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Primordial magnetic fields from pseudo-Goldstone bosons

arXiv:hep-ph/9209238v1 15 Sep 1992Primordial magnetic fields from pseudo-Goldstone bosonsW. Daniel Garretson, George B.

Field and Sean M. CarrollHarvard-Smithsonian Center for Astrophysics60 Garden Street, Cambridge, Massachussetts 02138 USA(Received 21 July 1992)The existence of large-scale magnetic fields in galaxies is well established, butthere is no accepted mechanism for generating a primordial field which could growinto what is observed today. We discuss a model which attempts to account for thenecessary primordial field by invoking a pseudo-Goldstone boson coupled to electro-magnetism.

The evolution of this boson during inflation generates a magnetic field;however, it seems difficult on rather general grounds to obtain fields of sufficientstrength on astrophysically interesting scales.98.80.Cq, 98.60.Jk, 95.30.Cq, 14.80.GtTypeset Using REVTEX1

I. INTRODUCTIONThe existence of a magnetic field of ∼10−6 gauss in our galaxy and other galaxies is wellestablished [1]. The explanation of how such a magnetic field arose is, however, far fromcertain.

While the creation and evolution of stellar magnetic fields is fairly well understood,the extension of these theories to galaxies suffers from problems relating to both scale lengthand time scales.Zeldovich, Ruzmaikin, and Sokoloff[1] and Parker [2] discuss the origin and effects ofmagnetic fields in the universe and draw the conclusion that the galactic field arises froma dynamo mechanism.The dynamo model requires a seed field at the epoch of galaxyformation which is coherent over a scale of ∼1 Mpc. We can parameterize the strengthof a primordial field by r = ρB/ργ, the ratio of the energy density ρB= B2/8π in themagnetic field relative to that of the background radiation ργ.

(This ratio is constant whilethe universe is a good conductor, which is almost always [3].) Then the field required toseed a galactic dynamo satisfies r >∼10−31, corresponding to an intergalactic field at theepoch of galaxy formation (z ∼3–5) of >∼10−20 gauss.

The implications of this requirementin terms of the origin of such fields and their possible effect on star formation (or the earlyhistory of the galaxy in general) are discussed by Rees [4].Kulsrud [5] has argued that the galactic dynamo explanation is fundamentally flawedin that the mean square deviations of the magnetic field will grow much faster than themean field itself, resulting in a disordered field with a much smaller mean field strengththan would naively be expected. Kulsrud then discusses the possibility that the magneticfield originated in the early universe and was embedded in the medium out of which thegalaxies ultimately formed.

In this case, Kulsrud argues that the intergalactic field wouldneed to be >∼10−12 gauss at the epoch of galaxy formation (giving r >∼0−15) in order toaccount for the observed interstellar field.Any dynamo theory requires a mechanism for generating the required seed field; how-ever, compelling mechanisms have been elusive. The hot plasma in the early universe is2

highly conducting and thus should strongly inhibit the growth of a magnetic field, even tor ∼10−31. Furthermore, any hypothetical process at work in the very early universe mustbe able to produce fields with characteristic length scales much larger then the horizon atthat time, in order to correspond to galactic scales today.The advent of inflation [6] has opened the door to new possibilities for generating aprimordial magnetic field.

There are two key features of an inflationary universe that makethe possibility of creating a magnetic field during this time particularly attractive. First,if there were an inflationary epoch at very early times in the universe, the exponentialexpansion would have reduced the conductivity to a negligible value by reducing the chargedparticle density, thus allowing the creation of a substantial magnetic field.

If this field werethen frozen into the plasma created during the subsequent reheating of the universe, it willbe supported by the effectively infinite conductivity of the plasma so that its strength willdecay only as inverse square of the scale factor.Second, if inflation did occur, then the entire observable universe today was, at somepoint in the early universe, contained entirely within the particle horizon. It would thenbe possible to use physical mechanisms operating on a scale smaller than the horizon togenerate magnetic fields that are coherent over macroscopic scales today, an opportunitywhich is not available in models of the early universe without inflation.Turner and Widrow [3] (TW) investigated the possibility that quantum fluctuationsduring an inflationary epoch might have generated a magnetic field that could be sustainedafter the wavelength of interest crossed beyond the horizon and thus give the observed fieldtoday.

TW considered coupling the electromagnetic field to the curvature tensor so as toamplify the fluctuation-induced field, but found satisfactory results could be obtained onlyat the expense of breaking gauge invariance.Ratra [7] has argued that it is possible to generate a magnetic field with present fieldstrength of >∼10−10 gauss on a scale of 1/1000 the present Hubble radius by coupling ascalar field Φ to the electromagnetic potential Aµ through a term of the form eΦFµνF µν,where Fµν = ∂µAν −∂νAµ is the electromagnetic field strength. This would be sufficient to3

explain the galactic magnetic field, but it remains to be seen whether or not this couplingcould arise naturally in realistic particle physics models.TW also suggested that the magnetic field could be sustained by coupling the EM fieldstrength to a pseudoscalar axion field φ via an interaction term in the Lagrange density ofthe formL = gφγγ4 φFµν eF µν ,(1.1)whereeF µν ≡12ǫµνρσFρσ is the dual of Fµν, and gφγγ = (α/2π)/f, where f is a couplingconstant with units of mass and α is the fine structure constant. However, they did notcomplete the necessary analysis to show whether or not this might indeed be the case.An interaction of this form has been studied by Carroll and Field [8] (see also [9]), whofound that the evolution of a Fourier mode of the magnetic field with wavenumber k isgoverned byd2F±dη2 + k2 ± gφγγkdφdη!F± = 0 ,(1.2)where F± = a2(By ± iBz) (the ± refers to different circular polarization modes of themagnetic field), dη = dt/a, and a is the scale factor of the universe (normalized so thata0 = 1 where a0 is the value of a today).

One (or both) of the polarization modes will beunstable for k < gφγγ|dφ/dη|, where both polarization modes can be unstable to exponentialgrowth if φ is oscillatory. Thus, if such a scalar field exists during inflation (perhaps theinflaton itself [10]) with the above coupling, this might provide a mechanism for generatinga substantial magnetic field.Here we will consider a generalization of the possibility suggested by TW [3], couplingthe photon to an arbitrary pseudo-Goldstone boson (PGB) rather than the axion of QCD.The PGB is characterized by a spontaneous symmetry breaking scale f (as above) and asoft explicit symmetry breaking scale Λ (see Sec.

II). We find that significant growth occursonly at a temperature near Λ, and that the magnetic field strength thus generated cannotgive an astrophysically interesting field at the end of inflation.4

We should point out that, while our notation throughout this paper suggests that we areworking with the photon of the standard U(1)em symmetry, the photon as a separate U(1)gauge boson will not exist at high temperature, since the SU(2) × U(1) gauge symmetrywill not have been broken.Nevertheless, our results should be correct up to factors oforder unity simply because the U(1) hypercharge symmetry projects onto the photon witha multiplicative factor of cos θw ≈0.88 at the electroweak phase transition.We will use units in which ¯h = c = kB = 1, such that G = m−2pl , where mpl ≈1.22 × 1019GeV is the Planck mass.II. THE SET-UPIn this section we briefly review the essentials of inflation, as well as the physics ofpseudo-Goldstone bosons and their couplings.Throughout this paper, we will assume that the universe is in a spatially flat FRWcosmology in which the metric is given byds2 = a2(η)(−dη2 + dx2) ,(2.1)where x represents the standard Cartesian 3-space (comoving) coordinates.

In addition, wewill assume that the universe is a perfect fluid with the equation of state p = γρ, wherep is the pressure, ρ is the total energy density, and γ is a constant. Using this equationand energy-momentum conservation, it is straightforward to show that ρ ∝a−3(1+γ) which,from Einstein’s equation, gives a ∝t2/3(1+γ).

In order to explain the horizon problem, werequire that, at some time in the past, the scale factor was growing faster than the horizon(H−1, where H = ˙a/a is the Hubble parameter and an overdot denotes differentiation withrespect to physical time t). Thus, we require −1 ≤γ < −13.

For simplicity in the followingdiscussion we will restrict ourselves to inflation in which γ = −1. This value for γ gives thebest possible conditions for generating a magnetic field, since the amount of inflation fromthe time that a given comoving wavelength crosses outside the horizon is minimal in thiscase, so this does not limit the validity of our results.5

At the end of inflation, the universe enters a reheating phase, in which the energy densityis matter-dominated. As a simplifying assumption, we take the process of reheating to beinstantaneous, such that the universe goes directly from inflation to radiation domination.Once again this is a best-possible assumption, since the magnetic field will decay morerapidly (relative to the total energy density) during a matter-dominated phase, in whichρ ∝a−3.Standard inflation is characterized by two parameters: the mass scale for the total energydensity M = ρ1/4 (note that this is a constant since ρ is constant during inflation withγ = −1); and the temperature TRH to which the universe reheats at the end of inflation.

His then given byH2 =4π3m2plM4 . (2.2)If we assume that the universe expands adiabatically after inflation so that the entropy percomoving volume element remains constant, it can be shown (see e.g.

[11]) that the totalexpansion from the time a given comoving wavelength λ (which is equal to the physicalwavelength today due to the normalization of a) crosses outside the horizon until the endof inflation is given byainfaλ≃1026λMpc M2TRHm3pl! 13,(2.3)where ainf is the value of a at the end of inflation (but before reheating), and aλ is the valueof a at the time λ crosses outside the horizon.

Expressing this in terms of the number ofe-foldings Nλ (ainf/aλ = eNλ) in the expansion, we haveNλ ≃48 + ln λMpc!+ 23 lnM1014 GeV+ 13 lnTRH1014 GeV. (2.4)Our assumption that reheating lasts for a negligible time amounts to setting M = TRH.In this paper we are concerned with the pseudo-Goldstone boson φ of a spontaneouslybroken symmetry.

PGBs are characterized by two mass scales: a large mass f at which theglobal symmetry from which the PGBs arise is spontaneously broken, and a smaller scale6

Λ at which the symmetry is explicitly broken. For concreteness, we imagine the breakdownof a global U(1) symmetry, resulting in the familiar Mexican hat — the radial degree offreedom gets a vacuum expectation value of order f, and the angular degree of freedombecomes a massless boson φ.

The hat is tilted by a small term of order Λ; the formerlymassless scalar φ becomes a PGB with mass of orderm ≈Λ2/f . (2.5)Cosmological constraints on the parameters f and Λ have been studied in [12].In many models, PGB’s interact with fermions by coupling to the axial vector current(for a review of PGB’s and their couplings, see [13]):Lint = 14f¯ψγµγ5ψ∂µφ = 14f Jµ5 ∂µφ ;(2.6)well-known examples include pions and axions.

Since the symmetry associated with the axialcurrent (that is, chiral symmetry) is anomalously broken, the current itself is not conserved:∂µJµ5 = α2πFµν eF µν ,(2.7)where α is the fine structure constant. Integrating by parts, we find that the anomaly (2.7)induces a coupling between φ and the electromagnetic field of the form (1.1).A similar situation occurs in the low-energy limit of string theory, which involves anantisymmetric two-index tensor field Bµν [13,14].

The Lagrangian for Bµν includes a kineticterm HµνρHµνρ, where Hµνρ is an antisymmetric field strength tensor. The demand thatthe theory be free of anomalies requires that the definition of Hµνρ include a term involvinggauge bosons (which we take to be abelian for simplicity):Hµνρ = ∂[µBνρ] −A[µFνρ] ,(2.8)where square brackets denote antisymmetrization.

In four dimensions the equations of mo-tion for Bµν allow us to recast its dynamics (at least semiclassically) in terms of a pseu-doscalar φ by the identification7

∂µφ = 1f ǫµαβγHαβγ . (2.9)The interaction between this pseudoscalar and electromagnetism, as implied by (2.8) and(2.9), can be described by an effective Lagrangian of the form (1.1).

Thus, string theoryoffers the possibility of PGB’s of the type we discuss.III. BASIC REQUIREMENTSWe envision a scenario in which the desired magnetic field is entirely created duringinflation and then frozen into the post-reheat plasma, so that the field strength decays asthe inverse square of the scale factor after reheating.

Thus, under the assumption that thedominant component of background photons is created at reheating, both ρB and ργ willdecay as a−4, such that r = ρB/ργ is essentially constant after inflation. As mentioned inthe introduction, we seek r = r0 >∼10−31 (to seed a galactic dynamo) or >∼10−15 (to directlyaccount for the observed magnetic field) at scales of ∼1 Mpc today.For convenience we define r during inflation as the ratio of ρB to the total (vacuum-dominated) energy density.

From above, we can readily calculate the required value of r atthe time that the wavelength of interest (comoving scale 1 Mpc) crosses outside the horizonin order to generate astrophysically interesting fields today. This is given byrMpc = ainfaMpc!4r0 ,(3.1)or, using (2.3),rMpc = 10104 Mmpl!4r0 .

(3.2)Since r < 1, we can use this to calculate the maximum allowed value for M such thatthere would be any hope for meeting the above condition. Using the values given abovefor r0, this gives M <∼10 GeV for a galactic dynamo, or <∼1 MeV to directly account forthe field.

It is important to realize that these are extreme upper bounds that can only beobtained under ideal situations (i.e. the magnetic field energy being comparable to that in8

the vacuum at the time the wavelength of interest crosses the horizon, and with negligiblereheating). Even so, the upper bound for seeding the galactic dynamo is, at best, marginal(that is, while current constraints on the time at which inflation might occur could allow itto occur as late as a temperature of ∼1 GeV since we merely require the universe to beradiation dominated by the epoch of primordial nucleosynthesis [3], constraints arising frombaryogenesis will probably require M >∼200 GeV, at least).

Thus, even if the magnetic fieldhas an energy density comparable to the background energy density when 1 Mpc crossesthe horizon during inflation, it will be too weak to explain the observed magnetic field if itsenergy decays according to the normal a−4.In other words, in order to generate a significant magnetic field during inflation, werequire “superadiabatic growth” — that is, a mechanism that will continue to increase theenergy density (or, at least, decrease the rate of decay) of the magnetic field at a wavelengthof 1 Mpc after 1 Mpc becomes super-horizon sized. At first sight, this may seem impossibledue to the fact that such a mechanism must apparently act in an acausal way.

However, itis possible that inflation may create a field that is coherent over scales much larger than thehorizon, and that this field can subsequently generate a magnetic field that is also coherentat super-horizon scales simply by classical field interactions.IV. EQUATIONS OF MOTIONThe Lagrange density for the photon and scalar field isL = −√g12∇µφ∇µφ + V (φ) + 14FµνF µν + gφγγ4 φFµν eF µν,(4.1)where g = −det(gµν) and ∇µ denotes the covariant derivative.

As mentioned previously, weare considering only the U(1) fields and ignoring any possible effects from the nonabeliangauge fields.The equations of motion for φ are−∇µ∇µφ + dV (φ)dφ= −gφγγ4 Fµν eF µν ,(4.2)9

and the equations of motion for Fµν are∇µF µν = −gφγγ(∇µφ) eF µν ,(4.3)along with the Bianchi identity∇µ eF µν = 0 . (4.4)The equations of motion are more transparent if we define the E and B fields byF µν = a−20ExEyEz−Ex0Bz−By−Ey−Bz0Bx−EzBy−Bx0.

(4.5)Then (4.2) becomes, after expanding the covariant derivatives,∂2φ∂η2 + 2aH ∂φ∂η −∇2φ + a2dV (φ)dφ= gφγγa2E · B ,(4.6)where ∇represents the usual 3-space gradient (for comoving coordinates). Similarly, (4.3)becomes∂∂η(a2E) −∇× (a2B) = −gφγγ∂φ∂η a2B −gφγγ(∇φ) × a2E ,(4.7)with∇· E = −gφγγ(∇φ) · B ,(4.8)while the Bianchi identity becomes∂∂η(a2B) + ∇× (a2E) = 0 ,(4.9)with∇· B = 0 .

(4.10)Since we are interested in the specific case where the background space-time is inflating,we make the assumption that the spatial derivatives of φ are negligible compared to the10

other terms (if this is not the case at the beginning of inflation, any spatial inhomogeneitieswill quickly be inflated away and this assumption will quickly become very accurate). Then,eliminating E in the above equations, we have ∂2∂η2 −∇2 −gφγγdφdη ∇×!

(a2B) = 0 . (4.11)Taking the spatial fourier transform of this equation so thatB(η, k) = 12πZeik·xB(η, x)d3x ,(4.12)and writing F = a2B, we then have∂2F∂η2 + k2F −gφγγdφdη ik × F = 0 .

(4.13)Finally, if we take k to point along the x-axis and define F± = Fy ± iFz, this becomes∂2F±∂η2 + k2 ± gφγγdφdη k!F± = 0 . (4.14)We can similarly manipulate the equations in an attempt to produce an expression for theevolution of E. It turns out, however, that we cannot uncouple the E field from the B field.We have, in short, ∂2∂η2 −∇2 −gφγγdφdη ∇×!

(a2E) = −gφγγd2φdη2 a2B ,(4.15)which, after taking the space fourier transform and defining G = a2E and G± = Gy ± iGz,becomes∂2G±∂η2 + k2 ± gφγγdφdη k!G± = −gφγγd2φdη2 F± . (4.16)In order to determine the evolution of E and B, we need to know how φ evolves.

We lookfirst at the case where E and B make a negligible contribution to the equation of motion forφ. Furthermore, we will consider the evolution after the time when the explicit symmetrybreaking for the PGB becomes important (at a temperature scale Λ).

The potential for theangular degree of freedom in a tilted Mexican hat is11

V (φ) = Λ4[1 −cos(φ/f)] . (4.17)Since the details of the potential do not affect our results, we will expand to lowest order:V (φ) ∼(Λ4/2f 2)φ2.

We again ignore spatial derivatives in φ, as well as any back reactionfrom the electromagnetic fields, to gived2φdt2 + 3H dφdt + Λ4f 2 φ = 0 ,(4.18)where we have used dη = dt/a to write this in terms of physical time rather than conformaltime. The general solution to (4.18) will be approximatelyφ(t) ≈f exp"−12 3H ±s9H2 −4Λ4f 2!

(t −t0)#∝a−3/2 sinΛ2f tfor Λ2f ≫H,exp−Λ43Hf2tfor Λ2f ≪H,(4.19)where we have used the fact that a ∝exp Ht.Looking again at (4.14) we see that we will only have a growing mode for the magneticfield ifdφdη = adφdt > k . (4.20)Furthermore, since we want this growth to be as large as possible, we will choose k such thatk ∼gφγγa ˙φmax at the time of interest.

For our initial analysis, we will assume Λ2/f ≫H sothat φ is oscillating rapidly compared to changes in a. Also, this implies that a is essentiallyconstant over several oscillations in φ, and thus we can write ∆η ≈∆t/a where a is constantfor time intervals ∆t ∼f/Λ2.In order to estimate the total growth in F±, we note that, for a fraction ǫ (where ǫ is notnecessarily small, although numerical integration of these equations for some cases indicatesthat ǫ ∼0.1) of each period, we can write F± ∝eα∆η where α =qgφγγkdφ/dη ∼agφγγΛ2 and∆η ≈∆t/a ∼ǫf/aΛ2.

Furthermore, this will continue for a time ∼H−1 (since this is thetime scale on which a and the amplitude of φ are changing), or for a total of n ∼H−1Λ2/foscillations, from which we can estimate the total growth in F± as12

F±,fF±,i∼exp ǫgφγγΛ2H!. (4.21)Of course, the exponent may contain other factors of order unity, but this estimate allowsus to understand the dependence of the growth in the magnetic field on the parameters inthe problem.

Note that the analysis is similar if Λ2/f ∼H, but then ∆η ≈∆t/a where ais constant is only true for time intervals ∆t <∼H−1. So using ∆η ∼H−1/a, we see that weget exponential growth with the same expression in the exponent, but now this exponent ismuch smaller.Since we are interested specifically in long wavelength magnetic fields, we would also liketo know the wavelength at which we get the most growth by this mechanism.

This followsfrom our assumption that the maximum growth occurs at k ≈agφγγ ˙φmax ≈agφγγΛ2, whichgivesλ ≈2πagφγγΛ2 ,(4.22)where λ is the wavelength today. More importantly, at the time the growth occurs the ratioof the wavelength to the horizon length is given byaλH−1 = 2πaHk≈2πHgφγγΛ2 .

(4.23)But, looking back at (4.21), we see that this ratio is, essentially, just the inverse of the factorin the exponent, i.e. the larger the growth in the magnetic field, the smaller the wavelengthat which it occurs!

Further, since an increase in the wavelength by a factor β will result ina decrease in the exponent for the growth by a factor β−1/2, it is apparent that significantgrowth in the magnetic field will only occur for wavelengths that are sub-horizon sized.We have shown that we cannot have growth in the magnetic field for super-horizon sizedwavelengths when the back effects of the E and B fields on φ are small, but there is stillthe possibility that the interaction between the fields and φ could allow the magnetic fieldto be sustained such that it does not decrease as a−2. However, from (4.7) and (4.8), wesee that, if φ decays as a−3/2 (that is, φ behaves as a non-relativistic fluid), then the RHS’s13

of these equations will rapidly become negligible compared to the individual terms on theLHS’s. We then recover the source-free Maxwell equations, from which B still decreases asa−2.

(The situation is only exacerbated if φ behaves as a relativistic fluid.) In short, thereseems to be no way of sustaining the magnetic field at super-horizon sized wavelengths.V.

SUMMARYIn this paper, we have considered a mechanism for creating a large scale magnetic fieldduring inflation, proposed by TW, in which the magnetic field is coupled to a pseudo-Goldstone boson. We showed first that the scale required of the magnetic field in order toexplain the galactic magnetic field (∼1 Mpc) implied that the growth had to occur at super-horizon sized wavelengths since the uncoupled equations of motion for a U(1) gauge fieldimply that the energy density in the field would simply decay too quickly to be significantat the end of inflation if there was no enhancement in the field for wavelengths larger thanthe horizon.We then considered the classical evolution of a U(1) gauge field coupled to a PGB whenthe back-effects of the field on the PGB were negligible and showed that such a couplingcan, in fact, produce growth in the field.

However, such growth can occur only at sub-horizon wavelengths, and thus does not provide a solution to the above problem. In themore general case when we allow the back-effects of the U(1) field to be significant, we stillhave the problem that any natural decay of the φ field ultimately allows the U(1) field touncouple from the PGB leaving us once again with a free U(1) field.

Hence, it seems to beimpossible to create a significant magnetic field by simply coupling the magnetic field to aPGB during inflation.ACKNOWLEDGEMENTSWe thank Bill Press for helpful discussions. Support for this work was provided by NASAunder grants NAGW-931 and NGT-50850, and by the National Science Foundation.14

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