Gravitational Baryogenesis Mechanisms

• Gravitational baryogenesis is a mechanism where time-dependent spacetime curvature, through derivative couplings to the baryon current, induces an effective chemical potential in thermal equilibrium.
• The approach employs a canonical (∂μR)J_B^μ interaction (or its alternatives) to link cosmological dynamics with baryon asymmetry, with the final asymmetry fixed at the decoupling (freeze-out) temperature.
• Various models—including vacuum inflation, running vacuum, Gauss-Bonnet, and teleparallel formulations—address practical issues like the radiation-era obstruction and higher-derivative instabilities.

Gravitational baryogenesis denotes a class of baryon-asymmetry mechanisms in which the expanding spacetime itself supplies the bias between baryons and antibaryons. In its canonical form, the baryon current couples derivatively to a gravitational scalar—most often the Ricci scalar—so that a time-dependent background curvature induces an effective baryon chemical potential. If baryon-number-violating reactions remain active until a decoupling temperature $T_D$, a net baryon asymmetry is frozen into the plasma. The mechanism has been embedded in backgrounds ranging from vacuum inflation and cosmological-constant-dominated FRW cosmologies to anisotropic, braneworld, teleparallel, and higher-curvature models, but it also faces a major theoretical challenge: the standard $(\partial_\mu R)J_B^\mu$ coupling generically feeds back into the gravitational equations and can produce higher-order instabilities (Huang et al., 2017, Arbuzova et al., 2017).

1. Canonical interaction and thermodynamic implementation

The standard operator used in gravitational baryogenesis is

$$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$

or equivalently, in some conventions,

$${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$

Here $J^\mu$ is the baryon current, $R$ is the Ricci scalar, and $M_*$ or $m_0$ is an effective cutoff scale. In a homogeneous cosmological background only the temporal derivative survives, so the interaction acts as an effective chemical potential,

$\mu_B \sim \pm \frac{\dot R}{M_*^2},$

with opposite sign for baryons and antibaryons. The same logic is used in extensions where $R$ is replaced by $(\partial_\mu R)J_B^\mu$0, $(\partial_\mu R)J_B^\mu$1, $(\partial_\mu R)J_B^\mu$2, $(\partial_\mu R)J_B^\mu$3, $(\partial_\mu R)J_B^\mu$4, $(\partial_\mu R)J_B^\mu$5, or a model-dependent function $(\partial_\mu R)J_B^\mu$6 of such quantities (Srivastava et al., 2020, Agrawal et al., 2021, Odintsov et al., 2016).

In thermal equilibrium, the induced asymmetry is computed from the usual small-chemical-potential expansion. Several papers write

$(\partial_\mu R)J_B^\mu$7

while a more detailed expression is

$(\partial_\mu R)J_B^\mu$8

whose linear term dominates for $(\partial_\mu R)J_B^\mu$9. With entropy density

$$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$0

many treatments obtain the standard estimate

$$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$1

or closely related expressions with $$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$2 and slightly different temperature conventions. The asymmetry is therefore controlled by the geometric time derivative evaluated when baryon-violating processes decouple (Saaidi et al., 2010, Agrawal et al., 2021, Atazadeh, 2020).

A notable feature, emphasized repeatedly in the literature, is that this mechanism can operate in thermal equilibrium. The curvature-induced bias modifies the equilibrium number densities themselves, so the usual intuition that baryogenesis strictly requires conventional out-of-equilibrium dynamics is softened, although freeze-out remains essential because it fixes the final yield when baryon-number violation becomes inefficient (Srivastava et al., 2020, Arbuzova et al., 2017).

2. Background dependence, freeze-out, and inflationary realizations

The efficacy of gravitational baryogenesis is highly background dependent. In standard FRW cosmology the familiar obstruction is that for a perfect radiation fluid with $$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$3, one has $$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$4 and hence $$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$5; during matter domination $$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$6 is typically too small to generate a large asymmetry. This motivates embeddings in nonstandard early-universe backgrounds where curvature remains nonzero and time dependent (Srivastava et al., 2020, Fukushima et al., 2016).

A prominent example is vacuum inflation, where inflation is driven by a quantum potential rather than an inflaton field. In that setting the matter/radiation density produced by Hawking radiation obeys

$$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$7

with

$$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$8

The curvature is

$$\frac{1}{M_*^2}\int d^4x\,\sqrt{-g}\,(\partial_\mu R)J^\mu,$$9

and the observationally favored value $${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$0 implies $${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$1 during inflation. Because entropy continues to increase until the end of inflation, the generated asymmetry is later diluted; the paper finds a numerical dilution factor of approximately $${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$2, leading to

$${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$3

and, for the reduced Planck scale choice of $${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$4, obtains

$${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$5

close to the quoted observed value $${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$6 (Huang et al., 2017).

A distinct FRW realization uses an exact solution with cosmological constant $${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$7,

$${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$8

which interpolates from decelerated to accelerated expansion. Its Ricci derivative is

$${\cal L}_{GBG}=\frac{f}{m_0^2}(\partial_\mu R)J_B^\mu.$$9

This immediately shows that $J^\mu$0 for $J^\mu$1 and $J^\mu$2, so neither pure radiation nor pure de Sitter generates asymmetry in that model. For the benchmark choice

$J^\mu$3

the derived $J^\mu$4 is reported to agree with the quoted observational level $J^\mu$5 over an appropriate time range (Srivastava et al., 2020).

A third inflationary construction attributes the departure from $J^\mu$6 to gravitationally induced particle production. In that scenario

$J^\mu$7

and the Ricci scalar becomes

$J^\mu$8

with

$J^\mu$9

The source vanishes both at the exact de Sitter start and in the late radiation limit, so baryogenesis is localized in the transition epoch. For $R$0, $R$1, and $R$2, the paper finds $R$3 and infers an inflationary scale $R$4 from the observed asymmetry (Lima et al., 2016).

3. The radiation-era obstruction and mechanisms that evade it

The central phenomenological question is how to maintain a nonzero source when the background is close to conformal radiation. A large fraction of the literature can be read as a sequence of attempts to evade the standard GR result $R$5 for $R$6.

Several distinct strategies recur. Some change the geometric invariant in the baryogenesis operator, replacing $R$7 by $R$8, torsional scalars, or boundary terms. Others keep the Ricci-scalar operator but modify the cosmological dynamics so that $R$9 or $M_*$0 remains nonzero even for radiation-like matter. A further strategy uses anisotropy or particle-creation effects to introduce additional sources of curvature evolution (Odintsov et al., 2016, Oikonomou et al., 2016, Atazadeh, 2020, Saaidi et al., 2010).

Framework	Effective source	Radiation/conformal behavior
Standard GR GBG	$M_*$1	vanishes for $M_*$2
Gauss-Bonnet GBG	$M_*$3	generically nonzero in radiation
Running vacuum models	$M_*$4 for $M_*$5	nonzero if $M_*$6 runs
DGP brane cosmology	brane-modified $M_*$7	nonzero for $M_*$8
Anisotropic Bianchi I	shear-dependent $M_*$9	anisotropy can sustain a source

In Bianchi I cosmology the Ricci scalar can be written as

$m_0$0

so shear enters explicitly. The corresponding $m_0$1 then depends on $m_0$2, $m_0$3, $m_0$4, $m_0$5, and $m_0$6. This means that anisotropy can enhance the baryon asymmetry and, in the simplest Bianchi-I analysis, can keep the source nonzero in situations where the isotropic contribution would be suppressed (Saaidi et al., 2010). However, when anisotropy is generated by anisotropic inflation, the enhancement is constrained by the CMB bound

$m_0$7

and by the gravitino bound $m_0$8. The resulting conclusion is that anisotropy does enhance $m_0$9, but generally not enough to explain the observed asymmetry without fine tuning (Fukushima et al., 2016).

Running vacuum models evade the conformal obstruction by promoting the vacuum energy to $\mu_B \sim \pm \frac{\dot R}{M_*^2},$0. For $\mu_B \sim \pm \frac{\dot R}{M_*^2},$1,

$\mu_B \sim \pm \frac{\dot R}{M_*^2},$2

so gravitational baryogenesis becomes directly sensitive to the running vacuum sector. Two benchmark running-vacuum models yield

$\mu_B \sim \pm \frac{\dot R}{M_*^2},$3

and

$\mu_B \sim \pm \frac{\dot R}{M_*^2},$4

respectively, for representative choices of $\mu_B \sim \pm \frac{\dot R}{M_*^2},$5, $\mu_B \sim \pm \frac{\dot R}{M_*^2},$6, and the running parameters $\mu_B \sim \pm \frac{\dot R}{M_*^2},$7 or $\mu_B \sim \pm \frac{\dot R}{M_*^2},$8 (Oikonomou et al., 2016). DGP brane cosmology achieves a similar effect through braneworld corrections to the Friedmann equation. In the $\mu_B \sim \pm \frac{\dot R}{M_*^2},$9 branch, the paper reports

$R$0

or

$R$1

for $R$2 on two decoupling branches, while standard GR would give zero (Atazadeh, 2020).

Replacing $R$3 by the Gauss-Bonnet invariant $R$4 is the most direct geometric workaround. Since

$R$5

one generally has $R$6 even during radiation domination. In $R$7 gravity with

$R$8

the baryon asymmetry scales as $R$9, and for

$(\partial_\mu R)J_B^\mu$00

the bound $(\partial_\mu R)J_B^\mu$01 implies

$(\partial_\mu R)J_B^\mu$02

The same paper notes that derivative couplings to $(\partial_\mu R)J_B^\mu$03 or $(\partial_\mu R)J_B^\mu$04 can also remain active in the conformal limit (Odintsov et al., 2016).

4. Modified-gravity generalizations

Beyond these background modifications, the literature generalizes gravitational baryogenesis by replacing the Ricci scalar with alternative gravitational or matter-geometry quantities, or by coupling the baryon current to $(\partial_\mu R)J_B^\mu$05 rather than directly to $(\partial_\mu R)J_B^\mu$06. The result is a large model space in which the freeze-out formula is structurally similar, but the source term is computed from the modified dynamics (Sahoo et al., 2019, Oikonomou et al., 2016).

In curvature-based modified gravity, a representative example is nonminimal $(\partial_\mu R)J_B^\mu$07 theory with

$(\partial_\mu R)J_B^\mu$08

In that framework the paper studies three derivative couplings: $(\partial_\mu R)J_B^\mu$09, $(\partial_\mu R)J_B^\mu$10, and $(\partial_\mu R)J_B^\mu$11. For the scale factor $(\partial_\mu R)J_B^\mu$12, it finds that the $(\partial_\mu R)J_B^\mu$13 and $(\partial_\mu R)J_B^\mu$14 cases can remain nonzero even in a radiation-dominated universe and can yield

$(\partial_\mu R)J_B^\mu$15

for $(\partial_\mu R)J_B^\mu$16, $(\partial_\mu R)J_B^\mu$17, and $(\partial_\mu R)J_B^\mu$18, whereas the $(\partial_\mu R)J_B^\mu$19 coupling is either zero or phenomenologically unacceptable in the same model (Sahoo et al., 2019). In anisotropic $(\partial_\mu R)J_B^\mu$20 gravity on a Bianchi-I background, the asymmetry is enhanced by anisotropy; Model I rises roughly linearly with the parameter $(\partial_\mu R)J_B^\mu$21, whereas Model II begins at a higher positive value and then decreases or stabilizes with $(\partial_\mu R)J_B^\mu$22, both using the target level $(\partial_\mu R)J_B^\mu$23 as the phenomenological benchmark (Agrawal et al., 2021).

Teleparallel generalizations replace curvature by torsion. In $(\partial_\mu R)J_B^\mu$24 gravity,

$(\partial_\mu R)J_B^\mu$25

and one studies either $(\partial_\mu R)J_B^\mu$26 or $(\partial_\mu R)J_B^\mu$27. The TEGR limit performs poorly, with

$(\partial_\mu R)J_B^\mu$28

for a benchmark radiation-era choice, but power-law models such as

$(\partial_\mu R)J_B^\mu$29

can reach

$(\partial_\mu R)J_B^\mu$30

while a generalized $(\partial_\mu R)J_B^\mu$31 coupling can give

$(\partial_\mu R)J_B^\mu$32

for suitable parameter choices (Oikonomou et al., 2016). Further teleparallel extensions include $(\partial_\mu R)J_B^\mu$33 gravity, where parameter sets reproducing

$(\partial_\mu R)J_B^\mu$34

are reported, albeit sometimes with very large or very small couplings (Mishra et al., 2023), and $(\partial_\mu R)J_B^\mu$35 or $(\partial_\mu R)J_B^\mu$36 models, where generalized couplings to $(\partial_\mu R)J_B^\mu$37, $(\partial_\mu R)J_B^\mu$38, or $(\partial_\mu R)J_B^\mu$39 are compared directly with the benchmark $(\partial_\mu R)J_B^\mu$40 (Azhar et al., 2020).

Matter-geometry extensions broaden the source space further. In energy-momentum squared gravity,

$(\partial_\mu R)J_B^\mu$41

the operator can involve $(\partial_\mu R)J_B^\mu$42 or $(\partial_\mu R)J_B^\mu$43. The $(\partial_\mu R)J_B^\mu$44 case is identified as the most promising; the $(\partial_\mu R)J_B^\mu$45 case has $(\partial_\mu R)J_B^\mu$46 in radiation domination, but $(\partial_\mu R)J_B^\mu$47 can still generate asymmetry. The same paper also notes that some parameter choices imply implausibly low baryon-violating scales, such as $(\partial_\mu R)J_B^\mu$48 GeV in one $(\partial_\mu R)J_B^\mu$49 example, which would already have been observed (Pereira et al., 2024).

Quantum-gravity-inspired and vector-tensor realizations have also been explored. In Hořava–Lifshitz gravity, modified FRW evolution leads to viable baryogenesis for quintessence-like and matter-dominated fluids, while radiation still gives zero asymmetry in the flat case (Maity et al., 2018). In Extended Proca-Nuevo gravity, where a massive vector field couples nonminimally to curvature, three cosmological histories—power-law, exponential, and modified exponential—are analyzed, and all are reported to yield asymmetry consistent with the benchmark $(\partial_\mu R)J_B^\mu$50 for suitable ranges of the interaction scale $(\partial_\mu R)J_B^\mu$51 (Sultan, 8 Apr 2025).

5. Instability, stabilization, and alternative formulations

The most serious criticism of standard gravitational baryogenesis is that the same derivative-curvature coupling that generates the chemical potential also modifies the gravitational field equations. Because the interaction contains derivatives of $(\partial_\mu R)J_B^\mu$52, varying the action with respect to the metric produces higher-order terms, so the usual algebraic trace relation of GR is replaced by a differential equation for the curvature scalar itself (Arbuzova et al., 2017, Arbuzova et al., 2023).

In the critical analysis of the standard $(\partial_\mu R)J_B^\mu$53 operator, the trace equation becomes fourth order. In a bosonic model, after thermal averaging and neglecting subleading terms, one obtains

$(\partial_\mu R)J_B^\mu$54

or, in the $(\partial_\mu R)J_B^\mu$55-stabilized formulation,

$(\partial_\mu R)J_B^\mu$56

The homogeneous solutions have exponentially growing modes, $(\partial_\mu R)J_B^\mu$57, with positive real parts for some roots. The conclusion is that the instability can develop on timescales much shorter than the Hubble time and can destroy standard cosmology unless additional physics regulates it (Arbuzova et al., 2017).

A proposed cure is to add a Starobinsky-type term,

$(\partial_\mu R)J_B^\mu$58

which introduces a scalaron mass $(\partial_\mu R)J_B^\mu$59. In that case the stability condition becomes

$(\partial_\mu R)J_B^\mu$60

For $(\partial_\mu R)J_B^\mu$61, one estimate given is

$(\partial_\mu R)J_B^\mu$62

This stabilizes the runaway in both bosonic and fermionic realizations, but the paper emphasizes that the resulting cosmology is still noticeably modified and the stabilized curvature amplitude remains much larger than in ordinary Friedmann evolution (Arbuzova et al., 2023).

Two important alternative directions attempt to sidestep these problems rather than repair them. One is “gravitational baryogenesis without CPT violation,” based on nonminimal curvature-matter couplings rather than a direct $(\partial_\mu R)J_B^\mu$63 term. There the baryon current obeys

$(\partial_\mu R)J_B^\mu$64

CP is violated by an $(\partial_\mu R)J_B^\mu$65 term, and out-of-equilibrium evolution is supplied by gravitational particle creation. The mechanism can generate $(\partial_\mu R)J_B^\mu$66 for

$(\partial_\mu R)J_B^\mu$67

while explicitly avoiding higher derivatives of $(\partial_\mu R)J_B^\mu$68 in the baryogenesis operator (Antunes et al., 2019).

The second is a recent unified framework in which the chemical potential is tied to entropy production rather than to $(\partial_\mu R)J_B^\mu$69,

$(\partial_\mu R)J_B^\mu$70

and CP violation is supplied by a gravitational theta term,

$(\partial_\mu R)J_B^\mu$71

The formalism highlights adiabatic cancellation of oscillatory sources through a universal low-pass factor

$(\partial_\mu R)J_B^\mu$72

and derives

$(\partial_\mu R)J_B^\mu$73

A dilaton UV completion with $(\partial_\mu R)J_B^\mu$74 is proposed, and the same framework predicts stochastic-gravitational-wave circular polarization

$(\partial_\mu R)J_B^\mu$75

for small $(\partial_\mu R)J_B^\mu$76 in a Loop Quantum Cosmology bounce (Mandel, 16 Jan 2026).

6. Phenomenology, parameter dependence, and present status

Across the literature, the target baryon asymmetry is consistently of order $(\partial_\mu R)J_B^\mu$77, but specific papers quote somewhat different observational reference values: $(\partial_\mu R)J_B^\mu$78 This spread reflects differing conventions and observational inputs rather than a substantive disagreement about the required scale (Huang et al., 2017, Srivastava et al., 2020, Agrawal et al., 2021, Mishra et al., 2023).

The parametric structure is simple in most models. Larger $(\partial_\mu R)J_B^\mu$79 or its generalized analog enhances the asymmetry, larger $(\partial_\mu R)J_B^\mu$80 suppresses it, and the decoupling temperature enters through the freeze-out relation. Some backgrounds add further dilution or enhancement factors. In vacuum inflation, entropy continues to grow and the frozen asymmetry is diluted by a factor $(\partial_\mu R)J_B^\mu$81 before the end of inflation (Huang et al., 2017). In anisotropic models, shear enhances $(\partial_\mu R)J_B^\mu$82, but CMB and reheating constraints severely limit how much that enhancement can matter in practice (Fukushima et al., 2016). In running-vacuum, DGP, and Gauss-Bonnet settings, the key advantage is precisely that the source does not vanish for radiation-like matter (Oikonomou et al., 2016, Atazadeh, 2020, Odintsov et al., 2016).

The phenomenological status is correspondingly mixed. Many constructions can be tuned to produce the correct order of magnitude: vacuum inflation, running vacuum models, DGP braneworld cosmology, several $(\partial_\mu R)J_B^\mu$83, $(\partial_\mu R)J_B^\mu$84, $(\partial_\mu R)J_B^\mu$85, $(\partial_\mu R)J_B^\mu$86, and EPN scenarios all report acceptable benchmark values (Huang et al., 2017, Sahoo et al., 2019, Oikonomou et al., 2016, Azhar et al., 2020, Sultan, 8 Apr 2025). Other cases are disfavored because the sign is wrong, the asymmetry is far too small, or the required parameters are implausible. Examples include the TEGR benchmark with $(\partial_\mu R)J_B^\mu$87, the $(\partial_\mu R)J_B^\mu$88 coupling in the specific $(\partial_\mu R)J_B^\mu$89 model, the $(\partial_\mu R)J_B^\mu$90 energy-momentum-squared example requiring $(\partial_\mu R)J_B^\mu$91 GeV, and several $(\partial_\mu R)J_B^\mu$92 fits that use extreme parameter magnitudes (Oikonomou et al., 2016, Sahoo et al., 2019, Pereira et al., 2024, Mishra et al., 2023).

The current research landscape therefore supports two simultaneous conclusions. First, gravitational baryogenesis remains a productive effective framework for linking baryon asymmetry to background geometry, and modified gravity can provide many mechanisms for keeping the source active when standard GR would suppress it. Second, the canonical $(\partial_\mu R)J_B^\mu$93 formulation is not automatically consistent as a dynamical theory of gravity, and a credible model must address higher-derivative backreaction, cosmological stability, and the plausibility of the required parameter region. Recent work has accordingly shifted part of the emphasis from merely obtaining $(\partial_\mu R)J_B^\mu$94 to formulating versions that evade adiabatic cancellation, survive stability tests, and offer external observables such as stochastic-gravitational-wave polarization (Arbuzova et al., 2017, Arbuzova et al., 2023, Mandel, 16 Jan 2026).