Magnetic Field Transfer From A Hidden Sector

Primordial magnetic fields in the dark sector can be transferred to magnetic fields in the visible sector due to a gauge kinetic mixing term. We show that the transfer occurs when the evolution of magnetic fields is dominated by dissipation due to finite electric conductivity, and does not occur at later times if the magnetic fields evolve according to magnetohydrodynamics scaling laws. The efficiency of the transfer is suppressed by not only the gauge kinetic mixing coupling but also the ratio between the large electric conductivity and the typical momentum of the magnetic fields. We find that the transfer gives nonzero visible magnetic fields today. However, without possible dynamo amplifications, the field transfer is not efficient enough to obtain the intergalactic magnetic fields suggested by the gamma-ray observations, although there are plenty of possibilities for efficient dark magnetogenesis, which are experimentally unconstrained.


I. INTRODUCTION
Primordial magnetic fields have been of interest for many years since they may explain the observed galaxy and galaxy cluster magnetic fields through the dynamo mechanism during structure formation [1].Moreover, the presence of intergalactic magnetic fields is also indicated by the recent observations of TeV blazars [2][3][4][5][6][7][8][9], which provide a lower bound on the magnetic field strength, B 10 −19 G at Mpc coherence scales and B 10 −16 G ×(λ/pc) −1/2 at smaller length scales [9].However, it is difficult to come up with astrophysical origins of these magnetic fields in the cosmic voids, and the challenge motivates the consideration of these intergalactic magnetic fields as remnants from the very early universe [10].Such magnetic fields can even be related to the matter-antimatter asymmetry of the Universe [11][12][13][14][15][16] or the production of dark matter density [17].
For example, inflationary magnetogenesis models are strongly constrained by observations of cosmic microwave background (CMB), which make it difficult to generate the required magnetic fields in these scenarios (see, however, Ref. [34]).The electroweak and QCD phase transitions are known to be crossovers within the Standard Model (SM) [35,36], and it is not clear if magnetic fields could be generated by the SM phase transitions.
However, once we consider particle physics beyond the SM, there are much more possibilities of magnetogenesis from the existence of additional U(1) symmetries, which can be preserved at an earlier time universe and therefore suffer from weaker experimental constraints.The additional U(1) can be a gauged U(1) B−L , other U(1) symmetries arising in grand unified theories [37], or simply a dark U (1) field that couples weakly to the visible sector.We can imagine a "dark magnetogenesis" mechanism in a hidden sector, for example, from a much stronger first order cosmological phase transition than the SM symmetry breaking.The dark symmetry breaking into a dark U(1) gauge symmetry can produce a strong dark magnetic field, which is mildly constrained if the process happens after inflation but before Big-bang nucleosynthesis (BBN), and the dark photon later obtains a mass from the dark U(1) breaking and decays into SM particles before BBN.Although dark magnetic monopoles may also be generated during the phase transition, we will not consider the case by assuming that the larger gauge symmetry group has a nontrivial first homotopy.If the dark magnetic fields are transferred to the SM magnetic fields after their production, they may provide seeds for the galaxy and galaxy cluster magnetic fields and explain the TeV blazar observation.In this article we examine the evolution of dark magnetic fields and how a transfer from dark to electromagnetic (or hypermagnetic in the SM) magnetic fields can occur.A related idea in which background dark photon generated through an oscillating axion-like particle gets converted into visible magnetic field is also discussed recently in Ref. [38].
Regardless of the details of the model, the dark U(1) D gauge field D µ and the visible U(1) Y gauge field Y µ will interact via a gauge kinetic mixing term, − D µν Y µν , with being the gauge kinetic mixing parameter.Such a gauge kinetic mixing can be removed by field redefinition but generally only when there are no couplings to matter.Once we introduce couplings to matter fields, the visible and dark gauge fields as well as the gauge kinetic mixing are uniquely defined.Here we define the gauge fields so that the SM matter fields are not charged under the dark U(1) symmetry in the basis with the nontrivial gauge kinetic mixing.
To study the cosmological evolution of dark and visible magnetic fields, we must account for the plasma in which these magnetic fields are embedded.Therefore, instead of solving the classical field theory equations, we study magnetohydrodynamic (MHD) equations that have been extended to include the dark sector fields.With some simplifying assumptions, notably ignoring turbulence, we find that dark magnetic fields are transferred to the visible sector at early times.The transfer efficiency is suppressed by a factor of k 2 c ∆t s /σ Y with k c being the typical momentum of the dark magnetic fields, ∆t s being the duration of the transfer, and σ Y being the (hyper)electric conductivity.At late times, once the magnetic fields evolve according to scaling laws indicated by MHD simulations, no further transfer occurs if there is no dynamo amplification.Unless further amplification of the magnetic field occurs, e.g. at the time of dark U(1) symmetry breaking, the suppression factor implies that the visible fields are too weak to explain the blazar observations.The paper is organized as follows.In Sec.II we describe the model and derive the evolution equations of visible and dark magnetic fields.In Sec.III we examine how the transfer from dark to visible magnetic fields occurs.In Sec.IV we adopt the formalism developed in the previous sections and evaluate the present properties of the intergalactic magnetic fields.We summarize our findings in Sec.V.

II. MODEL AND EVOLUTION EQUATIONS
We focus on the case where there are no light matter fields in the plasma that are simultaneously charged under both the visible and dark U(1) symmetries in the basis with gauge kinetic mixing.(The case with particles in the plasma that are charged under both U(1) symmetries is discussed in Appendix A.) The Lagrangian is now written as Here Y µ is the SM hypercharge gauge field, D µ is the dark U(1) gauge field, and Y µν and D µν are their field strengths.1 is the gauge kinetic mixing parameter, which can come from a loop-induced process with heavy mediators connecting the two sectors.J Y µ and J D µ are the visible and dark U(1) current carried by matter fields with the associated U(1) charges.We assume both U(1) symmetries remain unbroken throughout the B-field transferring process.Depending on the mass of the dark photon, there are constraints on [39].However, for a high dark U(1) breaking scale, much higher than the electroweak scale, there are no strong bounds on and hence ∼ O(0.1) is allowed.Since we focus on the dynamics at a scale higher than the electroweak scale, the visible magnetic fields are identified as the SM hyper U(1) magnetic fields.Hyper magnetic fields are subsequently transformed into (electro)magnetic fields at the electroweak phase transition [16].
Here we consider the case where the Universe is filled with thermal fluids, in which both U(1) charged particles are thermalized.In such an environment, the evolution of magnetic fields with a spatial scale larger than the intrinsic scale of the fluids can be described by the MHD equations [40], which consist of the Navier-Stokes equations and Maxwell's equations.
We modify the Maxwell's equations to include both gauge fields with kinetic mixing.
The focus of this work is the transition between dark and SM magnetic fields inside thermal fluids.Instead of giving a specific B-field generation model in the dark sector, we simply assume the existence of a dark magnetic field, B D (t i ), from the initial conformal time t i .The dark B-field can come from various field generation models but with a larger parameter space for phase transitions or chiral instability that are not directly constrained by SM physics.
The Lagrangian Eq. ( 1) leads to the equations of motion for the gauge fields, In terms of the electric and magnetic fields, we obtain modified Ampére's laws as where B Y and B D are the magnetic fields for the visible and dark gauge fields.We work in the conformal frame so that the effects of the cosmic expansion in the Friedmann Universe do not appear explicitly.The time should be understood as the conformal time and the electric conductivities should be rescaled by the scale factor a, σ a = a σ a,phys .The physical electric and magnetic fields, E phys and B phys , are obtained by We have adopted the nonrelativistic MHD approximation and neglected the displacement currents ĖY and ĖD , where E Y and E D are the visible and dark electric fields, since they are suppressed by factors of the fluid velocity v 1 compared to the total currents [41].
Faraday's laws as well as Gauss's laws for magnetism take the standard form, since they are derived from the definition of the field strength tensor.
Assuming the chiral magnetic current [42] as well as the chiral vortical current [43] are negligibly small, the currents obey Ohm's law, where v is the local velocity of both the SM and dark fluids, and σ ab is the electric conductivity tensor.To justify the treatment of the medium as a single fluid, we note that t-channel scattering between dark and SM particles, assuming similar masses, keeps the visible and dark fluids in thermal equilibrium as long as the scattering rate Γ ∼ N scat 2 α 2 Y T , is larger than the Hubble expansion rate H ∼ T 2 /m Pl where N scat , α Y and m Pl = 1.22×10 19 GeV are the number of particles that are involved in the scattering, the hyper fine structure constant and the Planck mass, respectively.Thus in order for the single fluid approximation to be justified, the temperature of the fluid must be smaller than At much lower temperatures, either the dark U(1) breaking or recombinations in the two sectors makes the system depart from thermal equilibrium.
In the high temperature phase, the conductivity tensor is evaluated by the Kubo formula as with the bracket being the one-boson irreducible correlation function [44,45].In our setup, since there are no fields that carry both the visible and dark U(1) charges, the off-diagonal components of the electric conductivity tensor vanishes at tree level and is suppressed by the kinetic mixing at higher order.Neglecting the off-diagonal components (see Appendix A for details), we write the visible and dark electric currents in terms of the visible and dark electric and magnetic fields as The visible and dark electric conductivities (σ Y ≡ σ Y Y and σ D ≡ σ DD ) are evaluated as [44,45] with g being the gauge coupling of the dominant thermal fluid particles, and e is the SM electric charge.The equation can be qualitatively understood as arising from the classical Drude model, σ ∼ n g 2 τ /m, with number density n ∼ T 3 , typical energy scale m ∼ T , and the characteristic time scale for large angle scattering τ ∼ (g 4 T ln g −1 ) −1 in thermal bath.
The coefficient C depends on the number of charged particle species, and in the SM ranges from 15, when only the electron is included, to 12, when all charged fermions besides top are included [45].Here the scale factor a is included since we define the electric conductivities in the conformal frame.As a result, the electric conductivities are invariant under the cosmic expansion in the limit we neglect the change of the number of relativistic particles.
We can eliminate the electric fields from Ohm's and Ampére's laws, so that the evolution equations for the magnetic fields read By redefining with α ≡ σ D /σ Y , the evolution equations for the magnetic fields are decoupled as We can see that the nonvanishing gauge kinetic mixing and/or α = 1 generate a difference in the effective electric conductivities and therefore a different time evolution of the two magnetic fields.This will be the source for the transfer from the dark to visible magnetic fields as we show in the next section by solving the evolution equations.Note that the field redefinition (Eqs.( 12) and ( 13)) makes sense only when = 0 and hence nonvanishing is essential for the magnetic field transfer.

III. TRANSFER OF MAGNETIC FIELDS
We explore the transfer of dark magnetic fields to visible magnetic fields in two steps.
First, at early times, soon after the dark B D is generated, we assume that the fluid velocity is negligible.At such early times, energy in the B-fields has not been transferred to kinetic flows and this assumption is justified.Eventually the velocity fields are emerged through the Lorentz force and the eddy turnover scale catches up with the coherence scale of the B-fields.At that time we can no longer ignore the fluid velocity.In this second stage, however, we can use the scaling laws derived using numerical MHD simulations [46,47].
A. First stage: v ≈ 0 Setting v → 0 in Eqs. ( 14) and ( 15), the equations linearize.Then it is convenient to go to Fourier space, with Q ± (k) being the circular polarization vectors.The mode functions B s (k, t) then obey, with the solutions, where t is the conformal time, and t i is the time of B D generation.The exponential decay corresponds to dissipation of the B-fields caused by the finite conductivity.Bs Y and Bs D decay with different rates due to the different effective conductivities.Since we are interested in having both the initial dark B-field generation and the later time dark U(1) breaking in the un-hatted basis (B Y,D ), we write the solutions in the original basis as in Eq. ( 1) and take Thus, even if magnetogenesis only comes from the dark sector, nonvanishing visible magnetic fields are still produced from the magnetic field transfer between two sectors.
The dark-to-visible transfer is a consequence of the difference of the effective electric conductivities in the basis BY and BD .As we discuss below, although both B-fields dissipate through Eq. ( 19) and (20), the incomplete cancellation between them lead to the linear growth of the visible field when k 2 (t − t i )/σ Y 1 with a size that is inverse proportional to the conductivity.We have been deriving results by assuming particles charged either under the SM or dark U(1) but not both.The result can alternatively be understood qualitatively in a different charge basis.From the original basis in the Lagrangian of Eq. ( 1), we can go to the basis without gauge kinetic mixing but with mixed currents that are charged under both dark and visible U(1) symmetries.Then the dark magnetic fields are associated with the current charged under both the dark and visible U(1), and the nonzero visible U(1) charge carried by the current in this basis sources the visible magnetic fields.
We show an example of the visible B-field evolution in Fig. 1.We are interested in scenarios with photon mixing 1 and conductivity ratio α ∼ 1.At early times, k 2 (t − while after k 2 (t the usual diffusion effects (blue curve).The change from growth to decay will occur at time However, in Sec.IV we will see that our assumption v ≈ 0 breaks down before the dissipative regime can start for the case with relatively small coherence length, and we have to use the full MHD solution that takes the fluid velocity into account (red curve).The efficiency of the transfer, is the same for both helicity modes.Since there is no transfer to the opposite helicity mode, the helicity-to-energy ratio for each k mode is conserved during the dark to visible B-field transfer.However, unless the magnetic field spectrum is dominated by a single k mode, the total helicity-to-energy ratio in the visible magnetic fields obtained by integrating over all k modes may differ from that in the dark magnetic fields if the helicity-to-energy ratio is k-dependent.In the case of maximally helical fields, only one of B ± is nonvanishing.Maximally helical B Y emerges from maximally helical B D with identical polarizations, independent of the spectrum since all the k modes are maximally helical.
l i n e a r g r o w t h dis sip ati on s c a li n g re g im e Blue curve shows the growth and dissipation of the visible B-field, B s Y , if the comoving eddy scale of the turbulence, (t − t i )B D (t i )/T 2 D , is much smaller than the coherence length 2π/k c of the dark B-field.However, before B s Y grows to its maximum size B s D permitted by the kinetic mixing, the fluid velocity cannot be ignored and turbulence becomes important.Then if there is no dynamo amplification, the B-field decays following the scaling law discussed in Sec.IV and is shown by the red curve in the plot.

B. Second stage: v = 0
Through the Lorentz force that acts on the charged particles in the fluids, velocity fields are eventually generated from the magnetic fields, and the fluid becomes turbulent.At that stage, the standard MHD studies (without dark magnetic fields) have shown that the magnetic fields evolve according to a scaling law that depends on whether there is an inverse cascade [46][47][48][49], direct cascade [40,46,47,50], or inverse transfer [51][52][53][54].In any case, as a first approximation, the magnetic fields are described by the comoving field strength B c (t) at the coherence length λ c (t) or the peak scale k(t) = 2π/λ c (t) in the conformal frame, and they evolve as Here n B and n λ are positive constants, which are determined by the helicity of the magnetic fields and properties of the turbulence [53,54].Supposing that (i) the equilibration of the magnetic fields and velocity fields, v ∼ B c / √ ρ c , where ρ c ∼ T 4 c denotes the comoving fluid energy density (T c is the temperature when the scale factor a = 1), is established when the system enters the scaling regime1 , and (ii) the coherence length is determined by the eddy scale of the turbulence, λ c ∼ vt ∝ B c t, we have n B = 1 − n λ , which is also seen in the MHD simulations.Analytical explanations of the scaling behavior, such as those given in Refs.[46,47,50,55], suggest that these exponents are insensitive to the values of MHD parameters.
We assume Eq. ( 6) is satisfied and take the single fluid approximation.This means the coherence lengths in two sectors are determined by the same eddy turnover scale and velocity field, λ Y, Ŷ λ D, D v t.In our setup, when B D B Y in the un-hatted frame, we have the following relations in the hatted frame, For α 1, both of the hatted magnetic fields drive the plasma velocity, v B D /T 2 c = B D,phys /T 2 with BD BY B D , and the hatted magnetic fields individually evolve according to the scaling laws, Eq. ( 24).For α 1, the velocity fields are driven by the BD fields, and the BD field strength as well as the coherence length evolve according to the scaling law.
Similarly, for α 1, the velocity fields are driven by the BY fields, and BY field strength as well as the coherence length evolve according to the scaling law.In these latter two cases, since the coherence length for the weaker hatted field, BY for α 1 and BD for α 1, is determined by the same eddy turnover scale, which is the same as the coherence length of the dominant hatted field, evolving with the scaling exponent n λ , we expect that both magnetic fields BY and BD evolve or decay with a scaling law of the same exponent once the turbulence is fully developed (see Appendix B for more detailed explanation).Since the visible and dark magnetic fields in the original basis (Eq.( 1)) are linear combination of those with the basis BY and BD (Eqs.( 12) and ( 13)), the ratio between the magnitude of dark and visible magnetic fields are fixed during turbulent evolution, and there is no further field transfer at the order of our approximation.
In scenarios of magnetogenesis such as those from the first order phase transitions, the system enters the scaling regime before the dissipation starts to erase the magnetic fields exponentially, k 2 c (t − t i )/σ Y < 1, with k c being the characteristic scale of the magnetic fields.In the next section we discuss a concrete setup and evaluate the visible magnetic field surviving until today.
Before we end the section, we want to comment on a possible amplification of B Y .It should be noticed that there is no numerical study yet for the case when where ρ D, Ŷ and ρ v are the energy densities in the hatted B-fields and fluid velocity field.
It remains as a possibility that the weaker BY field (for α 1) can experience dynamoamplification and/or will enter the scaling regime at some time after the stronger BD field starts scaling (and vice versa for α 1).The two B fields may therefore evolve with different scaling exponents for some time, which may result in additional amplification of the visible B Y fields.A quantitative estimation of such an amplification requires detailed numerical simulations.Here we take a conservative position where we assume that such amplifications of visible B-field are negligible.

IV. ESTIMATE OF PRESENT INTERGALACTIC MAGNETIC FIELD
Now we evaluate the strength and coherence length of the present visible magnetic fields in a concrete setup.Here we assume that the dark magnetic fields are generated at an early time before the electroweak phase transition and transferred to the SM hyper U(1) magnetic fields.
Then the hyper magnetic fields smoothly turn into the (electro)magnetic fields at the electroweak symmetry breaking without any decay or amplification, which evolve according to the scaling law and remain until today.Let us write the temperature at dark magnetogenesis as T (t i ) = T D and parameterize the typical momenta (or the inverse of the coherence length) of the dark magnetic fields as k c ∼ γH D .Here H D = 8π 3 g * /90 T2 D /m Pl 1.66 g 1/2 * T 2 D /m Pl is the Hubble parameter at the dark magnetogenesis with g * being the number of relativistic degrees of freedom. 2 γ is the ratio between the Hubble radius and initial magnetic field coherence length, which we take as a free parameter that parameterizes the magnetogenesis models.If we specify a magnetogenesis model, γ can be obtained e.g., from numerical simulations.For instance, γ ∼ 10 2 [32] if the dark magnetogenesis comes from a first order phase transition, and the initial coherence length of the magnetic fields is the order of the size of the largest bubbles at coalescence.The magnetic field strength at this time is denoted by B D (t i ) and has energy density D , supposing that the velocity fields gets equilibrated to the magnetic fields at a sufficiently earlier time.∆t s is the time interval of the first stage, and we take the scale factor to be a = 1 at t = t i to write it as The dissipative evolution starts at t the first stage ends before the system reaches the regime of dissipative evolution (∆t s t − t i ).Since we are interested in an efficient magnetogenesis ∼ B D (t D )/T 2 D ∼ 10 −2 and focus on scenarios with T D ≤ 10 14 GeV for the single fluid approximation, the inequality becomes γ < ∼ 10 2 after taking σ Y 100 T D from Eq. ( 9).The equality can be naturally satisfied, e.g., in magnetogenesis from the first order phase transition at T D 10 14 GeV that has typical γ 10 2 [32].We assume the inequality holds for the following derivation.Now let us evaluate the properties of the visible magnetic fields at the present epoch.In order to compare them with the observations here we move to the physical frame.Combining Eqs. ( 23), (26) and the relations k c = γH D , the coherence length at the time of dark magnetogenesis λ Y = 2π/k c , and the Hubble redshift factor (H D ∆t s ) −2 of the magnetic field strength (note that t denotes conformal time), the physical visible magnetic field strength and coherence length when the first stage ends and the system enters the scaling regime Here we have taken into account the redshift from the magnetic field generation to the onset of the scaling law using, which also gives the temperature at the onset of the scaling evolution (when fluid velocity cannot be ignored), Note that this expression applies only for B D (t i ) < 2πT 2 D /γ.For B D (t i ) > ∼ 2πζ 1/2 T 2 D /γ, magnetic fields will be entering the scaling regime in a Hubble time but we do not consider such cases.
Assuming that the visible magnetic fields evolve according to the scaling law without experiencing significant dynamo amplifications until recombination and afterwards evolve adiabatically until today, we can estimate the present strength and coherence length of the intergalactic magnetic fields using the scalings, where B 0 phys and λ 0 phys are the physical magnetic field strength and coherence strength today, respectively, and t rec is the conformal time at recombination.Since we are interested in getting maximal B-field and coherence length, we assume the dark B-field is generated with maximal helicity, in which case the exponent of the scaling exponent is known as n B = 1/3 and n λ = 2/3 [46,47].(See also Appendix B.) For example, maximally helical B-field can be generated by pseudo scalar inflation [20,21] and chiral instability [28,29] (see also Refs. [56,57]).Besides taking it as an assumption for generating large magnetic fields, the existence of helical intergalactic magnetic fields may even be indicated by data from the parity-violating correlations of the diffuse gamma ray flux [58][59][60][61].This gives a motivation to study the maximally-helical scenario, while our derivations can be easily adapted to different (n B , n λ ) assumptions.
From Eqs. ( 28), (31), and ( 32) we obtain the present magnetic field strength and coherence length in terms of the temperature and field strength at the dark magnetogenesis as, In Fig. 2, the field strength and coherence length of the present magnetic fields for different choices of T D are depicted with the blue thick solid line.We can see that such fields are too weak and incoherent to explain the observed deficit of secondary GeV cascade photons from blazars even if we take relatively extreme parameters such as T D 10 14 GeV (comparing to Eq. ( 6)) and a large γ.The main reason for the weak field strength is that the transfer efficiency factor k 2 c ∆t s /σ Y is very small, ∼ 1.0 × 10 −7 ( /0.1)(γ/10 2 )((B D (t i )/T 2 D )/0.01) −1 (T D /10 8 GeV) due to the large electrical conductivity (k c /σ Y ∼ H D /T D 1).One might think that taking γT D > 10 16 GeV and B D (t i )/T 2 D < 10 −2 can increase the size of B 0 phys in (33).But this violates the condition in Eq. ( 27), and the exponential dissipative decay happens too early and eliminate the magnetic fields before the turbulent plasma develops, so Eqs.( 33) and (34) do not apply.Moreover, since magnetic fields decay faster in the nonhelical or partially helical case, the maximally helical case we study should give the largest visible B-field.
In order to avoid all possible collider and cosmological constraints, we have in mind that the dark U(1) D symmetry breaks down at a high temperature e.g., above the electroweak symmetry breaking.It is not quite clear if the scaling relation with the exponents in Eq. (32) holds after the U(1) D symmetry breaks.When deriving Eq. ( 33) and (34) with the exponents can explain blazars  In this article, we have examined how dark magnetic fields can be transferred to the visible magnetic fields through the gauge kinetic mixing.We have considered the system with the Lagrangian Eq. ( 1) where there are independent dark and visible U(1) currents in the basis with gauge kinetic mixing.We have found that in such a system the visible magnetic fields emerge due to the transfer from the dark magnetic fields in thermal fluids through the gauge kinetic mixing, when the velocity fields are small.The efficiency is suppressed by the gauge kinetic mixing parameter as well as the large electric conductivity σ Y and the duration of the transfer.At some later time the velocity fields develop turbulence and the transfer terminates when the magnetic fields enter the scaling regime.The ratio between the visible and dark magnetic field strength is fixed at that time.Due to the shortness of the duration for the system to enter the scaling regime, the visible-to-dark magnetic field strength ratio is generally very small, say ∼ 10 −7 .As a result, it is not possible to explain the TeV blazar observations by the visible intergalactic magnetic fields generated by this mechanism without further dynamo amplification.
We have not considered the dark U(1) symmetry breaking in detail, which should be associated with a possible dynamo amplification of the visible magnetic fields as well as the dark cosmic string production.We assume the symmetry breaking does not affect the abundance of relic visible magnetic fields.However, we do not exclude the possibility that this would change the strength of the visible magnetic fields.This is left for future work.

Maximally helical case
In the maximally helical case, the reason why the exponents of two hatted fields ( BB and BY in our discussion) become the same can be explained as follows.In the conformal frame, we have the helicity conservations for both fields (denoted as BD for the dominant field and BS for the weaker field), Thus the exponents of the scaling laws for both fields are the same if both of them are maximally helical.

Nonhelical case with direct cascade
The direct cascade for the nonhelical magnetic fields can be derived as follows.Suppose k M b y Z + 8 4 k p z W N 5 j 6 O E + R H p S x 5 y S t B I j 8 P O S R V P s c u P u + W K U 3 O m s B e J m 5 M K 5 G h 0 y 1 + d X k z T i E m k g m j d d p 0 E / Y w o 5 F S w c a m T a p Y Q O i R 9 1 j Z U k o h p P 5 t e P L a P j N K z w 1 i Z k m h P 1 d 8 T G Y m 0 H k W B 6 Y w I D v S 8 N x H / 8 9 o p h p d + x m W S I p N 0 k M b y Z + 8 4 k p z W N 5 j 6 O E + R H p S x 5 y S t B I j 8 P O S R V P s c u P u + W K U 3 O m s B e J m 5 M K 5 G h 0 y 1 + d X k z T i E m k g m j d d p 0 E / Y w o 5 F S w c a m T a p Y Q O i R 9 1 j Z U k o h p P 5 t e P L a P j N K z w 1 i Z k m h P 1 d 8 T G Y m 0 H k W B 6 Y w I D v S 8 N x H / 8 9 o p h p d + x m W S I p N 0 k M b y Z + 8 4 k p z W N 5 j 6 O E + R H p S x 5 y S t B I j 8 P O S R V P s c u P u + W K U 3 O m s B e J m 5 M K 5 G h 0 y 1 + d X k z T i E m k g m j d d p 0 E / Y w o 5 F S w c a m T a p Y Q O i R 9 1 j Z U k o h p P 5 t e P L a P j N K z w 1 i Z k m h P 1 d 8 T G Y m 0 H k W B 6 Y w I D v S 8 N x H / 8 9 o p h p d + x m W S I p N 0 k M b y Z + 8 4 k p z W N 5 j 6 O E + R H p S x 5 y S t B I j 8 P O S R V P s c u P u + W K U 3 O m s B e J m 5 M K 5 G h 0 y 1 + d X k z T i E m k g m j d d p 0 E / Y w o 5 F S w c a m T a p Y Q O i R 9 1 j Z U k o h p P 5 t e P L a P j N K z w 1 i Z k m h P 1 d 8 T G Y m 0 H k W B 6 Y w I D v S 8 N x H / 8 9 o p h p d + x m W S I p N 0 1 s e s t W D l M / v w B 9 b n D 7 T q j 9 M = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " P M t uI + C o d v g 9 A K 9 K N H t 9 I O 7 0 0 Z I = " > A A A C C H i c b V D L S g N B E J y N r x h f q x 6 9 D A Y h g o b d I K g H I c Q c P E Z I T C A b l 9 n J b D J k 9 s F M r x C W X L 3 4 K 1 4 8 q H j 1 E 7 z 5 N 0 4 e B 0 0 s a C i q u u n u 8 m L B F V j W t 5 F Z W l 5 Z X c u u 5 z Y 2 t 7 Z 3 z N 2 9 O x U l k r I G j U Q k W x 5 R T P C Q N Y C D Y K 1 Y M h J 4 g j W 9 w f X Y b z 4 w q X g U 1 m E Y s 0 5 A e i H 3 O S W g J d f E g w K c g s u P r x x f E p q W n J g 7 J 3 W 3 e l 8 a p R W 3 O n L N v F W 0 J s C L x J 6 R P J q h 5 p p f T j e i S cB C o I I o 1 b a t G D o p k c C p Y K O c k y g W E z o g P d b W N C Q B U 5 1 0 8 s k I H 2 m l i / 1 I 6 g o B T 9 T f E y k J l B o G n u 4 M C P T V v D c W / / P a C f g X n Z S H c Q I s p N N F f i I w R H g c C + 5 y y S i I o S a E S q 5 v x b R P d C K g w 8 v p E O z 5 l x d J o 1 S 8 L N q 3 l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " P M t u I + C o d v g 9 A K 9 K N H t 9 I O 7 0 0Z I = " > A A A C C H i c b V D L S g N B E J y N r x h f q x 6 9 D A Y h g o b d I K g H I c Q c P E Z I T C A b l 9 n J b D J k 9 s F M r x C W X L 3 4 K 1 4 8 q H j 1 E 7 z 5 N 0 4 e B 0 0 s a C i q u u n u 8 m L B F V j W t 5 F Z W l 5 Z X c u u 5 z Y 2 t 7 Z 3 z N 2 9 O x U l k r I G j U Q k W x 5 R T P C Q N Y C D Y K 1 Y M h J 4 g j W 9 w f X Y b z 4 w q X g U 1 m E Y s 0 5 A e i H 3 O S W g J d f E g w K c g s u P r x x f E p q W n J g 7 J 3 W 3 e l 8 a p R W 3 O n L N v F W 0 J s C L x J 6 R P J q h 5 p p f T j e i S c B C o I I o 1 b a t G D o p k c C p Y K O c k y g W E z o g P d b W N C Q B U5 1 0 8 s k I H 2 m l i / 1 I 6 g o B T 9 T f E y k J l B o G n u 4 M C P T V v D c W / / P a C f g X n Z S H c Q I s p N N F f i I w R H g c C + 5 y y S i I o S a E S q 5 v x b R P d C K g w 8 v p E O z 5 l x d J o 1 S 8 L N q 3 l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " P M t u I + C o d v g 9 A K 9 K N H t 9 I O 7 0 0Z I = " > A A A C C H i c b V D L S g N B E J y N r x h f q x 6 9 D A Y h g o b d I K g H I c Q c P E Z I T C A b l 9 n J b D J k 9 s F M r x C W X L 3 4 K 1 4 8 q H j 1 E 7 z 5 N 0 4 e B 0 0 s a C i q u u n u 8 m L B F V j W t 5 F Z W l 5 Z X c u u 5 z Y 2 t 7 Z 3 z N 2 9 O x U l k r I G j U Q k W x 5 R T P C Q N Y C D Y K 1 Y M h J 4 g j W 9 w f X Y b z 4 w q X g U 1 m E Y s 0 5 A e i H 3 O S W g J d f E g w K c g s u P r x x f E p q W n J g 7 J 3 W 3 e l 8 a p R W 3 O n L N v F W 0 J s C L x J 6 R P J q h 5 p p f T j e i S c B C o I I o 1 b a t G D o p k c C p Y K O c k y g W E z o g P d b W N C Q B U5 1 0 8 s k I H 2 m l i / 1 I 6 g o B T 9 T f E y k J l B o G n u 4 M C P T V v D c W / / P a C f g X n Z S H c Q I s p N N F f i I w R H g c C + 5 y y S i I o S a E S q 5 v x b R P d C K g w 8 v p E O z 5 l x d J o 1 S 8 L N q 3 l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " P M t u I + C o d v g 9 A K 9 K N H t 9 I O 7 0 0Z I = " > A A A C C H i c b V D L S g N B E J y N r x h f q x 6 9 D A Y h g o b d I K g H I c Q c P E Z I T C A b l 9 n J b D J k 9 s F M r x C W X L 3 4 K 1 4 8 q H j 1 E 7 z 5 N 0 4 e B 0 0 s a C i q u u n u 8 m L B F V j W t 5 F Z W l 5 Z X c u u 5 z Y 2 t 7 Z 3 z N 2 9 O x U l k r I G j U Q k W x 5 R T P C Q N Y C D Y K 1 Y M h J 4 g j W 9 w f X Y b z 4 w q X g U 1 m E Y s 0 5 A e i H 3 O S W g J d f E g w K c g s u P r x x f E p q W n J g 7 J 3 W 3 e l 8 a p R W 3 O n L N v F W 0 J s C L x J 6 R P J q h 5 p p f T j e i S c B C o I I o 1 b a t G D o p k c C p Y K O c k y g W E z o g P d b W N C Q B U5 1 0 8 s k I H 2 m l i / 1 I 6 g o B T 9 T f E y k J l B o G n u 4 M C P T V v D c W / / P a C f g X n Z S H c Q I s p N N F f i I w R H g c C + 5 y y S i I o S a E S q 5 v x b R P d C K g w 8 v p E O z 5 l x d J o 1 S 8 L N q 3

FIG. 1 :
FIG. 1: An illustration of the evolution of mean visible B-field strength from the dark B-field transfer as a function of conformal time that is divided by the coherence length of the dark B-field

4 D
which can be comparable to the energy density of the thermal fluids ρ = ζT with ζ = π 2 g * /30.At first the dark magnetic fields evolve adiabatically except for the slight decay due to dissipation.The first stage terminates when the coherence length 2π/k c is caught up by the eddy turnover scale v∆t s with

ϵ 2 = 10 - 2 FIG. 2 :
FIG. 2: The magnetic field properties today for the maximally helical case.The blue thick solid line represents the magnetic field properties assuming T D < 10 14 GeV and constant scaling exponents.Blue shaded region represents the parameter space if we take into account the possible change of the scaling exponents after dark U(1) symmetry breaking.The black dashed lines represent how the final magnetic field properties differ for T D = 10 8 GeV (lower line) and T D = 10 14 GeV (upper line) if the scaling exponents change at the dark U(1) symmetry breaking while keeping the helicity conservation.The other parameters are fixed as = 0.1, γ = 10 2 andB D (t i )/T 2 D = 10 −2 .Green region is the region where the blazar observations are explained[9].The region inconsistent with the MHD evolution and the one in conflict with CMB observations are depicted in the red colored region and yellow colored region, respectively.

n B = 1 /
3 and n λ = 2/3, we implicitly assume helicity conservation for each of the B fields, λ B2Y,B /2π=const., with the same coherence length that is comparable to the eddy turnover scale λ ∼ vt, where the velocity fields are in equilibrium with the dark magnetic fieldsv ∼ v A ∼ B D,phys / √ ρ [14,15].(See also Appendix B.) However, these assumptions do not hold after dark U(1) D symmetry breaking.The Alfvén velocity evaluated with the B Y field V. SUMMARY