⚛️ phenomenology

Gravitational Dark Matter Production in Einstein-Cartan Theory

This paper investigates gravitational dark matter production within Einstein-Cartan theory by incorporating s-channel graviton exchange and the Holst term to refine calculations of the total cross section, collision term, and resulting dark matter abundance arising from torsion-induced four-fermion interactions.

Original authors: Kamil Mudrunka, Kazunori Nakayama

Published 2026-10-20

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Kamil Mudrunka, Kazunori Nakayama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, the most persistent mystery in cosmology has been the nature of dark matter. We know it exists because its gravity holds galaxies together, yet it refuses to reveal itself through light or any other familiar force. While many experiments hunt for dark matter by looking for it to bump into ordinary atoms, a quieter possibility remains: that dark matter interacts with our universe only through the force of gravity. Gravity is the weakest of nature's forces, so producing dark matter this way seemed unlikely to yield significant amounts. However, the rules of gravity themselves might be more complex than the standard model suggests.

Einstein's theory of general relativity describes gravity as the curvature of space and time caused by mass and energy. But there is a natural extension of this theory, known as Einstein-Cartan theory, which allows space-time to have a property called "torsion." Imagine space-time not just as a flexible sheet that bends, but as a fabric that can also twist. In this framework, that twist is directly linked to the intrinsic spin of particles, much like how mass links to curvature. While this twisting effect is negligible in our everyday world, it becomes powerful at the extreme energies of the early universe. When scientists integrate out this twist from the equations, it leaves behind a new, direct contact force between particles. This force acts like a handshake between fermions, a type of fundamental particle that includes electrons and the proposed dark matter candidates.

In a recent study, researchers revisited how dark matter might have been created in the first moments after the Big Bang, specifically within this Einstein-Cartan framework. They focused on a scenario where dark matter particles are fermions. Previous calculations had estimated how much dark matter would be produced if it arose solely from this new contact force between particles. The researchers in this study asked a crucial question: what happens if we also include the standard, purely gravitational process where two particles collide and exchange a graviton, the hypothetical particle that carries the force of gravity? They wanted to see if these two different pathways—the new contact force and the standard gravity exchange—would work together to create more dark matter, or if they would cancel each other out.

The team performed a detailed calculation of the collision rates, considering not just fermions but also the scalar and vector particles that make up the hot soup of the early universe. They discovered that the two processes do not interfere with each other in a way that changes the total amount of dark matter produced. Instead, the contributions simply add up. The new contact force remains the dominant driver, producing far more dark matter than gravity alone would. However, the inclusion of the standard gravitational exchange does make a small but measurable difference, increasing the total production rate by about seven percent compared to the contact force alone. More importantly, while the total number of particles remains the sum of the two effects, the way those particles are distributed by energy changes. The standard gravitational process tends to boost the production of lower-energy dark matter particles, altering the shape of the energy spectrum.

The researchers also explored a further refinement of the theory involving a specific mathematical term known as the Holst term. This term introduces a parameter that can change the strength of the new contact force and even create two additional types of interactions between particles. They found that depending on the value of this parameter, the total amount of dark matter produced could be either reduced or significantly increased. In the simplest case, the Holst term acts to weaken the production, but with a specific non-minimal setup, it could enhance the yield. The study concludes that if dark matter is indeed a fermion and if Einstein-Cartan theory describes the early universe correctly, the total abundance of dark matter today would be roughly 7.17 times greater than what would be predicted by pure General Relativity (without torsion) when both theories include the standard gravitational production mechanisms. This suggests that the subtle geometry of space-time, including its potential to twist, plays a vital role in populating the universe with the invisible mass that shapes our cosmos.

Technical Summary: Gravitational Dark Matter Production in Einstein-Cartan Theory

Problem Statement
The nature of dark matter remains unresolved, with the possibility that it interacts solely through gravity. While gravitational particle production during reheating or via thermal processes has been studied within General Relativity (GR), the nature of gravity itself is not fully constrained. GR assumes a torsion-free Levi-Civita connection. However, a natural generalization, Einstein-Cartan (EC) theory, treats the antisymmetric part of the connection (torsion) as a dynamical degree of freedom linked to spin. In EC theory, torsion acts as an algebraic constraint that, when integrated out, generates universal four-fermion contact interactions. While negligible at low energies, these interactions dominate over standard gravitational coupling at high energies. Previous estimates of fermionic dark matter production in the early universe via these EC interactions (e.g., Ref. [21]) did not fully account for the competing and potentially interfering s-channel graviton exchange processes, nor did they generalize to include the Holst term, which modifies the torsion-fermion coupling structure.

Methodology
The authors revisit the calculation of dark matter production in the early universe by simultaneously considering two mechanisms:

EC Four-Fermion Interaction: The contact interaction arising from integrating out torsion.
s-Channel Graviton Exchange: The standard gravitational production process mediated by a graviton.

The study assumes a thermalized bath of Standard Model (SM) particles (fermions, scalars, and gauge bosons) producing Dirac fermion dark matter. The authors calculate the squared matrix elements for the process $\bar{\psi}_{SM} + \psi_{SM} \to \bar{\psi}_{DM} + \psi_{DM}$ (and analogous scalar/vector channels) in the massless limit for SM particles, while retaining the dark matter mass where necessary for phase space integration.

Key methodological steps include:

Amplitude Calculation: Deriving matrix elements for pure EC, pure graviton exchange, and their interference terms.
Angular Integration: Analyzing the interference terms over the full solid angle to determine their contribution to the total cross-section versus their effect on the differential phase space distribution.
Boltzmann Equation: Computing the collision term $C(q)$ for the Boltzmann equation to determine the momentum-dependent production rate, accounting for the boost of the center-of-mass frame relative to the thermal bath.
Holst Term Extension: Introducing the Holst term (parity-odd curvature term) parameterized by the Barbero-Immirzi parameter $\gamma$ and a non-minimal coupling parameter $\alpha$ . This modifies the EC coupling and introduces new vector-vector ( $VV$ ) and axial-vector ( $AV$ ) four-fermion interactions.

Key Contributions and Results

Interference and Total Cross Section:
The paper demonstrates that for the minimal EC theory, the interference term between the s-channel graviton exchange and the EC four-fermion interaction vanishes upon integration over all scattering angles. Consequently, the total cross-section is simply the sum of the individual contributions.
- The EC interaction dominates the fermion channel, exceeding the pure graviton exchange contribution by a factor of 15 in the integrated squared matrix element.
- Including the s-channel graviton exchange enhances the total fermion-initiated production rate by approximately 6.7% compared to the EC-only result.
- When summing over all SM species (fermions, scalars, and vectors), the total dark matter abundance is increased by a factor of approximately 7.17 compared to the pure GR case (without torsion).
Phase Space Distribution:
While the total rate is additive, the interference term is non-zero for specific scattering angles. The authors find that the interference significantly modifies the momentum distribution of the produced dark matter. Specifically, the low-energy tail of the spectrum is boosted by approximately 12% compared to the EC-only scenario.
Holst Term Effects:
The inclusion of the Holst term introduces two new interaction types ( $AV$ and $VV$ ) alongside the modified axial-axial ( $AA$ ) interaction.
- The interference between these new terms and the s-channel graviton exchange also vanishes upon full angular integration, meaning the total abundance remains a sum of independent contributions.
- The Holst term allows for the tuning of the effective coupling strength via the parameters $\gamma$ and $\alpha$ . In the minimal fermion action case ( $\alpha=0$ ), the Holst term can only suppress the total cross-section relative to pure EC theory.
- However, with a non-minimal fermion action ( $\alpha \neq 0$ ), the production rate can be enhanced beyond the pure EC prediction.
- The authors show that by tuning the Immirzi parameter, it is theoretically possible to cancel the quadrupole anisotropy (the $P_2(\cos\theta)$ term) in the angular distribution of the produced dark matter, although this does not affect the total abundance.
Dark Matter Abundance Estimates:
The paper provides analytical estimates for the dark matter number-to-entropy ratio ( $Y_\infty$ ) and the density parameter ( $\Omega_c h^2$ ).
- With EC interactions, $\Omega_c h^2 \approx 0.068 (m_{DM}/10^{10} \text{ GeV}) (T_{RH}/10^{13} \text{ GeV})^3$ .
- Without EC interactions (pure GR), the value is significantly lower: $\Omega_c h^2 \approx 9.5 \times 10^{-3} (m_{DM}/10^{10} \text{ GeV}) (T_{RH}/10^{13} \text{ GeV})^3$ .

Significance
The paper claims to improve existing calculations of gravitational dark matter production by rigorously including the s-channel graviton exchange alongside the EC four-fermion interaction. The primary significance lies in quantifying the enhancement of dark matter abundance due to torsion-induced interactions, showing that EC theory can increase the predicted abundance by roughly an order of magnitude compared to standard GR. Furthermore, the work clarifies that while interference effects do not alter the total production rate, they are crucial for understanding the momentum distribution of the dark matter. Finally, the study generalizes these results to include the Holst term, demonstrating how the Barbero-Immirzi parameter and non-minimal couplings can modulate the production rate and angular anisotropies, offering a broader theoretical framework for torsion-induced dark matter generation.

More like this