Pith. sign in

REVIEW 5 major objections 4 minor 48 references

Rainbow cosmology's energy-dependent metric, acting on a complex scalar with softly broken U(1) symmetry, dynamically generates the observed baryon asymmetry from a symmetric start, giving η ≈ 29.29 λ (E_GUT/M)^4 (M/M_P)^(3/2).

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 05:21 UTC pith:FBPBEQU2

load-bearing objection A worked scalar-field calculation in rainbow cosmology that never connects its U(1) charge to baryon number and has internal arithmetic problems; as a baryogenesis claim it does not stand. the 5 major comments →

arxiv 2607.22252 v1 pith:FBPBEQU2 submitted 2026-07-24 gr-qc hep-ph

Dynamical Baryogenesis in Rainbow Cosmology

Surendra Kumar Gour , Malay K. Nandy This is my paper
classification gr-qc hep-ph
keywords baryogenesisrainbow gravitymodified dispersion relationcomplex scalar fieldU(1) symmetry breakingNoether chargebaryon-to-photon ratioradiation-dominated epoch
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that baryon asymmetry can be produced dynamically during the radiation-dominated era in rainbow cosmology, a quantum-gravity-inspired setting where the metric depends on probe energy. It analyzes a complex scalar field whose global U(1) symmetry is softly broken, so the two real components get slightly different masses. The modified Friedmann equation and field equations have analytic early- and late-time solutions, and the field's Noether charge density, time-averaged, decays as t^-3/2 just like the photon density, leaving a constant baryon-to-photon ratio. Matching that ratio to the observed 6.1e-10 constrains the symmetry-breaking coupling λ and scalar mass M to values like λ ~ 1e-7 with M ~ 1.6e16 GeV, showing no supersymmetry is needed.

Core claim

Rainbow cosmology with f=1+E/2E_P, g=1 modifies the Friedmann and scalar-field equations so that the scale factor grows as t early and t^{1/2} late, and the solutions pass from damped oscillations to Bessel functions of order 1/4. Matching at t_c=1/M fixes constants; time-averaging the U(1) Noether charge gives ⟨ρ⟩=(3/4)λ A² M^{3/2} sin²(1/√3) t^{-3/2}. Photon density also decays as t^{-3/2}, so η→29.29 λ(E_GUT/M)^4(M/M_P)^(3/2). Setting η=6.1×10⁻¹⁰ constrains λ∈10⁻¹⁸–10⁻⁷ and M∈10⁻⁴E_GUT–E_GUT, a viable non-SUSY mechanism if the scalar's U(1) charge represents baryon number.

What carries the argument

The load-bearing object is the U(1) Noether charge density ρ = (f/g³)(u v̇ - v u̇) of the complex scalar Φ = u + iv. The softly broken potential V = ½M²|Φ|² - (λ/4M²)(Φ² + Φ*²) splits the component masses, M_u² = M²(1-λ) and M_v² = M²(1+λ); this frequency mismatch makes ρ nonzero even when the initial state has u and v with equal amplitudes. The rainbow metric enters through f(ε) = 1 + E/(2E_P), g(ε)=1, which changes the Friedmann equation to H² = (8πG/3)ρ_r/f² and the friction term in the scalar equation of motion; the late-time solutions are Bessel functions J_{1/4}, Y_{1/4}, and the constants are fixed by matching at t_c = 1/M. It is the non-oscillatory, λ-proportional part of ρ after tim

Load-bearing premise

The argument rests on treating the Noether charge of the scalar's internal U(1) as the baryon asymmetry: if there is no coupling or decay that transfers this charge to Standard Model baryons, the computed rho/n_gamma is a scalar charge-to-photon ratio, not the cosmological baryon-to-photon ratio.

What would settle it

Compute the Standard-Model baryon number of the Phi field, e.g., the commutator [B_hat, Phi] in an effective theory; if it vanishes, the Noether charge is not baryon number and the mechanism cannot produce baryon asymmetry. Alternatively, an experiment or bound that excludes a Phi-quark coupling (or shows Phi decays before transferring the charge) would falsify the transfer step.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Baryon asymmetry is generated dynamically from an initially baryon-symmetric state, with no supersymmetry invoked.
  • The baryon-to-photon ratio approaches a constant at late times, given by η ≈ 29.29 λ (E_GUT/M)^4 (M/M_P)^(3/2).
  • Requiring η = 6.1×10⁻¹⁰ constrains the model to λ ~ 10⁻¹⁸–10⁻⁷ for M between 10⁻⁴ E_GUT and E_GUT, i.e., GUT-scale scalar masses with small symmetry-breaking couplings.
  • The rainbow correction to the photon number density, n_γ = (2ζ(3)/π²)T³ + (2π²/15)T⁴/E_P, is negligible for T ~ E_GUT << E_P, so the standard radiation scaling is preserved.
  • The mechanism ties the baryon asymmetry to the existence of energy-dependent (rainbow) metric modifications near the Planck scale.

Where Pith is reading between the lines

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

  • The identification of the scalar U(1) charge with baryon number is assumed, not derived; the cleanest way to upgrade this result to a full baryogenesis mechanism is to add a concrete transfer operator (e.g., Φ coupling to quark or lepton fields) and check that the asymmetry survives annihilation and sphaleron processes.
  • If the scalar field is long-lived, its late-time energy density could contribute to dark matter or affect BBN; the model's parameter space could be tested by such constraints even before any direct coupling is specified.
  • The same dynamical-charge mechanism should work for lepton number, suggesting a rainbow-cosmology variant of leptogenesis with the same λ(E_GUT/M)^4 scaling, which would be testable via neutrino-mass or washout constraints.
  • The prediction η constant in time comes from the matching of t^{-3/2} decays of both ρ and n_γ; any deviation in the equation of state (e.g., a matter-dominated episode) would break this and could be looked for in the early-universe expansion history.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper proposes a dynamical baryogenesis mechanism in the context of rainbow cosmology. A complex scalar field with a softly broken global U(1) symmetry is evolved in a modified FLRW background, and the U(1) Noether charge density is computed in the late-time limit. The authors derive a baryon-to-photon ratio, Eq. (96), that asymptotes to a constant proportional to the symmetry-breaking coupling λ and inversely proportional to M^{5/2}. They then use the observed η = 6.1×10^{-10} to constrain λ and M, claiming a viable non-SUSY baryogenesis scenario.

Significance. If the central identification of the scalar Noether charge with baryon number were justified, and if the analytic calculations were internally consistent, this would be an interesting addition to baryogenesis mechanisms, connecting quantum-gravity-inspired dispersion relations to cosmological asymmetry. The paper provides a transparent analytic treatment, with early- and late-time asymptotic solutions and a matching calculation in Appendix A. However, the core physical identification is not justified, and there are quantitative inconsistencies (Table 1 vs. Eq. (96); amplitude normalization; ad hoc matching time) that undermine the reported constraints. No reproducible code or machine-checked proof is provided.

major comments (5)
  1. [§3, §8, Eq. (78), Eq. (95)] The Noether charge density of the complex scalar's global U(1) is not the baryon asymmetry. The action (6) contains only a scalar and gravity; the potential (12) explicitly breaks the U(1) but contains no coupling to Standard Model fermions, no baryon-number-violating operator, and no decay or transfer process. The statement that the real and imaginary components 'represent baryons and antibaryons' is a modeling assignment, not a derived relation. Consequently ρ defined in Eq. (78) and η in Eq. (95) are scalar-charge-to-photon ratios, not the cosmological baryon-to-photon ratio. This is the load-bearing flaw: even if all the mathematics were correct, the comparison with observed η is unsupported.
  2. [Table 1 vs. Eq. (96)] For λ=10^{-7} and M=1.59×10^{16} GeV, Eq. (96) gives η ≈ 2.4×10^{-10}, not 6.1×10^{-10}. The same discrepancy appears for every row of Table 1. Thus the table does not reproduce the central formula it is meant to constrain, and the claimed λ–M ranges are not self-consistent.
  3. [Eqs. (93)–(94)] The normalization of the initial amplitude A is inconsistent. At t=0, Eq. (26) gives E=2E_P, so f(0)=2, but Eq. (94) uses f ≃ 1+E_GUT/(2E_P). Using f(0)=2 in Eq. (93) yields A = √6 (E_GUT/M)^2, not the √3 coefficient of Eq. (94) (with the EGUT-dependent factor evaluated near unity). The paper does not reconcile the energy scale E in the rainbow function with the potential-energy scale E_GUT.
  4. [Appendix A, t_c matching] The crossover time is set to t_c = K^2/Γ and then identified with 1/M. This imposes an arbitrary relation among K, Γ, and M. Since K is a free integration constant, this matching choice is not derived from the dynamics. The resulting expression for C_uD_v−C_vD_u, Eq. (116), and hence the final η, depend directly on this ad hoc choice.
  5. [Eq. (27) and Eq. (93)] The modified Friedmann equation (27) includes only the radiation energy density, but the scalar field's initial potential energy is set to E_GUT^4 (Eq. 93), which is comparable to the GUT-scale radiation energy density. The backreaction of the scalar field on the scale factor is neglected in the derivation of H(t) and T(t), which are used to compute n_γ. This affects the quantitative value of η.
minor comments (4)
  1. [§6.1, Eq. (55)] The setting of B_u=B_v=0 to avoid a 1/t divergence at t→0 is physically reasonable but should be stated explicitly as a boundary condition, and its relation to the 'baryon-symmetric initial state' should be clarified.
  2. [§8, text after Eq. (91)] Typo: 'procesed' should be 'proceed'.
  3. [§8, Eq. (80)] The leading-order expression for ρ(t) in Eq. (80) and the subsequent small-time expansion in Eq. (81) merit rechecking; the coefficient in Eq. (81) appears inconsistent with a direct Taylor expansion of Eq. (80).
  4. [Throughout] The notation E_GUT is used for both the initial potential energy density (Eq. 93) and the energy scale appearing in the rainbow functions (Eq. 94). These should be clearly distinguished, as the mismatch is related to Major Comment 3.

Circularity Check

0 steps flagged

No significant circularity: Eq. (96) is derived algebraically from stated inputs; the observed η enters only as a post-hoc constraint, and no load-bearing self-citations are present.

full rationale

The derivation is self-contained. Starting from the externally motivated rainbow ansatz (14), the paper derives the modified Friedmann equation (27), the scalar equations of motion (41–46), early- and late-time solutions (55, 63–64), the integration constants by matching at tc = 1/M (Appendix A), the Noether charge density (78–92), and the MDR photon density (73, 77). The central formula (95–96) is obtained by direct substitution η = ⟨ρ⟩/nγ, with no use of the observed η as an input anywhere in the chain. The observed value (6.1×10^-10) is used only afterward to define a constraint surface in the (λ, M) plane (Table 1); with one equation and two free parameters this yields a one-parameter family, a standard parameter-constraint exercise rather than a fitted quantity renamed as a prediction. The time-independence of η follows because both ⟨ρ⟩ and nγ dilute as t^-3/2 (∝ a^-3); that is kinematics, not an input. The identification of the scalar U(1) Noether charge with baryon asymmetry (Section 1: 'the real and imaginary components represent baryons and antibaryons, respectively' and Eq. 78) is a stated modeling assumption rather than a derived relation; this is a genuine physical-correctness risk—no coupling or transfer operator to Standard Model baryons is specified—but it is an assumption on which the derivation is conditional, not a step that reduces a prediction to its own inputs. Other weaknesses (two-parameter degeneracy, ad hoc crossover time, and the internal rounding mismatch between Table 1 and Eq. 96) are correctness or presentation concerns, not circularity. There are no load-bearing self-citations. Hence: no circularity, score 0.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 1 invented entities

The derivation rests on a speculative spacetime framework and several hand-chosen parameters: the rainbow functions, the potential, the initial energy scale, and the matching time. The most serious ledger entry is the unproven identification of the scalar's global-U(1) charge with baryon number, without which the entire eta calculation is not a baryogenesis calculation.

free parameters (5)
  • lambda (U(1)-breaking coupling) = 10^-18 to 10^-7 (Table 1)
    Free parameter of the potential; chosen after the fact to make eta match the observed 6.1e-10.
  • M (scalar mass) = 10^12 to 10^16 GeV (Table 1)
    Free parameter; together with lambda it forms a one-parameter family of fits, not a prediction.
  • E_GUT / initial potential energy U0 = 10^16 GeV
    Assumed initial energy scale; normalizes the amplitude A via Eq. 94. Not derived, and inconsistent with E(t->0)=2E_P from Eq. 26.
  • Crossover time t_c = 1/M
    Chosen by hand to match early and late solutions; the final coefficient C_uD_v - C_vD_u depends on this choice.
  • Initial amplitudes A_u = A_v = A = A = sqrt(3)(E_GUT/M)^2 (Eq. 94)
    Baryon-symmetric initial condition; A is fixed by the U0 assumption, not by independent data.
axioms (8)
  • domain assumption Rainbow gravity modifies the FLRW metric to ds^2 = -dt^2/f^2 + a^2/g^2 dx^2 with f,g depending on probe energy (Section 2, Eq. 5)
    Taken from Magueijo-Smolin literature; no derivation in this paper; speculative framework.
  • domain assumption Rainbow functions f(epsilon) = 1 + E/(2E_P), g(epsilon) = 1 (Eq. 14)
    Adopted from Amelino-Camelia et al.; affects every subsequent equation.
  • ad hoc to paper The U(1)-breaking potential V = (1/2)M^2|Phi|^2 - (lambda/4)M^2(Phi^2 + Phi*^2) with lambda << 1 (Eq. 12)
    Chosen to produce mass splitting between u and v; neither lambda nor M is independently fixed.
  • ad hoc to paper The scalar field charge density is baryon number; no coupling/decay to Standard Model baryons is introduced (Sections 3, 8)
    The model never couples Phi to quarks/leptons, so rho is a U(1) field charge, not necessarily baryon asymmetry.
  • ad hoc to paper Initial energy density of the scalar is E_GUT^4 with E_GUT = 10^16 GeV (Eq. 93)
    Normalizes amplitude A; arbitrary and inconsistent with E(0) = 2E_P from Eq. 26.
  • ad hoc to paper Early and late approximate solutions are matched at t_c = 1/M (Appendix A)
    The matching time is chosen by hand; the asymptotic coefficient C_uD_v - C_vD_u depends on this choice.
  • domain assumption Radiation energy density scales as rho_r proportional to a^-4 and scalar backreaction is neglected
    Radiation-dominated Friedmann equation is used; the scalar field's own energy density is not included in H.
  • standard math Bessel function identities and Wronskian (Eqs. 107, 115)
    Standard mathematical background; not at issue.
invented entities (1)
  • Complex scalar field Phi carrying baryon number no independent evidence
    purpose: Source of baryon asymmetry via soft U(1) breaking
    Beyond-Standard-Model field; no production/decay operator connecting it to quarks or leptons, and no observable signature is given.

pith-pipeline@v1.3.0-alltime-deepseek · 15865 in / 29513 out tokens · 256127 ms · 2026-08-01T05:21:11.898987+00:00 · methodology

0 comments
read the original abstract

We investigate baryogenesis in the framework of rainbow cosmology employing a complex scalar field with a softly broken global $U(1)$-symmetry. The modified dispersion relation of rainbow cosmology leads to energy-dependent modification to the FLRW metric components, that modifies the Friedmann equation and the scalar field dynamics. We thereby obtain analytical solutions for the scalar field evolution in the radiation-dominated epoch, and show that baryon asymmetry is generated dynamically even from an initially baryon-symmetric state. We find that the baryon-to-photon ratio asymptotically approaches a constant value in the long-time limit, which is proportional to the symmetry-breaking coupling strength $\lambda$, and scales with the scalar field mass as $M^{-5/2}$. Our results demonstrate that requiring consistency with the observed baryon asymmetry constrains the $\lambda$ and $M$ values within reasonable ranges, thus providing a viable dynamical mechanism for baryogenesis during the radiation-dominated era without invoking supersymmetry.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

48 extracted references · 6 canonical work pages

  1. [1]

    Nollett, and Michael S

    Scott Burles, Kenneth M. Nollett, and Michael S. Turner. What is the big-bang-nucleosynthesis prediction for the baryon density and how reliable is it?Phys. Rev. D, 63:063512, Feb 2001. URL: https://link.aps.org/doi/10.1103/PhysRevD.63.063512,doi:10. 1103/PhysRevD.63.063512

    doi:10.1103/physrevd.63.063512 2001

  2. [2]

    Nollett, and Michael S

    Scott Burles, Kenneth M. Nollett, and Michael S. Turner. Big bang nucleosynthesis predictions for precision cosmology.The Astrophysical Journal, 552(1):L1, apr 2001.doi:10.1086/320251

    doi:10.1086/320251 2001

  3. [3]

    C. L. Bennett et al. First-year wilkinson microwave anisotropy probe (wmap) observations: Preliminary maps and basic results.The Astro- physical Journal Supplement Series, 148(1):1, sep 2003.doi:10.1086/ 377253

    2003

  4. [4]

    Aghanim et al

    N. Aghanim et al. Planck 2018 results - VI. Cosmological parameters. Astronomy & Astrophysics, 641:A6, 2020.doi:10.1051/0004-6361/ 201833910

    doi:10.1051/0004-6361/ 2018

  5. [5]

    Tristram et al

    M. Tristram et al. Cosmological parameters derived from the final Planck data release (PR4).A&A, 682:A37, 2024.doi:10.1051/ 0004-6361/202348015

    2024

  6. [6]

    Navas et al

    S. Navas et al. Review of particle physics.Phys. Rev. D, 110:030001, Aug 2024. URL:https://link.aps.org/doi/10.1103/PhysRevD. 110.030001,doi:10.1103/PhysRevD.110.030001

    doi:10.1103/physrevd 2024

  7. [7]

    A. D. Sakharov. Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe.Pisma Zh. Eksp. Teor. Fiz., 5:32–35, 1967. doi:10.1070/PU1991v034n05ABEH002497. 25

    doi:10.1070/pu1991v034n05abeh002497 1967

  8. [8]

    Howard Georgi and S. L. Glashow. Unity of all elementary- particle forces.Phys. Rev. Lett., 32:438–441, Feb 1974. URL: https://link.aps.org/doi/10.1103/PhysRevLett.32.438,doi:10. 1103/PhysRevLett.32.438

    doi:10.1103/physrevlett.32.438 1974

  9. [9]

    Baryon number of the uni- verse.Phys

    Savas Dimopoulos and Leonard Susskind. Baryon number of the uni- verse.Phys. Rev. D, 18:4500–4509, Dec 1978. URL:https://link. aps.org/doi/10.1103/PhysRevD.18.4500,doi:10.1103/PhysRevD. 18.4500

    doi:10.1103/physrevd.18.4500 1978

  10. [10]

    Unified gauge theories and the baryon num- ber of the universe.Phys

    Motohiko Yoshimura. Unified gauge theories and the baryon num- ber of the universe.Phys. Rev. Lett., 41:281–284, Jul 1978. URL: https://link.aps.org/doi/10.1103/PhysRevLett.41.281,doi:10. 1103/PhysRevLett.41.281

    doi:10.1103/physrevlett.41.281 1978

  11. [12]

    Cosmological production of baryons.Phys

    Steven Weinberg. Cosmological production of baryons.Phys. Rev. Lett., 42:850–853, Mar 1979. URL:https://link.aps.org/doi/10.1103/ PhysRevLett.42.850,doi:10.1103/PhysRevLett.42.850

    doi:10.1103/physrevlett.42.850 1979

  12. [13]

    Kolb and Stephen Wolfram

    Edward W. Kolb and Stephen Wolfram. Baryon Number Generation in the Early Universe.Nucl. Phys. B, 172:224, 1980. [Erratum: Nucl.Phys. B 195, 542 (1982)].doi:10.1016/0550-3213(82)90012-8

    doi:10.1016/0550-3213(82)90012-8 1980

  13. [14]

    Kolb.The Early Universe, volume 69

    Edward W. Kolb.The Early Universe, volume 69. Taylor and Francis, 5 2019.doi:10.1201/9780429492860

    doi:10.1201/9780429492860 2019

  14. [15]

    Oxford University Press, Oxford, 2008

    Steven Weinberg.Cosmology. Oxford University Press, Oxford, 2008

    2008

  15. [16]

    V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov. On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe.Phys. Lett. B, 155:36, 1985.doi:10.1016/0370-2693(85) 91028-7. 26

    doi:10.1016/0370-2693(85 1985

  16. [17]

    M. E. Shaposhnikov. Possible Appearance of the Baryon Asymmetry of the Universe in an Electroweak Theory.JETP Lett., 44:465–468, 1986

    1986

  17. [18]

    James M. Cline. Baryogenesis. InLes Houches Summer School - Session 86: Particle Physics and Cosmology: The Fabric of Spacetime. Oxford University Press, 9 2006.arXiv:hep-ph/0609145

    Pith/arXiv arXiv 2006

  18. [21]

    A mini review on Affleck- Dine baryogenesis.New J

    Rouzbeh Allahverdi and Anupam Mazumdar. A mini review on Affleck- Dine baryogenesis.New J. Phys., 14:125013, 2012.doi:10.1088/ 1367-2630/14/12/125013

    2012

  19. [22]

    Cohen and David B

    Andrew G. Cohen and David B. Kaplan. Thermodynamic genera- tion of the baryon asymmetry.Physics Letters B, 199(2):251–258,

  20. [23]

    Cohen and David B

    Andrew G. Cohen and David B. Kaplan. Spontaneous baryogene- sis.Nuclear Physics B, 308(4):913–928, 1988. URL:https://www. sciencedirect.com/science/article/pii/0550321388901344,doi: 10.1016/0550-3213(88)90134-4

    arXiv 1988

  21. [25]

    Kribs, Hitoshi Mu- rayama, and Paul J

    Hooman Davoudiasl, Ryuichiro Kitano, Graham D. Kribs, Hitoshi Mu- rayama, and Paul J. Steinhardt. Gravitational baryogenesis.Phys. 27 Rev. Lett., 93:201301, Nov 2004. URL:https://link.aps.org/ doi/10.1103/PhysRevLett.93.201301,doi:10.1103/PhysRevLett. 93.201301

    doi:10.1103/physrevlett.93.201301 2004

  22. [26]

    Lorentz invariance with an invariant energy scale.Phys

    Jo˜ ao Magueijo and Lee Smolin. Lorentz invariance with an invariant energy scale.Phys. Rev. Lett., 88:190403, Apr 2002. URL:https:// link.aps.org/doi/10.1103/PhysRevLett.88.190403,doi:10.1103/ PhysRevLett.88.190403

    doi:10.1103/physrevlett.88.190403 2002

  23. [27]

    Gravity’s rainbow.Classical and Quan- tum Gravity, 21(7):1725, mar 2004.doi:10.1088/0264-9381/21/7/ 001

    Jo˜ ao Magueijo and Lee Smolin. Gravity’s rainbow.Classical and Quan- tum Gravity, 21(7):1725, mar 2004.doi:10.1088/0264-9381/21/7/ 001

    work page doi:10.1088/0264-9381/21/7/ 2004

  24. [28]

    Mena Marug´ an

    Pablo Gal´ an and Guillermo A. Mena Marug´ an. Quantum time un- certainty in a gravity’s rainbow formalism.Phys. Rev. D, 70:124003, Dec 2004. URL:https://link.aps.org/doi/10.1103/PhysRevD.70. 124003,doi:10.1103/PhysRevD.70.124003

    doi:10.1103/physrevd.70 2004

  25. [29]

    Gravitational baryogenesis in F(R) gravity’s rain- bow.Journal of High Energy Physics, 2:29, 2026.doi:10.1007/ JHEP02(2026)029

    Parviz Goodarzi. Gravitational baryogenesis in F(R) gravity’s rain- bow.Journal of High Energy Physics, 2:29, 2026.doi:10.1007/ JHEP02(2026)029

    2026

  26. [30]

    Testable scenario for relativity with minimum length.Physics Letters B, 510(1):255–263, 2001

    Giovanni Amelino-Camelia. Testable scenario for relativity with minimum length.Physics Letters B, 510(1):255–263, 2001. URL:https://www.sciencedirect.com/science/article/pii/ S0370269301005068,doi:10.1016/S0370-2693(01)00506-8

    work page doi:10.1016/s0370-2693(01)00506-8 2001

  27. [31]

    Doubly special relativity.Nature, 418:34– 35, 2002.arXiv:gr-qc/0207049,doi:10.1038/418034a

    Giovanni Amelino-Camelia. Doubly special relativity.Nature, 418:34– 35, 2002.arXiv:gr-qc/0207049,doi:10.1038/418034a

    Pith/arXiv arXiv 2002

  28. [32]

    Relativity in space-times with short dis- tance structure governed by an observer independent (Planckian) length scale.Int

    Giovanni Amelino-Camelia. Relativity in space-times with short dis- tance structure governed by an observer independent (Planckian) length scale.Int. J. Mod. Phys. D, 11:35–60, 2002.arXiv:gr-qc/0012051, doi:10.1142/S0218271802001330

    Pith/arXiv arXiv 2002

  29. [33]

    Iochum, T

    B. Iochum, T. Masson, T. Sch¨ ucker, and A. Sitarz. Compactκdeforma- tion and spectral triples.Reports on Mathematical Physics, 68(1):37– 28 64, 2011. URL:https://www.sciencedirect.com/science/article/ pii/S0034487711600268,doi:10.1016/S0034-4877(11)60026-8

    work page doi:10.1016/s0034-4877(11)60026-8 2011

  30. [34]

    An introduction toκ deformed symmetries, phase spaces and field theory.Symmetry, 13(6),

    Michele Arzano and Jerzy Kowalski-Glikman. An introduction toκ deformed symmetries, phase spaces and field theory.Symmetry, 13(6),

  31. [36]

    Generalized Lorentz invariance with an invariant energy scale.Phys

    Jo˜ ao Magueijo and Lee Smolin. Generalized Lorentz invariance with an invariant energy scale.Phys. Rev. D, 67:044017, Feb 2003. URL: https://link.aps.org/doi/10.1103/PhysRevD.67.044017,doi:10. 1103/PhysRevD.67.044017

    doi:10.1103/physrevd.67.044017 2003

  32. [37]

    Amelino-Camelia, John R

    G. Amelino-Camelia, John R. Ellis, N. E. Mavromatos, Dimitri V. Nanopoulos, and Subir Sarkar. Tests of quantum gravity from ob- servations of gamma-ray bursts.Nature, 393:763–765, 1998.arXiv: astro-ph/9712103,doi:10.1038/31647

    Pith/arXiv arXiv 1998

  33. [38]

    Modifications to Lorentz invariant dispersion in relatively boosted frames.Phys

    Uri Jacob, Flavio Mercati, Giovanni Amelino-Camelia, and Tsvi Pi- ran. Modifications to Lorentz invariant dispersion in relatively boosted frames.Phys. Rev. D, 82:084021, Oct 2010. URL:https://link.aps. org/doi/10.1103/PhysRevD.82.084021,doi:10.1103/PhysRevD.82. 084021

    work page doi:10.1103/physrevd.82.084021 2010

  34. [39]

    Quantum-Spacetime Phenomenology

    Giovanni Amelino-Camelia. Quantum-Spacetime Phenomenology. Living Rev. Rel., 16:5, 2013.arXiv:0806.0339,doi:10.12942/ lrr-2013-5

    Pith/arXiv arXiv 2013

  35. [40]

    Friedmann

    A. Friedmann. ¨Uber die Kr¨ ummung des Raumes.Zeitschrift fur Physik, 10:377–386, jan 1922.doi:10.1007/BF01332580. 29

    doi:10.1007/bf01332580 1922

  36. [41]

    Lema ˆ ıtre

    G. Lema ˆ ıtre. Un Univers homog` ene de masse constante et de rayon croissant rendant compte de la vitesse radiale des n´ ebuleuses extra- galactiques.Annales de la Soci´ et´ e Scientifique de Bruxelles, 47:49–59, jan 1927

    1927

  37. [42]

    H. P. Robertson. Kinematics and World-Structure. 3.Astrophys. J., 83:257–271, 1936.doi:10.1086/143726

    doi:10.1086/143726 1936

  38. [43]

    A. G. Walker. On Milne’s Theory of World-Structure.Proceedings of the London Mathematical Society, 42:90–127, jan 1937.doi:10.1112/ plms/s2-42.1.90

    1937

  39. [44]

    John D. Barrow. Some inflationary Einstein-aether cosmologies.Phys. Rev. D, 85:047503, Feb 2012. URL:https://link.aps.org/doi/10. 1103/PhysRevD.85.047503,doi:10.1103/PhysRevD.85.047503

    doi:10.1103/physrevd.85.047503 2012

  40. [45]

    Rainbow universe.Journal of Cosmology and Astroparticle Physics, 2007(08):017, aug 2007.doi:10.1088/1475-7516/2007/08/ 017

    Yi Ling. Rainbow universe.Journal of Cosmology and Astroparticle Physics, 2007(08):017, aug 2007.doi:10.1088/1475-7516/2007/08/ 017

    work page doi:10.1088/1475-7516/2007/08/ 2007

  41. [46]

    Academic Press, San Diego, CA, 1995

    George B Arfken and Hans J Weber.Mathematical methods for physi- cists. Academic Press, San Diego, CA, 1995

    1995

  42. [47]

    Stegun.Handbook of Mathematical Functions

    Milton Abramowitz and Irene A. Stegun.Handbook of Mathematical Functions. National Bureau of Standards, 1964

    1964

  43. [48]

    Compari- son of grand unified theories with electroweak and strong coupling constants measured at LEP.Physics Letters B, 260(3):447–455,

    Ugo Amaldi, Wim de Boer, and Hermann F¨ urstenau. Compari- son of grand unified theories with electroweak and strong coupling constants measured at LEP.Physics Letters B, 260(3):447–455,

  44. [49]

    Supersymmetric dynamical generation of the grand unification scale.Physics Letters B, 410(1):45–51,

    Hsin-Chia Cheng. Supersymmetric dynamical generation of the grand unification scale.Physics Letters B, 410(1):45–51,

  45. [50]

    Strong Coupling Running, Gauge Coupling Unifi- cation and the Scale of New Physics.JHEP, 11:117, 2015.arXiv: 1508.04176,doi:10.1007/JHEP11(2015)117

    Dimitri Bourilkov. Strong Coupling Running, Gauge Coupling Unifi- cation and the Scale of New Physics.JHEP, 11:117, 2015.arXiv: 1508.04176,doi:10.1007/JHEP11(2015)117. 31

    Pith/arXiv arXiv 2015

  46. [1991]

    URL:https://www.sciencedirect.com/science/article/ pii/0370269391916418,doi:10.1016/0370-2693(91)91641-8

    arXiv

  47. [1997]

    URL:https://www.sciencedirect.com/science/article/ pii/S0370269397009246,doi:10.1016/S0370-2693(97)00924-6. 30

    work page doi:10.1016/s0370-2693(97)00924-6

  48. [2021]

    3390/sym13060946

    URL:https://www.mdpi.com/2073-8994/13/6/946,doi:10. 3390/sym13060946

    2073