Cosmic shear nulling as a geometrical cosmological probe: Methodology and sensitivity to cosmological parameters and systematics

Technical Summary: Cosmic Shear Nulling as a Geometrical Cosmological Probe

Problem Statement
Modern weak lensing surveys (e.g., Euclid, LSST, DES) aim to constrain cosmological parameters, particularly dark energy and matter density, by measuring cosmic shear. However, extracting precise information is complicated by the need to model complex astrophysical effects, such as non-linear gravitational clustering, baryonic physics, and intrinsic alignments. Furthermore, standard analyses often rely on assumptions about the matter power spectrum and the evolution of structure. The authors propose a method to exploit the intrinsic geometric symmetries of the Universe to create a cosmological probe that is independent of these physical modeling assumptions, while simultaneously serving as a robust test for systematic errors, specifically photometric redshift calibration.

Methodology
The paper introduces a "nulling" test based on the Bernardeau-Nishimichi-Taruya (BNT) transform applied to tomographic cosmic shear maps.

1. The BNT Transform: This is a linear transformation of convergence ($\kappa$) or shear ($\gamma$) maps across $N_z$ redshift bins. The transformation coefficients ($p_{ai}$) are constructed solely from the geometric properties of space-time (specifically the angular diameter distance and the comoving distance kernel). The transform is designed such that the resulting "nulled" maps ($\hat{\kappa}_a$) have lensing kernels with compact support, effectively canceling contributions from low-redshift lenses (specifically, $\hat{\kappa}_a$ receives no contribution from lenses in bins $i \leq a-3$).
2. The Null Test: The core hypothesis is that if the background geometry of the Universe is correctly described, the nulled shear maps ($\hat{\kappa}_a$) should exhibit zero correlation with any low-redshift tracer of large-scale structure (e.g., galaxy density fields $\delta^g_i$ in bins $i \leq a-3$). The observable is the cross-spectrum $C^{\hat{\kappa}g}_{ai}(\ell)$. Under the null hypothesis, $\langle C^{\hat{\kappa}g}_{ai}(\ell) \rangle = 0$.
3. Systematics and Corrections: The authors account for higher-order effects that break the linear relation between shear and density.
   • Reduced Shear and Magnification Bias: These effects introduce non-linear corrections. The authors model these using bispectrum corrections (specifically the $Q_3$ parameter) within the Limber approximation.
   • Photometric Redshift Errors: The sensitivity of the test to the mean redshift of source bins ($z_b$) is analyzed. The coefficients of the BNT transform depend on the redshift distribution; thus, errors in $z_b$ can mimic cosmological deviations.
4. Statistical Framework: The constraining power is evaluated using Fisher matrix forecasts and Monte Carlo Markov Chain (MCMC) sampling. The analysis assumes a Euclid-like survey configuration (15,000 deg$^2$, 10 redshift bins, galaxy density of 30 arcmin$^{-2}$).

Key Contributions

• Geometric Independence: The paper demonstrates that the BNT transform coefficients depend only on the background geometry (specifically the redshift dependence of the angular diameter distance) and are independent of the matter power spectrum, the Hubble constant ($H_0$), and the amplitude of matter fluctuations ($\sigma_8$). This allows the test to probe cosmology without modeling non-linear structure growth.
• Degeneracy Identification: The authors identify a specific geometric degeneracy where two cosmological models cannot be distinguished if they share the same ratio of the second to the first derivative of the inverse comoving distance function ($\xi''_K / \xi'_K$). This limits the test's ability to fully determine the expansion history in isolation but highlights its specific sensitivity to the geometry.
• Systematic Control: The study quantifies the sensitivity of the null test to photometric redshift errors. It establishes that the test is highly sensitive to the mean redshift of source bins, making it a powerful diagnostic for calibration.

Results

• Constraining Power: In a Euclid-like configuration, the null test provides complementary constraints on cosmological parameters, specifically $\Omega_m$ and the dark energy equation of state ($w_0$). The constraints are roughly perpendicular to those obtained from standard $3\times2$pt (cosmic shear + galaxy clustering + galaxy-galaxy lensing) analyses, offering a unique handle on parameter space.
• Bispectrum Impact: To utilize scales up to $\ell_{max} \approx 16,000$, corrections for the bispectrum (due to reduced shear and magnification bias) are necessary. Without these corrections, the signal is biased; with them, the test remains robust, though it requires marginalizing over a nuisance parameter ($Q_3$).
• Redshift Precision Requirement: A critical finding is the precision requirement for photometric redshifts. To preserve the constraining power on cosmological parameters, the mean redshift of the source bins must be known to a precision of order $10^{-3}$. If the uncertainty is larger (e.g., $10^{-2}$), the cosmological constraints degrade significantly due to degeneracies between redshift errors and cosmological parameters.
• Comparison to Shear Ratios: The paper distinguishes this global nulling approach from local "shear ratio" methods. The BNT transform provides an exact, global nulling property based on theoretical assumptions, whereas shear ratios rely on local approximations and specific low-redshift samples.

Significance and Claims
The authors conclude that while cosmic shear nulling offers a theoretically clean, geometry-based probe that is independent of small-scale physics modeling, its standalone constraining power on cosmological parameters is subdominant compared to standard $3\times2$pt analyses.

Consequently, the paper argues that the primary value of this method lies not in replacing standard analyses, but in its utility as:

1. A Consistency Check: A robust test to verify the validity of the background cosmological model and the internal consistency of the survey data.
2. A Photometric Redshift Calibration Probe: Given its high sensitivity to mean redshift errors, the null test is proposed as a powerful tool to calibrate photometric redshifts, particularly at high redshifts where external calibration is difficult. By controlling these systematics, the method indirectly improves the precision of cosmological measurements derived from standard weak lensing and galaxy clustering analyses.

The methodology is presented as a "virtually cost-free" addition to current analysis toolkits, requiring no new data but offering a robust diagnostic for systematic effects and a complementary geometric constraint.