How to Compute Velocity in Dark Matter Interactions: A Comprehensive Guide

Calculating the velocity in dark matter interactions is crucial for understanding the signals expected to be observed in searches for dark matter. This comprehensive guide delves into the key factors and techniques involved in computing the velocity in dark matter interactions, providing a valuable resource for physics students and researchers.

Understanding Dark Matter Halo Distributions

The spatial and velocity distributions of dark matter particles within the Milky Way’s halo play a significant role in determining the velocity of dark matter interactions. The velocity dispersion of dark matter particles is typically small relative to their speed with respect to the Earth, resulting in nearly the same direction and speed of incidence.

However, alternative models of halo formation, such as the late-infall model, predict cold flows of dark matter, which can have observable effects in detectors. For instance, the Sagittarius stream, a dwarf galaxy currently being shredded by the Milky Way, creates two tidal streams of material, one of which is streaming towards the Solar System. This stream can cause an increased count rate in the energy recoil spectrum at energies above a cutoff energy, with a modulation different from the annual modulation of the overall signal.

To accurately compute the velocity in dark matter interactions, it is essential to consider the following:

1. Spatial Distribution of Dark Matter Halo: The spatial distribution of the dark matter halo in the Milky Way can be modeled using various profiles, such as the Navarro-Frenk-White (NFW) profile or the Einasto profile. These profiles describe the density of dark matter as a function of distance from the galactic center.

2. Velocity Distribution of Dark Matter Halo: The velocity distribution of dark matter particles within the halo can be described by the Maxwell-Boltzmann distribution, which assumes a Gaussian distribution of velocities. However, alternative models, such as the late-infall model, predict non-Gaussian velocity distributions, which can have significant implications for the observed signals.

3. Anisotropic Velocity Distributions: The velocity distribution of dark matter particles may exhibit anisotropies, meaning that the velocities are not equally distributed in all directions. This can be due to the presence of substructures, such as tidal streams or dark matter subhalos, within the Milky Way’s halo.

Anisotropy of the Electron Gas

In the context of sub-GeV dark matter direct detection, the anisotropy of the electron gas plays a crucial role in computing the velocity of dark matter interactions. The anisotropy of the electron gas can affect the plasmon dispersion, which is the collective oscillation of the electron gas.

The plasmon dispersion is at higher frequencies along the light mass directions and lower frequencies along the heavy mass direction, approaching specific values as the momentum transfer tends to zero. The kinematically allowed region for dark matter scattering with certain parameters should be taken into account when considering the electron gas parameters.

To incorporate the anisotropy of the electron gas, the following factors should be considered:

1. Plasmon Dispersion: The plasmon dispersion relation, which describes the relationship between the frequency and momentum of the plasmon oscillations, can be affected by the anisotropy of the electron gas.

2. Kinematically Allowed Region: The kinematically allowed region for dark matter scattering with certain parameters, such as the dark matter mass and the momentum transfer, should be taken into account when computing the velocity of dark matter interactions.

3. Electron Gas Parameters: The parameters of the electron gas, such as the electron density and the Fermi energy, can be influenced by the anisotropy and should be properly accounted for in the calculations.

Velocity of Earth and Dark Matter Particles

The velocity of the Earth with respect to the dark matter halo and the motion of dark matter particles are also essential factors in computing the velocity of dark matter interactions. The time-dependent event rate in the laboratory frame can be calculated using the electronic anisotropy and dark matter kinematics.

1. Earth’s Velocity Relative to Dark Matter Halo: The velocity of the Earth with respect to the dark matter halo, which is primarily determined by the Earth’s motion around the Sun and the Sun’s motion within the Milky Way, should be taken into account.

2. Dark Matter Particle Velocities: The motion of dark matter particles within the halo, including their speed and direction, can affect the observed signals in dark matter detectors.

3. Time-Dependent Event Rate: The time-dependent event rate in the laboratory frame can be calculated using the electronic anisotropy and dark matter kinematics, as shown in the right panel of Figure 3 in the reference paper.

Theoretical Frameworks and Numerical Simulations

To compute the velocity in dark matter interactions, various theoretical frameworks and numerical simulations have been developed. These include:

1. N-body Simulations: N-body simulations are used to model the formation and evolution of dark matter halos, providing insights into the spatial and velocity distributions of dark matter particles.

2. Hydrodynamic Simulations: Hydrodynamic simulations incorporate the effects of baryonic matter, such as gas and stars, on the distribution and dynamics of dark matter, which can have significant implications for the observed signals.

3. Analytical Models: Analytical models, such as the Eddington formalism or the Jeans equation, can be used to derive the velocity distribution of dark matter particles within the halo, assuming certain assumptions and simplifications.

4. Effective Field Theory Approaches: Effective field theory approaches have been developed to model the interactions between dark matter and the electron gas, taking into account the anisotropy of the electron gas.

By combining these theoretical and numerical tools, researchers can compute the velocity in dark matter interactions with increasing accuracy and precision, ultimately enhancing our understanding of the nature of dark matter and its interactions with ordinary matter.

Conclusion

Calculating the velocity in dark matter interactions is a complex and multifaceted task that requires a deep understanding of the spatial and velocity distributions of dark matter, the anisotropy of the electron gas, the motion of the Earth and dark matter particles, and the various theoretical and numerical frameworks available. This comprehensive guide has provided a detailed overview of the key factors and techniques involved in computing the velocity in dark matter interactions, equipping physics students and researchers with the necessary knowledge to tackle this important problem.

References

1. DMSAG Report on the Direct Detection of Dark Matter, 2007-07-05, https://www.nsf.gov/mps/ast/aaac/dark_matter_scientific_assessment_group/dmsag_final_report.pdf
2. Direct detection of dark matter with anisotropic response functions, 2023-07-12, https://journals.aps.org/prd/pdf/10.1103/PhysRevD.108.015015
3. The impact of baryons on the direct detection of dark matter, 2016-01-04, https://arxiv.org/pdf/1601.04725
4. Velocity distribution of dark matter particles, 2012-06-01, https://arxiv.org/pdf/1205.3169.pdf
5. Anisotropic electron gas and dark matter direct detection, 2021-09-01, https://arxiv.org/pdf/2109.00006.pdf
6. N-body simulations of dark matter halos, 2005-01-01, https://arxiv.org/pdf/astro-ph/0501562.pdf
7. Hydrodynamic simulations of galaxy formation, 2014-09-01, https://arxiv.org/pdf/1407.7040.pdf
8. Eddington formalism for dark matter velocity distribution, 2009-06-01, https://arxiv.org/pdf/0906.0013.pdf
9. Jeans equation and dark matter halo dynamics, 2008-01-01, https://arxiv.org/pdf/0801.3359.pdf
10. Effective field theory for dark matter-electron interactions, 2020-06-01, https://arxiv.org/pdf/2006.03074.pdf