Inflationary magnetogenesis in the perturbative regime
IInflationary magnetogenesis in the perturbative regime
Massimo Giovannini Department of Physics, CERN, 1211 Geneva 23, SwitzerlandINFN, Section of Milan-Bicocca, 20126 Milan, Italy
Abstract
While during inflation a phase of increasing gauge coupling allows for a scale-invariant hyperelectric spec-trum, when the coupling decreases a flat hypermagnetic spectrum can be generated for typical wavelengthslarger than the effective horizon. After the gauge coupling flattens out the late-time hypermagnetic powerspectra outside the horizon in the radiation epoch are determined by the hyperelectric fields at the end ofinflation whereas the opposite is true in the case of decreasing coupling. Instead of imposing an abruptfreeze after inflation, we consider a smooth evolution of the mode functions by positing that the gauge cou-plings and their conformal time derivatives are always continuous together with the background extrinsiccurvature. The amplified gauge power spectra are classified according to their transformation propertiesunder the duality symmetry. After clarifying the role of the comoving and of the physical spectra in theformulation of the relevant magnetogenesis constraints, the parameter space of the scenario is scrutinized.It turns out that a slightly blue hyperelectric spectrum during inflation may lead to a quasi-flat hypermag-netic spectrum prior to matter radiation equality and before the relevant wavelengths reenter the effectivehorizon. In this framework the gauge coupling is always perturbative but the induced large-scale magneticfields can be of the order of a few hundredths of a nG and over typical length scales between a fraction ofthe Mpc and 100 Mpc prior to the gravitational collapse of the protogalaxy. e-mail address: [email protected] a r X i v : . [ g r- q c ] D ec Introduction
Besides the invariance under local gauge transformations, the Weyl [1] and the duality [2, 3] symmetries areparticularly relevant for the dynamics of the gauge fields in general relativity and in scalar-tensor theoriesof gravity. If the governing equations of a given field are invariant under Weyl rescaling, the correspondingquantum fluctuations are not parametrically amplified by the evolution of the geometry [4, 5, 6]. Inthe absence of sources duality rotates field strengths into their duals (i.e. tensors into pseudotensors) andconstrains the gauge power spectra potentially amplified during conventional or unconventional inflationaryphases.The symmetries of the evolution equations of the gauge fields in curved space-times are one of themain handles on the origin of the large-scale magnetism, a perplexing problem originally posed by Fermi[7] in connection with the propagation of cosmic rays within the galaxy (see also [8]) and subsequentlyscrutinized by Hoyle [9] in a cosmological context in view of the comparatively large correlation scales ofthe fields. Indeed, while the typical diffusion scale in the interstellar medium is of the order of the AU(1 AU = 1 . × cm) magnetic fields are observed over larger scales ranging between the 30 kpc andfew Mpc (1 pc = 3 . × cm).Three general classes of suggestions have been proposed through the years. S oon after Hoyle’s obser-vations [9] Zeldovich [10] and of Thorne [11] suggested, in the context of anisotropic (but homogeneous)Bianchi-type models [12], that the large-scale magnetic fields could be a fossil remnant of a primordial fieldthat originated with the Universe. In a complementary perspective we could imagine that the large-scalemagnetic fields have been produced at some point during the radiation epoch and inside the Hubble radiusby the vorticity associated with the turbulent dynamics; the first one suggesting this possibility was prob-ably Harrison [13, 14] whose idea found direct applications in the context of phase transitions. Finally thethird class of scenarios implies that a spectrum of gauge fields is produced because of the breaking of Weylinvariance during a standard stage of inflationary expansion; this is the perspective discussed in the presentpaper. The collection of themes related to the origin and to the early time effects of large-scale magnetismhas been dubbed some time ago magnetogenesis [15]. While this terminology was so far quite successful,the problem itself has a long history as a number of inspiring monographs demonstrates [16, 17, 18]; seealso Refs. [19, 20, 21] for some more recent reviews.It has been repeatedly argued through the years that magnetic fields with a sufficiently large correlationscale could be generated during a phase of accelerated expansion in full analogy with what happens forthe scalar and tensor modes of the geometry (see e.g. [22]). The gauge fields parametrically amplified inthe early Universe are in fact divergenceless vector random fields that do not break the spatial isotropy(as it happens instead in the case of the fossil remnants discussed in Refs. [10, 11]). To avoid theconstraints imposed by Weyl invariance the gauge fields might directly couple to one or more scalar fields(see [23, 24, 25, 26] for an incomplete list of references). The scalar fields may coincide with one or (moreinflatons) or even with multiple spectator fields. This class of models is based on the following action: S gauge = − π (cid:90) d x √− G (cid:20) λ ( ϕ, ψ ) Y αβ Y αβ + λ pseudo ( ϕ, ψ ) Y αβ (cid:102) Y αβ (cid:21) . (1.1)Within the present notations Y µν and (cid:102) Y µν are, respectively, the gauge field strength and its dual; G =det G µν is the determinant of the four-dimensional metric with signature mostly minus . In Eq. (1.1) ϕ denotes the inflaton field while ψ represents a generic spectator field. During a stage of conventional(slow-roll) inflation the variation of λ is associated with the variation of the gauge coupling whose evolutionmay reach into the strong coupling regime (see the third paper of Ref. [23]). The presence of λ ( ϕ, ψ ) inEq. (1.1) is more relevant than the pseudoscalar (axion-like [27, 28]) coupling which will be ignored even The Greek indices run over the four space-time dimensions while the Latin (lowercase) indices run over the three spatialdimensions. The signature of the four-dimensional metric will be mostly minus i.e. (+ , , − , − , − ). Finally the relation betweenthe Riemann and the Ricci tensors will be chosen to be R µν = R α µαν .
2f it has been studied by many authors [29, 30, 31] in the context of the magnetic field generation. Forthe amplification of the magnetic field itself the pseudoscalar vertex is not essential but it may lead tohypermagnetic flux lines are linked or twisted as originally discussed in [32]. The produced Chern-Simonscondensates leads to a viable (but unconventional) mechanism for baryogenesis via hypermagnetic knots[32, 33] which are characterized by an average magnetic gyrotropy [34]. These gyrotropic and helical fieldsplay also a role in anomalous magnetohydrodynamics where the evolution of the magnetic fields at finiteconductivity is analyzed in the presence of anomalous charges [35]. In the collisions of heavy ions thisphenomenon is often dubbed chiral magnetic effect [36].In this paper duality will be explicitly used to deduce the gauge power spectra not only during inflationbut also during the subsequent decelerated expansion. We will show, in particular, that after the gaugecoupling flattens out the late-time hypermagnetic power spectra outside the horizon in the radiation epochare determined by the hyperelectric fields at the end of inflation whereas the opposite is true in the caseof decreasing coupling. The obtained results suggest that a slightly blue hyperelectric spectrum duringinflation may lead to a quasi-flat hypermagnetic spectrum prior to matter radiation equality and beforethe relevant wavelengths reenter the effective horizon. From the technical viewpoint these results will arisefrom the discussion of an appropriate transition matrix whose elements have well defined transformationproperties under the duality symmetry and control the form of the late-time spectra. Using these resultswe shall investigate the magnetogenesis requirements as well as all other pertinent constraints; we shallconclude that large-scale magnetic fields can be generated during a quasi-de Sitter stage of expansion whilethe gauge coupling remains perturbative throughout all the stages of the dynamical evolution.The layout of this paper is in short the following. In section 2 after introducing the necessary generalities,we shall discuss the duality symmetry both for the field equations and for the power spectra. At the end ofthe section we shall discuss two complementary (and dual) profiles for the evolution of the gauge couplings.In section 3 the gauge power spectra during inflation will be computed for typical wavelengths largerthan the effective horizon and related via duality transformations. Using the overall continuity of thewhole description the electric and the magnetic mode functions at the end of inflation will be explicitlyrelated, in section 4, to the gauge spectra in the radiation epoch via a transition matrix whose elementstransform in a well defined way under duality. The gauge spectra will then be computed in severalapproximations schemes with particular attention to the relevant phenomenological regimes. In section 5the magnetogenesis requirements will be examined in conjunction with the physical constraints. Section 6contains our concluding remarks. To avoid lengthy digressions various technical results have been relegatedto the appendices A, B and C.
With the purpose of making the whole discussion self-contained, the essential notations will be introducedin the first part of this section while in the second part we shall examine the gauge power spectra in thelight of the duality symmetry. In the last part of the section the physical aspects of the evolution of thegauge couplings will be specifically addressed.
The field content of the model may be different but it generally follows from a total action of the type: S tot = S grav + S scalar + S gauge , (2.1)where S gauge has been already introduced in Eq. (1.1) and, as already mentioned, Y µν and (cid:102) Y µν denote,respectively, the gauge field strength and its dual: (cid:102) Y µν = 12 E µνρσ Y ρσ , E µνρσ = (cid:15) µνρσ √− G . (2.2)3n Eq. (2.2) (cid:15) µνρσ denotes the totally antisymmetric symbol of Levi-Civita in four dimensions. In moreexplicit terms the gravitational and the scalar actions of Eq. (2.1) can be expressed as: S grav = − (cid:96) P (cid:90) d x √− G R,S scalar = (cid:90) d x √− G (cid:20) G αβ ∂ α ϕ∂ β ϕ + 12 G αβ ∂ α ψ∂ β ψ − W ( ϕ, ψ ) (cid:21) , (2.3)where (cid:96) P = 1 /M P = 8 π/M P and M P = 1 . × GeV is the Planck mass. The notations of Eq. (2.3) arepurposely schematic and ϕ denotes the inflaton while ψ is a spectator field. Various magnetogenesis modelshave been discussed in various contexts where W ( ϕ, ψ ) has a well defined expression. For instance in themodels of Ref. [23] W is only function of ϕ while an explicit magnetogenesis model based on spectatorfields can be found in the last paper of Ref. [24]. With these specifications, the general equations derivedfrom the actions (2.1)–(2.3) are: R νµ = (cid:96) P (cid:20) ∂ µ ϕ∂ ν ϕ − W δ νµ + T νµ (cid:21) , (2.4) G αβ ∇ α ∇ β ϕ + ∂W∂ϕ + 116 π ∂λ∂ϕ Y αβ Y αβ = 0 , (2.5) ∇ α (cid:18) λY αβ (cid:19) = 0 , ∇ α (cid:101) Y αβ = 0 . (2.6)In Eq. (2.4) T νµ denotes the (traceless) energy-momentum tensor of the gauge fields whose explicit form isgiven by: T νµ = λ π (cid:20) − Y µα Y νβ + 14 δ νµ Y αβ Y αβ (cid:21) . (2.7)We shall be mostly concerned with conformally flat background geometries whose associated line elementis: ds = G αβ dx α dx α = a ( τ )[ dτ − d (cid:126)x ] , (2.8)where a ( τ ) denotes the scale factor. In the geometry (2.8) the various components of T νµ defined in Eq.(2.7) are: T = ρ B + ρ E , T i = 14 πa ( (cid:126)E × (cid:126)B ) i , T ji = − ( p E + p B ) δ ji + Π jE i + Π jB i . (2.9)In Eq. (2.9) we introduced the energy density, the pressure and the anisotropic stresses of the hypermagneticand hyperelectric fields: ρ B = B πa , ρ E = E πa , p B = ρ B , p E = ρ E , (2.10)Π jE i = 14 πa (cid:18) E i E j − E δ ji (cid:19) , Π jB i = 14 πa (cid:18) B i B j − B δ ji (cid:19) , (2.11)where E = (cid:126)E · (cid:126)E and B = (cid:126)B · (cid:126)B . Equations (2.9), (2.10) and (2.11) are expressed in terms of (cid:126)E and (cid:126)B, i.e. the comoving hyperelectric and hypermagnetic fields. These rescaled quantities are actually thenormal modes of the system and are related to the physical fields as: (cid:126)E = a √ λ (cid:126)E ( phys ) , (cid:126)B = a √ λ (cid:126)B ( phys ) . (2.12)4he components of the field strengths can be directly expressed in terms of the components of the physicalfields; so for instance, in terms of (cid:15) i j k (i.e. the Levi-Civita symbol in three dimensions) we have Y i = − a E ( phys ) i , Y i j = − (cid:15) i j k B ( phys ) k /a , (2.13)and similarly for the dual strength. The physical fields are essential for the discussion of the actualmagnetogenesis constraints but the evolution equations of (2.6) are simpler in terms of the comoving fields: (cid:126) ∇ · ( √ λ (cid:126)E ) = 0 , (cid:126) ∇ · (cid:18) (cid:126)B √ λ (cid:19) = 0 , (2.14) ∂ τ (cid:18) √ λ (cid:126)E (cid:19) = (cid:126) ∇ × (cid:18) √ λ (cid:126)B (cid:19) , ∂ τ (cid:18) (cid:126)B √ λ (cid:19) = − (cid:126) ∇ × (cid:18) (cid:126)E √ λ (cid:19) . (2.15)Equations (2.14)–(2.15) have been written in the general case where λ can be inhomogeneous even if, as weshall see in the last part of this section, the gauge coupling will always be considered to be time-dependentbut homogeneous. Furthermore Eqs. (2.14)–(2.15) under the following duality tranformation: √ λ → √ λ , (cid:126)B → (cid:126)E , (cid:126)E → − (cid:126)B . (2.16)All the mode functions and power spectra obtained by the simultaneous evolution of the geometry andof the gauge coupling must be consistent with Eq. (2.16) both during the inflationary phase and in thesubsequent decelerated stages of expansion before the given scale reenters the effective horizon. The gaugecoupling e ( λ ) is related to the inverse of √ λ : S gauge = − (cid:90) d x √− Ge Y µν Y µν , e = 4 πλ . (2.17)Thus the gauge coupling increases when λ decreases and vice versa. Since we are going to discuss the parametric amplification of the quantum fluctuations of the gauge fieldsthe classical evolution summarized by Eqs. (2.14)–(2.15) must be complemented by the correspondingquantum treatment. From the semi-classical viewpoint this process can be viewed as the conversion oftraveling waves into standing waves; the same phenomenon occurs for the scalar and tensor modes of thegeometry [22] and it leads to the so-called Sakharov oscillations [37] (see also [38, 39, 40]). In the presentcontext the duality symmetry explicitly relates the Sakharov oscillations of the hypermagnetic [i.e. f k, α ( τ )]and hyperelectric mode functions [i.e. g k, α ( τ )] entering the expressions of the corresponding field operators:ˆ B i ( τ, (cid:126)x ) = − i (cid:15) mni (2 π ) / (cid:88) α = ⊕ , ⊗ (cid:90) d k k m e ( α ) n (ˆ k ) (cid:20) f k, α ( τ ) ˆ a (cid:126)k,α e − i(cid:126)k · (cid:126)x − f ∗ k, α ( τ )ˆ a † (cid:126)k,α e i(cid:126)k · (cid:126)x (cid:21) , (2.18)ˆ E i ( τ, (cid:126)x ) = − π ) / (cid:88) α = ⊕ , ⊗ (cid:90) d k e ( α ) i (ˆ k ) (cid:20) g k α ( τ )ˆ a (cid:126)k,α e − i(cid:126)k · (cid:126)x + g ∗ k, α ( τ )ˆ a † (cid:126)k,α e i(cid:126)k · (cid:126)x (cid:21) . (2.19)In Eqs. (2.18)–(2.19) the two vector polarizations are directed along the orthogonal unit vectors ˆ e ⊕ and ˆ e ⊗ that are also orthogonal to ˆ k (i.e. ˆ k · ˆ e α = 0). In Eq. (2.19) ˆ a (cid:126)k,α and ˆ a † (cid:126)k,α are the creation and annihilationoperators obeying, within the present notations, [ˆ a (cid:126)q,α , ˆ a † (cid:126)p,β ] = δ (3) ( (cid:126)q − (cid:126)p ) δ αβ . In Eqs. (2.18) and (2.19)the sum is performed over the physical polarizations e ( α ) i (ˆ k ) while the mode functions f k,α and g k α obey,in the absence of conductivity, the following pair of equations: f (cid:48) k, α = g k, α + F f k, α , g (cid:48) k, α = − k f k, α − F g k, α , F = √ λ (cid:48) √ λ , (2.20)5here the prime denotes a derivation with respect to the conformal time coordinate τ . The mode functionsin Eq. (2.20) must also be correctly normalized so that the Wronskian of any solution must satisfy for eachpolarization: f k, α ( τ ) g ∗ k,α ( τ ) − f ∗ k, α ( τ ) g k,α ( τ ) = i. (2.21)The field operators of Eqs. (2.18)–(2.19) can be represented in Fourier space as ˆ B j ( (cid:126)p, τ ) = − i p m (cid:15) mnj (cid:88) β (cid:20) e βn (ˆ p ) ˆ a (cid:126)p, β f q,β ( τ ) + e αn ( − ˆ p ) ˆ a †− (cid:126)p, β f ∗ p,β ( τ ) (cid:21) , ˆ E i ( (cid:126)q, τ ) = (cid:88) α (cid:20) e αi (ˆ q ) ˆ a (cid:126)q, α g q,α ( τ ) + e αi ( − ˆ q ) ˆ a †− (cid:126)q, α g ∗ q,α ( τ ) (cid:21) . (2.22)Therefore the corresponding two-point functions in Fourier space become: (cid:104) ˆ B i ( (cid:126)k, τ ) ˆ B j ( (cid:126)p, τ ) (cid:105) = 2 π k P B ( k, τ ) p ij (ˆ k ) δ (3) ( (cid:126)k + (cid:126)p ) , (2.23) (cid:104) ˆ E i ( (cid:126)k, τ ) ˆ E j ( (cid:126)p, τ ) (cid:105) = 2 π k P E ( k, τ ) p ij (ˆ k ) δ (3) ( (cid:126)k + (cid:126)p ) , (2.24)where p (ˆ k ) = ( δ ij − ˆ k i ˆ k j ); P B ( k, τ ) and P E ( k, τ ) are the comoving magnetic and electric power spectra ,respectively : P B ( k, τ ) = k π (cid:88) α = ⊕ , ⊗ | f k, α ( τ ) | ≡ k π | f k ( τ ) | , (2.25) P E ( k, τ ) = k π (cid:88) α = ⊕ , ⊗ | g k, α ( τ ) | ≡ k π | g k ( τ ) | . (2.26)For the present purposes it will be important to distinguish the comoving from the physical power spectra:while the comoving power spectra are obtained from the comoving field operators ˆ E i and ˆ B i , the physicalpower spectra follow from the the corresponding physical fields ˆ E ( phys ) i and ˆ B ( phys ) i , defined in Eq. (2.12);the relation between the physical and the comoving power spectra is therefore given by: P ( phys ) B ( k, τ ) = P B ( k, τ ) λ ( τ ) a ( τ ) , P ( phys ) E ( k, τ ) = P E ( k, τ ) λ ( τ ) a ( τ ) . (2.27)The phenomenological requirements (e.g. the magnetogenesis constraints to be discussed in section 5) mustbe typically expressed in terms of the physical spectra. Occasionally this distinction has not been clearlyspelled out and the consequences have been confusing, as we shall remark later on. Let us conclude thisdiscussion by recalling that the two coupled first-order equations given in Eq. (2.20) can be transformedin two (decoupled) second-order differential equations: f (cid:48)(cid:48) k, α + (cid:20) k − √ λ (cid:48)(cid:48) √ λ (cid:21) f k, α = 0 , g (cid:48)(cid:48) k, α + (cid:20) k − √ λ (cid:18) √ λ (cid:19) (cid:48)(cid:48) (cid:21) g k, α = 0 , (2.28)which have the same content of Eq. (2.20) provided the initial conditions are correctly imposed by takinginto account that g k, α is ultimately determined from f k,α and its derivative as g k,α = f (cid:48) k α −F f k α . Equations(2.20)–(2.28) are invariant under the following duality transformations : √ λ → / √ λ, f k, α → g k, α /k, g k, α → − kf k, α . (2.29) Equation (2.22) follows from Eqs. (2.18) and (2.19) by recalling that ˆ B j ( (cid:126)p, τ ) = (cid:82) d x ˆ B j ( (cid:126)x, τ ) e i(cid:126)p · (cid:126)x / (2 π ) / and thatˆ E i ( (cid:126)q, τ ) = (cid:82) d x ˆ E i ( (cid:126)x, τ ) e i(cid:126)q · (cid:126)x / (2 π ) / . If the mode functions for the two polarizations coincide the sums appearing in Eqs. (2.25)–(2.26) can be performed triviallysince f k ⊕ = f k ⊗ = f k and similarly for the hyperelectric mode function. In connection with Eqs. (2.20) and (2.29) we recall that F = √ λ (cid:48) / √ λ ; this means that for √ λ → / √ λ , F → −F . .3 Spectral energy density ad backreaction constraint Since the amplification of the gauge fields takes place in a homogeneous and isotropic background geometry(see Eq. (2.8)) the corresponding energy density must not exceed the critical energy density ρ crit = 3 M P H where H is the Hubble rate. The averaged energy density of the gauge fields follows from Eqs. (2.23) and(2.24) and from the definitions of ρ B and ρ E appearing in Eqs. (2.9), (2.10) and (2.11). Thanks to Eqs.(2.25)–(2.26) the final result is: (cid:104) ˆ ρ Y (cid:105) = (cid:104) ˆ ρ B (cid:105) + (cid:104) ˆ ρ E (cid:105) = 14 πa ( τ ) (cid:90) dkk (cid:20) P E ( k, τ ) + P B ( k, τ ) (cid:21) . (2.30)In terms of the physical power spectra of Eq. (2.27) the result (2.30) becomes: (cid:104) ˆ ρ Y (cid:105) = λ ( τ )4 π (cid:90) dkk (cid:20) P ( phys ) E ( k, τ ) + P ( phys ) B ( k, τ ) (cid:21) . (2.31)To compare energy density of the parametrically amplified gauge fields with the energy density of thebackground geometry we introduce the spectral energy density in critical units:Ω Y = 1 ρ crit d (cid:104) ˆ ρ (cid:105) d ln k = 23 H M P a (cid:20) P E ( k, τ ) + P B ( k, τ ) (cid:21) . (2.32)To guarantee the absence of dangerous backreaction effects Ω Y ( k, τ ) must always be much smaller 1throughout all the stages of the evolution and for all relevant scales; this requirement must be separatelyverified both during and after inflation. In Fig. 1 the profile describing the evolution of the gauge coupling is illustrated together with the mainnotations employed throughout the discussion. During the inflationary phase (i.e. for τ ≤ − τ in Fig. 1) γ (- τ / τ ) - γ [ q ( δ , γ ) ( τ / τ + ) + ] δ e << e < O ( ) - - - - - - - - τ / τ l og e / ( π ) Increasing gauge coupling e ( τ ) = π / λ ( τ ) Figure 1: The logic and the main notations employed for the dynamical description of an increasing gaugecoupling. The two different curves for τ ≥ − τ correspond to different values of δ (cid:28)
1. Note that q ( δ, γ ) = δ/γ [see also Eq. (A.3)]. 7escribes the rate of increase of the gauge coupling in the conformal time parametrization while δ controlsthe evolution during the post-inflationary stage of expansion when the gauge coupling flattens out: √ λ = (cid:112) λ (cid:18) − ττ (cid:19) γ , τ ≤ − τ , (2.33) √ λ = (cid:112) λ (cid:20) γδ (cid:18) ττ + 1 (cid:19) + 1 (cid:21) − δ , τ ≥ − τ . (2.34)The explicit form of Eqs. (2.33) and (2.34) is dictated by the continuity of √ λ and of √ λ (cid:48) ; furthermorethe physical range of γ and δ is given by: γ > , and 0 ≤ δ (cid:28) γ. (2.35)For short the limit δ → ≤ δ (cid:28) γ . As we shall see it will always be possible to derivethe results of the sudden approximation by taking the limit δ → √ λ and its derivative. Ifwe would simply assume that √ λ is constant the conformal time derivative will have a jump discontinuityand Eq. (2.28) will develop a singularity either in √ λ (cid:48)(cid:48) / √ λ or in (1 / √ λ ) (cid:48)(cid:48) √ λ . Since the present approachguarantees the continuity of gauge mode functions the related power spectra will also be continuous. Notethat the numerical value of λ may well coincide with 1 but it could also be slightly larger than 1 sincethis range is compatible with a gauge coupling that is perturbative for τ = − τ , as illustrated in Fig. 1.In the geometry of Eq. (2.8) the background equations derived from Eqs. (2.4)–(2.5) are:3 M P H = 12 ϕ (cid:48) + a W ( ϕ ) , M P ( H − H (cid:48) ) = ϕ (cid:48) , ϕ (cid:48)(cid:48) + 2 H ϕ (cid:48) + ∂W∂ϕ a = 0 , (2.36)where, as already mentioned, the prime denotes a derivation with respect to the conformal time coordinate τ ; furthermore H = (ln a ) (cid:48) = aH where H = ˙ a/a is the conventional Hubble rate and the overdot denotes aderivation with respect to the cosmic time coordinate t . The slow roll approximation specifies the evolutionduring the inflationary phase where the parameters (cid:15) , η and η are all much smaller than 1 and eventuallyget to 1 when inflation ends. The definitions of the slow roll parameters within the notations of this paperare as follows: (cid:15) = − ˙ HH = M P (cid:18) W , ϕ W (cid:19) , η = ¨ ϕH ˙ ϕ , η = M P (cid:18) W , ϕϕ W (cid:19) , (2.37)note that W , ϕ and W , ϕϕ are shorthand notations for the first and second derivatives of the potential W ( ϕ )with respect to ϕ . The slow roll parameters η , η and (cid:15) are not independent and their mutual relation, i.e. η = (cid:15) − η , follows from the slow roll version of Eqs. (2.36) written in the cosmic time coordinate t :3 H ˙ ϕ + ∂W∂ϕ = 0 , M P H = W, M P ˙ H = − ˙ ϕ . (2.38)We shall be assuming that the inflationary stage of expansion takes place for τ ≤ − τ : H = aH = − − (cid:15) ) τ , (cid:15) = − ˙ HH (cid:28) , (2.39)whereas after inflation (i.e. for τ ≥ − τ ) the background will be decelerated and dominated by radiation.The continuity of the scale factor demands: a inf ( τ ) = (cid:18) − ττ (cid:19) − β , τ ≤ − τ ,a rad ( τ ) = τ + ( β + 1) τ τ , τ ≥ − τ . (2.40)8ith β (cid:39) / (1 − (cid:15) ). Both Eqs. (2.39) and (2.40) assume that (cid:15) changes very slowly during inflation. Thecontinuity of the scale factor and of its conformal time derivative also imply the continuity of the extrinsiccurvature whose background value is given by K i j = − a H δ i j . For the purposes of the present analysis it is useful to complement the timeline of Fig. 1 with Fig. 2describing the dual situation where the gauge coupling is initially much larger than 1. In the situation (- τ / τ ) γ ˜ q δ ˜ , γ ˜ ( τ / τ + ) + - δ ˜ e ∼ O ( ) e >> - - - τ / τ l og e / ( π ) Decreasing gauge coupling e ( τ ) = π / λ ( τ ) Figure 2: We illustrate the main notations employed when discussing the case of the decreasing gaugecoupling. As in the case of Fig. 1 the two different curves for τ ≥ − τ correspond to different values of (cid:101) δ (cid:28)
1. Note that q ( (cid:101) δ , (cid:101) γ ) = (cid:101) δ / (cid:101) γ [see also Eq. (B.4)].illustrated by Fig. 2 the continuity of √ λ and √ λ (cid:48) imply the following parametrization √ λ = (cid:112) λ (cid:18) − ττ (cid:19) − (cid:101) γ , τ < − τ , (2.41) √ λ = (cid:112) λ (cid:20) (cid:101) γ (cid:101) δ (cid:18) ττ + 1 (cid:19) + 1 (cid:21)(cid:101) δ , τ ≥ − τ , (2.42)The physical region of the parameters is given by: (cid:101) γ > , (cid:101) δ ≥ , and 0 < (cid:101) δ (cid:28) (cid:101) γ . (2.43)The limit (cid:101) δ → (cid:101) δ (cid:28) (cid:101) δ (cid:28) (cid:101) γ since we shall assume throughout that (cid:101) γ is of order 1. As in the case ofincreasing gauge coupling (see Eqs. (2.33)–(2.34)), Eqs. (2.41)–(2.42) will be complemented by the smoothevolution of the geometry illustrated in Eq. (2.40). The dual profiles of Figs. 1 and 2 are not physicallyequivalent. If we consider a certain reference time τ = − τ i close to the onset of the inflationary phase, wewill have, according to Eq. (2.41) that (cid:112) λ i = (cid:112) λ (cid:18) a i a f (cid:19) (cid:101) γ (cid:28) ⇒ e ( τ i ) = √ π √ λ i (cid:29) , (2.44)where, by definition, λ i = λ ( − τ i ); in Eq. (2.44) we traded the conformal time for the scale factors by usingEq. (2.40) in the limit (cid:15) (cid:28)
1. Note in fact that ( a i /a f ) (cid:101) γ = e − N (cid:101) γ (cid:28) N is the total number of9nflationary e -folds. Since N = O (60) (or larger) we have that √ λ i will be O (10 − ) (or smaller). Equation(2.44) implies that the evolution of the gauge coupling starts from a non-perturbative regime unless √ λ isextremely large: only in this way we would have √ λ i = O (1). Whenever √ λ (cid:29) e = O (10 − ) but not much smaller. Furthermore, as we shallsee in section 5, the physical power spectra are suppressed as λ − and this will make their contributionmarginal for the phenomenological implications. A possibility suggested in Ref. [44] has been that the √ λ increases during inflation, decreases sharply during reheating, and then flattens out again. This suggestionis often assumed by various authors but rarely justified. The qualitative description of large-scale cosmological perturbations suggests that a given wavelengthexits the Hubble radius at some typical conformal time during an inflationary stage of expansion andapproximately reenters at τ k ∼ /k , when the Universe still expands but in a decelerated manner. Bya mode being beyond the horizon we only mean that the physical wavenumber is much less than theexpansion rate: this does not necessarily have anything to do with causality [22, 41]. Similarly the physicalwavenumbers of the hyperelectric and hypermagnetic fields can be much smaller than the rate of variationof the gauge coupling which now plays the role of the effective horizon. During inflation the relevant regimewill be the one where the wavelengths of the gauge fluctuations are inside the effective horizon which meansthat k/ F (cid:39) kτ (cid:28)
1. As long as kτ < kτ k ∼ τ = − τ ) but throughout the whole radiation phase. Thus the gauge spectra forwavelengths larger than the effective horizon during inflation do not necessarily coincide with the gaugespectra when the given mode reenters (see, in this respect, section 4).The logic of this section will be to compute the gauge spectra during inflation for the profiles schemat-ically introduced in Figs. 1 and 2. We shall then demonstrate that the gauge power spectra in the twocases are explicitly related by duality. This will mean, for instance, that during inflation the generation ofpotentially scale-invariant hyperelectric spectrum is only compatible with a phase where the gauge couplingdecreases while a flat hypermagnetic spectrum may only arise when the gauge coupling decreases. When √ λ is given by (2.33) the solution of Eq. (2.20) compatible with the Wronskian normalizationdictated by Eq. (2.21) is given by : f k ( τ ) = N µ √ k √− kτ H (1) µ ( − kτ ) , µ = (cid:12)(cid:12)(cid:12)(cid:12) γ − (cid:12)(cid:12)(cid:12)(cid:12) , (3.1) g k ( τ ) = N µ (cid:115) k √− kτ H (1) µ +1 ( − kτ ) , γ > , (3.2) g k ( τ ) = − N µ (cid:115) k √− kτ H (1) µ − ( − kτ ) , < γ < . (3.3) For γ → / µ → H (1) − ( z ) = e i π H (1)1 ( z ). See also Eq.(3.5).
10n general H (1) ν ( z ) will denote the Hankel functions first kind [42, 43] with argument z and index ν ; N ν isa complex number whose phase is required for a correct asymptotic normalization of the mode functions: H (1) ν ( z ) = J ν ( z ) + iY ν ( z ) , N ν = (cid:114) π e iπ (2 ν +1) / . (3.4)Throughout the whole discussion we shall assume, as in the standard theory of Hankel functions [42, 43]that ν is real and positive semi-definite:Re ν ≥ , Im ν = 0 , H (1) − ν ( z ) = e i π ν H (1) ν ( z ) , H (2) − ν ( z ) = e − i π ν H (2) ν ( z ) . (3.5)Inserting the correctly normalized mode functions of Eqs. (3.1), (3.2) and (3.3) into Eqs. (2.25)–(2.26) thegauge spectra for τ ≤ − τ turn out to be : P B ( k, τ ) = a H π ( − kτ ) | H (1) µ ( − kτ ) | → a H D ( µ ) | kτ | − µ (3.6) P E ( k, τ ) = a H π ( − kτ ) | H (1) µ +1 ( − kτ ) | → a H D ( µ + 1) | kτ | − µ +1) , γ > / , (3.7) P E ( k, τ ) = a H π ( − kτ ) | H (1) µ − ( − kτ ) | → a H D ( | µ − | ) | kτ | − | µ − | , < γ < / , (3.8)where we introduced the function: D ( x ) = 2 x − Γ ( x ) π , (3.9)that will be used throughout the whole paper exactly with the same meaning; the function D ( x ) ariseswhen the gauge spectra are evaluated outside the effective horizon (i.e. for | kτ | <
1) and the correspondingHankel functions are estimated using their limit for small arguments [42, 43]. Note, in this respect, that( − kτ ) can also be expressed as ( − kτ ) = k (1 − (cid:15) ) a H (cid:39) ka H = ka H e − N , (cid:15) (cid:28) , N = ln (cid:18) aa (cid:19) . (3.10)Since, by definition, µ = | γ − / | in terms of γ the comovong power spectra of Eqs. (3.6), (3.7) and (3.8)become: P B ( k, τ ) = a H D ( | γ − / | ) | kτ | −| γ − | , (3.11) P E ( k, τ ) = a H D ( γ + 1 / | kτ | − γ . (3.12)The derivation of Eq. (3.11) from Eq. (3.6) is immediate from the definition of µ in terms of γ (i.e. µ = | γ − / | ). On the contrary Eq. (3.12) is the common expression of Eqs. (3.7) and (3.8). In fact, inEq. (3.7) µ = γ − / µ + 1 = γ + 1 / µ = 1 / − γ and | µ − | always equals γ + 1 /
2. After direct insertion of Eqs. (3.11)–(3.12) into Eq. (2.32) the spectral energydensity for τ ≤ − τ isΩ Y ( k, τ ) = 23 (cid:18) HM P (cid:19) (cid:20) D B ( | γ − / | ) | kτ | −| γ − | + D E ( γ + 1 / | kτ | − γ (cid:21) . (3.13)The spectral energy density of Eq. (3.13) must be always subcritical (i.e. Ω Y ( k, τ ) (cid:28)
1) for τ ≤ − τ and | kτ | ≤
1; this requirement is not always satisfied even if, during the inflationary phase, H (cid:28) M P . The firstterm inside the square bracket at the right hand side of Eq. (3.13) denotes the magnetic contribution whilethe second term is the electric result . By looking together at Eqs. (3.11), (3.12) and (3.13) the followingconclusions naturally emerge: A term | − (cid:15) | has been omitted in the prefactors since it coincides with 1 for (cid:15) (cid:28) N denotes the total number of inflationary e -folds and has nothing to do with the normalization N µ of the mode functions. To stress this we just added a subscript to the function D ( x ) (by writing D B ( x ) and D E ( x )) even if the definition of D ( x )(see after Eq. (3.8)) is the same in both cases. if γ = 2 the hyperelectric spectrum is exactly scale-invariant while the magnetic spectrum is steeplyincreasing [i.e. P B ( k, τ ) ∝ | kτ | for kτ (cid:28)
1] so that the condition Ω Y ( k, τ ) (cid:28) • if 1 / < γ ≤ Y ( k, τ ) (cid:28) | kτ (cid:28) • if γ > kτ (cid:28)
1: in this case the bound Ω Y (cid:28) γ > γ → / : P B ( k, τ ) = a H π ( − kτ ) | H (1)0 ( − kτ ) | → a H π | kτ | ln | kτ | , (3.14) P B ( k, τ ) = a H π ( − kτ ) | H (1)1 ( − kτ ) | → a H π | kτ | , (3.15)Ω Y ( k, τ ) = 13 π (cid:18) HM P (cid:19) | kτ | (cid:20) | kτ | ln | kτ | (cid:21) . (3.16)Therefore, also in the case γ → / < γ ≤ γ > γ = 2the conventional wisdom is that it will also be minute at the galactic scale after the gauge coupling flattensout. This swift conclusion is only true provided the hypermagnetic magnetic power spectrum at the end ofinflation is not modified for τ ≥ − τ . In section 4 the gauge power spectra will be explicitly computed inthe regime where the gauge coupling flattens out (i.e. for τ ≥ − τ ) and it will be shown that the late-timehypermagnetic spectrum does not coincide with the hypermagnetic spectrum at the end of inflation whenthe gauge coupling increases. When √ λ evolves as in Eq. (2.41) (see also Fig. 2) the correctly normalized solution of Eqs. (2.20) and(2.21) are now given by f k ( τ ) = N (cid:101) µ √ k √− kτ H (1) (cid:101) µ ( − kτ ) , g k ( τ ) = N (cid:101) µ (cid:115) k √− kτ H (1) (cid:101) µ − ( − kτ ) , (3.17)where (cid:101) µ = (cid:101) γ + 12 , N (cid:101) µ = (cid:114) π e iπ (2 (cid:101) µ +1) / . (3.18)Inserting Eqs. (3.17) and (3.18) into Eqs. (2.25) and (2.26) the comoving power spectra are: (cid:102) P B ( k, τ ) = a H π ( − kτ ) | H (1) (cid:101) µ ( − kτ ) | → a H D ( (cid:101) µ ) | kτ | − (cid:101) µ , (3.19) (cid:102) P E ( k, τ ) = a H π ( − kτ ) | H (1) (cid:101) µ − ( − kτ ) | → a H D ( | (cid:101) µ − | ) | kτ | − | (cid:101) µ − | , (3.20) When γ → / f k ( τ ) = N √− kτ H (1)0 ( − kτ ) / √ k and g k ( τ ) = N (cid:112) k/ √− kτ H (1)1 ( − kτ ),where N = (cid:112) π/ e iπ/ . D ( x ) has been already defined in Eq. (3.9) for a generic argument x . To stress that thepower spectra (3.19)–(3.20) correspond to the case of decreasing gauge coupling a tilde has been added ontop of each expression. Since, according to Eq. (3.18), (cid:101) µ = (cid:101) γ + 1 /
2, Eqs. (3.19) and (3.20) are directlyexpressible in terms of (cid:101) γ and the result is: (cid:102) P B ( k, τ ) = a H D ( (cid:101) γ + 1 / | kτ | − (cid:101) γ , (3.21) (cid:102) P E ( k, τ ) = a H D ( | (cid:101) γ − / | ) | kτ | −| (cid:101) γ − | . (3.22)Equations (3.21)–(3.22) will now be inserted into Eq. (2.32); the explicit expression of the spectral energydensity is: (cid:102) Ω Y ( k, τ ) = 23 (cid:18) HM P (cid:19) (cid:20) D B ( (cid:101) γ + 1 / | kτ | − (cid:101) γ + D E ( | (cid:101) γ − / | ) | kτ | −| (cid:101) γ − | (cid:21) , (3.23)where in analogy with Eq. (3.13) the subscripts E and B have been added to the function D ( x ) with thepurpose of reminding that the origin of the corresponding terms. As in the case of Eq. (3.13) the spectralenergy density of Eq. (3.23) should always be subcritical (i.e. (cid:101) Ω Y ( k, τ ) (cid:28) τ ≤ − τ and | kτ | (cid:28) • when (cid:101) γ = 2 the hypermagnetic spectrum is flat while the hyperelectric spectrum is violet and it goesas (cid:102) P E ( k, τ ) ∝ | kτ | ; the spectral energy density is always subcritical for | kτ | (cid:28) • if (cid:101) γ = 3 the hyperelectric spectrum is flat however the hypermagnetic power spectrum diverges as | kτ | − in the limit | kτ | (cid:28) • in the 0 < (cid:101) γ < / < (cid:101) γ ≤ (cid:101) γ → / (cid:102) P B ( k, τ ) = a H π ( − kτ ) | H (1)1 ( − kτ ) | → a H π | kτ | , (3.24) (cid:102) P E ( k, τ ) = a H π ( − kτ ) | H (1)0 ( − kτ ) | → a H π | kτ | ln | kτ | , (3.25) (cid:102) Ω Y ( k, τ ) = 13 π (cid:18) HM P (cid:19) | kτ | (cid:20) | kτ | ln | kτ | (cid:21) . (3.26) A flat hypermagnetic spectrum is only consistent with the critical density bound provided (cid:101) γ = 2 (i.e.only when the gauge coupling is decreasing). Conversely a flat hyperelectric spectrum is only viable whenthe gauge coupling increases and γ = 2. Overall the only intervals where the critical density constraint issatisfied are given by 0 ≤ γ ≤ ≤ (cid:101) γ ≤
2. These results follow in fact from the duality symmetry ofEqs. (2.16) and (2.29). For the gauge spectra during inflation the duality symmetry implies, in general: √ λ → √ λ , P B ( k, τ ) → (cid:102) P E ( k, τ ) , P E ( k, τ ) → (cid:102) P B ( k, τ ) (3.27)If now Eqs. (3.11)–(3.12) are compared with Eqs. (3.21)–(3.22) the spectra with increasing and decreasinggauge coupling during inflation are explicitly related by the following transformation: γ → (cid:101) γ , P B ( k, τ ) → (cid:102) P E ( k, τ ) , P E ( k, τ ) → (cid:102) P B ( k, τ ) . (3.28) When (cid:101) γ = 1 / f k ( τ ) = N √− kτ H (1)1 ( − kτ ) / √ k and by g k ( τ ) = − N (cid:112) k √− kτ H (1)0 ( − kτ ). √ λ is actually inverted when γ → (cid:101) γ . Thanks Eq.(3.28) the spectral energy density is left invariant [i.e. Ω Y ( k, τ ) → (cid:101) Ω Y ( k, τ )] since the hyperelectric andthe hypermagnetic power spectra are interchanged. All in all the standard lore stipulates that the onlyphenomenologically relevant case is the one (cid:101) γ = 2 since the other cases lead to an hypermagnetic spectrumthat is either too steep or anyway inconsistent with the critical density bound. As already mentioned, thisstatement assumes that the gauge spectra are unmodified when coupling flattens out and the relevantwavelengths are still larger than the effective horizon. When the gauge coupling flattens out as illustrated in Figs. 1 and 2 the background geometry enters astage of decelerated expansion after the end of the inflationary phase. The post-inflationary gauge fieldsare determined by the continuous evolution of the corresponding mode functions whose late-time behaviourfollows from the elements of an appropriate transition matrix mixing together the hyperelectric and thehypermagnetic mode functions at the end of inflation and leading to specific standing oscillations. TheseSakharov phases are different for the ( γ, δ ) transition illustrated in Fig. 1 and in the case of the ( (cid:101) γ , (cid:101) δ )profile of Fig. 2. ( γ, δ ) transition The continuous parametrization of √ λ given in Eqs. (2.33) and (2.34) implies that the late-time valuesvalues of f k ( τ ) and g k ( τ ) for τ ≥ − τ are given by: (cid:18) f k ( τ ) g k ( τ ) /k (cid:19) = (cid:18) A f f ( k, τ, τ ) A f g ( k, τ, τ ) A g f ( k, τ, τ ) A g g ( k, τ, τ ) (cid:19) (cid:18) f k g k /k (cid:19) , (4.1)where f k = f k ( − τ ) and g k = g k ( − τ ) denote the values of the mode functions at end of the inflationaryphase and the matrix elements at the right hand side of Eq. (4.1) are determined from the continuity ofthe mode functions as described in appendix A: A f f ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) Y ν − ( qx ) J ν ( ky ) − J ν − ( qx ) Y ν ( ky ) (cid:21) ,A f g ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) J ν ( qx ) Y ν ( ky ) − Y ν ( qx ) J ν ( ky ) (cid:21) ,A g f ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) Y ν − ( qx ) J ν − ( ky ) − J ν − ( qx ) Y ν − ( ky ) (cid:21) ,A g g ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) J ν ( qx ) Y ν − ( ky ) − Y ν ( qx ) J ν − ( ky ) (cid:21) . (4.2)Since the inflationary mode functions f k and g k obey the Wronskian normalization of Eq. (2.21), also f k ( τ ) and g k ( τ ) must obey the same condition for τ ≥ τ and this happens provided : A f f ( k, τ, τ ) A g g ( k, τ, τ ) − A f g ( k, τ, τ ) A g f ( k, τ, τ ) = 1 . (4.3)The arguments of the Bessel functions appearing in (4.2) depend on qx and k y while the correspondingindices depend on δ ; the explicit expressions of q , y and ν are: q ( δ, γ ) = δγ , y ( τ, δ, γ ) = τ + τ [1 + q ( δ, γ )] , ν ( δ ) = δ + 1 / . (4.4) The validity of this condition can also be verified by plugging the explicit matrix elements of Eq. (4.2) into Eq. (4.3) andby using the standard recurrence relations involving the Bessel functions and their Wronskians [42, 43]. y ( − τ ) = qτ which also implies (by definition of x ) that ky ( − τ ) = q k τ = qx . Consequently, as expected from the continuity of the mode functions, A f g ( k, − τ , τ ) = A g f ( k, − τ , τ ) = 0 , A f f ( k, − τ , τ ) = A g g ( k, − τ , τ ) = 1 . (4.5)Since all the expression entering Eq. (4.2) ultimately depend on the dimensionless variables x = kτ , x = kτ and ν , the matrix appearing in Eq. (4.1) is in fact a function of δ , x and x for any fixed valueof γ : M ( δ, x, x ) = (cid:18) A f f ( δ, x, x ) A f g ( δ, x, x ) A g f ( δ, x, x ) A g g ( δ, x, x ) (cid:19) . (4.6)We stress that the variable x = kτ ≤ k in units of the maximal wavenumber of the spectrum(i.e. 1 /τ = a H ) and this is why it cannot be larger than O (1). When the mode functions f k ( τ ) and g k ( τ ) are deduced from Eq. (4.1) the gauge power spectra of Eqs.(2.25)–(2.26) become: P B ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) A f f f k + A f g g k k (cid:12)(cid:12)(cid:12)(cid:12) , (4.7) P E ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) k A g f f k + A g g g k (cid:12)(cid:12)(cid:12)(cid:12) . (4.8)One of the two terms inside each of the squared moduli appearing in Eqs. (4.7) and (4.8) will be alternativelydominant. For the sake of concreteness we shall now verify that the first term inside the squared modulusof Eq. (4.7) dominates against the second: (cid:12)(cid:12)(cid:12)(cid:12) A f g ( δ, x, x ) g k k (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) A f f ( δ, x, x ) f k (cid:12)(cid:12)(cid:12)(cid:12) . (4.9)Since all the variables have been explicitly defined, the validity of the condition (4.9) could be investigatednumerically . For a more general proof of Eq. (4.9) it is sufficient to consider Eq. (4.9) in the limit x < ≤ δ (cid:28) γ . An equivalent form of Eq. (4.9) and the result is: (cid:12)(cid:12)(cid:12)(cid:12) J ν ( qx ) Y ν ( ky ) − Y ν ( qx ) J ν ( ky ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) H (1) | γ − / | ( x ) H (1) γ +1 / ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) Y ν − ( qx ) J ν ( ky ) − J ν − ( qx ) Y ν ( ky ) (cid:12)(cid:12)(cid:12)(cid:12) . (4.10)If x (cid:28) x (cid:28) x /x = τ /τ (cid:28) (cid:18) qx (cid:19) − δ (cid:29) Γ(1 / − δ )Γ( | γ − / | )Γ(1 / δ )Γ( γ + 1 / (cid:18) x (cid:19) γ +1 / −| γ − / | , (4.11)where we used the explicit result of Eq. (A.9). When γ > / qx / − δ > ( x /
2) which is always verified as long as x < δ ≥ This analysis has been performed by fixing γ to a reference value and by scanning the relative weight of the two terms ofEq. (4.9) for different values of δ (cid:28) γ . For the sake of conciseness this numerical analysis will not be explicitly discussed. In the limit δ →
0, since q = δ/γ , the condition ( qx / − δ (cid:29) ( x /
2) is also verified as long as x <
1. Since thegauge coupling freezes (either partially or totally) for τ > − τ the physical situation discussed here corresponds to a range ofparameters where 0 ≤ δ (cid:28) / ≤ δ (cid:28) γ ; these two conditions will be used interchangeably. < γ < /
2, Eq. (4.11) implies ( qx / − δ > ( x / γ which is also verified in the physicalrange of the parameters.So far we demonstrated that Eq. (4.9) holds in the range x (cid:28) x (cid:28)
1. When x (cid:28) x (cid:29) ky (cid:39) x (cid:29) : J ν ( ky ) = M ν cos θ ν , Y ν ( ky ) = M ν sin θ ν , (4.12)where, for x (cid:29) θ ν ( x ) → x while M ν ( x ) → (cid:112) /πx − / [1 + O ( x − )]; this is the so-called modulus-phaseapproximation for the Bessel functions [42, 43]. Inserting Eq. (4.12) into Eq. (4.10) and expanding theremaining functions for x (cid:28) P B ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) A f g ( δ, x , x ) g k k (cid:12)(cid:12)(cid:12)(cid:12) . (4.13)The results of Eqs. (4.9)–(4.13) show that the hyperelectric field at the end of inflation determines thelate-time hypermagnetic field for τ (cid:29) − τ . This is of course not a general truism but it happens providedthe gauge coupling first increases during inflation and then flattens out in the radiation-dominated epoch.For the hyperelectric spectrum of Eq. (4.8) the inequality of Eq. (4.9) is in fact replaced by the followingcondition (cid:12)(cid:12)(cid:12)(cid:12) A g g ( δ, x, x ) g k (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) A g f ( δ, x, x ) k f k (cid:12)(cid:12)(cid:12)(cid:12) , (4.14)which can be verified explicitly by using the same strategy illustrated in the case of Eq. (4.9); for the sakeof conciseness these details will not be explicitly discussed. Therefore, thanks to Eq. (4.14), the late-timeexpression of the comoving hyperelectric spectrum is: P E ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) A g f ( δ, x x ) g k (cid:12)(cid:12)(cid:12)(cid:12) . (4.15)Equation (4.15) mirrors the result of Eq. (4.13) and it shows that the the hyperelectric power spectrumfor τ (cid:29) − τ is determined by the hyperelectric power spectrum at τ = − τ . As we shall see in a momentwhen the gauge coupling decreases the dual result will hold. To obtain a more explicit form of the gauge power spectra in the decelerated stage of expansion the matrixelements of Eq. (4.2) can be expanded in powers of x (cid:28)
1. The strategy will be to fix ky and expand thevarious terms in powers of of x with the subsidiary conditions 0 ≤ δ (cid:28) /
2. With these specifications Eq.(4.2) implies : A f f ( δ, x , x ) = (cid:18) qx (cid:19) δ (cid:20)(cid:114) x / − δ ) J − δ − / ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) − δ (cid:21) , (4.16) A f g ( δ, x , x ) = (cid:18) qx (cid:19) − δ (cid:20)(cid:114) x / δ ) J δ +1 / ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) δ (cid:21) , (4.17) A g f ( δ, x , x ) = (cid:18) qx (cid:19) δ (cid:20) − (cid:114) x / − δ ) J / − δ ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) − δ (cid:21) , (4.18) A g g ( δ, x , x ) = (cid:18) qx (cid:19) − δ (cid:20)(cid:114) x / δ ) J δ − / ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) δ (cid:21) . (4.19) Recall, in this respect, Eq. (4.4). Even if the value of x can be either smaller or larger than 1, as soon as x = kτ = O (1)the conductivity cannot be neglected and this situation will be more specifically discussed in section 5. For the moment weshall just consider the case x (cid:29) According to Eq. (4.4), ky = x + x ( q + 1) and, to lowest order in x , we have that ky = x + O ( x ). f k and g k inthe case of increasing coupling (see e.g. Eq. (A.8)) the hypermagnetic power spectrum becomes: P B ( k, τ ) = a H D ( γ + 1 / (cid:18) ka H (cid:19) − γ − δ F B ( kτ, δ ) ,F B ( x, δ ) = (cid:18) q (cid:19) − δ (cid:18) x (cid:19) Γ ( δ + 1 / J δ +1 / ( x ) . (4.20)Similarly, from Eq. (4.15) and (4.19) the hyperelectric spectrum turns out to be P E ( k, τ ) = a H D ( γ + 1 / (cid:18) ka H (cid:19) − γ − δ F E ( kτ, δ ) ,F E ( x, δ ) = (cid:18) q (cid:19) − δ (cid:18) x (cid:19) Γ ( δ + 1 / J δ − / ( x ) . (4.21)The results of Eqs. (4.20)–(4.21) only assume x < ≤ δ (cid:28) γ and can be evaluated either for kτ (cid:28) kτ (cid:29)
1. As long as kτ (cid:28) J α ( z ) (cid:39) ( z/ α / Γ( α + 1) [42, 43]; inthe opposite regime (i.e. kτ (cid:29)
1) Eq. (4.12) provides instead the valid approximation scheme .Another interesting limit is the sudden approximation which is not well defined a priori but only asthe δ → x and x are kept fixed and the matrix elements of Eq. (4.2) assume a rathersimple form implying: P B ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) cos ( x + x ) f k + sin ( x + x ) g k k (cid:12)(cid:12)(cid:12)(cid:12) , (4.22) P E ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) − k sin ( x + x ) f k + cos ( x + x ) g k (cid:12)(cid:12)(cid:12)(cid:12) . (4.23)Using Eq. (A.8) the gauge power spectra will be: P B ( k, τ ) = a H D ( γ + 1 / (cid:18) ka H (cid:19) − γ sin kτ , (4.24) P E ( k, τ ) = a H D ( γ + 1 / (cid:18) ka H (cid:19) − γ cos kτ . (4.25)The same results of Eqs. (4.24)–(4.25) follow immediately from Eqs. (4.20)–(4.21) by recalling that q − δ = ( δ/γ ) − δ → δ →
0. All in all, in the sudden approximation x and x are kept fixedwhile δ →
0; in the smooth limit δ may be very small (i.e. δ (cid:28)
1) but it is always different from zero.These two complementary approximations commute since it can be shown that:lim δ → (cid:20) lim x (cid:28) x (cid:28) M ( δ, x , x ) (cid:21) = lim x (cid:28) x (cid:28) (cid:20) lim δ → M ( δ, x , x ) (cid:21) = (cid:18) x − x (cid:19) . (4.26) ( (cid:101) γ , (cid:101) δ ) transition When the gauge coupling decreases and the dynamics of the transition follows the timeline of Fig. 2 theanalog of Eq. (4.1) can be written as: (cid:18) f k ( τ ) g k ( τ ) /k (cid:19) = (cid:18) (cid:102) A f f ( k, τ, τ ) (cid:102) A f g ( k, τ, τ ) (cid:102) A g f ( k, τ, τ ) (cid:102) A g g ( k, τ, τ ) (cid:19) (cid:18) f k g k /k (cid:19) , (4.27) Equations (4.20) and (4.21) hold for any value of kτ ; however, as we shall argue in section 5, for τ > τ k ∼ /k the powerspectra will be modified by the finite value of the conductivity. f k = f k ( − τ ) and g k = g k ( − τ ) follow from Eq. (3.17) and their explicit expression canbe found in Eq. (B.9). As usual we also added a tilde on top of the various matrix elements to stress thatthey are computed in the framework of the ( (cid:101) γ , (cid:101) δ ) transition. Thanks to the Wronskian normalization(2.21) the analog of Eq. (4.3) is now: (cid:102) A f f ( k, τ, τ ) (cid:102) A g g ( k, τ, τ ) − (cid:102) A f g ( k, τ, τ ) (cid:102) A g f ( k, τ, τ ) = 1 . (4.28)The various entries of the transition matrix appearing in Eq. (4.27) are: (cid:102) A f f ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) Y (cid:101) ν − ( qx ) J (cid:101) ν ( ky ) − J (cid:101) ν − ( qx ) Y (cid:101) ν ( ky ) (cid:21) , (cid:102) A f g ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) J (cid:101) ν ( qx ) Y (cid:101) ν ( ky ) − Y (cid:101) ν ( qx ) J (cid:101) ν ( ky ) (cid:21) , (cid:102) A g f ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) Y (cid:101) ν − ( qx ) J (cid:101) ν − ( ky ) − J (cid:101) ν − ( qx ) Y (cid:101) ν − ( ky ) (cid:21) , (cid:102) A g g ( k, τ, τ ) = π √ qx (cid:112) ky (cid:20) J (cid:101) ν ( qx ) Y (cid:101) ν − ( ky ) − Y (cid:101) ν ( qx ) J (cid:101) ν − ( ky ) (cid:21) . (4.29)The matrix elements of Eq. (4.29) and the values of f k and g k are determined from the continuity of themode functions as described in appendix B. The variables y , q and (cid:101) ν are now functions of (cid:101) δ and (cid:101) γ in fullanalogy with Eq. (4.4): q ( (cid:101) δ , (cid:101) γ ) = (cid:101) δ (cid:101) γ , y ( τ, (cid:101) δ , (cid:101) γ ) = τ + τ (cid:20) q ( (cid:101) δ , (cid:101) γ ) (cid:21) , (cid:101) ν ( (cid:101) δ ) = | (cid:101) δ − / | . (4.30)Note that Eq. (4.29) has the same form of Eq. (4.2); formally Eq. (4.29) can be obtained from Eq. (4.2)by replacing ν with (cid:101) ν . With the same notations of Eq(4.6) we can define the matrix (cid:103) M ( (cid:101) δ , x, x ) = (cid:18) (cid:102) A f f ( (cid:101) δ , x, x ) (cid:102) A f g ( (cid:101) δ , x, x ) (cid:102) A g f ( (cid:101) δ , x, x ) (cid:102) A g g ( (cid:101) δ , x, x ) (cid:19) . (4.31)Inserting the expressions of Eqs. (4.27) into Eqs. (2.25)–(2.26) the comoving power spectra for the ( (cid:101) γ , (cid:101) δ )transition are: (cid:102) P B ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12)(cid:102) A f f f k + (cid:102) A f g g k k (cid:12)(cid:12)(cid:12)(cid:12) , (4.32) (cid:102) P E ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12) k (cid:102) A g f f k + (cid:102) A g g g k (cid:12)(cid:12)(cid:12)(cid:12) . (4.33)As already argued for the ( γ, δ ) transition, the terms appearing inside the squared moduli at the righthand side of Eqs. (4.32) and (4.33) are not of the same order but one of the two terms will be alternativelydominant. Owing to the specific form of Eqs. (4.31) and (4.29)–(B.9) the hierarchies between the differentterms contributing to Eqs. (4.32)–(4.33) turn out to be: | (cid:102) A f f ( (cid:101) δ , x, x ) f k | (cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:102) A f g ( (cid:101) δ , x, x ) g k k (cid:12)(cid:12)(cid:12)(cid:12) , (4.34) | (cid:102) A g f ( (cid:101) δ , x, x ) f k | (cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:102) A g g ( (cid:101) δ , x, x ) g k k (cid:12)(cid:12)(cid:12)(cid:12) . (4.35)The inequalities Eqs. (4.34) and (4.35) imply that the late-time hypermagnetic power spectra are deter-mined by the hypermagnetic power spectra at τ and they can be explicitly verified by following the samestrategy illustrated in the case of the ( γ, δ ) transition. So for instance we can rewrite Eq. (4.34) as: (cid:12)(cid:12)(cid:12)(cid:12) Y (cid:101) ν − ( qx ) J (cid:101) ν ( ky ) − J (cid:101) ν − ( qx ) Y (cid:101) ν ( ky ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) H (1) (cid:101) µ − ( x ) H (1) (cid:101) µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) J (cid:101) ν ( qx ) Y (cid:101) ν ( ky ) − Y (cid:101) ν ( qx ) J (cid:101) ν ( ky ) (cid:12)(cid:12)(cid:12)(cid:12) . (4.36)18f both sides of Eq. (4.36) are expanded for x (cid:28) x < ≤ (cid:101) δ (cid:28) (cid:101) γ weobtain the following condition: (cid:18) qx (cid:19) (cid:101) δ (cid:29) Γ(3 / (cid:101) δ ) Γ( | (cid:101) γ − / | )Γ(1 / − (cid:101) δ ) Γ( (cid:101) γ + 1 / (cid:18) x (cid:19) (cid:101) γ +1 / −| (cid:101) γ − / | . (4.37)When (cid:101) γ > / x (cid:101) δ > x : this requirement is always verified for x < ≤ (cid:101) δ (cid:28) /
2. Similarly, in the range (cid:101) γ < /
2, Eq. (4.37) demands x (cid:101) δ > x (cid:101) γ which is alsotrue since (cid:101) δ (cid:28) (cid:101) γ . The same logic can be applied to Eq. (4.35) but this discussion will be omitted for thesake of conciseness. All in all Eqs. (4.34) and (4.35) imply that the comoving power spectra become (cid:101) P B ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12)(cid:102) A f f ( (cid:101) δ, x , x ) f k (cid:12)(cid:12)(cid:12)(cid:12) , (cid:101) P E ( k, τ ) = k π (cid:12)(cid:12)(cid:12)(cid:12)(cid:102) A g f ( (cid:101) δ, x , x ) k f k (cid:12)(cid:12)(cid:12)(cid:12) . (4.38) The spectra of Eq. (4.38) can be evaluated in various limits and, for this purpose, it is useful to expandEq. (4.29) for x with the subsidiary conditions 0 ≤ δ < /
2. The result of this stem can be expressed as: (cid:102) A f f ( (cid:101) δ , x , x ) = (cid:18) qx (cid:19) − (cid:101) δ (cid:20)(cid:114) x / (cid:101) δ ) J (cid:101) δ − / ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) (cid:101) δ (cid:21) , (4.39) (cid:102) A f g ( (cid:101) δ , x , x ) = (cid:18) qx (cid:19)(cid:101) δ (cid:20)(cid:114) x / − (cid:101) δ ) J − (cid:101) δ +1 / ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) − (cid:101) δ (cid:21) , (4.40) (cid:102) A g f ( (cid:101) δ , x , x ) = (cid:18) qx (cid:19) − (cid:101) δ (cid:20) − (cid:114) x / (cid:101) δ ) J / (cid:101) δ ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) (cid:101) δ (cid:21) , (4.41) (cid:102) A g g ( (cid:101) δ , x , x ) = (cid:18) qx (cid:19)(cid:101) δ (cid:20)(cid:114) x / − (cid:101) δ ) J − (cid:101) δ − / ( x ) + O ( x ) (cid:21) + O (cid:20) ( q x ) − (cid:101) δ (cid:21) . (4.42)Again using Eqs. (4.39)–(4.40) and (4.41)–(4.42) the inequalities (4.34)–(4.35) are verified and the corre-sponding power spectra turn out to be: (cid:102) P B ( k, τ ) = a H D ( (cid:101) γ + 1 / (cid:18) ka H (cid:19) − (cid:101) γ − (cid:101) δ (cid:102) F B ( k τ, (cid:101) δ ) , (cid:102) F B ( x, (cid:101) δ ) = (cid:18) q (cid:19) − (cid:101) δ (cid:18) x (cid:19) Γ ( δ + 1 / J (cid:101) δ − / ( x ) . (4.43)From Eq. (4.15) and (4.19) the hyperelectric spectrum is (cid:102) P E ( k, τ ) = a H D ( (cid:101) γ + 1 / (cid:18) ka H (cid:19) − (cid:101) γ − (cid:101) δ (cid:102) F E ( k τ, (cid:101) δ ) , (cid:102) F E ( x, (cid:101) δ ) = (cid:18) q (cid:19) − (cid:101) δ (cid:18) x (cid:19) Γ ( (cid:101) δ + 1 / J (cid:101) δ +1 / ( x ) . (4.44)Equations (4.43) and (4.44) can be evaluated in the sudden approximation by taking the limit (cid:101) δ → x and x are kept fixed; the gauge power spectra are, in this case (cid:102) P B ( k, τ ) = a H D ( (cid:101) γ + 1 / (cid:18) ka H (cid:19) − (cid:101) γ cos kτ , (4.45) Note that the same results of Eqs. (4.45)–(4.46) follow immediately from Eqs. (4.43)–(4.44) by recalling that q − (cid:101) δ =( (cid:101) δ / (cid:101) γ ) − (cid:101) δ → (cid:101) δ → P E ( k, τ ) = a H D ( (cid:101) γ + 1 / (cid:18) ka H (cid:19) − (cid:101) γ sin kτ . (4.46)By comparing Eqs. (4.45)–(4.46) with the analog results given in Eqs. (4.24)–(4.25) we note that thephases of the Sakharov oscillations are exchanged: while P B ( k, τ ) oscillates like sin kτ , the oscillations of (cid:102) P B ( k, τ ) go like cos kτ . The opposite is true for the hyperelectric power spectra. The differences in thephases of oscillation are a particular consequence of duality which will now be analyzed in general terms. During inflation duality relates the gauge power spectra in different dynamical situations (see Eqs. (3.27)–(3.28)). When the gauge coupling flattens out the gauge power spectra are also related by duality whichis equivalent to the following transformation:( γ, δ ) → ( (cid:101) γ, (cid:101) δ ) . (4.47)The action of Eq. (4.47) transforms the elements of the transition matrix according to A f f → (cid:102) A g g , A g g → (cid:102) A f f , A f g → − (cid:102) A g f , A g f → − (cid:102) A f g . (4.48)Equation (4.48) follows from the expressions of the matrix elements reported in Eqs. (4.2) and (4.29). Forthe sake of accuracy the results of Eq. (4.48) have been explicitly derived in appendix C. Moreover f k and g k will transform under duality as f k → g k /k and g k → − k f k . Thanks to Eqs. (2.29) and (4.48) theaction of Eq. (4.47) transforms the gauge power spectra as: P B ( k, τ ) → (cid:102) P E ( k, τ ) , P E ( k, τ ) → (cid:102) P B ( k, τ ) . (4.49)Also the approximate expressions of the power spectra, if they are correct, must transform according toEq. (4.49) since they must be consistent with duality; in particular: • the approximate power spectrum P B ( k, τ ) of Eq. (4.20) gives the approximate form of (cid:102) P E (i.e. Eq.(4.44)); • the transformation (4.47) applied to Eq. (4.21) gives the approximate form of the hyeprmagneticpower spectrum in the dual description (i.e. P E ( k, τ ) → (cid:102) P B ( k, τ )); • in the sudden approximation (when δ → (cid:101) δ →
0) the gauge power spectra of Eqs. (4.7)–(4.8)and (4.32)–(4.33) are related by dualityAll in all during inflation duality implies that when the gauge coupling increases the magnetic spectrumis never flat and if the gauge coupling is instead decreasing the electric spectrum is never flat. Afterinflation duality constrains the explicit forms of the gauge power spectra and the phases of the Sakharovoscillations. If the gauge coupling flattens out after a phase of increasing coupling the late-time gaugespectra are determined by the hyperelctric spectrum at the end of inflation. In the dual situation thegauge coupling freezes after a stage of decreasing coupling. Barring for the possible physical justificationsof a strongly coupled phase at the beginning of inflation, the late-time gauge power spectra follow fromthe hypermagnetic spectrum at the end of inflation.
The hypercharge field projects on the electromagnetic fields through the cosine of the Weinberg angle(i.e. cos θ W ) so that the late-time gauge power spectra could be compared both with the magnetogenesis20equirements and with other phenomenological constraints. In this section we adopt the standard notationscommonly employed in the context of the concordance paradigm. So for instance Ω R denotes presentcritical fraction of relativistic particles, Ω M is total critical fraction of massive species, r T defines the ratiobetween the tensor and scalar power spectra at the pivot scale k p = 0 .
002 Mpc − and so on and so forth.The present value of the scale factor will be normalized to 1 (i.e. a = 1) and, thanks to this widely usedconvention, the physical and the comoving frequencies coincide at the present time. Defining with τ k = 1 /k the reentry time of a generic wavelength, the ratio between τ k and the time ofmatter-radiation equality τ eq is: τ k τ eq = 1 . × − (cid:18) h Ω M . (cid:19)(cid:18) h Ω R . × − (cid:19) − / (cid:18) k Mpc − (cid:19) − , (5.1)since ( τ k /τ eq ) = √ H /k )Ω M / Ω R and H = 100 h (km / Mpc) Hz denotes the present value of the Hubblerate. Consequently the relevant wavenumbers for magnetogenesis considerations [i.e. k = O (Mpc − )]reentered prior to matter radiation equality.After the gauge coupling flattens out theamplitude of the comoving power spectra is O ( H ) (see, forinstance, Eqs. (4.20)–(4.21) and (4.43)–(4.44)). The inflationary Hubble rate H is related to the Planckscale as : H M P = (cid:112) π (cid:15) A R = 2 . × − (cid:18) r T . (cid:19) / (cid:18) A R . × − (cid:19) / , (5.2)where A R denotes the amplitude of the scalar power spectrum at the pivot scale k p = 0 .
002 Mpc − . Thanksto Eq. (5.2) it is then possible to express H in units of nG by recalling that M P = 4 . × nG . Thescale H also enters the explicit expression of x = kτ = k/ ( a H ): ka H = ka H (cid:18) a H a H (cid:19) = kH ζ / − αr (cid:18) π(cid:15) Ω R A R (cid:19) − / (cid:115) H M P , (5.3)where the first equality is just true by definition while the second one is the explicit estimate. As alreadymentioned, in Eq. (5.3) Ω R is the (present) critical fraction of massless species (in the concordanceparadigm h Ω R = 4 . × − ); note that H is usefully expressed in Planck units as ( H /M P ) = 1 . × − ( h / . ka H = 10 − . (cid:18) k Mpc − (cid:19) (cid:18) r T . (cid:19) − / (cid:18) h Ω R . × − (cid:19) − / (cid:18) A R . × − (cid:19) − / ζ / − αr . (5.4)In Eqs. (5.3)–(5.4) ζ r = H r /H and H r is the value of the Hubble rate at the moment when the radiationstarts dominating; in the concordance paradigm ζ r = 1 (and this is the value we shall assume most ofthe time). However it is sometimes useful to check for the stability of the obtained results with respectto a change of the reference scenario. This is why Eqs. (5.3)–(5.4) have been derived by considering ageneralized post-inflationary evolution where the radiation epoch is preceded by a phase expanding at arate parametrized by α . Finally, throughout the explicit numerical evaluation it will be useful to bear inmind the following two general expressions: (cid:18) a a (cid:19) = ζ α − / r (cid:18) R π(cid:15) A R (cid:19) / (cid:115) H M P , (cid:18) a H a H (cid:19) = ζ / − αr (cid:18) π(cid:15) Ω R A R (cid:19) − / (cid:115) H M P . (5.5) In Eq. (5.2) we traded (cid:15) for the tensor scalar ratio r T by using the consistency relation r T = 16 (cid:15) . If this intermediate stage of expansion is dominated by a perfect fluid with barotropic index w , we have that α =2 / [3( w + 1)]. .2 Physical spectra prior to reentry For τ (cid:28) τ k the power spectra follow from Eqs. (4.20)–(4.21) and (4.43)–(4.44) evaluated in the limit kτ (cid:28)
1. When the gauge coupling increases the physical spectra (2.27) associated with the comovingexpressions of Eqs. (4.20)–(4.21) are : P ( phys ) B ( k, τ ) = H λ (cid:18) a a (cid:19) D ( γ + 1 / δ + 1) q δ (cid:18) ka H (cid:19) − γ (cid:18) a H a H (cid:19) δ +2 , (5.6) P ( phys ) E ( k, τ ) = H λ (cid:18) a a (cid:19) D ( γ + 1 / q δ (cid:18) ka H (cid:19) − γ (cid:18) a H a H (cid:19) δ , τ < τ k . (5.7)When the gauge coupling decreases the comoving spectra of Eqs. (4.43)–(4.44) must be inserted into Eq.(2.27) and the result is: (cid:102) P ( phys ) B ( k, τ ) = H λ (cid:18) a a (cid:19) D ( (cid:101) γ + 1 / q (cid:101) δ (cid:18) ka H (cid:19) − (cid:101) γ (cid:18) a H a H (cid:19) (cid:101) δ , (5.8) (cid:102) P ( phys ) E ( k, τ ) = H λ (cid:18) a a (cid:19) D ( (cid:101) γ + 1 / (cid:101) δ + 1) q (cid:101) δ (cid:18) ka H (cid:19) − (cid:101) γ (cid:18) a H a H (cid:19) (cid:101) δ +2 , τ < τ k . (5.9)The explicit formulas of Eqs. (5.2)–(5.3) and (5.5) allow for an explicit evaluation of Eqs. (5.6)–(5.7) and(5.8)–(5.9). If λ = O (1) the only spectra consistent with the perturbative evolution of the gauge couplingare the ones of Eqs. (5.6)–(5.7). In the case of decreasing gauge coupling λ must be instead extremelylarge (see also Eq. (2.44) and discussion therein). This means, however, that the magnetic spectrum (5.8)will be suppressed as λ − (cid:39) e − (cid:101) γ N where N now denotes the total number of inflationary e -folds. Let usfinally remark that in the sudden limit (i.e. δ → (cid:101) δ →
0) the physical power spectra will alwaysfollow from Eq. (2.27) but the related comoving spectra will be given, respectively, by Eqs. (4.24)–(4.25)and (4.45)–(4.46). For instance, in the case of increasing gauge coupling the result will be: P ( phys ) B ( k, τ ) = H λ (cid:18) a a (cid:19) D ( γ + 1 / (cid:18) ka H (cid:19) − γ (cid:18) a H a H (cid:19) , (5.10) P ( phys ) E ( k, τ ) = H λ (cid:18) a a (cid:19) D ( γ + 1 / (cid:18) ka H (cid:19) − γ , τ < τ k . (5.11) For τ ≥ τ k the evolution of the mode functions is modified by the presence of the conductivity and it isapproximately given by : g (cid:48) k = − k f k − σ g k , f (cid:48) k = g k . (5.12)These equations can be systematically solved as an expansion in ( k/σ ) by setting initial conditions at τ = τ k . To lowest order the solution of Eq. (5.12) for τ ≥ τ k is: f k ( τ ) = A g f ( k, τ , τ k ) g k k e − k k σ , g k ( τ ) = (cid:18) kσ (cid:19) A g g ( k, τ , τ k ) g k e − k k σ , (5.13)where the magnetic diffusivity scale k σ has been defined as k − σ = (cid:82) ττ k dz/σ ( z ). While the estimate of k σ can be made accurate by computing the transport coefficients of the plasma in different regimes, for Concerning Eqs. (5.6) and (5.7) we note that λ = λ ( − τ ). For the sake of simplicity we shall assume λ = 1 even if λ could also be slightly larger than 1. Note that in a different system of units (where the gauge coupling is defined without the √ π factor) we would have 4 πσ (and not simply σ ) in Eq. (5.12). We are assuming here that the gauge coupling does not evolve anymore and it is frozen toits constant value. k/k σ ) is so small, for the phenomenologicallyinteresting scales, that the negative exponentials in Eq. (5.13) evaluate to 1. In fact by taking τ = τ eq wehave that k σ can be estimated as: (cid:18) kk σ (cid:19) = 4 . × − (cid:113) h Ω M ( z eq + 1) (cid:18) k Mpc − (cid:19) , (5.14)where Ω M is the present critical fraction in matter and z eq + 1 = a /a eq (cid:39) O (3200) is the redshift ofmatter-radiation equality. The physical power spectra after for τ (cid:29) τ k will then be given by: P ( phys ) B ( k, τ ) = H (cid:18) a a (cid:19) D ( γ + 1 / (cid:18) ka H (cid:19) − γ − δ F B ( kτ k , δ ) e − k k σ , (5.15) P ( phys ) E ( k, τ ) = (cid:18) kσ (cid:19) H (cid:18) a a (cid:19) D ( γ + 1 / (cid:18) ka H (cid:19) − γ − δ F E ( kτ k , δ ) e − k k σ . (5.16)where F B ( x, δ ) and F E ( x, δ ) coincide with the ones defined in Eqs. (4.20)–(4.21). According to Eqs. (5.15)–(5.16) the magnetic spectrum for k < k σ is practically not affected by the conductivity while the electricpower spectrum is suppressed by k /σ (cid:28)
1. Since prior to decoupling the electron-photon and electron-proton interactions tie the temperatures close together, the conductivity scales as σ ∼ (cid:112) T /m e T /α em where m e is the electromagnetic mass, T is the temperature and α em is the fine structure constant. For instance,if we take T = O (eV) we get k/σ = O (10 − ) for k = O (Mpc − ). The electric power spectrum of Eq.(5.16) will then be suppressed, in comparison with the magnetic spectrum, by a factor ranging between40 and 60 orders of magnitude. This also demonstrates, in practical terms, that the duality symmetry isbroken for τ > τ k . According to the standard lore the observed large-scale fields in galaxies (and to some extent in clus-ters) should have been much smaller before the gravitational collapse of the protogalaxy. Compressionalamplification typically increases the initial values of the magnetic seeds by 4 or even 5 orders of magni-tude and the logic underlying this statement is in short the following (see e. g. [19, 20, 21]). When theprotogalactic matter collapsed by gravitational instability over a typical scale O (Mpc) the mean matterdensity before collapse was of the order of ρ crit . whereas right after the collapse the mean matter densitybecame, approximately, six orders of magnitude larger than the critical density. Since the physical size ofthe patch decreases from 1 Mpc to 30 kpc the magnetic field increases, because of flux conservation, of afactor ( ρ a /ρ b ) / ∼ where ρ a and ρ b are, respectively the energy densities right after and right beforegravitational collapse.Most of the work in the context of the dynamo theory focuses on reproducing the correct features of themagnetic field of our galaxy. The dynamo term may be responsible for the origin of the magnetic field of thegalaxy. The galaxy has a typical rotation period of 3 × yrs and comparing this figure with the typicalage of the galaxy, O (10 yrs), it can be appreciated that the galaxy performed about 30 rotations sincethe time of the protogalactic collapse. The achievable amplification produced by the dynamo instabilitycan be at most of 10 , i.e. e . Thus, if the present value of the galactic magnetic field is O ( µ G), its valueright after the gravitational collapse of the protogalaxy might have been as small as O (10 − ) nG over atypical scale of 30–100 kpc.The compressional amplification and the dynamical action are typically combined together so that themagnetogenesis requirements roughly demand that the magnetic fields at the time of the gravitationalcollapse of the protogalaxy should be approximately larger than a (minimal) power spectrum which canbe estimated between O (10 − ) nG and O (10 − ) nG :log (cid:18) (cid:113) P ( phys ) B nG (cid:19) ≥ − ξ, < ξ < . (5.17)23 = 1 . , δ = 0 . γ = 1 . , δ = 0 . γ = 2 , , δ = 0 . k = 1 Mpc − (cid:113) P ( phys ) B = 10 − . nG (cid:113) P ( phys ) B = 10 − . nG (cid:113) P ( phys ) B = 10 − . nG k = 0 . − (cid:113) P ( phys ) B = 10 − . nG (cid:113) P ( phys ) B = 10 − . nG (cid:113) P ( phys ) B = 10 − . nG k = 0 .
01 Mpc − (cid:113) P ( phys ) B = 10 − . nG (cid:113) P ( phys ) B = 10 − . nG (cid:113) P ( phys ) B = 10 − . nGTable 1: Numerical values of the magnetic power spectrum at different scales and in the framework of the( γ, δ ) transition where the gauge coupling is perturbative throughout all the stages of its evolution.The least demanding requirement of Eq. (5.17) (i.e. (cid:113) P ( phys ) B ≥ − nG) follows by assuming that, aftercompressional amplification, every rotation of the galaxy increases the initial magnetic field of one e -fold.According to some this requirement is not completely since it takes more than one e -fold to increase thevalue of the magnetic field by one order of magnitude and this is the rationale for the most demandingcondition of Eq. (5.17), i.e. (cid:113) P ( phys ) B ≥ − nG. Let us now consider a simplified estimate where ζ r → α →
0. This is is the situation of the con-cordance paradigm and for the typical values of the parameters given above the results for (cid:113) P ( phys ) B ( k, τ )are reported in Tab. 1 for different values of γ and k . - - - - - - - - - - ( k / Mpc - ) γ log [ P B ( phys ) / nG ] ( ζ r = r T = δ = ) - - - - - - - - - - - ( k / Mpc - ) γ log [ P B ( phys ) / nG ] ( ζ r = r T = δ = ) Figure 3: The parameter space is illustrated in the ( γ, k ) plane. The numbers appearing on the variouscontours correspond to the common logarithm of the physical power spectrum. The common logarithm of k in units of Mpc − is reported on the horizontal axis.Since the non-screened vector modes of the hypercharge field project on the electromagnetic fieldsthrough the cosine of the Weinberg angle the estimates of Tab. 1 follow from Eq. (5.15) after multiplyingthe obtained result by cos θ W (recall, in this respect, that sin θ W (cid:39) . δ (cid:28) γ ) and the sudden (i.e. δ → δ = 0 . δ = 0. In Fig. 3 the physical power spectrum is illustrated in the ( k, γ ) plane. The - - - - - - - - - - γ l og r T log [ P B ( phys ) / nG ] ( ζ r =
1, k = - , δ = ) - - - - - - - - - - - γ l og r T log [ P B ( phys ) / nG ] ( ζ r =
1, k = - , δ = ) Figure 4: We illustrate the common logarithm of the power spectrum in the plane ( γ, r T ). We remind that r T denotes, within the present notations, the tensor to scalar ratio.black blobs in both plots indicate three reasonable regions of the parameter space where all the pertinentphenomenological constraints are satisfied. The numbers on the various curves denote, as explained ontop of the plot, the common logarithms of the power spectrum (expressed in nG ) along that curve. Bylooking at the intercept on the γ axis we see that the phenomenologically reasonable values of γ correspondto spectra that are blue or, at most, quasi-flat but always slightly increasing with k (as already suggestedby the results of Tab. 1). The results of Fig. 3 are complemented by Fig. 4 where the power spectra - - - - - - - - γ l og ζ r log [ P B ( phys ) / nG ] ( α = / r T = = - , δ = ) - - - - - - - - - γ l og ζ r log [ P B ( phys ) / nG ] ( α = / r T = = - , δ = ) Figure 5: We illustrate the parameter space in the ( γ, ζ r ) plane.are illustrated in the ( γ, r T ) plane for fixed wavenumber k = Mpc − . The black blobs correspond this25ime to the regions where r T = O (0 . r T < .
07 while in Ref. [46] the range of values r T < .
064 is quoted By looking together at Figs.3 and 4 we can appreciate that the viable range of γ roughly extends between 1 . - - - - - - - - - - α l og ζ r log [ P B ( phys ) / nG ] ( γ = / r T = = - , δ = ) - - - - - - - - - α l og ζ r log [ P B ( phys ) / nG ] ( γ = / r T = = - , δ = ) Figure 6: The allowed region of the parameter space is illustrated with a shaded area for different valuesof r T and for the two complementary sets of magnetogenesis requirements.swiftly analyze the ( γ, ζ r ) plane by choosing α = 1 /
3. We remind that ζ r ≤ α is the corresponding rate ofexpansion that has been chosen to be smaller than in the case of radiation.This is equivalent, in practice, to a long post-inflationary phase dominated by a stiff equation of state.Since ζ r = H r /H we see that for long durations of this phase the viable range of γ increases: it is enough tocompare the case ζ r = 1 on top of each plot with the case ζ r (cid:28) γ . In Fig. 6 we also considered the ( α, ζ r ) plane for γ = 3 / δ and in the( γ, k ) plane. The different shading corresponds to the two extreme values of ξ appearing in Eq. (5.17).26e could go on slicing the parameter space almost indefinitely but, for the sake of conciseness, it isnow useful to illustrate the relevant region of the parameter space where all the various constraints andrequirements deduced so far are consistently met.Figure 8: The allowed region of the parameter space is illustrated with a shaded area for fixed k and inthe ( γ, δ ) plane. As in Fig. 7 the different shadings correspond to the two extreme values of ξ appearingin Eq. (5.17).In Fig. 7 the shaded areas denote the region where the spectral energy density is subcritical both duringand after inflation while the magnetogenesis and the Cosmic Microwave Background (CMB) constraintsare all satisfied. While the discussion of the CMB constraints would probably require a separate analysis,some general results could be used and these are essentially the ones reproduced in Fig. 5: they amountto requiring that the physical power spectrum after equality but before decoupling is smaller than 10 − nG for typical wavenumbers comparable with the pivot scale k p = 0 .
002 Mpc − at which the scalar andtensor power spectra are customarily assigned. The proper analysis of the CMB effects associated with themagnetic random fields started already after the first releases of the WMAP data; through the years variousdirect constraints have been derived both from the temperature and from the polarization anisotropies (see,for instance, [47] for a recent review). The main difference between the two plots of Fig. 7 stems fromthe magnetogenesis requirements: the larger area in the right plot corresponds to the situation where eachgalactic rotation amplifies the initial magnetic field value by one e -fold while the smaller area follows byrequiring that the physical power spectrum exceed 10 − nG (see Eq. (5.17) and discussion therein). InFig. 8 we consider the same constraints of Fig. 7 but in the ( γ, δ ) plane. The physical region obviouslycorresponds to the case δ (cid:28) O (1 . ≤ γ <
2. If the magnetogenesisrequirement are relaxed γ can be as small as O (1 . Concluding considerations
Duality implies that the hypermagnetic power spectra parametrically amplified from quantum fluctuationsduring a quasi-de Sitter stage of expansion are scale-invariant (or slightly blue) if the gauge couplingdecreases while an increase of the gauge coupling is only compatible with a flat hyperelectric spectrumfor wavelengths larger than the effective horizon at the corresponding epoch. The same duality symmetrydemands that the late-time gauge spectra do not always coincide with the results obtained at the end ofinflation: the late-time hypermagnetic spectra follow directly from the hypermagnetic mode functions atthe end of inflation whenever the gauge coupling decreases. Conversely if the gauge coupling increasesthe late-time hypermagnetic spectra are determined by the hyperelectric mode functions at the end of theinflationary phase. On a technical ground these results follow from the specific analysis of an appropriatetransition matrix whose elements have well defined transformation properties under the duality symmetryand control the form of the late-time spectra.Form a semiclassical viewpoint the mechanism analyzed here is the gauge analog of the Sakharovoscillations where travelling waves are transformed into standing waves with well defined phases that dependon the dynamics of the underlying geometry. The standing oscillations associated with the hyperelectricand with the hypermagnetic mode functions follow from the dynamics of the gauge couplings. Only whenthe gauge coupling decreases it is therefore reasonable to compute the post-inflationary physical spectraby a simple redshift of the comoving result during inflation. Conversely such a procedure would lead to anincorrect result in the case of an increasing gauge coupling where the late time hypermagnetic spectrum isdetermined by the hyperelectric spectrum at the end of inflation.In summary, all the magnetogenesis constraints can be successfully satisfied when the gauge couplingremains perturbative throughout all the stages of its evolution. More precisely a slightly blue hyperelectricspectrum during inflation may lead to a quasi-flat hypermagnetic spectrum at late times. The inducedlarge-scale magnetic fields turn out to be of the order of a few thousands of a nG over typical length scalesbetween few Mpc and 100 Mpc. The magnetogenesis requirements are therefore satisfied together with allthe backreaction constraints both during and after inflation. For the sweet spots of the parameter spacethe only further amplification required to seed the galactic magnetic fields is the one associated with thecompressional amplification.
Acknowledgments
It is a pleasure to thank T. Basaglia and S. Rohr of the CERN Scientific Information Service for their kindhelp throughout various stages of this investigation.28
Transition matrix and increasing gauge coupling
In this first part of this appendix we shall consider a gauge coupling that sharply increases during theinflationary phase and then flattens out in the post-inflationary era. The profile of √ λ corresponds, inthis case, to Eqs. (2.33)–(2.34) with 0 ≤ δ (cid:28) γ (see also Fig. 1 and discussion therein). Since theparametrization of √ λ is continuous and differentiable the evolution of the mode functions during thepost-inflationary stage (i.e. for τ ≥ − τ ) follows from the solution of Eqs. (2.20) and (2.21). For τ ≥ − τ the mode functions f k ( τ ) and g k ( τ ) are given in terms of certain coefficients A ± ( k, τ ): f k ( τ ) = √ ky √ k (cid:20) A − ( k, τ ) N ν H (1) ν ( ky ) + A + ( k, τ ) N ∗ ν H (2) ν ( ky ) (cid:21) , (A.1) g k ( τ ) = (cid:115) k (cid:112) ky (cid:20) A − ( k, τ ) N ν H (1) ν − ( ky ) + A + ( k, τ ) N ∗ ν H (2) ν − ( ky ) (cid:21) , (A.2)where the following auxiliary quantities have been introduced: N ν = (cid:114) π e iπ (2 ν +1) / , y ( τ ) = τ + τ (1 + q ) ,q = q ( δ, γ ) = δγ , ν = (cid:18) δ + 12 (cid:19) . (A.3)The separate continuity of Eqs. (3.1)–(3.2) and (A.1)–(A.2) across τ = − τ , determines the explicit formsof A ± ( k, τ ) in the case γ > / A − ( k, τ ) = iπN µ N ν √ qx (cid:20) H (1) µ ( x ) H (2) ν − ( q x ) − H (1) µ +1 ( x ) H (2) ν ( q x ) (cid:21) ,A + ( k, τ ) = iπN µ N ∗ ν √ qx (cid:20) H (1) µ +1 ( x ) H (1) ν ( q x ) − H (1) µ ( x ) H (1) ν − ( q x ) (cid:21) , γ > / , (A.4)where x = kτ . For 0 < γ < / A ± ( k, τ ) are instead obtained by matching Eqs. (3.1)and (3.3) with Eqs. (A.1)–(A.2): A − ( k, τ ) = iπN µ N ν √ qx (cid:20) H (1) µ ( x ) H (2) ν − ( q x ) + H (1) µ − ( x ) H (2) ν ( q x ) (cid:21) ,A + ( k, τ ) = − iπN µ N ∗ ν √ qx (cid:20) H (1) µ − ( x ) H (1) ν ( q x ) + H (1) µ ( x ) H (1) ν − ( q x ) (cid:21) , < γ < / . (A.5)Since the electric and the magnetic mode functions must obey the Wronskian normalization of Eq. (2.21), A ± ( k, τ ) must satisfy the condition | A + ( k, τ ) | − | A − ( k, τ ) | = 1. If the mixing coefficients A ± ( k, τ ) areredefined as A + = N ν A + and A − = N ∗ ν (cid:101) A − the general expressions of Eqs. (A.1) and (A.2) become f k ( τ ) = 1 √ k (cid:112) ky (cid:20)(cid:18) A + + A − (cid:19) J ν ( ky ) + i (cid:18) A − − A + (cid:19) Y ν ( ky ) (cid:21) , (A.6) g k ( τ ) = (cid:115) k (cid:112) ky (cid:20)(cid:18) A + + A − (cid:19) J ν − ( ky ) + i (cid:18) A − − A + (cid:19) Y ν − ( ky ) (cid:21) . (A.7)Equations (A.6) and (A.7) can now be referred to the electric and magnetic mode functions of Eqs. (3.1)and (3.2)–(3.3) evaluated at the end of inflation i.e. f k = N µ √ k √ x H (1) µ ( x ) , g k = (cid:115) k N µ √ x H (1) µ +1 ( x ) , γ > / ,f k = N µ √ k √ x H (1) µ ( x ) , g k = − (cid:115) k N µ √ x H (1) µ − ( x ) , < γ < / , (A.8) The inflationary mode functions depend on the range of γ (see, in in this respect, Eqs. (3.1) and (3.2)–(3.3)). Thereforethat Eq. (A.4) only holds for γ > / < γ < / f k = f k ( − τ ) and g k = g k ( − τ ). Since x = kτ < (cid:12)(cid:12)(cid:12)(cid:12) kf k g k (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) H (1) | γ − / | ( x ) H (1) γ +1 / ( x ) (cid:12)(cid:12)(cid:12)(cid:12) → Γ( | γ − / | )Γ( γ + 1 / (cid:18) x (cid:19) γ +1 / −| γ − / | . (A.9)The values of f k ( τ ) and g k ( τ ) for τ ≥ − τ will then be expressible in terms of f k and g k in the followingmanner (cid:18) f k ( τ ) g k ( τ ) /k (cid:19) = (cid:18) A f f ( k, τ, τ ) A f g ( k, τ, τ ) A g f ( k, τ, τ ) A g g ( k, τ, τ ) (cid:19) (cid:18) f k g k /k (cid:19) . (A.10)The entries of the matrix appearing at the right hand side of Eq. (A.10) have been given in Eq. (4.2);they follow directly from Eqs. (A.6)–(A.7) once the various coefficients are made explicit in terms of Eqs.(A.4) and (A.5). B Transition matrix and decreasing gauge coupling
When the gauge coupling decreases Eqs. (2.41) and (2.42) describe the evolution of √ λ and of √ λ (cid:48) interpolating between the inflationary stage and the subsequent radiation epoch (see also Fig. 2). In fullanalogy with Eqs. (A.1) and (A.2) the mode functions for τ ≥ − τ are determined in terms of a set of newcoefficients (cid:101) A ± ( k, τ ): f k ( τ ) = √ ky √ k (cid:20) (cid:101) A − ( k, τ ) N (cid:101) ν H (1) (cid:101) ν ( ky ) + (cid:101) A + ( k, τ ) N ∗ (cid:101) ν H (2) (cid:101) ν ( ky ) (cid:21) , (B.1) g k ( τ ) = − (cid:115) k (cid:112) ky (cid:20) (cid:101) A − ( k, τ ) N (cid:101) ν H (1) (cid:101) ν +1 ( ky ) + (cid:101) A + ( k, τ ) N ∗ (cid:101) ν H (2) (cid:101) ν +1 ( ky ) (cid:21) , (cid:101) δ > , (B.2) g k ( τ ) = (cid:115) k (cid:112) ky (cid:20) (cid:101) A − ( k, τ ) N (cid:101) ν H (1) (cid:101) ν − ( ky ) + (cid:101) A + ( k, τ ) N ∗ (cid:101) ν H (2) (cid:101) ν − ( ky ) (cid:21) , < (cid:101) δ < / , (B.3)where, in analogy with Eq. (A.3), the following auxiliary quantities have been introduced: N (cid:101) ν = (cid:114) π e iπ (2 (cid:101) ν +1) / , y ( τ ) = τ + τ (1 + q ) ,q = q ( (cid:101) δ , (cid:101) γ ) = (cid:101) δ (cid:101) γ , (cid:101) ν = (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) δ − (cid:12)(cid:12)(cid:12)(cid:12) . (B.4)The expression of (cid:101) ν now depends on the range of (cid:101) δ . More precisely for (cid:101) δ > / (cid:102) A − ( k, τ ) = − iπN (cid:101) µ N (cid:101) ν √ qx (cid:20) H (1) (cid:101) µ ( x ) H (2) (cid:101) ν +1 ( q x ) − H (1) (cid:101) µ − ( x ) H (2) (cid:101) ν ( q x ) (cid:21) , (cid:102) A + ( k, τ ) = − iπN (cid:101) µ N ∗ (cid:101) ν √ qx (cid:20) H (1) (cid:101) µ − ( x ) H (1) (cid:101) ν ( q x ) − H (1) (cid:101) µ ( x ) H (1) (cid:101) ν +1 ( q x ) (cid:21) , (cid:101) δ > / , (B.5)where, as in Eqs. (A.4)–(A.5), x = kτ . In the range 0 < (cid:101) δ < / (cid:102) A ± ( k, τ ) are: (cid:102) A − ( k, τ ) = iπN (cid:101) µ N (cid:101) ν √ qx (cid:20) H (1) (cid:101) µ ( x ) H (2) (cid:101) ν − ( q x ) + H (1) (cid:101) µ − ( x ) H (2) (cid:101) ν ( q x ) (cid:21) , (cid:102) A + ( k, τ ) = − iπN (cid:101) µ N ∗ (cid:101) ν √ qx (cid:20) H (1) (cid:101) µ − ( x ) H (1) (cid:101) ν ( q x ) + H (1) (cid:101) µ ( x ) H (1) (cid:101) ν − ( q x ) (cid:21) , ≤ (cid:101) δ < / . (B.6)30s in the case of the results of appendix A, because of the Wronskian normalization of Eq. (2.21), themixing coefficients of Eqs. (B.5)–(B.6) must satisfy | (cid:101) A + ( k, τ ) | − | (cid:101) A − ( k, τ ) | = 1. We can thereforesummarize the situation by saying that while A ± ( k, τ ) of Eqs. (A.4)–(A.5) depend upon the range of γ ,the (cid:102) A ± ( k, τ ) of Eqs. (B.5)–(B.6) depend on the range of (cid:101) δ . Inserting the expressions of the coefficients(B.5) and (B.6) back into Eqs. (B.1), (B.2) and (B.3) the values of f k ( τ ) and g k ( τ ) for different valuesof (cid:101) δ can then be referred to the inflationary mode functions at τ = − τ directly obtainable from Eqs.(3.17)–(3.18): f k = N (cid:101) µ √ k √ x H (1) (cid:101) µ ( x ) , g k = − N (cid:101) µ (cid:115) k √ x H (1) (cid:101) µ − ( x ) , (B.7)where (cid:101) µ = (cid:101) γ + 1 /
2. As in the case of Eq. (A.9) the ratio of the mode functions of Eq. (B.7) can alwaysbe evaluated in the small argument limit and and the result is: (cid:12)(cid:12)(cid:12)(cid:12) kf k g k (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) H (1) (cid:101) γ +1 / ( x ) H (1) | (cid:101) γ − / | ( x ) (cid:12)(cid:12)(cid:12)(cid:12) → Γ( (cid:101) γ + 1 / | (cid:101) γ − / | ) (cid:18) x (cid:19) | (cid:101) γ − / |− (cid:101) γ − / . (B.8)The magnetic and the electric mode functions for τ > − τ can therefore be related to f k and g k of Eqs.(B.7) as: (cid:18) f k ( τ ) g k ( τ ) /k (cid:19) = (cid:18) (cid:102) A f f ( k, τ, τ ) (cid:102) A f g ( k, τ, τ ) (cid:102) A g f ( k, τ, τ ) (cid:102) A g g ( k, τ, τ ) (cid:19) (cid:18) f k g k /k (cid:19) , (B.9)where the explicit form of the various coefficients appearing at the right hand side of Eq. (B.9) has beenreported in Eq. (4.29) which is the analog of Eq. (4.2). C Transition matrix and duality transformations
The transition matrices of Eqs. (A.10) and (4.29) are related by duality transformations. As it can beexplicitly verified from Eqs. (2.33)–(2.34) and (2.41)–(2.42) a duality transformation implies that ( γ, δ )are transformed into ( (cid:101) γ , (cid:101) δ ): √ λ → / √ λ ⇒ γ → (cid:101) γ , δ → (cid:101) δ . (C.1)It has been shown [see Eq. (3.28) and discussion therein] that during inflation the transformation γ → (cid:101) γ exchanges electric into magnetic power spectra and vice versa. We shall now demonstrate that afterinflation the corresponding duality transformation exchanges the elements of the transition matrices (A.10)and (4.29). Let us first start by noting that when δ → (cid:101) δ the corresponding Bessel indices ν and (cid:101) ν arerelated as: δ → (cid:101) δ ⇒ ν → − (cid:101) ν . (C.2)Equation (C.2) is a consequence of the definitions of ν ( δ ) = δ +1 / (cid:101) ν ( (cid:101) δ ) = | (cid:101) δ − / | given, respectively,in Eqs. (4.4) and (4.30). By now applying the transformation Eq. (C.2) to the matrix elements of M ( δ, x , x ) and (cid:103) M ( (cid:101) δ , x , x ) the following transformations are easily deduced: A f f → (cid:102) A g g , A g g → (cid:102) A f f , (C.3) A f g → − (cid:102) A g f , A g f → − (cid:102) A f g . (C.4) Equations (4.29) and (4.2) do not coincide since ν and (cid:101) ν are different. Under duality, the elements of the transition matriceshave a well defined transformation law given in Eq. (4.48) (see also appendix C hereunder).Along a similar perspective, unlikethe case of Eqs. (3.2) and (3.3), the expression of g k ( τ ) is unique for all the range of (cid:101) γ >
0; this occurrence follows from thedifference between µ and (cid:101) µ . A f f ( δ, x , x ) which we write, for immediate convenience, in terms of Hankel functions of first and secondkinds: A f f ( δ, x , x ) = i √ qx (cid:112) ky (cid:20) H (1) ν ( k y ) H (2) ν − ( qx ) − H (1) ν − ( qx ) H (2) ν ( k y ) (cid:21) . (C.5)By now performing the transformation γ → (cid:101) γ and δ → (cid:101) δ (i.e. ν → − (cid:101) ν ) we obtain, from Eq. (C.5), A f f ( (cid:101) δ , x , x ) = i √ qx (cid:112) ky (cid:20) H (1)1 − (cid:101) ν ( k y ) H (2) − (cid:101) ν ( qx ) − H (1) − (cid:101) ν ( qx ) H (2)1 − (cid:101) ν ( k y ) (cid:21) . (C.6)It is understood that while doing the transformation also the arguments of q and y will change accordingto Eqs. (4.4) and (4.30): q ( δ, γ ) = δ/γ → (cid:101) δ / (cid:101) γ = q ( (cid:101) δ , (cid:101) γ ) , (C.7)and similarly for y . We now recall that the Hankel functions of generic index µ and generic argument z obey [42, 43] H (1) − µ ( z ) = e iπµ H (1) µ ( z ) , H (2) − µ ( z ) = e − iπµ H (2) µ ( z ) . (C.8)Inserting Eq. (C.8) into Eq. (C.6) we therefore obtain A f f ( (cid:101) δ , x , x ) = − i √ qx (cid:112) ky (cid:20) H (1) (cid:101) ν − ( k y ) H (2) (cid:101) ν ( qx ) − H (1) (cid:101) ν ( qx ) H (2) (cid:101) ν − ( k y ) (cid:21) , (C.9)which coincides with (cid:102) A g g ( (cid:101) δ , x , x ), as anticipated in Eq. (C.3). With a similar procedure also theremaining transformation rules of Eqs. (C.3) and (C.4) can be easily deduced.32 eferences [1] A. Lichnerowicz, Magnetohydrodynamics: waves and shock waves in curved space-time , (Kluwer aca-demic publisher, Dordrecht, 1994).[2] S. Deser and C. Teitelboim, Phys. Rev. D , 1592 (1976).[3] S. Deser, J. Phys. A , 1053 (1982).[4] L. Parker, Phys. Rev. Lett. , 562 (1968).[5] N. D. Birrell and P. C. W. Davies, Quantum fields in curved space , (Cambridge University Press,Cambridge, UK, 1982).[6] L. Parker and D. Toms,
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