Flipped Rotating Axion Non-minimally Coupled to Gravity: Baryogenesis and Dark Matter

Anish Ghoshal; Chao Chen; Konstantinos Dimopoulos; Suruj Jyoti Das

REVIEW 3 major objections 5 minor 1 cited by

Flipped Rotating Axion Non-minimally Coupled to Gravity: Baryogenesis and Dark Matter

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single spectator axion, non-minimally coupled to gravity, can rotate after inflation and generate both the baryon asymmetry and dark matter.

desk verdict A well-motivated cogenesis idea with an honest but unresolved constant-xi window; the running-coupling bridge needs to be made concrete before the mechanism is viable. read the letter →

arxiv 2502.08720 v2 pith:MVGZWKMW submitted 2025-02-12 hep-ph astro-ph.COgr-qchep-th

Chao Chen , Suruj Jyoti Das , Konstantinos Dimopoulos , Anish Ghoshal This is my paper · ORCID

classification hep-phastro-ph.COgr-qchep-th

keywords flippedvacuummanifoldaxionrotationspontaneousbaryogenesisnon-minimalcouplingtogravitykinationdarkmatterMajorongravitationalwaves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that a single spectator axion-like particle can explain both the baryon asymmetry and the dark matter, without ever introducing explicit symmetry-breaking operators. The trick is a periodic non-minimal coupling to gravity: when the Universe switches from inflation to a kination-dominated era after inflation, the Ricci scalar changes sign and the effective axion potential flips, turning the old minimum into a peak. The axion then rolls and keeps rotating because the potential barrier shrinks at the same rate as its kinetic energy. That rotation sources the baryon excess through spontaneous baryogenesis, and at late times the same field oscillates and behaves as dark matter. The paper spells out the parameter relations, checks fragmentation and Kibble issues, and illustrates the scenario with the Majoron of the Type-I seesaw mechanism, where it predicts sub-eV Majoron masses and right-handed neutrinos above about $10^8$ GeV.

What carries the argument

The load-bearing object is the periodic non-minimal coupling $\gamma^2(\phi)=1+\xi[1-\cos(\phi/f)]$ in Eq. (2), inserted into $\mathcal{L}=\frac12 m_P^2\gamma^2 R-\frac12(\partial\phi)^2-M^4[1-\cos(\phi/f)]$. This coupling respects the discrete shift symmetry while tying the height and sign of the axion's cosine potential to the Ricci scalar; when $R$ changes sign at the inflation-to-kination transition, the minimum and maximum swap, and the matched $a^{-6}$ scaling of the barrier height and of the rolling kinetic energy lets the axion cross the barrier each cycle and rotate. The same object sets the condition (Eq. (9)) for rolling, the baryon yield (Eq. (26)) through $\dot\theta_m \simeq \sqrt{12\xi}\,m_P H_{\rm end}/f$, and the constraints from fragmentation (Eqs. (60), (65)) and the Kibble problem (Eq. (67)).

What would settle it

Numerically evolve the homogeneous axion together with its fluctuation modes using a concrete running coupling $\xi(\sigma)$ from Eq. (68) that interpolates between $\xi<\frac14(f/m_P)^2$ at the end of inflation and $\xi>\frac34(f/m_P)^2$ during kination. If for all choices of $\beta$ and $\mu$ the angular velocity $\dot\theta$ drops below $\sqrt{|V_{\rm eff}|}/f$ before reheating, or the fluctuation modes grow enough to stop the rotation, then the cogenesis mechanism fails in the regime the paper needs.

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Extended reading notes

Core claim

The paper's claim is that cogenesis can be driven by a 'flipped rotating axion': the effective potential appearing in Eq. (5), $V_{\rm eff} = (M^4 - \frac12 \xi m_P^2 R)[1-\cos(\phi/f)]$, changes phase between inflation and kination because $R=3(1-3w)H^2$ flips sign. During inflation the minimum sits at $\phi=\pi f$; during kination that point becomes the top of the barrier and the new minimum is at zero. Because both the barrier height and the axion's kinetic energy scale as $a^{-6}$ in kination, the field slides over the diminishing barrier and enters sustained rotation. The rotation gives a baryon yield $Y_B \simeq (3\sqrt{30}\,\xi c_B / 2\pi\sqrt{g_*})\, T_{B-L}^2/(f T_{\rm reh})$ through spontaneous baryogenesis, while later freezing and thawing in the bare potential $M^4[1-\cos(\phi/f)]$ produces the dark matter abundance with $M\sim 10^{-9}\,{\rm GeV}\,(m_P/f)^{3/2}$. In the concrete Type-I seesaw realization the rotating axion is the Majoron, with sub-eV mass and right-handed neutrino masses above $3\times10^8$ GeV, and the kination era makes the inflationary gravitational-wave background blue-tilted and constrained by BBN.

Load-bearing premise

The load-bearing premise is that a running non-minimal coupling $\xi(\sigma)$ can smoothly grow from below $\frac14(f/m_P)^2$ during inflation to above $\frac34(f/m_P)^2$ during kination, so that the axion both avoids the Kibble problem and acquires enough kick to rotate; the paper offers this as a possibility (Eq. (68)) without a concrete model or simulation showing that the rotation starts and persists under the running coupling.

Editorial extensions

If this is right

Cogenesis needs only one spectator field: the axion's rotation generates the baryon asymmetry, and its later oscillations in the bare potential produce the dark matter, with no separate dark sector.
The axion mass decouples from the baryon asymmetry (cf. Eq. (26) vs Eq. (35)), so the Majoron can be lighter than sub-eV while the seesaw's right-handed neutrinos sit above about $10^8$ GeV; this is the concrete prediction of the Type-I seesaw realization.
The kination era turns the inflationary gravitational-wave spectrum blue-tilted; avoiding overproduction during BBN forces $T_{\rm reh}\gtrsim2\times10^7$ GeV, and near-future CMB and GW experiments can probe the resulting $\Delta N_{\rm eff}$ and peak frequency.
Both fragmentation and the Kibble problem push $\xi$ toward $(f/m_P)^2$, which in the GUT-scale example means $\xi\sim10^{-4}$ and ties the baryogenesis condition to $T_{B-L}^2/T_{\rm reh}\sim Y_B m_P$.
If the reheating temperature is too low for the field to freeze (violating Eq. (36)), the axion can still become dark matter by switching from rotation to coherent oscillations once its mass catches up with the Hubble rate.

Reading between the lines

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

The paper leaves the running of $\xi$ as a loose end; a concrete UV model that fixes $\beta$ and $\mu$ in Eq. (68) would either close the gap between the rolling bound and the Kibble bound or reveal that the rotation window is empty.
Because the sign of the rotation is set by quantum diffusion during inflation, the mechanism predicts a single, coherent rotation direction inside our horizon; a dedicated calculation of the induced isocurvature power spectrum could turn that prediction into a CMB polarization test.
The fragmentation estimates neglect backreaction; a lattice simulation at $\xi\sim(f/m_P)^2$ would show whether the baryon yield in Eq. (26) needs an order-one efficiency correction, which would shift the inferred right-handed neutrino mass scale.
Nothing in the mechanism binds the axion mass to the baryon asymmetry; applying the same flipped-potential kick to other pNGBs, such as a QCD-axion-like state whose potential appears at a late phase transition, could open a new route to kinetic misalignment, though the paper only gestures at this extension.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the core idea is good and the authors are honest about many of its limitations, but the central parameter window is not yet demonstrated. The genuinely new ingredient is a periodic non-minimal coupling xi[1 - cos(phi/f)] that flips the axion effective potential when the Universe transitions from inflation to kination, so the axion can roll over a shrinking barrier and begin rotating without explicit U(1)-breaking operators. Applying that rotation to spontaneous baryogenesis plus late-time axion DM is a real combination, and the Majoron seesaw example makes it concrete. The GW section is standard but useful: kination makes the inflationary GW background blue-tilted, and the BBN bound on Delta Neff gives Treh greater than about 10^7 GeV, which is a sharp, testable constraint. They also cite the relevant prior work on the flip and on periodic non-minimal coupling, and they do not hide the fragmentation and Kibble issues. The fragmentation analysis is approximate but gives a sensible upper bound, xi less than about O(100)(f/mP)^2.

The elephant in the room is the disjoint xi windows. Rotation needs xi > 3/4 (f/mP)^2 (Eq. 9); avoiding the Kibble problem needs xi < 1/4 (f/mP)^2 (Eq. 67). The paper sees this and invokes a running xi(sigma), but there is no concrete model and no numerical evolution with a time-varying xi. The plots in Fig. 2 use constant xi at or above (f/mP)^2, which sits above the Kibble-safe bound, and the baryon yield in Eq. (26) is derived with the constant-large-xi estimate for the maximum velocity, so it is not established in the running-coupling regime. The statement that xi needs to grow by a couple of orders of magnitude is also overstated: from the upper bound 1/4 to the lower bound 3/4 is only a factor of three. None of this is a category error; it is a missing piece of model-building. The paper would be much stronger with a concrete beta, mu, and a simulation showing rotation, baryon yield, and DM abundance at one consistent parameter point.

I also think the 'mass predictions' are really constraints after imposing the observed YB and Omega_DM, not independent predictions. That is standard model-building and the paper mostly presents them that way, but the abstract's wording is a bit strong. The circularity concern flagged by the reader is not the real problem.

Bottom line: this paper deserves a serious referee. The idea is novel enough, the literature handling is careful, and the GW and neutrino tests give it bite. I would send it to review, but the referee should push for a concrete running-xi model and numerical verification. As it stands the mechanism is conditional, not established.

Referee Report

3 major / 5 minor

Summary. The paper proposes a cogenesis mechanism in which an axion-like particle non-minimally coupled to gravity acquires a rotating expectation value when the Ricci scalar flips sign at the transition from inflation (w = -1) to kination (w = 1). The effective potential changes phase, giving the axion a kinetic kick, and the decreasing potential barrier in kination sustains the rotation. The rotating axion generates the baryon asymmetry via spontaneous baryogenesis (Eq. (26)) and later freezes and oscillates as dark matter when the bare mass potential dominates (Eqs. (32)-(35)). The authors derive constraints on the non-minimal coupling ξ and reheating temperature from gravitational waves, axion fragmentation, and the Kibble mechanism, and apply the setup to the Majoron in a Type-I seesaw model, finding Majoron masses below sub-eV and right-handed neutrino masses above roughly 10^8 GeV.

Significance. If the mechanism is realized, it offers a novel way to source axion rotation without explicit U(1) breaking operators, potentially achieving both baryogenesis and dark matter from a single spectator field, with testable gravitational-wave spectra and neutrino-mass relations. The paper contains useful analytic estimates for the baryon yield, dark matter scale, and gravitational-wave spectrum, and it includes numerical background solutions for rotation and fluctuations. However, the central viability is currently weakened by the unresolved tension between the rolling condition and the Kibble bound, and by the unquantified running coupling invoked to bridge them; the claimed parameter space therefore lacks a demonstrated point where all constraints are simultaneously satisfied.

major comments (3)

[Section VIII, Eqs. (9), (67), (68)] The constant-ξ windows in Eqs. (9) and (67) are disjoint: Eq. (9) requires ξ > (3/4)(f/mP)^2 for rolling, while Eq. (67) requires ξ < (1/4)(f/mP)^2 to avoid the Kibble problem. The rescuing running coupling ξ(σ) of Eq. (68) is only an ansatz; the paper gives no concrete model or numerical demonstration that ξ rises by the required factor of a few during kination while preserving the assumptions used to derive the rotation, the baryon yield, and the dark matter scale. The text near Eq. (69) claiming ξ ∼ (f/mP)^2 is 'near the edge of both ranges' is inaccurate, since (f/mP)^2 is a factor of 4 above the Eq. (67) upper bound.
[Section III, Eq. (26) and Section VIII text] The baryon yield Eq. (26) uses θ̇_m from Eq. (21), derived for ξ ≫ (f/mP)^2. In the parameter space judged viable in Section VIII (ξ ∼ (f/mP)^2, Eq. (69)), the paper states that Eq. (21) is unreliable and appeals to the average θ̇ remaining close to the barrier height; however, no quantitative estimate or simulation of this average for ξ ∼ (f/mP)^2 is given. Fig. 2 shows that θ̇ oscillates, and the average may differ from Eq. (21) by an O(1) or larger factor, which directly enters the central YB prediction through Eq. (22).
[Section II, Fig. 2] The numerical demonstration of rotation in Fig. 2 is performed for constant ξ ≥ (f/mP)^2, i.e., values that violate the Kibble-safe bound of Eq. (67). The paper does not present a numerical evolution with the running ξ(σ) of Eq. (68), so the sustained rotation in the argued consistent parameter space is not demonstrated. A benchmark simulation with ξ starting below (1/4)(f/mP)^2 and growing to above (3/4)(f/mP)^2 during kination would directly address this gap.

minor comments (5)

[Eq. (22)] The phrase 'entropy energy density' should read 'entropy density'.
[Section VII, after Eq. (60)] The phrase 'compared to blue the bound' should read 'compared to the bound'.
[Abstract and throughout] The spelling of 'co-genesis' in the abstract and 'cogenesis' elsewhere should be harmonised.
[Section VIII, after Eq. (66)] The statement that the lower bound in Eq. (9) is 'mildly violated' is misleading; the upper bound of Eq. (67) is a factor of 3 below the lower bound of Eq. (9), which is not a mild violation.
[Figs. 2 and 6] The numerical setup is not fully specified; a short paragraph describing the time-stepping, initial conditions, and any back-reaction treatment would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: rotation, Y_B, and Omega_DM are computed from explicit dynamics, with observed values used as constraints; the running-xi bridge between Eqs. (9) and (67) is an acknowledged modeling gap, not a circular step.

full rationale

The paper's derivation chain is self-contained. The periodic non-minimal coupling is an explicit model input (Eq. (2)), and the potential flip (Eqs. (6)-(7)) follows from R = 3(1 - 3w)H^2 with w = -1 to w = +1, not from the baryon or dark-matter abundances. Rotation is established by numerically solving Eq. (18) with the quantum-diffusion initial condition (19); Fig. 2 shows barrier crossing for xi = (f/m_P)^2 and larger, so the central dynamical claim is not assumed as an output. The baryon yield (26) is derived from the standard spontaneous-baryogenesis formula using the rotation velocity and its redshift; the observed Y_B is then used to constrain T_{B-L}^2/T_reh (Eq. (28)), which is normal model-building, not a fitted-input-as-prediction. The dark-matter relation (35) follows from matching the rotating-axion energy density to the observed dark-matter abundance at equality, again a constraint rather than a circular definition. Self-citations (e.g., Refs. [93], [116]) are pointers to earlier Ricci-reheating and gravitational-wave calculations whose relevant results are reproduced or stated in the text; they are not load-bearing. The paper itself flags the unresolved issues: to connect the Kibble-safe bound xi < (1/4)(f/m_P)^2 (Eq. (67)) with the rolling bound xi > (3/4)(f/m_P)^2 (Eq. (9)) it introduces a running xi(sigma) and says 'changing xi by an order of magnitude seems quite realistic' (Sec. VIII), while also stating 'We leave a detailed numerical simulation for future studies'; in the low-xi regime it admits 'our estimate of thetadot_m given by Eq. (21) is unreliable' with only order-of-magnitude validity claimed. These are honestly acknowledged modeling and viability gaps, not input-output equivalence. No step reduces a 'prediction' to its own input by construction; hence no significant circularity.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard FRW cosmology plus several model choices: the periodic non-minimal coupling is treated as an effective potential, the axion is a spectator, M is negligible during kination, a thermal bath exists during kination, B-L interactions are in equilibrium until TB-L, and an unquantified running xi(sigma) reconciles disjoint bounds. The free parameters xi, f, M, Treh, TB-L (or MN), cB, zfo and the running-coupling constants are not measured elsewhere.

free parameters (8)

xi, non-minimal coupling strength = ~ (f/mP)^2, e.g. 1e-4 for f ~ 1e-2 mP
Chosen inside the paper's allowed ranges Eqs. (9), (60)-(67); it sets the rotation kick and the baryon yield.
f, axion decay constant = >= 1e11 GeV; benchmark f ~ 1e-2 mP
Free scale with a lower bound from symmetry restoration, Eq. (16); the dark matter relation Eq. (35) ties M to f.
M, bare symmetry-breaking scale (axion mass m_phi = M^2/f) = M ~ 1e-9 GeV (mP/f)^{3/2}
Fixed by requiring the axion to supply all dark matter at matter-radiation equality, Eqs. (34)-(35).
Treh, reheating temperature = >~ 2.2e7 GeV from Planck DeltaNeff bound
Free parameter with a lower bound from gravitational wave overproduction, Eq. (49); enters the baryon yield and the GW frequency.
TB-L, decoupling temperature of B-L interactions (or MN/zfo) = Constrained by observed YB; benchmark TB-L ~ 8e7 GeV
Inferred from the observed baryon asymmetry via Eqs. (26)-(28), not independently measured.
cB, O(1) transport coefficient = O(1)
Unspecified coefficient in the spontaneous baryogenesis yield, Eq. (22).
zfo, freeze-out parameter for RHN inverse decays = O(10)
Used to translate MN to TB-L in Sec. VI; not computed for this non-standard cosmology.
xi0, beta, mu in the running coupling xi(sigma) = Not specified
Ad hoc running invoked to reconcile the disjoint xi ranges; no concrete values or UV model are given.

assumptions (8)

standard math FRW background with radiation, matter and kination, and R = 3(1-3w)H^2.
Used throughout, e.g. in Eqs. (6)-(7), to compute the sign flip of the effective potential.
domain assumption The periodic non-minimal coupling gamma^2 = 1 + xi(1-cos(phi/f)) preserves the discrete shift symmetry and can be treated as a purely effective potential term.
Central to the flip mechanism; stated in Sec. II and used in the equation of motion, Eq. (18).
domain assumption The axion remains a spectator with subdominant energy density during rotation, rho_phi/rho = 2xi << 1.
Used in Eqs. (8) and (30) to justify ignoring backreaction on the background.
domain assumption The bare mass term M^4 is negligible during inflation and kination compared with the xi-dependent term.
M is dropped in Eqs. (6)-(7), which is required for the flip to be controlled by xi.
domain assumption A thermal bath is present during kination with a maximum temperature bounded by Tmax ~ (sqrt(lambda)/g) f, and the U(1) symmetry remains broken.
Required for the axion to exist and for spontaneous baryogenesis to operate; Eqs. (10)-(16).
domain assumption B-L violating inverse decays are in equilibrium down to TB-L and act as a wash-in source, with no other significant source or washout of the asymmetry.
Assumed in Sec. III and VI; vanilla thermal leptogenesis is deliberately neglected.
ad hoc to paper The running non-minimal coupling xi(sigma) in Eq. (68) can increase xi by a couple of orders of magnitude during kination without spoiling inflation or the rotation.
Invoked in Sec. VIII to reconcile Eq. (9) with Eq. (67); no concrete realization with explicit beta and mu is provided.
ad hoc to paper Quantum diffusion during inflation gives the mean-squared displacement in Eq. (66) and homogenizes the direction of rotation within our observable universe.
Used to evade the Kibble problem in Sec. VIII; this relies on a stochastic averaging assumption.

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Cite this review

Pith. "Pith review of Flipped Rotating Axion Non-minimally Coupled to Gravity: Baryogenesis and Dark Matter." pith.science (2026). https://pith.science/paper/MVGZWKMW

@misc{pith2026250208720,
  author       = {Pith},
  title        = {Pith review of: Flipped Rotating Axion Non-minimally Coupled to Gravity: Baryogenesis and Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MVGZWKMW}},
  note         = {Machine review of arXiv:2502.08720}
}

abstract

We demonstrate that the co-genesis of baryon asymmetry and dark matter can be achieved through the rotation of an axion-like particle, driven by a flip in the vacuum manifold's direction at the end of inflation. This can occur if the axion has a periodic non-minimal coupling to gravity, while preserving the discrete shift symmetry. In non-oscillating inflation models, after inflation there is typically a period of kination (with $w = 1$). In this case, it is shown that the vacuum manifold of the axion is flipped and the axion begins rotating in field space, because it can slide across the decreasing potential barrier as in Ricci reheating. Such a rotating axion can generate the baryon asymmetry of the Universe through spontaneous baryogenesis, while at later epochs it can oscillate as dark matter. The period of kination makes the primordial gravitational waves (GW) generated during inflation sharply blue-tilted which constrains the parameter space due to GW overproduction, while being testable by next generation CMB experiments. As a concrete example, we show that such a cogenesis of baryon asymmetry and dark matter can be realized for the axion as the Majoron in the Type-I seesaw setup, predicting mass ranges for the Majoron below sub eVs, with right-handed neutrino mass above $\mathcal{O}(10^{8})$ GeV. We also show that in order to avoid fragmentation of the axion condensate during the rotation, we require the non-minimal coupling $\xi \sim (f/m_P)^2 $ or somewhat larger, where $f$ is the axion decay constant.

Figures

Figures reproduced from arXiv: 2502.08720 by the authors.

**Figure 2.** Left panel: The rotation of the axion (θ), with the red dashed lines indicating the potential barriers at −π, π, 3π, 5π, ... . The different colors denote different values of ξ, with rotation taking place even for ξ = (f /mP ) 2 . Right panel: Evolution of ˙θ, H, p ||Veff ||/f from the end of inflation until reheating. The black line denotes the Hubble parameter H, while the colored lines indicate ˙θ and p ||Veff ||… view at source ↗

**Figure 3.** Schematic showing the evolution of the energy densities of the different components of the Universe [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The current GW spectrum (blue), choosing values of reheating temperature [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Predictions for cogenesis of dark matter and baryon asymmetry in the Type I seesaw setup with [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Left panel: Evolution of the axion fluctuations for different values of ξ, exceeding f 2 /m2 P . Substantial growth can be seen with the increase in the value of ξ. Right panel: Evolution of the instability band along with the physical momentum (see Eq. (57)). Note tha…

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