On the magnetic evolution in Friedmann universes and the question of cosmic magnetogenesis

We analyse the evolution of primordial magnetic fields in spatially flat Friedmann universes and reconsider the belief that, after inflation, these fields decay adiabatically on all scales. Without~abandoning classical electromagnetism or standard cosmology, we demonstrate that this is not necessarily the case for superhorizon-sized magnetic fields. The underlying reason for this is causality, which confines the post-inflationary process of electric-current formation, electric-field elimination and magnetic-flux freezing within the horizon. As a result, the adiabatic magnetic decay is not a~priori guaranteed on super-Hubble scales. Instead, after inflation, large-scale magnetic fields obey a~power-law solution, where one of the modes drops at a rate slower than the adiabatic. Whether this slowly decaying mode can dominate and dictate the post-inflationary magnetic evolution depends on the initial conditions. These are determined by the evolution of the field during inflation and by the nature of the transition from the de Sitter phase to the reheating era and then to the subsequent epochs of radiation and dust. We discuss two alternative and complementary scenarios to illustrate the role and the implications of the initial conditions for cosmic magnetogenesis. Our main claim is that magnetic fields can be superadiabatically amplified after inflation, as long as they remain outside the horizon. This means that inflation-produced fields can reach astrophysically relevant residual strengths without breaking away from standard physics. Moreover, using the same causality arguments, one can constrain (or in some cases assist) the non-conventional scenarios of primordial magnetogenesis that amplify their fields during inflation. Finally, we show that our results extend naturally to the marginally open and the marginally closed Friedmann universes.


Introduction
The origin of cosmic magnetism remains an essentially open question despite the efforts and the established widespread presence of magnetic (B) fields in the universe [1]. Recent reports of the first ever detection of intergalactic fields, with strengths close to 10 −15 G, have added to the mystery [2]. Provided they are verified, these claims also support the case for primordial magnetism [3]. The latter is an attractive proposition because it could potentially explain all the large-scale B-fields of the universe. Nevertheless, there are serious theoretical problems in producing such primordial fields. These mainly stem from the long standing belief that (conventional) magnetic fields in spatially flat Friedmann-Robertson-Walker (FRW) universes decay adiabatically throughout the evolution of these models and on all scales.
The structure of the magnetic fields in galaxies seems to support the galactic-dynamo idea [4]. Depending on the efficiency of the amplification, dynamos generally require B-seeds stronger than ∼ 10 −22 G at the time of completed galaxy formation. It has also been claimed that this lower limit could be pushed down to ∼ 10 −30 G in spatially open, or in Λ-dominated FRW models [5]. The size of the initial magnetic seed is also an issue, since it should not be smaller that ∼ 100 pc after the collapse of the protogalaxy, which implies a comoving scale of approximately 10 Kpc before the collapse. Seeds generated after inflation, during the radiation era for example, are typically to small in size because their coherence length can never exceed that of the causal horizon at the time. Inflation naturally achieves superhorizon correlations, so it can easily produce primordial fields of the required length. Nevertheless, in all the standard scenarios of inflationary magnetogenesis, B-fields decay adiabatically as soon as they cross outside the Hubble horizon. This practically means that B ∝ a −2 , with a = a(t) representing the cosmological scale factor, essentially from the beginning of inflation until today. The result is astrophysically irrelevant magnetic fields. In particular, the residual strength of a field with comoving (pre-collapse) size close to 10 Kpc today can be as low as 10 −53 G (see [3] and references therein). Having said that, the aforementioned numerical result assumes that the adiabatic magnetic decay persists on all scales after inflation. This is believed to reflect the high electrical conductivity of the post-inflationary universe, which in turn is thought to guarantee that magnetic fields remain frozen into the cosmic medium. Magnetic-flux freezing, however, cannot be achieved without the presence of electric currents. Therefore, applying the adiabatic decay-law on all scales, implicity assumes the existence of superhorizon-sized electric currents. Such a hypothesis, however, violates causality.
During inflation there are no electric currents and their formation starts once the universe enters its reheating phase. The process is causal, which implies that the coherence length of the newly formed currents never exceeds that of the horizon. After inflation, the latter coincides with the Hubble radius. Hence, the same causality arguments that confine the post-inflationary Bfields within the Hubble scale (see above), also forbid the electric currents from achieving super-Hubble correlations. Without superhorizon-sized currents, there can be no (causal) electric-field elimination on those scales and this changes the evolution of the associated magnetic fields. More specifically, beyond the Hubble radius, the magnetic flux is no longer conserved and the adiabatic (B ∝ a −2 ) decay-law is not a priori guaranteed. In fact, a straightforward calculation shows that, after inflation, superhorizon-sized B-fields obey a power-law solution, where one of the modes drops slower than the adiabatic. This slowly decaying mode can dominate and dictate the magnetic evolution depending on the initial conditions. The latter are decided by the evolution of the field during the de Sitter phase and by the nature of the transition from inflation to reheating (as well as to the radiation and the dust epochs). Following Israel's work on "junction conditions", we discuss two typical and complementary scenarios, illustrating how superhorizon-sized magnetic fields can be superadiabatiaclly amplified after inflation.
The rate of the aforementioned slowly decaying magnetic mode depends on the equation of state of the matter that fills the universe at the time. Throughout reheating, in particular, one finds that B ∝ a −3/2 , which slows down further to B ∝ a −1 in the radiation era, before returning to the B ∝ a −3/2 -law during the subsequent dust epoch. Thus, as long as they remain outside the Hubble radius, magnetic fields are superadiabatically amplified all along their postinflationary evolution. Once back inside the horizon, however, the electric currents take over and quickly eliminate the accompanying electric fields. This freezes the B-fields into the highly conductive plasma and restores their adiabatic decay-law. The time of the second horizon entry depends on the scale of the magnetic mode in question. Clearly, the larger the wavelength of the mode the longer its stays outside the Hubble radius and the stronger its superadiabatic amplification. Assuming a magnetic seed with current comoving (pre-collapse) scale around 10 Kpc, for example, one can show that its present magnitude is approximately 10 −33 G. This is far stronger than the previously quoted value of 10 −53 G. Further amplification is expected to occur during the collapse of the magnetised protogalactic cloud, which can bring the final strength of the field closer to, or even within, the galactic-dynamo range.
Although our analysis is primarily focused on the spatially flat FRW models, our conclusions and results extend naturally to their marginally closed and marginally open counterparts. This is intuitively plausible and it can also be shown analytically. Moreover, in marginally open Friedmann models, the superadiabatic amplification extends to the (physically unambiguous) subcurvature magnetic modes. Consequently, classical electromagnetism and conventional FRW cosmology can produce cosmological B-fields with residual strengths much larger than generally expected. Moreover, using the same arguments, one can constraint essentially all the non-conventional scenarios of primordial magnetogenesis that amplify their magnetic fields during inflation and then allow them to decay adiabatically to the present. In fact, the more efficient the inflationary amplification is, the stronger the constraints. The main message, however, is that causality and the inferred absence of superhorizon-sized electric currents make the post-inflationary evolution of large-scale B-fields a matter of initial conditions. These dot not always guarantee the adiabatic decay-law, but also allow for the superadiabatic amplification of primordial magnetic fields on super-Hubble lengths. A development that could put the question of cosmic magnetism under an entirely new perspective.

The question of cosmic magnetogenesis
We distinguish the scenarios of cosmological magnetic generation according to whether they operate before or after recombination. Here, we will briefly outline the main problems faced by the early-universe mechanisms, which are classified further into inflationary and post-inflationary.

The scale question
The main drawback of primordial magnetic fields generated after inflation is their scale. Typical dynamos require seeds with coherence lengths no less than 100 pc by the time galaxy formation has been completed. This translates into a comoving scale of approximately 10 Kpc before the collapse of the proto-galactic cloud. Post-inflationary magnetic fields, however, are generally much smaller in size [3]. The reason is causality, which always confines the correlation length of the generated B-field within that of the causal horizon (i.e. the Hubble radius). Put another way, given that no physical process propagates faster than the speed of light, all causally produced magnetic fields have sizes smaller than the Hubble length at the time of their creation. 1 The latter is typically too small. For instance, assuming that the magnetic field is produced at the electroweak phase transition, its present size will be close to that of our solar system.
Theoretically, the scale problem can be solved, if there is an amount of magnetohydrodynamic (MHD) turbulence in the plasma and the initial B-seed is highly helical. In that case magnetic helicity cascades inversely from smaller to larger scales, shifting magnetic energy to larger wavelengths and increasing the size of the original seed [6]. Nevertheless, the "inverse cascade" scenario seems unlikely to deliver the desired results, unless the amount of primordial magnetic helicity is unrealistically large.

The strength question
There is no scale issue whatsoever for magnetic fields generated during inflation. What the de Sitter phase does, is to stretch quantum fluctuations in the Maxwell field from subhorizon scales far beyond the Hubble radius, where they freeze out as classical electromagnetic fields. The main problem of inflationary magnetogenesis is the extreme weakness of the residual B-field, which is believed to have no astrophysical significance. Recall that galactic dynamos typically need magnetic seeds between ∼ 10 −22 G and ∼ 10 −12 G at the time the galaxy has been formed, although it might be possible to push the lower limit down to ∼ 10 −30 in open or in Λ-dominated FRW universes. This can happen because galaxies are older in the latter models, which gives the dynamo more time to produce the observed µGauss-order fields [5].
The problem is that magnetic fields that have survived the de Sitter phase are largely expected to have residual strengths far below 10 −30 G. This has been attributed to the so-called adiabatic magnetic decay. The belief, in other words, that conventional B-fields decay as B ∝ a −2 (a is the scale factor of the universe) at all times and on all scales. As a result, the typical strengths of inflationary magnetic fields quoted in the literature are below 10 −50 G, unless classical Maxwellian electromagnetism or standard cosmology are abandoned (e.g. see [3] and also § 3.2.2 here). In the following sections we will demonstrate that this is not necessarily the case.

Magnetic fields in flat FRW universes
Cosmological magnetic fields in spatially flat FRW universes are widely believed to decay adiabatically on all scales and at all times, during both their inflationary and post-inflationary life. Here, we will take another look at the evolution of large-scale B-fields after inflation.

The linearised Maxwell's equations
Let us start by considering an FRW spacetime, with Euclidean spatial geometry, permeated by weak electromagnetic perturbations. Then, introduce a group of observers with 4-velocity u a (normalised so that u a u a = −1). Assuming that g ab is the spacetime metric, the symmetric tensor h ab = g ab + u a u b (with h ab u b = 0, h ac h c b = h ab and h a a = 3) projects orthogonal to u a and onto the observers' spatial hypersurfaces (i.e. their rest spaces -e.g. see [7]). We can use u a and h ab to decompose all the variables, the equations and the operators into their timelike and spacelike parts. For example, the electromagnetic (Faraday) tensor splits into an electric (E a ) disguise" for their inflationary counterparts (see § 4 below). and a magnetic (B a ) field as F ab = 2u [a E b] + ε abc B c , with ε abc representing the 3-dimensional Levi-Civita tensor. Similarly, Maxwell's formulae decompose into two propagation and two constraint equations. Linearised around our FRW background, these formulae reduce to [7] and Note that overdots are proper-time derivatives (e.g.
is the spatial covariant-derivative operator and ∇ a is its 4-dimensional analogue. Also, H =ȧ/a is the background Hubble parameter and J a is the spatial electric current. The latter is related to the electric field via Ohm's law, which takes the simple covariant form with ς representing the electrical conductivity of the cosmic medium [8].

The role of the electric currents
Maxwell's equations monitor the evolution of the electromagnetic field, but its overall behaviour also depends on the electric properties of the supporting medium. More specifically, as we will see next, the decisive feature is the presence or not of coherent electric currents.

Low conductivity limit
When the conductivity is low, ς → 0 and Ohm's law ensures that J a → 0. This happens during inflation, for example, when the universe is expected to be a very poor electrical conductor. In the absence of spatial currents, both components of the Maxwell field obey identical equations, while (1) and (2) combine to provide the following wave-like equation for the magnetic field Clearly, an exactly analogous expression monitors the evolution of the electric counterpart. Note that the above holds on spatially flat FRW backgrounds only (for the open and the closed Friedmann models see § 7.1 and § 7.2 respectively). Also, w = p/ρ is the barotropic index of the matter (with ρ and p representing its energy density and isotropic pressure respectively), while D 2 = D a D a is the 3-dimensional covariant Laplacian operator. Introducing the rescaled magnetic field B a = a 2 B a and using conformal time (η withη = 1/a), expression (4) reduces to the familiar compact form [3] B ′′ a − a 2 D 2 B a = 0 , where primes denote conformal-time derivatives. The above is (formalistically) independent of the type of matter that fills the universe. This means that Eq. (5) can be used throughout the evolution of the universe, provided that there are no electric currents and matter corresponds to a single barotropic fluid. Finally, by means of the harmonic splitting B a = n B (n) Q This differential equation accepts the standard oscillatory solution with n > 0 representing the comoving wavenumber of the n-th magnetic mode.

High conductivity limit
At the other end of the spectrum, one may assume that the conductivity of the cosmic medium is very large. This is supposed to happen after inflation, during reheating and throughout the subsequent radiation and dust epochs of the universe. When ς → ∞ Ohm's law (see Eq. (3)) guarantees that the electric field vanishes in the frame of the fluid, leaving its magnetic component frozen into the highly conductive medium. This is the well known ideal-MHD approximation. It is crucial to emphasise, however, that, in addition to the high conductivity of the matter, the ideal-MHD limit also requires the presence of electric currents with coherence scales equal to those of the corresponding electric and magnetic fields. After all, it is the currents that eliminate the electric fields and freeze their magnetic counterparts into the matter. Without the electric field, Maxwell's formulae (see Eqs. (1) and (2) in § 3.1) reduce to one propagation and two constraint equations, respectively given bẏ Recalling that H =ȧ/a, the former of the above integrates immediately to give B ∝ a −2 independent of the type of matter that fills the universe. This result has been long thought to guarantee that B-fields decay adiabatically throughout their post-inflationary evolution on all scales. Combining the above with the adiabatic decay of the field during inflation, means that the residual strength of conventional B-fields is given by (e.g. see [3] and references therein) with λ B representing the current comoving (pre-collapse) scale of the magnetic mode in question (measured in Mpc). In other words, the final magnetic strength is independent of the particulars of the adopted inflationary scenario and depends only on the coherence size of the field. Setting λ B ≃ 10 Kpc, which is the minimum required length for the dynamo to work (see § 2.1 earlier), we find that B ≃ 10 −53 G today. Magnetic seeds of such strength are twenty orders of magnitude below the lowest dynamo limit quoted in the literature (see § 2.2 earlier), which makes them astrophysically irrelevant for all practical purposes.

Large-scale magnetic fields in flat FRW universes
All the astrophysically relevant magnetic fields produced during inflation remain outside the Hubble horizon for most of their post-inflationary evolution. This implies that Eq. (9), as well as the numerical estimates resulting from it, require that the ideal-MHD limit applies on superhorizon scales. As we will argue next, however, such an assumption violates causality.

Causality and magnetic-flux freezing
During inflation the electrical conductivity of the universe is very low and we are essentially dealing with a source-free electromagnetic field. In other words, there are no electric currents during the inflationary phase and both components of the Maxwell field are present. The conductivity of the universe starts rising once we have entered reheating. As the electrical resistivity drops the currents establish themselves, eliminate the electric fields and freeze their magnetic counterparts into the matter. However, the processes of current formation, electricfield elimination and magnetic-flux freezing are causal. Since no physics propagates faster than the speed of light, the range and the domain of influence of above processes can never exceed the particle horizon. After inflation the latter essentially coincides with the Hubble radius. Without superhorizon-sized currents and in the presence of the large-scale electric fields, which were also produced during inflation, we cannot apply the ideal-MHD approximation on super-Hubble scales. After all, the ideal-MHD limit is a condition that results from causal microphysical processes and for this reason its domain of influence can never exceed the horizon. Therefore, magnetic fields with superhorizon correlations cannot be treated as frozen into the cosmic medium, even after inflation is over. One can also demonstrate this mathematically by looking at the nonlinear expression of Faraday's law, which takes the 1+3 covariant forṁ relative to an observer comoving with the fluid [7]. Note thatḂ a = h a bḂ b by construction. Also, Θ is the expansion scalar, σ ab is the shear tensor, ω ab is the vorticity tensor and A a =u a the 4-acceleration vector. 2 We should also point out that, when linearised around an FRW background, the above reduces to Eq. (1b). At the ideal-MHD limit, when the electric field vamishes in the frame of the fluid, we may drop the last two terms from the right-hand side of (10). Then, setting Θ/3 =ȧ/a, we arrive at This result guarantees that the spacelike vector r a = a 3 B a is a relative position vector and as such it connects the same particles at all times (e.g. see [9] for further details). Consequently, the same particles are always aligned along the same magnetic-field line, which is another way of saying that the B-field is frozen into the matter. When the electric terms are included into Faraday's law, however, the right-hand side of (12) is no longer zero (even at the linear level). This means that r a = a 3 B a is not a connecting vector and that the magnetic field cannot be treated as frozen into the matter (i.e. the ideal-MHD limit does not apply). 2 The irreducible kinematic variables, namely the expansion scalar, the shear and the vorticity tensors and the 4-acceleration vector, are obtained after decomposing the covariant derivative of the observer's 4-velocity as In the process we have also used the projection tensor h ab = g ab + uau b (with g ab being the spacetime metric), which projects orthogonal to the ua-field and onto the observers instantaneous 3-D rest-space (e.g. see [7]).

Post-inflationary magnetic evolution
Following our discussion so far, causality guarantees that beyond the horizon we are still dealing with the (source-free) electromagnetic fields that have been left there from the inflationary era, even after the end of inflation. Therefore, as long as they remain outside the Hubble radius, magnetic fields do not obey Eq. (8a), but the wave-like formula (5), just like their inflationary predecessors. Recalling that B = a 2 B, the associated solution (see Eq. (7) in § 3.2.1) reads where n > 0 and nη = λ H /λ n . This ratio measures the physical size of the magnetic mode in question (λ n = a/n) relative to that of the Hubble horizon (λ H = 1/H). Here we will focus on wavelengths much larger than the Hubble radius (i.e. those with λ H /λ n ≪ 1), which in conformal-time terms are characterised by nη ≪ 1. On these scales a simple Taylor expansion reduces solution (13) to the power law with a = a(η). This transition from oscillation to power-law evolution at the Hubble threshold is nothing new to cosmological perturbation theory. It happens to linear density perturbations, for example, during the radiation era (e.g. see § 4.3 in [10]). Physically, the change from oscillatory behaviour to power law at the Hubble length, simply reflects the fact that superhorizon-sized perturbations have oscillation periods longer than the lifetime of the universe. 3 Cosmological magnetic fields evolve in line with the power law (14) from the moment they cross outside the Hubble horizon during inflation, until the time of their re-entry (in the radiation epoch, or later in the dust era). Before proceeding to examine the implications of solution (14) for the magnetic evolution after inflation, let us take a brief look at the inflationary phase first. Recall that during that period there are no electric currents and all the astrophysically relevant lengths are well outside the Hubble horizon. Assuming exponential (de Sitter-type) expansion, we may set a ∝ −1/η with η < 0. Then, after calculating the two integration constants on the right-hand side of (14), the latter recasts into Consequently, large-scale (conventional) magnetic fields on spatially flat FRW backgrounds decay adiabatically (i.e. B ∝ a −2 ) throughout the de Sitter phase of the expansion. Let us now look at evolution of superhorizon-sized cosmological B-fields after inflation. Solution (14) contains two modes, the second of which is not a priori negligible despite the smallness of nη. In fact, depending on the values of the integration constants (i.e. on the initial conditions), this mode can dictate the post-inflationary magnetic evolution on super-Hubble scales (see § 4.3 below). To demonstrate this one simply needs to evaluate the two integration constants on the right-hand side of Eq. (14) and rewrite the latter as Note that we have dropped the mode index (n) for the economy of the discussion and used the relation H = a ′ /a 2 for the Hubble parameter (recall that the primes indicate differentiation with respect to the conformal time). The above monitors the linear evolution of superhorizon-sized magnetic fields on spatially flat FRW backgrounds. We should also point out that the barotropic index of the matter is not necessarily constant but it can vary with time (i.e. w = w(t)). This means that solution (16) applies continuously throughout the lifetime of the universe, provided the cosmological expansion is entirely smooth and the matter can always be treated as a single barotropic medium. Under this proviso, expression (16) also monitors the magnetic evolution through the various cosmological transitions (e.g. the one leading from inflation to reheating). The precise physics of the early transitions and the exact nature of the cosmic medium during those periods are still ambivalent. Nevertheless, the barotropic index of the matter is believed to maintain constant value during prolonged periods in the lifetime of the universe. As long as w remains invariant, the cosmological scale factor and the conformal time are related by where w = −1/3 and the zero suffix indicates a given initial time. On using the above, it is straightforward to show that H = a ′ /a 2 = 2/(1 + 3w)aη and then recast solution (16) into The latter also monitors the linear evolution of superhorizon-sized magnetic fields on spatially flat FRW backgrounds filled with a single barotropic medium. In contrast to solution (16), however, here the barotropic index of the matter has been treated as a constant. Consequently, solution (18) does not apply continuously throughout the evolution of the universe, but only to periods during which w = constant = −1/3 (e.g. to the reheating and the radiation eras when w = 0 and w = 1/3 respectively). In other words, solution (18) is a special case of (16).

The epochs of reheating, radiation and dust
Looking at solutions (16) and (18), we immediately notice that the first of the two magnetic modes on their right-hand side always decays adiabatically. The rate of the second mode, however, is not a priori fixed but depends on the equation of state of the cosmic medium. The latter also determines the relation between the cosmological scale factor and the conformal time.
In particular, as long as w = constant > −1/3 the second mode on the right-hand side of (18) decays at a rate slower than the adiabatic. The same behaviour can also be seen in solution (16). Therefore, when dealing with conventional matter, supperhorizon sized magnetic fields on spatially flat FRW backgrounds are superadiabatically amplified. This, under the proviso that the initial conditions allow the second modes in (16) and (18) to survive and dominate. With these in mind, let us take a closer look at the post-inflationary magnetic evolution. During the reheating phase, as well as during the dust era later, w = 0 , a ∝ η 2 and H = 2/aη. Then, solutions (16) and (18) reduce to Therefore, as long as reheating lasts (as well as after equipartition) superhorizon-sized magnetic fields deplete as B ∝ a −3/2 , instead of following the standard adiabatic (B ∝ a −2 ) decay-law. During the intermediate epoch of radiation w = 1/3, which means that a ∝ η and H = 1/aη. Then, throughout that period solutions (16) and (18) take the form ensuring that large-scale magnetic fields drop as B ∝ a −1 when radiation dominates the energy density of the universe. Finally, let us also consider a phase of stiff-matter domination. In that case, w = 1, a ∝ η 1/2 , H = 1/2aη and with the dominant mode remaining constant. Note that towards the end of inflation, when the inflaton rolls down the slope of the potential, the effective equation of state of the cosmic medium is that of stiff matter. In summary, after the end of the de Sitter phase, large-scale B-fields on spatially flat FRW backgrounds obey solutions which always contain modes with decay rates slower than the adiabatic. This occurs naturally, without the need to break away from conventional electromagnetic theory or to abandon standard cosmology. Whether these slowly-decaying magnetic modes can dominate over the adiabatic one depends on their associate coefficients. When the latter are of roughly the same order of magnitude, the slowly decaying modes quickly take over and dictate the subsequent evolution of the B-field. The initial conditions at the beginning of the post-inflationary epoch are therefore crucial.

The role of the initial conditions
The initial conditions of the post-inflationary magnetic evolution are decided by the field's behaviour in the de Sitter phase and by the nature of the transitions to the eras of reheating and radiation. Based on Israel's work on junction conditions [11], we will discuss two typical and complementary initial-condition scenarios. Alternative approaches may also be possible.
Scenario A: Let us consider the typical scenario, where the background barotropic index undergoes an abrupt change from w − * before the transition to w + * afterwards (with w + * = w − * ). 4 Let us also assume that the matching spatial hypersurface is that of constant conformal time. This translates into a "jump" in the expansion rate of the background universe, namely in the Hubble parameter, on either side of the transit surface (i.e. . The latter implies a discontinuity in the extrinsic curvature of the matching hypersurface, which requires the presence of a "thin shell" there with finite energy-momentum tensor. Practically speaking, we assume that the width of the shell is too small compared to the scales of interest. In that case, the aforementioned shell can be replaced by a spacelike hypersurface. Discontinuities of this nature can be used to bypass the (as yet ambivalent) details of early cosmological transitions, like the one leading from inflation to reheating (e.g. see [12] and references therein).
Conventional scenarios of inflationary magnetogenesis demand that the magnetic field decays adiabatically throughout the de Sitter regime (i.e. B ∝ a −2 ). Typical non-conventional mechanisms of primordial magnetic generation, on the other hand, amplify their B-fields superadiabatically during inflation (i.e. B ∝ a −m with 0 ≤ m < 2) [3,13]. With these in mind, let us assume that all along the de Sitter phase the magnetic field obeys the power law where 0 ≤ m ≤ 2 and the zero suffix indicates the beginning of the exponential expansion. Note that the second equality reflects the fact that a ∝ −1/η, with η < 0, during de Sitter-type inflation (i.e. for w = −1 -see Eq. (17) in § 4.3). Differentiating (22) with respect to the conformal time gives B ′ = mB/η, which ensures that at the end of inflation proper. Once into reheating the barotropic index changes from w − * = −1 to w + * = 0. Then, according to solution (18), throughout reheating superhorizon-sized magnetic fields evolve as with a ≥ a + * (see also Eq. (19) in § 4.2). When the transit hypersurface is that of constant conformal time, we may set η + * = −η − * (recall that η + * > 0 and η − * < 0). 5 This implies a "jump" in the expansion rate of the background universe and a discontinuity in its extrinsic curvature of the matching hypersurface, which can be compensated by the presence of a thin layer there [12]. Consequently, assuming that there is no magnetic discontinuity at the linear level, namely that The above sets the initial conditions for the evolution of the B-field during reheating and combines with solution (24) to give where a ≥ a + * . 6 Therefore, as long as m = 4, the dominant magnetic mode of (26) drops as B ∝ a −3/2 and the B-field is superadiabatically amplified throughout reheating. We remind the reader that almost all the scenarios of inflationary magnetogenesis assume that 0 ≤ m ≤ 2.
5 Setting η + * = −η − * on either side of the transit hypersurface does not constitute a real discontinuity in the conformal time. The jump ([η * ] + − = 2η + * ) is only an apparent one, since it can be removed by replacing η with the variable x = |η|. Then, during the de Sitter phase a = 1/x, with x < x * and x * ≥ 0. After the end of inflation, throughout reheating for example, we have the relation a ∝ x 2 , where now x > x * . 6 We can obtain the evolution law (26) starting from solution (16) as well. This requires calculating the jump in the value of the Hubble parameter caused by the abrupt change of the barotropic index on the matching hypersurface, which is that of constant conformal time. In order to do that recall first that H − * = −1/a − * η − * at the end of the de Sitter regime and H + * = 2/a + * η + * at the start of reheating. Then, using conditions (23) and (25) Let us look at the magnetic evolution in the subsequent epochs of radiation and dust. Following solution (26) and keeping in mind that a ∝ η 2 during reheating (see Eq. (17)), we deduce that B ∝ η −3 throughout that period. Then, just before the transition to the radiation era. Once there, the barotropic index of the background matter changes from w − * = 0 to w + * = 1/3 and solution (18) reads with a ≥ a + * (see also Eq. (20) in § 4.2). As before, suppose that the matching hypersurface is that of constant conformal time and assume that the magnetic evolution through the transition is smooth. Then, demanding η + and sets the initial conditions for the magnetic evolution in the radiation era. Substituting the above into the right-hand side of (28), we arrive at where a ≥ a + * . Consequently, superhorizon-sized magnetic fields are superadiabatically amplified (i.e. B ∝ a −1 ) all along the radiation epoch as well.
Similarly, we find that η − * B ′ − * = −B − * prior to the equilibrium time, since a ∝ η when w = 1/3 (see Eq. (17) in § 4.3). At the time of matter-radiation equality the background barotropic index changes from w − * = 1/3 to w + * = 0. Then, when the matching hypersurface is that of constant conformal time and the B-field evolves smoothly through the transit, we have at the start of matter domination. Finally, setting w = w + * = 0 into the right-hand side of solution (18) and using the above initial conditions, leads to with a ≥ a + * . Therefore, as long as the magnetic field remains outside the Hubble radius, B ∝ a −3/2 and its superadiabatic amplification continues into the dust era as well. Moreover, the effect is independent of the magnetic evolution during the de Sitter phase. On whether, in particular, the B-field depleted adiabatically throughout inflation or not.
Scenario B: Suppose that the background equation of state undergoes an abrupt change, as the universe crosses from one epoch to the next, but this time do not allow for a thin shell on the transition hypersurface. Then, there can be no discontinuity in the extrinsic curvature of the matching surface. When dealing with a Friedmann universe, this means no jump in the value of the background Hubble parameter there (i.e. [H * ] + − = H + * − H − * = 0). In such a case, the transit hypersurface is that of constant energy density, though not necessarily of constant conformal time. Discontinuities of this nature can also be used to cope with early universe transitions.
In line with the literature on inflationary magnetogenesis and with Scenario A before, let us assume that large-scale primordial magnetic fields obey the power law during the de Sitter phase. Again, the zero suffix indicates the onset of the exponential expansion and 0 ≤ m ≤ 2. Differentiating the above with respect to the conformal time, guarantees that at the start of reheating and sets the initial conditions for the subsequent evolution of the B-field. Following solution (16), throughout the reheating phase (when a ∝ η 2 ), superhorizon-sized magnetic fields are monitored by with a ≥ a + * . Inserting condition (35) into the right-hand side of the above and keeping in mind that H = 2/aη during reheating, we obtain where a ≥ a + * . When m = 2 the second term on the right-hand side vanishes, leaving the adiabatic (i.e. B ∝ a −2 ) mode only. For m = 2, however, the second mode of solution (37) survives and the magnetic decay-rate slows down to B ∝ a −3/2 . A straightforward calculation confirms that this pattern is repeated at the subsequent transitions to the radiation and the dust eras. Consequently, in the absence of thin shells on the transition hypersurfaces, only magnetic fields that decay adiabatically during a certain cosmological epoch will continue to do so for their subsequent evolution. When there is no adiabatic decay prior to the transit, the B-field is superadiabatically amplified after the transition (provided w + * > −1/3). The implications of scenario B for inflationary magnetogenesis are fairly straightforward to deduce. Primordial magnetic fields that happen to decay adiabatically throughout inflation will continue to do so for the rest of the lifetime. This is essentially the conventional scenario of primordial magnetogenesis, which produces B-fields with astrophysically irrelevant residual strengths. However, large-scale magnetic fields that did not obey the B ∝ a −2 law during the de Sitter phase will experience superadiabatic amplification after the end of inflation. This can affect the non-conventional scenarios of primordial magnetogenesis that superadiabatically amplify their B-fields during inflation (see § 6.2 below).

The residual magnetic field
Following our discussion so far, depending on the initial conditions, conventional large-scale B-fields can be superadiabatically amplified throughout the post-inflationary evolution of a flat FRW universe. Next, we will estimate the residual strength of such fields.

The time of second horizon crossing
To begin with, recall that imposing the adiabatic decay-law at all times and on all scales has lead to magnetic fields of approximately 10 −53 G today, assuming their current comoving size is close to 10 Kpc (see Eq. (9) in § 3.2.2). Also, during the de Sitter regime, superhorizon-sized magnetic fields decay adiabatically as expected (see solution (15) in § 4.2). The situation changes after inflation, when the magnetic decay-rate slows down (see solutions (19) and (20) in § 4.3). Throughout reheating, in particular, we have B ∝ a −2/3 . This slows down further (to B ∝ a −1 ) in the radiation era, before returning to the B ∝ a −2/3 law during the subsequent dust epoch. 7 As a result, the residual magnetic strength can be considerably larger than expected. The overall amplification depends on the scale of the magnetic mode in question, which determines the time of horizon entry. Recall that once inside the Hubble radius the adiabatic decay is restored. This occurs because on subhorizon scales the electric currents take over, eliminate the electric fields and freeze their magnetic counterparts into the highly conductive medium. Put another way, we can apply the ideal-MHD limit only after the second horizon crossing. Then onwards, the magnetic flux remains conserved and the B-field decays adiabatically.
Suppose that the current comoving scale of the magnetic seed is λ B ≃ 10 Kpc, which is the minimum required for the dynamo to work. Fields of this size have (λ H /λ B ) † ≃ 3 × 10 5 , where the †-suffix denotes the present, assuming that λ H ≃ 3 × 10 3 Mpc is the Hubble radius today. Given that λ H ∝ t and λ B ∝ a, we deduce that λ H /λ B ∝ a 1/2 during the dust era (when t ∝ a 3/2 ) and λ H /λ B ∝ a throughout the preceding radiation epoch (when t ∝ a 2 ). Putting these together, one finds that scales close to 10 Kpc today entered the horizon at a HC ≃ 1/3 × 10 −3 a EQ . The latter translates into T HC ≃ 3 × 10 −6 GeV, since T ∝ a −1 at all times and T EQ ≃ 10 −9 GeV. Until then, the B-field was lying outside the Hubble radius and it was superadiabatically amplified.

The final magnetic strength
As mentioned above, magnetic fields decay adiabatically during the de Sitter phase and once they are back inside the horizon after inflation. Therefore, the superadiabatic amplification occurs from the end of inflation proper until the second horizon crossing. Suppose that ρ B = B 2 is the magnetic energy density and ρ that of the dominant matter component. Then, at the end 7 At the end of inflation the scalar field rolls down the slope of the potential to its minimum. Then, the effective equation of state of the inflaton field (φ) is that of stiff matter, with p φ ≃ ρ φ ≃φ/2. During these final stages we have a ∝ √ η, with η > 0, which substituted into Eq. (13) leads to B = C3(a0/a) + C4 (see also solution (21)).
In other words, for the brief period between the de Sitter phase and reheating, the dominant magnetic mode of (13) remains constant. We are not going to consider the implications of the aforementioned epoch here.
of the de Sitter regime we have Note that M is the scale of inflation, T RH is the reheat temperature (both measured in GeV) and λ B is the current physical scale (measured in Mpc) of the magnetic mode in question. During reheating, ρ B ∝ a −3 and ρ ∝ a −3 as well. Therefore, throughout this phase, the dimensionless ratio ρ B /ρ remains unchanged, which means that (ρ B /ρ) RH ≃ (ρ B /ρ) DS . Once into the radiation era, however, ρ B ∝ a −2 and ρ ≃ ρ γ ∝ a −4 , with ρ γ representing the energy density of the radiative component. Hence, for a magnetic mode that crosses inside the Hubble horizon before equipartition, we have After horizon crossing ρ B , ρ γ ∝ a −4 , ensuring that their ratio remains constant until today. In other words today (recall that the †-suffix corresponds to the present). As we have seen in the previous section, magnetic fields with current comoving size close to 10 Kpc, re-enter the horizon at T HC ≃ 3 × 10 6 GeV. Substituting this value into the right-hand side of Eq. (40) and recalling that (ρ γ ) † ≃ 10 −51 GeV 4 , gives Therefore, assuming that M ≃ 10 17 GeV and T RH ≃ 10 10 GeV, the present magnitude of a cosmological magnetic field with current physical size close to 10 Kpc is approximately 10 −33 G, instead of 10 −53 G. In other words, by simply appealing to causality, one can increase the final strength of conventional inflationary magnetic seeds by roughly twenty orders of magnitude.

Implications for cosmic magnetogenesis
Our results solely affect superhorizon-sized magnetic fields, which means that they do not interfere at all with the mechanisms of post-inflationary magnetogenesis. Due to causality reasons, the latter produce subhorizon-sized B-fields only. There are potentially pivotal consequences, however, for the inflationary scenarios, both conventional and non-conventional.

Conventional scenarios
Conventional inflationary magnetic fields decay adiabatically during the de Sitter phase, but deplete at a slower pace after inflation. Here, this happens within scenario A (see § 4.4 earlier).
In general, any scenario that allows the second magnetic mode on the right-hand side of solution (19) to survive at the start of reheating will lead to the same result. Then, the residual comoving (pre-collapse) magnitude of a B-field (with physical scale close to 10 Kpc today) will be approximately 10 −33 G. The magnetic strengths required for the dynamo to work are estimated at the time of completed galaxy formation (see § 2.2 earlier). The magnitude quoted above is comoving, which means that it does not include the magnetic amplification that occurs during the collapse of the proto-galactic cloud. Assuming an idealistic spherically symmetric collapse, we may add up to four orders of magnitude to the comoving magnetic strength. Adopting the more realistic scenario of anisotropic protogalactic collapse leads to further increase by one or two orders of magnitude [14]. All these can bring the final magnetic strength close to 10 −27 G by the time the galaxy is formed. This is stronger than 10 −30 G, which is the minimum magnetic strength quoted in the literature as capable of seeding the dynamo [5]. Hence, astrophysically relevant magnetic fields are theoretically possible without violating conventional electromagnetism or abandoning standard cosmology. Additional magnetic amplification may be possible as well. The literature contains mechanisms that could enhance cosmological B-fields during both the earlier and the later stages of their evolution. Here, we would like to draw the reader's attention to an alternative possibility, which is directly related to our discussion. In line with solution (18)), the stiffer the equation of state of the cosmic medium, the slower the magnetic decay and the stronger its superadiabatic ampolification. In fact, at the stiff-matter limit (i.e. when p = ρ and w = 1) we obtain for the dominant magnetic mode (see also solutions (18) and (21)). This result implies that a relatively brief period of stiff-matter domination during the early stages of the radiation era could lead to substantial magnetic amplification without necessarily affecting the observational constraints. 8 In terms of temperature, the radiation epoch typically spans from T RH ≃ 10 10 GeV up to T eq ≃ 10 −9 GeV. Therefore, if the universe is dominated by stiff matter between, say, T RH ≃ 10 10 GeV and T ≃ 10 7 GeV, the final magnetic strength will increase from 10 −27 G to 10 −21 G. The latter lies within what one might call typical galactic-dynamo requirements.

Non-conventional scenarios
The vast majority of the inflationary magnetogenesis mechanisms operate outside conventional electromagnetic theory, or introduce some other kind of new physics. There is a very long list of non-conventional scenarios and for this reason we direct the reader to [3] for recent reviews and specific references, while a relatively brief discussion can be found in [13]. In most of the proposed mechanisms the B-field is superadiabatically amplified (i.e. B ∝ a −m , with m < 2) during the de Sitter phase. After that, standard electromagnetism is usually restored and the final magnetic strength is estimated by assuming that B-fields decay adiabatically until today. This assumption does not a priori hold, however, given that all the astrophysically relevant modes remain outside the horizon at least until late into the radiation era. On these scales, the aforementioned magnetic fields are superadiabatically amplified throughout their post-inflationary evolution within both of our initial-condition scenarios (see § 4.4 earlier). Therefore, residual magnitudes that were based on the adiabatic-decay law after inflation need to be revised. As we will argue next, the revision will affect (to a larger or lesser degree) essentially all the mechanisms of primordial magnetogenesis that amplify their B-fields during inflation. Scenarios of inflationary magnetic amplification are often susceptible to backreaction problems. In other words, the Maxwell field can get strong enough to start interfering with the background kinematics. Even when there are no backreaction issues, however, there might be problems with the observational constrains. The strongest constrain so far on cosmological magnetic fields comes the Cosmic Microwave Background (CMB), the high isotropy of which prohibits homogeneous B-fields with current magnitudes larger than ∼ 10 −9 G [16]. None of the aforementioned scenarios of primordial magnetogenesis violates the CMB constraint, but only after assuming that the adiabatic decay-law holds from the end of inflation until today. When the B-field remains superadiabatically amplified throughout its entire post-inflationary evolution, however, one should probably revise the residual magnetic strengths and check whether they comply or not with the CMB observations. In what follows we will consider two characteristic alternative scenarios to illustrate our argument. The first will allow for a rather strong superadiabatic amplification during the de Sitter regime (i.e. B ∝ a −m , with m < 1), while in the second the amplification will be relatively mild (i.e. B ∝ a −m , with 1 < m < 2).
Suppose that B ∝ a −1/2 throughout inflation, which implies relatively strong amplification during that period. Then, at the end of the de Sitter phase, the relative magnetic strength will be given by the ratio where M and T RH are the energy scale and the reheat temperature of the inflationary model respectively (both measured in GeV), while λ B is the current comoving scale of the field (in Mpc). For a magnetic mode that crosses inside the horizon at recombination we may set λ B ≃ 3×10 3/2 Mpc at present. Ignoring reheating for simplicity, the above magnetic mode is superadiabatically amplified during the radiation era and for the brief period between equipartition and decoupling. In that case we have today. For typical values of the inflationary parameters, for instance when M ∼ 10 17 GeV and T RH ∼ 10 10 GeV, the above gives B † ≫ 10 −9 G, in violation of the CMB constraints. Therefore, causality and the resulting absence of superhorizon-sized electric currents can essentially rule out a host of primordial magnetogenesis mechanisms.
The situation changes drastically when the inflationary amplification of the B-field is relatively weak. For instance, let us assume that B ∝ a −3/2 throughout the de Sitter phase. Then, proceeding as before we find and subsequently for a magnetic mode that crossed the horizon around the time of decoupling. Fields with the above (comoving) strength today are too weak to affect the CMB isotropy but strong enough to seed the galactic dynamo. Recall that a comoving magnitude of approximately 10 −25 G can increase to roughly ∼ 10 −19 G by the time the galaxy is formed. So, in this case, the absence of large-scale electric currents and the inferred superadiabatic magnetic amplification on super-Hubble lengths appears to assist the associated scenarios of cosmic magnetogenesis, thus making them more promising candidates.
Overall, mechanisms of magnetic generation where the primordial B-field undergoes substantial (superadiabatic) amplification during inflation are likely to be in conflict with the observations. On the other hand, scenarios that achieve relatively mild amplification during the de Sitter phase can produce magnetic seeds of real astrophysical relevance. According to our analysis, the threshold rate is close to B ∝ a −1 , or m ≃ 1. The bottom line is that, to lesser or larger degree, one should revise the final magnetic strengths of essentially all the scenarios that amplify their B-fields during inflation and then allow them to decay adiabatically until today.
The new values should be then compared to the upper limits set by the observations.

The case of nearly flat FRW universes
Although we have so far confined our analysis to Friedmann universes with Euclidean spatial geometry, the same results also apply to FRW models with nearly flat spacelike hypersurfaces. To a large extent this may be intuitively obvious, but it can be shown analytically as well.

Marginally open FRW models
We begin by recalling that, at the ideal-MHD limit, magnetic fields decay adiabatically irrespective of the background spatial curvature. Thus, in the presence of highly conductive electric currents, B ∝ a −2 at all times and in all three FRW spacetimes. When there are no currents, however, the magnetic evolution also depends on the geometry of the background universe. Throughout inflation, for example, or on superhorizon scales after the end of the accelerated expansion phase, the magnetic field obeys the wave-like equation [17] B ′′ (n) + n 2 + 2K B (n) = 0 , where K = 0, ±1 is the 3-curvature index. 9 In Friedmann models with negative spatial curvature (i.e. for K = −1), the above takes the form with the comoving eigenvalue being positive and continuous (i.e. n > 0). Equation (48) accepts two qualitatively different families of solutions, depending on the range of the associated eigenvalues. When n 2 < 2, in particular, we find hyperbolic behaviour with On the other hand, we recover the more familiar oscillatory evolution, as we move to smaller scales with n 2 > 2.
When dealing with open Friedmann models, the scale factor and the curvature contribution to the total energy density are conveniently expressed in terms of the conformal time as respectively. Note that the β-parameter is decided by the equation of state of the matter and is given by β = (1 + 3w)/2 = 0. Here, we will consider the post-inflationary evolution of the universe, which means that w ≥ 0 and β ≥ 1/2 always. We are also interested in magnetic fields with super-Hubble correlations. Following (51a), if n is the (comoving) eigenvalue of a mode, its (physical) size relative to the Hubble scale is determined by the ratio since H = a ′ /a 2 . Let us now confine to marginally open FRW universes. According to Eq. (51b), these spectimes are characterised by very small values of the conformal time (i.e. Ω K ≪ 1 implies η ≪ 1 and vice versa). It is then straightforward to show that, during the reheating and the dust eras (i.e. when β = 1/2), marginally open Friedmann models have a ∝ η 2 , Ω K ≃ −η 2 /4 and λ H /λ n ≃ nη/2. Throughout the radiation epoch, on the other hand, β = 1 and relations (51), (52) lead to a ∝ η, Ω K ≃ η and λ H /λ n ≃ nη respectively. All these mean that superhorizon-sized modes in marginally open FRW universes satisfy the constraint nη ≪ 1, just like in their spatially flat counterparts. The difference is that now η ≪ 1 as well. This ensures that, in marginally open Friedmann models, even small-scale modes with fairly large eigenvalues (i.e. with n ≫ 1) can lie outside the Hubble radius (i.e. satisfy the condition nη ≪ 1). (50), depending on their wavelength (i.e. on the range of the associated eigenvalues). On these wavelengths, nη ≪ 1 and η ≪ 1, when the FRW background is marginally open. Then, both (49) and (50) reduce to the power-law 10

Superhorizon-sized magnetic fields evolving on spatially open Friedmannian backgrounds obey solution (49) or
as long as Given that η ≪ 1 in marginally open FRW universes, there is an extensive range of wavelengths that satisfy both nη ≪ 1 and ( |n 2 − 2| )η ≪ 1 at the same time. These include all the modes with n 2 < 2 as well as many having n 2 > 2. For example, a magnetic mode with n 2 = 10 2 lies outside the Hubble radius (i.e. has nη ≪ 1) and also satisfies condition (54), as long as η ≪ 1/10. In that case, the associated B-field evolves according to solution (53). We have therefore arrived to an evolution law identical to that of the flat FRW case (compare Eq. (53) to solution (14) in § 4.2). Moreover, evaluating the integration constants of (53) and dropping the mode-index (n) for simplicity, we obtain which is identical to solution (16). Hence, by simply repeating the process of § 4.2, we find that B ∝ a −3/2 during the reheating and dust eras and B ∝ a −1 throughout the radiation epoch (see solutions (19) and (20) in § 4.3). More specifically, recalling that a ∝ η 2 and H = 2/aη during reheating and dust, we may recast (55) into Similarly, when radiation dominates we have a ∝ η and H = 1/aη, in which case solution (55) becomes These results confirm that magnetic fields in marginally open Friedmann universes can be superadiabatically amplified throughout their post-inflationary evolution. This happens as long as the B-fields remain outside the Hubble horizon and the initial conditions allow the second modes on the right-hand side of (56) and (57) to dictate the magnetic evolution (see § 4.4 for further discussion). It is then straightforward to show that the residual comoving strength of such a magnetic field, with current coherence scale around 10 Kpc, is close to 10 −33 G (see § 5.2 and Eq. (41) there). This value can increase to 10 −27 G by the time the galaxy is fully formed, while additional growth might possible as well (see § 6.1 earlier). Following a series of articles on the magnetic evolution in open FRW universes [18], is was recently claimed that superadiabatic amplification is impossible in marginally open Friedmann models [19]. The authors reached this conclusion after arriving at Eq. (53). Once there, however, they dropped the second mode from the right-hand side of that solution. The authors did so based on the fact that η ≪ 1, but without evaluating the integration constants first. As we have shown here, that was not the right decision. This oversight prevented the authors from noticing the importance of their "redundant" mode and then from realising that marginally open FRW universes can superadiabatically amplify magnetic fields during their post-inflationary evolution.
Before closing this section, we should briefly comment on the nature of the magnetic modes involved. As we have explained above, solution (53) applies to magnetic modes with n 2 < 2 and also to those having n 2 > 2, provided they simultaneously satisfy the conditions nη ≪ 1 and ( |n 2 − 2| )η ≪ 1. In quantum mechanical terms, modes with n 2 < 2 are termed supercurvature and they have been claimed to suffer from normalisation problems that make them physically ambiguous. Modes with n 2 > 2, on the other hand, are known as subcurvature and are physically unambiguous. The range of the subcurvature magnetic modes that experience superadiabatic amplification depends on the value of the conformal time, which in turn is decided by the "amount" of spatial curvature. For example, current observations indicate a nearly flat universe with |Ω K | 10 −3 today. Assuming negative curvature and setting Ω K ≃ 10 −4 at present, suggests that η ≃ 2 × 10 −2 today (see Eq. (51b) and recall that β = 1/2 for dust). Then, at least some of the (subcurvature) magnetic modes with 2 < n 2 50 2 are currently superadiabatically amplified (i.e. they satisfy both nη ≪ 1 and ( |n 2 − 2| )η ≪ 1 simultaneously). Clearly, as we go back in time the values of Ω K and η drop significantly, ensuring that many more subcurbature magnetic modes were superadiabatically amplified in the past.

Marginally closed FRW models
Let us now turn our attention to Friedmann models with positive spatial curvature. In terms of conformal time, the scale factor and the curvature contribution to the total energy density of a spatially closed FRW universe are given by a = a 0 sin(βη) sin(βη 0 ) 1/β and respectively. Note that β = (1 + 3w)/2 and w ≥ 0, as with the open models discussed in the previous section. Similarly, the relative size of a mode is decided by the ratio λ H /λ n = n/aH = n tan(βη). Here, however, the commoving eigenvalue is discrete with n 2 ≥ 3. On this background and in the absence of electric currents, magnetic fields obey the differential equation which accepts the oscillatory solution B (n) = C 1 cos n 2 + 2 η + C 2 sin n 2 + 2 η .
The above monitors the post-inflationary evolution of large-scale B-fields on FRW backgrounds with positive spatial curvature.
Let us now focus upon the marginally closed Friedmann universes. Following (58b), these models are characterised by very small values of the conformal time (i.e. |Ω K | ≪ 1 ⇔ η ≪ 1), in exact analogy with their marginally open counterparts. In this case we have a ∝ η 2 , Ω K ≃ −η 2 /4 and λ n /λ H ≃ nη/2 during the reheating and the dust eras (i.e. when β = 1/2). Throughout the radiation epoch, on the other hand, β = 1, a ∝ η, Ω K = −η 2 and λ n /λ H ≃ nη. Consequently, superhorizon-sized modes on marginally closed Friedmann backgrounds have nη ≪ 1 and η ≪ 1 simultaneously. Therefore, for magnetic modes that also satisfy the constraint ( √ n 2 + 2 )η ≪ 1, solution (60) reduces to the power-law 11 As with the marginally open Friedmann universes of the previous section, the above is essentially identical to solution (14) of the flat FRW models. In fact, after evaluating the integration constants of (61), one recovers solutions (16), (18) and all the evolution laws obtained in § 4.3 earlier. More specifically, we find that the dominant magnetic mode decays as B ∝ a −3/2 during both the reheating and the dust eras and like B ∝ a −1 when radiation dominates the energy density of the universe. Consequently, large-scale B-fields in marginally closed Friedmann universes can be superadiabatically amplified throughout their post-inflationary evolution, just like their counterparts in the spatially flat and the marginally open models.

Discussion
Finding an answer to the question of cosmic magnetism has proved a rather difficult theoretical task. Over the years, the efforts have mainly focused on slowing down the so-called adiabatic magnetic decay and thus increase the residual strength of the initial seed-field to astrophysically relevant values. So far, almost all of the proposed solutions work outside what we might call "standard physics". By breaking away from Maxwellian electromagnetism, for example, it is possible to achieve magnetic magnitudes much larger that the "conventional" final strength of ∼ 10 −53 G. The latter value, however, has been obtained after assuming that primordial magnetic fields decay adiabatically during the whole of their post-inflationary evolution on all scales. Here, we have taken another look into this assumption. The adiabatic magnetic decay after inflation has been attributed to the high electrical conductivity of the matter during most of reheating and throughout the subsequent eras of radiation and dust. This has been thought enough to guarantee that the magnetic flux remains conserved at all times and on all scales. However, magnetic flux-freezing cannot be achieved without the electric currents. It is the currents that eliminate the electric fields and freeze their magnetic counterparts into the highly conductive matter, which then leads to the adiabatic depletion of the B-fields. Therefore, applying the adiabatic decay-law on all scales implicitly assumes the presence of electric currents with superhorizon correlations. Recall that all the astrophysically relevant B-fields of inflationary origin remain outside the Hubble radius until late into the radiation era (or even later). The electric currents, however, are formed after inflation by local physical processes and their size can never exceed that of the Hubble horizon. The same is also true for the processes of electric-field elimination and magnetic-flux freezing. Both processes are 11 Given that η ≪ 1 always in marginally closed Friedmann models, it is straightforward to verify that at any given time there is a whole range of eigenvalues that satisfy both nη ≪ 1 and ( √ n 2 + 2 )η ≪ 1 at the same time.
causal and the domains they operate are therefore confined within the horizon at the time. All these mean that applying the ideal-MHD approximation on super-Hubble scales violates causality. After all, the ideal-MHD limit (and the inferred post-inflationary magnetic-flux freezing) is the result of causal physical processes, none of which operates on superhorizon scales. Motivated by the above, we have adopted a current-free treatment for the evolution of superhorizon-sized magnetic fields after inflation. The absence of electric currents on super-Hubble lengths means that large-scale B-fields obey the wave-like formula of their inflationary predecessors. Beyond the horizon, however, the period of the oscillation is longer than the lifetime of the universe. Consequently, as long as they stay outside the Hubble radius, the magnetic modes do not oscillate but obey a power-law solution. The latter contains two modes, the second of which decays at a pace slower than the adiabatic after inflation. Depending on the initial conditions, this slowly decaying mode can dominate and thus dictate the post-inflationary magnetic evolution. When this happens superhorizon-sized magnetic fields deplete as B ∝ a −3/2 throughout the reheating and the dust eras. During the intermediate radiation epoch, on the other hand, the decay rate slows down further to B ∝ a −1 . In general, the "stiffer" the equation of state of the matter, the slower the magnetic decay rate.
The initial conditions for the post-inflationary magnetic evolution are set by the field's evolution during the de Sitter phase and by the nature of the transition from inflation to reheating and later to the radiation and the dust eras. Here, following Israel's work on junction conditions, we have employed two qualitatively different but complementary initial-condition scenarios. Alternative approaches are also likely. Scenario A allows for an abrupt change in the background equation of state on the transition hypersurface and for the presence of a thin shell there. Within this scenario, primordial B-fields are superadiabatically amplified throughout their post-inflationary evolution as long as they remain outside the Hubble radius. Moreover, for all practical purposes, the amplification occurs irrespectively of the magnetic evolution during the de Sitter phase. This scenario can in principle produce astrophysically relevant B-fields, with residual strengths close (or even within) the typical galactic dynamo requirements, without abandoning neither classical electromagnetism nor standard cosmology. Scenario B also allows for a sudden change in the cosmic equation of state, but assumes that there is no thin shell on the matching hypersurface. Here, we found that primordial B-fields that decayed adiabatically during the de Sitter regime will continue to do so for the rest of their lifetime. This essentially reproduces the typical scenario of conventional magnetogenesis that leads to astrophysically irrelevant B-fields at present. However, scenario B also allows for the superadiabatic amplification after inflation of primordial magnetic fields that did not decay adiabatically in the de Sitter phase. This can have serious implications for the mechanisms of cosmic magnetogenesis that amplify their B-fields during inflation. More specifically, in connection with the CMB limits on the anisotropy of the universe, scenario B severely constrains models that achieve relatively strong inflationary amplification for their magnetic fields. On the other hand, when the de Sitter enhancement is mild, scenario B can help to produce astrophysically promising B-fields.
To summarise, causality ensures that there is no a priori flux-freezing on super-Hubble scales, even after inflation. On these wavelengths, primordial B-fields can be superadiabatically amplified throughout their post-inflationary evolution depending on the initial conditions. Here, we have discussed two simple but complementary initial-condition scenarios. In general, the superadiabatic amplification will (sooner or later) occur, as long as the initial conditions at the start of reheating allow the slowly decaying (i.e. the second) modes in solutions (16) and (18) to survive. Naturally, once back inside the horizon, the electric currents take over, eliminate the electric fields and freeze their magnetic counterparts into the highly conductive cosmic medium. Then onwards, the magnetic flux remains conserved and the adiabatic decay-law is restored.
The aforementioned phase of superadiabatic amplification can increase the residual strength of conventional inflationary produced B-fields by many orders of magnitude. For example, a magnetic mode with current comoving (pre-collapse) scale close to 10 Kpc, which is the minimum required by the galactic dynamo, re-enters the horizon a little before equipartition. This mode has been superadiabatically amplified during reheating and most of the radiation era. As a result, the residual magnetic strength is not the "standard" ∼ 10 −53 G but the much larger ∼ 10 −33 G. The latter increases further during the protogalactic collapse and can reach strengths within the general dynamo requirements. Additional amplification may also occur during the earlier or the later stages of the field's evolution. Consequently, conventional electromagnetism and standard cosmology can produce magnetic fields of astrophysically relevant magnitudes. Non-conventional scenarios of cosmic magnetogenesis, on the other hand, which amplify the Bfields during inflation and adopt the adiabatic decay-law for the whole of their post-inflationary evolution, may require revision to avoid potential conflict with the CMB constrains.
It is also worth pointing out that our analysis and our results are not confined to the spatially flat Friedmann models, but extend naturally to their marginally open and marginally closed counterparts. Thus, superhorizon-sized magnetic fields of cosmological origin can be superadiabatically amplified in FRW universes with mildly curved spatial sections as well. To a large extent, this is intuitively plausible. What is less straightforward and more interesting is that the marginally open models can superadiabatically amplify both supercurvature and subcurvature magnetic modes. The former have been claimed to suffer from normalisation problems that make them physically ambiguous, although the whole issue may merely reflect the absence as yet of a quantum theory of gravity. Nevertheless, the fact that a wide range of the physically unambiguous subcurvature magnetic modes are also superadiabatically amplified, means that the mechanism discussed here works in all the cosmologically relevant Friedmann models We would like to close with some thoughts on the question of cosmic magnetism and the efforts made so far to resolve it. As we have already pointed out, the overwhelming majority of the proposed solutions operate outside classical Maxwellian theory, or conventional cosmology (or both). Over the years, this has established in the community the belief that it is not possible to produce cosmological magnetic fields of astrophysical relevance within what we call standard physics. This is to be expected, since long standing beliefs develop their own inertia as time goes by. Nevertheless, although still relatively sparse and incomplete, there has been work suggesting that the issue of classical electromagnetism on conventional FRW backgrounds has not been exhausted yet and the present paper takes another step in this direction. The underlying point is that through a mere appeal to causality the magnetic decay-rate after inflation is not a priori adiabatic, but becomes a matter of the initial conditions. If these can increase the final strength of inflationary B-fields by twenty or so orders of magnitude, then it might be worth reconsidering the necessity of introducing new physics to address the question of cosmic magnetism.