Effect of the Cubic Torus Topology on Cosmological Perturbations
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Date
2021
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Abstract
We study the effect of the cubic torus topology of the Universe on scalar cosmological
perturbations which define the gravitational potential. We obtain three alternative forms of the
solution for both the gravitational potential produced by point-like masses, and the corresponding
force. The first solution includes the expansion of delta-functions into Fourier series, exploiting
periodic boundary conditions. The second one is composed of summed solutions of the Helmholtz
equation for the original mass and its images. Each of these summed solutions is the Yukawa
potential. In the third formula, we express the Yukawa potentials via Ewald sums. We show that
for the present Universe, both the bare summation of Yukawa potentials and the Yukawa-Ewald
sums require smaller numbers of terms to yield the numerical values of the potential and the force
up to desired accuracy. Nevertheless, the Yukawa formula is yet preferable owing to its much
simpler structure.
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Keywords
spatial topology, gravitational potential, Yukawa interaction
Citation
Universe