Primordial Magnetic Fields, Right Electrons, and the Abelian Anomaly

M. Joyce; M. Shaposhnikov

REVIEW

Primordial Magnetic Fields, Right Electrons, and the Abelian Anomaly

M. Joyce, M. Shaposhnikov

Astrophysics · Submitted 28 Feb 1997 · 18 references

Pith's verdict: A correct and important mechanism paper: the linear instability is clean and checkable, but the headline numbers rest on an unshown numerical integration whose homogeneous-μ assumption is acknowledged and never quantified.

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arXiv IDastro-ph/9703005 (version 2)

Pith IDZDSBVBP2, Pith's permanent record of this paper and its review

CitationM. Joyce, M. Shaposhnikov. "Primordial Magnetic Fields, Right Electrons, and the Abelian Anomaly." arXiv:astro-ph/9703005 (1997). https://pith.science/paper/ZDSBVBP2

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Editor's note

read the letter

The thing to know up front: this is the paper that put the chiral magnetic instability into cosmology, and the core physics is right. Above T_R ~ 80 TeV the right-electron Yukawa is out of equilibrium, so e_R number is effectively conserved, and a finite μ_R makes the hypercharge field unstable. The derivation is short enough to check by hand: anomaly equation plus Ohm's law gives Eqs. (7)-(9), an unstable circularly polarized mode growing for k < 4μa. That part is solid and has held up in the later literature. What is genuinely new is the cosmological realization, not the anomalous physics itself — the cold-matter analogue is Rubakov-Tavkhelidze [9], properly cited. Identifying e_R as the relevant charge above T_R is the real insight, and it is a good one. Soft spots, in proportion. The nonlinear numbers — the Eq. (13) scaling, the δ > 2×10^-4 survival threshold, H ~ 4×10^22 G — come from numerically integrating Eq. (12), and no figure, scheme, or convergence check is shown. For a letter that is forgivable, but it means those numbers are order-of-magnitude claims you cannot verify from the text. The homogeneous-μ assumption stated after Eq. (12) is acknowledged and is a reasonable first approximation, since local backreaction is second order in the field; the authors do not estimate the error, and the worry that local depletion could shift the threshold is fair as a caveat, not a refutation. One thing the stress test gets wrong: it says the abstract's 10^22 G is for δ = 10^-2, which it claims corresponds to δ_R ~ 2×10^-7, far below the GUT range. That is an arithmetic slip. From the paper's own relation, μ/T ≈ 3.3 δ_R, so δ = 10^-2 is δ_R ≈ 3×10^-3, squarely inside the GUT range the abstract mentions. The abstract's juxtaposition is fine. The B-L bound evasion is one qualitative paragraph with no calculation, and the turbulence discussion leans on cited work the authors themselves call suggestive. Both are flagged as such. Who it is for: anyone working on primordial magnetogenesis or anomalous transport in the early universe. It is a foundational reference. Recommendation: send it out. It deserves a referee, and the referee should ask for the numerical details behind Eq. (13).

Referee report

Significance

If correct, the result identifies a magnetogenesis channel that requires no new physics, no conformal-symmetry breaking introduced by hand, and no free parameters beyond the right-electron asymmetry itself: the anomaly coefficient, the hypercharge of e_R, and the conductivity are all standard-model inputs. That is an attractive and economical structure, and it links three otherwise unrelated items — the smallness of the electron Yukawa coupling, a possible lepton asymmetry, and primordial magnetic seeds — in a way that is falsifiable in principle. I checked the linear sector by hand: the growth exponent (9), the condition k < 4μa, and the numerical value δ_crit ≈ 10^-6 following from Eq. (10) with σ ≈ 68T and M_0 ≈ 7.1×10^17 GeV all reproduce, and the quoted 6×10^26 gauss at T_R for δ ≈ 5×10^-6 is consistent with the parametric form H ~ 10^2 √ε δ T^2. The paper is also commendably explicit about which numbers come from the linear approximation and which from the nonlinear integration. The weaknesses are in the framing of the headline number and in the unquantified spatial averaging used for the nonlinear results, not in the underlying mechanism.

Major objections (3)

[Abstract (s2) and the paragraph following Eq. (11)] The abstract's headline number and the asymmetry required to produce it are not reconciled anywhere in the text. Using the paper's own equilibrium relation μ_R = (245π² N_eff)[78/388 δ_R − 20/188 δ_1 + 15/22(δ_2+δ_3)]T with N_eff = 106.75, one finds δ ≡ μ/T ≈ 5×10^4 δ_R, so the value δ = 10^-2 quoted for H ≃ 4×10^22 gauss corresponds to δ_R ≈ 2×10^-7. That is two to five orders of magnitude below the GUT-motivated range δ_R ~ 10^-2–10^-4 invoked in the third paragraph and echoed in the abstract's phrase 'possibly arising due to asymmetries produced at the GUT scale'. For δ_R in the GUT range, μ/T is of order 10^2–10^6, i.e. deep in the degenerate regime, where the linear-in-μ effective term (3) and the thermal-equilibrium relation for μ_R quoted above are both outside the regime the authors say they assume ('assumed only small enough to be treated perturbatively'). The abstract should state the δ_R (or μ/T) to which the 10^22 gauss figure applies, and the text should say explicitly what happens for δ_R ~ 10^-4–10^-2, even if only to note that the calculation does not apply there.
[Eq. (12) and the sentence immediately following it] The nonlinear results — the scaling laws (13), the survival threshold δ > 2×10^-4, and the amplitudes quoted at the electroweak scale — all rest on the stated 'simplifying assumption that the distribution of right electron number is homogeneous in space'. The authors acknowledge the assumption but give no estimate of its accuracy. This matters because the anomaly equation (1) is local: the Chern-Simons density, and hence the local depletion of μ_R, is sourced by E·H and therefore varies on the same scale as the field, which the instability concentrates at k ~ 2μ. A volume-averaged μ_R cannot capture the shutting-off of growth in the regions where the field is strongest, which is precisely the backreaction that terminates the instability. Since the threshold δ > 2×10^-4 and the final amplitudes are the paper's central quantitative claims, at least a bound on the size of the effect is needed — for example a spatially resolved one-dimensional integration of Eqs. (4)-(6), or an argument showing that local depletion lags the volume average by a controlled factor. If the effect is small, saying so with an estimate would considerably strengthen the letter.
[Final paragraph, and abstract sentence on B−L bounds] The claim that 'any bound on the strength of exotic interactions can be evaded' is asserted rather than shown. It requires that a significant Chern-Simons number survive in the condensate until the electroweak scale, which is exactly the quantity the nonlinear integration of Eq. (12) is meant to determine, yet no value for the surviving CS fraction is given as a function of δ, and no comparison is made with the size of the condensate needed to reprocess into the observed baryon asymmetry. The abstract's 'may also be evaded' inherits this gap. Either a quantitative estimate of the surviving CS number at T_ew should be supplied, or the claim should be explicitly downgraded to a qualitative remark and removed from the abstract.

Minor objections (6)

[Notation, throughout] The symbol δ is used both for the dimensionless chemical potential μ/T (defined after Eq. (10)) and, with subscripts, for the charge-to-entropy ratios δ_R, δ_1, δ_2, δ_3 in the equilibrium relation for μ_R. Because the two uses differ by a factor of order 5×10^4, the reader can easily conflate them; a distinct symbol for μ/T would remove the ambiguity.
[Paragraph following Eq. (6)] The equilibrium relation μ_R = (245π² N_eff)[78/388 δ_R − 20/188 δ_1 + 15/22(δ_2+δ_3)]T is quoted without derivation or citation. Reference [18] (Harvey and Turner; Campbell et al.; Fischler et al.) appears in the bibliography but is not cited at this point; a citation there would let the reader check the numerical coefficients.
[Abstract and paragraph following Eq. (11)] The abstract quotes a 'typical inhomogeneity scale ~10^6/T', but the text gives ~10^5/T for δ ≈ 5×10^-6 at T_R and ~2×10^6/T at the electroweak scale for the large-δ case. The abstract should specify which regime the quoted scale refers to.
[Eq. (10)] As printed, the definition of T_g is difficult to parse. Writing it explicitly as T_g = 8(μ/T)² M_0/(σ/T) would make the derivation of δ_crit ≈ 10^-6 transparent and allow the reader to verify it against σ ≈ 68T.
[Eq. (5) and the preceding sentence] The step from j = σE to the anomalous Ohm's law (5), and the justification for dropping ∂E/∂t, would benefit from one more sentence; the sign of the μ_R H term relative to Eq. (3) is the kind of detail a reader will want to check, and the neglect of displacement current is only justified a posteriori by the growth time-scale.
[Paragraph on turbulence, following Eq. (13)] The estimate that the magnetic Reynolds number is large for bulk velocities above ~5×10^-9 should be accompanied by the values of σ, L, and v used, so that the reader can see how much margin there is; the conclusion that the fields may seed a galactic dynamo is appropriately hedged, but the intermediate numbers are not shown.

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The paper

The paper claims

Joyce and Shaposhnikov argue that a primordial excess of right-handed electrons makes the hypercharge magnetic field unstable, generating fields of order $10^{22}$ gauss by the electroweak scale.

The reading

The paper asks whether the standard model by itself can manufacture magnetic fields in the early universe, without new physics and without inventing a coupling that breaks conformal invariance by hand. The authors answer yes: if the plasma carries a net number of right-handed electrons, a charge violated only by the abelian anomaly and erased too slowly by chirality-flipping processes above $T_R$ of about $80$ TeV, then the hypercharge field is unstable. The instability works because fermion number can be traded for Chern-Simons number of the gauge field at lower total energy, so field modes with wavenumber below $4mu a$ grow exponentially and eat the fermion excess. Evolving the growth down to the electroweak scale, the authors estimate fields near $10^{22}$ gauss at $T$ around $100$ GeV on an inhomogeneity scale of roughly $10^{6}/T$, which is a candidate seed population for galactic magnetism. A secondary payoff is that the Chern-Simons number stored in the condensate survives to the electroweak transition, so bounds on exotic $B-L$ violating operators can be evaded.

What carries the argument

The engine is the anomalous Maxwell system: the anomaly equation $partial_mu j^mu_R = -(g'^2 y_R^2/64pi^2) f_{mu nu} ilde f^{mu nu}$ combined with an Ohmic closure $vec{E} = (1/sigma)(vec{nabla} imes vec{H} + 4alpha' mu_R vec{H}/pi)$. In a Fourier basis this reduces to two coupled equations for the two transverse polarizations, one of which grows with exponent $lambda_+ = -(k/sigma a)(k au - 4int mu a, d au')$ while the orthogonal mode always decays. The chemical potential is then evolved by a kinetic equation that balances the anomalous term proportional to $vec{E}cdotvec{H}$ against the perturbative chirality-flip rate $Gamma_R = T_R T/M_0$. The nonlinear result is a set of scaling laws, $mu_R/T propto k_{ m phy}/T propto H_{ m phy}^2/T^4 propto (T/T_g)^{1/2}$, which the authors explain as the chemical potential and the maximally growing mode continually sliding to longer wavelengths as each scale saturates.

Extended reading notes

Core claim

The central claim is that a state at finite chemical potential for a charge with only an abelian anomaly, right-handed electron number being the standard-model example, is unstable to the generation of hypercharge magnetic field. The anomaly equation ties the right-electron number to the Chern-Simons number of the hypercharge configuration, so a plasma with excess right electrons can lower its energy by winding gauge fields rather than by keeping the fermions. Solving Maxwell's equations with an anomalous current term, the authors find a circularly polarized mode that grows whenever $k < 4mu a$, and they follow that growth through the electroweak scale. For asymmetries above about $2 imes 10^{-4}$ the damping never catches up before $T$ reaches $100$ GeV, and the amplitude scales as $H propto sqrt{delta}$, giving roughly $4 imes 10^{22}$ gauss for $delta = 10^{-2}$ at a scale of order $10^{6}/T$.

Load-bearing premise

The nonlinear numerical evolution that produces the headline amplitudes assumes the right-electron number density is uniform in space, even though the field being generated is not.

Editorial extensions

If this is right

If the hypercharge fields survive to the electroweak phase transition, electroweak mixing converts them into ordinary magnetic fields, supplying a seed that a galactic dynamo could later amplify.
For asymmetries above about $2 imes 10^{-4}$ the damping never overtakes the growth before the electroweak scale, and the field amplitude scales as $H propto sqrt{delta}$, reaching roughly $4 imes 10^{22}$ gauss at $delta = 10^{-2}$.
Below $delta_{ m crit} approx 10^{-6}$ nothing happens, because the unstable modes are frozen on the relevant timescales and the anomaly plays no role at all.
The Chern-Simons number left in the condensate is converted into quarks and leptons at the electroweak transition, so previously derived bounds on exotic $B-L$ violating operators can be evaded when $delta$ is large enough.
The growing configuration has a non-zero $vec{H}cdotvec{nabla} imes vec{H}$, so the fields it produces break parity, a handedness that could in principle leave an observational imprint.

Reading between the lines

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

The reduced model integrates a spatially homogeneous chemical potential, while the field it generates is by construction inhomogeneous; a full three-dimensional treatment in which $mu_R$ develops structure on the same scales as the field would test whether the survival threshold and the $H propto sqrt{delta}$ scaling are robust or artifacts of the reduction.
The same anomalous term that drives this instability is the one invoked for chiral magnetic effects in heavy-ion plasmas, so the two settings may be mutually constraining even though their densities and temperatures differ by many orders of magnitude.
Whether these seeds actually reach galactic scales depends on turbulence at the electroweak transition; a crossover rather than a first-order transition would remove the stirring the authors rely on and weaken the cosmological case considerably.
In GUT models where the right-electron asymmetry is tied to the baryon asymmetry, the field amplitude would become an indirect handle on the baryogenesis sector, and conversely a firm upper bound on primordial fields would constrain the initial $delta$.

Original abstract

In the standard model there are charges with abelian anomaly only (e.g. right-handed electron number) which are effectively conserved in the early universe until some time shortly before the electroweak scale. A state at finite chemical potential of such a charge, possibly arising due to asymmetries produced at the GUT scale, is unstable to the generation of hypercharge magnetic field. Quite large magnetic fields ($\sim 10^{22}$ gauss at $T\sim 100$ GeV with typical inhomogeneity scale $ \sim \frac{ 10^6}{T}$) can be generated. These fields may be of cosmological interest, potentially acting as seeds for amplification to larger scale magnetic fields through non-linear mechanisms. Previously derived bounds on exotic $B-L$ violating operators may also be evaded.

The compiled record

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

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,348 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.58 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

In the standard model there are charges with abelian anomaly only (e.g. right-handed electron number) which are effectively conserved in the early universe until some time shortly before the electroweak scale. A state at finite chemical potential of such a charge, possibly arising due to asymmetries produced at the GUT scale, is unstable to the generation of hypercharge magnetic field. Quite large magnetic fields ($\sim 10^{22}$ gauss at $T\sim 100$ GeV with typical inhomogeneity scale $ \sim \frac{ 10^6}{T}$) can be generated. These fields may be of cosmological interest, potentially acting as seeds for amplification to larger scale magnetic fields through non-linear mechanisms. Previously derived bounds on exotic $B-L$ violating operators may also be evaded.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In the standard model there are charges with abelian anomaly only (e.g. right-handed electron number) which are effectively conserved in the early universe until some time shortly before the electroweak scale.	background 1.00	0.08	0.08	asserted 0.99
s1	A state at finite chemical potential of such a charge, possibly arising due to asymmetries produced at the GUT scale, is unstable to the generation of hypercharge magnetic field.	result 0.82	0.50	0.38	asserted 0.53
s2	Quite large magnetic fields ($\sim 10^{22}$ gauss at $T\sim 100$ GeV with typical inhomogeneity scale $ \sim \frac{ 10^6}{T}$) can be generated.	result 1.00	0.68	0.36	asserted 0.89
s3	These fields may be of cosmological interest, potentially acting as seeds for amplification to larger scale magnetic fields through non-linear mechanisms.	interpretation 1.00	0.25	0.20	hedged possibility 1.00
s4	Previously derived bounds on exotic $B-L$ violating operators may also be evaded.	interpretation 0.88	0.15	0.06	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.65.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

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

levels run

P abstract

not run

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

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

References

Works cited (18)

[9]

V. Rubakov and A. Tavkhelidze, Phys. Lett. 165B:109, 1985; V. Rubakov, Prog. Theor. Phys. 75:366, 1986

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M.S. Turner and L.M. Widrow, Phys. Rev. D37:2743, 1988

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B. Ratra, The Astronomical J., 391:L1, 1992; A. Dolgov and J. Silk, Phys. Rev. D47:3144, 1993; M. Gasperini, M. Giovannini and G. Veneziano, P hys. Rev. Lett. 75:3796, 1995; D. Lemoine and M. Lemoine, Phys. Rev. D52:1955, 1995

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V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. L ett., 155B:36, 1985

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[5]

B. Campbell, S. Davidson, J. Ellis and K. Olive, Phys. Let t. 297B :118, 1992

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[6]

In alternative cosmologies such as that considered in M. Joyce, Phys. Rev. D55:1875, 1997, in which the expansion rate before nucleosynthesis is diﬀerent to the standard one, TR can be well below the electroweak scale

View on Pith 1997
[7]

L. E. Ibanez and F. Quevedo, Phys. Lett., B283:261, 1992

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J. Cline, K. Kainulainen and K. Olive, Phys. Rev. Lett. 71:2372, 1993; Phys. Rev. D49:6394, 1994

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E. S. Fradkin, Proc. Lebedev Inst. 29:1, 1965

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K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnik ov, Nucl. Phys. B493:413, 1997

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E.W. Kolb and M.S. Turner, The Early Universe, Addison- Wesley, Reading, MA, 1990

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A.N. Redlich and L.C.R. Wijewardhana, Phys. Rev. Lett. 54:970, 1985

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M. Joyce, T. Prokopec and N. Turok, Phys. Rev. D53:2930, 1996; G. Baym and H. Heiselberg, astro-ph/9704214 11

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P. Olesen, astro-ph/9610154; A. Brandenburg, K. Enqvi st and P. Olesen, hep- ph/9608422; Phys. Rev. D54:1291, 1996

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J. Harvey and M. Turner, Phys. Rev. D 42 : 3344 (1990); B. Campbell, S. Davidson, J. Ellis and K. Olive, Phys. Lett. 256B :457, 1991; W. Fischler, G.F. Giudice, R.G. Leigh, and S. Raban, Phys. Lett. B258: 45 (1991) 12

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nucl-th 2016-08 conditional novelty 4.0 of 10

Colliding same-mass, different-charge nuclei at RHIC can isolate the magnetic-field-dependent part of charge separation and test the chiral magnetic effect.
Progress on Cosmological Magnetic Fields
astro-ph.CO 2020-10 accept novelty 3.0 of 10

Cosmological magnetic fields generated at the electroweak phase transition survive to today only if they carry helicity, and the helicity needed exceeds what Standard Model baryogenesis supplies, pointing to new CP-vi...
The Chiral Magnetic Effect and Anomaly-Induced Transport
hep-ph 2013-12 accept novelty 2.0 of 10

The review argues that the chiral magnetic effect is a topologically protected, non-dissipative transport phenomenon of chiral fermions, linking the quark-gluon plasma, the early universe, and Weyl semimetals.

How later papers used it

134 later papers cite it

Evidence: substantial. 130 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 263 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 compared their own results with it.

Compares with it 2 of 4 papers with this label

Reported that its own estimate of magnetic field strength in the collision dominated regime broadly agrees with this paper.

Primordial magnetic field and kinetic theory with Berry curvature (2015) · 0.99 · paraphrased: the licence does not allow quoting
Reported that its own estimate of the magnetic field strength in the collision dominated regime broadly agrees with this paper.

Origin and dynamics of the Primordial Magnetic field in a parity violating plasma (2017) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

103 arXiv papers cite this in 201 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 144 sentences without a confident label

Contests it 1 paper

However, such a constraint is not applicable in our case because it requires the electroweak symmetry to be restored at high temperatures [ref] .

Sphaleron freeze-in baryogenesis with gravitational waves from the QCD transition · Introduction · contest 0.72

Compares with it 3 papers

These results are in broad agreement with Ref. [ref] .

Origin and dynamics of the Primordial Magnetic field in a parity violating plasma · Discussion · 0.99
The power law decay $∝t^-1/2$ of the chemical potential reported in section [ref] was also predicted in the context of MHD in several works [ref] .

Chiral charge dynamics in Abelian gauge theories at finite temperature · Self-similarity and inverse cascade · 0.94
This behavior is quite different from the electromagnetic case where the chiral plasma instability is caused by the non-zero chemical potential itself but not its time variation [ref] .

Chiral gravitational effect in time-dependent backgrounds · Propagation of gravitational waves · 0.99

Cites it as support 1 paper

The free energy decreases with decreasing $k_h$ [ref] , and the inverse cascade indeed drives $k_h$ to small values [ref] , so we drop the second term.

Baryogenesis from Decaying Magnetic Helicity in Axiogenesis · Helicity density at equilibrium · 0.98

Cites it as background 41 papers

This phenomenon is called the chiral plasma instability (CPI) [ref] , and has to be avoided for a successful baryogenesis.

Baryogenesis from combined Higgs - scalar field inflation · The chiral plasma instability · 0.81
Moreover, the presence of electric currents from CME results in unstable modes for dynamically growing magnetic fields, known as chiral plasma instability (CPI) [ref] , which has multiple applications particularly in cosmology and astrophysics [ref] .

Effective Chiral Magnetic Effect from Neutrino Radiation · Introduction · 0.80
First proposed in Joyce:1997uy, various studies have explored the development of an instability leading to the growth of helical magnetic fields, known as the chiral plasma instability (CPI) [ref] , using both analytical methods and direct numerical simulation of the nonlinear equations of motion of chiral magnetohydrodynamics () [ref] .

Primordial magnetic field from chiral plasma instability with sourcing · Introduction · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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