We study the amplification of electromagnetic vacuum fluctuations induced by
the evolution of scalar metric perturbations at the end of inflation. Such
perturbations break the conformal invariance of Maxwell equations in
Friedmann-Robertson-Walker backgrounds and allow the growth of magnetic fields
on super-Hubble scales. We relate the strength of the fields generated by this
mechanism with the power spectrum of scalar perturbations and estimate the
amplification on galactic scales for different values of the spectral index.
Finally we discuss the possible effects of finite conductivity during
reheating.Comment: 10 pages, 1 figure. Final version to appear in Phys. Rev. 

Maroto, Antonio L.

English

arXiv

We study the amplification of electromagnetic vacuum fluctuations induced by the evolution of scalar metric perturbations at the end of inflation. Such perturbations break the conformal invariance of Maxwell equations in Friedmann-Robertson-Walker backgrounds and allow the growth of magnetic fields on super-Hubble scales. We estimate the strength of the fields generated by this mechanism on galactic scales and compare the results with the present bounds on the galactic dynamo seed fields.We study the amplification of electromagnetic vacuum fluctuations induced by the evolution of scalar metric perturbations at the end of inflation. Such perturbations break the conformal invariance of Maxwell equations in Friedmann-Robertson-Walker backgrounds and allow the growth of magnetic fields on super-Hubble scales. We relate the strength of the fields generated by this mechanism with the power spectrum of scalar perturbations and estimate the amplification on galactic scales for different values of the spectral index. Finally we discuss the possible effects of finite conductivity during reheating

Maroto, Antonio Lopez

CERN Document Server

CERN-TH/2000-258Primordial magnetic fields from metric perturbationsAntonio L. MarotoCERN Theory Division,CH-1211 Geneva 23, Switzerland,andDept. F´ısica Teo´rica,Universidad Complutense de Madrid,28040 Madrid, SpainABSTRACTWe study the amplification of electromagnetic vacuum fluctuations induced by the evolutionof scalar metric perturbations at the end of inflation. Such perturbations break the conformalinvariance of Maxwell equations in Friedmann-Robertson-Walker backgrounds and allow thegrowth of magnetic fields on super-Hubble scales. We relate the strength of the fields generatedby this mechanism with the power spectrum of scalar perturbations and estimate the amplifica-tion on galactic scales for different values of the spectral index. Finally we discuss the possibleeffects of finite conductivity during reheating.1 IntroductionThe existence of cosmic magnetic fields with large coherence lengths (> 10 kpc) and typicalstrength of 10−6 G, still remains an open problem in astrophysics [1]. A partial explanation,widely considered in the literature, is based on the amplification of seed fields by means ofthe so called galactic dynamo mechanism. In this mechanism, the differential rotation of thegalaxy is able to transfer energy into the magnetic field, but nevertheless it still requires apre-exisiting field to be amplified. The present bounds on the necessary seed fields to complywith observations are in the range Bseed >∼ 10−17 − 10−22 G (h = 0.65 − 0.5) at decouplingtime, coherent on a comoving scale of λG ∼ 10 kpc, for a flat universe without cosmologicalconstant. For a flat universe with nonvanishing cosmological constant, the limits can be relaxedup to Bseed >∼ 10−25 − 10−30 G (h = 0.65− 0.5) at decoupling for ΩΛ = 0.7 and Ωm = 0.3 [2].The observations of micro-Gauss magnetic fields in two high-redshift objects (see [1, 2] andreferences therein) could, if correct, impose more stringent conditions on the seeds fields oreven on the dynamo mechanism itself.The cosmological origin of the seed fields is one of the most interesting possibilities, althoughsome other mechanisms at the astrophysical level, such as the Biermann battery process, havealso been considered [3, 4]. In the cosmological case, in which we will be mainly interestedin this work, it is natural to expect [5] that the same mechanism that gave rise to the large-scale galactic structure, i.e. amplification of quantum fluctuations during inflation, was alsoresponsible for the generation of the primordial magnetic fields. However, it was soon noticed[5] that the gravitational amplification does not operate in the case of electromagnetic (EM)fields. This is because of the conformal triviality of Maxwell equations in Friedmann-Robertson-Walker (FRW) backgrounds, i.e. conformally invariant equations in a conformally flat space-time. In order to avoid this difficulty, several production mechanisms have been proposed inwhich Maxwell equations are modified in different ways. Thus for example, the addition ofmass terms to the photon or higher-curvature terms in the Lagrangian was studied in [5]. Thecontribution of the conformal anomaly was included in [6]. In the context of string cosmology,the effects of a dynamical dilaton field were taken into account in [7]. Other examples includenon-minimal gravitational-electromagnetic coupling [8], inflaton coupling to EM Lagrangian [9],spontaneous breaking of Lorentz invariance [10] or backreaction of minimally coupled chargedscalars [11, 12, 13]. Some of them are able to generate fields of the required strength to seedthe galactic dynamo or even to account for the observations without further amplification.In this paper we explore the alternative possibility, i.e. we avoid conformal triviality by con-sidering deviations from the FRW metric (see [14] for a suggestion along these lines). This ap-proach is rather natural since we know that galaxies formed from small metric inhomogeneitiespresent at large scales and, in addition, it does not require any modification of Maxwell electro-magnetism. In the inflationary cosmology, metric perturbations are generated when quantumfluctuations become super-Hubble sized and thereafter evolve as classical fluctuations, reenter-ing the horizon during radiation or matter dominated eras [15]. The same mechanism wouldoperate on large-scale EM fluctuations. However, if conformal invariance is not broken, eachpositive or negative frequency EM mode will evolve independently, without mixing. This im-1plies that photons cannot be created and therefore magnetic fields are not amplified. However,in the presence of an inhomogeneous background, we will show that the mode-mode couplingbetween EM and metric perturbations generates the mixing. This in turn will allow us to relatethe strength of the magnetic field created by this mechanism and the particular form of themetric perturbations described by the corresponding power spectrum. Those photons producedin the inflation-radiation transition with very long wavelengths can be seen as static electricor magnetic fields. Because of the high conductivity of the Universe in the radiation era, theelectric components are rapidly damped whereas, thanks to magnetic flux conservation, themagnetic fields will remain frozen in the plasma and their subsequent evolution will be trivial,Ba2 =const [5, 9]. The paper is organized as follows. In section 2 we obtain the Maxwell equa-tions in the presence of an inhomogeneous background and calculate the occupation numberof the photons produced. In section 3 we apply these results to calculate the correspondingmagnetic field generated at galactic scales. Section 4 is devoted to the analysis of the effectsof finite conductivity in those results and finally, section 5 includes the main conclusions of thepaper.2 Maxwell equations and photon productionAlthough there are previous works on the production of scalar and fermionic particles in inho-mogeneous backgrounds [16, 17], in this paper we will need to extend the analysis to the caseof gauge fields. Let us then consider Maxwell equations∇µF µν = 0, (1)in a background metric that can be splitted as gµν = g0µν + hµν , whereg0µνdxµdxν = a2(η)(dη2 − δijdxidxj) (2)is the flat FRW metric in conformal time andhµνdxµdxν = 2a2(η)Φ(dη2 + δijdxidxj) (3)is the most general form of the linearized scalar metric perturbation in the longitudinal gaugeand where it has been assumed that the spatial part of the energy-momentum tensor is diagonal,as indeed happens in the inflationary or perfect fluid cosmologies [15]. In this expression Φ(η, ~x)is the gauge invariant gravitational potential. The equation (1) can be written as:1√g∂∂xµ(√ggµαgνβ(∂αAβ − ∂βAα))= 0, (4)which leads in this background to the following linearized equations∂∂xi((1− 2Φ)(∂iA0 − ∂0Ai)) = 0, (5)2for ν = 0 and∂∂η((1− 2Φ)(∂iA0 − ∂0Ai))+∂∂xj((1 + 2Φ)(∂jAi − ∂iAj)) = 0, (6)for ν = i. In addition, we will use the Coulomb gauge condition ~∇ · ~A = 0.In order to study the amplification of vacuum fluctuations, let us consider a particularsolution of the above equations that we will denote by A~k,λµ (x) such that asymptotically in thepast it behaves as a positive frequency plane wave with momentum ~k and polarization λ, i.e,A~k,λµ (x)η→−∞→ A(0)~k,λµ (x) =1√2kVµ(~k, λ)ei(~k~x−kη) (7)where k2 = ~k2. For the two physical polarization states we have, ~(~k, λ) ·~k = 0 and 0(~k, λ) = 0.We will work in a finite box with comoving volume V and we will take the continuum limit atthe end of the calculation. We are assuming that metric perturbations vanish before inflationstarts, so that we can define an appropriate initial conformal vacuum state. Because of thepresence of the inhomogeneous background, in the asymptotic future, this solution will behaveas a linear superposition of positive and negative frequency modes with different momenta anddifferent polarizations, i.e.,A~k,λµ (x)η→∞→ ∑λ′∑q(αkqλλ′µ(~q, λ′)√2qVei(~q~x−qη) + βkqλλ′∗µ(~q, λ′)√2qVe−i(~q~x−qη))(8)It is possible to obtain an expression for the Bogolyubov coefficients αkqλλ′ and βkqλλ′ to firstorder in the metric perturbations. With that purpose, we look for solutions of the equations ofmotion in the form:A~k,λµ (x) = A(0)~k,λµ (x) + A(1)~k,λµ (x) + ... (9)where A(0)~k,λµ (x) is the solution in the absence of perturbations given by (7). Introducing thisexpansion in (5) and Fourier transforming, we obtain for the temporal component of the EMfield to first order in the perturbations:A(1)~k,λ0 (~q, η) = −√2kV~q · ~(~k, λ)q2Φ(~k + ~q, η)e−ikη (10)where, as usual, Φ(~q, η) = (2pi)−3/2∫d3xei~q~xΦ(~x, η). The zeroth order equation implies A(0)~k,λ0 (~q, η) =0. The spatial equations (6) can be written to first order as:2Φ′A(0)′i + ∂iA(1)′0 − A(1)′′i + 2~∇Φ · ~∇A(0)i − 2~∇Φ · ∂i ~A(0) + ~∇2A(1)i + 4Φ~∇2A(0)i = 0 (11)Inserting again expansion (9), these equations can be rewritten in Fourier space as:d2dη2A(1)~k,λi (~q, η) + q2A(1)~k,λi (~q, η)− J~k,λi (~q, η) = 0 (12)3where:J~k,λi (~q, η) = −√2kViΦ′(~k + ~q, η) + k2 − ~k · ~qkΦ(~k + ~q, η) i(~k, λ)e−ikη+ (~(~k, λ) · ~q) Φ(~k + ~q, η)kike−ikη − i~(~k, λ) · ~qq2ddη(Φ(~k + ~q, η)e−ikη)qi (13)Solving these equations we find, up to first order in the perturbations:A~k,λi (~q, η) =i(~k, λ)√2kVδ(~q − ~k)e−ikη + 1q∫ ηη0J~k,λi (~q, η′) sin(q(η − η′))dη′ (14)where η0 denotes the starting time of inflation. Comparing this expression with (8), it isstraightforward to obtain the Bogolyubov coefficients βkqλλ′, they are given by:βkqλλ′ =−i√2qV∫ η1η0~ (~q, λ′) · ~J ~k,λ(~q, η)e−iqηdη (15)where η1 denotes the present time. The total number of photons created with comovingwavenumber kG = 2pi/λG, corresponding to the relevant coherence length, is therefore given by[18]:NkG =∑λ,λ′∑k|βkkGλλ′ |2 (16)We will concentrate only in the effect of super-Hubble scalar perturbations whose evolution isrelatively simple [15]:Φ(~k, η) = Ck1addη(1a∫a2dη)+ Dka′a3, (17)the second term decreases during inflation and can soon be neglected. Thus, it will be useful torewrite the perturbation as: Φ(~k, η) = CkF(η). During inflation or preheating, these perturba-tions evolve in time, whereas they are practically constant during radiation or matter eras. Wewill neglect the effects of the perturbations once they reenter the horizon. This is a good ap-proximation for modes reentering right after the end of inflation since they are rapidly damped.In addition, we will show that those modes are the more relevant ones in the calculation.The power spectrum corresponding to (17) is given by:PΦ(k) = k3|Ck|22pi2V= A2S(kkC)n−1(18)where for simplicity we have taken a power-law behaviour with spectral index n and we have setthe normalization at the COBE scale λC ' 3000 Mpc with AS ' 5 · 10−5. In the case of a bluespectrum, with positive tilt (n > 1), perturbations will grow at small scales and it is necessary4to introduce a cut-off kmax in order to avoid excessive primordial black hole production [19].Accordingly, only below the cut-off the perturbative method will be reliable. For negative tiltor scale-invariant spectrum there will be also a small scale cut-off related to the minimum sizeof the horizon kmax <∼ aIHI, where the I subscript denotes the end of inflation.We can obtain an explicit expression for the total number of photons (16) in terms of thepower spectrum. Taking the continuum limit∑k → (2pi)−3/2V∫d3k, we get:NkG =∑λ,λ′V∫ d3k(2pi)3/2|βkkGλλ′ |2=∑λ,λ′V∫d3k(2pi)3/2|C|k+kG||22kGV 2∣∣∣∣∣∣∫dη√2kiF ′ + k2 − ~k · ~kGkF (~(~k, λ) · ~(~kG, λ′))+ (~(~k, λ) · ~kG) (~(~kG, λ′) · ~k) Fk)e−i(kG+k)η)∣∣∣∣2(19)Notice that the last term in (13) does not contribute to βkqλλ′ because of the transversalitycondition of the polarization vectors. The integration in d3k is dominated by the upper limit,i.e. k  kG and accordingly we can ignore the effect of the terms proportional to ~kG. Inaddition, for those modes k which are outside the Hubble radius at the end of inflation, wehave kη  1. With these simplifications we obtain:NkG '∑λ,λ′∫dk dΩ(2pi)3/2|Ck|2k22kGV∣∣∣∣∫dη(√2k((iF ′ + kF) (~(~k, λ) · ~(~kG, λ′))))∣∣∣∣2 (20)Performing the integration in the angular variables and using the definition of the power spec-trum in (18), we obtain:NkG '4(2pi)3/23kG∫dkA2S(kkC)n−1 ∣∣∣∣∫dη (iF ′ + kF)∣∣∣∣2 (21)Finally, we will estimate the time integral. The behaviour of scales that reenter the horizonduring the radiation dominated era is oscillatory with a decaying amplitude [15], therefore,there is no long-time contribution to the integral that could spoil the perturbative method.Thus, for simplicity we will assume that the function F vanishes for η ≥ 1/k, and accordinglywe estimate, |∫ dη (iF ′ + kF)|2 ∼ O(1). Our final expression for the occupation number is:NkG '4(2pi)3/2A2S3kG(kC)n−1∫ kmaxkCdkkn−1 ' 4(2pi)3/2A2S3 nknmaxkG kn−1C(22)3 Magnetic field generationThe energy density stored in a magnetic field mode Bk with wavenumber k is given by:ρB(ω) = ωdρBdω=|Bk|22, (23)5

Primordial magnetic fields from metric perturbations

© 2001 The American Physical Society. This work has been partially supported by the Ministerio de Educación y Ciencia (Spain) (CICYT AEN 97-1693).We study the amplification of electromagnetic vacuum fluctuations induced by the evolution of scalar metric perturbations at the end of inflation. Such perturbations break the conformal invariance of Maxwell equations in Friedmann-Robertson-Walker backgrounds and allow the growth of magnetic fields on super-Hubble scales. We relate the strength of the fields generated by this mechanism with the power spectrum of scalar perturbations and estimate the amplification on galactic scales for different values of the spectral index. Finally we discuss the possible effects of finite conductivity during reheating.Ministerio de Educación y Ciencia (Spain)CICYTDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Primordial magnetic fields from metric perturbations

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