Perturbation theory for modeling galaxy bias: validation with simulations of the Dark Energy Survey

Standard perturbation theory expands the evolved dark matter density field, $\delta_{\mathrm{m}}(\textbf{x})$ in terms of the extrapolated linear density field, shear field, the divergence of the velocity field and rotational invariants constructed using the gravitational potential. In Fourier space, this expansion can be written as Bernardeau et al. (2002)

Here $F_{n}(\mathbf{k}_{1},..,\mathbf{k}_{n})$ are the mode coupling kernels constructed out of correlations between the scalar quantities mentioned above and $\delta_{\mathrm{D}}$ is the Dirac delta function. The form of the $F_{n}$ kernels can be derived by solving the perturbative fluid equations. For example under the assumptions of the spatially flat, cold dark matter model of cosmology, $F_{2}$ is well approximated by

For $\Omega_{\mathrm{m}}<1$, $\alpha=\frac{3}{7}(\Omega_{\mathrm{m}})^{-2/63}$ and $\mu=\frac{\textbf{k}\cdot\textbf{k}^{\prime}}{k\cdot k^{\prime}}$. In this analysis, we use the Einstein de-Sitter limit and assume $\alpha=\frac{3}{7}$.

In the above section, the non-local terms included in the expansion of galaxy overdensity comes only from shear and velocity divergences. However, those terms are still local in the spatial sense, meaning that the formation of biased tracers only depends on the scalar quantities discussed above at the same position as the tracer. A short-range non-locality due to non-linear effects in halo and galaxy formation within some some scale $R$, will change Eq. 3 to: (McDonald and Roy, 2009)

where, generally $|\textbf{x}-\textbf{x}^{\prime}|<R$ and $R$ is usually of the order of halo radius. Taylor expanding this function we can see that lowest order gradient-type term that can contribute to $\delta_{\mathrm{g}}$ is proportional to $\nabla^{2}\delta_{\mathrm{m}}$. Hence, we can further generalize our Eq. 5 to include this gradient-type term as

Note that in Fourier space, this term would scale as $k^{2}\delta_{\mathrm{m}}(k)$.

Moreover, as discussed in Carrasco et al. (2012), it is theoretically inconsistent to use small scale modes in the integration over Fourier space. So we use effective integrated ultra-violet (UV) terms in the final expansion for the power spectrum. This effective term also enters as a $k^{2}$ contribution in the large-scale limit. For example, if we expand the non-linear matter power spectrum in terms of the linear power spectrum ($P_{\rm L}(k)$) using the PT framework, we have to include this $k^{2}$ piece usually written as $c^{2}_{\rm s}k^{2}P_{\rm L}(k)$, where $c_{\rm s}$ is the effective adiabatic sound speed.

Note that the bias parameters that will appear in the expansion of $\delta_{g}$ in Eq. 7 will be un-observable “bare bias” parameters and need not have the physical meaning usually attributed to the large scale tracer bias (for example,  the measurable responses of galaxy statistics to a given fluctuation). We refer the reader to McDonald and Roy (2009) for the details on the renormalization of these “bare bias” parameters by combining all the parameters with similar power spectrum kernels. After renormalizing, we can write the tracer-matter cross spectrum ($P_{\mathrm{gm}}$) and auto power spectrum of the tracer ($P_{\mathrm{gg}}$) as:

Here the bias parameters like $b_{1}$, $b_{2}$, $b_{\mathrm{s}}$ and $b_{\rm 3nl}$ are the renormalized bias parameters which are physically observable. The bias parameter $b^{\rm hd}_{\nabla^{2}\delta}$ is the higher-derivative bias parameter and $c^{2}_{\rm s}$ is the sound speed term as described by EFT (§II.3). As for the kernels, $P_{\rm b_{1}b_{2}}(k)$ is generated from ensemble average of $\langle\delta_{\mathrm{m}}\delta^{2}_{\mathrm{m}}\rangle$, $P_{\rm b_{1}s^{2}}(k)$ is generated from $\langle\delta_{\mathrm{m}}s^{2}\rangle$ and $P_{\rm b_{1}b_{\rm 3nl}}$ is generated from a combination of ensemble average between $\delta_{\mathrm{m}}$ and arguments of $f^{(3)}$ (see Eq. 7) that contribute at 1-loop level (Saito et al., 2014). For the exact form of above kernels, see the Appendix A of Saito et al. (2014).

Instead of expanding the Eulerian galaxy overdensity field directly as we have done above, we can also predict the galaxy overdensity by evolving the Lagrangian galaxy overdensity (see Matsubara (2013) for detailed calculations). These two approaches should evaluate to the same galaxy overdensity at a given loop order (Fry, 1996; Baldauf et al., 2012; Chan et al., 2012; Matsubara, 2013; Saito et al., 2014). By equating the two approaches and neglecting shear-like terms in the Lagrangian overdensity as they are small for bias values of our interest (see §V and Modi et al. (2017)), we get the prediction of the co-evolution value of the renormalized bias parameters: $b_{\mathrm{s}}=(-4/7)\times(b_{1}-1)$ and $b_{\rm 3nl}=(b_{1}-1)$¹¹1note that our co-evolution value of $b_{\rm 3nl}$ differs from Saito et al. (2014) as we include their prefactor of 32/315 in our definition of $P_{\rm b1b3nl}$ (Matsubara, 2013; Saito et al., 2014). This co-evolution picture naturally describes how gravitational evolution generates the non-local biasing even from the local biased tracers in high redshift Lagrangian frame.

We use different choices of $P_{\mathrm{mm}}$ and $P^{\rm grad}_{\mathrm{mm}}$ in our analysis. These choices will be detailed in the §IV.1.

We are interested in the cosmological applications of imaging surveys via projected correlation functions. Projections of the 3D correlation functions $\xi_{\mathrm{gg}}$ and $\xi_{\mathrm{gm}}$, to angular coordinates in finite redshift bins give the projected correlations known as $w_{\mathrm{gg}}(\theta)$ and $\gamma_{t}(\theta)$ respectively. We estimate the covariance of these projected statistics for the DES-Y3 like survey. This allows us to estimate the angular scales for which our perturbation theory model is a good description for DES-Y3 like sensitivity.

Our main analysis involves the auto and cross-correlations functions for galaxies and matter: $\xi_{\mathrm{mm}}$, $\xi_{\mathrm{gm}}$ and $\xi_{\mathrm{gg}}$. Our focus is on galaxy bias, so we would like to minimize artifacts that are specific to the clustering of matter, in particular sampling effects due to the finite volume of the simulations (see Appendix A). Therefore, we fit our theory models to the ratios: $\xi_{\mathrm{gg}}/\xi_{\mathrm{mm}}$ and $\xi_{\mathrm{gm}}/\xi_{\mathrm{mm}}$ so that the galaxy two-point functions are analyzed relative to the matter-matter correlation (see Appendix B and Fig. 12 for an analysis on correlation functions $\xi_{\mathrm{gm}}$ and $\xi_{\mathrm{gg}}$ directly). We consider three models to describe these measured ratios:

where, $\mathcal{F}$ denotes the Fourier transform and $P^{\rm 1-Loop}_{\mathrm{gm}}(k)$ is the effective sum of all the terms dependent on $b_{2},b_{\mathrm{s}}$ and $b_{\rm 3nl}$ in Eq. 8. An analogous form of this expansion can be derived for $P_{\mathrm{gg}}(k)$. The term $P^{\rm 1-Loop}_{\mathrm{mm}}(k)$ is the 1-Loop PT estimate of the matter-matter correlation function. Model A is the linear bias model and the numerator in Model B is similar to the model considered by previous analyses using the EFT description of clustering (Goldberger and Rothstein, 2006; Senatore, 2015; Baumann et al., 2012; Ivanov et al., 2019; D’Amico et al., 2019; Perko et al., 2016; Chudaykin and Ivanov, 2019). In this study, we also analyze Model C, which differs from Model B in the use of the full nonlinear matter power spectrum using halofit (as opposed to 1-loop PT in Model B) in the numerator. This model is motivated by completely re-summing the matter-matter auto-correlation term to all orders as it uses the fully non-linear fits to simulations such as halofit (Takahashi et al., 2012): $P^{\rm NL}_{\mathrm{mm}}=P^{\rm HF}_{\mathrm{mm}}$. We make similar a choice for $P^{\rm grad}_{\mathrm{mm}}(k)$ (Baldauf et al., 2015). The bias term, $b_{\nabla^{2}\delta}$ is the sum of both the higher-derivative bias term ($b^{\rm hd}_{\nabla^{2}\delta}$) and the sound speed term ($c^{2}_{\rm s}$) for $P_{\mathrm{gm}}(k)$. The sound speed term is zero in Model C as the fully non-linear matter power spectra include any correction from the UV divergent integrals. Hence in Model C, $b_{\nabla^{2}\delta}=b^{\rm hd}_{\nabla^{2}\delta}$. Unlike Model C, in Model B the sound speed term is not zero, so there we denote $b_{\nabla^{2}\delta}=b^{\rm hd}_{\nabla^{2}\delta}+c^{2}_{\rm s}$.

The choice of different power spectra for the three models are given in Table 1.

Note that the denominator of Models B and C implicitly assumes that halofit is a good description of the matter-matter correlation on the scales we are interested in. We check this assumption using the matter density field from the MICE simulations. The residuals of the matter-matter correlation functions for both halofit and EFT are shown in Fig. 2. The EFT theory curve is predicted by fitting the measured $\xi_{\mathrm{mm}}$ on scales larger than 4 Mpc/$h$ with the model: $\xi_{\mathrm{mm}}=\mathcal{F}(P^{\rm 1-Loop}_{\mathrm{mm}}(k)+c^{2}_{\rm s}k^{2}P_{\rm lin}(k))$. We can see that EFT shows deviations at the 5% level while halofit is a good description of $\xi_{\mathrm{mm}}$ over all scales and redshifts – typically within 2% for the bins with percent level error bars on the measurement.

To assess the goodness of fit of the models, we use the reduced $\chi^{2}$. For a good fit to $n_{\mathrm{d}}$ number of data-points, using a model with $n_{\mathrm{v}}$ free parameters, we expect the $\chi^{2}/{\rm d.o.f}$ to have a mean of $1$ and standard deviation of $\sqrt{2/{\rm d.o.f}}$, where ${\rm d.o.f}=n_{\mathrm{d}}-n_{\mathrm{v}}$ is the total number of degrees of freedom.

The mode coupling kernels that appear in perturbative terms, such as the higher-order bias contributions in Eq. 8, in Fourier space take the form of convolution integrals. For example in Standard Perturbation Theory, we expand the evolved over-density field of tracers in terms of the linear overdensity, up to third order. This results in terms in the power spectrum that are proportional to $P_{22}(k)$ (given by the ensemble average $\langle\delta^{(2)}\delta^{(2)}\rangle$) and $P_{13}(k)$ (given by $\langle\delta^{(1)}\delta^{(3)}\rangle$). These kernels can be efficiently evaluated using fast Fourier transform techniques presented in McEwen et al. (2016); Fang et al. (2016); Schmittfull et al. (2016), if one transforms these convolution integrals to the prescribed general form. We use the publicly available Python code FAST-PT as detailed in McEwen et al. (2016) to evaluate all the PT kernels, which is also tested against a C version of the code CFASTPT²²2FAST-PT is available at https://github.com/JoeMcEwen/FAST-PT, and CFASTPT is available at https://github.com/xfangcosmo/cfastpt.

We estimate a covariance for the data vector by applying the jackknife method (Qenouille, 1956; Tukey, 1958) to the simulation split into $N_{\rm jk}$ number of patches. We use the k-means clustering algorithm to get the patches, which roughly divides the octant of sky occupied by our galaxy samples into $N_{\rm jk}$ equal-area patches. We use these same patches for covariance calculation in each of our tomographic bins. The accuracy of the estimated covariance increases with increasing $N_{\rm jk}$ and for scales much smaller than the size of an individual patch (Norberg et al., 2009; Friedrich et al., 2016). As the total area of the mock catalogs is fixed, changing the number of jackknife patches changes each patch’s size.

In order to provide constraints on both non-linear and linear bias parameters, the analysis requires a covariance estimate that correctly captures the auto and cross-correlations between radial bins over both small and large scales to provide constraints on both non-linear and linear bias parameters. We find that we need to limit the analysis to $z>0.3$ to achieve stable covariance estimates. For this reason, we do not analyze the MICE catalog over the first tomographic bin used in the DES-Y1 analysis ($0.15<z<0.3$).

We estimate the jackknife covariance using $N_{\rm jk}=300$ patches. For the lowest redshift bin ($0.3<z<0.45$), this results in an individual jackknife patch with a side length of approximately $100$Mpc/$h$. We determine the maximum scale included in our analysis by varying the number of patches and comparing the estimated errors at different scales. We find the covariance estimate to be stable below 40 Mpc/$h$ and use this as our maximum scale cut. These tests are detailed in Appendix. A.

We explicitly remove the cross-covariance between tomographic bins as there is negligible overlap in the galaxy samples of two different redshift bins, and as length scales of interest are much smaller than the radial extent of the tomographic bins. We correct for biases in the inverse covariance (when calculating the reduced $\chi^{2}$) due to the finite number of jackknife patches using the procedure described in Hartlap et al. (2006).

Note that Fig. 11 shows the signal to noise for these 3D statistics for each radial bin for our fiducial covariance.

We split the galaxy sample into four tomographic bins, following the DES Year-1 analysis DES Collaboration et al. (2017). The redshift ranges for the four bins are: $0.3<z<0.45$, $0.45<z<0.6$, $0.6<z<0.75$ and $0.75<z<0.9$ .

The auto and cross-correlations measured with the galaxy and matter catalogs in the MICE simulations are shown in Fig. 3. We use the Landy-Szalay estimator (Landy and Szalay, 1993) to estimate the correlation functions $\xi_{\rm gg}$, $\xi_{\rm gm}$ and $\xi_{\rm mm}$ for all the $N_{\rm jk}$ jackknife patches (see §IV.4). We create a random catalog with 10 times the number of galaxies in each tomographic bin and with number densities corresponding to smoothed galaxy number density. We then use the ratios $\xi_{\rm gg}/\xi_{\rm mm}$ and $\xi_{\rm gm}/\xi_{\rm mm}$ to create our datavector and jackknife covariance. We use the public code Treecorr Jarvis et al. (2004) to measure the cross correlations. We jointly fit these ratios $\xi_{\mathrm{gg}}/\xi_{\mathrm{mm}}$ and $\xi_{\mathrm{gm}}/\xi_{\mathrm{mm}}$ with PT models mentioned in §IV.1, as described next.

As a first analysis step, we fit the correlation function ratios measured from the simulation with the three models, Model A, B and C (Eq. 23) described in §IV.1. Model A only has one free parameter, linear bias $b_{1}$, while Model B and C in principle have $b_{1},b_{2},b_{\mathrm{s}},b_{\rm 3nl}$ and $b_{\nabla^{2}\delta}$ as free parameters. Here $b_{\nabla^{2}\delta}$ is the higher-derivative bias parameter. Among these parameters, by using the equivalence of Lagrangian and Standard Eulerian perturbation theory (see §II.4), we can write $b_{\mathrm{s}}$ and $b_{\rm 3nl}$ in terms of $b_{1}$ as their co-evolution value. Therefore, the simplest complete 1Loop model has $b_{1}$, $b_{2}$ and $b_{\nabla^{2}\delta}$ as free parameters. We fit our measurements while varying the number of free parameters in both Model B and Model C, to find the minimum number of parameters needed to describe the measured correlation function for different scale cuts.

We analyze the MICE data-vector with two different minimum scale cuts: 8Mpc/$h$ and 4Mpc/$h$. In Fig. 4, we compare the marginalized constraints on $b_{\nabla^{2}\delta}$ for Model B and C for each redshift bin. The marginalized constraints on $b_{\nabla^{2}\delta}$ are consistent with zero for Model C, for all redshift bins, and both scale cuts. In contrast, Model B shows significant detection of the $b_{\nabla^{2}\delta}$ term. It appears that the EFT term mostly captures the departure of the matter correlation function model from the truth.

Figure 5 compares the goodness of fit of different models by showing the reduced $\chi^{2}$ estimated from the best-fit of various model choices (as given in the $x$-axis). We find that using Model C with only $b_{1}$ and $b_{2}$ as free parameters gives a reduced $\chi^{2}$ consistent with 1 for all redshift bins (with $b_{\mathrm{s}}$ & $b_{\rm 3nl}$ fixed to their co-evolution value and $b_{\nabla^{2}\delta}=0$). Hence, we conclude that adding these as free parameters is not needed to model the measurements on the scales considered here. In what follows, we consider this model choice of using 1Loop PT with free $b_{1}$ and $b_{2}$ as our fiducial model. We also compare our fits to Model A, with free linear bias parameter $b_{1}$. The residuals of the observables, i.e., the ratios $\xi_{\mathrm{gg}}/\xi_{\mathrm{mm}}$ and $\xi_{\mathrm{gm}}/\xi_{\mathrm{mm}}$, are shown in Fig. 6 for a scale cut of 8Mpc/$h$, and in Fig. 7 for a scale cut of 4Mpc/$h$. Note that halofit describes the matter-matter autocorrelation above scales of 4Mpc/$h$ at about the 2% level (see Fig. 2). In these and following figures, we refer to $\xi^{\rm model}_{\rm gg}=\xi_{\rm gg}/\xi_{\rm mm}$ and $\xi^{\rm model}_{\rm gm}=\xi_{\rm gm}/\xi_{\rm mm}$. Our fiducial model fits the simulations on scales above 4Mpc/$h$ and $z<1$ within 2%, while the linear bias model performs significantly worse.

We also show the residuals of our fits to the Maglim sample in Fig. 7. We find that similar to the redMaGiC sample results, the fiducial model describes the measurements within about 2% above scales of 4Mpc/$h$.

In this section we revisit the approximation that the non-linear bias parameters $b_{\mathrm{s}}$ and $b_{\rm 3nl}$ follow the co-evolution relation. The equivalence of the local Lagrangian and non-local Eulerian description predicts $b_{\mathrm{s}}=-4/7(b_{1}-1)$ and $b_{\rm 3nl}=(b_{1}-1)$ (see §II.4). We test this assumption by freeing up these parameters in addition to $b_{1}$ and $b_{2}$ and re-fitting the measurements with these extended models. Figure 8 shows the relation between the non-linear bias parameters and $b_{1}$ at the two scale cuts and for both redMaGiC and Maglim galaxy samples. The points in each panel for each scale cut corresponds to the four tomographic bins. The top panel shows the relation between $b_{1}$ and $b_{2}$ (when the parameters $b_{\mathrm{s}}$ and $b_{\rm 3nl}$ are fixed to their co-evolution value), the middle panel shows the relation between $b_{1}$ and $b_{\mathrm{s}}$ (when $b_{\rm 3nl}$ is fixed to its co-evolution value) and the bottom panel shows the relation between $b_{1}$ and $b_{\rm 3nl}$ when ($b_{\mathrm{s}}$ is fixed to its co-evolution value). The fits obtained when all the parameters are free have bigger uncertainty but are consistent with the other approaches: the relation between the parameters $b_{\mathrm{s}}-b_{1}$ and $b_{\rm 3nl}-b_{1}$ are consistent with the expected co-evolution value. We also note that the recovered relation with $b_{1}$ is consistent for the two scale cuts, which is a further test that the 1Loop PT is a sufficient and complete model for the scales of interest in this analysis.

It is possible to predict the relation between $b_{2}$ and $b_{1}$ for our galaxy samples (the measurements are shown in the top panel of Fig. 8). However, unlike the $b_{\rm s}-b_{1}$ and $b_{\rm 3nl}-b1$ relation, predicting $b_{2}-b_{1}$ relation requires knowledge of the HOD of galaxy samples. Since an accurate HOD of the galaxy sample in data is challenging and not yet available for DES, we have treated $b_{2}$ as a free parameter. Therefore, only the measurements of the $b_{2}-b_{1}$ relation from simulations are shown in Fig. 8.

As described in §II.5, we can convert our measurements and fits for the 3D correlation functions to the projected statistics typically used by the imaging surveys. We show such a conversion in Fig. 9 for galaxy number densities in MICE simulations corresponding to the redMaGiC galaxies satisfying $0.3<z_{\rm l}<0.45$ and fourth source tomographic bin as used in the DES Y1 analysis. Note that Fig. 9 does not show direct measurements of $w(\theta)$ and $\gamma_{t}$, but a transformation of the measured and best-fit datavector to angular statistics. Since our analysis is based on the ratios $\xi_{\rm gg}/\xi_{\rm mm}$ and $\xi_{\rm gm}/\xi_{\rm mm}$, we first convert our measured datavector and best-fit theory curves to $\xi_{\rm gg}$ and $\xi_{\rm gm}$ and then apply Eq. 11 and Eq. 22 to estimate angular correlation functions. We use halofit prediction of $\xi_{\rm mm}$, which is a good fit to the matter-matter autocorrelation for our scales of interest (see Fig. 2) to convert the ratios to $\xi_{\rm gg}$ and $\xi_{\rm gm}$.

The error bars in Fig. 9 are calculated from Gaussian covariance³³3We use the COSMOSIS package Zuntz et al. (2015) https://bitbucket.org/joezuntz/cosmosis/wiki/Home as we do not expect significant non-gaussian contribution to the covariance of the angular statistics (see (Krause et al., 2017)). The covariance is estimated using all the galaxies satisfying the redshift criteria mentioned above in the MICE simulation. Explicitly, we generate this covariance with lens and source galaxies covering 5156.6 square degrees with number densities (per square arc-minutes) of lens galaxies in four tomographic bins corresponding to 0.039, 0.058, 0.045 and 0.028 respectively. The number density and shape noise of source galaxies is assumed to be the same as DES Y3 (Friedrich, O. et al., prep). Due to a similar area and number densities, this covariance is comparable to the expected DES Year-3 covariance (Friedrich, O. et al., prep). Note that the shaded region corresponds to scales below 4Mpc/$h$, which are not used in the 3D fits. The top panel shows the projected galaxy correlation function, $w(\theta)$ and bottom panel shows galaxy-galaxy lensing signal, $\gamma_{t}(\theta)$. Note that to estimate $\gamma_{t}$, we fit for the point-mass term as described in §II.5. This best-fit value of the point-mass term is obtained by fitting for the coefficient $B$ in Eq. 22.

Figure 9 demonstrates that our model describes the projected angular correlation functions well above scales of 4Mpc/$h$. The error bars in that figure provide a DES Y3 like benchmark for such an agreement. Note that the fractional statistical uncertainties for projected statistics are much larger than their 3D counterparts. Hence the 3D tests presented in §V.2 are substantially more stringent than the projected statistics require.

The analysis of measured $w(\theta)$ and $\gamma_{t}(\theta)$ is detailed in Appendix C.

There have been multiple studies in the literature probing the validity of PT models using simulations Saito et al. (2014); Angulo et al. (2015); de la Bella et al. (2018); Werner and Porciani (2019); Eggemeier et al. (2020). Most of these studies have focused on Fourier space rather than configuration space. One reason for this choice is that non-linear and linear scales are better separated in Fourier space while in configuration space, even large scales receive a contribution from non-linear Fourier modes. However, many cosmic surveys perform their cosmological parameter analysis in configuration space as it is easier to take into account a non-contiguous mask and depth variations. Hence an understanding of the validity of PT models is required in real space to get unbiased cosmology constraints.

The Fourier space studies conducted by Saito et al. (2014) and Angulo et al. (2015) focus only on dark-matter halos and do not aim to reduce the number of free parameters required to explain the auto and cross-correlations between dark matter halos and dark matter particles. de la Bella et al. (2018) and Werner and Porciani (2019) probe this question on the minimum number of bias parameters but again focus on dark matter halos as the biased tracers. Recently Eggemeier et al. (2020) have conducted a study similar to ours in Fourier space using three different galaxy samples (mock SDSS and BOSS catalogs) and four halo samples. For a most general case, they find that a four-parameter model (linear, quadratic, cubic non-local bias, and constant shot noise with fixed quadratic tidal bias) can describe correlations between galaxies and matter catalogs, with the inclusion of scale-dependent noise from halo exclusion being particularly beneficial for the combination of auto and cross spectra. They also explore the restriction to a two-parameter model by imposing co-evolution relations, as done in this paper, and find that in general, this reduces the highest Fourier mode for which the model is robust, but it can result in higher constraining power compared to the five parameter model. However, this particular scenario is not general across samples and requires careful validation with simulations, as done here. The main differences in our study are: we work in configuration space with two different galaxy samples that have a higher number density, cover a wider redshift range, and probe smaller host halo masses. Our galaxy samples also have a significantly larger satellite fraction (for example, the first two redMaGiC bins have a $\sim 50\%$ satellite fraction) compared to SDSS and BOSS catalogs.

These crucial differences make our study complementary to the above studies. Ours is especially relevant for imaging surveys as it is tailored to DES. The consistency of our conclusions with Eggemeier et al. (2020) suggests that a two-parameter model may have wide applicability, particularly for surveys with different galaxy selections. This would be an extremely useful result and is worth investigating in detail for the next generation of surveys.