Ohmic currents and predecoupling magnetism

Massimo Giovannini1,2 and Nguyen Quynh Lan3

1Department of Physics, Theory Division, CERN, 1211 Geneva 23, Switzerland2INFN, Section of Milan-Bicocca, 20126 Milan, Italy

3Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam(Received 21 May 2009; published 28 July 2009)

Ohmic currents induced prior to decoupling are investigated in a standard transport model accounting

both for the expansion of the background geometry as well as of its relativistic inhomogeneities. The

relative balance of the Ohmic electric fields in comparison with the Hall and thermoelectric contributions

is specifically addressed. The impact of the Ohmic currents on the evolution of curvature perturbations is

discussed numerically and it is shown to depend explicitly upon the evolution of the conductivity.

DOI: 10.1103/PhysRevD.80.027302 PACS numbers: 98.70.Vc, 78.20.Ek, 98.62.En, 98.80.Cq

Prior to photon decoupling the plasma is electricallyneutral, the static (Coulomb) potential is exponentiallysuppressed beyond the Debye length (see, e.g. [1]) whilethe concentration of the electric charges is 1010 timessmaller than the concentration of the photons (see e.g. [2]).This effect would naively seem to increase the role of theHall and thermoelectric terms whose magnitude is in-versely proportional to the charge concentration [3]. Theaim of the present paper is to clarify the situation andinvestigate more quantitatively the different contributionsresponsible of Ohmic currents especially in the light of theongoing attempt of a consistent inclusion of large-scalemagnetic fields in the calculation of the cosmic microwavebackground (CMB) observables [4,5]. Consider, to beginwith, the Vlasov-Landau system of equations for electronsand ions1

@fe;i@

þ ~v ~r ~xfe;i e½ ~Eþ ~v ~B ~r ~qfe;i ¼@fe;i@

coll

;

(1)

where ~E ¼ a2 ~E and ~B ¼ a2 ~B are, respectively, the comov-

ing electric and magnetic fields; ~v ¼ ~q=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2 þm2a2

pis the

velocity and ~q is the comoving three-momentum. In theultrarelativistic limit (i.e. q ma), ~v ¼ ~q=j ~qj and, there-fore, Eq. (1) is invariant under a Weyl rescaling of thegeometry g. Conversely, ~q ¼ ma ~v when the given spe-

cies are nonrelativistic. In terms of the distribution func-tions of Eq. (1), the evolution equations of theelectromagnetic fields are given by

~r ~E ¼ 4eZ

d3v½fið ~x; ~v; Þ feð ~x; ~v; Þ;~r ~B ¼ 0;

(2)

~r ~Eþ ~B0 ¼ 0;

~r ~B ~E0 ¼ 4eZ

d3v ~v½fið ~x; ~v; Þ feð ~x; ~v; Þ;(3)

where the prime denotes a derivation with respect to theconformal time coordinate . The evolution equations ofthe comoving concentrations of electrons and ions (i.e.respectively ne and ni) can be written, in explicit terms,

as2 n0e;i þ ~r ðne;i ~ve;iÞ 3ne;ic0 ¼ 0, where c is the (sca-

lar) fluctuation of spatial components of the metric in thelongitudinal gauge [6] defined by the conditions sg00 ¼2a2, sgij ¼ 2a2cij. Introducing the global charge and

the total current, i.e. q ¼ eðni neÞ and ~J ¼ eðni ~vi ne ~veÞ, the difference of the two equations for the charge

concentrations implies that 0q þ ~r ~J 3c 0q ¼ 0. The

relevant Maxwell equations become then ~r ~E ¼ 4q

and ~r ~B ¼ 4 ~J þ ~E0. The fiducial values of the cosmo-logical parameters employed to illustrate the present esti-mates correspond to the best fit of the WMAP 5 yr dataalone [7,8]. The conductivity (and the related mobility) canbe computed in the customary framework of the Krookmodel [9,10] which holds for weakly ionized plasmas, and,with some numerical differences, also in the fully ionized

case. The collision terms of Eq. (1) are eiðfe feÞand ieðfi fiÞ, where ei and ie are the collisionrates of electrons and ions and where fe;i are two

Maxwellian distributions, i.e. fðvÞ ¼ ½ma=ð2TÞ3=2 exp½mav2a=ð2TÞ. The induced electric field slightlyperturb the Maxwellian distributions and, therefore, theexplicit form of the conductivity can be derived fromEq. (3) by following exactly the same steps of the standard

1The conformal time coordinate will be denoted by and thegeometry will be assumed to be conformally flat, i.e. g ¼a2ðÞ, where ¼ diagð1;1;1;1Þ is the Minkowskimetric.

2Throughout the paper the physical quantities will be denotedby a tilde while the comoving quantities will appear without thetilde. For instance the (comoving) concentrations, energy den-sities, and pressures of electrons and ions will be denoted,respectively, by ne;i ¼ a3~ne;i, e;i ¼ a3e;i, and by pe;i ¼ne;iTe;i ¼ a4 ~pe;i. Similarly, for photons, ¼ ð2=15ÞT4

¼a4 ~. When needed these two notations will be employedwithout further explanations.

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1550-7998=2009=80(2)=027302(4) 027302-1 2009 The American Physical Society

plasma calculation (see e.g. [1,3]) with the important dif-ference that, because of the breaking of Weyl invariance,the scale factors aðÞ appear ubiquitously:

¼ 9

8ffiffiffi3

p T

e2

ffiffiffiffiffiffiffiffiffiT

mea

s½lnCðTÞ1

¼ 4:35 107 eV

T0

2:725 K

3=2

h20M0

0:1326

1=2

ffiffiffiffiffiffiffiaeqa

r;

CðTÞ ¼ 3

2e3

T3

n0

1=2 ¼ 1:102 108

h20b0

0:02273

1=2; (4)

where n0 is the common value of the (comoving) electronand ion concentrations;C is the argument of the Coulomblogarithm and M0 is the critical fraction of matter in theCDM model. We are now interested in the evolution

equation of the Ohmic current ~J whose explicit form canbe derived by combining the governing equations for elec-

trons and ions:

~v0i þH ~vi ¼ eni

ia½ ~Eþ ~vi ~B ~r

~rpi

ai

þ aie

e

i

ð ~ve ~viÞ þ 4

3

~

~i

aið ~v ~viÞ;(5)

~v 0e þH ~ve ¼ ene

ea½ ~Eþ ~ve ~B ~r

~rpe

ae

þ aeið ~vi ~veÞ þ 4

3

~

~e

aeð ~v ~veÞ:(6)

By taking the difference of Eq. (5) (multiplied by eni) andof Eq. (6) (multiplied by ene) the following equation can beobtained:

@ ~J

@þH ~J ¼ !2

pe þ!2pi

4~E eðni neÞ ~r eni

~rpi

ai

þ ene~rpe

ae

þ eneaei

1þme

mi

ðni neÞðme þmiÞ

mine þ nime

~vb ðmi þmeÞeðnime þ nemiÞ

~J

þ e2neniðme þmiÞ

ameðnime þ nemiÞ1þme

mi

~vb ~Bþ e

ðnime þ nemiÞa

nime

mi

nemi

me

~J ~Bþ 4

3e

i

mi

e

me

~v þ

eniðme þmiÞmeðmine þmeniÞ

ineðme þmiÞmiðmine þmeniÞ

~vb

e

eðmine þ nimeÞmi

me

þ i

eðmine þ nimeÞme

mi

~J

; (7)

where the plasma frequencies and the baryonic velocity

have been introduced, i.e. !pe;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ne;ie

2=ðme;iaq

Þ and~vb ¼ ðme ~ve þmp ~viÞ=ðme þmpÞ. The evolution of ~vb iscoupled to the velocity of the photons and it is obtainedby summing up (instead of subtracting) Eq. (5) (multipliedby mi) and Eq. (6) (multiplied by me):

~v0b þH ~vb ¼

~J ~B

a4 ~bð1þme=miÞ ~r

þ 4

3

~

~b

aeð ~v ~vbÞ; (8)

~v0 ¼ 1

4~r ~rþ aeð ~vb ~vÞ: (9)

The result of this double expansion implies, from Eq. (7),

@ ~J

@þ

H þ aie þ

4e

3n0me

~J

¼ !2pe

4

~Eþ ~vb ~Bþ

~rpe

en0 ~J ~B

en0

þ 4ee

3me

ð ~vb ~vÞ: (10)

The terms ~J0 andH ~J are comparable in magnitude and areboth smaller than ie and e, i.e. H ~J ’ ~J0 < ð4=3Þ

½=ðn0meÞe < aie. While it is important to solve theevolution of ~J during all the predecoupling regime, theprevious chain of inequalities implies that, asymptotically,the form of Eq. (10) is dominated by the term containingie. At the right-hand side the term containing ð ~vb ~vÞcan be estimated by subtracting Eqs. (8) and (9). Thedifference ð ~vb ~vÞ is driven exponentially to zero at arate controlled by aeð1þ R1

b Þ where Rb ¼ð3=4Þ~b=~. The asymptotic form of the Ohm’s law canbe written, for large conformal times as

~J ¼ !2pe

4faie þ ð4=3Þ½=ðn0meÞeg

~Eþ ~vb ~Bþ

~rpe

en0 ~J ~B

n0e

: (11)

The term containing the gradient of the electron pressure isthe curved-space counterpart of the thermoelectric term [3]while the term proportional to the vector product of thecurrent and of the magnetic field is the curved-space coun-terpart of the Hall term. The displacement current can beneglected in comparison with the Ohmic current (i.e.4 ~J ~E0) provided the left hand side of Eq. (10) issubleading in comparison with the induced electric field(i.e. ~J0 !2

pe;i~E). The latter requirement demands, after

derivation with respect to the conformal time , the fulfill-

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027302-2

ment of the condition ~J00 !2pe~J: the one-fluid description

correctly captures the dynamics in the low-frequencybranch of the spectrum of plasma excitations, i.e. ! !pe. If the thermoelectric and Hall terms are neglected,then the electromagnetic fields obey the following pair ofequations, i.e.:

@ ~B

@¼ ~r ð ~vb ~BÞ þ 1

4r2 ~B;

@ ~E

@¼ @

@ð ~vb ~BÞ þ 1

4r2 ~E;

(12)

where ¼ !2pe=ð4eiÞ is given by Eq. (4); Eq. (12)

implies that wave numbers k2 > k2 ¼ H are dissipatedbecause of the finite value of the conductivity. The explicitvalue of the diffusivity scale k is given by

kT

¼ 2:177 1018

g

10:75

1=4

h20M0

0:1326

3=4

3=4; (13)

where ¼ a=aeq. In comoving temperature units, theHubble wave number is H =T ¼ 1:0471031

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig=10:75

q(where g is the effective number of

relativistic degrees of freedom); as expected not onlyH k but also H kDebye ’ 5:26 106T wherekDebye is the wave number corresponding to

Debye ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT=ð8n0e2Þ

pi.e. the screening length of the

Coulomb potential between two charges in the plasma.The diffusivity scale, on the contrary, sets a (lower) limitin the coherence scale of the Ohmic fields. The dominanceof the drift term (i.e. j ~vb ~Bj) over the thermoelectric andHall terms demands, from Eq. (11), the fulfillment of thefollowing pair of relations:

B

nG

> 108:27

k

T

h20M0

0:1326

1=2 ffiffiffiffi

p;

B

nG

< 1011:19

T

k

h20M0

0:1326

1=2 1ffiffiffiffi

p :

(14)

In Fig. 1 (plot at the left) the contribution of the Hall andthermoelectric terms to the Ohmic current is illustrated.

For comparison the diffusivity, Debye, and Hubble scalesare also reported. The shaded area denotes the regionwhere the conditions of Eq. (14) are approximately ful-filled, i.e. for typical amplitudes of the comoving magneticfield in the range 105nG<B< 105nG. Because of thevalue of the charge concentration, the Hall contributiondominates over the thermoelectric term of the electrons

j ~J ~Bjj ~rpej

’ B2

4n0T’ 8:19 102

B

nG

2h20b0

0:02273

: (15)

We remind that, in spite of our plot, thermoelectric termsare often invoked to generate large-scale magnetic fields(see e.g. [2]). When the plasma contains a magnetic fieldwhose Fourier modes are stochastically distributed withpower spectrum PBðkÞ, the asymptotic form of the Ohmlaw (11), within the shaded region of the parameter spaceof Fig. 1, induces an effective electric field

hEið ~k; ÞEjð ~p; Þi ¼ 22

k3PijðkÞPEðk; Þð3Þð ~kþ ~pÞ;

PEðk; Þ ¼g21ðÞ þ

g22ðÞ

PBðkÞg1

¼ 4:29 1010

k

T

h20M0

0:1326

1=2;

g2 ¼ 4:89 105

h20M0

0:1326

1=2

; (16)

where PijðkÞ ¼ ðij kikj=j ~kj2Þ is the transverse projec-tor. The stochastic electromagnetic fields as well as theinduced Ohmic currents, being inhomogeneous, affect thecurvature perturbations3 whose evolution, in the absence ofnonadiabatic pressure fluctuations, is given by [4,5]

−8 −6 −4 −2 0 2 4 6 8−35

−30

−25

−20

−15

−10

−5

0

log(B/nG)

log(

k/T

)

h02 Ω

M0 =0.1326, a = a

eq

Thermoelectric terms

Hall termsk=kσ

Hubble wavenumberk= k

Debye

−5 −4 −3 −2 −1 0 1 2−9

−8

−7

−6

−5

−4

−3

−2

−1

0

log(τ/τ1)

log|

ζ k(τ)|

h02 Ω

M 0 = 0.1326

magnetic contribution (units ΩB=1)

Ohmic contribution (units ΩB=1)

FIG. 1 (color online). Relative contribution of Hall, thermoelectric and drift terms in the Ohmic current (plot at the left).Contribution of Ohmic currents to the evolution of curvature perturbations (plot at the right). In the left plot (as well as in Fig. 2)1 sets the equality time scale [more precisely eq ¼ ð ffiffiffi

2p 1Þ1].

3We shall denote by the density contrast on uniform curva-ture hypersurfaces (see, e. g. [11]) while R is the curvatureperturbation on comoving orthogonal hypersurfaces.

BRIEF REPORTS PHYSICAL REVIEW D 80, 027302 (2009)

027302-3

0 ¼ ~E ~J3a4ð~pt þ ~tÞ

þHsBð3c2st 1Þ3ð~t þ ~ptÞ

þ HsE

3ð~t þ ~ptÞ½3c2stg212 þ g22ð3c2st 2Þ

g212 þ g22

ð ~r ~vtÞ3

;

(17)

where sB ¼ B2=ð8a4Þ and sE ¼ E2=ð8a4Þ Theevolution of as well as the evolution of c can beintegrated and the results are reported in Fig. 1 (plot atthe left) and in Fig. 2. Note that, in Figs. 1 and 2 B ¼sB=ð8~Þ and E ¼ sE=ð8~Þ. The magneticfield intensity is always assessed after regularization overa given scale L, i.e., following the notation of [5] BL ¼9:536 108ðBL=nGÞ2 where BL is the comoving inten-sity prior to the gravitational collapse of the protogalaxy[4,5]. In what follows, as often done in numerical integra-tions, units B ¼ 1 will be used and we will be interestedin the relative weight, at late times, of the Ohmic and of themagnetic components. The dynamical evolution sup-presses (by roughly 9 orders of magnitude) the electriccontribution in comparison with the magnetic one (seeFig. 1, right plot) and the dominant contribution is givenby the drift term of Eq. (11) [see also Eq. (16)]. In thebaryon rest frame, by definition, the drift term vanishesand, therefore, the Ohmic contribution is dramatically sup-pressed in comparison with the ~E ~J term (see Fig. 2 plot atthe left). The suppression can be understood in terms of thefollowing orders of magnitude estimates

~E ~Ja4 ~

’k

k

2BðkÞ; E2

a4 ~

’k

2BðkÞ; (18)

both estimates can be obtained by going to Fourier spaceand by recalling that, in the shaded region of Fig. 1 (plot atthe right) the total current is just ~J ’ ~r ~B. Moreover, ifwe are in the shaded region of Fig. 1 (left plot), in thebaryon rest frame ~E ’ ~J=. In conclusion the drift termgives the dominant contribution to the curvature inhomo-geneities induced by the Ohmic currents. Conversely, inthe baryon rest frame the dominant contribution from theOhmic fields stems from ~E ~J which is, roughly speaking,20 orders of magnitude smaller than the magnetic contri-bution for scales of the order of the Hubble radius (seeFig. 2, plot at the right, where the evolution of c isillustrated). The remaining terms proportional to E areeven more suppressed and are, typically, 60 orders ofmagnitude smaller than the magnetic contribution and 30orders of magnitude smaller than ~E ~J (see Fig. 2, plot atthe left). The present result improves on the usual approx-imations posited in the Boltzmann integrators accountingfor the effects of large-scale magnetic fields [4,5] on CMBobservables. In this paper only the very large length-scaleshave been considered.

N.Q. L wishes to thank the CERN physics departmentand, in particular, Professor L. Alvarez-Gaume for kindhospitality and financial support.

[1] T. J. M Boyd and J. J. Sanderson, The Physics of Plasmas(Cambridge University Press, Cambridge, 2003).

[2] M. Giovannini, Classical Quantum Gravity 23, R1 (2006).[3] L. Spitzer, Physics of Fully Ionized Plasmas (J. Wiley and

Sons, New York, 1962).[4] M. Giovannini, Phys. Rev. D 70, 123507 (2004); 74,

063002 (2006); M. Giovannini and K. Kunze, Phys. Rev.D 77, 061301 (2008).

[5] M. Giovannini, Phys. Rev. D 79, 103007 (2009).

[6] J.M. Bardeen, Phys. Rev. D 22, 1882 (1980).[7] G. Hinshaw et al. (WMAP Collaboration), Astrophys. J.

Suppl. Ser. 180, 225 (2009).[8] E. Komatsu et al. (WMAP Collaboration), Astrophys. J.

Suppl. Ser. 180, 330 (2009).[9] P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94,

511 (1954).[10] E. P. Gross and M. Krook, Phys. Rev. 102, 593 (1956).[11] J. Hwang, Astrophys. J. 375, 443 (1991).

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−30

−29

−28

−27

−26

−25

−24

−23

−22

−21

log(τ/τ1)

log|

ζ k(τ)|

h0

2 ΩM0

= 0.1326

E ⋅ J (in units ΩB =1)

Ohmic contribution in the baryonrest frame (rescaled by a factor 1030)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−9

−8

−7

−6

−5

−4

−3

−2

−1

log(τ/τ1)

log|

ψk(τ

)|

h02 Ω

M0 = 0.1326

purely magnetic contribution(units Ω

B=1)

E⋅J (rescaled by a factor 1021)

FIG. 2 (color online). The contribution of the Ohmic fields to the curvature perturbations (plot at the left) and to the metricfluctuations (plot at the right).

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