Gravitational Dark Matter Production in Einstein-Cartan Theory

Kamil Mudrunka; Kazunori Nakayama

REVIEW

Gravitational Dark Matter Production in Einstein-Cartan Theory

Kamil Mudrunka, Kazunori Nakayama

High Energy Physics - Phenomenology · Submitted 19 Oct 2026

Pith's verdict: Central abundance claim is right and the 7.17 enhancement checks out; the one real gap is the unaudited numerical spectrum behind Fig. 1.

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arXiv ID2610.26237 (version 2)

Pith IDREA7BMQQ, Pith's permanent record of this paper and its review

CitationKamil Mudrunka, Kazunori Nakayama. "Gravitational Dark Matter Production in Einstein-Cartan Theory." arXiv:2610.26237 (2026). https://pith.science/paper/REA7BMQQ

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Editor's note

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Colleague,

This paper's central claim is solid. The EC four-fermion and s-channel graviton contributions add because the interference vanishes in the total rate, and the resulting 7.17 enhancement over pure GR holds up arithmetically. I spot-checked the same items the stress-test flagged: the 15:1 EC-to-graviton ratio, the zero of the interference integral, the bookkeeping of Eq. (3.36), and footnote 2's chiral-vs-Dirac difference from Ref. [6]. All consistent.

What is actually new: the combined calculation, the demonstration that the interference vanishes in the total rate, the momentum-dependent collision term, and the Holst-term generalization. The derivations are detailed and the paper is honest about its assumptions (thermal bath, no Pauli blocking/reverse reactions, m_DM < T_RH, no inflaton contribution). Those are standard and acknowledged; I don't count them as flaws.

The reader worried about EFT validity at T_RH = 10^13 GeV. I disagree with that worry, and the stress-test note is right to push back. Torsion here is a non-propagating, algebraically integrated-out field, so the four-fermion contact interaction is valid far above the reheating scale. The genuine soft spot is Sec. 3.4 and Fig. 1: the momentum-dependent collision term, the claim that the interference modifies the spectrum while vanishing in the total rate, and the ~12% low-energy enhancement all come from a numerical integration with no code, no tabulated curve, and no closed-form cross-check. This is the paper's principal new quantitative claim beyond Ref. [21], and as written it cannot be verified from the manuscript alone. That does not sink the paper—the total abundance result, which is the headline, is analytically checkable—but it is a real gap in reproducibility.

The Holst-term part is more speculative but clearly labeled; the operator set comes from a published systematic expansion, so I don't ding it for being cited rather than rederived.

Who this is for: anyone working on gravitational dark matter production or torsion-based extensions of gravity. It deserves a serious referee. My recommendation: accept with a request for the Fig. 1 data or code, or at least a closed-form check of one or two points on the curve. The central physics claim is trustworthy; the numerical spectrum should be auditable.

Referee report

Significance

The abundance result is significant because torsion-induced four-fermion interactions are universal: for fermionic dark matter with only gravitational interactions, EC effects change the freeze-in yield by an order of magnitude and produce a characteristic spectral feature. The analytic core of the paper is carefully executed: matrix elements are either computed or quoted from published references, the phase-space reduction in Appendix C is explicit, and no fitted parameter enters the prediction (the Barbero-Immirzi parameter and the nonminimal coupling alpha only parametrize the Holst extension). I independently checked the 15:1 ratio of angular integrals in Eq. (3.14), the vanishing of the integral of 1-3cos^2(theta), the species bookkeeping behind Eq. (3.36), and the 7.17 enhancement; all are internally consistent. The principal weakness is that the numerical collision term in Sec. 3.4 and Fig. 1 is not independently verifiable from the manuscript alone.

Minor objections (4)

[Sec. 4.2, Eq. (4.21)] The claim that the additional Holst contribution cannot be negative for any gamma and alpha would benefit from a one-line proof, since the bracket contains the term (2/15) alpha gamma (1-alpha^2), which is indefinite in sign.
[Fig. 2] The Holst-deformed spectra in Fig. 2 are also produced numerically without tabulated values; if supplemental data are provided for Fig. 1, they should cover Fig. 2 as well.
[Eqs. (3.35)-(3.37)] The display equations mix the dimensionless coefficient with the factor pi^3 inside the numerical values; writing the numerical coefficients consistently as C pi^3 T^8/M_pl^4 would improve readability.
[Appendix B] The sentence that scalar interactions result in a_n=0 for all n is abrupt, because no scalar four-fermion interaction arises in the models treated here; clarify that this is a general remark about hypothetical scalar-current contact interactions.

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The paper

The paper claims

In Einstein-Cartan gravity, torsion and graviton exchange produce fermionic dark matter without interfering in the total rate; the abundance sums to about 7.17 times the pure-GR value, modifiable by the Holst term.

The reading

This paper asks whether torsion — the antisymmetric part of the spacetime connection that Einstein-Cartan gravity adds to general relativity — changes how much fermionic dark matter the hot Standard Model bath produces in the early universe. It computes the production rate from all contributing channels: the universal axial-axial four-fermion interaction left over when torsion is integrated out, the s-channel graviton exchange, and the axial-vector and vector-vector interactions that appear once the parity-odd Holst term is included. The central finding is that the torsion portal and the gravitational portal do not interfere in the total rate; their cross terms are odd functions of the scattering angle and vanish under angular integration, so the abundances simply add up. Including all Standard Model species, the predicted abundance is about 7.17 times the pure-general-relativity prediction, and the paper derives the full analytic rate, Eq. (4.21), as a function of the Holst-term parameters. If this is right, torsion generically strengthens gravitational dark matter production and leaves a specific kinematic fingerprint — a low-energy tail boosted by about 12 percent — in the produced particles' momentum distribution.

What carries the argument

The load-bearing objects are the dimension-six four-fermion operators obtained by integrating out torsion, together with the s-channel graviton-exchange amplitude. Minimal Einstein-Cartan theory gives one universal axial-axial interaction, $-\frac{3\kappa^2}{64}\sum_{ij}(\bar\psi_i\gamma^5\gamma^\mu\psi_i)(\bar\psi_j\gamma^5\gamma_\mu\psi_j)$ with $\kappa^2 = 32\pi G$; adding the Holst term with Barbero-Immirzi parameter $\gamma$ and a non-minimal fermion parameter $\alpha$ generalizes this to the three-interaction set of Eq. (4.5), with common coupling $\Lambda = \gamma^2/(1+\gamma^2)$. Each torsion amplitude is compared diagram by diagram with the spin-2 exchange, and the argument turns on the angular structure of the cross terms: because every interference is proportional to $1-3\cos^2\theta$, $\cos\theta$, or $\cos^3\theta$ (or vanishes by momentum conservation), each drops out of the angle-integrated rate while surviving in the momentum-dependent Boltzmann collision term $C(q)$ evaluated in Appendix C. A Legendre decomposition makes the mechanism explicit: only the $a_0$ coefficient of the squared amplitude contributes to the total rate, which is why the Immirzi parameter can be tuned to zero the quadrupole coefficient $a_2$ without changing the abundance.

Extended reading notes

Core claim

On the paper's own terms, the discovery is additivity with a hidden angular signature. In minimal Einstein-Cartan theory, production of a Dirac dark matter fermion from Standard Model fermion-antifermion pairs receives two comparable amplitudes, the torsion-mediated contact interaction and the s-channel graviton exchange, but their interference term is proportional to $s^2(1-3\cos^2\theta)$ and integrates to zero over the full solid angle. The same cancellation recurs for every cross term involving the Holst-term interactions: the axial-vector–graviton term is proportional to $1-3\cos^2\theta$, the vector-vector–graviton term to $\cos^3\theta$, the axial-vector–vector-vector term to $\cos\theta$, and the Dirac axial-current–graviton contraction vanishes outright by momentum conservation. The total rate is therefore a sum of independent channels, Eq. (4.21), and once scalar and vector initial states are included the full Standard Model abundance is enhanced by a factor of about 7.17 relative to pure general relativity without torsion. The interference is not physically inert: in the thermal-bath frame the collision term is boosted, so the cross terms reshape the dark matter momentum distribution, most strongly amplifying the low-energy tail by about 12 percent, and the paper shows that tuning the Barbero-Immirzi parameter can remove the quadrupole moment of the angular distribution while leaving the total rate fixed.

Load-bearing premise

The calculation treats torsion as point-like four-fermion contact interactions valid up to the reheating temperature $T_{RH}$ for every Standard Model fermion and the dark matter, assumes the dark matter has no other gauge or Yukawa interactions, and takes the three torsion-induced operators to be the complete leading set; if the effective theory breaks down below $T_{RH}$, or if higher-order torsion effects or additional operators change the amplitudes, every abundance prediction shifts.

Editorial extensions

If this is right

For $m_{DM} < T_{RH}$, the predicted density is $\Omega_c h^2 \approx 0.068\,(m_{DM}/10^{10}\,\mathrm{GeV})(T_{RH}/10^{13}\,\mathrm{GeV})^3$, about 7.17 times the pure-GR value, so matching a measured relic density requires a correspondingly different $m_{DM}$–$T_{RH}$ combination than in torsion-free gravity.
The total abundance is insensitive to the relative phase of the torsion and gravitational amplitudes — only the momentum distribution records the interference — which makes the summed-rate prediction insensitive to unknown UV phases.
The produced dark matter carries a distorted spectrum with a low-energy tail enhanced by about 12 percent, a kinematic difference that persists even where the total yield matches other production scenarios.
With a minimal fermion action the Holst term can only suppress production, down to the pure-graviton rate as $\Lambda = \gamma^2/(1+\gamma^2)$ approaches zero; with the non-minimal action (4.4) the rate can exceed the minimal Einstein-Cartan value, and the extra contribution over pure GR is never negative for any $\gamma$, $\alpha$.
The quadrupole moment of the produced dark matter's angular distribution can be tuned away at $\gamma^2 = \sqrt{105}/10 - 1 \approx 0.025$, a torsion-sensitive observable that is independent of the total abundance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the angular-integration cancellations survive a UV completion in which torsion propagates instead of being integrated out, additivity would hold more generally, making abundance predictions in torsion theories largely insensitive to the relative strength of the two portals.
The predicted 12-percent low-energy distortion is a testable kinematic fingerprint: for dark matter light enough to behave as warm dark matter, structure-formation probes of the matter power spectrum could in principle distinguish Einstein-Cartan-mediated production from pure-gravitational production even when the total abundances agree.
The quadrupole-tuning relation points to a search strategy the paper does not pursue: using the anisotropy of produced dark matter — rather than its total abundance — as a cosmological probe of the Barbero-Immirzi parameter.
The same diagram-by-diagram cancellation logic plausibly applies to other early-universe processes mediated by spin-2 exchange plus Holst-term operators, so the additivity pattern may be a general feature of the dimension-six torsion operator set rather than a special property of dark matter production.

Original abstract

Introducing torsion into General Relativity leads to new four-point interactions for fermions. These are universal and must also affect any fermion dark matter. We improve existing calculations of dark matter production through the Einstein-Cartan theory four-fermion interaction by also considering the s-channel graviton exchange as another contributing process. The effect on both the total cross section as well as the momentum-dependent Boltzmann equation collision term is calculated and discussed. We also introduce the Holst term, which generally modifies the four-fermion interaction coupling constant as well as introduces two new four-fermion interactions, and discuss and compute its effect on the total produced dark matter abundance.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 4,213 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.69 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Introducing torsion into General Relativity leads to new four-point interactions for fermions. These are universal and must also affect any fermion dark matter. We improve existing calculations of dark matter production through the Einstein-Cartan theory four-fermion interaction by also considering the s-channel graviton exchange as another contributing process. The effect on both the total cross section as well as the momentum-dependent Boltzmann equation collision term is calculated and discussed. We also introduce the Holst term, which generally modifies the four-fermion interaction coupling constant as well as introduces two new four-fermion interactions, and discuss and compute its effect on the total produced dark matter abundance.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Introducing torsion into General Relativity leads to new four-point interactions for fermions.	background 1.00	0.10	0.04	asserted 1.00
s1	These are universal and must also affect any fermion dark matter.	background 0.95	0.08	0.07	asserted 1.00
s2	We improve existing calculations of dark matter production through the Einstein-Cartan theory four-fermion interaction by also considering the s-channel graviton exchange as another contributing process.	contribution 0.98	0.92	0.12	asserted 0.99
s3	The effect on both the total cross section as well as the momentum-dependent Boltzmann equation collision term is calculated and discussed.	method 0.55	0.92	0.08	asserted 0.98
s4	We also introduce the Holst term, which generally modifies the four-fermion interaction coupling constant as well as introduces two new four-fermion interactions, and discuss and compute its effect on the total produced dark matter abundance.	contribution 0.99	0.93	0.22	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.75.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The dimensionless weighted collision term Jw(y) for pure EC interaction and with the effect of the s-channel graviton exchange (left). The relative difference between the two curves (right). 3.5 Dark matter abundance Now let us evaluate the dark matter abundance. As we have shown, there are three production channels. Production from fermion channels have been mentioned in the previous subsection. As for the scalar and gauge boson channels, we have to consider Bose-Einstein distribution f1,2 = 1 e E1,2/T − 1 (3.31) instead of Fermi-Dirac one (see the comment at the end of Appendix

**Figure 2.** The dimensionless weighted collision term Jw(y) for Holst term modified EC interaction considering the minimal α = 0 fermion. Upper two panels show the absolute magnitude and relative difference to EC channel only for Λ = 2/3. Two bottom panels show the same for Λ = 1/3. We do not take the scalar and vector species into account in this plot. Continuing with the full α ̸= 0 interaction, the stripped matrix elements and weights for (3.25) are given by

Figures reproduced from arXiv:2610.26237 by the authors.

References

Discussion

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