraints using the truncated data vector; the results for the CDFs are tabulated in Table 2. The table also shows constraints from generic scale cuts, where we set a fixed minimum angular scale for all tomographic bins and all thresholds. For the fixed angular scale cuts of 3.2', 10', 20', baryonic effects cause a parameter bias of $\delta = 1.2$, 0.6, and 0.3, respectively. Cutting all scales below 20' causes a fractional change of $\approx 30$ per cent in the constraints. At these scale cuts, the CDFs are comparable to combining second and third Moments, and we have verified that combining the CDFs with the

moments still leads to a 30 per cent improvement in the constraints. The PSFs and reduced shear effect have no impact on scale cuts so we do not show them here. Note that, as was discussed in Section 5.6, the impact of baryonic effects is an underestimate given the baryonified COSMOGRID maps used to estimate the effect have a 6.4' minimum resolution scale. Baryon effects are more impactful at smaller scales and will be more than 10 per cent of the signal if the resolution limit is corrected. However, for our goal of consistently comparing the impacts on CDFs and Moments, this suppression is not a limiting factor.

Table 2 shows that baryon imprints and source clustering both cause notable differences in the parameter constraints, especially in $\sigma_8$ and $w_0$. The FoM in the 3D parameter space drops by a factor of nearly 10 after implementing these scale cuts, which highlights the growing need to improve modelling of these effects instead of robustly trimming the data vector to be insensitive to the effects. Note that while the impact of source clustering on determining the scale cuts is larger than that of the baryonic imprints – which is counter to the standard expectation – this is once again because of the suppression of baryon effects on the small scales due to the resolution scale of the COSMOGRID data products.

Fig. 12 and Table 3 also show the results from defining scale cuts using both baryon imprints and source clustering, and doing so for CDFs and for the second and third moments. This provides a self-consistent reference to compare the two data vectors. The combination of scale cuts is done by looking at both baryonic effects and source clustering, and at each data point we pick the amplitude of the effect that is highest, i.e. $E = \max|\text{Baryons, SC}|$ for each data point. We find that the moments' constraints are comparable to the CDFs' after these scale cuts. Once we remove $w_0$ from the analysis the scale cuts cause only a factor of 3 degradation of the FoM as opposed to the factor of 10 if we include $w_0$.

Generally, one may expect the CDFs to be less sensitive to these effects than the moments; reduced shear, source clustering, and baryon imprints are all effects that grow with the amplitude of the

<figure><img src="image_12.png" /><figcaption>**Figure 12.** The Fisher constraints from CDFs (left) and second + third moments (right) measured on simulations. We present four cases, where we either have no scale cuts or cut the data vector so the parameter bias in the $\Omega_m - \sigma_8 - w_0$ contour is below a certain value; see equation (21). The CDFs and the moments have comparable constraints, which are denoted in Table 3.</figcaption></figure>
<table><caption>**Table 2.** The Fisher information constraints presented in this work for CDFs measured on simulations and for a joint analysis of $\Omega_m$, $\sigma_8$, and $w_0$, but after implementing various types of scale cuts. From top to bottom, we do (i) simple, fixed angular scale cuts, and then cuts based on (ii) baryonic imprints and (iii) source clustering. The cuts are made by removing data points until $\delta < X$, where $\delta$ – defined in equation (21) – is the total parameter bias in a full N-D parameter space. We show the size of the modified data vector in the rightmost column. The FoM is quoted relative to the FoM of the CDFs constraints with no scale cuts.</caption><thead><tr><th>Scale Cut</th><th>σ(Ω<sub>m</sub>)</th><th>σ(σ<sub>8</sub>)</th><th>σ(w<sub>0</sub>)</th><th>FoM</th><th>N<sub>dof</sub></th></tr></thead><tbody><tr><td colspan="6" style="text-align: center; font-style: italic;">Fixed angular cuts</td></tr><tr><td>θ > 3'</td><td><strong>0.018</strong></td><td><strong>0.025</strong></td><td><strong>0.15</strong></td><td><strong>1.0</strong></td><td><strong>460</strong></td></tr><tr><td>θ > 10'</td><td>0.022</td><td>0.032</td><td>0.18</td><td>0.85</td><td>410</td></tr><tr><td>θ > 20'</td><td>0.025</td><td>0.035</td><td>0.21</td><td>0.59</td><td>270</td></tr><tr><td colspan="6" style="text-align: center; font-style: italic;">Baryonic imprints cuts</td></tr><tr><td>δ < 0.3</td><td>0.033</td><td>0.053</td><td>0.33</td><td>0.14</td><td>129</td></tr><tr><td>δ < 0.2</td><td>0.035</td><td>0.061</td><td>0.39</td><td>0.10</td><td>100</td></tr><tr><td>δ < 0.1</td><td>0.036</td><td>0.062</td><td>0.42</td><td>0.08</td><td>92</td></tr><tr><td colspan="6" style="text-align: center; font-style: italic;">Source clustering cuts</td></tr><tr><td>δ < 0.3</td><td>0.038</td><td>0.063</td><td>0.44</td><td>0.07</td><td>84</td></tr><tr><td>δ < 0.2</td><td>0.038</td><td>0.078</td><td>0.57</td><td>0.05</td><td>71</td></tr><tr><td>δ < 0.1</td><td>0.042</td><td>0.109</td><td>0.82</td><td>0.03</td><td>34</td></tr></tbody></table>

Once the scale-cuts have been defined, we recompute the Fisher constraints using the truncated data vector; the results for the CDFs are tabulated in Table 2. The table also shows constraints from generic scale cuts, where we set a fixed minimum angular scale for all tomographic bins and all thresholds. For the fixed angular scale cuts of 3.2', 10', 20', baryonic effects cause a parameter bias of $\delta = 1.2$, 0.6, and 0.3, respectively. Cutting all scales below 20' causes a fractional change of $\approx 30$ per cent in the constraints. At these scale cuts, the CDFs are comparable to combining second and third Moments, and we have verified that combining the CDFs with the

moments still leads to a 30 per cent improvement in the constraints. The PSFs and reduced shear effect have no impact on scale cuts so we do not show them here. Note that, as was discussed in Section 5.6, the impact of baryonic effects is an underestimate given the baryonified COSMOGRID maps used to estimate the effect have a 6.4' minimum resolution scale. Baryon effects are more impactful at smaller scales and will be more than 10 per cent of the signal if the resolution limit is corrected. However, for our goal of consistently comparing the impacts on CDFs and Moments, this suppression is not a limiting factor.

Table 2 shows that baryon imprints and source clustering both cause notable differences in the parameter constraints, especially in $\sigma_8$ and $w_0$. The FoM in the 3D parameter space drops by a factor of nearly 10 after implementing these scale cuts, which highlights the growing need to improve modelling of these effects instead of robustly trimming the data vector to be insensitive to the effects. Note that while the impact of source clustering on determining the scale cuts is larger than that of the baryonic imprints – which is counter to the standard expectation – this is once again because of the suppression of baryon effects on the small scales due to the resolution scale of the COSMOGRID data products.

Fig. 12 and Table 3 also show the results from defining scale cuts using both baryon imprints and source clustering, and doing so for CDFs and for the second and third moments. This provides a self-consistent reference to compare the two data vectors. The combination of scale cuts is done by looking at both baryonic effects and source clustering, and at each data point we pick the amplitude of the effect that is highest, i.e. $E = \max|\text{Baryons, SC}|$ for each data point. We find that the moments' constraints are comparable to the CDFs' after these scale cuts. Once we remove $w_0$ from the analysis the scale cuts cause only a factor of 3 degradation of the FoM as opposed to the factor of 10 if we include $w_0$.

Generally, one may expect the CDFs to be less sensitive to these effects than the moments; reduced shear, source clustering, and baryon imprints are all effects that grow with the amplitude of the

<figure><img src="image_14.png" /><figcaption>**Figure 12.** The Fisher constraints from CDFs (left) and second + third moments (right) measured on simulations. We present four cases, where we either have no scale cuts or cut the data vector so the parameter bias in the $\Omega_m - \sigma_8 - w_0$ contour is below a certain value; see equation (21). The CDFs and the moments have comparable constraints, which are denoted in Table 3.</figcaption></figure>
<table><caption>**Table 3.** The Fisher information constraints presented in this work for a joint analysis of either $\Omega_m$, $\sigma_8$, and $w_0$ (top two) or just $\Omega_m$ and $\sigma_8$ (bottom two), but after implementing scale cuts to reduce the parameter bias. We show the constraints, after scale cuts, for both the CDFs and for the combination of second and third moments. We show the size of the modified data vector in the rightmost column. The FoM is quoted relative to the FoM of the CDFs constraints with no scale cuts.</caption><thead><tr><th>Scale Cut</th><th>σ(Ω<sub>m</sub>)</th><th>σ(σ<sub>8</sub>)</th><th>σ(w<sub>0</sub>)</th><th>FOM</th><th>N<sub>dof</sub></th></tr></thead><tbody><tr><td colspan="6" style="text-align: center; font-weight: bold;">CDFs, All cuts (Ω<sub>m</sub>, σ<sub>8</sub>, w<sub>0</sub>)</td></tr><tr><td>δCDF &lt; 0.3</td><td><strong>0.037</strong></td><td><strong>0.063</strong></td><td><strong>0.44</strong></td><td>0.07</td><td>84</td></tr><tr><td>δCDF &lt; 0.2</td><td>0.037</td><td>0.074</td><td>0.52</td><td>0.05</td><td>75</td></tr><tr><td>δCDF &lt; 0.1</td><td>0.040</td><td>0.102</td><td>0.75</td><td>0.03</td><td>49</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">2nd and 3rd moments, all cuts (Ω<sub>m</sub>, σ<sub>8</sub>, w<sub>0</sub>)</td></tr><tr><td>δMoments &lt; 0.3</td><td>0.037</td><td>0.050</td><td>0.34</td><td>0.13</td><td>109</td></tr><tr><td>δMoments &lt; 0.2</td><td>0.040</td><td>0.076</td><td>0.50</td><td>0.06</td><td>86</td></tr><tr><td>δMoments &lt; 0.1</td><td>0.045</td><td>0.105</td><td>0.74</td><td>0.03</td><td>58</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">CDFs, all cuts (Ω<sub>m</sub>, σ<sub>8</sub>)</td></tr><tr><td>δCDF &lt; 0.3</td><td>0.033</td><td>0.050</td><td>—</td><td>0.27</td><td>99</td></tr><tr><td>δCDF &lt; 0.2</td><td>0.035</td><td>0.053</td><td>—</td><td>0.20</td><td>79</td></tr><tr><td>δCDF &lt; 0.1</td><td>0.038</td><td>0.057</td><td>—</td><td>0.16</td><td>52</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">2nd and 3rd moments, all cuts (Ω<sub>m</sub>, σ<sub>8</sub>)</td></tr><tr><td>δMoments &lt; 0.3</td><td>0.031</td><td>0.044</td><td>—</td><td>0.34</td><td>118</td></tr><tr><td>δMoments &lt; 0.2</td><td>0.036</td><td>0.051</td><td>—</td><td>0.22</td><td>91</td></tr><tr><td>δMoments &lt; 0.1</td><td>0.044</td><td>0.058</td><td>—</td><td>0.12</td><td>4</td></tr></tbody></table>

density field and/or the convergence field. This means they impact the tails of the density/lensing distribution the most and leave the ‘bulk’ of the PDF – roughly the 68 per cent or the 95 per cent region centred around the median – relatively unaffected. The moments are defined as an integral over the whole distribution and so cannot isolate just parts of it. The CDFs on the other hand can perform such an isolation. They fundamentally only probe whether or not a pixel’s convergence is above a given threshold; thus, if the convergence is well above/below the threshold, the measurement of the CDFs is unaffected by that pixel value shifting around due to various effects. For example, the negative thresholds $k < 0$ will be unaffected by the baryon imprints in massive haloes, as massive haloes exist in $\kappa > 0$ regions and baryon imprints reduce the $\kappa$ value but always keep it positive, and so the convergence around haloes will always be above the $k < 0$. Of course, if the $\kappa$ values of interest are near a threshold, then any shifts will have a stronger impact on the CDF measurements at that threshold. This argument also suggests there are a particular choice of thresholds that balance constraining power while alleviating such systematics. We have not explored such an optimal selection. In Table 3, we also redo the scale cuts but now leave out $w_0$ when computing the total parameter bias, as this is a closer match to the procedures used in Stage III surveys (e.g. Krause et al. 2021). Our qualitative findings remain the same even in this case.

## 6 CONCLUSIONS

In this work, we have explored the use of the Cumulative Distribution Functions (CDFs) of the convergence field as a summary statistic for extracting cosmological information, drawing on the development of the kNN distributions for the discrete fields. The CDFs are a convenient, succinct summary of the field that approximately capture all higher moments of the field in a significantly shorter data vector that is also quicker to compute. We explore the theoretical advantages of using these CDFs and check their sensitivity to the relevant practical challenges in extracting robust cosmology constraints from Y3-like data. The conclusions of this work are as follows:

(i) For scales of $3' < \theta < 200'$ and tomographic bins of DES Y3, the CDFs have better constraints on $\Omega_m$, $\sigma_8$ and $w$ when compared to those from the combination of both second and third Moments (Fig. 4). This improvement is modest, but the CDFs still have a slightly different degeneracy direction to the moments, and combining the CDFs and moments leads to the constraints improving by 20–30 per cent.

(ii) The CDFs measured on a Gaussian field provide Fisher constraints that are completely consistent with the angular power spectra and second Moments computed on the fully nonlinear, non-Gaussian field (Fig. 4). The CDFs and moments all have Gaussian likelihoods as well (Fig. B1).

(iii) The DES Y3 noise field is highly non-Gaussian, with a very significant fourth moment (Fig. 6). There is some cosmological signal at large scales in the fourth moment, but none in the fifth moment.

(iv) We create a PSF ‘mass map’ for testing PSF contributions at the map level, and show the signal from PSF shapes is 2–3 orders of magnitude below the cosmological signal (Fig. 7). This validates not only the CDFs, but also indirectly validates the minimal impact of the PSFs on information beyond the third moment (existing works have already validated them at the second and third moment level).

(v) The presence or lack of spatial correlations in the source galaxy number counts, i.e. ‘source clustering’, impacts the convergence field model at the 1–10 per cent level (Fig. 8).

(vi) The CDFs are sensitive to correlations between the convergence field and the shape noise field, induced by source clustering. We detect these correlations at $13\sigma$, and can adequately model them in the simulated maps (Fig. 9).

(vii) The reduced shear approximation changes the cosmological signal at the 1–5 per cent level (Fig. 10), while baryon imprints are 1–10 per cent of the cosmological signal (Fig. 11).

(viii) We perform scale cuts that limit the parameter bias due to systematic effects under a certain level. The cut CDF data vector has comparable constraining power to the cut data vector of the second and third Moments (Table 2 and Fig. 12).

Optimizing the summary of fields is a rich area of study, with a variety of approaches and outcomes. The CDFs, through their sensitivity to all moments of the field, probe both the cosmological signal at all these orders as well as any potential modelling challenges that surface at these orders (e.g. the high kurtosis of the noise field that does not impact 2-point and 3-point functions). This sensitivity to all orders becomes a more relevant trait as we extend our analyses to smaller scales, which are more nonlinear and thus more non-Gaussian. It may also become relevant in constraining – and/or marginalizing over – the impact of baryons on the density field; these effects happen pre-dominantly within haloes, and so are localized around the most nonlinear regions of the density field and thus will have non-Gaussian signatures. The CDFs might also be one of the few ways to probe the highest orders of information in the field. They are more robust given they can isolate specific parts of the distribution, and this is in contrast to the higher order moments which will be increasingly sensitive to noise/outliers in the tails of the distribution. Thus, if there is significant, usable higher-order information in the cosmological field (for example, in future surveys with different noise levels and sensitivities), the CDF may be one of the only ways to robustly access it.

While efforts have already been made to obtain cosmology from up to the third moment, we show there remains some information beyond the third moment that can likely be accessed in a robust manner, i.e. without worrying about systematics. Effects like reduced shear, source clustering, and baryons have some impact that is at
<table><caption>**Table 3.** The Fisher information constraints presented in this work for a joint analysis of either $\Omega_m$, $\sigma_8$, and $w_0$ (top two) or just $\Omega_m$ and $\sigma_8$ (bottom two), but after implementing scale cuts to reduce the parameter bias. We show the constraints, after scale cuts, for both the CDFs and for the combination of second and third moments. We show the size of the modified data vector in the rightmost column. The FoM is quoted relative to the FoM of the CDFs constraints with no scale cuts.</caption><thead><tr><th>Scale Cut</th><th>σ(Ω<sub>m</sub>)</th><th>σ(σ<sub>8</sub>)</th><th>σ(w<sub>0</sub>)</th><th>FOM</th><th>N<sub>dof</sub></th></tr></thead><tbody><tr><td colspan="6">CDFs, All cuts (Ω<sub>m</sub>, σ<sub>8</sub>, w<sub>0</sub>)</td></tr><tr><td>δCDF &lt; 0.3</td><td><strong>0.037</strong></td><td><strong>0.063</strong></td><td><strong>0.44</strong></td><td>0.07</td><td>84</td></tr><tr><td>δCDF &lt; 0.2</td><td>0.037</td><td>0.074</td><td>0.52</td><td>0.05</td><td>75</td></tr><tr><td>δCDF &lt; 0.1</td><td>0.040</td><td>0.102</td><td>0.75</td><td>0.03</td><td>49</td></tr><tr><td colspan="6">2nd and 3rd moments, all cuts (Ω<sub>m</sub>, σ<sub>8</sub>, w<sub>0</sub>)</td></tr><tr><td>δMoments &lt; 0.3</td><td>0.037</td><td>0.050</td><td>0.34</td><td>0.13</td><td>109</td></tr><tr><td>δMoments &lt; 0.2</td><td>0.040</td><td>0.076</td><td>0.50</td><td>0.06</td><td>86</td></tr><tr><td>δMoments &lt; 0.1</td><td>0.045</td><td>0.105</td><td>0.74</td><td>0.03</td><td>58</td></tr><tr><td colspan="6">CDFs, all cuts (Ω<sub>m</sub>, σ<sub>8</sub>)</td></tr><tr><td>δCDF &lt; 0.3</td><td>0.033</td><td>0.050</td><td>—</td><td>0.27</td><td>99</td></tr><tr><td>δCDF &lt; 0.2</td><td>0.035</td><td>0.053</td><td>—</td><td>0.20</td><td>79</td></tr><tr><td>δCDF &lt; 0.1</td><td>0.038</td><td>0.057</td><td>—</td><td>0.16</td><td>52</td></tr><tr><td colspan="6">2nd and 3rd moments, all cuts (Ω<sub>m</sub>, σ<sub>8</sub>)</td></tr><tr><td>δMoments &lt; 0.3</td><td>0.031</td><td>0.044</td><td>—</td><td>0.34</td><td>118</td></tr><tr><td>δMoments &lt; 0.2</td><td>0.036</td><td>0.051</td><td>—</td><td>0.22</td><td>91</td></tr><tr><td>δMoments &lt; 0.1</td><td>0.044</td><td>0.058</td><td>—</td><td>0.12</td><td>4</td></tr></tbody></table>

density field and/or the convergence field. This means they impact the tails of the density/lensing distribution the most and leave the ‘bulk’ of the PDF – roughly the 68 per cent or the 95 per cent region centred around the median – relatively unaffected. The moments are defined as an integral over the whole distribution and so cannot isolate just parts of it. The CDFs on the other hand can perform such an isolation. They fundamentally only probe whether or not a pixel’s convergence is above a given threshold; thus, if the convergence is well above/below the threshold, the measurement of the CDFs is unaffected by that pixel value shifting around due to various effects. For example, the negative thresholds $k < 0$ will be unaffected by the baryon imprints in massive haloes, as massive haloes exist in $\kappa > 0$ regions and baryon imprints reduce the $\kappa$ value but always keep it positive, and so the convergence around haloes will always be above the $k < 0$. Of course, if the $\kappa$ values of interest are near a threshold, then any shifts will have a stronger impact on the CDF measurements at that threshold. This argument also suggests there are a particular choice of thresholds that balance constraining power while alleviating such systematics. We have not explored such an optimal selection. In Table 3, we also redo the scale cuts but now leave out $w_0$ when computing the total parameter bias, as this is a closer match to the procedures used in Stage III surveys (e.g. Krause et al. 2021). Our qualitative findings remain the same even in this case.

## 6 CONCLUSIONS

In this work, we have explored the use of the Cumulative Distribution Functions (CDFs) of the convergence field as a summary statistic for extracting cosmological information, drawing on the development of the kNN distributions for the discrete fields. The CDFs are a convenient, succinct summary of the field that approximately capture all higher moments of the field in a significantly shorter data vector that is also quicker to compute. We explore the theoretical advantages of using these CDFs and check their sensitivity to the relevant practical challenges in extracting robust cosmology constraints from Y3-like data. The conclusions of this work are as follows:

(i) For scales of $3' < \theta < 200'$ and tomographic bins of DES Y3, the CDFs have better constraints on $\Omega_m$, $\sigma_8$ and $w$ when compared to those from the combination of both second and third Moments (Fig. 4). This improvement is modest, but the CDFs still have a slightly different degeneracy direction to the moments, and combining the CDFs and moments leads to the constraints improving by 20–30 per cent.

(ii) The CDFs measured on a Gaussian field provide Fisher constraints that are completely consistent with the angular power spectra and second Moments computed on the fully nonlinear, non-Gaussian field (Fig. 4). The CDFs and moments all have Gaussian likelihoods as well (Fig. B1).

(iii) The DES Y3 noise field is highly non-Gaussian, with a very significant fourth moment (Fig. 6). There is some cosmological signal at large scales in the fourth moment, but none in the fifth moment.

(iv) We create a PSF ‘mass map’ for testing PSF contributions at the map level, and show the signal from PSF shapes is 2–3 orders of magnitude below the cosmological signal (Fig. 7). This validates not only the CDFs, but also indirectly validates the minimal impact of the PSFs on information beyond the third moment (existing works have already validated them at the second and third moment level).

(v) The presence or lack of spatial correlations in the source galaxy number counts, i.e. ‘source clustering’, impacts the convergence field model at the 1–10 per cent level (Fig. 8).

(vi) The CDFs are sensitive to correlations between the convergence field and the shape noise field, induced by source clustering. We detect these correlations at $13\sigma$, and can adequately model them in the simulated maps (Fig. 9).

(vii) The reduced shear approximation changes the cosmological signal at the 1–5 per cent level (Fig. 10), while baryon imprints are 1–10 per cent of the cosmological signal (Fig. 11).

(viii) We perform scale cuts that limit the parameter bias due to systematic effects under a certain level. The cut CDF data vector has comparable constraining power to the cut data vector of the second and third Moments (Table 2 and Fig. 12).

Optimizing the summary of fields is a rich area of study, with a variety of approaches and outcomes. The CDFs, through their sensitivity to all moments of the field, probe both the cosmological signal at all these orders as well as any potential modelling challenges that surface at these orders (e.g. the high kurtosis of the noise field that does not impact 2-point and 3-point functions). This sensitivity to all orders becomes a more relevant trait as we extend our analyses to smaller scales, which are more nonlinear and thus more non-Gaussian. It may also become relevant in constraining – and/or marginalizing over – the impact of baryons on the density field; these effects happen pre-dominantly within haloes, and so are localized around the most nonlinear regions of the density field and thus will have non-Gaussian signatures. The CDFs might also be one of the few ways to probe the highest orders of information in the field. They are more robust given they can isolate specific parts of the distribution, and this is in contrast to the higher order moments which will be increasingly sensitive to noise/outliers in the tails of the distribution. Thus, if there is significant, usable higher-order information in the cosmological field (for example, in future surveys with different noise levels and sensitivities), the CDF may be one of the only ways to robustly access it.

While efforts have already been made to obtain cosmology from up to the third moment, we show there remains some information beyond the third moment that can likely be accessed in a robust manner, i.e. without worrying about systematics. Effects like reduced shear, source clustering, and baryons have some impact that is at
the 0.1 per cent–10 per cent level depending on the effect and the angular scale. After enacting scale cuts to reduce the bias on cosmological constraints to be within 0.3σ, the CDF data vector still provides constraints better than those of the second and third moment data vector. We have identified that accurate modelling of the noise field at higher orders is the current limiting factor in robustly inferring cosmology from statistics like the CDFs. Alternatively, an accurate way of denoising the CDFs – which effectively bypasses requirements in modelling the noise field by removing its contribution from the data vector – would enable robust cosmology constraints with the CDFs.

Finally, we note that even though this work has specifically focused on validating the CDF as a summary statistic, the validation results have significant implications for the broader range of lensing convergence statistics discussed in the literature. The key underlying information is the distribution of convergence as a function of scale, $P(\kappa_{\theta})$, and the CDFs are a convenient and compact way of summarizing this distribution/information. Other statistics summarize this distribution in different ways, such as lensing-in-cells<fnref n="17" /> and Minkowski Functionals.<fnref n="18" /> As has been discussed above, another closely connected statistic is the moments of the field, $\langle\kappa_{\theta}^n\rangle$, which are a further summary of the distribution, $P(\kappa_{\theta}|\theta)$, and computing moments to an arbitrarily high order is equivalent to computing the CDF to arbitrarily many thresholds.

As we move towards Stage IV surveys with wider survey areas and deeper observations – both leading to higher precision measurements – other systematics could become relevant. As a rough example, the LSST Year 10 data set will have ~3 times the survey area as DES Y3 and ~5 times the source galaxy number density as DES Y3 (The LSST Dark Energy Science Collaboration 2018), which leads to a factor of 4 increase in precision of the data vector and in the significance of any systematic we discuss in this work. The reduced shear effect (Fig. 10) – which can be safely ignored in Stage III surveys – will likely need to be included in the model for Stage IV, especially for LSST’s highest redshift bins as the amplitude of the effect grows with redshift. However, this component can be trivially included via simulation-based modelling using the same approach we used to include its effects in our simulations (Section 5.5). Source clustering will also be a necessary modelling ingredient for Stage IV surveys as its signal-to-noise will exceed 1 for LSST. While this modelling can also be done through simulation-based modelling, it requires some galaxy bias prescription (equation (17)) which would introduce a modelling uncertainty that has yet to be quantified. Additionally, we discussed that the Born approximation is adequate for modelling the weak lensing field under DES-like uncertainties. However, previous works have shown that for Stage IV data quality we will require ray-tracing when using higher-order statistics (Petri, Haiman & May 2017).

These effects above – reduced shear, source clustering, and Born approximation – impact all statistical summaries of the lensing field, including the standard 2pt and 3pt functions. Systematics that will uniquely impact the CDFs are then effects that generate a fourth moment and beyond. We have already found in this work that the

fourth moment of the noise field is a highly relevant modelling component for the CDFs. In DES Y3, this was primarily sourced by the survey depth fluctuations as well as the intrinsic, cosmological clustering of source galaxies. In general, however, any process that spatially modifies the shape noise per galaxy or the number of galaxies per pixel will generate the fourth moment. For Stage IV surveys, the precision will be high enough that effects such as spatially varying multiplicative bias – which impacts the measured variance of the shape distribution – could also be a required modelling component, but we must first quantify how much this bias will actually vary across the sky.

The validation steps performed in this work have implications for the statistics mentioned above – lensing-in-cells, Minkowski Functionals, field moments etc. For example, it is likely that PSF ellipticity correlations will be a few orders of magnitude below the cosmological signal for all of these statistics. A similar case can be made for the impact of source clustering and the reduced shear approximation. Of course, it is still ideal to perform a separate validation for those statistics to explicitly verify their robustness to these effects, but the results of this work indicate – given the statistics all summarize the same underlying distribution, $P(\kappa_{\theta}|\theta)$ – that it is likely these other statistics will also be robust to these. By using the CDFs, which are approximately summarizing all higher order moments, we have tested these systematics at the map level and beyond the third moment. We hope the methodologies for map-level tests that we employed and/or introduced in this work enable more checks of the large library of higher-order statistics that are being developed for the convergence field, and thus enhance the trustworthiness of these newer statistics.

## ACKNOWLEDGEMENTS

DA is supported by NSF grant No. 2108168. CC is supported by the Henry Luce Foundation and DOE grant DE-SC0021949. We thank the referee for their insightful comments that helped improve this work.

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.

<footnote n="17">¹⁷This is the lensing-focused analogue of counts-in-cells, where the latter is the distribution of tracer counts within a given volume, $P(k_{\text{tr}}|V)$. If we replace trace counts with lensing convergence, then we obtain lensing-in-cells.</footnote>

<footnote n="18">¹⁸The CDFs are the same as the zeroth-order Minkowski functional, though in our formalism we also introduce a cross-correlation method – inspired by the formalism for kNNs in Banerjee & Abel (2021b) – which is traditionally not used/defined for the Minkowski Functionals.</footnote>
the 0.1 per cent–10 per cent level depending on the effect and the angular scale. After enacting scale cuts to reduce the bias on cosmological constraints to be within 0.3σ, the CDF data vector still provides constraints better than those of the second and third moment data vector. We have identified that accurate modelling of the noise field at higher orders is the current limiting factor in robustly inferring cosmology from statistics like the CDFs. Alternatively, an accurate way of denoising the CDFs – which effectively bypasses requirements in modelling the noise field by removing its contribution from the data vector – would enable robust cosmology constraints with the CDFs.

Finally, we note that even though this work has specifically focused on validating the CDF as a summary statistic, the validation results have significant implications for the broader range of lensing convergence statistics discussed in the literature. The key underlying information is the distribution of convergence as a function of scale, $P(\kappa_{\theta})$, and the CDFs are a convenient and compact way of summarizing this distribution/information. Other statistics summarize this distribution in different ways, such as lensing-in-cells<fnref n="17" /> and Minkowski Functionals.<fnref n="18" /> As has been discussed above, another closely connected statistic is the moments of the field, $\langle\kappa_{\theta}^n\rangle$, which are a further summary of the distribution, $P(\kappa_{\theta}|\theta)$, and computing moments to an arbitrarily high order is equivalent to computing the CDF to arbitrarily many thresholds.

As we move towards Stage IV surveys with wider survey areas and deeper observations – both leading to higher precision measurements – other systematics could become relevant. As a rough example, the LSST Year 10 data set will have ~3 times the survey area as DES Y3 and ~5 times the source galaxy number density as DES Y3 (The LSST Dark Energy Science Collaboration 2018), which leads to a factor of 4 increase in precision of the data vector and in the significance of any systematic we discuss in this work. The reduced shear effect (Fig. 10) – which can be safely ignored in Stage III surveys – will likely need to be included in the model for Stage IV, especially for LSST’s highest redshift bins as the amplitude of the effect grows with redshift. However, this component can be trivially included via simulation-based modelling using the same approach we used to include its effects in our simulations (Section 5.5). Source clustering will also be a necessary modelling ingredient for Stage IV surveys as its signal-to-noise will exceed 1 for LSST. While this modelling can also be done through simulation-based modelling, it requires some galaxy bias prescription (equation (17)) which would introduce a modelling uncertainty that has yet to be quantified. Additionally, we discussed that the Born approximation is adequate for modelling the weak lensing field under DES-like uncertainties. However, previous works have shown that for Stage IV data quality we will require ray-tracing when using higher-order statistics (Petri, Haiman & May 2017).

These effects above – reduced shear, source clustering, and Born approximation – impact all statistical summaries of the lensing field, including the standard 2pt and 3pt functions. Systematics that will uniquely impact the CDFs are then effects that generate a fourth moment and beyond. We have already found in this work that the

fourth moment of the noise field is a highly relevant modelling component for the CDFs. In DES Y3, this was primarily sourced by the survey depth fluctuations as well as the intrinsic, cosmological clustering of source galaxies. In general, however, any process that spatially modifies the shape noise per galaxy or the number of galaxies per pixel will generate the fourth moment. For Stage IV surveys, the precision will be high enough that effects such as spatially varying multiplicative bias – which impacts the measured variance of the shape distribution – could also be a required modelling component, but we must first quantify how much this bias will actually vary across the sky.

The validation steps performed in this work have implications for the statistics mentioned above – lensing-in-cells, Minkowski Functionals, field moments etc. For example, it is likely that PSF ellipticity correlations will be a few orders of magnitude below the cosmological signal for all of these statistics. A similar case can be made for the impact of source clustering and the reduced shear approximation. Of course, it is still ideal to perform a separate validation for those statistics to explicitly verify their robustness to these effects, but the results of this work indicate – given the statistics all summarize the same underlying distribution, $P(\kappa_{\theta}|\theta)$ – that it is likely these other statistics will also be robust to these. By using the CDFs, which are approximately summarizing all higher order moments, we have tested these systematics at the map level and beyond the third moment. We hope the methodologies for map-level tests that we employed and/or introduced in this work enable more checks of the large library of higher-order statistics that are being developed for the convergence field, and thus enhance the trustworthiness of these newer statistics.

## ACKNOWLEDGEMENTS

DA is supported by NSF grant No. 2108168. CC is supported by the Henry Luce Foundation and DOE grant DE-SC0021949. We thank the referee for their insightful comments that helped improve this work.

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.

<footnote n="17">¹⁷This is the lensing-focused analogue of counts-in-cells, where the latter is the distribution of tracer counts within a given volume, $P(k_{\text{tr}}|V)$. If we replace trace counts with lensing convergence, then we obtain lensing-in-cells.</footnote>

<footnote n="18">¹⁸The CDFs are the same as the zeroth-order Minkowski functional, though in our formalism we also introduce a cross-correlation method – inspired by the formalism for kNNs in Banerjee & Abel (2021b) – which is traditionally not used/defined for the Minkowski Functionals.</footnote>
gan, NSF's NOIRLab, 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 Na-
tional Science Foundation under Grant Numbers AST-1138766
and AST-1536171. The DES participants from Spanish institutions
are partially supported by MICINN under grants ESP2017-89838,
PGC2018-094773, PGC2018-102021, 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) do e-Universo (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.

DATA AVAILABILITY

All DES Y3 data used in this work are publicly available at
https://des.ncsa.illinois.edu/releases/y3a2/. The ULAGAM simulation
suite, which was denoted as A23 throughout this work for brevity, is
available at https://ulagam-simulations.readthedocs.io.

REFERENCES

Adelberger K. L., Steidel C. C., Giavalisco M., Dickinson M., Pettini M., Kellogg M., 1998, ApJ, 505, 18

Allys E., Marchand T., Cardoso J. F., Villaescusa-Navarro F., Ho S., Mallat S., 2020, Phys. Rev. D, 102, 103506

Amara A., Réfrégier A., 2008, MNRAS, 391, 228

Amodeo S. et al., 2021, Phys. Rev. D, 103, 063514

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

Anbajagane D., Evrard A. E., Farahi A., Barnes D. J., Dolag K., McCarthy I. G., Nelson D., Pillepich A., 2020, MNRAS, 495, 686

Anbajagane D., Evrard A. E., Farahi A., 2022a, MNRAS, 509, 3441

Anbajagane D. et al., 2022b, MNRAS, 510, 2980

Anbajagane D. et al., 2022c, MNRAS, 514, 1645

Anbajagane D. et al., 2023a, preprint (arXiv:2310.00059)

Anbajagane D., Chang C., Lee H., Gatti M., 2023b, preprint
(arXiv:2310.02349)

Asgari M., Friswell I., Yoon M., Heymans C., Dvornik A., Joachimi B., Simon P., Zuntz J., 2021a, MNRAS, 501, 3003

Asgari M. et al., 2021b, A&A, 645, A104

Banerjee A., Abel T., 2021a, MNRAS, 500, 5479

Banerjee A., Abel T., 2021b, MNRAS, 504, 2911

Banerjee A., Abel T., 2023, MNRAS, 519, 4856

Barthelemy A., Halder A., Gong Z., Uhlemann C., 2023, preprint
(arXiv:2307.09468)

Baugh C. M., Gaztanaga E., Efstathiou G., 1995, MNRAS, 274, 1049

Beltz-Mohrmann G. D., Berlind A. A., 2021, preprint (arXiv:2103.05076)

Bernardeau F., 1998, A&A, 338, 375

Blake C., James J. B., Poole G. B., 2014, MNRAS, 437, 2488

Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301, 27

Boyle A., Uhlemann C., Friedrich O., Barthelemy A., Codis S., Bernardeau
F., Giocoli C., Baldi M., 2021, MNRAS, 505, 2886

Carlstrom J. E., Holder G. P., Reese E. D., 2002, ARA&A, 40, 643

Cataneo M., Uhlemann C., Arnold C., Gough A., Li B., Heymans C., 2022,
MNRAS, 513, 1623

Chang C. et al., 2018, MNRAS, 475, 3165

Cheng S., Ménard B., 2021, MNRAS, 507, 1012

Chisari N. E. et al., 2018, MNRAS, 480, 3962

Cui W. et al., 2022, MNRAS, 514, 977

Dalal R. et al., 2023, preprint (arXiv:2304.00701)

Davies C. T., Cautun M., Giblin B., Li B., Harnois-Déraps J., Cai Y.-C., 2021,
MNRAS, 507, 2267

Doux C. et al., 2022, MNRAS, 515, 1942

Duffy A. R., Schaye J., Kay S. T., Dalla Vecchia C., Battye R. A., Booth C.
M., 2010, MNRAS, 405, 2161

Euclid Collaboration, 2023, preprint (arXiv:2301.12890)

Fluri J., Kacprzak T., Refregier A., Amara A., Lucchi A., Hofmann T., 2018,
Phys. Rev. D, 98, 123518

Fluri J., Kacprzak T., Lucchi A., Refregier A., Amara A., Hofmann T.,
Schneider A., 2019, Phys. Rev. D, 100, 063514

Fluri J., Kacprzak T., Lucchi A., Schneider A., Refregier A., Hofmann T.,
2022, Phys. Rev. D, 105, 083518

Friedrich O. et al., 2018, Phys. Rev. D, 98, 023508

Friedrich O., Uhlemann C., Villaescusa-Navarro F., Baldauf T., Manera M.,
Nishimichi T., 2020, MNRAS, 498, 464

Fu L. et al., 2014, MNRAS, 441, 2725

Gatti M. et al., 2020, MNRAS, 498, 4060

Gatti M. et al., 2021, MNRAS, 504, 4312

Gatti M. et al., 2022, Phys. Rev. D, 106, 083509

Gatti M. et al., 2023, preprint (arXiv:2307.13860)

Giannantonio T., Scranton R., Crittenden R. G., Nichol R. C., Boughn S. P., Myers A. D., Richards G. T., 2008, Phys. Rev. D, 77, 123520

Giblin B., Cai Y.-C., Harnois-Déraps J., 2023, MNRAS, 520, 1721

Gnedin O. Y., Kravtsov A. V., Klypin A. A., Nagai D., 2004, ApJ, 616, 16

Gong Z., Halder A., Barreira A., Seitz S., Friedrich O., 2023, preprint
(arXiv:2304.01187)

Gough A., Uhlemann C., 2022, Universe, 8, 55

Gruen D. et al., 2018, Phys. Rev. D, 98, 023507

Halder A., Friedrich O., Seitz S., Varga T. N., 2021, MNRAS, 506, 2780

Hamana T., Colombi S. T., Thion A., Devriendt J. E. G. T., Mellier Y., Bernardeau F., 2002, MNRAS, 330, 365

Hartlap J., Simon P., Schneider P., 2007, A&A, 464, 399

Heydenreich S., Brück B., Harnois-Déraps J., 2021, A&A, 648, A74

Heydenreich S., Brück B., Burger P., Harnois-Déraps J., Unruh S., Castro T., Dolag K., Martinet N., 2022, A&A, 667, A125

Heydenreich S., Linke L., Burger P., Schneider P., 2023, A&A, 672, A44

Hill J. C., Baxter E. J., Lidz A., Greco J. P., Jain B., 2018, Phys. Rev. D, 97,
083501

Hinshaw G. et al., 2013, ApJS, 208, 19

Hoffmann K. et al., 2022, Phys. Rev. D, 106, 123510

Jain B., Seljak U., White S., 1998, preprint (arXiv:astro-ph/9804238)

Jarvis M. et al., 2021, MNRAS, 501, 1282

Jeffrey N., Lanusse F., Lahav O., Starck J.-L., 2020, MNRAS, 492, 5023

Jeffrey N. et al., 2021, MNRAS, 505, 4626

Kacprzak T., Fluri J., Schneider A., Refregier A., Stadel J., 2023, J. Cosmol. Astropart. Phys., 2023, 050

Kaiser N., Squires G., 1993, ApJ, 404, 441

Kratochvil J. M., Haiman Z., May M., 2010, Phys. Rev. D, 81, 043519

Krause E., Hirata C. M., 2010, A&A, 523, A28

Krause E. et al., 2021, preprint (arXiv:2105.13548)

Kruse G., Schneider P., 2000, MNRAS, 318, 321

Lanzieri D., Lanusse F., Modi C., Horowitz B., Harnois-Déraps J., Starck
J.-L., The LSST Dark Energy Science Collaboration, 2023, preprint
(arXiv:2305.07531)

Lee E. et al., 2022, MNRAS, 517, 5303

Li X. et al., 2023, preprint (arXiv:2304.00702)
gan, NSF's NOIRLab, 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 MICINN under grants ESP2017-89838, PGC2018-094773, PGC2018-102021, 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) do e-Universo (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.

## DATA AVAILABILITY

All DES Y3 data used in this work are publicly available at https://des.ncsa.illinois.edu/releases/y3a2/. The ULAGAM simulation suite, which was denoted as A23 throughout this work for brevity, is available at https://ulagam-simulations.readthedocs.io.

## REFERENCES

* Adelberger K. L., Steidel C. C., Giavalisco M., Dickinson M., Pettini M., Kellogg M., 1998, ApJ, 505, 18

* Allys E., Marchand T., Cardoso J. F., Villaescusa-Navarro F., Ho S., Mallat S., 2020, Phys. Rev. D, 102, 103506

* Amara A., Réfrégier A., 2008, MNRAS, 391, 228

* Amodeo S. et al., 2021, Phys. Rev. D, 103, 063514

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

* Anbajagane D., Evrard A. E., Farahi A., Barnes D. J., Dolag K., McCarthy I. G., Nelson D., Pillepich A., 2020, MNRAS, 495, 686

* Anbajagane D., Evrard A. E., Farahi A., 2022a, MNRAS, 509, 3441

* Anbajagane D. et al., 2022b, MNRAS, 510, 2980

* Anbajagane D. et al., 2022c, MNRAS, 514, 1645

* Anbajagane D. et al., 2023a, preprint (arXiv:2310.00059)

* Anbajagane D., Chang C., Lee H., Gatti M., 2023b, preprint (arXiv:2310.02349)

* Asgari M., Friswell I., Yoon M., Heymans C., Dvornik A., Joachimi B., Simon P., Zuntz J., 2021a, MNRAS, 501, 3003

* Asgari M. et al., 2021b, A&A, 645, A104

* Banerjee A., Abel T., 2021a, MNRAS, 500, 5479

* Banerjee A., Abel T., 2021b, MNRAS, 504, 2911

* Banerjee A., Abel T., 2023, MNRAS, 519, 4856

* Barthelemy A., Halder A., Gong Z., Uhlemann C., 2023, preprint (arXiv:2307.09468)

* Baugh C. M., Gaztanaga E., Efstathiou G., 1995, MNRAS, 274, 1049

* Beltz-Mohrmann G. D., Berlind A. A., 2021, preprint (arXiv:2103.05076)

* Bernardeau F., 1998, A&A, 338, 375

* Blake C., James J. B., Poole G. B., 2014, MNRAS, 437, 2488

* Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301, 27

* Boyle A., Uhlemann C., Friedrich O., Barthelemy A., Codis S., Bernardeau F., Giocoli C., Baldi M., 2021, MNRAS, 505, 2886

* Carlstrom J. E., Holder G. P., Reese E. D., 2002, ARA&A, 40, 643

* Cataneo M., Uhlemann C., Arnold C., Gough A., Li B., Heymans C., 2022, MNRAS, 513, 1623

* Chang C. et al., 2018, MNRAS, 475, 3165

* Cheng S., Ménard B., 2021, MNRAS, 507, 1012

* Chisari N. E. et al., 2018, MNRAS, 480, 3962

* Cui W. et al., 2022, MNRAS, 514, 977

* Dalal R. et al., 2023, preprint (arXiv:2304.00701)

* Davies C. T., Cautun M., Giblin B., Li B., Harnois-Déraps J., Cai Y.-C., 2021, MNRAS, 507, 2267

* Doux C. et al., 2022, MNRAS, 515, 1942

* Duffy A. R., Schaye J., Kay S. T., Dalla Vecchia C., Battye R. A., Booth C. M., 2010, MNRAS, 405, 2161

* Euclid Collaboration, 2023, preprint (arXiv:2301.12890)

* Fluri J., Kacprzak T., Refregier A., Amara A., Lucchi A., Hofmann T., 2018, Phys. Rev. D, 98, 123518

* Fluri J., Kacprzak T., Lucchi A., Refregier A., Amara A., Hofmann T., Schneider A., 2019, Phys. Rev. D, 100, 063514

* Fluri J., Kacprzak T., Lucchi A., Schneider A., Refregier A., Hofmann T., 2022, Phys. Rev. D, 105, 083518

* Friedrich O. et al., 2018, Phys. Rev. D, 98, 023508

* Friedrich O., Uhlemann C., Villaescusa-Navarro F., Baldauf T., Manera M., Nishimichi T., 2020, MNRAS, 498, 464

* Fu L. et al., 2014, MNRAS, 441, 2725

* Gatti M. et al., 2020, MNRAS, 498, 4060

* Gatti M. et al., 2021, MNRAS, 504, 4312

* Gatti M. et al., 2022, Phys. Rev. D, 106, 083509

* Gatti M. et al., 2023, preprint (arXiv:2307.13860)

* Giannantonio T., Scranton R., Crittenden R. G., Nichol R. C., Boughn S. P., Myers A. D., Richards G. T., 2008, Phys. Rev. D, 77, 123520

* Giblin B., Cai Y.-C., Harnois-Déraps J., 2023, MNRAS, 520, 1721

* Gnedin O. Y., Kravtsov A. V., Klypin A. A., Nagai D., 2004, ApJ, 616, 16

* Gong Z., Halder A., Barreira A., Seitz S., Friedrich O., 2023, preprint (arXiv:2304.01187)

* Gough A., Uhlemann C., 2022, Universe, 8, 55

* Gruen D. et al., 2018, Phys. Rev. D, 98, 023507

* Halder A., Friedrich O., Seitz S., Varga T. N., 2021, MNRAS, 506, 2780

* Hamana T., Colombi S. T., Thion A., Devriendt J. E. G. T., Mellier Y., Bernardeau F., 2002, MNRAS, 330, 365

* Hartlap J., Simon P., Schneider P., 2007, A&A, 464, 399

* Heydenreich S., Brück B., Harnois-Déraps J., 2021, A&A, 648, A74

* Heydenreich S., Brück B., Burger P., Harnois-Déraps J., Unruh S., Castro T., Dolag K., Martinet N., 2022, A&A, 667, A125

* Heydenreich S., Linke L., Burger P., Schneider P., 2023, A&A, 672, A44

* Hill J. C., Baxter E. J., Lidz A., Greco J. P., Jain B., 2018, Phys. Rev. D, 97, 083501

* Hinshaw G. et al., 2013, ApJS, 208, 19

* Hoffmann K. et al., 2022, Phys. Rev. D, 106, 123510

* Jain B., Seljak U., White S., 1998, preprint (arXiv:astro-ph/9804238)

* Jarvis M. et al., 2021, MNRAS, 501, 1282

* Jeffrey N., Lanusse F., Lahav O., Starck J.-L., 2020, MNRAS, 492, 5023

* Jeffrey N. et al., 2021, MNRAS, 505, 4626

* Kacprzak T., Fluri J., Schneider A., Refregier A., Stadel J., 2023, J. Cosmol. Astropart. Phys., 2023, 050

* Kaiser N., Squires G., 1993, ApJ, 404, 441

* Kratochvil J. M., Haiman Z., May M., 2010, Phys. Rev. D, 81, 043519

* Krause E., Hirata C. M., 2010, A&A, 523, A28

* Krause E. et al., 2021, preprint (arXiv:2105.13548)

* Kruse G., Schneider P., 2000, MNRAS, 318, 321

* Lanzieri D., Lanusse F., Modi C., Horowitz B., Harnois-Déraps J., Starck J.-L., The LSST Dark Energy Science Collaboration, 2023, preprint (arXiv:2305.07531)

* Lee E. et al., 2022, MNRAS, 517, 5303

* Li X. et al., 2023, preprint (arXiv:2304.00702)
Lim S. H., Barnes D., Vogelsberger M., Mo H. J., Nelson D., Pillepich A., Dolag K., Marinacci F., 2021, MNRAS, 504, 5131

Lovell M. R. et al., 2018, MNRAS, 481, 1950

MacCrann N. et al., 2022, MNRAS, 509, 3371

Martinet N. et al., 2018, MNRAS, 474, 712

Mecke K. R., Buchert T., Wagner H., 1994, A&A, 288, 697

Munshi D., Jung G., Kitching T. D., McEwen J., Liguori M., Namikawa T., Heavens A., 2023, Phys. Rev. D, 107, 043516

Myles J. et al., 2021, MNRAS, 505, 4249

Omori Y., 2022, preprint (arXiv:2212.07420)

Osato K., Liu J., Haiman Z., 2021, MNRAS, 502, 5593

Pandey S. et al., 2022, Phys. Rev. D, 105, 123526

Park C. F., Allys E., Villaescusa-Navarro F., Finkbeiner D. P., 2022, preprint (arXiv:2204.05435)

Parroni C., Cardone V. F., Maoli R., Scaramella R., 2020, A&A, 633, A71

Peacock J. A., 1983, MNRAS, 202, 615

Peel A., Pettorino V., Giocoli C., Starck J.-L., Baldi M., 2018, A&A, 619, A38

Petri A., Liu J., Haiman Z., May M., Hui L., Kratochvil J. M., 2015, Phys. Rev. D, 91, 103511

Petri A., Haiman Z., May M., 2017, Phys. Rev. D, 95, 123503

Planck Collaboration, 2016a, A&A, 594, A13

Planck Collaboration, 2016b, A&A, 594, A16

Planck Collaboration, 2020, A&A, 641, A7

Potter D., Stadel J., Teyssier R., 2017, Comput. Astrophys. Cosmol., 4, 2

Schneider A., Teyssier R., Stadel J., Chisari N. E., Le Brun A. M. C., Amara A., Refregier A., 2019, J. Cosmol. Astropart. Phys., 2019, 020

Secco L. F. et al., 2022a, Phys. Rev. D, 105, 023515

Secco L. F. et al., 2022b, Phys. Rev. D, 105, 103537

Sellentin E., Loureiro A., Whiteway L., Lafaurie J. S., Balan S. T., Olamaie M., Jaffe A. H., Heavens A. F., 2023, preprint (arXiv:2305.16134)

Sevilla-Noarbe I. et al., 2021, ApJS, 254, 24

Shan H. et al., 2018, MNRAS, 474, 1116

Shao M., Anbajagane D., Chang C., 2022, preprint (arXiv:2212.05964)

Springel V., 2005, MNRAS, 364, 1105

Stiskalek R., Bartlett D. J., Desmond H., Anbajagane D., 2022, MNRAS, 514, 4026

Sunseri J., Li Z., Liu J., 2023, Phys. Rev. D, 107, 023514

Sunyaev R. A., Zeldovich Y. B., 1972, Comments on Astrophys. Space Phys., 4, 173

Takahashi R., Hamana T., Shirasaki M., Namikawa T., Nishimichi T., Osato K., Shiroyama K., 2017, ApJ, 850, 24

The Dark Energy Survey Collaboration, 2005, preprint (arXiv:astro-ph/0510346)

The LSST Dark Energy Science Collaboration, 2018, preprint (arXiv:1809.01669)

Uhlemann C., Friedrich O., Villaescusa-Navarro F., Banerjee A., Codis S., 2020, MNRAS, 495, 4006

Van Waerbeke L. et al., 2013, MNRAS, 433, 3373

Villaescusa-Navarro F. et al., 2020, ApJS, 250, 2

Wang Y., Banerjee A., Abel T., 2022, MNRAS, 514, 3828

White M., Hu W., 2000, ApJ, 537, 1

Zürcher D., Fluri J., Sgier R., Kacprzak T., Refregier A., 2021, J. Cosmol. Astropart. Phys., 2021, 028

Zürcher D. et al., 2022, MNRAS, 511, 2075

## APPENDIX A: 3-FIELD CDFS AND BEYOND

Formally, in the Gaussian limit, the 3-field CDF contains no new information beyond those from the 2-field CDFS, since they can also be described completely by the multivariate normal in equation (8). Thus, the 3-field CDFS can be predicted exactly using the covariance of the fields as a function of smoothing scale.

We show this explicitly in Fig. A1. We make measurements of the 3-field and 4-field CDFS on Gaussian fields, and then exactly predict the measurements given the covariance matrix as a function of smoothing scale. The covariance matrix is measured directly on

<figure><img src="image_13.png" /><figcaption>**Figure A1.** Measurements for the 3-field and 4-field CDFS on the noiseless DES Y3-like simulations (coloured lines), and a theoretical prediction in the limit of the field being Gaussian (black, dashed lines). The latter follows the same procedure of Section 2.3. The Gaussian model fits the data well, as is expected in this limit. The bin indices show the different tomographic bins used in the measurement.</figcaption></figure>

the map. We have verified the residuals between the measured N-field CDFS and the prediction is within $0.1\sigma$, where $\sigma$ comes solely from cosmic variance. This test is an extension of Fig. 2 for N-field CDFS of higher N.

## APPENDIX B: GAUSSIANITY OF COVARIANCE MATRIX

The process of performing a Fisher forecast, or obtaining constraints using likelihood minimization, assumes the likelihood of the data vector is Gaussian, i.e. the measurement uncertainty in the data vector is distributed as a multivariate Gaussian. We test here the validity of that assumption. We do so by first transforming every realization $i$ of a data vector by removing its mean, $S_i = D_i - \langle D \rangle$, where the mean is computed over all $i$ realizations. We then compute $\chi^2 = S_i C^{-1} S_i$, where $C$ is the covariance matrix estimated using all realizations of $D$. In the limit that the likelihood is Gaussian, the distribution of $\chi^2$ must follow a standard $\chi^2$ distribution.

In Fig. B1, we show the measured and expected distributions for four different data vectors, and in all cases we find the measured distributions match the expected Gaussian-limit distributions. We also compute a Kolmogorov-Smirnov statistic to quantify the level of agreement between the measured and expected distribution (Peacock 1983). This validates that the Fisher formalism is an accurate way to estimate potential constraints from the statistics considered in this work. Some additional techniques can also be used to quantify this Gaussianity of the likelihood (Park et al. 2022; Euclid Collaboration 2023), and they are roughly similar to the approach we have taken here.
Lim S. H., Barnes D., Vogelsberger M., Mo H. J., Nelson D., Pillepich A., Dolag K., Marinacci F., 2021, MNRAS, 504, 5131

Lovell M. R. et al., 2018, MNRAS, 481, 1950

MacCrann N. et al., 2022, MNRAS, 509, 3371

Martinet N. et al., 2018, MNRAS, 474, 712

Mecke K. R., Buchert T., Wagner H., 1994, A&A, 288, 697

Munshi D., Jung G., Kitching T. D., McEwen J., Liguori M., Namikawa T., Heavens A., 2023, Phys. Rev. D, 107, 043516

Myles J. et al., 2021, MNRAS, 505, 4249

Omori Y., 2022, preprint (arXiv:2212.07420)

Osato K., Liu J., Haiman Z., 2021, MNRAS, 502, 5593

Pandey S. et al., 2022, Phys. Rev. D, 105, 123526

Park C. F., Allys E., Villaescusa-Navarro F., Finkbeiner D. P., 2022, preprint (arXiv:2204.05435)

Parroni C., Cardone V. F., Maoli R., Scaramella R., 2020, A&A, 633, A71

Peacock J. A., 1983, MNRAS, 202, 615

Peel A., Pettorino V., Giocoli C., Starck J.-L., Baldi M., 2018, A&A, 619, A38

Petri A., Liu J., Haiman Z., May M., Hui L., Kratochvil J. M., 2015, Phys. Rev. D, 91, 103511

Petri A., Haiman Z., May M., 2017, Phys. Rev. D, 95, 123503

Planck Collaboration, 2016a, A&A, 594, A13

Planck Collaboration, 2016b, A&A, 594, A16

Planck Collaboration, 2020, A&A, 641, A7

Potter D., Stadel J., Teyssier R., 2017, Comput. Astrophys. Cosmol., 4, 2

Schneider A., Teyssier R., Stadel J., Chisari N. E., Le Brun A. M. C., Amara A., Refregier A., 2019, J. Cosmol. Astropart. Phys., 2019, 020

Secco L. F. et al., 2022a, Phys. Rev. D, 105, 023515

Secco L. F. et al., 2022b, Phys. Rev. D, 105, 103537

Sellentin E., Loureiro A., Whiteway L., Lafaurie J. S., Balan S. T., Olamaie M., Jaffe A. H., Heavens A. F., 2023, preprint (arXiv:2305.16134)

Sevilla-Noarbe I. et al., 2021, ApJS, 254, 24

Shan H. et al., 2018, MNRAS, 474, 1116

Shao M., Anbajagane D., Chang C., 2022, preprint (arXiv:2212.05964)

Springel V., 2005, MNRAS, 364, 1105

Stiskalek R., Bartlett D. J., Desmond H., Anbajagane D., 2022, MNRAS, 514, 4026

Sunseri J., Li Z., Liu J., 2023, Phys. Rev. D, 107, 023514

Sunyaev R. A., Zeldovich Y. B., 1972, Comments on Astrophys. Space Phys., 4, 173

Takahashi R., Hamana T., Shirasaki M., Namikawa T., Nishimichi T., Osato K., Shiroyama K., 2017, ApJ, 850, 24

The Dark Energy Survey Collaboration, 2005, preprint (arXiv:astro-ph/0510346)

The LSST Dark Energy Science Collaboration, 2018, preprint (arXiv:1809.01669)

Uhlemann C., Friedrich O., Villaescusa-Navarro F., Banerjee A., Codis S., 2020, MNRAS, 495, 4006

Van Waerbeke L. et al., 2013, MNRAS, 433, 3373

Villaescusa-Navarro F. et al., 2020, ApJS, 250, 2

Wang Y., Banerjee A., Abel T., 2022, MNRAS, 514, 3828

White M., Hu W., 2000, ApJ, 537, 1

Zürcher D., Fluri J., Sgier R., Kacprzak T., Refregier A., 2021, J. Cosmol. Astropart. Phys., 2021, 028

Zürcher D. et al., 2022, MNRAS, 511, 2075

## APPENDIX A: 3-FIELD CDFS AND BEYOND

Formally, in the Gaussian limit, the 3-field CDF contains no new information beyond those from the 2-field CDFS, since they can also be described completely by the multivariate normal in equation (8). Thus, the 3-field CDFS can be predicted exactly using the covariance of the fields as a function of smoothing scale.

We show this explicitly in Fig. A1. We make measurements of the 3-field and 4-field CDFS on Gaussian fields, and then exactly predict the measurements given the covariance matrix as a function of smoothing scale. The covariance matrix is measured directly on

<figure><img src="image_15.png" /><figcaption>**Figure A1.** Measurements for the 3-field and 4-field CDFS on the noiseless DES Y3-like simulations (coloured lines), and a theoretical prediction in the limit of the field being Gaussian (black, dashed lines). The latter follows the same procedure of Section 2.3. The Gaussian model fits the data well, as is expected in this limit. The bin indices show the different tomographic bins used in the measurement.</figcaption></figure>

the map. We have verified the residuals between the measured N-field CDFS and the prediction is within $0.1\sigma$, where $\sigma$ comes solely from cosmic variance. This test is an extension of Fig. 2 for N-field CDFS of higher N.

## APPENDIX B: GAUSSIANITY OF COVARIANCE MATRIX

The process of performing a Fisher forecast, or obtaining constraints using likelihood minimization, assumes the likelihood of the data vector is Gaussian, i.e. the measurement uncertainty in the data vector is distributed as a multivariate Gaussian. We test here the validity of that assumption. We do so by first transforming every realization $i$ of a data vector by removing its mean, $S_i = D_i - \langle D \rangle$, where the mean is computed over all $i$ realizations. We then compute $\chi^2 = S_i C^{-1} S_i$, where $C$ is the covariance matrix estimated using all realizations of $D$. In the limit that the likelihood is Gaussian, the distribution of $\chi^2$ must follow a standard $\chi^2$ distribution.

In Fig. B1, we show the measured and expected distributions for four different data vectors, and in all cases we find the measured distributions match the expected Gaussian-limit distributions. We also compute a Kolmogorov-Smirnov statistic to quantify the level of agreement between the measured and expected distribution (Peacock 1983). This validates that the Fisher formalism is an accurate way to estimate potential constraints from the statistics considered in this work. Some additional techniques can also be used to quantify this Gaussianity of the likelihood (Park et al. 2022; Euclid Collaboration 2023), and they are roughly similar to the approach we have taken here.
<figure><img src="image_14.png" /></figure>

**Figure B1.** The chi-squared distributions of the data vectors (solid lines), compared with a theoretical chi-squared distribution (dotted black line) with $N_{\text{dof}}$ given by the size of the data vector. In the Gaussian likelihood limit, the theoretical distributions will match the measured distribution. A Kolmogorov–Smirnov test shows the probability that the observed and expected distributions are similar exceeds $p > 0.1$. The data vectors considered in this work have a sufficiently Gaussian likelihood.

## APPENDIX C: DEPENDENCE OF DATA VECTOR ON COSMOLOGY

In Fig. C1, we show the derivative of the CDF measurement with the three cosmology parameters we have varied in Section 4.2. For brevity, we only show the derivative for the 1-field CDF of the fourth tomographic bin. At fixed threshold, the scale-dependence of the derivatives varies across the parameters, particularly at larger scales. At smaller scales, the derivatives with respect to $\Omega_m$ and $\sigma_8$ have larger amplitudes for the negative tail ($k = -0.02$) than the positive tail ($k = 0.02$). The derivative for $k = 0$ (green line) is near-zero in the 1-field CDFs, but we have checked that it is significantly non-zero for 2-field CDFs; this difference between the 1-field and 2-field behaviour is similar to that seen in Fig. 3. Any change in the $k = 0$ line for 1-field CDFs means the median of the distribution (and thus, the shape of the distribution) is being altered.

<figure><img src="image_15.png" /></figure>

**Figure C1.** The derivative of the CDFs data vector with respect to $\sigma_8$, $\Omega_m$, and $w_0$. For brevity, we only show results for the 1-field CDF of the highest redshift bin. The derivatives are shown as dotted lines when the CDFs fall into the 99.7 per cent tail. The dashed grey line shows the zero-derivative mark as a reference. The derivatives with different parameters have noticeable scale differences, leading to the degeneracy breaking in the parameter constraints. The negative tail ($k = -0.02$) has a higher derivative amplitude for $\Omega_m$ and $\sigma_8$ compared to that of the positive tail ($k = 0.02$).

<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">³Indian Institute of Science Education and Research, Pune 411008, India</footnote>

<footnote n="4">⁴Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA</footnote>

<footnote n="5">⁵Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA</footnote>

<footnote n="6">⁶SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA</footnote>

<footnote n="7">⁷Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA</footnote>

<footnote n="8">⁸Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland</footnote>

<footnote n="9">⁹Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA</footnote>

<footnote n="10">¹⁰Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="11">¹¹Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="12">¹²Institute for Astronomy, University of Hawaii'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA</footnote>

<footnote n="13">¹³Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150 University Avenue, Madison, WI 53706-1390, USA</footnote>

<footnote n="14">¹⁴Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15312, USA</footnote>

<footnote n="15">¹⁵Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="16">¹⁶Laboratório Interinstitucional de e-Astronomia – LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ-20921-400, Brazil</footnote>

<footnote n="17">¹⁷Dpto. Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="18">¹⁸Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA</footnote>

<footnote n="19">¹⁹Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="20">²⁰Department of Physics, Duke University, Durham, NC 27708, USA</footnote>

<footnote n="21">²¹NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA</footnote>

<footnote n="22">²²Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA</footnote>

<footnote n="23">²³Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA</footnote>

<footnote n="24">²⁴NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA</footnote>

<footnote n="25">²⁵Université Grenoble Alpes, CNRS, LPSC-IN2P3, F-38000 Grenoble, France</footnote>

<footnote n="26">²⁶Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada</footnote>

<footnote n="27">²⁷Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA</footnote>

<footnote n="28">²⁸University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany</footnote>
<figure><img src="image_16.png" /></figure>

**Figure B1.** The chi-squared distributions of the data vectors (solid lines), compared with a theoretical chi-squared distribution (dotted black line) with $N_{\text{dof}}$ given by the size of the data vector. In the Gaussian likelihood limit, the theoretical distributions will match the measured distribution. A Kolmogorov-Smirnov test shows the probability that the observed and expected distributions are similar exceeds $p > 0.1$. The data vectors considered in this work have a sufficiently Gaussian likelihood.

## APPENDIX C: DEPENDENCE OF DATA VECTOR ON COSMOLOGY

In Fig. C1, we show the derivative of the CDF measurement with the three cosmology parameters we have varied in Section 4.2. For brevity, we only show the derivative for the 1-field CDF of the fourth tomographic bin. At fixed threshold, the scale-dependence of the derivatives varies across the parameters, particularly at larger scales. At smaller scales, the derivatives with respect to $\Omega_m$ and $\sigma_8$ have larger amplitudes for the negative tail ($k = -0.02$) than the positive tail ($k = 0.02$). The derivative for $k = 0$ (green line) is near-zero in the 1-field CDFs, but we have checked that it is significantly non-zero for 2-field CDFs; this difference between the 1-field and 2-field behaviour is similar to that seen in Fig. 3. Any change in the $k = 0$ line for 1-field CDFs means the median of the distribution (and thus, the shape of the distribution) is being altered.

<figure><img src="image_17.png" /></figure>

**Figure C1.** The derivative of the CDFs data vector with respect to $\sigma_8$, $\Omega_m$, and $w_0$. For brevity, we only show results for the 1-field CDF of the highest redshift bin. The derivatives are shown as dotted lines when the CDFs fall into the 99.7 per cent tail. The dashed grey line shows the zero-derivative mark as a reference. The derivatives with different parameters have noticeable scale differences, leading to the degeneracy breaking in the parameter constraints. The negative tail ($k = -0.02$) has a higher derivative amplitude for $\Omega_m$ and $\sigma_8$ compared to that of the positive tail ($k = 0.02$).

<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">³Indian Institute of Science Education and Research, Pune 411008, India</footnote>

<footnote n="4">⁴Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA</footnote>

<footnote n="5">⁵Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA</footnote>

<footnote n="6">⁶SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA</footnote>

<footnote n="7">⁷Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA</footnote>

<footnote n="8">⁸Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland</footnote>

<footnote n="9">⁹Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA</footnote>

<footnote n="10">¹⁰Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="11">¹¹Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="12">¹²Institute for Astronomy, University of Hawaii'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA</footnote>

<footnote n="13">¹³Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150 University Avenue, Madison, WI 53706-1390, USA</footnote>

<footnote n="14">¹⁴Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15312, USA</footnote>

<footnote n="15">¹⁵Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="16">¹⁶Laboratório Interinstitucional de e-Astronomia – LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ-20921-400, Brazil</footnote>

<footnote n="17">¹⁷Dpto. Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="18">¹⁸Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA</footnote>

<footnote n="19">¹⁹Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA</footnote>

<footnote n="20">²⁰Department of Physics, Duke University, Durham, NC 27708, USA</footnote>

<footnote n="21">²¹NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA</footnote>

<footnote n="22">²²Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA</footnote>

<footnote n="23">²³Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA</footnote>

<footnote n="24">²⁴NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA</footnote>

<footnote n="25">²⁵Université Grenoble Alpes, CNRS, LPSC-IN2P3, F-38000 Grenoble, France</footnote>

<footnote n="26">²⁶Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada</footnote>

<footnote n="27">²⁷Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91109, USA</footnote>

<footnote n="28">²⁸University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany</footnote>
<footnote n="29"><sup>29</sup>School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA,
UK</footnote>

<footnote n="30"><sup>30</sup>Department of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290
Versoix, Switzerland</footnote>

<footnote n="31"><sup>31</sup>Department of Physics & Astronomy, University College London, Gower
Street, London WC1E 6BT, UK</footnote>

<footnote n="32"><sup>32</sup>Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK</footnote>

<footnote n="33"><sup>33</sup>Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas,
13083-859 Campinas, SP, Brazil</footnote>

<footnote n="34"><sup>34</sup>Department of Physics, University of Genova and INFN, Via Dodecaneso
33, I-16146 Genova, Italy</footnote>

<footnote n="35"><sup>35</sup>Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK</footnote>

<footnote n="36"><sup>36</sup>Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), E-28040 Madrid, Spain</footnote>

<footnote n="37"><sup>37</sup>Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA</footnote>

<footnote n="38"><sup>38</sup>Department of Physics and Astronomy, Stony Brook University, Stony
Brook, NY 11794, USA</footnote>

<footnote n="39"><sup>39</sup>Département de Physique Théorique et Center for Astroparticle Physics,
Université de Genève, 24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland</footnote>

<footnote n="40"><sup>40</sup>Institut d'Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona,
Spain</footnote>

<footnote n="41"><sup>41</sup>Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, F-31400 Toulouse,
France</footnote>

<footnote n="42"><sup>42</sup>Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile</footnote>

<footnote n="43"><sup>43</sup>Department of Astronomy, University of Michigan, Ann Arbor, MI 48109,
USA</footnote>

<footnote n="44"><sup>44</sup>Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="45"><sup>45</sup>Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK</footnote>

<footnote n="46"><sup>46</sup>Department of Physics, Northeastern University, Boston, MA 02115, USA</footnote>

<footnote n="47"><sup>47</sup>Physics Department, William Jewell College, Liberty, MO 64068, USA</footnote>

<footnote n="48"><sup>48</sup>Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-
21029 Hamburg, Germany</footnote>

<footnote n="49"><sup>49</sup>School of Mathematics and Physics, University of Queensland, Brisbane,
QLD 4072, Australia</footnote>

<footnote n="50"><sup>50</sup>Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India</footnote>

<footnote n="51"><sup>51</sup>Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, NO-0315 Oslo, Norway</footnote>

<footnote n="52"><sup>52</sup>Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, E-08193 Bellaterra (Barcelona),
Spain</footnote>

<footnote n="53"><sup>53</sup>Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA</footnote>

<footnote n="54"><sup>54</sup>Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA</footnote>

<footnote n="55"><sup>55</sup>Department of Physics, The Ohio State University, Columbus, OH 43210,
USA</footnote>

<footnote n="56"><sup>56</sup>Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA</footnote>

<footnote n="57"><sup>57</sup>Australian Astronomical Optics, Macquarie University, North Ryde, NSW
2113, Australia</footnote>

<footnote n="58"><sup>58</sup>Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA</footnote>

<footnote n="59"><sup>59</sup>George P. and Cynthia Woods Mitchell Institute for Fundamental Physics
and Astronomy, and Department of Physics and Astronomy, Texas A&M
University, College Station, TX 77843, USA</footnote>

<footnote n="60"><sup>60</sup>Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona,
Spain</footnote>

<footnote n="61"><sup>61</sup>Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ-
20921-400, Brazil</footnote>

<footnote n="62"><sup>62</sup>School of Physics and Astronomy, University of Southampton, Southampton
SO17 1BJ, UK</footnote>

<footnote n="63"><sup>63</sup>National Center for Supercomputing Applications, 1205 West Clark St.,
Urbana, IL 61801, USA</footnote>

<footnote>This paper has been typeset from a TeX/LATeX file prepared by the author.</footnote>
<footnote n="29"><sup>29</sup>School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA,
UK</footnote>

<footnote n="30"><sup>30</sup>Department of Astronomy, University of Geneva, ch. d'Écogia 16, CH-1290
Versoix, Switzerland</footnote>

<footnote n="31"><sup>31</sup>Department of Physics & Astronomy, University College London, Gower
Street, London WC1E 6BT, UK</footnote>

<footnote n="32"><sup>32</sup>Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK</footnote>

<footnote n="33"><sup>33</sup>Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas,
13083-859 Campinas, SP, Brazil</footnote>

<footnote n="34"><sup>34</sup>Department of Physics, University of Genova and INFN, Via Dodecaneso
33, I-16146 Genova, Italy</footnote>

<footnote n="35"><sup>35</sup>Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK</footnote>

<footnote n="36"><sup>36</sup>Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), E-28040 Madrid, Spain</footnote>

<footnote n="37"><sup>37</sup>Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA</footnote>

<footnote n="38"><sup>38</sup>Department of Physics and Astronomy, Stony Brook University, Stony
Brook, NY 11794, USA</footnote>

<footnote n="39"><sup>39</sup>Département de Physique Théorique et Center for Astroparticle Physics,
Université de Genève, 24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland</footnote>

<footnote n="40"><sup>40</sup>Institut d'Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona,
Spain</footnote>

<footnote n="41"><sup>41</sup>Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, F-31400 Toulouse,
France</footnote>

<footnote n="42"><sup>42</sup>Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile</footnote>

<footnote n="43"><sup>43</sup>Department of Astronomy, University of Michigan, Ann Arbor, MI 48109,
USA</footnote>

<footnote n="44"><sup>44</sup>Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="45"><sup>45</sup>Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK</footnote>

<footnote n="46"><sup>46</sup>Department of Physics, Northeastern University, Boston, MA 02115, USA</footnote>

<footnote n="47"><sup>47</sup>Physics Department, William Jewell College, Liberty, MO 64068, USA</footnote>

<footnote n="48"><sup>48</sup>Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-
21029 Hamburg, Germany</footnote>

<footnote n="49"><sup>49</sup>School of Mathematics and Physics, University of Queensland, Brisbane,
QLD 4072, Australia</footnote>

<footnote n="50"><sup>50</sup>Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India</footnote>

<footnote n="51"><sup>51</sup>Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, NO-0315 Oslo, Norway</footnote>

<footnote n="52"><sup>52</sup>Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, E-08193 Bellaterra (Barcelona),
Spain</footnote>

<footnote n="53"><sup>53</sup>Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA</footnote>

<footnote n="54"><sup>54</sup>Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA</footnote>

<footnote n="55"><sup>55</sup>Department of Physics, The Ohio State University, Columbus, OH 43210,
USA</footnote>

<footnote n="56"><sup>56</sup>Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA</footnote>

<footnote n="57"><sup>57</sup>Australian Astronomical Optics, Macquarie University, North Ryde, NSW
2113, Australia</footnote>

<footnote n="58"><sup>58</sup>Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA</footnote>

<footnote n="59"><sup>59</sup>George P. and Cynthia Woods Mitchell Institute for Fundamental Physics
and Astronomy, and Department of Physics and Astronomy, Texas A&M
University, College Station, TX 77843, USA</footnote>

<footnote n="60"><sup>60</sup>Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona,
Spain</footnote>

<footnote n="61"><sup>61</sup>Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ-
20921-400, Brazil</footnote>

<footnote n="62"><sup>62</sup>School of Physics and Astronomy, University of Southampton, Southampton
SO17 1BJ, UK</footnote>

<footnote n="63"><sup>63</sup>National Center for Supercomputing Applications, 1205 West Clark St.,
Urbana, IL 61801, USA</footnote>

<footnote>This paper has been typeset from a TeX/LATeX file prepared by the author.</footnote>