REVIEW
Large-scale magnetic fields from hydromagnetic turbulence in the very early universe
Astrophysics · Submitted 7 Feb 1996 · 23 references
Pith's verdict: A clean conformal-MHD derivation plus an honest but under-controlled shell-model estimate that primordial fields inverse-cascade to larger scales.
T0 · AI review
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Referee report
The paper
The paper claims
Turbulence in the early-universe plasma may stretch a primordial magnetic field from particle-physics scales to scales of parsecs, growing its coherence length by four to five orders of magnitude.
The reading
What carries the argument
Two pieces of machinery carry the argument. The first is a scaling identity: in a flat expanding universe the relativistic MHD equations, written in the tilde variables $R^4 S$, $R^4 p$, $R^2 B$, $R^3 J$, $R^2 E$ and conformal time $dtilde{t} = dt/R$, are formally identical to the static-universe equations, so a numerical solution on a box of size $L$ also solves the expanding problem at a rescaled time. The second is the cascade (shell) model, in which velocity and magnetic field become complex scalars $v_n$ and $b_n$ on wavenumbers $k_n = 2^n$ and the nonlinear terms are approximated by nearest- and next-nearest-neighbour couplings with free coefficients $A$, $B$, $C$; the coefficients are fixed by demanding conservation of total energy together with either the three-dimensional helicity-like quantity (hydrodynamic choice $A=1$, $B=-1/2$, $C=0$) or the magnetic-helicity-like quantity $H_M = sum (-1)^n k_n^{-1} b_n^* b_n$ (MHD choice $A=7/5$, $B=-1/10$, $C=1$). The conserved quantity associated with the inverse cascade is $int d^3x, A dot B$ in three dimensions and $int d^2x, A^2$ in two, and it is this conservation law that the shell coefficients are built to respect.
Extended reading notes
Core claim
The central claim is that magnetohydrodynamic turbulence in the early universe drives an inverse cascade that moves magnetic energy from small to large scales, so that a primordial field generated at, say, the electroweak scale ends up with a comoving coherence length four to five orders of magnitude larger after about $10^9$ initial times. The authors state the consequence directly: at large scales primordial magnetic fields are considerably stronger than expected from considerations that ignore MHD turbulence. Supporting evidence comes in three tiers. First, a derivation showing that relativistic MHD in a Robertson-Walker background with $p = 3^{-1}$ (i.e. $p = 4^{-1}$ in their notation) maps onto the non-expanding equations under the rescalings $R^4 S$, $R^4 p$, $R^2 B$, $R^3 J$, $R^2 E$ and conformal time, with $R^4 E_{tot}$ conserved. Second, a $64^2$ and $128^2$ two-dimensional simulation with hyperdiffusion, in which the integral scale rises from $l_0 = 0.09$ to $0.95$ (low resolution) and from $0.04$ to $0.76$ (high resolution). Third, the shell model, where the integral scale climbs by four to five orders of magnitude and is fitted by $l_0(t) = r_0 (t/t_0)^{1/4}$, equivalently $l_0(T) = r_0 (T/T_0)^{-1/2}$. The worked example is concrete: taking $r_0 = 10^{-3}$ cm at $T = 100$ GeV and an initial field of $10^{18}$ G gives roughly 2 pc and $10^{-11}$ G today.
Load-bearing premise
The four-to-five-orders-of-magnitude growth claim rests entirely on a scalar shell model with hand-chosen coupling coefficients standing in for real three-dimensional relativistic magnetohydrodynamics over nine orders of magnitude in time, and the paper never shows that the toy model's cascade rate matches that of the full equations.
Editorial extensions
If this is right
- Estimates of the seed field required for galactic dynamos can be revised: a field that reaches parsec scales with the amplitude the paper quotes, about $10^{-11}$ G today from $10^{18}$ G at the electroweak transition, is enough to feed dynamo amplification in high-redshift galaxies.
- The stochastic dilution factor $N$ in area-averaged primordial field estimates, quoted as $N \sim 10^{24}$ for 100 kpc today, is reduced to $N \le 10^{19}$ once turbulence is included, so the field decreases less steeply with scale than a pure flux-conservation argument gives.
- The coherence length of a primordial field becomes a turbulence output rather than an input: it is set by the inverse-cascade rate, not by the particle-physics mechanism that generated the field.
- The shift of magnetic energy toward large scales also shifts magnetic energy density, which the authors note may be relevant to density fluctuations sourced by magnetic pressure.
- Because the inverse cascade is tied to conservation of magnetic helicity, any primordial field with non-zero helicity should show the effect more strongly than a non-helical one, making helicity a diagnostic of how far the cascade has run.
Reading between the lines
- The scaling identity in Sec. II is the paper's most durable result independent of the turbulence claim: it says any static-box relativistic MHD simulation can be reinterpreted as an expanding-universe simulation by a rescaling of time, which is a reusable tool beyond this paper's specific conclusions.
- A sharper test than the integral scale would be the shape of the magnetic energy spectrum at late times: a Kolmogorov-like $k^{-5/3}$ or Iroshnikov-Kraichnan $k^{-3/2}$ slope, as opposed to the initial white-noise $k^{+2}$, would be a spectral fingerprint that a future three-dimensional simulation or an observational constraint on primordial field spectra could check.
- If the coherence scale grows as $t^{1/4}$ in conformal time, then the growth is slow enough that most of the effect must accumulate before recombination, which suggests the predicted present-day coherence is sensitive to the unknown equation of state and turbulence level during the QCD and electroweak epochs rather than to late-time physics.
- The negative correlation the authors find at large separations, which they attribute to the dipole-like closure of field loops required by $\nabla \cdot B = 0$, is a testable morphological prediction: a primordial field that has cascaded should show anticorrelation at a scale comparable to its coherence length.
Original abstract
We investigate hydromagnetic turbulence of primordial magnetic fields using magnetohydrodynamics (MHD) in an expanding universe. We present the basic, covariant MHD equations, find solutions for MHD waves in the early universe, and investigate the equations numerically for random magnetic fields in two spatial dimensions. We find the formation of magnetic structures at larger and larger scales as time goes on. In three dimensions we use a cascade (shell) model, that has been rather successful in the study of certain aspects of hydrodynamic turbulence. Using such a model we find that after ${\cal O}(10^9)$ times the initial time the scale of the magnetic field fluctuation (in the comoving frame) has increased by 4-5 orders of magnitude as a consequence of an inverse cascade effect (i.e. transfer of energy from smaller to larger scales). Thus {\it at large scales} primordial magnetic fields are considerably stronger than expected from considerations which do not take into account the effects of MHD turbulence.
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.
The abstract, sentence by sentence
We investigate hydromagnetic turbulence of primordial magnetic fields using magnetohydrodynamics (MHD) in an expanding universe. We present the basic, covariant MHD equations, find solutions for MHD waves in the early universe, and investigate the equations numerically for random magnetic fields in two spatial dimensions. We find the formation of magnetic structures at larger and larger scales as time goes on. In three dimensions we use a cascade (shell) model, that has been rather successful in the study of certain aspects of hydrodynamic turbulence. Using such a model we find that after ${\cal O}(10^9)$ times the initial time the scale of the magnetic field fluctuation (in the comoving frame) has increased by 4-5 orders of magnitude as a consequence of an inverse cascade effect (i.e. transfer of energy from smaller to larger scales). Thus {\it at large scales} primordial magnetic fields are considerably stronger than expected from considerations which do not take into account the effects of MHD turbulence.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate hydromagnetic turbulence of primordial magnetic fields using magnetohydrodynamics (MHD) in an expanding universe. | contribution 0.87 | 0.90 | 0.03 | asserted 0.62 |
| s1 | We present the basic, covariant MHD equations, find solutions for MHD waves in the early universe, and investigate the equations numerically for random magnetic fields in two spatial dimensions. | contribution 0.57 | 0.94 | 0.03 | asserted 0.98 |
| s2 | We find the formation of magnetic structures at larger and larger scales as time goes on. | result 1.00 | 0.94 | 0.19 | asserted 1.00 |
| s3 | In three dimensions we use a cascade (shell) model, that has been rather successful in the study of certain aspects of hydrodynamic turbulence. | method 0.96 | 0.38 | 0.02 | asserted 0.92 |
| s4 | Using such a model we find that after ${\cal O}(10^9)$ times the initial time the scale of the magnetic field fluctuation (in the comoving frame) has increased by 4-5 orders of magnitude as a consequence of an inverse cascade effect (i.e. transfer of energy from smaller to larger scales). | result 1.00 | 0.95 | 0.31 | asserted 1.00 |
| s5 | Thus {\it at large scales} primordial magnetic fields are considerably stronger than expected from considerations which do not take into account the effects of MHD turbulence. | interpretation 0.91 | 0.72 | 0.86 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:astro-ph/9602031 by the authors.
References
How later papers used it
72 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.
Uses its method or tool 2 of 6 papers with this label · 1 by the same authors
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Moved to the conformal frame, replacing variables by conformal or comoving quantities as in this paper, while the bulk velocity field is unchanged.
Chiral effects in astrophysics and cosmology (2022) · 0.84 · paraphrased: the licence does not allow quoting
By the same authors
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It is solved together with [ref] [equation] [equation] where $=(·+·lnρ)/3 -[·(×)+^2/σ]/ρ$, and $H=[·(×)+^2/σ]ρ$ are additional terms that are retained in the calculation, and $_ν=2·(ρν)/ρ$ is the viscous force, where $S_ij=(u_i,j+u_j,i)-δ_ij·$ are the components of the rate-of-strain tensor with commas denoting partial derivatives, and $ν$ is the kinematic viscosity.
Simulating relic gravitational waves from inflationary magnetogenesis (2021) (same authors) · The model · 0.92
How the literature uses it
52 arXiv papers cite this in 94 sentences
Papers citing it · 72 sentences without a confident label
Contests it 2 papers
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In particular, we also show that previous work assuming the $γ^2 ≈1$ limit in the subrelativistic regime (cf. the original work [ref] , or the reviews [ref] and references therein) ignores time derivatives of the Lorentz factor, $γ$, which can lead to additional non-linear terms of order $U^2/ℓ$, being $U$ and $ℓ$ a characteristic velocity and length scale of the fluid, respectively, when the fluid is dominated by radiation.
Relativistic magnetohydrodynamics in the early Universe · Introduction · contest 0.76
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[ref] is not strictly consistent, as derivatives of the Lorentz $γ$ factor are discarded even though their effects are not generally subleading.
Plasma heating during the chiral plasma instability · Chiral magnetohydrodynamics · contest 0.79
Uses its method or tool 5 papers
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Turbulent flows in the early universe can be modeled by solving the hydromagnetic equations for the density $ρ$, the velocity $$, and the magnetic field $$ with $·=0$, adopting an ultrarelativistic equation of state in an expanding universe using conformal time and comoving variables [ref] with a forcing term $$ in the induction equation for $$ [equation] We recall that the conformal time $t$ is normalized to unity at the time $t_*$ of magnetic field generation, $ρ$ is in units of the initial value, $$ is in units of the speed of light, and the magnetic energy density $^2/2$ is measured in units of the radiation density at the time of generation.
Can we observe the QCD phase transition-generated gravitational waves through pulsar timing arrays? · Turbulent Sources · 0.89
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We seek to address these questions using the theory of chiral magnetohydrodynamics () [ref] .
Plasma heating during the chiral plasma instability · Introduction · 0.86
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One can now write the Maxwell's equations in the fiducial frame as [ref] :
Origin and dynamics of the Primordial Magnetic field in a parity violating plasma · Generation of magnetic fields above EW scale · 0.80
Cites it as support 1 paper
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First of all, they are conservative: whatever the initial generation mechanism, the magnetic field is expected to enter a phase of fully developed and freely decaying turbulence [ref] .
Gravitational wave signal from primordial magnetic fields in the Pulsar Timing Array frequency band · Introduction · 0.96
Cites it as background 13 papers
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More difficult is to address the issue of whether the magnetic field spectrum experiences inverse cascade, i.e. transference of energy from high to low momentum modes [ref] - [ref] .
Primordial magnetic fields from preheating at the electroweak scale · Late time evolution · 0.89
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Many attempts to study the evolution of primordial magnetic fields have been performed in the past [ref] .
Time Evolution of the Large-Scale Tail of Nonhelical Primordial Magnetic Fields with Back-Reaction of the Turbulent Medium · Introduction · 1.00
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The application to the early Universe focusses particularly on the decay of magnetic fields and their associated increase of typical length scales during the radiation-dominated epoch from microphysical to galactic scales [ref] ; see also [ref] and [ref] for reviews.
Scaling of the Hosking integral in decaying magnetically-dominated turbulence · Introduction · 0.95
Discussion
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