Novel geometrical test of cosmological expansion from photometric data

1. Problem Statement

Standard weak lensing analyses (such as the "3x2pt" method combining cosmic shear, galaxy clustering, and their cross-correlation) rely heavily on accurate modeling of the non-linear growth of cosmic structure. This dependence introduces significant theoretical uncertainties, particularly at small scales where baryonic physics and non-linear gravitational instability are difficult to model precisely. Consequently, constraints on cosmological parameters, specifically the dark energy equation of state ($w_0, w_a$), are often degenerate with parameters describing structure growth (e.g., $\sigma_8$).

The authors aim to develop a method that:

• Probes the expansion history of the Universe (geometry) independently of the growth of structure.
• Utilizes data at arbitrary, even deeply non-linear, scales without requiring complex non-linear modeling.
• Provides a robust "null test" to validate cosmological models.

2. Methodology

The BNT Transform

The core of the methodology is the Bernardeau-Nishimichi-Taruya (BNT) transform. This is a linear transformation applied to tomographic cosmic shear maps (convergence maps, $\kappa$).

• Mechanism: The transform constructs linear combinations of convergence maps from different redshift bins ($\hat{\kappa}_a = p_{ai} \kappa_i$) such that the contribution from low-redshift lenses is mathematically nulled (zeroed out).
• Dependence: The transformation coefficients ($p_{ai}$) depend only on the redshift distribution of source galaxies and the angular distance-redshift relation ($D_A(z)$). They are explicitly independent of the matter power spectrum, the amplitude of matter fluctuations ($\sigma_8$), and the Hubble constant ($H_0$).
• Result: The transformed kernels are more localized in redshift, effectively separating physical scales along the line of sight.

The Null Test

The authors propose a specific null test based on the statistical independence of the transformed data:

• Hypothesis: If the cosmological parameters used to calculate the BNT coefficients ($p_{ai}$) are correct, the cross-correlation between the nulled convergence maps ($\hat{\kappa}$) and low-redshift galaxy density fields ($\delta_g$) should be zero.
• Observable: The cross-spectrum $\tilde{C}^{\hat{\kappa}g}_{ai}(\ell)$.
• Condition: $\langle \tilde{C}^{\hat{\kappa}g}_{ai}(\ell) \rangle = 0$ for all multipoles $\ell$ and valid bin pairs where the source redshifts of the nulled map do not overlap with the lens redshifts of the galaxy density field.

Handling Systematics and Biases

• Non-linear Corrections: The authors acknowledge that at small scales, non-linear effects (reduced shear, magnification bias) break the linear relation. They introduce a nuisance parameter $Q_3$ (related to the Bispectrum normalization) to model these second-order corrections.
• Likelihood Construction: A Gaussian likelihood is constructed using the observed cross-spectra and a bias correction template. The covariance matrix is derived directly from the data (auto-spectra of the transformed maps and galaxy catalogs), avoiding reliance on theoretical non-linear models.
• Analysis Tools: The study employs both a Fisher Matrix analysis for initial performance estimation and Markov Chain Monte Carlo (MCMC) sampling (using the cobaya package) to explore parameter degeneracies and posterior distributions.

Simulation Setup

• Survey: Modeled after the Euclid mission (Stage IV).
• Parameters: Flat universe ($\Omega_K=0$) with parameters $\{\Omega_m, w_0, w_a\}$. Fiducial values: $\{0.32, -1, 0\}$.
• Data: 10 photometric redshift bins, galaxy density $30 \text{ arcmin}^{-2}$, median redshift $z_m=0.9$.
• Noise: Includes shape noise ($\sigma_s = 0.3$) and shot noise.

3. Key Contributions

1. Geometric Probe Independent of Structure Growth: The paper demonstrates that the BNT transform allows for the extraction of cosmological information regarding the expansion rate ($D_A(z)$) that is completely decoupled from the amplitude of matter fluctuations ($\sigma_8$) and non-linear structure growth modeling.
2. Robust Null Test: It establishes a rigorous null test where a non-zero signal directly indicates a mismatch in the assumed angular distance-redshift relation, rather than uncertainties in the matter power spectrum.
3. Utilization of Non-linear Scales: Unlike standard methods that must cut small-scale data to avoid modeling errors, this method can theoretically harness statistical power from deeply non-linear scales because the test relies on geometry, not the specific shape of the power spectrum.
4. Identification of Degeneracies: The authors characterize the specific degeneracy between $\Omega_m$ and $w_0$ inherent to this method, showing it differs from standard weak lensing degeneracies.

4. Results

• Constraining Power (Fisher Analysis):

  • The nulling test alone has weak constraining power on the dark energy equation of state ($w_0, w_a$).
  • However, when combined with Planck CMB priors on $\Omega_m$ (specifically $\sigma(\Omega_m) \approx 0.052$), the Figure-of-Merit (FoM) improves by more than a factor of 10.
  • The constraining power is currently limited by shape noise, not cosmic variance. Increasing the source galaxy density significantly boosts the FoM.

• MCMC Analysis:

  • Degeneracy: In the absence of external priors, the method exhibits a strong degeneracy between $\Omega_m$ and $w_0$. This is distinct from the standard weak lensing degeneracy because the method is insensitive to the growth rate of fluctuations.
  • Precision with Priors: When combined with Planck+BAO priors on $\Omega_m$ ($\sigma(\Omega_m) \approx 0.012$), the method achieves a precision on $w_0$ of $\sigma(w_0) \approx 0.095$.
  • Comparison: This precision is competitive with, and in some aspects complementary to, standard analyses, achieved with negligible computational cost (using standard summary statistics).

• Bias Correction: The inclusion of the $Q_3$ parameter successfully marginalizes over the bias introduced by reduced shear and magnification effects, allowing the test to remain valid down to smaller angular scales.

5. Significance and Future Outlook

• Theoretical Breakthrough: This work provides a "pure geometrical test" for cosmology using weak lensing. It proves that one can constrain the expansion history without needing to solve the complex problem of non-linear structure formation.
• Stage IV Surveys: For future missions like Euclid, LSST, and Roman, this method offers a way to utilize the full statistical power of the data, including small scales, which are currently discarded or heavily modeled in standard analyses.
• Limitations & Solutions: The primary limitation identified is shape noise. The authors suggest that future techniques, such as Line Intensity Mapping (LIM) combined with CMB lensing, could drastically reduce shape noise by increasing the number of tracers, thereby making this null test a highly powerful cosmological probe.
• Complementarity: The method is highly complementary to standard 3x2pt analyses. While standard methods are sensitive to growth, this method is sensitive to geometry, allowing for a powerful cross-check of the standard cosmological model ($\Lambda$CDM).

In summary, the paper presents a novel, computationally efficient, and theoretically robust framework to test cosmological expansion using weak lensing data, effectively bypassing the uncertainties associated with non-linear structure growth.