Dark Energy Survey Year 6 Results: Magnification modeling and its impact on galaxy clustering and galaxy-galaxy lensing cosmology

Technical Summary: Dark Energy Survey Year 6 Results: Magnification modeling and its impact on galaxy clustering and galaxy-galaxy lensing cosmology

Problem Statement
Gravitational lensing magnification alters the observed spatial distribution of galaxies by modifying their apparent positions, fluxes, and sizes. This effect introduces a bias in the observed number density of galaxies, known as magnification bias. In large-scale structure (LSS) surveys, neglecting this effect can lead to systematic shifts in cosmological parameter inference derived from galaxy clustering and galaxy-galaxy lensing (the "2×2pt" analysis). As surveys like the Dark Energy Survey (DES) achieve greater depth and precision (Year 6, Y6, vs. Year 3, Y3), the impact of magnification becomes increasingly relevant, particularly for high-redshift lens bins and when combining with cosmic shear (the "3×2pt" analysis). The primary challenge lies in accurately quantifying the magnification coefficients ($C$) for complex, multi-criteria galaxy selection functions (MagLim++) to prevent biases in cosmological constraints such as $S_8$ and $\Omega_m$.

Methodology
The authors investigate the impact of lens magnification on the DES Y6 cosmology analysis using the fiducial lens sample, MagLim++. The methodology involves three main components:

1. Theoretical Framework: The observed galaxy density contrast is decomposed into an intrinsic component and a magnification-induced component, $\delta^{\text{mag}}_g = C \kappa$, where $\kappa$ is the lensing convergence and $C = C_{\text{area}} + C_{\text{sample}}$. $C_{\text{area}} = -2$ accounts for the dilution of number density due to area stretching, while $C_{\text{sample}}$ captures the response of the selection function to flux and size changes.
2. Estimation of Magnification Coefficients: The paper employs three approaches to estimate $C_{\text{sample}}$ for six tomographic redshift bins:
   • Balrog Synthetic Source Injection (Fiducial): Artificial galaxies are injected into real DES images with and without a simulated magnification ($\delta\kappa \approx 0.01$). The MagLim++ selection is applied to both runs, and the fractional change in selected counts yields $C_{\text{sample}}$. This method accounts for the full survey transfer function, including complex selection criteria (color, size, star-galaxy separation). A reweighting scheme is applied to match Balrog properties to the real data.
   • Flux Perturbation (Data): A constant magnitude offset is applied to real data magnitudes to simulate magnification, re-selecting the sample to estimate $C_{\text{sample}}$. This is tested on both real data and the Balrog catalog.
   • Flux Perturbation (Balrog): The flux-only method applied to the Balrog catalog to assess systematic differences between simulation and data.
3. Cosmological Impact Assessment: The authors perform Bayesian parameter inference on simulated noiseless data vectors and unblinded DES Y6 data. They test various priors on the magnification coefficients (Gaussian centered on Balrog estimates, fixed values, flat priors, and shifted Gaussians) and analyze both 2×2pt and 3×2pt statistics. They also investigate the inclusion of cross-clustering between lens redshift bins and the exclusion of lens bin 2 (a choice motivated by internal consistency tests in the broader Y6 analysis).

Key Contributions

• Precise Magnification Coefficients: The paper provides the first high-precision estimates of magnification coefficients for the DES Y6 MagLim++ sample using the Balrog synthetic injection framework. The derived coefficients for the six redshift bins are $[3.16 \pm 0.08, 2.76 \pm 0.21, 4.09 \pm 0.15, 4.42 \pm 0.16, 4.90 \pm 0.29, 4.83 \pm 0.25]$.
• Improved Systematics Control: Compared to Y3, the Y6 analysis utilizes a larger Balrog area and a reweighting scheme that significantly reduces systematic uncertainties between the simulation and the data. The flux-only estimates from Balrog and real data show much better agreement in Y6 than in Y3.
• Validation of Modeling Choices: The study validates the fiducial modeling choice of using Gaussian priors centered on Balrog-derived estimates. It demonstrates that this approach yields unbiased cosmological constraints in simulated data.
• Quantification of Bias: The paper quantifies the systematic shifts introduced by neglecting magnification or using incorrect priors.

Results

• Impact of Neglecting Magnification: Neglecting magnification (fixing $C_{\text{sample}} = 2$) introduces significant systematic shifts in cosmological parameters relative to the fiducial analysis with Gaussian priors. In the 2×2pt analysis, this results in a shift of $1.37\sigma$ in $S_8$ and $-0.84\sigma$ in $\Omega_m$. When cosmic shear is included (3×2pt), the shifts are $-0.61\sigma$ in $S_8$ and $-0.71\sigma$ in $\Omega_m$.
• Robustness to Priors: The cosmological results are robust when tested against various prior choices. Marginalizing over magnification coefficients with Gaussian priors (fiducial) recovers constraints nearly identical to fixing the coefficients to their true values. Using uninformative flat priors slightly degrades constraining power (increasing $S_8$ uncertainty by 15% in 2×2pt) but does not introduce significant biases in the parameter means.
• Cross-Clustering: Including cross-clustering between lens redshift bins helps self-calibrate the magnification signal, slightly improving constraints on $C_{\text{sample}}$ at high redshifts and reducing biases when flat priors are used.
• Lens Bin 2: When lens bin 2 is included in the analysis with flat priors, the inferred magnification coefficient shifts to unphysical negative values. This behavior, combined with other consistency tests, justifies the exclusion of bin 2 from the fiducial Y6 analysis. The exclusion of bin 2 does not bias the magnification analysis for the remaining bins.
• Data vs. Simulation: The inferred magnification coefficients from unblinded DES Y6 data are generally consistent with Balrog estimates, though some deviations occur in high-redshift bins when flat priors are used. These deviations are attributed to unmodeled systematics rather than incorrect magnification modeling, as the resulting cosmological shifts align with projection effects seen in simulations.

Significance
The paper establishes that weak lensing magnification is a critical systematic effect that must be modeled to avoid biases in precision cosmology with Stage-III and Stage-IV surveys. The DES Y6 analysis demonstrates that using synthetic source injection (Balrog) to derive informative priors on magnification coefficients provides a reliable and conservative treatment of these effects. While the impact on current DES Y6 constraints is manageable, the study highlights that accurate modeling of magnification is essential for future surveys with greater depth and statistical power, where these effects will become more dominant and where uninformative priors may lead to larger biases. The work confirms that the fiducial Y6 modeling choices successfully mitigate magnification-induced biases, ensuring the robustness of the survey's cosmological results.