Relaxing the Bounds on Primordial Magnetic Seed Fields

a r X i v :a s t r o -p h /9904022v 2 2 A p r 1999

DAMTP-1999-39astro-ph/9904022

Relaxing the Bounds on Primordial Magnetic Seed Fields

Anne-Christine Davis ?,Matthew Lilley ?and Ola T¨o rnkvist ?

Department of Applied Mathematics and Theoretical Physics,University of Cambridge,

Silver Street,Cambridge CB39EW,United Kingdom

(March 27,1999)We point out that the lower bound on the primordial magnetic ?eld required to seed the galactic dynamo is signi?cantly relaxed in an open universe or in a universe with a positive cosmological constant.It is shown that,for reasonable cosmological parameters,primordial seed ?elds of strength 10?30Gauss or less at the time of galaxy formation could explain observed galactic magnetic ?elds.As a consequence,mechanisms of primordial magnetic seed-?eld generation that have previously been ruled out could well be viable.We also comment on the implications of the observation of micro-Gauss magnetic ?elds in galaxies at high redshift.

PACS numbers:98.80.-k,98.35.Eq,98.62.En,98.80.Es

Magnetic ?elds pervade most astrophysical systems [1],but their origin is unknown.Spiral galaxies are observed to possess large-scale magnetic ?elds with strength of the order of 10?6G and direction aligned with the rotational motion.A plausible explanation is that galactic mag-netic ?elds result from the exponential ampli?cation of an initially weak seed ?eld by a mean-?eld dynamo [2,3].Many proposals have been put forward regarding the ori-gin of such a seed ?eld.One suggestion is that it might arise spontaneously from non-parallel gradients of pres-sure and charge-density during galaxy formation [4].A wider range of possibilities is o?ered if the seed ?eld is of primordial origin.This category includes cosmological magnetic ?elds [5]as well as magnetic ?elds created by any of a number of early-universe particle-physics mech-anisms [6]such as collisions of bubbles in a ?rst-order phase transition [7]or false-vacuum in?ation [8,9].

The seed-?eld strength required at the time of com-pleted galaxy formation (t gf )for a galactic dynamo to produce the present magnetic ?eld strength B 0～10?6G is usually quoted in the range ～10?23–10?19G.Such lower bounds are obtained by considering the dynamo ampli?cation in a ?at universe with zero cosmological constant for “typical”values of the parameters of the αω–dynamo.The seed ?eld must also be coherent on a scale at least as large as the size of the largest turbulent eddy,～100pc [2].Most proposed models of primordial seed-?eld generation fail to meet these requirements as formulated above.

In this paper we address the issue in the light of re-cent developments in cosmology.Observations of distant type-IA supernovae [10]and of anisotropies in the cosmic microwave background (CMB)[11]in combination have made it increasingly likely that the universe is less dense than the critical density and has a positive cosmological constant Λ.Most previous studies of magnetic ?elds have assumed a Λ=0universe with critical matter density.

frequently in the literature,Γ?1

=

0.3Gyrs[13]and

Γ?1=0.5Gyrs[2].

Any magnetic seed?eld is exponentially ampli?ed until

it reaches the equipartition energy(B～fewμG)when

further growth is suppressed by dynamical back reaction

of the Maxwell stresses on the turbulence.Assuming

that the dynamo mechanism begins to operate around

the time of completed galaxy formation t gf,the lower

bound on the strength of the seed?eld at this epoch is

given by

B gf≥B0e?Γ(t0?t gf),(1)

where t0is the age of the universe.Expressions for t0in

di?erent cosmologies are found in e.g.Refs.[14]and[15].

In particular,for a given value of H0,open universes and

universes with a positive cosmological constant are signif-

icantly older than the?=1Einstein-de Sitter universe.

The time of galaxy formation t gf can be estimated from

a spherical collapse model.As we shall show presently,

galaxies of a given average densityˉρgal have collapsed

at approximately the same time after the Big Bang for

all realistic cosmological models.Galaxies in an open or

Λ>0universe are therefore older,giving the dynamo

mechanism more time t0?t gf to operate.Consequently,

a much smaller magnetic?eld B gf can seed the dynamo

and still give the observed micro-Gauss?eld B0.

The spherical collapse model[16,17]describes the non-

linear collapse of a bounded spherical region with average

local densityˉρi larger than the critical density at some

initial time t i in the matter-dominated era.This over-

density causes the sphere to break away from the Hubble

expansion,reach a maximal(turn-around)radius r m,and

eventually collapse to form a gravitationally-bound sys-

tem.The general equation of motion for a shell of radius

r enclosing mass M is[15]

1

dt

2

?GM6Λr2=E,(2)

where E is a constant.This equation can be separated

to yield

dt=±

r3/2

m

2GM

√

2

(1?cosθ),(4)

t+T=

1

GM r m

96

(66θ?93sinθ+15sin2θ?sin3θ)+O(β2) ,(5)

where T is a constant which can be neglected[17].We

see that turn-around occurs at a time t m correspond-

ing toθ=π.As the spherical region recollapses for

t>t m,random non-radial particle velocities become

important;the simple collapse model breaks down and

the collapse is halted at a?nal radius r vir given by the

virial theorem.For universes with zero cosmological con-

stant r vir=r m/2.For0≤β<1/2(which is re-

quired for collapse to occur)Lahav et al.[15]showed

that1/2≥r vir/r m>0.366and also obtained the ap-

proximate relation

r vir

2?β

.(6)

We can estimateβfor a galaxy:Taking0≤λ0<1,

M=1011M⊙≈2×1045g and r vir<～15kpc we get

0≤β<

H20

(0.366)3

～3×10?5.(7)

The small value ofβsigni?es that the vacuum energy

density plays a negligible role compared to the matter

density in the collapse of objects as small and dense

as galaxies.From Eq.(6)it follows that we can set

r vir=r m/2for all realistic values of the cosmological con-

stant.Moreover,we can neglect theβ-dependent terms

in Eq.(5).

It is generally assumed[17]that gravitational collapse

is complete at the time t vir>t m when r approaches zero

in Eq.(4),corresponding toθ≈2π.1This assumption

is supported by N-body simulations and,because of the

small value ofβ,remains valid in any realistic Friedmann

cosmology.From Eq.(5)we then have t vir≈2t m as well

as t2m=3π/(32Gˉρm),whereˉρm≡3M/(4πr3m)is the

average density of the spherical region at turn-around.

It follows from r vir=r m/2thatˉρgal=8ˉρm.Finally,

with t gf=t vir,all these relations combine to give

ˉρgal=

3π

1The naive estimate,that collapse is complete when the ra-

dius r in Eq.(4)reaches the virial radius r vir(corresponding

toθ=3π/2),is unrealistic as the radius decreases more slowly

during virialisation than the spherical collapse model would

imply.

0.2

0.4

0.6

0.8

1

?0?40?30

?20

?10L o g 10(B g f /1 G a u s s )

Γ ?1

= 0.5 Gyrs, h = 0.65Γ ?1

= 0.5 Gyrs, h = 0.5Γ ?1

= 0.3 Gyrs, h = 0.65Γ ?1

= 0.3 Gyrs, h = 0.5

0.2

0.4

0.6

0.8

1

?0 = 1 ? λ0

?70

?50

?30

?10

L o g 10(B g f /1 G a u s s )

Γ ?1

= 0.5 Gyrs, h = 0.65Γ ?1

= 0.5 Gyrs, h = 0.5Γ ?1

= 0.3 Gyrs, h = 0.65Γ ?1

= 0.3 Gyrs, h = 0.5

a)

b)

FIG.1.Lower bound on the seed ?eld at galaxy formation B gf vs ?0:(a)universe with Λ=0,(b)?at Λuniverse.

Refs.[14]and [15]).It may at ?rst seem mysterious that ?0does not enter in Eq.(8)or any of the derivations leading to it.The reason is that (for Λ=0)the same average local density (larger than the critical density)is required for a spherical region to collapse regardless of the density of the surrounding universe.By Birkho?’s theorem the evolution therefore proceeds in an identical manner.

We are now in a position to calculate bounds on mag-netic seed ?elds B gf at the time of completed galaxy for-mation in di?erent cosmologies.We take B 0=10?6G and ˉρgal =10?24g cm ?3.The latter value corresponds to the average density of the galactic halo rather than the central disc,whose density is ～10?23g cm ?3.The reason for this choice is that the spherical collapse model uses the simpli?ed assumption of a uniform-density “top-hat”pro?le of the galactic density distribution,and since the halo comprises most of the volume of the galaxy,this value seems more appropriate.The precise value of ˉρgal is of little importance as our results are quite insensitive to it.

The results are displayed in Fig.1(a)for a Λ=0uni-verse and in Fig.1(b)for a ?at Λuniverse (?0+λ0=1).The quantity h is the Hubble parameter H 0in units of 100km sec ?1Mpc ?1.For comparison,the straight hor-izontal line in each plot shows the constraint of B gf ≥10?20G given by Rees [19].It can be seen on these graphs that in an open universe,and particularly in a universe with a signi?cant cosmological constant Λ,this requirement is too strong.For reasonable cosmological parameters and the same value of Γ,the dynamo mecha-nism could generate currently observed galactic magnetic ?elds from a seed ?eld of the order of 10?30G or less at the completion of galaxy formation provided that the seed ?eld is coherent on a scale ξgf >～100pc.

We shall now evolve these bounds back to the time of radiation decoupling,taking the conservative view that there is no magnetohydrodynamic turbulence or

dynamo mechanism operating during gravitational col-lapse (see,however,Ref.[4]for more optimistic propos-als).The magnetic ?eld is assumed to be frozen into the plasma and its evolution is determined by ?ux conser-vation Br 2=const .,where r is a length scale evolving with the matter,i.e.r ～(ˉρ)?1/3.Care must be taken not to associate this length scale with the scale factor a (t ),as a collapsing galaxy is decoupled from the Hubble expansion.One obtains [8]

B gf

ρdec

2/3=

ˉρgal (1+z dec )2

,

(9)

where we have used the energy conservation relation ρ=ρ0(1+z )3for the matter component and the fact that the matter density ρdec at the epoch t dec is very nearly uniform.The redshift of radiation decoupling,z dec ,has a weak dependence on the product ?B h 2in-volving the fraction of critical density in baryons ?B and

is constrained to lie in the interval 1200<～z dec <～1400

[14].

Note that the magnetic ?eld will decrease between t dec and t gf ,since by virtue of the Hubble expansion the phys-ical volume of the galaxy is larger than the volume con-taining the same mass at t dec .The depletion depends on the cosmological parameters via the present matter density ρ0≈1.88×10?29?0h 2[g cm ?3].It can be seen that the depletion is somewhat smaller in universes with ?0<1.This further increases the e?ect of cosmo-logical parameters in relaxing the bounds on primordial seed ?elds.The resulting bounds for B dec are shown in Fig.2(a)and Fig.2(b)for a Λ=0universe and for a ?at Λuniverse,respectively.

We shall now address the issue of the correlation length of the magnetic ?eld.In order for the galactic dynamo to begin to operate,the correlation length of the seed ?eld at the time of completed galaxy formation must satisfy

0.2

0.4

0.6

0.8

1

?0?40?30

?20

?10

L o g 10(B d e c /1 G a u s s )

Γ ?1

= 0.5 Gyrs, h = 0.65Γ ?1

= 0.5 Gyrs, h = 0.5Γ ?1

= 0.3 Gyrs, h = 0.65Γ ?1

= 0.3 Gyrs, h = 0.5

0.2

0.4

0.6

0.8

1

?0 = 1 ? λ0

?70

?50

?30

?10

L o g 10(B d e c /1 G a u s s )

Γ ?1

= 0.5 Gyrs, h = 0.65Γ ?1

= 0.5 Gyrs, h = 0.5Γ ?1

= 0.3 Gyrs, h = 0.65Γ ?1

= 0.3 Gyrs, h = 0.5

a)

b)

FIG.2.Lower bound on the seed ?eld at radiation decoupling B dec vs ?0:(a)universe with Λ=0,(b)?at Λuniverse.

ξgf >～100pc [2].

2

Using the spherical

collapse model,one can calculate the physical scale r dec at the time of radiation decoupling that will evolve into the size of a galaxy.At any time before the onset of gravitational col-lapse the matter density follows the Hubble expansion and it makes sense to express r dec in the constant co-moving quantity x de?ned by r =a (t )x .The comoving scale x corresponding to a galaxy is given by [17]

x gal

=0.95 ?0h 2

?1/3M 1/3

12[Mpc],(10)

where M 12=M/1012M ⊙.

The correlation length ξcan be written as a fraction of the radius of the galaxy,ξ=ηr vir .With the sim-pli?ed assumption of the spherical collapse model that the collapsing region has uniform density,the collapse is homogeneous and isotropic and di?erent scales col-lapse proportionately.Assuming that the magnetic ?eld is frozen into the plasma between t dec and t gf we have x corr =ηx gal .For a galaxy,M 12≈0.1,and the typi-cal length scale of the turbulent motion,ξturb =100pc,corresponds to η≈1/150,giving the following bound on the comoving correlation length

x corr >～x turb =5–10kpc

(11)

for observationally realistic values 0.25>?0h 2>0.025.This bound is somewhat higher than that stated in Ref.[12].The bound should not be applied before t dec ,since the correlation length then evolves according to complicated magnetohydrodynamic processes and is not proportional to the scale factor a (t )[12].

x turb

d/2

B dec ,

(12)

where B rms is the quantity that must satisfy the bound given in Fig.2,with B dec and x dec being the strength and comoving correlation length,respectively,of the pri-mordial seed ?eld evolved from formation to t dec .The exponent d can equal 1,2,or 3depending on the aver-aging procedure used.This complicated issue [20]shall not be addressed in this paper.

There have been observations of micro-Gauss ?elds at redshifts of z =0.395[21]and z =2[22],although the latter has been criticised [23].If correct,these observa-tions are di?cult to explain in a ?at universe with Λ=0.They may,however,be easier to understand in an open or Λuniverse.Applying our model to the z =0.395case,with B 0.395=10?6G,we obtain for a ?at Λuniverse the bounds at t =t dec shown in Fig.3.Hence a seed ?eld of 10?20G at t dec ,or equivalently 10?23G at t gf ,could account for this observation.

If we attempt a similar analysis in the z =2case,the required seed ?eld is su?ciently high that it would have other cosmological implications,e.g.on the CMB [24]or structure formation [25].Consequently we conclude that,unless the dynamo parameters are radically di?erent for high column density Ly-αclouds (e.g.if they have fast-spinning cores and thereby have a higher angular-velocity gradient |r dω/dr |[26])these observations cannot be ex-

0.2

0.4

0.6

0.8

1

?0 = 1 ? λ0

?60

?40

?20

L o g 10(B d e c /1 G a u s s )

Γ ?1

= 0.5 Gyrs, h = 0.65Γ ?1

= 0.5 Gyrs, h = 0.5Γ ?1

= 0.3 Gyrs, h = 0.65Γ ?1

= 0.3 Gyrs, h = 0.5

FIG.3.Lower bound on B dec vs ?0for generating a ?eld strength of 10?6G at redshift z =0.395by the dynamo mech-anism in a universe with λ0+?0=1.

plained by ampli?cation of a primordial seed ?eld by a galactic dynamo.

In this paper we have reconsidered the constraints on the primordial magnetic ?eld required to seed the galac-tic dynamo in the light of recent cosmological advances.We have shown that,in an open universe or a universe with Λ>0,a much smaller seed ?eld is required to ex-plain the observed micro-Gauss ?elds in galaxies.As a consequence,mechanisms of primordial magnetic seed-?eld generation that had previously been ruled out,on the grounds of giving too small strength or correlation length,could well be viable.We have evolved the bounds back to the epoch of radiation decoupling t dec ,assuming that from t dec to the present the magnetic ?eld is frozen into the plasma and evolves ?rst via ?ux conservation and thereafter by ampli?cation via a dynamo mechanism.The remaining problem is to evolve primordial magnetic ?elds from the time of their generation to t dec taking into account various plasma e?ects.A step in this direc-tion has been taken in Ref.[12].This work needs to be generalised to di?erent cosmologies,although it can be expected that the main cosmological e?ects occur at late times.

We are grateful to K.Dimopoulos,L.Hui,M.J.Rees and N.Weiss for helpful discussions.This work was sup-ported in part by the U.K.PPARC.Support for M.L.was provided by a PPARC studentship and by Fitzwilliam College,Cambridge;for O.T.by the European Commis-sion’s TMR programme under Contract No.ERBFMBI-CT97-2697.