Weak Lensing Low Multipoles

TL;DR

This work analyzes the largest-angle information encoded in weak-lensing convergence by deriving exact full-sky expressions for the convergence multipoles $C_\ell^\kappa$ without the Limber approximation and studying how the dipole, quadrupole, and octupole saturate with source redshift $z_s$ in $\Lambda$CDM. It combines theory, horizon-scale $N$-body simulations (Quijote) with observer-environment variants and an observational reconstruction from the 2MRS survey to quantify how local structure and the observer’s environment modulate these low-$\ell$ modes. The study finds that low-$\ell$ convergence amplitudes saturate around $z_s\sim1$ with $d_\kappa^2 = \frac{9}{4\pi} C_1^\kappa$ in the range $10^{-5}$–$10^{-4}$, while nearby environments can boost signals at very low redshift; by $z\sim0.1$ all observer classes converge to $d_\kappa\sim(2-4)\times10^{-5}$. When translated to a number-count dipole, weak lensing contributes at most a few percent to the observed cosmic dipole anomaly, and leakage from higher multipoles under sky masks is insufficient to explain the discrepancy. The results underscore the need to model convergence low-multipole signals in precision cosmology and anticipate direct measurements with future wide-area surveys (Euclid, LSST, SPHEREx) to clarify their role in large-angle cosmological tensions.

Abstract

We analyse the low-multipole components of the weak-lensing convergence field in a FLRW universe. The low-multipole convergence field, encodes the largest-angle coherent potential gradients, essential for assessment of large-angle features in data. To study this large angle signal, we perform a combined analytical, numerical and observational study. Starting from exact analytical expressions for the convergence power spectrum, we quantify how the dipole, quadrupole and octupole build up with source redshift and show that, in $Λ$CDM, they saturate at an amplitude of order $10^{-4}$. We then use full-sky, horizon-scale $N$-body simulations (Quijote) to explore the dependence of this signal on the observer's environment, comparing random observers to ``Milky Way-like'' observers. In parallel, we reconstruct the convergence field due to our local Universe with the 2MASS Redshift Survey (2MRS), with proper treatment of incompleteness and galaxy bias. We find that the observed low multipoles from observation is above the $Λ$CDM mean predictions, but in full agreement with Milky Way-like observers in the simulation. Finally, by converting the convergence dipole into a number-count dipole, we test whether weak lensing can contribute to the cosmic dipole anomaly, an idea motivated by its natural alignment with the CMB dipole and by the fact that lensing, unlike clustering, cannot be removed by cross-matching surveys and thus survives in all high-redshift catalogues. We show that weak lensing by local structure contributes at most a few percent to this observed anomaly.

Figures (9)

• Figure 1: Evolution of the expected dipole ($\ell=1$), quadrupole ($\ell=2$), octopole ($\ell=3$), and $\ell=10$ contribution to the convergence power spectrum as a function of source redshift $z_s$, computed within $\Lambda$CDM. All low--$\ell$ modes exhibit a clear saturation at $z_s \simeq 1$--$1.5$.
• Figure 2: The weight function $w_\kappa(r,r_s)$ for sources at $z_s = 0.5$, $1.0$, and $1.5$. Although the peak of $w_\kappa$ lies roughly halfway between observer and source, this does not imply that structures at this distance dominate the low multipoles. Projection effects suppress the contribution of distant structures to the largest angular scales.
• Figure 3: Dipole amplitude $C_1^\kappa$ (top), quadrupole amplitude $C_2^\kappa$ (middle), octopole amplitude $C_3^\kappa$ (bottom), of the convergence field for the three classes of observers in the Quijote simulation. For each observer type, 50 independent realizations are analysed. Error bars represent the variance across observers.
• Figure 4: Completeness function $f_c(z)$ of the 2MRS survey after our cuts, along with the redshift distribution of the 42,580 surviving galaxies. The vertical dotted line marks the redshift at which the completeness falls below $0.1\%$. We take this strict limit to achieve a higher precision of the convergence map.
• Figure 5: Top: number of 2MRS sources per Healpix pixel. Bottom: convergence map from 2MRS, with dipole directions overlaid: weak-lensing dipole at $z_s=0.07$, CMB dipole planck_collaboration_planck_2020, and 2MRS clustering dipole. Grey region: Galactic mask ($|b|<10^\circ$).