# Non-conformal evolution of magnetic fields during reheating

###### Abstract

We consider the evolution of electromagnetic fields coupled to conduction currents during the reheating era after inflation, and prior to the establishing of the proton-electron plasma. We assume that the currents may be described by second order causal hydrodynamics. The resulting theory is not conformally invariant. The expansion of the Universe produces temperature gradients which couple to the current and generally oppose Ohmic dissipation. Although the effect is not strong, it suggests that the unfolding of hydrodynamic instabilities in these models may follow a different pattern than in first order theories, and even than in second order theories on non expanding backgrounds.

## I Introduction

The existence of magnetic fields on galactic and larger scales is one of the main puzzles in present day cosmology KKT11 ; DurNer13 ; KleFle15 . Neither of the two major paradigms proposed to attack this question, namely dynamo amplification and primordial origin, seems to be able to provide a solution by itself FuyYok14 . It therefore seems likely that both mechanisms are at work, i.e., a seed field is generated early in the cosmic evolution and then subjected to one or several amplification stages CT-AK . This calls for a careful analysis of the cosmological history of magnetic fields BanJed04 .

Lots of efforts have been made to understand the evolution of primordial fields in the proton-electron plasma during the radiation dominated epoch. Special mention deserves the studies that address turbulent evolution, where fields with non-trivial topology i.e., with non zero magnetic helicity, would not be washed out by expansion as quickly as those with null magnetic helicity BanJed04 ; AK-EC ; Sigl .

If we accept the existence of Inflation, then there must be a stage between it and the establishing of the proton-electron plasma where non-equilibrium processes dominated. That epoch is known as ‘reheating’. Moreover, electroweak (EW) and quantum-chromodynamic (QCD) phase transitions could have taken place by the end of it. Little is known of this epoch, besides the fact that all matter is created by the oscillatory decay of the inflaton. For example, the typical relaxation times and correlation times of the different interactions are not known.

In this paper we shall perform a preliminary (see below) analysis of the evolution of magnetic fields during the reheating era AHKK14 ; MRV14 . To this end, we shall consider that, on top of the two dominant contributions to the energy density, namely the coherent oscillations of the inflaton Star80 and the incoherent radiation field, there is a charged fluid that may interact non-trivially with the electromagnetic field. We do not identify this fluid with the usual proton-electron plasma because we consider the evolution during an epoch well before quantum-chromodynamic phase transition.

Both the coherent electromagnetic fields and the charged fluid could be created as a side effect of reheating by the parametric amplification of vacuum fluctuations of a massive scalar field, as it has been discussed elsewhere AK-EC . A suitable candidate for the massive field could be the lightest supersymmetric partner, the - wag-14 . We shall assume that this fluid supports both viscous stresses and conduction currents, namely, electric currents without mass transport. For simplicity, we shall use Maxwell theory to describe the fields, in spite of the fact that the temperatures involved may be above the electroweak transition.

At those early epochs the temperature and curvature of the Universe are very high and consequently a generally relativistic treatment is mandatory. The theory of relativistic real fluids has a long history but only relatively recently it has been put to the test, through its application to relativistic heavy ion collisions (RHICs)RHICs . Simply put, the straightforward covariant generalization of the Navier-Stokes equations leads to the so-called first order theories (FOTs), of which the Eckart Eck40 and Landau-Lifshitz LL6 formulations are the best known. These theories have severe formal problems HisLin83 which may be solved (among several possible strategies sps ) by going over to the so-called second order theories (SOTs)SOTs . The performance of SOTs with respect to RHICs is analyzed in SOT-RHICs .

There is not a single SOT framework as compelling as the Navier-Stokes equations in the non-relativistic regime SOTs ; extended ; dtts . However, in the linearized regime they all agree in providing a set of Maxwell - Cattaneo equations MaxCat for the viscous stresses and conduction currents, while they differ in the way the transport coefficients in these equations are linked to the underlying kinetic theory description Jaiswal ; Kunihiro ; Denicol ; BRSSS ; TakInu10 ; HuaKoi12 ; Strickland . For this reason in this paper we shall consider only the linearized regime. This is what makes our analysis preliminary, because it is likely that the most important effects of the fluid - field interaction will be connected to nonlinear phenomena such as inverse cascades BKT14 ; BerLin14 , field - turbulence interactions Sigl and hydrodynamic instabilities inst . However, as we shall see, already in the linearized regime there are significant qualitative differences between SOTs and FOTs, and between SOTs on flat and expanding backgrounds.

There is a large literature on cosmological models based on SOTs BNK79 ; PJC82 ; PavZim ; Mar95 . This literature focused for the largest part on homogeneous models, where the interest was in how viscous effects modified the cosmic expansion and contributed to entropy generation. These analysis showed that there are meaningful differences between ideal, first and second order theories even at the largest scales. To our knowledge, the application of SOTs to inhomogeneous models is less developed than FOTs BBMR14 ; FTW14 . This consideration also contributed to make an analysis such as this paper a necessary first step. We note that a family of exact solutions for the Boltzmann equation in expanding backgrounds with a well defined hydrodynamic limit is known, which provides a helpful test bench for the theory exact .

In summary, we shall adopt the so-called divergence type theory supplemented by the entropy production variational principle (EPVP) as a representative SOT, but will regard the transport coefficients as free parameters, rather than attempting to derive them from an underlying kinetic description EC-PR . For this reason, our analysis is relevant to any SOT model.

The equations of the model are the conservation laws for energy - momentum and charge, the Maxwell equations, and the Maxwell - Cattaneo equations providing closure; for a detailed derivation see EC14 . In the linearized regime, these equations decouple in three sets of modes, sound waves, incompressible shear waves, and electromagnetic waves coupled to conduction currents. We shall consider only the latter.

We shall model the Universe during reheating as a spatially flat Friedman - Robertson - Walker (FRW) model , whose metric in conformal time is , being the conformal factor. We shall assume for the fluid an equation of state and vanishing bulk viscosity. Under these prescriptions, FOTs lead to conformally invariant equations Sigl . Therefore the electromagnetic fields are suppressed by a factor, on top of the hydrodynamic evolution. We shall consider the evolution of electromagnetic fields in an environment where the temperature is higher than the QCD phase transition temperature, i.e. a scenario where SOTs seem to correctly describe the state of the matter.

Unlike FOTs, the equations derived from SOTs are not conformally invariant: the expansion of the Universe creates temperature gradients which couple to the fluid velocity and conduction currents. This leads to a weaker suppression of the magnetic fields than expected from a FOT framework. This is the main conclusion of this paper. The effect is not large, but suggests that these SOTs models may be more sensitive to nonlinear effects, such as hydrodynamic instabilities, than FOTs or even SOTs on non-expanding backgrounds. This possibility will be investigated elsewhere.

The paper is organized as follows: In Section II we introduce the formalism and the covariant equations of second order hydrodynamics. We analyze the conformal invariance of the theory and derive the equations for the fields as well as for the viscous stress and conduction current, showing that the latter are explicitly non conformally invariant. In Section III we linearize the equations and propose a simple, toy model, to solve them. In Section IV we consider the homogeneous case , that permits to study the electric field and conduction current separately from the magnetic field. In section V we consider super-horizon modes of astrophysical interest, i.e., , and find that the magnetic fields evolves in a way clearly different than in FOT’s models. In Section VI we summarize and discuss our results. We leave for the Appendix A the analysis of sub-horizon modes as they are not as astrophysically interesting as super-horizon modes. In Appendices B and C we quote some secondary results and technicalities for the reader interested in those details. We work with signature and natural units , thus time and length have dimensions of , while wavenumbers, mass and temperature units are those of .

## Ii General Relativistic Fluid Equations

### ii.1 The equations in covariant form and their decomposition

We consider a system composed by a neutral plasma plus electromagnetic field in a flat FRW universe, whose
metric in conformal time is ,
being the conformal factor. This form of the metric is obtained from the one written in
physical time by defining , with the Hubble constant during Inflation^{1}^{1}1With this
definition, is already dimensionless..
If for Inflation we consider the de Sitter prescription,
, then with .
If for reheating we accept that during that period the Universe evolves as if it were
dominated by matter Star80 , then and consequently .
Observe that we have matched the two expressions at such that . As is a
fixed, characteristic energy scale, we can use it to build non-dimensional quantities, as we did with conformal
time, e. g. we define dimensionless lengths and corresponding wavenumbers as and .
Magnetic and electric field units are so we write and .
To complete, we quote the temperature and the electric conductivity .
We use greek letters to denote space-time indices, and latin letters when we deal with spatial-only components.
Besides, we use semicolons to express covariant derivatives and commas to denote partial derivatives; in
particular a ’prime’ will denote partial derivative with respect to conformal time, i.e.,
To evaluate the different covariant derivatives we need the
Christoffel symbols, whose only non-null components are ,
and .

Let be the fluid four-velocity. We decompose it as

(1) |

with . It is satisfied that and . is the velocity of fiducial observers and represents deviations from Hubble flow, i.e. peculiar velocities. Each of these velocities defines a congruence of time-lines, for which there is an orthonormal space-like surface defined through the projectors

(2) |

The matter is described by the energy momentum tensor, , which we decompose as

(3) |

with

(4) |

and

(5) |

the viscous stress tensor. In eq. (5), is a characteristic relaxation time and is a Lagrange multiplier whose evolution equation will be given below; for it reduces to the FOT dissipative shear viscous tensor. We write the electromagnetic field tensor in form relative to the fiducial observers as

(6) |

with . For future use, we define . Observe that the electric and magnetic fields are obtained from (6) as and respectively. The electric current is

(7) |

with

(8) |

where is another Lagrange multiplier whose evolution equation is also given below, and that for gives the usual Ohm’s law.

Although we shall regard in eqs. (5) and (8) as free parameters, we observe that these equations may be derived from a linearized Boltzmann equation EC14 , in which case they are seen to be

(9) |

with the one particle distribution function, the integration measure ( is the mass of the plasma particles), and where is a multiplicative factor in the linearized collision integral. Common choices for are Marle’s prescription marle1 ; marle2 , i.e. , and the Anderson-Witting proposal and-witt1 ; and-witt2 whereby .

Observe that in eq. (6) we defined the electromagnetic field relative to fiducial observers. It is also with respect to this velocity that we shall define the ‘total time derivative’ or ‘dot derivative’, namely . The ‘total spatial derivative’ is accordingly defined as .

The equations we have to solve are the conservation equations (matter coupled to the electromagnetic field plus charge conservation), Maxwell equations and two equations that describe the evolution of the Lagrange multipliers and . The conservation laws are

(10) |

(11) |

and Maxwell equations in covariant form read

(12) |

(13) |

To our purposes the best is to rewrite the previous equations in 3+1 form relative to fiducial observers. This is achieved by projecting each set along and onto its orthogonal surface described by . The projection along is defined as ellis-73 and the one onto the orthogonal surface as . For the set (10) we first replace expression (1) in eq. (4) and define

(14) | |||||

(15) | |||||

(16) | |||||

(17) |

We thus write eq. (3) as

(18) |

For eqs. (7) plus (8) we directly obtain

(19) |

To find the evolution equation for the plasma we assume the equation of state . For the projection along of eq. (10) we have

(20) |

while for the spatial projection we obtain

(21) |

For eq. (11) using (19) we have

(22) |

As Maxwell equations are already written in terms of the projection is straightforward. For the inhomogeneous Maxwell equations (12) we have

(23) | |||||

(24) |

while for the homogeneous ones (13) we obtain

(25) | |||||

(26) |

We now discuss the equations for the Lagrange Multipliers and , see EC-PR and EC14 for details. and are transverse with respect to and is also traceless, i.e. they satisfy

(27) |

Their evolution equations in covariant form are straightforwardly obtained from the corresponding Minkowski expressions given in Ref. EC14 . We obtain:

(28) |

(29) | |||||

with the shear tensor. In the derivation from linearized kinetic theory the functions are given by EC14 :

(30) |

We only mention this because it makes it easy to check the dimensions of and ; otherwise we shall regard them as free parameters. The dimensions of the different expressions under the integrals are , , , with meaning ’energy’ and consequently and . As the only energy scale of the plasma is its temperature, we rewrite eq. (4) as

(31) |

and eq. (8) as

(32) |

with dimensionless, coefficients.

### ii.2 Conformal Invariance

To analyze conformal invariance we begin by rewriting the coefficients in eq. (28) and (29) as

(33) |

where are again numerical, coefficients. Therefore the mentioned eqs. read

(34) |

(35) | |||||

We now transform the different quantities in the model according to

(36) |

(and similar rules for and )

(37) |

(38) |

and

(39) |

Replacing these transformations in eqs. (18), (7), (31) and (32) we find

(40) |

and the set of eqs. (20)-(21) becomes

(41) |

(42) |

while for eq. (22) we have

(43) |

It is a well known result that Maxwell equations are conformally invariant. For the homogeneous equations it is a trivial result, and for the inhomogeneous equations it is directly apparent from the transformation law for and the last of exprs. (40). Therefore transforming eqs. (23)-(26) we have

(44) | |||||

(45) | |||||

(46) | |||||

(47) |

Notwithstanding, when we apply the above conformal transformations to the evolution equations for and conformal invariance is lost. To see this, we replace and from eqs. (1) and (6), and use the conformal transformations defined above to obtain

(48) |

and

(49) | |||||

In both equations, the terms proportional to do not cancel out and this fact makes the two equations non conformal invariant. As the fields evolve coupled to this plasma, the conservation of the magnetic flux during their early evolution is lost. To have a glimpse of the effect of this coupling on the amplitude of the magnetic field, we shall solve the equations in the linear regime.

## Iii Linear Evolution

The system of equations that describe the evolution of the plasma is non linear. We shall study the linear regime, that is suitable for small amplitudes. We shall also consider that the plasma is neutral, i.e., we assume . First order quantities are , the linear equations read and the electromagnetic field. Writing

(50) | |||||

(51) | |||||

(52) | |||||

(53) | |||||

(54) | |||||

(55) |

where we see that at this level the plasma equations have separated from the electromagnetic equations, so from now on we concentrate only in the latter as our focus is the electromagnetic field evolution. Before going on, observe that if we set in eq. (53) we have that . Replacing this expression into Ampère law, eq. (54), the factor that multiplies in the last term of the r.h.s. becomes , and recalling the constitutive relation between electric field and density current, , we can read the expression for the (commoving) electric conductivity:

(56) |

Observe also, that due to the conformal scalings (39) the physical and commoving electric conductivities are related in the usual way, i.e. . We then rewrite eq. (54) as

(57) |

To go on we change the time dependence from to , whence and . Assuming incompressible evolution and transforming Fourier we get

(58) | |||||

(59) | |||||

(60) |

We shall not attempt to solve system (58)-(59) numerically, as this would oblige us to stick to a specific range of parameters. Instead, to have a glimpse of how the system behaves we assume a simple configuration given by

(61) |

Defining the matrices

(62) |

and

(63) |

the system of equations for the electromagnetic sector can be written in matrix form as

(64) |

where now a ’prime’ denotes derivative with respect to , i.e., . In spite of its simple form, it is rather difficult to solve eq. (64) exactly, except for the homogeneous mode, . We begin by solving this case and then consider perturbatively the case , that corresponds to modes well outside the particle horizon as e.g., the galactic scale. The solution for modes is given in Appendix A.

To appreciate the features of the SOT evolution, it is convenient to keep in mind their behavior in the limit, whereby the model reduces to a FOT. In that case system (58)-(60) plus (56) and model (61) reduces to

(65) | |||||

(66) |

and this (conformally invariant) system can be combined to give a wave equation whose solutions are the exponentials with . Observe that when , and . The second solution describes the “frozen” magnetic field, and the first the “discharge” of the electric field due to the resistivity of the plasma. If we have the well known pure exponential decay.

## Iv Analytic Solution for the Homogeneous Mode

In the case, eqs. (64) may be solved in closed form. We then begin by putting in matrix and the r.h.s. of eq. (64) equal to zero. Proposing as solution a time dependence of the form and imposing that the determinant of the resulting system be zero we obtain the eigenvalue equation:

(67) |

whose solutions are

(68) | |||||

(69) |

Observe that there exists a critical relaxation time, . Also, and more importantly, when we have that while blows out. Therefore and converge to the roots of the FOT model. The corresponding eigenvectors are

(70) |

To find the solution of the inhomogeneous equation we propose

(71) |

and substitute in eq. (64). For it is straightforwardly obtained that . Recalling that this coefficient corresponds to , and that this eigenvector represents the magnetic field, this means the obvious result that the commoving field remains constant and consequently the physical magnetic intensity will decay as . The other coefficients satisfy