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arxiv: 2512.16976 · v2 · submitted 2025-12-18 · ✦ hep-ph · astro-ph.CO· gr-qc· hep-th

Creation of spin-3/2 dark matter via cosmological gravitational particle production

Edward W. Kolb , Andrew J. Long , Evan McDonough , Jingyuan Wang This is my paper

Pith reviewed 2026-05-16 20:59 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COgr-qchep-th
keywords cosmological gravitational particle productionspin-3/2 dark matterraritroninflationBogoliubov formalismsound speedrelic density
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The pith

Gravitational particle production during and after inflation can generate the observed dark matter density from stable spin-3/2 particles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps how a free massive spin-3/2 field, minimally coupled to gravity, gets created through cosmological gravitational particle production. Production efficiency splits into two regimes depending on whether the particle mass exceeds or falls below the Hubble rate at the end of inflation. In the lighter regime the longitudinal mode sound speed drops toward zero, strongly boosting the number of high-momentum particles created. With stability assumed, this mechanism populates a broad swath of parameter space that can reproduce the measured dark matter abundance, and the same holds when the mass is allowed to vary with time.

Core claim

In the minimal model of a free massive spin-3/2 field, the Bogoliubov calculation of mode functions shows that cosmological gravitational particle production yields a relic density matching observations whenever the raritron mass is either well above or well below the Hubble parameter at the end of inflation, with the light-mass case receiving a large enhancement from the vanishing sound speed of the helicity-1/2 mode.

What carries the argument

The sound speed of the longitudinal (helicity-1/2) mode of the raritron, whose evolution with the mass-to-Hubble hierarchy controls the momentum spectrum of gravitationally produced particles.

If this is right

  • Stable raritrons produced this way can account for all dark matter across both heavy and light mass regimes relative to the inflationary Hubble scale.
  • The light-mass regime produces a high-momentum tail in the spectrum that is absent in the heavy-mass case.
  • Allowing the raritron mass to vary with time still enhances production relative to the constant heavy-mass case without removing the high-momentum particles.
  • The Bogoliubov results agree with the Boltzmann formalism in regimes where the latter applies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mechanism supplies a purely gravitational origin for spin-3/2 dark matter that requires no additional fields or couplings.
  • The enhanced high-momentum tail for light raritrons could leave distinct imprints on small-scale structure formation that differ from standard cold dark matter predictions.
  • If such particles exist, their gravitational-only interactions would make them difficult to detect by conventional particle-physics experiments but potentially accessible through future gravitational-wave or cosmological observables.

Load-bearing premise

The spin-3/2 particles remain stable and experience no interactions beyond minimal gravitational coupling that would cause them to decay or thermalize.

What would settle it

An observation that the dark matter relic density lies outside the range calculable from the Bogoliubov spectrum for any mass-to-Hubble ratio at the end of inflation, or direct evidence that the particles possess non-minimal couplings.

read the original abstract

We study the cosmological gravitational particle production (CGPP) of spin-3/2 particles during and after cosmic inflation, and map the parameter space that can realize the observed dark matter density in stable spin-3/2 particles. Originally formulated by Rarita and Schwinger, the relativistic theory of a massive spin-3/2 field later found a home in supergravity as the superpartner of the graviton, and in nuclear physics as baryonic resonances and nuclear isotopes. We study a minimal model realization, namely a free massive spin-3/2 field minimally coupled to gravity, and adopt the name raritron for this field. We demonstrate that CGPP of raritrons crucially depends on the hierarchy between the raritron mass $m_{3/2}$ and the Hubble parameter at the end of inflation $H_e$, with high-mass and low-mass cases distinguished by the evolution of the sound speed $c_s$ of the longitudinal (helicity-1/2) mode, which is approximately unity at all times for heavy (relative to Hubble) raritrons and can become small or vanish for lighter raritrons, leading to a dramatic enhancement of production of high momentum particles in the latter case. Assuming the raritrons are stable, this leads to a wide parameter space to produce the observed dark matter density. Finally, we consider a time-dependent raritron mass, which can be chosen to remove the vanishing sound speed of the longitudinal mode, but which nonetheless enhances the production relative to the constant high-mass case, and in particular does not necessarily tame the high momentum tail of the spectrum. We perform our calculations using the Bogoliubov formalism and compare, when applicable, to the Boltzmann formalism.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies cosmological gravitational particle production of a free massive spin-3/2 field (termed the raritron) minimally coupled to gravity. Using the Bogoliubov formalism, it computes the production spectra during and after inflation, distinguishing high-mass (c_s ≈ 1) and low-mass (c_s → 0 for the longitudinal helicity-1/2 mode) regimes, and shows that a wide range of m_{3/2} and H_e values can yield the observed dark-matter density assuming stability. A time-dependent mass extension is also analyzed, which enhances production relative to the constant high-mass case without necessarily suppressing the high-momentum tail. Results are compared to the Boltzmann approach where applicable.

Significance. If the stability assumption holds, the work provides a concrete, minimal realization of spin-3/2 dark matter via purely gravitational production, extending CGPP studies to higher spins and identifying novel dynamical features from the longitudinal-mode sound speed. The explicit mapping of viable parameter space and the Bogoliubov-based spectra constitute a technically solid contribution that could be directly tested against relic-density constraints.

major comments (2)
  1. [Results section on spectra computation] The high-momentum tail enhancement in the low-mass regime is load-bearing for the claim of a wide viable parameter space; the manuscript should supply the explicit numerical implementation details (e.g., integration limits, mode-equation solver convergence criteria, and cutoff handling) for the Bogoliubov coefficient computation to allow verification of the reported spectra.
  2. [Time-dependent mass extension] § on time-dependent mass: the statement that the time-dependent case 'does not necessarily tame the high momentum tail' is central to the comparison with the constant-mass low-mass regime; an explicit plot or table quantifying the tail slope for representative time-dependent m(t) profiles would strengthen this point.
minor comments (3)
  1. [Introduction] The introduction of the name 'raritron' would benefit from a single sentence noting its relation to the Rarita-Schwinger field and prior usage in the literature.
  2. [Figures] Figure captions should list the specific m_{3/2}/H_e ratios and integration ranges used for each panel to improve reproducibility.
  3. [Formalism] Notation for the sound speed c_s of the longitudinal mode should be defined once in the text before its first use in the mode-equation discussion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive comments, which we address point by point below. We will incorporate the requested clarifications and additions in the revised manuscript.

read point-by-point responses

  1. Referee: [Results section on spectra computation] The high-momentum tail enhancement in the low-mass regime is load-bearing for the claim of a wide viable parameter space; the manuscript should supply the explicit numerical implementation details (e.g., integration limits, mode-equation solver convergence criteria, and cutoff handling) for the Bogoliubov coefficient computation to allow verification of the reported spectra.

    Authors: We agree that additional numerical details will improve reproducibility. In the revised manuscript we will add a new subsection (or appendix) in the Results section that explicitly states the integration limits over comoving momentum, the convergence criteria and tolerances employed in the numerical solution of the mode equations, and the precise cutoff procedure used to extract the high-momentum tail of the Bogoliubov coefficients. These additions will allow independent verification of the reported spectra. revision: yes

  2. Referee: [Time-dependent mass extension] § on time-dependent mass: the statement that the time-dependent case 'does not necessarily tame the high momentum tail' is central to the comparison with the constant-mass low-mass regime; an explicit plot or table quantifying the tail slope for representative time-dependent m(t) profiles would strengthen this point.

    Authors: We thank the referee for this suggestion. In the revised version we will add a new figure (or table) in the time-dependent-mass section that displays the high-momentum spectral index (or slope) for several representative m(t) profiles. The figure will directly compare these slopes to the constant-mass low-mass case, thereby quantifying the statement that the tail is not necessarily tamed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation independent of target DM density

full rationale

The paper computes raritron production spectra directly from the Rarita-Schwinger mode equations in FRW using the Bogoliubov formalism, with the high-mass vs. low-mass distinction arising from the explicit evolution of the longitudinal-mode sound speed c_s. The observed DM density is matched only after the fact by scanning parameters under the stability assumption; no spectra are fitted to the target abundance during the derivation, and no load-bearing steps reduce to self-citations or ansatzes imported from prior work by the same authors. The comparison to the Boltzmann approach is presented as an independent cross-check. The central result is therefore a parameter-space mapping, not a tautological reproduction of its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the stability of the raritron and the assumption of minimal gravitational coupling; the production calculation itself uses standard quantum-field methods in curved spacetime.

free parameters (2)
  • raritron mass m_3/2
    Scanned across values relative to H_e to map the dark-matter density; not fitted to a specific data point but explored to find viable regions.
  • Hubble parameter at end of inflation H_e
    Key scale that sets the hierarchy with m_3/2 and controls the production efficiency.
axioms (2)
  • ad hoc to paper The raritron is stable and does not decay or thermalize after production.
    Explicitly stated as an assumption required for the particles to constitute dark matter today.
  • domain assumption Minimal coupling to gravity with no additional interactions.
    The model is defined as a free massive spin-3/2 field minimally coupled to gravity.
invented entities (1)
  • raritron no independent evidence
    purpose: Name for the minimal massive spin-3/2 field under study.
    Purely a naming convention for the free field; no new physical entity is postulated beyond the standard Rarita-Schwinger field.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Decaying spin-3/2 dark matter from baryon number violation

    hep-ph 2026-05 unverdicted novelty 7.0

    Non-supersymmetric spin-3/2 dark matter with baryon-violating portals can explain the relic abundance through UV and Boltzmann-suppressed freeze-in, with viable parameter space constrained by indirect detection, direc...

  2. Graviton Production from Inflaton Condensate: Boltzmann vs Bogoliubov

    hep-ph 2026-04 unverdicted novelty 6.0

    For quadratic inflaton potentials Boltzmann and Bogoliubov spectra agree at short wavelengths, but for steeper potentials non-adiabatic transition effects captured only by Bogoliubov are sizable across a broad momentum range.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 2 Pith papers · 12 internal anchors

  1. [1]

    Kolb and A.J

    E.W. Kolb and A.J. Long,Cosmological gravitational particle production and its implications for cosmological relics,Rev. Mod. Phys.96(2024) 045005 [2312.09042]

    work page arXiv 2024

  2. [2]

    Rarita and J

    W. Rarita and J. Schwinger,On a theory of particles with half integral spin,Phys. Rev.60(1941) 61

    work page 1941

  3. [3]

    Inflatino-less Cosmology

    J.J.M. Carrasco, R. Kallosh and A. Linde,Minimal supergravity inflation,Phys. Rev. D93(2016) 061301 [1512.00546]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  4. [4]

    String Theory Origin of Constrained Multiplets

    R. Kallosh, B. Vercnocke and T. Wrase,String Theory Origin of Constrained Multiplets,JHEP09 (2016) 063 [1606.09245]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  5. [5]

    Garcia, Y

    M.A.G. Garcia, Y. Mambrini, K.A. Olive and S. Verner,Case for decaying spin- 3/2 dark matter,Phys. Rev. D102(2020) 083533 [2006.03325]

    work page arXiv 2020

  6. [6]

    Gravitino Production After Inflation

    R. Kallosh, L. Kofman, A.D. Linde and A. Van Proeyen,Gravitino production after inflation,Phys. Rev. D61(2000) 103503 [hep-th/9907124]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  7. [7]

    Superconformal Symmetry, Supergravity and Cosmology

    R. Kallosh, L. Kofman, A.D. Linde and A. Van Proeyen,Superconformal symmetry, supergravity and cosmology,Class. Quant. Grav.17(2000) 4269 [hep-th/0006179]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  8. [8]

    Non-Thermal Production of Dangerous Relics in the Early Universe

    G.F. Giudice, I. Tkachev and A. Riotto,Nonthermal production of dangerous relics in the early universe,JHEP08(1999) 009 [hep-ph/9907510]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  9. [9]

    Thermal and Non-Thermal Production of Gravitinos in the Early Universe

    G.F. Giudice, A. Riotto and I. Tkachev,Thermal and nonthermal production of gravitinos in the early universe,JHEP11(1999) 036 [hep-ph/9911302]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  10. [10]

    Gravitino production in hybrid inflationary models

    M. Bastero-Gil and A. Mazumdar,Gravitino production in hybrid inflationary models,Phys. Rev. D62 (2000) 083510 [hep-ph/0002004]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  11. [11]

    Gravitino Problem in Minimal Supergravity Inflation

    F. Hasegawa, K. Mukaida, K. Nakayama, T. Terada and Y. Yamada,Gravitino Problem in Minimal Supergravity Inflation,Phys. Lett. B767(2017) 392 [1701.03106]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  12. [12]

    Kolb, A.J

    E.W. Kolb, A.J. Long and E. McDonough,Catastrophic production of slow gravitinos,Phys. Rev. D 104(2021) 075015 [2102.10113]. – 20 –

    work page arXiv 2021

  13. [13]

    Kolb, A.J

    E.W. Kolb, A.J. Long and E. McDonough,Gravitino Swampland Conjecture,Phys. Rev. Lett.127 (2021) 131603 [2103.10437]

    work page arXiv 2021

  14. [14]

    Terada,Minimal supergravity inflation without slow gravitino,Phys

    T. Terada,Minimal supergravity inflation without slow gravitino,Phys. Rev. D103(2021) 125022 [2104.05731]

    work page arXiv 2021

  15. [15]

    Dudas, M.A.G

    E. Dudas, M.A.G. Garcia, Y. Mambrini, K.A. Olive, M. Peloso and S. Verner,Slow and Safe Gravitinos,Phys. Rev. D103(2021) 123519 [2104.03749]

    work page arXiv 2021

  16. [16]

    Antoniadis, K

    I. Antoniadis, K. Benakli and W. Ke,Salvage of too slow gravitinos,JHEP11(2021) 063 [2105.03784]

    work page arXiv 2021

  17. [17]

    Y. Ema, K. Mukaida, K. Nakayama and T. Terada,Nonthermal Gravitino Production after Large Field Inflation,JHEP11(2016) 184 [1609.04716]. [18]Planckcollaboration,Planck 2018 results. X. Constraints on inflation,Astron. Astrophys.641(2020) A10 [1807.06211]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  18. [18]

    Freedman and A

    D.Z. Freedman and A. Van Proeyen,Supergravity, Cambridge Univ. Press, Cambridge, UK (5, 2012), 10.1017/CBO9781139026833

    work page doi:10.1017/cbo9781139026833 2012

  19. [19]

    Kolb, A.J

    E.W. Kolb, A.J. Long, E. McDonough and G. Payeur,Completely dark matter from rapid-turn multifield inflation,JHEP02(2023) 181 [2211.14323]

    work page arXiv 2023

  20. [20]

    Garcia, M

    M.A.G. Garcia, M. Pierre and S. Verner,New window into gravitationally produced scalar dark matter, Phys. Rev. D108(2023) 115024 [2305.14446]

    work page arXiv 2023

  21. [21]

    Capanelli, L

    C. Capanelli, L. Jenks, E.W. Kolb and E. McDonough,Gravitational production of completely dark photons with nonminimal couplings to gravity,JHEP09(2024) 071 [2405.19390]

    work page arXiv 2024

  22. [22]

    Krauss,New Constraints on Ino Masses from Cosmology

    L.M. Krauss,New Constraints on Ino Masses from Cosmology. 1. Supersymmetric Inos,Nucl. Phys. B 227(1983) 556

    work page 1983

  23. [23]

    Ellis, J.E

    J.R. Ellis, J.E. Kim and D.V. Nanopoulos,Cosmological Gravitino Regeneration and Decay,Phys. Lett. B145(1984) 181

    work page 1984

  24. [24]

    Gravitational Fermion Production in Inflationary Cosmology

    D.J.H. Chung, L.L. Everett, H. Yoo and P. Zhou,Gravitational Fermion Production in Inflationary Cosmology,Phys. Lett. B712(2012) 147 [1109.2524]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  25. [25]

    Hashiba, S

    S. Hashiba, S. Ling and A.J. Long,An analytic evaluation of gravitational particle production of fermions via Stokes phenomenon,JHEP09(2022) 216 [2206.14204]

    work page arXiv 2022

  26. [26]

    Y. Ema, K. Nakayama and Y. Tang,Production of Purely Gravitational Dark Matter,JHEP09(2018) 135 [1804.07471]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  27. [27]

    Kaneta, S.M

    K. Kaneta, S.M. Lee and K.-y. Oda,Boltzmann or Bogoliubov? Approaches compared in gravitational particle production,JCAP09(2022) 018 [2206.10929]

    work page arXiv 2022

  28. [28]

    Chakraborty, S

    A. Chakraborty, S. Clery, M.R. Haque, D. Maity and Y. Mambrini,Generalizing the Bogoliubov vs Boltzmann approaches in gravitational production,Phys. Rev. D112(2025) 043511 [2503.21877]

    work page arXiv 2025

  29. [29]

    Kaneta, W

    K. Kaneta, W. Ke, Y. Mambrini, K.A. Olive and S. Verner,Gravitational production of spin-3/2 particles during reheating,Phys. Rev. D108(2023) 115027 [2309.15146]

    work page arXiv 2023

  30. [30]

    Gravitational production of super-Hubble-mass particles: an analytic approach

    D.J.H. Chung, E.W. Kolb and A.J. Long,Gravitational production of super-Hubble-mass particles: an analytic approach,JHEP01(2019) 189 [1812.00211]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  31. [31]

    Basso, D.J.H

    E. Basso, D.J.H. Chung, E.W. Kolb and A.J. Long,Quantum interference in gravitational particle production,JHEP12(2022) 108 [2209.01713]

    work page arXiv 2022

  32. [32]

    Y. Ema, K. Nakayama and Y. Tang,Production of purely gravitational dark matter: the case of fermion and vector boson,JHEP07(2019) 060 [1903.10973]. [34]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys.641 (2020) A6 [1807.06209]

    work page arXiv 2019

  33. [33]

    Gondolo and G

    P. Gondolo and G. Gelmini,Cosmic abundances of stable particles: Improved analysis,Nucl. Phys. B 360(1991) 145. – 21 –

    work page 1991