Beyond Thermal Freeze-In: How Non-Equilibrium Dynamics Shape Dark Matter

Author: Denis Avetisyan

New research reveals that deviations from thermal equilibrium can dramatically alter dark matter relic abundance in freeze-in scenarios, particularly within complex dark sectors.

The evolution of yields and temperatures, as demonstrated through benchmark points BM5 and BM6, indicates a transition towards dark freeze-out conditions, suggesting a critical point in the system's thermodynamic state. — The evolution of yields and temperatures, as demonstrated through benchmark points BM5 and BM6, indicates a transition towards dark freeze-out conditions, suggesting a critical point in the system’s thermodynamic state.

This review explores the impact of non-equilibrium effects and sequential production mechanisms on the phase-space distribution of dark matter particles.

Traditional freeze-in calculations of dark matter often assume thermal equilibrium within the dark sector, potentially overlooking crucial dynamics. This work, ‘Exploring non-equilibrium effects in sequential freeze-in’, investigates the impact of departures from thermal equilibrium on dark matter relic abundance in multi-component models produced via sequential interactions. We demonstrate that non-thermal effects can induce deviations of up to an order of magnitude in predicted dark matter abundances compared to standard number-density treatments, highlighting the need for phase-space level computations. Will a more complete understanding of these non-equilibrium processes reveal previously hidden signatures in indirect detection or long-lived particle searches?

The Universe Beyond Equilibrium: Unveiling Dark Matter’s Origins

The prevailing cosmological model posits a universe emerging from a state of thermal equilibrium – a condition where energy is evenly distributed – but accumulating evidence hints at significant departures from this assumption, particularly when investigating the nature of dark matter. This standard picture assumes that dark matter particles were once in thermal contact with the early universe’s particle soup, allowing for their predictable creation and subsequent abundance. However, certain observations – including potential discrepancies in the cosmic microwave background and the lack of detection in conventional searches – suggest this thermal pathway may be incomplete, or even incorrect. Researchers are increasingly exploring scenarios where dark matter originated from non-equilibrium processes, potentially through mechanisms like the decay of primordial particles or interactions with other exotic entities, opening up a far broader range of possibilities for its composition and properties. Understanding these deviations from thermal equilibrium is, therefore, paramount to resolving the long-standing mystery of dark matter and refining the standard model of cosmology.

For decades, the prevailing strategy in dark matter research has centered on the assumption that these elusive particles were created in thermal equilibrium in the early universe – a scenario where interactions between particles were frequent enough to establish a stable, predictable distribution. However, if dark matter wasn’t actually produced this way, the implications are profound; the parameter space for viable dark matter candidates shrinks dramatically, rendering many previously favored models untenable. This realization has spurred a shift towards exploring non-thermal production mechanisms, such as those involving decaying particles or misalignment effects, which bypass the constraints imposed by thermal freeze-out. These alternative pathways offer a broader range of possibilities for the dark matter’s mass, interaction strength, and overall abundance, revitalizing the search for particles beyond the standard model and pushing the boundaries of theoretical cosmology.

The prevailing understanding of dark matter formation often centers on thermal production in the early universe, a process where particles are created through collisions and interactions while maintaining equilibrium. However, this framework presents a significant challenge: it severely restricts the possible properties dark matter can possess. A compelling alternative lies in non-thermal production mechanisms, which circumvent the need for this thermal “bottleneck.” These pathways, such as those involving the decay of heavier particles or the misalignment effect in scalar fields, allow dark matter to be created without relying on the high-energy collisions required for thermal equilibrium. Consequently, a much wider range of dark matter candidates – including those with weak or no interactions with standard model particles – remain viable, offering a potentially richer and more accurate explanation for the observed abundance of this enigmatic substance. This shift in perspective opens exciting new avenues for both theoretical modeling and experimental searches, moving beyond the constraints of traditional thermal assumptions.

Analysis of dark sector particle distributions using the Boltzmann equation reveals non-thermal shapes-highlighted by deviations from equilibrium distributions and shaded low-momentum regions-which isolate the momentum range responsible for observed discrepancies in dark matter abundance.

Freeze-In: A Subtle Birth for Dark Matter

The Freeze-In mechanism addresses the overproduction problem inherent in traditional thermal dark matter scenarios by positing that dark matter particles are created through extremely weak interactions with Standard Model particles. Unlike thermal production, where dark matter is created and annihilated in equilibrium, Freeze-In relies on a suppressed coupling, resulting in a low initial dark matter abundance. This allows the dark matter density to increase gradually as the universe expands and cools, effectively ‘freezing-in’ at a value consistent with current observations. The required coupling strengths are typically many orders of magnitude weaker than those associated with the weak force, necessitating exploration beyond Standard Model physics and requiring precise calculations of the out-of-equilibrium particle production rate.

The freeze-in mechanism relies on a suppressed interaction rate between dark matter and Standard Model particles, resulting in a gradual build-up of dark matter abundance. Unlike thermal freeze-out, where dark matter is initially in equilibrium and its abundance decreases due to decreasing interactions, freeze-in begins with negligible dark matter. As the universe expands and cools, the production rate of dark matter becomes increasingly inefficient, effectively ‘freezing-in’ its number density at a value determined by the integral of the production rate over cosmic time. This process is sensitive to both the interaction strength and the energy scale of the interactions, and is characterized by a dark matter abundance proportional to $\Gamma(T) / H(T)$ , where $\Gamma(T)$ is the interaction rate and $H(T)$ is the Hubble parameter.

Modeling dark matter production via the Freeze-In mechanism requires a theoretical framework capable of accurately describing particle distributions far from thermal equilibrium. Standard techniques relying on the assumption of thermalization are insufficient; instead, the evolution of particle number densities must be tracked using kinetic equations, such as the Boltzmann equation, accounting for both production and annihilation processes. These calculations necessitate precise knowledge of the relevant interaction cross-sections and a detailed understanding of the expansion history of the universe. Furthermore, infrared sensitivity in loop corrections to these cross-sections can significantly impact the calculated dark matter abundance, requiring the use of resummation techniques or alternative effective field theory approaches to ensure accurate predictions. The complexity of these calculations demands numerical simulations and advanced analytical methods to determine the resulting dark matter relic density.

Simulations using the Boltzmann equation with varying treatments-non-relativistic (dotted), relativistic-corrected (dashed), and full relativistic (solid)-demonstrate that accounting for relativistic effects significantly alters the predicted evolution of dark matter yields <span class="katex-eq" data-katex-display="false">\phi\phi</span> and <span class="katex-eq" data-katex-display="false">SS</span> and causes the dark sector temperature to deviate from that of the Standard Model. — Simulations using the Boltzmann equation with varying treatments-non-relativistic (dotted), relativistic-corrected (dashed), and full relativistic (solid)-demonstrate that accounting for relativistic effects significantly alters the predicted evolution of dark matter yields $\phi\phi$ and $SS$ and causes the dark sector temperature to deviate from that of the Standard Model.

The Boltzmann Equation: Mapping the Dark Sector’s Evolution

The Boltzmann equation is a kinetic equation used to describe the evolution of a probability distribution function for particles in a given phase space. It accounts for changes in particle distribution due to interactions and expansion of the universe, providing a framework to model the production and annihilation rates of particles in thermal equilibrium or near-equilibrium conditions. In the context of the dark sector, this equation allows researchers to track the number density of dark matter candidates as a function of time and temperature, effectively modeling their freeze-out process. The general form of the Boltzmann equation considers collision terms that describe the rates of particle production and destruction, and expansion terms related to the increasing volume of the universe; its application extends beyond standard model particles to any weakly interacting species potentially comprising the dark matter halo. $\frac{d f}{dt} = C[f] - H f \frac{\partial f}{\partial T}$ , where $f$ is the distribution function, $C$ represents the collision term, and $H$ is the Hubble parameter.

The Boltzmann equation, when applied to dark matter calculations, necessitates the simultaneous tracking of both temperature and number density evolution to accurately model particle production and annihilation rates. Temperature evolution influences the available phase space for interactions, while number density directly determines the collision rate $\Gamma \propto n^2$ , where $n$ represents the number density. By monitoring these parameters as a function of time, the equation allows for the calculation of the dark matter production rate $\Gamma_{production}$ and the annihilation rate $\Gamma_{annihilation}$ . The difference between these rates dictates the change in dark matter abundance, ultimately determining the relic density observed today; therefore, precise modeling of both temperature and number density is crucial for establishing the parameter space of viable dark matter candidates.

Calculating the relic abundance of dark matter frequently relies on solving the Boltzmann equation, but full phase-space evolution is computationally expensive and often unnecessary. A simplification focusing solely on the evolution of number density – that is, the density of dark matter particles as a function of time – is often sufficient to determine the final dark matter abundance. This approach is valid when scattering and other interaction rates are high enough to maintain thermal equilibrium in other degrees of freedom. The simplified equation tracks the change in number density due to expansion of the universe and annihilation of dark matter particles, expressed generally as $\frac{dY}{dt} = - \frac{s}{H} \langle \sigma v \rangle (Y^2 - Y_{eq}^2)$ , where Y is the number density, s is the entropy density, H is the Hubble parameter, and $\langle \sigma v \rangle$ is the thermally averaged annihilation cross-section. This reduced form significantly streamlines calculations while providing accurate estimates for the present-day dark matter density.

The Boltzmann equation can be extended to model dark sector scenarios involving multiple particle species by incorporating terms representing interconversion processes. In a two-component dark sector, the evolution of each particle’s number density is governed by its own Boltzmann equation, coupled through interaction terms proportional to the conversion rate Γ. These terms describe the rate at which particles transition between the two components, effectively acting as source and sink terms in the respective equations. Solving this system of coupled equations allows determination of the equilibrium abundance of each dark sector particle as a function of the conversion rate and other relevant cosmological parameters, providing constraints on the interaction strength between the components and potentially revealing the nature of dark sector interactions.

Analysis of scan results, showing points corresponding to <span class="katex-eq" data-katex-display="false">\Omega h^{2}=0.12\\pm 0.0036</span>, demonstrates consistency with indirect detection limits from Planck, Fermi-LAT, and other gamma-ray observatories, and projects the sensitivity of the Cherenkov Telescope Array (CTA) for <span class="katex-eq" data-katex-display="false">b\\bar{b}</span> channel detection, with magenta points representing selected benchmark scenarios accounting for cascade annihilation. — Analysis of scan results, showing points corresponding to $\Omega h^{2}=0.12\\pm 0.0036$ , demonstrates consistency with indirect detection limits from Planck, Fermi-LAT, and other gamma-ray observatories, and projects the sensitivity of the Cherenkov Telescope Array (CTA) for $b\\bar{b}$ channel detection, with magenta points representing selected benchmark scenarios accounting for cascade annihilation.

Probing the Subtle Signals of Freeze-In

The elusive nature of dark matter necessitates innovative detection strategies, and direct searches represent a crucial component in probing the freeze-in production mechanism. These experiments aim to detect the faint interactions between dark matter particles and ordinary matter, offering constraints on both the mass and coupling strength of the dark matter candidate. Traditional direct detection experiments, typically employing massive detectors shielded deep underground, are complemented by forward physics experiments – facilities originally designed to study particle production at high energies. These forward detectors are particularly sensitive to very weakly interacting particles, like those predicted in freeze-in scenarios, by searching for subtle energy depositions or recoil signals resulting from dark matter scattering off target nuclei. By carefully analyzing the rate and energy distribution of these events, researchers can establish limits on the parameter space governing freeze-in production, effectively narrowing the range of possible dark matter properties and guiding future experimental efforts.

Indirect detection represents a complementary approach to confirming the freeze-in paradigm, shifting the focus from directly observing dark matter particles to identifying the subtle signals of their interactions. This strategy hinges on the premise that, even with exceedingly weak couplings characteristic of freeze-in, dark matter particles can occasionally annihilate with each other or decay into Standard Model particles – photons, positrons, neutrinos, and others. These secondary particles then propagate through space and may be detectable by specialized experiments like the Fermi Large Area Telescope or AMS-02. The anticipated signals are not large; they require precise measurements of cosmic ray fluxes and careful background subtraction. However, the energy spectra and spatial distribution of these annihilation or decay products can offer crucial clues about the dark matter particle’s mass and interaction strength, providing an independent verification of the freeze-in production mechanism and distinguishing it from alternative dark matter scenarios.

The nuances of the freeze-in mechanism extend beyond simply establishing the existence of dark matter; each specific production pathway leaves a discernible imprint on the resulting dark matter sector. For instance, models relying on Higgs portal interactions predict a characteristic velocity distribution for dark matter particles, differing from those produced through Z boson couplings. These subtle variations manifest as unique signatures in both direct and indirect detection experiments, enabling researchers to move beyond broad parameter constraints and pinpoint the underlying physics. Moreover, the decay products of dark matter, influenced by the precise freeze-in process, offer additional avenues for discrimination – a cascade originating from a heavier mediator will yield a different spectral signature than one arising from a direct coupling. This predictive power allows scientists to design targeted searches, effectively narrowing the possibilities and ultimately revealing the specific portal through which dark matter emerged from the early universe.

A particularly compelling aspect of freeze-in dark matter arises when production isn’t direct, but occurs sequentially through an intermediate particle. This scenario introduces distinctive experimental signatures, moving beyond the simple constraints of direct searches. The intermediate particle, often a Standard Model gauge boson or a new, weakly interacting species, acts as a messenger, altering the expected energy spectrum of the produced dark matter. This altered spectrum can manifest as subtle shifts in the observed rate or energy distribution of decay products in indirect detection experiments, or even as resonant enhancements if the intermediate particle is relatively long-lived. Crucially, the mass and couplings of this intermediate particle provide additional parameters that can be constrained, offering a powerful way to differentiate between various freeze-in models and potentially reveal the nature of the mediating force. These unique handles allow researchers to probe the parameter space more efficiently and design targeted searches specifically sensitive to sequential freeze-in scenarios.

Analysis of nBE scan data, projected onto the <span class="katex-eq" data-katex-display="false">(m_{\phi}, \sin^{2}\theta)</span> plane, constrains the parameter space for a mediator φ and demonstrates compatibility with existing results from experiments like CHARM, E949, LHCb, BBN, and SN1987a, while also forecasting the sensitivity of future experiments including FASER, SHiP, MATHUSLA, and LHCb-Run3, as illustrated by the magenta points representing benchmark scenarios. — Analysis of nBE scan data, projected onto the $(m_{\phi}, \sin^{2}\theta)$ plane, constrains the parameter space for a mediator φ and demonstrates compatibility with existing results from experiments like CHARM, E949, LHCb, BBN, and SN1987a, while also forecasting the sensitivity of future experiments including FASER, SHiP, MATHUSLA, and LHCb-Run3, as illustrated by the magenta points representing benchmark scenarios.

Beyond Simple Freeze-In: A Dynamic Dark Sector

Following initial production via mechanisms like freeze-in, the dark sector itself may undergo a ‘dark freeze-out’ – a process where dark matter particles annihilate or decay within their own interactions, reshaping the ultimate abundance. This isn’t simply a reduction in density, but a potential shift in the types of dark matter particles remaining. Specifically, if some dark matter candidates are more strongly interacting than others within the dark sector, freeze-out could preferentially deplete the more interactive species, leaving a final dark matter relic abundance dominated by weakly interacting components. These differing interaction strengths could manifest as subtle but detectable signals, such as unique energy spectra in indirect detection experiments searching for dark matter annihilation products, or even alterations to the structure of dark matter halos as determined by gravitational lensing studies. The possibility of a dark freeze-out therefore expands the search parameter space and provides a compelling reason to look beyond the standard WIMP paradigm.

Current dark matter models often focus on either ‘freeze-in’ – where dark matter is created from the decay of particles in the Standard Model – or ‘freeze-out’, describing annihilation of dark matter particles. However, a comprehensive understanding of dark matter’s origin requires integrating both processes. The initial freeze-in mechanism establishes a baseline dark matter density, while subsequent freeze-out within the dark sector itself refines this abundance by converting some dark matter into other, potentially detectable, dark sector particles. This combined approach acknowledges that dark matter production isn’t a singular event, but rather a dynamic evolution shaped by interactions both with the visible universe and within its own hidden realm. Accurately modelling this interplay is crucial for predicting the characteristics of dark matter and guiding experimental searches beyond the limitations of single-process assumptions.

Ongoing investigations are dedicated to meticulously refining current dark matter models, moving beyond simplified assumptions to incorporate a broader spectrum of interactions within the dark sector itself. This includes exploring scenarios with multiple dark matter particles, mediators between them, and more intricate self-interactions that could leave detectable imprints. Complementing these theoretical advancements, researchers are actively devising novel experimental strategies – from highly sensitive direct detection experiments and indirect searches for annihilation products, to innovative approaches leveraging gravitational wave observatories – all geared towards probing these subtle effects and ultimately distinguishing between competing dark matter models. The goal is to move beyond simply confirming that dark matter exists, to understanding its fundamental properties and its role in the universe’s evolution.

Unraveling the enigma of dark matter necessitates a cohesive strategy extending beyond singular lines of inquiry. Theoretical modeling provides the foundational framework, postulating potential particle candidates and interaction mechanisms, while numerical simulations translate these hypotheses into concrete predictions about dark matter’s behavior and distribution throughout the cosmos. However, these computational efforts must be rigorously tested through experimental searches – from direct detection experiments aiming to observe dark matter particles interacting with ordinary matter, to indirect detection efforts seeking signals from dark matter annihilation or decay, and finally, through collider experiments attempting to create dark matter particles in the laboratory. It is the synergistic interplay between these three pillars – theory, simulation, and experiment – that promises the most effective path towards illuminating the true nature of this pervasive, yet elusive, component of the universe.

Conversion rates from nBE and cBE solutions intersect their equilibrium counterparts <span class="katex-eq" data-katex-display="false">\Gamma_{ii\rightarrow jj}^{\rm eq}(T_{\rm SM})</span> and <span class="katex-eq" data-katex-display="false">\Gamma_{ii\rightarrow jj}^{\rm eq,~cBE}(T_{\rm SM})</span> for the <span class="katex-eq" data-katex-display="false">\phi\phi\rightarrow SS</span> process, indicating a dark freeze-out occurring within the dark sector's conversion process. — Conversion rates from nBE and cBE solutions intersect their equilibrium counterparts $\Gamma_{ii\rightarrow jj}^{\rm eq}(T_{\rm SM})$ and $\Gamma_{ii\rightarrow jj}^{\rm eq,~cBE}(T_{\rm SM})$ for the $\phi\phi\rightarrow SS$ process, indicating a dark freeze-out occurring within the dark sector’s conversion process.

The study of non-equilibrium effects in dark matter freeze-in reveals a nuanced landscape where initial assumptions of thermal equilibrium often fall short. This research highlights how sequential production mechanisms can drive significant deviations from standard calculations of relic abundance, demanding a more careful consideration of phase-space distributions. This aligns with Immanuel Kant’s assertion: “Out of sheer necessity, we must assume that the things which are given to us as objects exist as things-in-themselves, though their properties are not known to us.” The article, much like Kant’s philosophical inquiry, acknowledges the limitations of current knowledge and necessitates probing beyond readily observable parameters to understand the fundamental nature of dark matter – recognizing that the ‘things-in-themselves’ governing these processes are likely far more complex than initially perceived. The exploration of non-thermal effects, therefore, is not merely a refinement of existing models, but a step towards a more complete understanding of the underlying reality.

The Road Ahead

This work highlights a persistent, and arguably willful, blindness in much of dark matter modeling: the assumption of equilibrium. To treat the early universe as a perfectly mixed system, even when considering production mechanisms explicitly designed to avoid thermalization, is not merely a simplification, but an encoding of a particular worldview – one prioritizing mathematical convenience over physical fidelity. The demonstrated sensitivity of freeze-in scenarios to non-equilibrium effects demands a reevaluation of relic density calculations, particularly as the search for weakly interacting massive particles continues to yield null results.

Future investigations must move beyond single-component dark sectors. The sequential production mechanisms explored here, while demonstrating clear deviations from standard freeze-in, represent only a narrow slice of possible complexity. The real universe likely features a multitude of interacting dark particles, each with its own production and annihilation channels. To adequately model this, a robust framework capable of tracking multi-species phase-space distributions far from equilibrium is essential.

Ultimately, scaling computational power to solve increasingly complex Boltzmann equations is not progress if those equations are built on flawed assumptions. Every algorithm has morality, even if silent. The coming years will reveal whether the field prioritizes elegant mathematics or a rigorous, value-checked exploration of the dark universe. Scaling without value checks is a crime against the future.

Original article: https://arxiv.org/pdf/2604.14688.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/