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Register a new userProper Fourier decomposition formalism for cosmological fields in spherical shells
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Lado Samushia
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Cosmological random fields are often analysed in spherical Fourier-Bessel basis. Compared to the Cartesian Fourier basis this has an advantage of properly taking into account some of the relevant physical processes (redshift-space distortions, redshift evolution). The observations usually come in redshift slices and have a partial sky coverage. These masking effects strongly correlate Fourier-Bessel modes that are meant for a perfect spherical geometry and result in a lot of redundant measurements. This work proposes a new Fourier basis that is better suited for measurements in redshift shells and results in fewer Fourier modes, with the radial modes strictly uncorrelated on large scales and the angular modes with significantly reduced redundancy. I argue that the spherical Fourier analysis of cosmological fields should always use these new modes instead of the historically established Fourier-Bessel eigenfunctions. The new angular modes on the other hand have number of practical advantages and disadvantages and whether or not to adopt them for a particular analysis should be made on a case by case basis.
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The spherical Fourier-Bessel (SFB) decomposition is a natural choice for the radial/angular separation that allows optimal extraction of cosmological information from large volume galaxy surveys. In this paper we develop a SFB power spectrum estimator that allows the measurement of the largest angular and radial modes with the next generation of galaxy surveys. The code measures the pseudo-SFB power spectrum, and takes into account mask, selection function, pixel window, and shot noise. We show that the local average effect is significant only in the largest-scale mode, and we provide an analytical covariance matrix. By imposing boundary conditions at the minimum and maximum radius encompassing the survey volume, the estimator does not suffer from the numerical instabilities that have proven challenging in the past. The estimator is demonstrated on simplified Roman-like, SPHEREx-like, and Euclid-like mask and selection functions. For intuition and validation, we also explore the SFB power spectrum in the Limber approximation. We release the associated public code written in Julia.
High-precision cosmology requires the analysis of large-scale surveys in 3D spherical coordinates, i.e. spherical Fourier-Bessel decomposition. Current methods are insufficient for future data-sets from wide-field cosmology surveys. The aim of this paper is to present a public code for fast spherical Fourier-Bessel decomposition that can be applied to cosmological data or 3D data in spherical coordinates in other scientific fields. We present an equivalent formulation of the spherical Fourier-Bessel decomposition that separates radial and tangential calculations. We propose the use of the existing pixelisation scheme HEALPix for a rapid calculation of the tangential modes. 3DEX (3D EXpansions) is a public code for fast spherical Fourier-Bessel decomposition of 3D all-sky surveys that takes advantage of HEALPix for the calculation of tangential modes. We perform tests on very large simulations and we compare the precision and computation time of our method with an optimised implementation of the spherical Fourier-Bessel original formulation. For surveys with millions of galaxies, computation time is reduced by a factor 4-12 depending on the desired scales and accuracy. The formulation is also suitable for pre-calculations and external storage of the spherical harmonics, which allows for additional speed improvements. The 3DEX code can accommodate data with masked regions of missing data. 3DEX can also be used in other disciplines, where 3D data are to be analysed in spherical coordinates. The code and documentation can be downloaded at http://ixkael.com/blog/3dex.
Cosmological weak lensing has been a highly successful and rapidly developing research field since the first detection of cosmic shear in 2000. However, it has recently been pointed out in Yoo et al. that the standard weak lensing formalism yields gauge-dependent results and, hence, does not meet the level of accuracy demanded by the next generation of weak lensing surveys. Here, we show that the Jacobi mapping formalism provides a solid alternative to the standard formalism, as it accurately describes all the relativistic effects contributing to the weak lensing observables. We calculate gauge-invariant expressions for the distortion in the luminosity distance, the cosmic shear components and the lensing rotation to linear order including scalar, vector and tensor perturbations. In particular, the Jacobi mapping formalism proves that the rotation is fully vanishing to linear order. Furthermore, the cosmic shear components contain an additional term in tensor modes which is absent in the results obtained with the standard formalism. Our work provides further support and confirmation of the gauge-invariant lensing formalism needed in the era of precision cosmology.
We introduce in this article a general formalism for Fourier based wave front sensing. To do so, we consider the filtering mask as a free parameter. Such an approach allows to unify sensors like the Pyramid Wave Front Sensor (PWFS) and the Zernike Wave Front Sensor (ZWFS). In particular, we take the opportunity to generalize this two sensors in terms of sensors class where optical quantities as, for instance, the apex angle for the PWFS or the depth of the Zernike mask for the ZWFS become free parameters. In order to compare all the generated sensors of this two classes thanks to common performance criteria, we firstly define a general phase-linear quantity that we call meta-intensity. Analytical developments allow then to split the perfectly phase-linear behavior of a WFS from the non-linear contributions making robust and analytic definitions of the sensitivity and the linearity range possible. Moreover, we define a new quantity called the SD factor which characterizes the trade-off between these two antagonist quantities. These developments are generalized for modulation device and polychromatic light. A non-exhaustive study is finally led on the two classes allowing to retrieve the usual results and also make explicit the influence of the optical parameters introduced above.
We report on the buckling and subsequent collapse of orthotropic elastic spherical shells under volume and pressure control. Going far beyond what is known for isotropic shells, a rich morphological phase space with three distinct regimes emerges upon variation of shell slenderness and degree of orthotropy. Our extensive numerical simulations are in agreement with experiments using fabricated polymer shells. The shell buckling pathways and corresponding strain energy evolution are shown to depend strongly on material orthotropy. We find surprisingly robust orthotropic structures with strong similarities to stomatocytes and tricolpate pollen grains, suggesting that the shape of several of Natures collapsed shells could be understood from the viewpoint of material orthotropy.
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