Inefficiency of chiral dynamos in protoneutron stars and the early universe

The chiral plasma instability (CPI) has been invoked as a possible mechanism for generating primordial magnetic fields in the universe and ultrastrong fields in neutron stars. We investigate chiral dynamos where the chirality imbalance is pumped by a source on a timescale $t_0$ and show that the CPI rate $γ$ is limited to $γ_0/(1+{\cal Q}^2)$, where ${\cal Q}= (γ_0 t_0)^{1/3}$ and $γ_0$ corresponds to models with instantaneously created chirality imbalance $(t_0=0)$. We then find that chiral flipping with rate $Γ_{\mathrm f}$ hinders the chiral dynamo if $Γ_{\mathrm f} >γ_0/(1+{\cal Q}^2)$ and completely suppresses it if $Γ_{\mathrm f} >γ_0/(1+{\cal Q}^{3/2})$. Realistic $t_0$ typically give ${\cal Q}\gg 1$, which makes the dynamo greatly vulnerable to the suppression by chiral flipping. The suppression is strong in protoneutron stars and may be (barely) avoided near the electroweak transition in the early universe.

💡 Research Summary

The paper revisits the chiral plasma instability (CPI) as a mechanism for generating strong magnetic fields in astrophysical settings, focusing on the realistic scenario where the chiral imbalance is not created instantaneously but is pumped continuously over a finite timescale t₀. Starting from the chiral magnetohydrodynamics (MHD) equations, the authors introduce a chiral chemical potential µ (proportional to the chiral density n₅) and show that the total helicity H = ⟨µ⟩ + λ H is conserved apart from source, flipping, and boundary terms.

In the classic “instantaneous‑µ₀” picture, the CPI growth rate is γ(k)=η k(µ−k), peaking at γ₀=η µ₀²/4. The instability converts µ₀ into magnetic helicity until both reach roughly µ₀/2, yielding a magnetic field B≈µ₀/√λ. However, when µ is supplied by a source S over a time t₀, the dynamics change dramatically. The authors define a dimensionless parameter

 𝒬 ≡ (γ₀ t₀)¹ᐟ³ = µ₀/µ_Q,

where µ_Q≈(4S/η)¹ᐟ³ is the saturation value of µ in the presence of continuous pumping. If γ₀ t₀ ≪ 1 (𝒬 ≪ 1), the system behaves like the instantaneous case. In realistic astrophysical environments, however, γ₀ t₀ ≫ 1, so 𝒬 ≫ 1. In this regime the CPI acts already while the source is still building up µ, preventing µ from ever reaching µ₀. Instead, most of the injected chiral charge is immediately transferred into magnetic helicity, and the final magnetic field strength is set by B_Q≈µ_Q√λ.

The authors then incorporate chiral flipping with rate Γ_f. They introduce another dimensionless quantity

 Γ ≡ Γ_f (1+𝒬²)/γ₀,

which measures the relative importance of flipping. For Γ ≪ 1 flipping is negligible; for Γ ≫ 1 the flipping term dominates the µ‑evolution, limiting µ to µ_Γ = S/Γ_f. The CPI then grows on a reduced rate γ_Γ≈η µ_Γ²/4. If Γ exceeds 𝒬¹ᐟ², the CPI growth becomes too slow to generate any appreciable magnetic field before the source shuts off. Consequently, the magnetic helicity generated over the whole pumping interval is

 H ≈ µ₀ λ × { 1 (Γ < 1), f_H (1 < Γ ≪ 𝒬¹ᐟ²), 0 (Γ ≫ 𝒬¹ᐟ²) },

where f_H (< 1) is the fraction of injected chirality that ends up as magnetic helicity when flipping is moderate.

To validate these analytic estimates, the authors perform pseudo‑spectral simulations (Dedalus, 256³ grid) of the full chiral MHD system with various Q and Γ values. A fiducial run with Q=15, Γ=0 reproduces the three‑stage evolution: (i) linear growth of µ, (ii) a quasi‑steady state where µ≈µ_Q and magnetic helicity grows at the source rate, and (iii) decay of µ after the source stops while the magnetic field remains. Spectra show an inverse cascade from the initial CPI wavenumber k≈µ_Q down to larger scales as the system relaxes. Simulations with non‑zero Γ confirm that moderate flipping reduces the final helicity fraction f_H, while large Γ (≫ Q¹ᐟ²) suppresses magnetic field growth entirely.

Finally, the paper applies the framework to two astrophysical contexts. In proto‑neutron stars (PNS) typical parameters (electron density ≈10³⁶ cm⁻³, temperature ≈30 MeV, conductivity ≈10²⁴ s⁻¹) give η ≈ 10⁻⁴ cm² s⁻¹, t₀ ≈ 10⁻³ s, and thus Q ≈ 10³–10⁴. The flipping rate Γ_f is estimated at ≈10⁴ s⁻¹, yielding Γ ≫ Q¹ᐟ². Hence the CPI is essentially quenched in PNS, contradicting earlier optimistic estimates of magnetar‑scale fields generated by the chiral dynamo.

In the early universe, near the electroweak transition (k_BT ≈ 100 GeV), the conductivity is much larger, t₀ is extremely short (≈10⁻¹² s), giving Q ≈ 10–30. The flipping rate at that temperature is tiny (Γ_f ≈ 10⁻³ s⁻¹), so Γ ≪ 1 and the CPI can operate, potentially producing magnetic fields of order 10¹⁴–10¹⁵ G on cosmological scales. However, as the universe cools, flipping rates rise quickly, so the window for efficient CPI is narrow.

In summary, the paper demonstrates that the efficiency of chiral dynamos is governed by the interplay of the pumping timescale (through Q) and the chiral flipping rate (through Γ). Realistic astrophysical environments with slow pumping (large Q) are highly susceptible to flipping, making the CPI an unlikely source of ultra‑strong magnetic fields in proto‑neutron stars, while it may still play a role in the early universe, but only in a brief epoch around the electroweak transition.