Weak Lensing Low Multipoles

1. Problem Statement

Weak gravitational lensing (WL) is typically analyzed on intermediate and small angular scales (cosmic shear). However, the low-multipole components (dipole, quadrupole, octupole) of the convergence field ($\kappa$) remain largely unexplored. These modes encode the largest-angle coherent potential gradients generated by local matter distributions (superclusters, voids, filaments).

The study addresses two primary motivations:

1. Cosmological Systematics: Low-$\ell$ modes leak into masked-sky reconstructions and perturb full-sky observables, potentially mimicking primordial non-Gaussianity (PNG) or Integrated Sachs-Wolfe (ISW) signals if not properly modeled.
2. The Cosmic Dipole Anomaly: Recent observations (e.g., Secrest et al.) show a discrepancy between the kinematic dipole inferred from the Cosmic Microwave Background (CMB) and the number-count dipole of distant galaxies. While weak lensing perturbs both solid angles and fluxes, creating a dipole in number counts, its magnitude has not been rigorously quantified. Previous studies provided only loose upper bounds, leaving open the question of whether local structure lensing could explain this anomaly.

2. Methodology

The authors employ a tripartite approach combining analytical theory, numerical simulations, and observational data reconstruction.

A. Theoretical Analysis

• Exact Formalism: They derive exact full-sky expressions for the convergence power spectrum ($C_\ell^\kappa$) for multipoles $\ell = 1, 2, 3$ without using the Limber approximation (which is invalid for low $\ell$).
• Redshift Evolution: They compute the evolution of the dipole, quadrupole, and octupole amplitudes as a function of source redshift ($z_s$) within a $\Lambda$CDM framework.
• Weight Function Analysis: They analyze the lensing weight function $w_\kappa(r, r_s)$ to understand why low multipoles saturate at high redshifts (projection effects shift distant structures to higher multipoles).

B. Numerical Simulations (Quijote)

• Environment Dependence: Using the Quijote N-body simulation suite, they place 50 observers in three distinct environments to test the impact of local density:
  1. Random: Uniformly distributed in the simulation volume.
  2. Milky Way-like: Located within $\sim 12$ Mpc of a Virgo-like halo with peculiar velocities matching the Local Group ($\sim 370$ km/s).
  3. Virgo-like: Centered on massive halos ($\sim 10^{15} h^{-1} M_\odot$).
• Measurement: They generate convergence maps up to $z \approx 0.2$ and measure the multipole amplitudes for each observer class.

C. Observational Reconstruction (2MRS)

• Data Source: They reconstruct the convergence field using the 2MASS Redshift Survey (2MRS) combined with distances from Cosmicflows-4 (CF4).
• Corrections: They apply rigorous corrections for:
  • Completeness: Modeling the redshift-dependent selection function ($f_c(z)$) due to the magnitude limit ($K_s < 11.75$).
  • Bias: Applying a linear galaxy bias ($b_g \approx 1.3$).
  • Masking: Handling the Galactic plane mask ($|b| < 10^\circ$).
• Comparison: They compare the reconstructed multipoles against $\Lambda$CDM predictions and the simulation results.

D. Cosmic Dipole Anomaly Test

• They derive the relationship between the convergence dipole ($D_\kappa$) and the number-count dipole ($D_{WL}$) using the Ellis-Baldwin formalism extended for weak lensing.
• They calculate the contribution of WL to the dipole for specific surveys (CatWISE quasars and NVSS radio galaxies) based on their flux distribution slopes ($x$).
• They perform a "toy model" analysis to estimate the potential contribution of massive structures beyond the 2MRS completeness limit (e.g., Shapley Supercluster).

3. Key Contributions & Results

Theoretical Findings

• Saturation of Low Multipoles: The convergence dipole, quadrupole, and octupole grow with source redshift but saturate at $z_s \sim 1$. The amplitude stabilizes at order $10^{-5}$ to $10^{-4}$.
• Physical Mechanism: This saturation occurs because distant structures project onto smaller angular scales on the sky, shifting their contribution to higher multipoles ($\ell > 3$). The low-$\ell$ modes are dominated by the local universe ($z \lesssim 0.2$).
• Fitting Function: The authors provide a fitting function for $\langle C_\ell^\kappa \rangle(z_s)$ for $\ell=1,2,3$ in $\Lambda$CDM.

Simulation Results (Environment Dependence)

• Local Enhancement: Observers in "Milky Way-like" environments (near massive overdensities) exhibit convergence multipoles up to an order of magnitude larger than the statistical mean at very low redshifts ($z \lesssim 0.05$).
• Convergence: By $z \approx 0.1$, all observer classes converge to similar amplitudes, indicating that the local environment effect diminishes rapidly with distance.
• Statistical Atypicality: The Milky Way's location is statistically atypical (only $\sim 1.4\%$ of particles in the simulation), explaining why local observations exceed the global $\Lambda$CDM mean.

Observational Results (2MRS)

• Reconstruction: The reconstructed low multipoles from 2MRS are above the $\Lambda$CDM mean but in perfect agreement with the "Milky Way-like" observers in the Quijote simulation.
• Robustness: The results are insensitive to the choice of distance indicator (redshift vs. CF4 distances), suggesting peculiar velocities do not significantly bias the low multipoles.

Impact on Cosmic Dipole Anomaly

• Magnitude: The weak lensing contribution to the number-count dipole is negligible compared to the observed anomaly.
  • For CatWISE ($x > 1$), WL enhances the dipole by $\sim 2 \times 10^{-4}$.
  • For NVSS ($x < 1$), WL suppresses the dipole by $\sim 6 \times 10^{-5}$.
  • The observed anomalies are $\sim 10^{-2}$, meaning WL contributes less than a few percent.
• Leakage: The leakage of higher multipoles into the dipole due to sky masking is also negligible if the dipole itself is small.
• Beyond 2MRS: Even assuming extreme, unobserved massive structures (e.g., a $5 \times 10^{18} M_\odot$ overdensity) beyond the 2MRS horizon, the induced dipole only increases by a factor of a few, remaining insufficient to explain the anomaly.

4. Significance and Conclusion

• Resolution of the Anomaly: The study definitively rules out weak lensing by local structure as the primary cause of the cosmic number-count dipole anomaly. The effect is real but too small to reconcile the discrepancy between the CMB dipole and galaxy counts.
• Necessary Correction: While not the cause of the anomaly, the weak lensing dipole is a physically unavoidable foreground/background effect. It must be modeled and subtracted in future precision cosmology to avoid biasing measurements of:
  • Primordial Non-Gaussianity ($f_{NL}$).
  • The Integrated Sachs-Wolfe (ISW) effect.
  • Tests of statistical isotropy.
• Observer Bias: The results highlight that observers in special environments (like the Local Group) will systematically measure higher low-$\ell$ convergence than the cosmic average, a crucial consideration for interpreting wide-angle surveys.
• Future Outlook: The authors emphasize that upcoming surveys (Euclid, LSST/Rubin, SPHEREx) with deep, wide-area coverage will be essential to directly measure these low multipoles at higher redshifts ($z > 1$) and fully map the lensing kernel, moving beyond the shallow limits of current catalogs like 2MRS.

In summary, this paper provides the first rigorous, combined theoretical and observational quantification of weak lensing low multipoles, establishing them as a significant local environmental effect that must be accounted for in precision cosmology, while confirming they do not explain the cosmic dipole anomaly.