arX

iv:1

507.

0051

0v3

[ast

ro-p

h.C

O]

10 O

ct 2

016

NORDITA-2015-83

The evolution of the primordial magnetic field since its generation

Tina Kahniashvili∗

The McWilliams Center for Cosmology and Department of Physics,

Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA

Department of Physics, Laurentian University,

Ramsey Lake Road, Sudbury, ON P3E 2C, Canada and

Abastumani Astrophysical Observatory, Ilia State University,

3-5 Cholokashvili Avenue, Tbilisi, 0162, Georgia

Axel Brandenburg†

Laboratory for Atmospheric and Space Physics,

University of Colorado, Boulder, CO 80303, USA

JILA and Department of Astrophysical and Planetary Sciences,

University of Colorado, Boulder, CO 80303, USA

Nordita, KTH Royal Institute of Technology and Stockholm University,

Roslagstullsbacken 23, 10691 Stockholm, Sweden and

Department of Astronomy, AlbaNova University Center,

Stockholm University, 10691 Stockholm, Sweden

Alexander G. Tevzadze‡

Faculty of Exact and Natural Sciences, Tbilisi State University,

3 Chavchavadze Avenue, Tbilisi, 0179, Georgia and

Abastumani Astrophysical Observatory, Ilia State University,

3-5 Cholokashvili Avenue, Tbilisi, 0162, Georgia

1

AbstractWe study the evolution of primordial magnetic fields in an expanding cosmic plasma. For this purpose we present

a comprehensive theoretical model to consider the evolution of MHD turbulence that can be used over a wide rangeof physical conditions, including cosmological and astrophysical applications. We model different types of decayingcosmic MHD turbulence in the expanding universe and characterize the large-scale magnetic fields in such a medium.Direct numerical simulations of freely decaying MHD turbulence are performed for different magnetogenesis scenar-ios: magnetic fields generated during cosmic inflation as well as electroweak and QCD phase transitions in the earlyuniverse.

Magnetic fields and fluid motions are strongly coupled due to the high Reynolds number in the early universe.Hence, we abandon the simple adiabatic dilution model to estimate magnetic field amplitudes in the expanding uni-verse and include turbulent mixing effects on the large-scale magnetic field evolution. Numerical simulations havebeen carried out for non-helical and helical magnetic field configurations. The numerical results show the possibil-ity of inverse transfer of energy in magnetically dominatednon-helical MHD turbulence. On the other hand, decayproperties of helical turbulence depend on whether the turbulent magnetic field is in a weakly or a fully helical state.

Our results show that primordial magnetic fields can be considered as a seed for the observed large-scale magneticfields in galaxies and clusters. Bounds on the magnetic field strength are obtained and are consistent with the upperand lower limits set by observations of extragalactic magnetic fields.

∗Electronic address: tinatin@andrew.cmu.edu†Electronic address: Axel.Brandenburg@Colorado.edu‡Electronic address: aleko@tevza.org

2

I. INTRODUCTION

Understanding the origin and evolution of cosmic magnetismis one of the challenging questions of mod-ern astrophysics. The major questions include theoreticalas well as observational aspects of the problem:when and how was the cosmic magnetic field generated? How did it evolve during the expansion of theuniverse? What are modern observational constraints on themagnetic fields at large scales? Are magneticfields observed at galactic and extragalactic scales of cosmological or astrophysical origin? The types ofturbulence considered here are characterized by a strong random initial magnetic field. The interaction withthe velocity field leads too inverse spectral transfer toward large scales that is unknown in non-magneticturbulence.1

The goal is to identify important properties of cosmic magnetic turbulence in the expanding universe.Properties of decaying MHD turbulence in primordial plasmalink magnetogenesis scenarios operating inthe early universe with the constraints on the large-scale magnetic fields set by present observations. Hence,studying the magnetic field evolution, we can identify likely magnetogenesis scenarios responsible for ex-citing seed fields in the early universe and exclude unlikelyones using constraints set by modern or futureobservations.

The problem of cosmological magnetogenesis is guided by recent observations of large-scale magneticfields. Indeed, galaxies are known to have magnetic fields that are partly coherent on the scale of the galaxywith field strengths reaching10−6 Gauss (G) (see Refs. [1–6] and references therein). These magneticfields are the result of amplification of initial weak seed fields of unknown nature. Moreover, it is now clearthatµG-strength magnetic fields were already present in normal galaxies (like our Milky Way) when theuniverse was less than half of its present age [7–9]. This poses strong limits on the seed magnetic fieldstrength and its amplification timescale.

From a theoretical point of view there are two scenarios thatcan lead to the generation of magneticfields at extragalactic scales [10]: a bottom-up (astrophysical) scenario, where the seed field is typicallyvery weak and the observed large-scale magnetic field is transported from local sources within galaxies tolarger scales [11], and a top-down (cosmological) scenariowhere a significant seed field is generated priorto galaxy formation in the early universe on scales that are large at the present time [12]. The major themeof this review is to discuss the evolution, structure, and effects of cosmic magnetic fields with the goal tobetter understand its origin and observational signatures.

We will briefly discuss cosmic magnetohydrodynamic (MHD) turbulence in order to understand themagnetic field evolution. MHD turbulence in the context of astrophysical plasma processes has been studiedfor a long time. On the other hand, the effects of MHD turbulence in cosmological contexts has receivedattention only in recent years [13]. Simulations show that the kinetic energy of turbulent motions in galaxyclusters can be as large as 5–10% of the thermal energy density [14]. This can influence the physics ofclusters [15], and at least should be modeled correctly whenperforming large-scale simulations [16–21].Turbulent motions can also affect cosmological phase transitions; see Refs. [22–24] and references therein).Turbulence can be generated by a small initial cosmologicalmagnetic fields. Understanding mechanisms forexciting primordial turbulence is an important goal. We argue that even if the total energy density present inturbulence is small, its effects might be substantial because of the strongly nonlinear nature of the relevantphysical processes.

1 The paper is based on the presentation by Tina KahniashviliCosmic Magnetic Fields: Origin, Evolution, and Signaturesat theTurbulent Mixing and Beyond Workshop 2014’Mixing in Rapidly Changing Environments - Probing Matter at the Extremes’.

3

Recent important observations Refs. [25–33] (also Ref. [34] for recent study, and Ref. [35] for discus-sions on possible uncertainties in the measurements of blazar spectra), suggest the existence of magneticfields in the universe at scales large enough to suggest a primordial origin [10]. This result is robust to po-tential plasma instabilities of the two-stream family [36–38]. Prior to these observations, there existed onlyupperlimits of the order of a few nG for the intergalactic magneticfield. These were obtained through Fara-day rotation of the cosmic microwave background (CMB) polarization plane [39–49] and Faraday rotationof polarized emission of distant quasars [50–53]. Other tests to deriveupperlimits on large-scale correlatedmagnetic fields are based on their effect on the CMB (see Ref. [54] and references therein), [55–82], CMBdistortions [83–92], the broken isotropy limits, [93–102], big bang nucleosynthesis (BBN) data [103–105],or large scale structure (LSS) formation [106–128]. Thelower limit on the intergalactic magnetic field invoids of order10−18 G on 1 Mpc scales is a puzzle of modern astrophysics (see Ref. [129]), and could verywell be the result of the amplification of a primordial cosmological field [31].

In what follows we review recent efforts which include the pioneering studies of primordial magneticfield evolution through cosmological phase transitions; see Refs. [130–140]. The decay of cosmic magneticfield in the universe has been analyzed through numerical simulations of decaying MHD turbulence. Majorfindings include (i) the possibility of the inverse transferof non-helicalcausally generated magnetic fields[135]; (ii) fast growth of vorticity in the magnetized universe [132]; (iii) growth of helical structures at largescales for partially helical magnetic fields generated at cosmological phase-transitions [133, 134, 136, 137],and more interestingly the absence of the inverse cascade for inflation-generated fully or partially helicalmagnetic fields [139, 140].

II. MODELING MHD TURBULENCE IN THE UNIVERSE

The origin of the cosmic magnetic field has been discussed fordecades, starting with Enrico Fermi’s pa-per of 1949 [141]. The approach presented below is novel in several ways. (i) Primordial magnetic fields aregenerally analyzed in the “frozen-in” approximation due toa high conductivity of cosmic plasma, when themagnetic field evolves only due to the dilution of field lines as the universe expands. In contrast, we accountfor the actual coupling between the magnetic field and the cosmic plasma, which leads to major differenceswith the frozen-in approximation at some epochs. (ii) Much work on MHD turbulence is focused on specificastrophysical objects (such as galaxies, clusters, interstellar medium, or stellar magnetosphere). Instead wehave developed a comprehensive theoretical framework to consider the evolution of MHD turbulence overa wide range of physical conditions, beyond any specific application. (iii) Cosmic MHD turbulence isusually studied within one of two limiting cases, the viscous (optically thick) or free-streaming (opticallythin) regimes. These two regimes differ in the form of viscous or drag forces. Realistic turbulent behavioris somewhere in between these two limits, and the numerical simulations have the capability to describeadequately a smooth transition between these two regimes.

As noted above, several astrophysical observations show the presence of a large-scale correlated mag-netic fields in the universe. The recent study by Dolag et al. [31] concludes that these magnetic fields aremost likely seeded by a field of primordial origin. In fact, many different mechanisms of cosmological seedmagnetic field generation have been proposed. Some of these employ symmetry breaking during phasetransitions (e.g. electroweak or QCD) [142–162]. On the other hand, if the magnetic field originated dur-ing a cosmological phase transition, its configuration is strongly limited by causality [163]: the correlationlength of the magnetic field cannot exceed the Hubble horizonat the moment of field generation. Thecausality condition combined with the divergence-free field condition implies a magnetic energy spectrum

4

at large scalesEM (k) ∝ k4 [164] (the so-called Batchelor spectrum [165]). Recent numerical simulations[131–135] confirm that cosmological turbulence produces a Batchelor spectrum completely independentlyof initial conditions present in the cosmic plasma. Combining thiscausalspectrum with the requirementthat the total energy density of the magnetic field be less than 10% of the radiation energy density (to beconsistent with standard BBN) leads to a strong limit on the smoothed amplitude of the magnetic fieldat large scales of the order of10−26 to 10−19 Gauss at 1 Mpc [166], although the effective value of themagnetic field derived through its total energy density is high enough, of the order of10−6 Gauss [130].(Note that this argument does not account for further evolution of the magnetic field in MHD turbulence).Taking into account that the magnetic field effects are mostly determined by effective values (i.e. the totalenergy density), and noticing that the extremely low limitsat large scales of causal fields are consequencesof normalization (smoothing procedure), the upper bounds have been re-determined in terms of the effec-tive strength of the magnetic field; see Refs. [44, 46, 122]. The BBN limits have also been re-analyzed byaccounting for the MHD evolution oftoymagnetic fields throughout expansion of the universe [134].

Our particular interest lies in helical magnetic fields thatcan be generated in the early universe; seeRefs. [167–179] and references therein. There are two main motivations for considering helical seed mag-netic fields: (i) the presence of helical magnetic fields in the early universe can be related to the lepto-and baryogenesis problems [180]; (ii) it sheds light on the evolution of helical magnetic fields in stellarmagnetospheres, AGNs, and voids [181, 182].

An exception to the Batchelor spectrum (spectral indexn = 4) is the possibility of inflationary magne-togenesis, in which the spectral index of the magnetic field could be less than+1, and the simplest optionis a scale-invariant spectrum withn → −1 [183–207]. Inflation-generated magnetic field scenarios shouldbe considered with some caution due to the possibility of significant backreaction [208–211], which is notan issue for the phenomenological, effective classical model; see Ref. [212] and references therein. Thefirst simulations describing the inflation-generated magnetic field coupled to the primordial plasma sug-gested that the presence of an initial magnetic field leads tolarge-scale turbulent motions in the rest plasma[132]. Ongoing research consists in the study of inflation-generated helical magnetic field (with a scale-invariantk−1 spectrum) evolution during the expansion of the universe [138–140]. Simulations show thatinflation-generated magnetic fields retain information about initial conditions. In other words, they decayvery slowly when compared with phase transition-generatedfields. Magnetic fields are almost “frozen-in”the primordial plasma at large scales, where causality allows interaction only at scales smaller than theHubble horizon and they correspondingly retain their initial spectral shape. On the other hand, within thecausal horizon, the magnetic seed field interacts with cosmic plasma leading to the excitation of kineticmotions (turbulent velocities); see below. We also discussan alternative approach where cosmic magneticfields originate during the late stages of the evolution of the universe [106, 213]. In this case the correlationlength is strongly limited by the causality requirement. Due to the sharp spectral shape at large scales, themagnetic field amplitude might be low. The simplest astrophysical magnetogenesis mechanism invokes theejection of magnetic flux from compact systems such as AGNs orsupernovae [214, 215]. In this scenariothe generation of a strong magnetic field is ensured by its extremely fast generation due to rapid rotationof the object [11]. Other mechanisms are based on the generation of a small seed by plasma processes[129, 216, 217], which are then amplified by MHD dynamo mechanisms [218, 219].

5

III. THE MODEL

As mentioned above, we focus on the magnetic field evolution during the expansion of the universefrom the moment of magnetic field generation until today. Over this lengthy period, the magnetic field isaffected by different physical processes that result in amplification as well as damping: the complexities ofthe problem are due to the strong coupling between magnetic field and turbulent motions. First, to accountfor the cosmological expansion we must reformulate the MHD equations in terms of comoving quantities[220]. Specific epochs most relevant to the final configuration of the primordial magnetic field are relatedto cosmological phase transitions, neutrino decoupling, nucleosynthesis, recombination, and reionization;see [10, 221] for reviews and Refs. [110, 222–227].

In the following, we discuss numerical simulations performed with the PENCIL CODE. This publicMHD code (https://github.com/pencil-code) (see also Ref. [228]) is particularly well suitedfor simulating turbulence owing to its high spatial (sixth order) and temporal (third order) accuracy, whilestill taking advantage of the finite difference in terms of speed and straightforward parallelization. Recentresults from the PENCIL CODE include MHD turbulence simulations at the electroweak or QCD phasetransitions [131–135].

a. Numerical Technique.By default, the PENCIL CODE solves the MHD equations for the logarith-mic densityln ρ, flow velocityv, and the magnetic vector potentialA as follows:

D

Dηln ρ = −∇ · v , (1)

D

Dηv = J ×B − c2s∇ ln ρ+ fvisc , (2)

∂

∂ηA = v ×B + fM + λ∇2A . (3)

Here, η is the conformal time andD/Dη ≡ ∂/∂η + v · ∇ is the advective time derivative,fvisc =

ν(

∇2v + 13∇∇ · v +G

)

is the compressible viscous force for constantν, Gi = 2Sij∇j ln ρ, and

Sij = 12(vi,j + vj,i) − 1

3δijvk,k is the traceless rate-of-strain tensor. The pressure is given byp = ρc2s,wherecs = 1/

√3 is the speed of sound in the case of an ultra-relativistic gas, andJ = ∇ ×B/4π is the

current density.In our simulations we use a vanishing magnetic forcing termfM = 0 everywhere, except for the purpose

of producing initial conditions, as explained below.b. Initial Conditions. To produce initial conditions, we run the simulation for a short time (∆t ≈

0.5λ1/cs) with a random (in time)δ-correlated magnetic forcefM in Eq. (3). The forcing term is composedof plane monochromatic waves pointing randomly in all possible directions with an average wavenumberk0 and fractional helicity〈fM ·∇× fM 〉/〈k0f2

M〉 = 2σ/(1 + σ2). Hereσ is the parameter characterizingthe initial forcing. Initial conditions for the magnetic and velocity fields produced from such a procedurehave the advantage of being turbulent, still self-consistent solutions of the MHD equations.

c. Effective Magnetic Field Characteristics.A magnetic field generated during phase transitionsthrough any magnetogenesis scenario should satisfy the causality condition [163, 164, 220]. The maxi-mal correlation lengthξmax of a causally generated primordial magnetic field should be shorter then theHubble radius at the time of generation,H−1

⋆ . We define the parameterγ = ξmax/H−1⋆ ≤ 1, which can

describe the number of primordial magnetic field bubbles inside the Hubble radius, and thusN ∝ γ−3.To account for the universe expansion we use comoving length, which is measured today and correspondsto the Hubble radius at the moment of magnetic field generation. Comoving length should be inversely

6

proportional to the phase transition temperature (T⋆):

λH⋆ = 5.8× 10−10 Mpc

(

100GeV

T⋆

)(

100

g⋆

)1/6

. (4)

For the QCD phase transition (g⋆ = 15 andT⋆ = 0.15 GeV), the comoving length equals 0.5 pc, whilefor the electroweak phase transition (g⋆ = 100 andT⋆ = 100 GeV) it should be equal to6× 10−4 pc. In allcases, the correlation length of the primordial magnetic field should not exceed the comoving value of theHubble radius:ξmax ≤ λH . Obviously, the latter condition accountsonly for the increase of the correlationlength due the expansion of the universe, and does not account for the effects of cosmic MHD turbulence(free decay or an inverse cascade in the case if primordial magnetic fields have nonzero helicity). Note, thatthe number of bubbles inside the Hubble radius is around 6 (γ ≃ 0.15) for the QCD phase transition andaround 100 (γ ≃ 0.01) for the electroweak phase transition. Thus, the maximal correlation length for theQCD and electroweak phase transitions should be 0.08 pc and6× 10−6 pc, respectively. On the other hand,the correlation length is unlimited in the case of inflation-generated magnetic fields.

As mentioned above, the primordial magnetic field contributes to the relativistic component and thus thetotal energy density of the primordial magnetic fieldρB(aN), whereaN is the scale factor during nucleosyn-thesis, is limited by the BBN bound: it cannot exceed10% of the radiation energy densityρrad(aN). It isstraightforward to see that the maximal value of the effective magnetic field defined through the total mag-netic energy does not depends on the temperature at the moment of generation (T∗), and depends weakly onthe relativistic degrees of freedom (γ) at the moment of the magnetic field generation.

The dominant contribution to the magnetic field energy density comes from the given length-scale,the so-calledintegral scale, where the magnetic field strength reaches its maximum. Thus, when dealingwith phase transition-generated magnetic fields, we adopt the following idealizing approximation: wegenerate initial conditions for freely decaying turbulence simulations by running a numerical simulation offorced MHD equations for a short time interval. The externalelectromagnetic force, intended to generatea turbulent state, is introduced in the form ofδ functions that peak at a characteristic wavenumber,k0 =

2π/ξ−10 . This yields random magnetic fields with correlation lengthξ0. Thus, the magnetic field strength

at the characteristic length scale isB(eff) =√8πρB . The characteristic length scale of the initial turbulent

stateξ0 is set by the size of the largest magnetic eddies, because theprimordial magnetic field evolves withthe primordial MHD turbulence. In this approach, the characteristic length scale of the magnetic field is setby the bubble size of the phase transition when magnetogenesis occurs.

d. Magnetic Field Spectrum.The interaction between magnetic field and plasma gives riseto kineticmotions, and the turbulent backreaction results in spreading of the spectral energy density of magnetic fieldover a range of wavenumbers. At scales longer then the integral scale of the turbulence (small wavenum-bers), the spectral energy density develops into the form ofa power lawEM = Akn, whereA is a normaliza-tion constant, andn is the spectral index. The spectrum of the turbulent magnetic field can be determined bythe spectral expansion of the two-point correlation function of the magnetic field〈Bi(x)Bj(x+ r)〉, whoseFourier transform with respect tor gives the spectral function

FMij (k) = Pij(k)

EM(k)

4πk2+ iεijlkl

HM(k)

8πk2. (5)

Here,Pij(k) = δij−kikj/k2, εijl is the antisymmetric tensor, andHM(k) is the magnetic helicity spectrum.

In this case, a white noise spectrum corresponds to the spectral indexn = 2 [163], while the Batchelorspectrum corresponds to the spectral indexn = 4 [164]. The power law of large-scale MHD turbulencespectrum extends down to the integral scaleξM , which is itself a time-dependent quantity throughout the

7

turbulence decay process. At scales shorter than the integral scale the spectral energy density of the magneticfield decreases rapidly due to the combined action of turbulent decay and viscous damping.

e. Decay Laws. The correlation length of the turbulent magnetic field evolves in time during the freedecay of turbulence. We may describe the decay laws of the magnetic correlation lengthξM(η) and thespectral energy densityEM (η) using two power law indicesnξ andnE:

ξM(η) = ξM(η0)

(

η

η0

)nξ

, (6)

EM (η) = EM (η0)

(

η

η0

)nE

. (7)

The spectral energy density of the primordial MHD turbulence spectrum can be split into its large-scale andshort-scale components, above and below the time dependentintegral scale:

EM (k, η) = E0(η)

{

k4 when k < kI(η)

k−5/3 when k > kI(η), (8)

wherek = k/kI andkI(η) = 2π/ξM (η). Hence, Eqs. (6) and (7) can be used to describe the time evolutionof spectral amplitude of magnetic fieldE0(η) for a given turbulent spectrum:

E0(η) =5

17πξM (η0)EM (η0)

(

η

η0

)nξ+nE

. (9)

IV. RESULTS

A. The Inverse Transfer for Non-Helical Fields

The inverse cascade is by now a well-known effect in helical magnetic turbulence [229]. One of theremarkable results is the presence ofnon-helical inverse transfer for magnetically dominated (causallygenerated) MHD turbulence; see Fig. 1, where we show spectral energy transfer rates, which demonstratethat the inverse transfer is about half as strong as with helicity. However, in both cases the magnetic gain atlarge scales results from velocity at similar scales interacting with smaller-scale magnetic fields [135]. Thisresult has not been emphasized in previous studies, see Refs. [230, 231] and references therein, and has nowbeen confirmed by independent research groups [232–234].

Recent high resolution simulations with different magnetic Prandtl numbers PrM = ν/λ [135] haveshown a cleark−2 spectrum in the inertial range. This is the first example of fully isotropic magneticallydominated MHD turbulence (governed by the phase transition-generated magnetic fields) exhibiting whatwe argue to be weak turbulence scaling [235]. On the other hand, the Kolmogorov scalingk−5/3 has beenrecovered for the case of the inflation-generated helical magnetic fields [139].

B. Inflation Generated Magnetic Fields

Magnetic fields generated during the inflation should be affected by cosmological phase transitionsoccurring at later times during the expansion of the universe. In this case, a separate study of the imprint ofphase transitions on cosmic magnetic fields is needed. For this purpose we adopt a general approach that can

8

FIG. 1: Spectral transfer functionTkpq, (a) as a function ofk and summed over allp andq, (b) as a function ofp andq for k/k1 = 4, and (c) as a function ofk andq for p/k1 = 4. The dashed line in (a) and the insets in (b) and (c) showthe corresponding case for a direct numerical simulations with helicity; both for PrM = 1. See Fig. 3 of Ref. [135].

be applied to both, QCD and electroweak phase transitions. In each case, turbulence forcing is determinedby the phase transition bubble size. Rapid phase transitions generate turbulence, which then decays slowlyat large scales. In contrast to previous studies, the inflation-generated magnetic field is not frozen into thecosmic plasma. Turbulence is generated during a short forcing period, which then is followed by slow decay(see Refs. [131, 132] for details). Recent simulations showed an increasing characteristic length scale ofthe velocity field and the establishment of ak2 (white noise) spectrum at large scales. This increase ofvorticity perturbations occurs until it reaches equipartition with the magnetic field [134]. Figure 2 showsthe evolution of kinetic and magnetic field spectra from those simulations. Numerical results show thatinflation-generated magnetic fields are not significantly modified at large scales by their coupling to theplasma during a cosmological phase transition. The coupling of cosmic magnetic field with the phasetransition-generated fluid turbulence leads to deviationsof the magnetic field spectrum from the initialscale-invariant shape only at intermediate scales.

Ongoing research consists in pursuing high resolution numerical simulations of helical inflation-generated magnetic field evolution. Such a field, being subject to inflationary expansion, is characterized bya scale-invariant spectrumn → −1, and its correlation length can be as large as Hubble horizontoday oreven larger (i.e., even when the total energy densityEM is finite, the correlation lengthξM ∝ ∫

dkEM (k)/k

divergences fork → 0). In contrast to well known helical magnetic field decay laws[13, 165, 226, 236–238, 240–244],an absenceof the inverse cascade has been found for inflation-generated magnetic fields.Furthermore, an unusually slow growth of the correlation length and conservation of helicity has been re-covered even for the case of partially helical magnetic fields. These unexpected and unknown featuresof magnetic helicity are the result of a substantial turbulent power at large scales and the impossibility ofthe redistribution of helical fields at small wavenumbers (only the forward cascade is possible). A morethorough investigation of this phenomenon will be performed through varying initial conditions and basicparameters of primordial plasma.

C. Growth of Helical Structures

It is long known that the magnetic helicity plays a crucial role in determining the evolution pattern ofMHD turbulence. Distinct evolution characteristics are known for helical and non-helical fields. In recentsimulations, a partially helical initial magnetic field wasused [133], assuming a tiny initial magnetic helicity

9

FIG. 2: Magnetic (solid lines) and kinetic (dashed lines) energy spectra in regular time intervals. Re≈ 170. Themagnetic and kinetic spectra at the last time are additionally marked in red and blue, respectively.

FIG. 3: Visualizations ofBx (upper row) andvx (lower row) at three times during the magnetic decay of a weaklyhelical field withσ = 0.03 generated during QCD phase transitions. See Fig. 2 of Ref. [133].

during the QCD phase transition. It was shown that at late times the resulting field attains the maximallyallowed magnetic helicity. This result is important since helicity crucially affects the MHD dynamics,and has very interesting consequences in astrophysical objects (e.g. galactic magnetic fields [245–247],for example). The resulting magnetic field has an amplitude of around 0.04 nG and a correlation lengthof order 20 kpc, which (assuming realistic scenarios of amplification [31]) serves as a seed for galacticmagnetic fields. At this point the electroweak phase transition-generated magnetic fields are less promisingdue to a smaller initial correlation length, but are not completely excluded [248], in particular for fullyhelical magnetic fields [134].

Magnetic helicity is a crucial factor that affects the evolution of primordial magnetic fields. The evo-

10

FIG. 4: Comparison ofBx (upper row) andln ρ (lower row) for an inflation-generated magnetic field withσ = 1

(left) andσ = 0.03 (right).

lution of the primordial magnetic field that has been produced with weak initial magnetic helicity thatundergoes two consecutive stages. During the first stage, the evolution of a partially helical magnetic fieldspectrum is much similar to that of non-helical magnetic fields, and is sometimes described as a directcascade. At this stage the spectral energy density cascadesfrom large to small scales, where it undergoesde-correlation and viscous damping. Magnetic helicity is conserved and hence its fractional value increasesduring the turbulent decay process. The second stage in the primordial magnetic field evolution sets in whenthe turbulent state with maximal helicity is reached. The maximal value of the magnetic helicity that canbe reached is limited by the realizability condition. Indeed, the conservation of magnetic helicity leads to adecay of the magnetic energy density inversely proportional to the correlation length of the turbulence:

ξM (η) ≥ ξminM (η) ≡ |HM (η)|/2EM (η), (10)

whereξminM (η) is the minimal correlation length of the turbulent state. Hence, an inverse cascade develops

during the second stage of the primordial magnetic field evolution. In Ref. [133], we have studied theξM(η)

andξminM (η) for σ = 1, 0.1, and0.03 in the case of the QCD phase transition. It seems that, especially at

lower σ, the increase ofξM is slow (∼ η1/2) while ξM(η) ≫ ξminM (η). However, since magnetic helicity

conservation implies thatEM decreases asη−1, the minimal correlation lengthξminM (η) soon reachesξM (η);

see Fig. (5). When the correlation length of the turbulent magnetic field reaches a minimal correlationlength, the turbulence reaches its fully helical state. Then the turbulence decays according to the helicalturbulent decay laws:ξM ∼ η2/3 andEM ∼ η−2/3. Hence, we identify two distinct phases in the MHDturbulence evolution: the phase ofweakly helicalturbulence decay withnξ = 1/2 andnE = −1, andthe phase offully helical turbulent decay withnξ = 2/3 andnE = −2/3. The fully helical case ischaracterized by an inverse cascade whereE0(η) ∝ ξM (η)EM (η) = const (see Eq. (9). These results arein full agreement with earlier works [13, 165, 236–238, 240,242]. The effective coupling of the primordialmagnetic fields and cosmic plasma ends at the time of recombination. At later stages, primordial magneticfields exhibit much slower developments [220].

11

FIG. 5: Evolution of turbulent correlation lengthξM (η) (solid) and minimal correlation lengthξminM (η) (dashed) of a

helical state forσ = 1 (black), 0.1 (blue) and 0.03 (red). (Fig. 3 of Ref. [133]).

Knowing the initial values of the turbulent magnetic correlation lengthξM(η0), the minimal correlationlengthξmin

M (η0) set by the realizability condition, we can calculate the time intervalηfully needed for the tur-bulence to reach its fully helical state during the decay process. Since these two scales approach each otherasη1/2, the result isηfully = η0[ξM (η0)/ξ

minM (η0)]

2. Hence, the time interval needed for the developmentof a fully helical state can be calculated using the initial values of turbulent energy and helicity:

ηfully = 4η0ξ2ME2

M/H2M . (11)

Note that this time increases inversely proportional to thesquare of the initial helicity of the turbulentstateHM . Assuming that the initial magnetic helicity isξM/λH⋆ times lower then the maximal helicity inthe case of strong CP violation during phase transition, we can calculate the time needed for the cosmicturbulence to reach its maximally helical state:ηfully = η0/γ

2.

V. DISCUSSION

Let us now discuss our results in the broader context of the workshop ‘Mixing in Rapidly ChangingEnvironments—Probing Matter at the Extremes’. Our work has demonstrated a rather generic trend of de-caying MHD turbulence to display an increase of energy at large length scales. This process is well-knownin helical MHD turbulence [229], but to a lesser extent it also occurs in non-helical MHD turbulence [135].Moreover, if there is small fractional helicity initially,this fraction will increase with time proportional tothe square root of time. At a certain time, the fractional helicity will be 100%, after which the decay ofenergy slows down and the increase of the correlation lengthspeeds up.

An increase of energy at large length scales is not a common phenomenon in hydrodynamic turbulenceand has never been found for passive scalars. This special behavior in MHD is likely to lead to uncon-ventional mixing properties, although this has not yet beenwell quantified for decaying MHD turbulence.However, for statistically stationary turbulence, an increase of energy at small wavenumbers is usually de-scribed as non-diffusive turbulent transport, which is particularly well known in mean-field dynamo theory[250]. In a sense, this is more reminiscent of what might looklike “anti-mixing”. This is to some extent due

12

to the fact that vector fields behave differently from scalarfields. This can in part be due to the presence ofadditional conservation laws. In particular, magnetic helicity provides an extremely powerful constraint.

The significance of studying decaying MHD turbulence is manifold. On the one hand, our results willhelp to better understand the nature of cosmic magnetism andwill gain insight into the decay laws of cosmicMHD turbulence. On the other hand, our analysis can be applied not only to cosmological scales, but tomolecular clouds or even protoplanetary disks where decaying magnetic turbulence can crucially affectthe global state or the formation of local structures. The results of our research have therefore importantimplications in many areas, including fluid dynamics, earlyuniverse physics, high energy astrophysics,MHD modeling, and large-scale structure formation in the universe.

VI. CONCLUSIONS

We have discussed the evolution of primordial magnetic fields during the expansion of the universeand have addressed some of the observational signatures. The coupling between the magnetic fields andcosmic turbulence leads to novel results as compared to previously adopted frozen-in approximations whenmagnetic field evolution was considered solely due to the field line dilution in the expanding universe. Forthis purpose, we have developed a comprehensive theoretical model to consider the evolution of MHDturbulence over a wide range of physical conditions, beyondany specific astrophysical application.

Numerical simulations have shown novel effects in the evolution of magnetic fields in cosmic turbulence.It seems that inverse transfer that normally occurs in helical MHD turbulence, can also take place for non-helical magnetic fields if the MHD turbulence is magnetically dominated. In this case, large-scale velocityperturbations power up the magnetic field, leading to substantial increase of magnetic power spectra at largescales and corresponding inverse transfer.

An analysis of the inflation-generated magnetic field is carried out to find out how they are affectedby the cosmic phase transitions (QCD and electroweak phase transition). Numerical results show thatlarge-scale magnetic fields survive phase transitions, andthus, phase transitions cannot rule out inflationarymagnetogenesis as the source of seed magnetic field in the universe.

On the other hand, magnetic fields generated during phase transitions can have tiny helicity. We haveshown that during the evolution in MHD turbulence magnetic field helicity grows until it reaches maximalhelical state. It seems that helicity growth rate is fast enough to reach maximal helicity well before theepoch of recombination, when the primordial magnetic field decouples from cosmic turbulence.

We have used results of high resolution simulations to set limits on the large-scale magnetic field gener-ated during early stages of the evolution of the universe (inflation or cosmological phase transitions). Indeed,it seems that magnetic fields produced during this epochs andsubsequently modified by cosmic MHD turbu-lence can reach amplitudes similar to the lower bounds of theobservational magnetic fields even accountingfor the effects of large-scale turbulent decay as well as additional Alfven wave damping. The extremely lowvalues derived for smoothed magnetic field [166] do not implythat the effective magnetic field is also smallin the1pc–1 kpc range and cannot lead to the observational signatures inblazar emission spectra. Usingthe effective magnetic field approach we obtain results thatdo not depend on the specific spectral shapeof the magnetized turbulence. Observational signatures ofthe magnetogenesis during electroweak phasetransition can come from future observations, if weak magnetic fields with10−14–10−15 G amplitude andfew pc correlation length are detected. While a somewhat stronger magnetic field with a correlation lengthof the order of kpc might indicate the presence of QCD phase transition magnetogenesis. In turn, magneticfields with extremely large correlation length (1Mpc or higher) will indicate inflationary magnetogenesis.

13

FIG. 6: Evolution of the effective magnetic fieldB(eff) and correlation lengthξM in the case of magnetogenesis atelectroweak phase transition (green) and QCD phase transition (orange). Arrows indicate the evolutionary path of thestrength and integral scale of helical and non-helical turbulent magnetic fields during the radiation-dominated era upto their final values. Thick solid line(s) show possible present day field strengths and integral scales of the magneticfield generated during phase transitions (see Fig. 7 of Ref. [134]).

Acknowledgments

T.K. would like to express her appreciation to Organizing Committee of Turbulent Mixing and BeyondWorkshop’Mixing in Rapidly Changing Environments - Probing Matter at the Extremes’(2014, The AbdusSalam International Centre for Theoretical Physics, ICTP)and the Center for hospitality. It is our pleasureto thank Marco Ajello, Nick Battaglia, Leonardo Campanelli, Rupert Croft, Ruth Durrer, Arthur Kosowsky,Francesco Miniati, Aravind Natarjan, Andrii Neronov, Yurii Maravin, and Tanmay Vachaspati for usefuldiscussions. We acknowledge partial support from the Swedish Research Council grants 621-2011-5076and 2012-5797, the European Research Council AstroDyn Research Project 227952, the FRINATEK grant231444 under the Research Council of Norway, the Swiss NSF SCOPES grant IZ7370-152581, the NASAAstrophysics Theory program grant NNXl0AC85G, the NSF Astrophysics and Astronomy Grant Programgrants AST-1109180, AST-1615100 and AST-1615940, and Shota Rustaveli Georgian National ScienceFoundation grant FR/264/6-350/14.

[1] L. M. Widrow, Rev. Mod. Phys.74, 775 (2002).[2] J. P. Vallee, New Astron. Rev.48, 763 (2004).[3] M. Giovannini, Lect. Notes Phys.737, 863 (2008).[4] R. Beck, AIP Conf. Proc.1085, 83 (2009); R. Beck, In Proc. of Magnetic Fields in the Universe: From

14

Laboratory and Stars to Primordial Structures, eds. M. Soida, K. Otmianowska-Mazur, E.M. de Gouveia DalPina and A. Lazarian; SKA design and timeline updates, arXiv:1111.5802 [astro-ph.CO].

[5] C. Van Eck,et al., Astrophys. J.728, 97 (2011).[6] A. Fletcher, R. Beck, A. Shukurov, E. M. Berkhuijsen and C. Horellou, Mon. Not. Roy. Astron. Soc.112, 2396

(2011).[7] R. Beck, AIP Conf. Proc.1381, 117 (2011).[8] M. L. Bernet, F. Miniati, S. J. Lilly, P. P. Kronberg and M.Dessauges-Zavadsky, Nature454, 302 (2008).[9] P. P. Kronberg, M. L. Bernet, F. Miniati, S. J. Lilly, M. B.Short and D. M. Higdon, Astrophys. J.676, 7079

(2008).[10] R. Durrer and A. Neronov, Astron. Astrophys. Rev.21, 62 (2013).[11] R. M. Kulsrud and E. G. Zweibel, Rept. Prog. Phys.71, 0046091 (2008).[12] A. Kandus, K. E. Kunze and C. G. Tsagas, Phys. Rept.505, 1 (2011).[13] D. Biskamp,Magnetohydrodynamic Turbulence(Cambridge: Cambridge Univ. Press) 2003.[14] A. Kravtsov and S. Borgani, Ann. Rev. Astron. Astrophys., 50, 353 (2012).[15] K. Subramanian, A. Shukurov and N. E. L. Haugen, Mon. Not. Roy. Astron. Soc.366, 1437 (2006).[16] F. Vazza, G. Tormen, R. Cassano, G. Brunetti and K. Dolag, Mon. Not. Roy. Astron. Soc.369, L14 (2006).[17] T. Fang, P. J. Humphrey and D. A. Buote, Astrophys. J.691, 1648 (2009).[18] T. H. Greif, J. L. Johnson, R. S. Klessen and V. Bromm, Mon. Not. Roy. Astron. Soc.387, 1021 (2008).[19] R. Pakmor, A. Bauer and V. Springel, Mon. Not. Roy. Astron. Soc.418, 1392 (2012).[20] F. Vazza, M. Brggen, C. Gheller and P. Wang, Mon. Not. Roy. Astron. Soc.445, no. 4, 3706 (2014).[21] J. Cho, Astrophys. J.797, no. 2, 133 (2014).[22] J. R. Espinosa, T. Konstandin, J. M. No and G. Servant, Phys. Rev. D76, 083002 (2007).[23] J. R. Espinosa, T. Konstandin, J. M. No and G. Servant, JCAP 1006, 028 (2010).[24] S.-N. X. Medina, S. J. Arthur, W. J. Henney, G. Mellema, A. Gazol, Mon. Not. Roy. Astron. Soc.445, 1797

(2014).[25] A. Neronov and I. Vovk, Science328, 73 (2010).[26] F. Tavecchio, G. Ghisellini, G. Bonnoli and L. Foschini, Mon. Not. Roy. Astron. Soc.414, 3566 (2011).[27] W. Essey, S. Ando and A. Kusenko, Astropart. Phys.35, 135 (2011).[28] A. M. Taylor, I. Vovk and A. Neronov, Astron. Astrophys.529, A144 (2011).[29] H. Huan, T. Weisgarber, T. Arlen and S. P. Wakely, Astrophys. J.735, L28 (2011).[30] I. Vovk, A. M. Taylor, D. Semikoz and A. Neronov, Astrophys. J.747, L14 (2012).[31] K. Dolag, M. Kachelriess, S. Ostapchenko and R. Tomas, Astrophys. J.727, L4 (2011).[32] C. D. Dermer, M. Cavadini, S. Razzaque, J. D. Finke and B.Lott, Astrophys. J.733, L21 (2011).[33] K. Takahashi, M. Mori, K. Ichiki and S. Inoue, Astrophys. J.744, L7 (2012).[34] J. D. Finke, L. C. Reyes, M. Georganopoulos, K. Reynolds, M. Ajello, S. J. Fegan and K. McCann, Astrophys.

J.814, no. 1, 20 (2015).[35] T. C. Arlen, V. V. Vassiliev, T. Weisgarber, S. P. Wakelyand S. Y. Shafi, Astrophys. J.796, 18 (2014).[36] A. E. Broderick, P. Chang and C. Pfrommer, Astrophys. J.752, 22 (2012).[37] F. Miniati and A. Elyiv, Astrophys. J.770, 54 (2013).[38] M. Lemoine, J. Plasma Phys.81, 4501 (2015).[39] A. Kosowsky and A. Loeb, Astrophys. J.469, 1 (1996).[40] D. D. Harari, J. D. Hayward and M. Zaldarriaga, Phys. Rev. D 55, 1841 (1997).[41] C. Scoccola, D. Harari and S. Mollerach, Phys. Rev. D70, 063003 (2004) [astro-ph/0405396].[42] L. Campanelli, A. D. Dolgov, M. Giannotti and F. L. Villante, Astrophys. J.616, 1 (2004).[43] A. Kosowsky, T. Kahniashvili, G. Lavrelashvili and B. Ratra, Phys. Rev. D71, 043006 (2005).[44] T. Kahniashvili, Y. Maravin and A. Kosowsky, Phys. Rev.D 80, 023009 (2009).[45] M. Giovannini and K. E. Kunze, Phys. Rev. D78, 023010 (2008).[46] T. Kahniashvili, A. G. Tevzadze, S. K. Sethi, K. Pandey and B. Ratra, Phys. Rev. D82, 083005 (2010).[47] L. Pogosian, A. P. S. Yadav, Y. F. Ng and T. Vachaspati, Phys. Rev. D84, 043530 (2011).[48] P. A. R. Ade,et al. [POLARBEAR Collaboration], arXiv:1509.02461 [astro-ph.CO].

15

[49] M. S. Pshirkov, P. G. Tinyakov and F. R. Urban, arXiv:1504.06546 [astro-ph.CO].[50] P. P. Kronberg, M. Simard-Normandin, Nature,263, 653 (1976).[51] P. P. Kronberg, J. J. Perry, Astrophys. J.263, 518 (1982).[52] P. Blasi, S. Burles, A. V. Olinto Astrophys. J.514, L79, (1999).[53] E. Komatsu,et al. [WMAP Collaboration], Astrophys. J. Suppl.180, 330 (2009).[54] et al. [Planck Collaboration], arXiv:1502.01594 [astro-ph.CO].[55] J. A. Adams, U. H. Danielsson, D. Grasso and H. Rubinstein, Phys. Lett. B388, 253 (1996).[56] K. Subramanian and J. D. Barrow, Phys. Rev. Lett.81, 3575 (1998).[57] R. Durrer, P. G. Ferreira and T. Kahniashvili, Phys. Rev. D 61, 043001 (2000).[58] A. Mack, T. Kahniashvili and A. Kosowsky, Phys. Rev. D65, 123004 (2002).[59] L. Pogosian, T. Vachaspati and S. Winitzki, Phys. Rev. D65, 083502 (2002).[60] K. Subramanian, T. R. Seshadri and J. D. Barrow, Mon. Not. Roy. Astron. Soc.344, L31 (2003).[61] K. Subramanian and J. D. Barrow, Mon. Not. Roy. Astron. Soc.335, L57 (2002).[62] C. Caprini, R. Durrer and T. Kahniashvili, Phys. Rev. D69, 063006 (2004).[63] A. Lewis, Phys. Rev. D70, 043011 (2004).[64] T. Kahniashvili, Astron. Nachr.327, 414 (2006).[65] T. Kahniashvili and B. Ratra, Phys. Rev. D71, 103006 (2005).[66] T. Kahniashvili and B. Ratra, Phys. Rev. D75, 023002 (2007).[67] J. R. Kristiansen and P. G. Ferreira, Phys. Rev. D77, 123004 (2008).[68] D. Paoletti, F. Finelli and F. Paci, Mon. Not. Roy. Astron. Soc.396, 523 (2009).[69] J. R. Shaw and A. Lewis, Phys. Rev. D81, 043517 (2010).[70] D. G. Yamazaki, K. Ichiki, T. Kajino and G. .J. Mathews, Phys. Rev. D81, 103519 (2010).[71] K. Ichiki, K. Takahashi and N. Sugiyama, Phys. Rev. D85, 043009 (2012).[72] D. G. Yamazaki, K. Ichiki, T. Kajino and G. J. Mathew, Adv. Astron.2010, 586590 (2010).[73] D. Paoletti and F. Finelli, Phys. Rev. D83, 123533 (2011).[74] K. E. Kunze, Phys. Rev. D83, 023006 (2011).[75] K. E. Kunze, Phys. Rev. D85, 083004 (2012).[76] P. Trivedi, T. R. Seshadri and K. Subramanian, Phys. Rev. Lett.108, 231301 (2012).[77] D. Paoletti and F. Finelli, Phys. Lett. B726, 45 (2013).[78] P. Chen and T. Suyama, Phys. Rev. D88, 123521 (2013).[79] D. G. Yamazaki, K. Ichiki and K. Takahashi, Phys. Rev. D88, no. 10, 103011 (2013).[80] P. Trivedi, K. Subramanian and T. R. Seshadri, Phys. Rev. D 89, 043523 (2014).[81] T. Kahniashvili, Y. Maravin, G. Lavrelashvili and A. Kosowsky, Phys. Rev. D90, 083004 (2014).[82] M. Ballardini, F. Finelli and D. Paoletti, JCAP1510, no. 10, 031 (2015).[83] K. Jedamzik, V. Katalinic and A. V. Olinto, Phys. Rev. Lett. 85, 700 (2000).[84] J. B. Dent, D. A. Easson and H. Tashiro, Phys. Rev. D86, 023514 (2012).[85] H. Tashiro, J. Silk and D. J. E. Marsh, Phys. Rev. D88, 125024 (2013). [arXiv:1308.0314 [astro-ph.CO]].[86] K. E. Kunze and E. Komatsu, JCAP1401, 009 (2014).[87] K. Miyamoto, T. Sekiguchi, H. Tashiro and S. Yokoyama, Phys. Rev. D89, no. 6, 063508 (2014).[88] K. Jedamzik and T. Abel, JCAP1310, 050 (2013).[89] M. A. Amin and D. Grin, Phys. Rev. D90, no. 8, 083529 (2014).[90] K. E. Kunze and E. Komatsu, JCAP1506, 027 (2015).[91] J. Chluba, D. Paoletti, F. Finelli and J. A. Rubio-Martn, Mon. Not. Roy. Astron. Soc.451, no. 2, 2244 (2015).[92] J. M. Wagstaff and R. Banerjee, arXiv:1508.01683 [astro-ph.CO].[93] R. Durrer, T. Kahniashvili and A. Yates, Phys. Rev. D58, 123004 (1998).[94] G. Chen, P. Mukherjee, T. Kahniashvili, B. Ratra, Y. Wang, Astrophys. J.611, 655 (2004).[95] I. Brown and R. Crittenden, Phys. Rev. D72, 063002 (2005).[96] M. Demianski and A. G. Doroshkevich, Phys. Rev. D75, 123517 (2007).[97] A. Bernui and W. S. Hipolito-Ricaldi, Mon. Not. R. Astron. Soc.389, 1453 (2008).[98] P. K. Samal, R. Saha, P. Jain and J. P. Ralston, Mon. Not. R. Astron. Soc.396, 511 (2009).

16

[99] T. Kahniashvili, G. Lavrelashvili and B. Ratra, Phys. Rev. D78, 063012 (2008).[100] P. Naselsky and J. Kim, arXiv:0804.3467 [astro-ph].[101] J. Kim and P. Naselsky, JCAP0907, 041 (2009).[102] P. A. R. Ade,et al. [Planck Collaboration], Astron. Astrophys.571, A23, (2014).[103] M. Kawasaki and M. Kusakabe, Phys. Rev. D86, 063003 (2012).[104] D. G. Yamazaki, M. Kusakabe, T. Kajino, G. J. Mathews and M. K. Cheoun, Phys. Rev. D90, 023001 (2014).[105] D. G. Yamazaki and M. Kusakabe, Phys. Rev. D86, 123006 (2012).[106] L. M. Widrow, D. Ryu, D. R. G. Schleicher, K. Subramanian, C. G. Tsagas and R. A. Treumann, Space Sci.

Rev.166, 37 (2012).[107] J. R. Shaw and A. Lewis, Phys. Rev. D86, 043510 (2012).[108] T. G. Cowling, Mon. Not. Roy. Astron. Soc., 116, 114 (1956).[109] L. Mestel and L. Spitzer, Mon. Not. Roy. Astron. Soc., 116, 503 (1956).[110] K. Jedamzik, V. Katalinic and A. V. Olinto, Phys. Rev. D57, 3264 (1998).[111] S. K. Sethi and K. Subramanian, Mon. Not. Roy. Astron. Soc.356, 778 (2005).[112] H. Tashiro and N. Sugiyama, Mon. Not. Roy. Astron. Soc.368, 965 (2006).[113] H. Tashiro, N. Sugiyama and R. Banerjee, Phys. Rev. D73, 023002 (2006).[114] H. Tashiro and N. Sugiyama, Mon. Not. Roy. Astron. Soc.372, 1060 (2006).[115] D. R. G. Schleicher, R. Banerjee and R. S. Klesser, Phys. Rev. D78, 083005 (2008).[116] D. R. G. Schleicher, R. Banerjee and R. S. Klessen, Astrophys. J.692, 236 (2009).[117] S. K. Sethi and K. Subramanian, JCAP0911, 021 (2009).[118] H. Tashiro and N. Sugiyama, Mon. Not. Roy. Astron. Soc.411, 1284 (2011).[119] S. K. Sethi, B. B. Nath and K. Subramanian, Mon. Not. Roy. Astron. Soc.387, 1589 (2008).[120] D. G. Yamazaki, K. Ichiki, T. Kajino and G. J. Mathews, Phys. Rev. D81, 023008 (2010).[121] D. R. G. Schleicher and F. Miniati, Mon. Not. Roy. Astron. Soc. Lett.,418, L143 (2011).[122] T. Kahniashvili, Y. Maravin, A. Natarajan, N. Battaglia and A. G. Tevzadze, Astrophys. J.770, 47 (2013).[123] C. Fedeli and L. Moscardini, JCAP1211, 055 (2012).[124] K. L. Pandey and S. K. Sethi, Astrophys. J.762, 15 (2013).[125] K. L. Pandey, T. R. Choudhury, S. K. Sethi and A. Ferrara, Mon. Not. Roy. Astron. Soc.451, no. 2, 1692

(2015).[126] T. Venumadhav, A. Oklopcic, V. Gluscevic, A. Mishra and C. M. Hirata, arXiv:1410.2250 [astro-ph.CO].[127] E. O. Vasiliev and S. K. Sethi, Astrophys. J.786, 142 (2014).[128] P. A. R. Ade,et al. [Planck Collaboration], arXiv:1502.04123 [astro-ph.GA].[129] F. Miniati and A. R. Bell, Astrophys. J.729, 73 (2011).[130] T. Kahniashvili, A. G. Tevzadze and B. Ratra, Astrophys. J.726, 78 (2011).[131] T. Kahniashvili, A. Brandenburg, A. G. Tevzadze and B.Ratra, Phys. Rev. D81, 123002 (2010).[132] T. Kahniashvili, A. Brandenburg, L. Campanelli, B. Ratra and A. G. Tevzadze, Phys. Rev. D86, 103005 (2012).[133] A. G. Tevzadze, L. Kisslinger, A. Brandenburg and T. Kahniashvili, Astrophys. J.759, 54 (2012).[134] T. Kahniashvili, A. G. Tevzadze, A. Brandenburg and A.Neronov, Phys. Rev. D87, no. 8, 083007 (2013).[135] A. Brandenburg, T. Kahniashvili and A. G. Tevzadze, Phys. Rev. Lett.114, no. 7, 075001 (2015).[136] T. Kahniashvili, A. Brandenburg and A. G. Tevzadze, arXiv:1507.00510 [astro-ph.CO].[137] A. Kahniashvili, A. Brandenburg, A. Tevzadze, T. Vachaspati, “Helical Magnetic Fields from Phase Transi-

tions”, 2016, in preparation.[138] A. Kahniashvili, R. Durrer, A. Brandenburg, A. Tevzadze, W. Yin, “Inflationary Magnetic Helicity: Evolution

and Signatures”, 2016, in preparation.[139] A. Brandenburg, T. Kahniashvili, A. Tevzadze, “Absence of the Inverse Cascade for Inflation-Generated Mag-

netic Fields”, 2016, in preparation.[140] A. Brandenburg and T. Kahniashvili, arXiv:1607.01360 [physics.flu-dyn].[141] E. Fermi, ”On the Origin of the Cosmic Radiation”, Phys. Rev.75, 1169 (1949).[142] E. R. Harrison, Mon. R. Astron. Soc.147, 279 (1970).[143] T. Vachaspati, Phys. Lett. B265, 258 (1991).

17

[144] J. M. Cornwall, Phys. Rev.D 56, 6146, (1997).[145] G. Sigl, A. V. Olinto and K. Jedamzik, Phys. Rev. D55, 4582 (1997).[146] M. Joyce and M. E. Shaposhnikov, Phys. Rev. Lett.79, 1193 (1997).[147] M. Hindmarsh and A. Everett, Phys. Rev. D58, 103505 (1998).[148] K. Enqvist, Int. J. Mod. Phys.D 7, 331 (1998).[149] D. Grasso and A. Riotto, Phys. Lett. B418, 258 (1998).[150] J. Ahonen and K. Enqvist, Phys. Rev.D 57, 664 (1998).[151] M. Giovannini, Phys. Rev.D 61, 063004 (2000).[152] T. Vachaspati, Phys. Rev. Lett.87, 251302 (2001).[153] A. D. Dolgov and D. Grasso, Phys. Rev. Lett.88, 011301 (2002).[154] D. Grasso and A. Dolgov, Nucl. Phys. Proc. Suppl.110, 189 (2002).[155] D. Boyanovsky, H. J.de Vega, and M. Simionato, Phys. Rev. D 67, 023502 (2003).[156] D. Boyanovsky, H. J. de Vega and M. Simionato, Phys. Rev. D 67, 123505 (2003).[157] A. Diaz-Gil, J. Garcia-Bellido, M. Garcia Perez, and A. Gonzalez-Arroyo, Phys. Rev. Lett.100, 241301 (2008).[158] T. Stevens, M. B. Johnson, L. S. Kisslinger, E. M. Henley, W.-Y. P. Hwang, and M. Burkardt, Phys. Rev.D 77,

023501 (2008).[159] J. M. Quashnock, A. Loeb and D. N. Spergel, Astrophys. J. 344, L49 (1989).[160] E. M. Henley, M. B. Johnson, and L. S. Kisslinger, Phys.Rev. D81, 085035 (2010).[161] F. R. Urban and A. R. Zhitnitsky, Phys. Rev. D82, 043524 (2010).[162] T. Stevens, M. B. Johnson, L. S. Kisslinger and E. M. Henley, Phys. Rev. D85, 063003 (2012).[163] C. J. Hogan, Phys. Rev. Lett.51, 1488 (1983).[164] R. Durrer and C. Caprini, JCAP0311, 010 (2003).[165] P. A. Davidson,Turbulence(Oxford: Oxford University Press, 2004).[166] C. Caprini and R. Durrer, Phys. Rev. D65, 023517 (2001).[167] J. M. Cornwall, Phys. Rev. D56, 6146 (1997).[168] M. Giovannini and M. E. Shaposhnikov, Phys. Rev. D57, 2186 (1998).[169] G. B. Field and S. M. Carroll, Phys. Rev. D62, 103008 (2000).[170] T. Vachaspati, Phys. Rev. Lett.87, 251302 (2001).[171] H. Tashiro, T. Vachaspati and A. Vilenkin, Phys. Rev. D86, 105033 (2012).[172] G. Sigl, Phys. Rev. D66, 123002 (2002).[173] K. Subramanian and A. Brandenburg, Phys. Rev. Lett.93, 205001 (2004).[174] L. Campanelli and M. Giannotti, Phys. Rev. D72, 123001 (2005).[175] V. B. Semikoz and D. D. Sokoloff, Astron. Astrophys.433, L53 (2005); “Large-scale cosmological magnetic

fields and magnetic helicity,” Int. J. Mod. Phys. D14, 1839 (2005).[176] A. Diaz-Gil, J. Garcia-Bellido, M. Garcia Perez and A.Gonzalez-Arroyo, Phys. Rev. Lett.100, 241301 (2008).[177] L. Campanelli, Int. J. Mod. Phys. D18, 1395 (2009).[178] L. Campanelli, Phys. Rev. Lett.111, no. 6, 061301 (2013).[179] R. K. Jain, R. Durrer and L. Hollenstein, J. Phys. Conf.Ser.484, 012062 (2014).[180] A. J. Long, E. Sabancilar and T. Vachaspati, JCAP1402, 036 (2014).[181] H. Tashiro, W. Chen, F. Ferrer and T. Vachaspati, Mon. Not. Roy. Astron. Soc.445, L41 (2014).[182] H. Tashiro and T. Vachaspati, Mon. Not. Roy. Astron. Soc. 448, 299 (2015).[183] M. S. Turner and L. M. Widrow, Phys. Rev. D37, 2743 (1988).[184] B. Ratra, Astrophys. J.391, L1 (1992).[185] K. Bamba and J. Yokoyama, Phys. Rev. D69, 043507 (2004).[186] L. Motta and R. R. Caldwell, Phys. Rev. D85, 103532 (2012).[187] R. R. Caldwell, L. Motta and M. Kamionkowski, Phys. Rev. D 84, 123525 (2011).[188] D. Lemoine and M. Lemoine, Phys. Rev. D52, 1955 (1995).[189] M. Gasperini, M. Giovannini and G. Veneziano, Phys. Rev. Lett. 75, 3796 (1995).[190] G. Lambiase, S. Mohanty and G. Scarpetta, JCAP0807, 019 (2008).[191] L. Campanelli, P. Cea, G. L. Fogli and L. Tedesco, Phys.Rev. D77, 043001 (2008).

18

[192] L. Campanelli and P. Cea, Phys. Lett. B675, 155 (2009).[193] K. E. Kunze, Phys. Rev. D81, 043526 (2010).[194] K. Bamba, C. Q. Geng and L. W. Luo, JCAP1210, 058 (2012).[195] J. B. Jimnez and A. L. Maroto, Springer Proc. Phys.137, 215 (2011).[196] F. A. Membiela and M. Bellini, Eur. Phys. J. C72, 2181 (2012).[197] T. Fujita and S. Mukohyama, JCAP1210, 034 (2012).[198] L. Motta and R. R. Caldwell, Phys. Rev. D85, 103532 (2012).[199] C. Bonvin, C. Caprini and R. Durrer, Phys. Rev. D86, 023519 (2012).[200] C. T. Byrnes, L. Hollenstein, R. K. Jain and F. R. Urban,JCAP1203, 009 (2012).[201] R. R. Caldwell, L. Motta and M. Kamionkowski, Phys. Rev. D 84, 123525 (2011).[202] K. Bamba, C. Q. Geng, S. H. Ho and W. F. Kao, Eur. Phys. J. C72, 1978 (2012).[203] J. Martin and J. Yokoyama, JCAP0801, 025 (2008).[204] J. Beltran Jimenez and A. L. Maroto, Phys. Rev. D83, 023514 (2011).[205] R. Durrer, L. Hollenstein and R. K. Jain, JCAP1103, 037 (2011).[206] M. Das and S. Mohanty, Int. J. Mod. Phys. A27, 1250040 (2012).[207] F. A. Membiela and M. Bellini, JCAP1010, 001 (2010).[208] V. Demozzi, V. Mukhanov and H. Rubinstein, JCAP0908, 025 (2009).[209] S. Kanno, J. Soda and M. a. Watanabe, JCAP0912, 009 (2009).[210] F. R. Urban, JCAP1112, 012 (2011).[211] V. Demozzi and C. Ringeval, JCAP1205, 009 (2012).[212] G. Tasinato, JCAP1503, 040 (2015).[213] A. Brandenburg and A. Lazarian, Space Sci. Rev.178, 163 (2013).[214] S. A. Colgate, H. Li and V. Pariev, astro-ph/0012484.[215] H. Xu, H. Li, D. C. Collins, S. Li and M. L. Norman, Astrophys. J.698, L14 (2009).[216] G. Gregori, A. Ravasio,et al.Nature,481, 480 (2012).[217] E. P. Alves, T. Grismayer, S. F. Martins, F. Fiuza, R. A.Fonseca and L. O. Silva, Astrophys. J.746. L14 (2012).[218] J. Schober, D. Schleicher, C. Federrath, R. Klessen and R. Banerjee, Phys. Rev. E.85, 026303 (2012).[219] S. Sur, C. Federrath, D. R. G. Schleicher, R. Banerjee and R. S. Klessen, Mon. Roy. Not. Astron. Sci.4233148

(2012).[220] A. Brandenburg, K. Enqvist and P. Olesen, Phys. Rev. D54, 1291 (1996).[221] A. Brandenburg and A. Nordlund, Rep. Prog. Phys.74, 046901 (2011).[222] R. Banarjee, Astron. Nach.334, 537 (2013).[223] K. Subramanian and J. D. Barrow, Phys. Rev. D58, 083502 (1998).[224] K. Dimopoulos and A. C. Davis, Phys. Lett. B390, 87 (1997).[225] J. M. Wagstaff, R. Banerjee, D. Schleicher and G. Sigl,Phys. Rev. D89, 103001 (2014).[226] R. Banerjee and K. Jedamzik, Phys. Rev. D70, 123003 (2004).[227] T. Tajima, S. Cable, K. Shibata, R. M. Kulsrud, Astrophys. J.390 309 (1992).[228] A. Brandenburg and W. Dobler, Comput. Phys. Commun.147, 471 (2002).[229] A. Pouquet, U. Frisch and J. Leorat, J. Fluid Mech.77. 321 (1976).[230] K. Jedamzik and G. Sigl, Phys. Rev. D83, 103005 (2011).[231] A. Saveliev, K. Jedamzik and G. Sigl, Phys. Rev. D86, 103010 (2012).[232] A. Berera and M. Linkmann, “Magnetic helicity and the evolution of decaying magnetohydrodynamic turbu-

lence,” Phys. Rev. E90, no. 4, 041003 (2014).[233] J. Zrake, Astrophys. J.794, no. 2, L26 (2014).[234] W. E. East, J. Zrake, Y. Yuan and R. D. Blandford, arXiv:1503.04793 [astro-ph.HE].[235] S. Galtier, S. V. Nazarenko, A. C. Newell and A. Pouquet, J. Plasma Phys.63, 447 (2000).[236] D. Biskamp,Nonlinear Magnetohydrodynamics(Cambridge: Cambridge Univ. Press) 2000.[237] D. Biskamp and W. C. Muller, Phys. Rev. Lett., 83, 2195(1999), D. Biskamp and W. C. Muller, Phys. Plasma,

7, 4889 (2000).[238] D. T. Son, Phys. Rev. D59, 063008 (1999).

19

[239] E. Blackman, Spa. Sci. Rev.188, 59 (2015).[240] A. Saveliev, K. Jedamzik and G. Sigl, Phys. Rev. D87, no. 12, 123001 (2013).[241] L. Campanelli, Phys. Rev.D70, 083009 (2004).[242] L. Campanelli, Phys. Rev. Lett.98, 251302 (2007).[243] M. Christensson, M. Hindmarsh and A. Brandenburg, Astron. Nachr.326, 393 (2005).[244] M. Christensson, M. Hindmarsh and A. Brandenburg, Phys. Rev. E64, 056405 (2001).[245] K. Subramanian and A. Brandenburg, Mon. Not. Roy. Astron. Soc.445, 2930 (2014).[246] L. Chamandy, A. Shukurov, K. Subramanian, Mon. Not. Roy. Astron. Soc.446, L6 (2015).[247] E. A. Mikhailov, Astron. Lett.40, 398 (2014).[248] J. M. Wagstaff and R. Banerjee, JCAP1601, 002 (2016).[249] E. C. Whipple, T. G. Northrop, and D. A. Mendis, J. Geophys. Res.90, 7405 (1985).[250] Blackman, E. G., Spa. Sci. Rev.188, 59 (2015).

20