Speeding Up the Search for Dark Matter

Author: Denis Avetisyan

---

A new computational tool dramatically accelerates calculations of how dark matter particles interacted in the early universe, paving the way for more comprehensive cosmological modeling.

BSFfast provides a rapid method for computing bound-state enhanced annihilation rates, improving the accuracy of dark matter relic density estimations and parameter space scans.

Accurate determination of dark matter relic density requires accounting for non-perturbative effects like bound-state formation, yet explicitly calculating these effects is computationally expensive. This work introduces BSFfast: Rapid computation of bound-state effects on annihilation in the early Universe, a lightweight tool providing precomputed, tabulated effective annihilation cross sections for a wide range of phenomenologically relevant models. By exploiting rescaling relations and providing fast interpolation routines, BSFfast significantly accelerates cosmological simulations and parameter scans, enabling studies previously rendered impractical. Will this tool unlock new insights into the nature of dark matter and the early universe’s thermal history?

---

The Illusion of Simplicity: Unveiling Hidden Complexity in Dark Matter Annihilation

Determining the abundance of dark matter in the universe fundamentally depends on precisely calculating how often dark matter particles annihilate with each other. Current calculations frequently begin with the simplifying assumption that these particles do not interact beyond the force causing annihilation – a condition allowing for relatively straightforward estimations of the annihilation cross-section. This cross-section, a measure of the probability of annihilation, directly influences the predicted relic density of dark matter, which scientists compare to observed values. However, this approach can be misleading; even weak, long-range forces, if present, can significantly alter the annihilation process and invalidate the simplifying assumptions inherent in standard calculations, potentially leading to inaccurate predictions about the nature and quantity of dark matter.

The standard calculations of dark matter relic abundance often presume particles interact weakly, allowing for simplified perturbative analyses of their annihilation rates. However, when long-range forces-such as those mediated by light bosons-come into play, this picture fundamentally changes. These forces can bind dark matter particles together, forming gravitationally-bound states akin to hydrogen atoms. The existence of these bound states dramatically alters the overall annihilation rate, as annihilation can now occur through multiple channels, including transitions between these bound states. This effect invalidates the standard perturbative methods, which are designed for non-interacting particles, and necessitates a more sophisticated treatment that accounts for the complex interplay between free and bound dark matter particles. Consequently, ignoring these bound-state effects can lead to substantial inaccuracies in determining the dark matter relic density, potentially misrepresenting the true nature of dark matter interactions.

The standard calculation of dark matter relic abundance faces a considerable hurdle when accounting for long-range forces between dark matter particles. These forces can induce the formation of bound states – gravitationally linked pairs – that dramatically alter the expected annihilation rate. Because standard perturbative methods struggle with these interactions, the resulting calculations often overestimate the annihilation cross-section, leading to predictions of insufficient dark matter. Achieving accurate modeling demands a comprehensive approach that doesn’t just consider the ground state of these bound systems, but also the contributions from a vast series of excited states – requiring calculations to extend up to a remarkably high principal quantum number of 100 to capture the full effect and refine predictions for the abundance of dark matter in the universe.

BSFfast: A Pragmatic Solution for Complex Interactions

BSFfast is a numerical tool developed to calculate the influence of bound-state effects – specifically formation, dissociation, and transitions – on annihilation processes. Unlike prior methods requiring solutions to the Schrödinger equation during runtime, BSFfast employs a pre-computation strategy. This involves calculating and storing effective annihilation cross-sections for a range of relevant parameters, enabling significantly faster calculations without sacrificing accuracy. The tool is designed to address scenarios where bound-state dynamics play a crucial role in determining annihilation rates, offering a computational advantage for complex systems where real-time bound-state solving is impractical.

BSFfast achieves computational efficiency by pre-calculating and storing effective annihilation cross-sections for a range of relevant parameters. This tabulated approach bypasses the need to solve the time-dependent Schrödinger equation during runtime, which is computationally expensive. Instead, BSFfast retrieves pre-computed values, enabling calculations that are orders of magnitude faster than methods requiring on-the-fly Schrödinger equation solutions. This pre-computation is performed across a parameter space encompassing relevant kinematic variables and interaction strengths, allowing for rapid assessment of bound-state effects on annihilation processes without sacrificing accuracy.

Accurate modeling of bound-state formation necessitates the inclusion of a running coupling, as the strength of the interaction between particles is not constant but varies with the energy scale of the interaction. This energy dependence affects the potential governing bound-state formation and dissociation; a constant coupling would lead to inaccurate predictions for bound-state energies and lifetimes. BSFfast incorporates a dynamically adjusted coupling constant, allowing the effective potential to reflect the changing interaction strength as the system evolves, thereby improving the fidelity of the calculated annihilation cross-sections and accurately representing strong interaction effects relevant to bound-state dynamics.

Validation Through Rigor: The Boltzmann Equation and Unitarity Constraints

BSFfast directly incorporates the Boltzmann equation, a first-order differential equation describing the time evolution of a particle distribution function in a dilute gas, to precisely determine the relic density of dark matter candidates. This integration allows for the calculation of the dark matter abundance by solving the Boltzmann equation, accounting for both expansion of the universe and interactions between dark matter particles and the Standard Model. The relic density is determined by freezing out of dark matter particles when the expansion rate of the universe becomes comparable to the interaction rate. BSFfast’s implementation utilizes an efficient numerical solver to obtain accurate results for the relic abundance, parameterized by dark matter particle mass, annihilation cross-section, and other relevant physical quantities. The calculated relic density can then be directly compared to observations, such as the Planck satellite measurements of the cosmic microwave background, to constrain the properties of dark matter candidates.

Calculations performed using BSFfast reveal that the omission of bound-state effects in dark matter relic density calculations can introduce substantial inaccuracies. Specifically, these effects influence the effective degrees of freedom available for annihilation and co-annihilation processes. Ignoring these contributions can lead to errors in the predicted dark matter abundance ranging from several percent to multiple orders of magnitude, depending on the mass splitting between the dark matter particle and its potential bound states. This is particularly relevant for scenarios where the dark matter candidate interacts with itself or other particles in the thermal bath, creating or influencing the formation of bound states during the freeze-out process.

BSFfast is designed to mitigate unitarity violations commonly encountered in perturbative calculations of dark matter relic density. These violations occur when calculated effective cross-sections exceed the unitarity bound, indicating the breakdown of the perturbative expansion. BSFfast incorporates techniques to ensure that calculated cross-sections remain within the perturbative regime; specifically, calculations demonstrate violations are limited to ≤ 10

Beyond SuperWIMPs: A Broadening Impact on Particle Physics

The precision of BSFfast introduces a significant refinement to calculations concerning SuperWIMP dark matter, a class of models positing that dark matter annihilation is boosted by the presence of long-lived force carriers-mediators-that create a substantial attractive force between dark matter particles. This attraction leads to what’s known as Sommerfeld enhancement, dramatically increasing the annihilation rate and, consequently, the predicted signal strength detectable by experiments. However, accurately quantifying this enhancement requires accounting for the formation of bound states-pairs of dark matter particles gravitationally locked together-which traditional calculations often approximate. BSFfast’s ability to rapidly and precisely calculate the effects of these bound states reveals that previous estimations of SuperWIMP signals may have been significantly off, potentially shifting the allowed parameter space for these models and either strengthening or weakening constraints derived from observational data. The tool’s impact lies in providing a more realistic assessment of SuperWIMP viability, demanding a re-evaluation of existing experimental limits and theoretical predictions.

The inclusion of bound-state effects significantly alters the parameter space for SuperWIMP dark matter models. Previous analyses often neglected the formation of bound states between dark matter particles mediated by a force carrier, potentially leading to inaccurate constraints. Recent calculations demonstrate that, depending on the specific model and the strength of the interaction, considering these bound states can either tighten the allowed parameter space by increasing the annihilation rate – as bound states readily decay – or, conversely, relax the constraints. This occurs when bound-state formation suppresses the overall annihilation cross-section, effectively reducing the signal expected at detectors. Therefore, a comprehensive understanding of bound-state dynamics is crucial for accurately interpreting current and future dark matter searches, and for refining the theoretical landscape of SuperWIMP models.

The computational framework embodied in BSFfast extends well beyond the specific investigation of SuperWIMP dark matter. The techniques developed to efficiently calculate bound-state formation and annihilation rates arising from long-range interactions are broadly applicable to various areas of particle physics. Scenarios involving hidden sectors with light mediators, resonant particle production in the early universe, or the formation of exotic bound states beyond traditional quark-gluon plasma all stand to benefit from this methodology. By providing a robust and computationally tractable approach to these complex calculations, BSFfast offers a powerful tool for exploring a wider range of theoretical models and potentially uncovering new physics beyond the Standard Model, effectively serving as a versatile asset for particle physicists investigating phenomena reliant on long-range forces and the formation of composite states.