[Paper Review] New Velocity Distribution in the Context of the Eddington Theory

J. D. Vergados, David A. Owen|arXiv (Cornell University)|Mar 19, 2002

Dark Matter and Cosmic Phenomena54 references30 citations

TL;DR

This paper proposes a new velocity distribution for cold dark matter (CDM) within the Eddington theory framework, deriving density and velocity profiles from a distribution function dependent on energy and angular momentum. Unlike traditional Maxwell-Boltzmann models, this approach yields a physically motivated, non-Maxwellian velocity distribution with an intrinsic cutoff, leading to significant, asymmetry-dependent variations in direct detection rates—particularly in directional and modulated signals—offering a more rigorous alternative to phenomenological assumptions in LSP detection studies.

ABSTRACT

Exotic dark matter together with the vacuum energy (associated with the cosmological constant) seem to dominate the Universe. Thus its direct detection is central to particle physics and cosmology. Supersymmetry provides a natural dark matter candidate, the lightest supersymmetric particle (LSP). One essential ingredient in obtaining the direct detection rates is the density and velocity distribution of the LSP. The detection rate is proportional to this density in our vicinity. Furthermore, since this rate is expected to be very low, one should explore the two characteristic signatures of the process, namely the modulation effect, i.e. the dependence of the event rate on the Earth's motion and the correlation of the directional rate with the motion of the sun. Both of these crucially depend on the LSP velocity distribution. In the present paper we study simultaneously density profiles and velocity distributions based on the Eddington theory.

Motivation & Objective

• To develop a self-consistent, physically motivated velocity distribution for cold dark matter using Eddington's theory.
• To derive density and velocity profiles from a distribution function dependent on energy and angular momentum.
• To examine the impact of velocity anisotropy on rotational curves and direct detection rates.
• To provide a framework for calculating direct detection rates that avoids ad hoc velocity cutoffs.
• To compare the new distribution to standard Gaussian models, especially in directional and modulated detection scenarios.

Proposed method

• Uses Eddington's formalism to relate the distribution function f(E, J) to the gravitational potential Φ(r), with f(E, J) = Kλ(−2E)λ[1 + αs J²/(rsυm)²].
• Solves Poisson's equation numerically to obtain the potential Φ(r) and density ρ(r) from the derived distribution function.
• Introduces an anisotropy parameter αs to model axially symmetric velocity distributions, replacing the traditional hand-tuned cutoff.
• Derives the velocity distribution function Ψ(x) that includes both isotropic and anisotropic contributions, with x = r/rs.
• Computes the non-directional and directional differential event rates using the new velocity distribution and nuclear form factors.
• Evaluates the dependence of total and modulated detection rates on the asymmetry parameter αs, particularly for 127I and 100 GeV LSPs.

Experimental results

Research questions

• RQ1 How does the Eddington-based velocity distribution with intrinsic cutoff compare to the standard Maxwell-Boltzmann distribution in direct detection predictions?
• RQ2 What is the impact of velocity anisotropy (parameterized by αs) on the rotational curves and local dark matter density profiles?
• RQ3 How do the directional and modulated detection rates depend on the new velocity distribution compared to Gaussian models?
• RQ4 Can the new distribution explain observed rotational curves while remaining consistent with direct detection constraints?
• RQ5 To what extent does the asymmetry parameter αs affect the total, non-directional detection rate, especially after including nuclear form factors?

Key findings

• The new velocity distribution is non-Maxwellian and features a physically derived upper velocity cutoff, unlike the hand-tuned cutoffs in standard models.
• The total direct detection rate shows strong dependence on the asymmetry parameter αs, especially when the nuclear form factor is included, with significant deviations from Gaussian model predictions.
• For λ = 1/2 and 100 GeV LSPs, the differential rate T(u) varies substantially with αs, indicating that asymmetry strongly influences detectability.
• The directional rate, proportional to Ψdir(x), depends critically on the observation direction relative to the Sun's motion, offering a strong experimental signature.
• The parameter κ = Rdir/R, which quantifies directional rate enhancement, is independent of LSP mass and nuclear parameters but strongly dependent on observation angle, providing a robust experimental handle.
• The model predicts that rotational curves are sensitive to αs, particularly when the parameter a = αs|Φ₀|/υ²m is large, suggesting that astrophysical data can constrain the asymmetry parameter.