3D weak lensing with spin wavelets on the ball

Boris Leistedt, Jason D. McEwen, Thomas D. Kitching, and Hiranya V. Peiris

Phys. Rev. D 92, 123010 – Published 29 December 2015

Abstract

We construct the spin flaglet transform, a wavelet transform to analyze spin signals in three dimensions. Spin flaglets can probe signal content localized simultaneously in space and frequency and, moreover, are separable so that their angular and radial properties can be controlled independently. They are particularly suited to analyzing cosmological observations such as the weak gravitational lensing of galaxies. Such observations have a unique 3D geometrical setting since they are natively made on the sky, have spin angular symmetries, and are extended in the radial direction by additional distance or redshift information. Flaglets are constructed in the harmonic space defined by the Fourier-Laguerre transform, previously defined for scalar functions and extended here to signals with spin symmetries. Thanks to various sampling theorems, both the Fourier-Laguerre and flaglet transforms are theoretically exact when applied to bandlimited signals. In other words, in numerical computations the only loss of information is due to the finite representation of floating point numbers. We develop a 3D framework relating the weak lensing power spectrum to covariances of flaglet coefficients. We suggest that the resulting novel flaglet weak lensing estimator offers a powerful alternative to common 2D and 3D approaches to accurately capture cosmological information. While standard weak lensing analyses focus on either real- or harmonic-space representations (i.e., correlation functions or Fourier-Bessel power spectra, respectively), a wavelet approach inherits the advantages of both techniques, where both complicated sky coverage and uncertainties associated with the physical modeling of small scales can be handled effectively. Our codes to compute the Fourier-Laguerre and flaglet transforms are made publicly available.

Received 15 August 2015

DOI:https://doi.org/10.1103/PhysRevD.92.123010

Authors & Affiliations

Boris Leistedt^1,*, Jason D. McEwen^2,†, Thomas D. Kitching^2,‡, and Hiranya V. Peiris^1,§

¹Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
²Mullard Space Science Laboratory, University College London, Surrey RH5 6NT, United Kingdom

^*boris.leistedt.11@ucl.ac.uk
^†jason.mcewen@ucl.ac.uk
^‡t.kitching@ucl.ac.uk
^§h.peiris@ucl.ac.uk

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Vol. 92, Iss. 12 — 15 December 2015

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Images

Figure 1
Real and imaginary parts of the Fourier-Laguerre basis function ${{}_{s}Y}_{ℓ m} (n) K_{p} (r)$ with $s = 0$ , $ℓ = 10$ , $m = 5$ , $p = 40$ , shown on the 3D Fourier-Laguerre sampling scheme with $L = P = 80$ , plotted up to $r = R / 2$ , with $R$ being the radius of the final sample of the radial sampling scheme [39]. This plot shows that the 3D basis functions successfully probe harmonic modes in the angular and radial directions separately. Furthermore, notice that the basis functions are not well localized in space, similar to standard Fourier bases.
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Figure 2
Tiling of the angular (left panel) and radial (right panel) harmonic lines employed to construct flaglets ${{}_{s}Ψ}^{i j}$ in Eq. (59). The left panel shows the kernels $κ_{λ} (\frac{ℓ}{λ^{i}})$ and $η_{λ} (\frac{ℓ}{λ^{I_{0}}})$ tiling the range $ℓ = 0, \dots, L - 1$ (zero is not shown but is included in $η_{λ}$ ), while the right panel shows $κ_{ν}$ and $η_{ν}$ tiling $p = 0, \dots, P - 1$ . Since $L = P = 243$ and $λ = ν = 3$ , six kernels are needed to fully tile $[0, L - 1]$ and $[0, P - 1]$ , i.e., the scales considered are $i = 0, \dots, 5$ and $j = 0, \dots, 5$ . The first two scales are incorporated into the scaling function by setting $I_{0} = J_{0} = 2$ . The thicker lines highlight the scales shown in Figs. 3 and 4. Note that we do not show the hybrid kernel $η_{λ ν}$ , included in the scaling function to satisfy the admissibility condition at low $p$ and $ℓ$ , see Eqs. (53) and (61). We also do not include the directional component $ζ_{ℓ m}$ in this figure, which is nevertheless shown in real space in Fig. 3.
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Figure 3
Real part of spin-0 directional flaglets, plotted up to $r = R / 2$ (with $R$ being the last node of the radial sampling scheme), showing their excellent localization and directional properties (even and odd for $N = 2$ , 3, respectively). This demonstrates that the flaglet transform can probe radial and angular modes well defined in both real and harmonic space. The radial and angular harmonic modes $ℓ$ and $p$ corresponding to the scales $i = j = 2$ , 3 are highlighted as thicker lines in Fig. 2. The dashed lines and the half sphere show the slices that are plotted in the various subpanels of this figure. Flaglets are localized in both pixel and frequency space, unlike the Fourier-Laguerre basis functions shown in Fig. 1, which are delta functions in harmonic space and therefore not localized in pixel space.
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Figure 4
Same as Fig. 3, but for spin-2 flaglets and showing their real and imaginary parts, as well as the complex modulus. This demonstrates that the flaglets natively support the symmetries of spin signals while being well localized in both real and harmonic space.
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