arXiv subjects

Geoff Vasil

Publications and source records attributed to Geoff Vasil.

5 recordsLinked to original sources

arXiv2026

Improved frequency hierarchy treatment for anisotropic spectral distortions

Spectral distortion anisotropies of the cosmic microwave background (CMB) provide a new probe of the early Universe that can be accessed using traditional CMB imaging techniques. It is possible to compute the creation and evolution of anisotropic signals for various scenarios using the frequency hierarchy method recently developed for CosmoTherm. However, the current treatment is not perfect and some approximations had to be made. Here, we carefully construct a modified form for the evolution equations that has the full equilibrium solutions built into the formulation. We improve the formalism to account for i) additional stimulated scattering effects, ii) kinematic corrections to the thermalization terms, iii) corrections to the standard perturbation variables and iv) direct photon sources. These effect could not be captured with the original formulation of the frequency hierarchy method but are indeed important for cleanly separating real distortions from temperature signals. However, we show that previous results are not altered significantly when compared to the improved formulation presented here. As a new worked example, which could indeed not be treated before, we also illustrate how possible changes in the temperature-redshift relation would create spectral distortion anisotropies in the pre-recombination era. The theoretical methods presented here are also an important step towards being able to consistently predict the CMB spectral distortion anisotropies in photon-dark photon and photon-axion conversion scenarios.

astro-ph.CO↗

arXiv2019

A sparse spectral method on triangles

In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse differentiation operators. This allows us to derive a practical spectral method for solving linear partial differential equations on triangles with sparse discretizations. We can thereby rapidly solve partial differential equations using polynomials with degrees in the thousands, resulting in sparse discretizations with as many as several million degrees of freedom.

math.NA↗

arXiv2018

On the magnetorotational instability and elastic buckling

This paper demonstrates an equivalence between rotating magnetised shear flows and a stressed elastic beam. This results from finding the same form of dynamical equations after an asymptotic reduction of the axis-symmetric magnetorotational instability (MRI) under the assumption of almost-critical driving. The analysis considers the MRI dynamics in a non-dissipative near-equilibrium regime. Both the magnetic and elastic systems reduce to a simple one-dimensional wave equation with a nonlocal nonlinear feedback. Under transformation, the equation comprises a large number of mean-field interacting Duffing oscillators. This system was the first proven example of a strange attractor in a partial differential equation. Finding the same reduced equation in two natural applications suggests the model might result from other applications and could fall into a universal class based on symmetry.

astro-ph.SR↗

arXiv2018

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as \textit{grad} and \textit{div}. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.

math.NA↗

arXiv2018

Recurrence relations for orthogonal polynomials on a triangle

This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partial derivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in a systematic fashion by introducing ladder operators that map an orthogonal polynomial to another by incrementing or decrementing its associated parameters by one.

math.CA↗