Magnetic Field Spectrum at Cosmological Recombination

0.53 MB·English

by  Kiyotomo Ichiki

Magnetic Field Spectrum at Cosmological Recombination
Kiyotomo Ichiki1 ∗, Keitaro Takahashi2, Naoshi Sugiyama3, Hidekazu Hanayama4, Hiroshi Ohno5
1Research Center for the Early Universe, University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
3Department of Physics and Astrophysics, Nagoya University, Nagoya 464-8602, Japan
4National Astronomical Observatory of Japan, Tokyo 181-8588, Japan and
5Corporate Research and Development Center, Toshiba Corporation, Kawasaki 212-8582, Japan
(Dated: February 5, 2008)
A generation of magnetic fields from cosmological density perturbations is investigated. In the
primordialplasmabeforecosmological recombination,allofthematerialsexceptdarkmatterinthe
universeexistintheformofphotons,electrons,andprotons(andasmallnumberoflightelements).
Due to the different scattering nature of photons off electrons and protons, electric currents and
7
electricfieldsareinevitablyinduced,andthusmagneticfieldsaregenerated. Wepresentadetailed
0
formalism of the generation of cosmological magnetic fields based on the framework of the well-
0
established cosmological perturbation theory following our previous works. We numerically obtain
2 the power spectrum of magnetic fields for a wide range of scales, from k ∼ 10−5 Mpc−1 up to
n k ∼ 2×104 Mpc−1 and provide its analytic interpretation. Implications of these cosmologically
a generated magnetic fields are discussed.
J

1 PACSnumbers: 91.25.Cw,98.80.-k
1

1 I. INTRODUCTION
v
9
Thereareobservationalevidenceswhichindicatethatmagneticfieldsexistnotonlygalaxies,butalsoinevenlarger
2
3 systems, such as cluster of galaxiesand extra-clusterspaces [1]. In addition to gravity,they wouldplay an important
1 role in the formation processes of various objects and their dynamical evolution in the universe. The effects of large
0 scale magnetic fields on galaxy formation and on formation of the first stars are extensively studied [2, 3, 4, 5, 6].
7 Yet, the origin of such large scale magnetic fields is still a mystery [1]. It is now widely believed that the magnetic
0
fieldsatlargescalesareamplifiedfromatinyfieldandmaintainedbythehydro-magneticprocesses,i.e.,the dynamo.
/
h However, the dynamo needs a seed field to act on and does not explain the origin of magnetic fields. As far as the
p magnetic fields in galaxies are concerned, the seed fields as large as 10−20∼−30 G are required in order to account for
- the observed fields [7] of order 1µ G at the present universe. Discoveries of magnetic fields at damped Lyα systems
o
r [8] in the early universe and an intervening galaxy at intermediate redshift [9] may require even larger seed fields.
t The question arises here is, thus, what is the origin of such seed fields? Most of previous theories to explain
s
a the origin of seed fields can be categorized into two, i.e., astrophysical and cosmological mechanisms. Almost all
:
v astrophysical mechanisms of generating seed magnetic fields are based on the Biermann battery effect [10]. This
i mechanism is operative when the gradients in thermodynamic quantities, such as temperature and density, are not
X
parallel to that of pressure. This mechanism has been applied to various astrophysical systems, which include stars
r [11], supernova remnants [12, 13], protogalaxies [14], large-scale structure formation [15], and ionization front at
a
cosmological recombination [16]. These studies show that magnetic fields with amplitude 10−16 10−21 G could
∼
be generated. Recently, it is proposed that the Weibel instability at the structure formation shocks can generate
ratherstrongmagneticfields 10−7 G[17, 18]. However,the coherence-lengthofseedfields generatedby astrophysical
mechanisms, especially the Weibel instability, tends to be too small to account for galaxy-scale magnetic fields.
Onthe other hand,cosmologicalmechanismsbasedoninflationcanproduce magneticfields witha largecoherence
lengthsince acceleratingexpansionduringinflationcanstretchesmall-scalefieldsto scalesthatcanexceedthe causal
horizon. However,inthesimplestmodelswiththeusualelectromagneticfield,whichisconformallycoupledtogravity,
the energy density in a fluctuation of the field diminishes as a−4 (where a is cosmic scale factor) to lead a negligible
amplitude of magnetic fields at the end of inflation. Therefore, to create enough amount of magnetic fields which
survive an expansion of the universe until today, some exotic couplings between the electromagnetic field to other
fields, such as dilaton [19, 20], Higgs type scalar particles [21], or gravity [22], must be introduced. Although some
models of them are consideredto be naturalextensions of standard particle model, the nature of generated magnetic
fieldsdependshighlyonthewayofextension,whosevaliditycanneverbetestedexperimentallybyterrestrialparticle
∗ E-mailaddress: [email protected]
2
accelerators. Moreover, it is recently argued that almost all models that generate magnetic fields during inflation at
the galactic scale (λ 0.1 Mpc) are severely constrained as B < 10−39 G [23, 24], for a blue primordial spectrum.
λ
∼
This is because anisotropicstressof magnetic fields wouldproduc∼e a largeamountofgravitationalwaves,which then
spoil the standard Big Bang nucleosynthesis by bringing an overproduction of helium nuclei (and deuterium).
Inadditiontothetwocategoriesdescribedabove,however,thereisathirdcategoryforthegenerationoflarge-scale
seed fields: cosmological fluctuations in the universe can create magnetic fields prior to cosmological recombination.
ThiscategorydatesbacktoHarrison(1970)[25]andhasbeenrigorouslystudiedinrecentyears. Theattractivepoint
is that mechanisms based on cosmological perturbations are much less ambiguous than the previous two categories
because cosmological pertrubations have now been well understood both theoretically and observationally through
cosmicmicrowavebackground(CMB)anisotropyandlargescalestructureoftheuniverse. Thusitispossibletomake
a robust quantitative evaluation of the generated magnetic fields.
Originally, Harrison found that the vorticity in a primordial plasma can generate magnetic fields. This is because
electronsandionswouldtendtospinatdifferentratesastheuniverseexpandsduetotheradiationdragonelectrons,
arising rotation-type electric current and thus inducing magnetic fields. It was found that magnetic fields of 10−21 G
would result if the vorticity is equivalent to galactic rotation by z = 10. More recently, magnetic field generation at
recombination was investigated by evaluating the induced electric field which arise in the matter flow dragged by a
dipole photon field, resulting in a seed field 10−26 G [26]. Following Harrison’s idea, Matarrese reported that 10−29
G seed fields can arise from the vorticity generated by second-order density perturbations at recombination [27]. In
a similar analysis, a larger value, 10−21 G, was obtained by considering earlier periods when the energy density of
photons was larger and the photon mean free path was smaller [28]. Other specific second order effects from the
coupling between density and velocity fluctuations on the seed field generation was evaluated in [29].
In [30], we presented a formalism to calculate the spectrum of seed fields based on the cosmological perturbation
theory. Therearethreeessentialpointsinordertoobtainthecorrectspectrum. Firstofall,wehavetotreatelectron,
proton and photon fluids separately to evaluate the amount of electric currentand electric field. Secondly, we need a
precise evaluation of collision terms between three fluids, which generate the difference in motion between them. At
this point, we found that not only the velocity difference between electron and photon fluids, which is conventionally
studied, but alsophotonanisotropicstresscontributes to the difference inmotionbetweenelectronandprotonfluids.
Finally, the second-order perturbation theory is necessary because vector-type perturbations, such as vorticity and
magneticfield,areknowntobe absentatthe firstorder. Basedonthis formalism,in[31],wecalculatedthespectrum
of generated magnetic fields at a range between k 10−5 Mpc−1 and k 10 Mpc−1. Then we showed that the
∼ ∼
magnetic field is contributed mainly from the velocity difference between electrons and photons, which we called the
”baryon-photonslipterm”,onlargescales(k .1Mpc−1),whilephotonanisotropicstressisdominantonsmallscalles
(k & 1 Mpc−1). As a whole, the spectrum is small-scale-dominant and the amplitude is about 10−20 G at k = 1
Mpc−1.
Inthispaper,wepresentdetailsofourformalismandextenditinseveraldirections. First,insectionII,wedescribe
the derivation of equations of motion for electron and proton fluids, carefully evaluating the collision term between
electrons and photons, and then obtain the evolution equation for magnetic fields. We calculate and interpret the
resultingspectrumnumericallyinsectionIII.InsectionIV,wederivetheanalyticalexpressionofthespectrum,which
allowsustoconfirmthenumericalresultsandextrapolatethespectrumintomuchsmallerscalesthancanbeobtained
numerically. In section V, we discuss about the magnetic felicity, which in fact explicitly vanishes. Finally, in section
VI, we give some discussions and conclusions.
II. BASIC EQUATIONS
Here we derive basic equations for the generation of magnetic fields, i.e., perturbation equations of photon, proton
and electron fluids. While protons and electrons are conventionally treated as a single fluid, however, it is necessary
to deal with proton and electron fluids separately in order to discuss the generation of magnetic fields. Let us begin
with the Euler equations. Those are given by
m nuµu enuµF =Cpe+Cpγ, (1)
p p pi;µ− p iµ i i
m nuµu +enuµF =Cep+Ceγ , (2)
e e ei;µ e iµ i i
where m is the proton (electron) mass, u is the bulk velocity of protons (electrons), F is the usual Maxwell
p(e) p(e) µi
tensor. The thermal pressure of proton and electron fluids are neglected. Here µ,ν = 0,1,2,3 and i = 1,2,3. The
r.h.s. of Eq. (1) and (2) represent the collision terms. The first terms in Eqs. (1) and (2) are collision terms for the
Coulomb scattering between protons and electrons, which is given by
Cpe = Cep = (u u )e2n2η , (3)
i − i − pi− ei
3
where
πe2m1/2 1+z −3/2 lnΛ
η = e lnΛ 9.4 10−16sec , (4)
(k T )3/2 ∼ × 105 10
B e (cid:18) (cid:19) (cid:18) (cid:19)
is the resistivity of the plasma and lnΛ is the Coulomb logarithm. As is well known, this term acts as the diffusion
term in the evolution equation of magnetic field. The importance of the diffusion effect can be estimated by the
diffusion scale,
τ 1/2
λ √ητ 100 AU, (5)
diff ≡ ∼ H−1
(cid:18) 0 (cid:19)
above which magnetic field cannot diffuse in the time-scale τ. Here H = 70km/s/Mpc is the present Hubble
0
parameter. Thus, at cosmologicalscales considered in this paper, this term can be safely neglected.
The other terms expressed by Cp(e)γ are the collision terms for Compton scattering of protons (electrons) with
i
photons. Since photons scatter off electrons preferentially compared with protons by a factor of (m /m )2, we can
e p
safely drop the term Cpγ from the Euler equation of protons. This difference in collision terms between protons and
i
electrons ensures that small difference in velocity between protons and electrons, that is, electric current, is indeed
generated once the Compton scattering becomes effective.
A. Compton Collision Term
Let us now evaluate the Compton scattering term. In the limit of completely elastic collisions between photons
and electrons, this term vanishes. Typically, in the regime of interest in this paper, very little energy is transfered
between electrons and photons in Compton scatterings. Therefore it is a good approximationto expand the collision
term systematically in powers of the energy transfer.
Let us demonstrate this specifically. We consider the collision process
γ(p )+e−(q ) γ(p′)+e−(q′), (6)
i i → i i
where the quantities in the parentheses denote the particle momenta. To calculate this process, we evaluate the
collision term in the Boltzmann equation of photons:
a d3q d3q′ d3p′
C[f(p~)] = (2π)4 M 2
p (2π)32E (q) (2π)32E (q′) (2π)32E(p′) | |
Z e Z e Z
δ(3)[p~+~q p~′ q~′]δ[E(p)+E (q) E(p′) E (q′)]
e e
× − − − −
[f (q~′)f(p~′)(1+f(p~)) f (~q)f(p~)(1+f(p~′))] , (7)
e e
× −
where f(p~) and f (~q) are the distribution functions of photons and electrons, E (q) = q2+m2 is the energy of
e e e
an electron, and the delta functions enforce the energy and momentum conservations. We have dropped the Pauli
p
blocking factor (1 f ). The Pauli blocking factor can be always omitted safely in the epoch of interest, because f
e e
−
is very small after electron-positron annihilations. Note that the stimulating factor can be also dropped because this
does not contribute to the Euler equation.
Integrating over q~′, we obtain
a d3q d3p′ (2π)
C[f(p~)] = M 2
pZ (2π)32Ee(q)Z (2π)32E(p′)2Ee(|~q+p~−p~′|)| |
δ[E(p)+E (q) E(p′) E (~q+p~ p~′ )]
e e
× − − | − |
[f (~q+p~ p~′)f(p~′) f (~q)f(p~)] . (8)
e e
× − −
In the regime of our interest, energy transfer through the Compton scattering is small and can be ignored in the
first order density perturbations. As we already discussed earlier, however, it is essential to take the second order
couplings in the Compton scattering term into consideration for generation of magnetic fields. Therefore we expand
the collisiontermuptothe firstorderinpowersofthe energytransfer†,andkeeptermsupto secondorderindensity
perturbations.
[†]However,weshallkeepuptothesecondordertermsforthepurposeofreference.
4
The expansion parameter is the energy transfer,
q2 (~q+p~ p~′)2 (p~′ p~) ~q (p~ p~′)2
E (q) E (q′)= − − · − , (9)
e e
− 2m − 2m ≈ m − 2m
e e e e
over the temperature of the universe. Employing p T, we can estimate the order of this expansion parameter as
( pq ) ( q ),whichissmallwhenelectronsaren∼on-relativistic. Notethat,inthecosmologicalThomsonregime,
O meT ∼O me
electrons in the thermal bath of photons are non-relativistic, p q2 , and the energy of photons is much smaller
∼ 2me
than the rest mass of a electron, p m . Thus, it also holds that q √2m p p, and the second term in Eq.(9) is
e e
≪ ∼ ≫
usually smaller than the first one.
Now let us divide the collision integralinto four parts, i.e., the denominators of the Lorentz volume, the scattering
amplitude,thedeltafunctionandthedistributionfunctions,andexpandthemduetotheexpansionparameterdefined
above. First of all, the denominator in the Lorentz invariant volume can be expanded to
1 1 −1 1 −1
= m + ~q+~p p~′ 2 m + ~q 2
Ee(q)Ee(|~q+p~−p~′|) (cid:18) e 2me| − | (cid:19) (cid:18) e 2me| | (cid:19)
1
≈ m2e 1−E(mqe)2 −E(mpq2e)−E(mpe)2 , (10)
(cid:16) (cid:17)
where
q2 (p~ p~′) ~q (p~ p~′)2
E(mqe)2 = m2e, E(mqp2e) = −m2e · , E(mpe)2 = 2−m2e . (11)
Secondly, we consider the matrix element. The matrix element for Compton scattering in the rest frame of the
electron is given by,
p˜′ p˜
M 2 = 2(4π)2α2 + sin2β˜ ,
| | "p˜ p˜′ − #
cosβ˜ = p˜ˆ p˜ˆ′ , (12)
·
wherep˜andp˜′ aretheenergiesofincidentandscatteredphotons,p˜ˆandp˜ˆ′ aretheunitvectorsofp~˜andp~˜′,respectively,
denotingthedirectionsofthephotonsinthisframe. TheLorentztransformationwithelectron’svelocity(q/m )gives
e
the following relations,
p 1 (q/m )2
e
= − , (13)
p˜ 1 ~p ~q/pm
p− · e
p pµ = p˜p˜µ. (14)
µ µ
Using these relations, we evaluate the matrix element in the CMB frame as [32]
|M|2 = 2(4π)2α2 M0+Mmqe +M(mqe)2 +M(mqp2e)+M(mpe)2 , (15)
= 1+cos2β(cid:16), (cid:17) (16)
0
M
~q
q = 2cosβ(1 cosβ) pˆ+pˆ′ , (17)
Mme − − (cid:20)me ·(cid:16) (cid:17)(cid:21)
q2
M(mqe)2 = 2cosβ(1−cosβ)m2e , (18)
~q 2 (~q pˆ)(~q pˆ′)
qp = (1 cosβ)(1 3cosβ) (pˆ+pˆ′) +2cosβ(1 cosβ) · · , (19)
Mm2e − − (cid:20)me · (cid:21) − m2e
p2
M(mpe)2 = (1−cosβ)2m2e . (20)
5
Thirdly, we expand the delta function to
δ[p p′+E (q) E (q′)]
e e
− −
∂δ(p p′+E (q) E (q′))
≈δ(p−p′)− − ∂Ee(q′)− e |Ee(q)=Ee(q′)(Ee(q)−Ee(q′))
e
1∂2δ(p p′+E (q) E (q′))
+ 2 − ∂pe2 − e |Ee(q)=Ee(q′)(Ee(q)−Ee(q′))2
2
∂δ(p p′) (p~ p~′) ~q (p p′)2 1 (p~ p~′) ~q ∂2δ(p p′)
=δ(p p′)+ − − · + − + − · −
− ∂p′ " me 2me # 2" me # ∂p′2
∂δ(p p′) 1 ∂2δ(p p′)
≡δ(p−p′)+ ∂p−′ D(mq)+D(mp) + 2D(2mq) ∂p′−2 . (21)
h i
where
(p~ p~′) ~q (p p′)2
D(mq) ≡ −me · , D(mp) ≡ 2−me . (22)
Finally, the distribution of the electron can be expanded to
∂f ∂2f
f (~q+~p p~′) f (~q)+ e (p~ p~′)+(pi p′i) e (pj p′j). (23)
e − ≈ e ∂~q · − − ∂qi∂qj −
We assume that the electrons are kept in thermal equilibrium and in the Boltzmann distribution:
2π 3/2 (~q m v~)2
e e
f (~q)=n exp − , (24)
e e
m T − 2m T
(cid:18) e e(cid:19) (cid:20) e e (cid:21)
where v is the bulk velocity of electrons. The derivatives of the distribution function with respect to the momentum
e
are given as
∂f ~q m v~
e e e
= f (~q) − , (25)
e
∂~q − m T
e e
∂2f ∂f (~q)qi m vi δ
e e e ij
= − f (~q) . (26)
e
∂q ∂q − ∂q m T − m T
i j j e e e e
By substituting above equations, Eq.(23) is written as
2
~q m v~ 1 (p~ p~′)(~q m v~) p~ p~′ 2
f (~q+p~ p~′) f (~q) f (~q) − e e (p~ p~′)+ f (~q) − − e e | − |
e e e e
− ≈ − meTe · − 2 " meTe # − meTe 
1  
≡ fe(~q) 1−F(mq)+ 2F(2mq)−F(mp) . (27)
(cid:20) (cid:21)
Therefore, we have
1
fe(~q+p~−p~′)f(p~′)−fe(~q)f(p~)=fe(~q) f(p~′)−f(p~) −f(p~′)fe(~q)F(mq)−f(p~′)fe(~q) F(mT)− 2F(2mq) . (28)
(cid:20) (cid:21)
h i
Fortunately, it has been known that the leading term (zeroth order term), obtained by multiplying together the
first term in the delta function and the zeroth order distribution functions, is zero. It means that we only have to
keep up to the first order terms when we expand the matrix element and the energies, in order to keep the collision
term up to the second order [33].
Therefore we have,
M 2
Ee(|q)E|e(q′) ≈6πσT(M0+Mmqe) (29)
6
Combining altogether, we obtain the collision term expanded with respect to the energy transfer as (note that this
expansion is not with respect to the density perturbations)
a π d3p′ d3q
C[f]= [(0th order term)+(1st order terms)+(2nd order terms)]+ (30)
p 4 (2π)3p′(2π)3 ···
(cid:16) (cid:17)Z
where
0th order term:
6πσ δ(p p′)f (~q) f(p~′) f(p~)) , (31)
T 0 e
M − −
1st order terms: h i
∂δ(p p′)
6πσTM0fe(~q) −δ(p−p′)f(p~′)Fmq + ∂p−′ Dmq f(p~′)−f(p~)
(cid:20) h i(cid:21)
+6πσT q fe(~q)δ(p p′) f(p~′) f(p~) , (32)
Mm − −
2nd order terms: h i
1 1∂2δ(p p′)
6πσTM0fe(~q)"−δ(p−p′)f(p~′)(cid:20)Fmp − 2F2mq (cid:21)+ 2 ∂p′−2 D2mq hf(p~′)−f(p~)i
∂δ(p p′) ∂δ(p p′)
+ − p f(p~′) f(p~) − q + p f(p~′) q
∂p′ Dm − − ∂p′ Dm Dm Fm#
h i (cid:16) (cid:17)
∂δ(p p′)
+6πσTMmq fe(~q) −δ(p−p′)f(p~′)Fmq + ∂p−′ Dmq +Dmp f(p~′−f(p~)) . (33)
(cid:20) (cid:16) h i(cid:17)(cid:21)
From now on, we omit the secondorder terms. These terms are not only much smaller than the firstorder terms but
also may not contribute to the Euler equation at all (see [34]). Evaluating the first moment of the above collision
term, we obtain the Compton scattering term in the Euler equation (41) as
d3p
Ceγ = p C[f]
i − (2π)3 i
Z
4σ ρ an 1
= T γ e (v v )+ v Π j . (34)
− 3 ei− γi 4 ej γi
(cid:20) (cid:21)
Here moments of the distribution functions are given by
d3p
pf (p~)=ρ , (35)
(2π)3 γ γ
Z
d3p 4
p f (p~)= ρ v , (36)
(2π)3 i γ 3 γ γi
Z
d3q
f (~q)=n , (37)
(2π)3 e e
Z
d3q
q f (~q)=ρ v , (38)
(2π)3 i e e ei
Z
d3p 1 1
p−1p p f (p~)= ρ Π + ρ δ , (39)
(2π)3 i j γ 3 γ γij 3 γ ij
Z
where ρ and ρ (= m n ) are energy densities of photons and electrons, vi and vi are their bulk three velocities
γ e e e γ e
defined by vi ui/u0, and Πij is anisotropic stress of photons. It should be noted that the collision term (34) was
≡ γ
obtained nonperturbatively with respect to density perturbations [30].
B. Evolution Equations of Magnetic Fields
Now we obtain the Euler equations for protons and electrons as
m nuµu enuµF =0, (40)
p p pi;µ− p iµ
4σ ρ an 1
m nuµu +enuµF = T γ (v v )+ v Π j , (41)
e e ei;µ e iµ − 3 ei− γi 4 ej γi
(cid:20) (cid:21)
7
where m is the proton mass. Here we ignore the pressure of proton and electron fluids. Also the Coulomb collision
p
termisneglectedasexplainedbelowEq. (5). Notethatthe collisiontermwasnotevaluatedinamanifestlycovariant
way. Here the left hand side in Eqs. (40) and (41) should be evaluated in conformal coordinate system. We also
assumed the local charge neutrality: n = n n . In the case without electromagnetic fields (F = 0), the sum of
e p iµ
∼
the equations (40) and (41) gives the Euler equation for the baryons in the standard perturbation theory. On the
other hand, subtracting Eq. (40) multiplied by m from Eq. (41) multiplied by m , we obtain
e p
m m j m m j
p e nuµ i +jµ p− e i u +en(m +m )uµF (m m )jµF
i p e iµ p e iµ
− e " (cid:18)n(cid:19);µ (cid:18)mp+meen − (cid:19);µ# − −
4m ρ anσ 1
= p γ T (v v )+ v Π j , (42)
− 3 ei− γi 4 ej γi
(cid:20) (cid:21)
where uµ and jµ are the center-of-mass 4-velocity of the proton and electron fluids and the net electric current,
respectively, defined as
m uµ+m uµ
uµ p p e e, (43)
≡ m +m
p e
jµ en(uµ uµ). (44)
≡ p − e
Employing the Maxwell equations Fµν =jµ, we see that the quantities in the squarebracketin the l.h.s. of Eq. (42)
;ν
is suppressed at the recombination epoch, compared to the second term, by a factor [35]
c2 103cm−3 1Mpc 2
3 10−40 , (45)
L2ω2 ∼ × n L
p (cid:18) (cid:19)(cid:18) (cid:19)
wherecis the speedoflight,Lisacharacteristiclengthofthe systemandω = 4πne2/m isthe plasmafrequency.
p e
The third termin the l.h.s. of Eq. (42), i.e., (m m )jµF , is the Hall termwhich canalso be neglected because
p e iµ
the Coulomb coupling between protons and electron−s is so tight that ui uipui . Then we obtain a generalized
| |≫| p− e|
Ohm’s law:
4σ ρ a 1
uµF = T γ (v v )+ v Π j C . (46)
iµ − 3e ei− γi 4 ej γi ≡ i
(cid:20) (cid:21)
Now we derive the evolution equation for the magnetic field, which can be obtained from the Bianchi identities
F =0, as
[µν,λ]
3
0 = ǫijkuµF
2 [jk,µ]
u0 u0
= uµ i ǫijk C + ,jC (ui j uj i)+ ,j( jui iuj), (47)
B ,µ− j,k u0 k!− ,jB − ,jB u0 B −B
whereǫijk isthe Levi-Civita`tensorand i (a2Bi)=ǫijkF /2isthe magneticfieldinthecomovingframe[36]. We
jk
B ≡
will now expand the photon energy density, fluid velocities and photon anisotropic stress with respect to the density
perturbation as
ρ (t,x )=(ρ0) (t)+(ρ1) (t,x )+ , u0(t,x )=a(t)−1+(u1)0(t,x )+ ,
γ i γ γ i i i
··· ···
ui(t,x )=(u1)i(t,x )+(u2)i(t,x )+ , v (t,x )=(v1) (t,x )+(v2) (t,x )+
i i i i i i i i i
··· ···
Πiγj(t,xi)=(Π1)iγj(t,xi)+··· , (48)
where the superscripts(0),(1), and (2)denote the order ofexpansionand t is the cosmic time. Remembering that i
is a second-orderquantity, we see that all terms involving i in Eq. (47), other than the first term, can be neglecteBd.
B
Thus we obtain
d i u0
B ǫijk C + ,jC
dt ∼ j,k u0 k!
= 4σT3e(ρ0)γaǫijk"(cid:18)δ(γ1,)j−2(Φ1),j(cid:19)(cid:16)(v1)ek−(v1)γk(cid:17)− 41(cid:18)(v1)el (Π1)lγj(cid:19),k−2(h1)lj,k(cid:16)(v1)el−(v1)γl(cid:17)−(cid:16)(v2)ej,k−(v2)γj,k(cid:17)#(4,9)
8
(1) (1)
where Φand hi are first-order curvature and tensor perturbations, respectively, in Poisson gauge, and we used the
j
density contrast of photons, (δ1)γ,k (ρ1)γ,k/ (ρ0)γ. Further, we employed the fact that there is no vorticity in the linear
≡
order: ǫijk (v1) =0. It should be noted that the velocity of electron fluid can be approximatedto the center-of-mass
j,k
velocityatthisorder,(v1)i (v1)i. Thephysicalmeaningofthisequationisthatelectronsgain(orlose)theirmomentum
e ∼
through scatterings due to the relative velocity to photons (baryon-photon slip), and the anisotropic pressure from
photons. The momentum transfer from the photons ensures the velocity difference between electrons and protons,
and thus eventually generates magnetic fields. We found that the contribution from the curvature perturbation is
alwaysmuchsmallerthanthatfromthe densitycontrastofphotons inthe firstterminEq.(49)(which willbe clearly
seen in Figs.1 and 2). Therefore we shall omit the curvature perturbation hereafter when considering the evolution
of magnetic fields.
The first term in Eq. (49) is exactly the same discussed in [29]. They have estimated contributions from these
terms by considering typical values at recombination. Here we solve the equation numerically and obtain a robust
prediction of the amplitude of magnetic fields.
Eq.(49)showsthatthe magneticfield cannotbe generatedin the first(linear)order. The r.h.s. ofEq.(49)contains
two kinds of source terms, i.e., intrinsic second order quantities and products of first order quantities. Since the first
order quantities can be exactly evaluated within the frame work of the standard cosmological linear perturbation
theory,wehereafterconcentrateonthe productsoffirstorderquantities. The evaluationofthe intrinsicsecondorder
term will be left for the future work.
III. COSMOLOGICAL MAGNETIC FIELDS
A. Spectrum of Cosmologically Generated Magnetic Fields
Inthissectionwederivethespectrumofmagneticfieldsgeneratedbycosmologicalperturbations. Thecosmological
perturbations can be decomposed into scalar, vector, and tensor modes. These modes physically correspond to
perturbations of density, vorticity and gravitational waves, respectively. These cosmological perturbations are most
likelytobegeneratedduringtheinflationepoch. Mostofthesimple singlefieldinflationmodels predictgenerationof
adiabatic scalar and tensor type perturbations but vector type perturbations. Even if the vector type perturbations
are generated, they are known to be damped away in the expanding universe. Observations of CMB temperature
fluctuations and large scale structure of the universe strongly suggest the adiabatic scalar perturbations while we
only have upper limits on tensor perturbations [37]. Therefore at most tensor perturbations are only sub-dominant
components. Hence we consider only scalar type perturbations for linear order quantities throughout this paper.
The linear-order scalar perturbations can be written as
1
δ = (ik )δ (~k,t)ei~k·~xd3~k, (50)
γ,k k γ
(2π)3
Z
1
v = p ( ikˆ)v (~k,t)ei~k·~xd3~k, (51)
el l e
(2π)3 −
Z
1
v = p kˆk v (~k,t)ei~k·~xd3~k, (52)
el,k l k e
(2π)3
Z
1 1
Πl = p ( kˆlkˆ + δl)Π(~k,t)ei~k·~xd3~k, (53)
j (2π)3 − j 3 j
Z
1 1
Πl = p (ik )( kˆlkˆ + δl)Π(~k,t)ei~k·~xd3~k. (54)
j,k (2π)3 k − j 3 j
Z
Here kˆi ki/k. By using these expressiopns, we can write the evolution equation of magnetic fields in Fourier space
≡
as,
4σ ρ 1 1
(a2Bi(K~,t))· = T γǫijk d3k~′ kˆ′K δ (K~ k~′,t)δv(k~′,t) α (K~,k~′)v (K~ k~′,t)Π(k~′,t) (55)
3e (2π)3 j k γ − − 4 jk e −
Z (cid:20) (cid:21)
p
9
where we have defined,
δv = v v , (56)
e γ
−
K 4
α (K~,k~′) = K(Kˆ kˆ′) k′ Kˆ kˆ′. (57)
jk K~ k~′ · − 3 k j
| − |(cid:18) (cid:19)
To obtain the spectrum, one needs to evaluate Bi(K~)B∗(K~′) =S(K)δ(K~ K~′). That is
i −
D E
4σ 2 1 t0 t0
a4Bi(K~)B∗(K~′) = T ǫijkǫlm d3k~′d3~k dt′ dt′′
i 3e i (2π)3
(cid:18) (cid:19) Z Z0 Z0
kˆ′K δ (K~ k~′,t′)δv(k~′,t′)a(t′)ρ (t′)kˆ K′ δ∗(K~′ ~k,t′′)δv∗(~k,t′′)a(t′′)ρ (t′′)
× j k γ − γ l m γ − γ
(cid:20)
1
kˆ′K δ (K~ k~′,t′)δv(k~′,t′)a(t′)ρ (t′)α (K~′,~k)v∗(K~′ ~k,t′′)Π∗(~k,t′′)a(t′′)ρ (t′′)
−4 j k γ − γ lm e − γ
1
α (K~,k~′)v (K~ k~′,t′)Π(k~′,t′)a(t′)ρ (t′)kˆ K′ δ∗(K~′ ~k,t′′)δv∗(~k,t′′)a(t′′)ρ (t′′)
−4 lm e − γ l m γ − γ
1
+ α (K~,k~′)v (K~ k~′,t′)Π(k~′,t′)a(t′)ρ (t′)α (K~′,~k)v∗(K~′ ~k,t′′)Π∗(~k,t′′)a(t′′)ρ (t′′)(5.8)
16 jk e − γ lm e − γ
(cid:21)
The final task we should do is to take an ensemble average of this expression. Assuming that perturbations are
random Gaussian variables and using the Wick’s theorem, we can expand the ensemble average of the products of
four Gaussian variables as, for example,
δ (K~ k~′,t′)δv(k~′,t′)δ∗(K~′ ~k,t′′)δv∗(~k,t′′) = δ (K~ k~′,t′)δv(k~′,t′) δ∗(K~′ ~k,t′′)δv∗(~k,t′′)
γ − γ − γ − γ −
D E D ED E
+ δ (K~ k~′,t′)δ∗(K~′ ~k,t′′) δv(k~′,t′)δv∗(~k,t′′)
γ − γ −
D ED E
+ δ (K~ k~′,t′)δv∗(~k,t′′) δv(k~′,t′)δ∗(K~′ ~k,t′′) .
γ − γ −
D ED E (59)
Since the evolution equations for linear order perturbation variables are independent of kˆ, we may write
δ (~k,t) = ψ (~k)δ (k,t) , (60)
γ i γ
δv(~k,t) = ψ (~k)δv(k,t) , (61)
i
Π(~k,t) = ψ (~k)Π(k,t) , (62)
i
where ψ (~k)is the initialperturbationandδ (k,t),δu(k,t), andΠ(k,t) arethe transferfunctions, whicharethe solu-
i γ
tionofthe Einstein-Boltzmannequationsunderthe adiabaticinitialconditionwithψ (~k)=1. Byvirtueoflinearized
i
equations, the overall amplitude and spectrum of the primordial fluctuations, P(k), can be given independently of
the transfer functions. Here P(k) is defined using the two-point correlation function of ψ in Fourier space,
i
ψ (~k)ψ∗(k~′) =P(k)δ(~k k~′) , (63)
i i −
D E
where δ is the Dirac delta function. According to the convention, we write P(k) kns−4 with ns being the spectral
∝
index of primordial density perturbations. Using these definitions, we finally obtain,
4σ 2 d3~k
S(K) = T K2 1 (kˆ Kˆ)2 P(K~ ~k )P(k)
3e (2π)3 − · | − |
(cid:18) (cid:19) Z h i
k
S2(K~ ~k,k) S (k, K~ ~k )S (K~ ~k,k)
× 1 | − | − K~ ~k 1 | − | 1 | − |
(cid:26) | − |
1 k
+ βS (k, K~ ~k )S (K~ ~k ,k) αS (k, K~ ~k)S (K~ ~k ,k)
4" 2 | − | 1 | − | − K~ ~k 1 | − | 2 | − | #
| − |
1
+ α αS2(K~ ~k ,k) βS (k, K~ ~k )S (K~ ~k,k) , (64)
16 2 | − | − 2 | − | 2 | − |
h i(cid:27)
10
4 4
10 10
102 δγ 102
e v e
d γ d
u u
plit 100 Πγ plit 100
m δ m
cdm
n a10-2 Φ n a 10-2 δ
o o
bati -4 bati 10-4 vvbb−v
ur10 ur γ b
ert ert 10-6 δcdm
p -6 p
10
-8
10
-8
10
-10
10
10-6 10-5 10-4 10-3 10-2 10-1 10-6 10-5 10-4 10-3 10-2 10-1
scale factor scale factor
FIG. 1: Evolution of perturbations relevant with magnetic field generation in a ΛCDM model with wavenumber k/h = 1.6
Mpc−1. Left: the five lines represent density contrast of CDM (δCDM; black dotted line), and density contrast (δγ; red
line) , velocity perturbation (vγ; blue dash-dotted line), anisotropic stress (Πγ; green dashed line) of photons, and curvature
perturbation(Φ;brownthinline)asindicated. Right: timeevolutionsofdensitycontrast(δ ;redline)andvelocityperturbation
b
(vb; blue dash-dotted line) of baryons. The baryon-photon slip (vγ −ve; blue dash-dotted line) is also shown in the figure. In
the deep radiation dominated era, baryon-photon slip and anisotropic stress of photons are severely suppressed compared to
the other perturbation variables. Perturbation variables are shown in the Newtonian gauge [38] while anisotropic stress and
slip terms are gauge invariant.
where
S (k,k′) = a(t′)ρ (t′)δ (k,t′)δv(k′,t′)dt′ , (65)
1 γ γ
Z
S (k,k′) = a(t′)ρ (t′)v (k,t′)Π (k′,t′)dt′ , (66)
2 γ e γ
Z
3K(Kˆ kˆ) 2k
α = · − , (67)
3K~ ~k
| − |
K K k(Kˆ kˆ)
2
− ·
β = . (68)
n K~ ~k2 o − 3
| − |
Here S and S describe effects of baryon-photon slip and photon anisotropic pressure. Specifically, the terms in the
1 2
second and fourth lines in Eq.(64) are what we will call in subsequent sections the contributions to the generationof
magnetic fields from baryon-photonslip and anisotropic stress of photons, respectively.
B. Cosmological Perturbation Theory and Numerical Calculations
Evolutions of perturbations relevant to the generation of magnetic fields can be solved within the frame work of
the standard cosmological perturbation theory. The standard theory describing evolutions of linear perturbations,
which was originally developed by Lifshiz [39] and summarized in his textbook [40] and other recent articles [41, 42],
is well tested by a number of observations and firmly established. As for an application to cosmological models, the
first realistic numerical calculation was done by Peebles & Yu [43], where the distribution of photons was solved by
directoryintegratingtheBoltzmannequation. Thecalculationsweresubsequentlyextendedtoincludeneutrinos[44],