[Paper Review] On Wave Dark Matter, Shells in Elliptical Galaxies, and the Axioms of General Relativity

Hubert L. Bray|arXiv (Cornell University)|Dec 23, 2012

Cosmology and Gravitation Theories35 references8 citations

TL;DR

This paper proposes that wave dark matter—modeled as a scalar field solving the Einstein-Klein-Gordon equations—can naturally produce interleaved, shell-like density structures in elliptical galaxies through superpositions of spherically symmetric and first-order spherical harmonic perturbations. The model demonstrates that phase interference between fundamental and first-excited modes generates alternating density maxima and minima at antipodal points, reproducing the observed interleaved shell morphology seen in real galaxies.

ABSTRACT

This paper is a sequel to the author's paper entitled "On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity" [arXiv:1004.4016] which explored a geometrically natural axiomatic definition for dark matter modeled by a scalar field satisfying the Einstein-Klein-Gordon wave equations which, after much calculation, was shown to be consistent with the observed spiral and barred spiral patterns in disk galaxies. We give an update on where things stand on this "wave dark matter" model of dark matter (aka scalar field dark matter and boson stars), an interesting alternative to the WIMP model of dark matter, and discuss how it has the potential to help explain the long-observed interleaved shell patterns, also known as ripples, in the images of elliptical galaxies.

Motivation & Objective

• To investigate whether wave dark matter, as a geometric alternative to WIMPs, can explain the formation of interleaved shell structures in elliptical galaxies.
• To analyze the role of scalar field solutions to the Einstein-Klein-Gordon equations in generating periodic density modulations resembling observed shells.
• To explore the gravitational and dynamical implications of such wave dark matter structures on visible matter distribution in elliptical galaxies.
• To provide a theoretical framework linking wave dark matter density profiles to the observed luminosity ripples in elliptical galaxies.

Proposed method

• Construct approximate solutions to the Einstein-Klein-Gordon equations using superpositions of spherically symmetric and first-order spherical harmonic modes.
• Model the dark matter density as a sum of radial functions multiplied by spherical harmonics, specifically using $ f_{\omega_{0},0}(r) $ and $ f_{\omega_{1},1}(r) $, representing ground and first-excited states.
• Express the total wave dark matter density $ \mu_{DM} $ in terms of spherical harmonics via homogeneous harmonic polynomials, enabling gravitational potential computation.
• Use the phase relationship between $ f_{\omega_{0},0}(r) $ and $ f_{\omega_{1},1}(r) $ to determine where constructive and destructive interference occur, forming shells.
• Generate simulated density profiles using a MATLAB function (shells.m) that computes radial and angular dependencies to visualize shell formation.
• Analyze the odd symmetry of the perturbation term to explain the interleaved nature of shells, where maxima on one side correspond to minima on the opposite side.

Experimental results

Research questions

• RQ1 Can wave dark matter produce density structures that qualitatively match the observed interleaved shell patterns in elliptical galaxies?
• RQ2 What is the role of phase interference between different scalar field modes in forming periodic, radially symmetric density modulations?
• RQ3 How does the odd symmetry of the perturbation term in the wave function lead to the alternating shell pattern observed on opposite sides of a galaxy?
• RQ4 To what extent can wave dark matter density fluctuations gravitationally influence the distribution of visible matter to form observable luminosity ripples?
• RQ5 What is the mathematical and physical basis for modeling wave dark matter using solutions to the Einstein-Klein-Gordon equations with specific spherical harmonic components?

Key findings

• Wave dark matter models with superposed $ \ell=0 $ and $ \ell=1 $ spherical harmonic modes produce density profiles with alternating maxima and minima that form interleaved shells.
• The perturbation term $ \tilde{U}_1(r)(r\cos\alpha\sin\theta) $ introduces an odd function in the angular variable, which ensures that a density maximum on one side of the galaxy corresponds to a minimum on the opposite side.
• Three major shells are formed on each side of the origin in the simulated density profiles, corresponding to three instances of phase alignment between the two radial functions in the interference pattern.
• The radial functions $ f_{\omega_{0},0}(r) $ and $ f_{\omega_{1},1}(r) $ go in and out of phase periodically, creating regions of enhanced and reduced density that manifest as visible shells.
• The model's predicted shell structure closely matches the observed interleaved morphology in galaxies like NGC 474 and NGC 4382, as shown in Figures 1 and 2.
• The wave dark matter density can be expressed in terms of spherical harmonics via homogeneous harmonic polynomials, enabling consistent computation of the gravitational potential and simulation of observable effects.