Reynolds numbers in the early Universe
aa r X i v : . [ a s t r o - ph . C O ] N ov CERN-PH-TH/2010-281
Reynolds numbers in the early Universe
Massimo Giovannini Department of Physics, Theory Division, CERN, 1211 Geneva 23, SwitzerlandINFN, Section of Milan-Bicocca, 20126 Milan, Italy
Abstract
After electron-positron annihilation and prior to photon decoupling the magnetic Reynolds num-ber is approximately twenty orders of magnitude larger than its kinetic counterpart which is, inturn, smaller than one. In this globally neutral system the large-scale inhomogeneities are providedby the spatial fluctuations of the scalar curvature. Owing to the analogy with the description ofMarkovian conducting fluids in the presence of acoustic fluctuations, the evolution equations of aputative magnetic field are averaged over the large-scale flow determined by curvature perturba-tions. General lessons are drawn on the typical diffusion scale of magnetic inhomogeneities. It isspeculated that Reynolds numbers prior to electron-positron annihilation can be related to the en-tropy contained in the Hubble volume during the various stages of the evolution of the conductingplasma. e-mail address: [email protected] n a conducting plasma, such as the early Universe, the kinetic and magnetic Reynolds numbersare defined as [1, 2, 3] R kin = v rms L v ν th , R magn = v rms L B ν magn , P r magn = R magn R kin , (1)where v rms estimates the bulk velocity of the plasma while ν th and ν magn are the coefficients of thermaland magnetic diffusivity; L v and L B are, respectively, the correlation scales of the velocity field andof the magnetic field. In Eq. (1) P r magn denotes the so-called magnetic Prandtl number [1, 2, 3].Prior to electron-positron annihilation (i.e. T ≥ MeV) the coefficient of thermal diffusivity canbe estimated as ν th ∼ ( α T ) − from the two-body scattering of relativistic species with significantmomentum transfer. The conductivity of the plasma is σ ∼ T /α em so that the magnetic diffusivitybecomes ν magn = α em (4 πT ) − . Assuming, for sake of simplicity, thermal and kinetic equilibrium ofall relativistic species (which is not exactly the case for T ∼ MeV) the kinetic Reynolds number turnsout to be R kin ≃ O (10 ), the magnetic Reynolds number is R magn ≃ π/α R kin ∼ O (10 ) and P r magn ∼ . The latter estimates have been obtained by assuming, in Eq. (1), L v ≃ L B ∼ H − (where H − is the Hubble radius at the corresponding epoch); when the evolution of the backgroundgeometry is decelerated (i.e. a ( t ) ∼ t ǫ with 0 < ǫ <
1) = the particle horizon coincides with the Hubbleradius up to an immaterial numerical factor (i.e. ǫ/ (1 − ǫ )) which shall be neglected throughout . In thesymmetric phase of the standard electroweak theory, picking up a temperature T >
100 GeV where allthe species (including the Higgs boson and the top quark) are in thermal and kinetic equilibrium, ν th and ν magn can be computed [4] in terms of the hypercharge coupling constant since the non-screenedvector modes at finite conductivity are associated with the hypercharge field. In the electroweak case R kin ∼ O (10 ) and R magn ∼ O (10 ).The hypothesis of primeval turbulence has been a recurrent theme since the first speculations onthe origin of the light nuclear elements. The implications of turbulence for galaxy formation have beenpointed out in the fifties by Von Weizs¨aker and Gamow [5]. They have been scrutinized in the sixtiesand early seventies by various authors [6] (see also [7, 8] and discussions therein). In the eighties ithas been argued [9] that first-order phase transitions in the early Universe, if present, can provide asource of kinetic turbulence and, hopefully, the possibility of inverse cascades which could lead to anenhancement of the correlation scale of a putative large-scale magnetic field, as discussed in in [10, 11](see also [12] and references therein). The limits R kin ≫ R magn ≫ In what follows we shall assume a conformally flat Friedmann-Robertson-Walker background geometry g µν = a ( τ ) η µν where η µν is the Minkowski metric, τ the conformal time coordinate and a ( τ ) the scale factor; the confor-mal time coordinate τ is related to the cosmic time t as a ( τ ) dτ = dt . ν magn ≪ ddτ Z Σ ~B · d~ Σ = − ν magn Z Σ ~ ∇ × ( ~ ∇ × ~B ) · d~ Σ , ddτ Z V d x ~A · ~B = − ν magn Z V d x ~B ( · ~ ∇ × ~B ) , (2)where V and Σ are a fiducial volume and a fiducial surface moving with the conducting fluid; ~B and ~A denote the comoving magnetic field and the comoving vector potential. In the ideal hydromagneticlimit (i.e. σ → ∞ , ν magn → R magn → ∞ ) the flux is exactly conserved and the number oflinks and twists in the magnetic flux lines is also preserved by the time evolution. If R kin ≫ R magn ≤ O (1) the system is still turbulent; however, since the total time derivative of the magneticflux and of the magnetic helicity are both O ( ν magn ) the terms at the right hand side of Eq. (2) cannotbe neglected. Finally, if R magn ≫ R kin ≪ T >
Mev the Reynolds numbers can be viewed as a measure the entropy stored in a givenHubble volume. The entropy stored within the Hubble volume V H ∼ πH − ( t ) / S H = 43 πsH − = 8 π N eff v α R , (3)where s is the entropy density of the plasma. The adiabatic expansion implies that the total entropyover a comoving volume is conserved. Conversely S H , i.e. the entropy stored in the Hubble volume,increases as the plasma cools down since the temperature redshifts as a − (where a is the scale factor)but the Hubble volume typically increases faster than a both during radiation (i.e. V H ∼ a ) andduring matter (i.e. V H ∼ a / ). Up to numerical factors S H ∼ O (10 ) right before electron-positronannihilation and S H ∼ O (10 ) in the symmetric phase of the electroweak theory. The maximalentropy stored in the Hubble volume today is obtained by integrating the entropy density of theCosmic Microwave Background radiation (CMB in what follows) over the present value of the Hubblevolume: S γ = 43 πs γ H − ≃ . × (cid:18) h . (cid:19) − , s γ = 445 π T γ , (4)where T γ = 2 .
725 K. In the standard lore the huge value of the entropy contained in the Hubblevolume is the result of an appropriate theory of the initial conditions, since the adiabaticity conditioncan only be mildly violated after inflation and for a standard thermal history.According to Eqs. (3)–(4) it would be tempting to establish a causal connection between the Hubbleentropy of the CMB and the largeness of the Reynolds numbers. While such a connection cannot beexcluded for
T >
MeV, Eqs. (3) and (4), taken at face value, would imply that the kinetic and magnetic In hydromagnetic turbulence it is customarily assumed that
P r magn ≃ We do not consider here the possibility of a gravitational entropy associated with a Hubble screen approximatelysaturating the Hawking-Bekenstein bound and implying S screen ∼ H − M ≃ O (10 ) ≫ S γ . T ≪ MeV. The latter conclusion is incorrect insofaras the thermal diffusion coefficient sharply increases after electron-positron annihilation while theconductivity is only suppressed as q T / ( m e a ) where T = a T is the comoving temperature and m e is the electron mass. While prior to electron-positron annihilation R kin ≫ R magn ≫
1, after e + - e − annihilation R magn is still very large, R kin gets smaller than 1 and P r magn sharply increases.Indeed, prior to last scattering, the thermal diffusivity is dominated by Thomson scattering and theconcentration of the charge carriers is not of the order of the photon concentration (as for
T >
MeV)but ten orders of magnitude smaller i.e. n e = η b n γ where n γ is the comoving concentration of thephotons and η b = O (10 − ) is the ratio of the baryonic concentration to the photon concentrationindirectly probed by big-bang nucleosynthesis and affecting the present abundances of light nuclearelements. The thermal diffusion coefficient is then given by: ν th ( τ ) = 45 c
2s b ( τ ) λ e γ ( τ ) , c s b ( τ ) = 1 p R b ( τ )] , (5)where the electron-photon mean free path λ γ e and the ratio between the baryonic matter density andthe photon energy density R b are defined as: λ γ e = a ˜ n e a ( τ ) σ γ e , R b ( τ ) = 34 (cid:18) ω b0 ω γ (cid:19)(cid:18) aa (cid:19) = 685 . z + 1 (cid:18) ω b0 . (cid:19) ; (6)the baryon matter density ρ b is the sum of the matter densities and of the ions and electrons; σ γ e isthe electron-photon cross section; ω b0 = h Ω b0 and Ω b0 is the critical fraction of baryons. For thesake of simplicity we shall adopt, in the explicit estimates, the following fiducial set of parameters(Ω b0 , Ω c0 , Ω de0 , h , n s , ǫ re ) ≡ (0 . , . , . , . , . , . , (7)which are determined from the WMAP 7yr data alone [13] in the light of the vanilla ΛCDM scenario.After electron-positron annihilation the conductivity given by binary collisions can be estimated as σ ( τ ) = σ Tα em s Tm e a C , Λ C ( T ) = 32 e (cid:18) T πn e (cid:19) / = 1 . × (cid:18) ω b0 . (cid:19) − / , (8)where σ = 9 / (8 π √
3) depends on the way multiple scattering is estimated and Λ C is the argument ofthe Coulomb logarithm.Let us finally come to a more detailed estimate of the velocity field prior to last scattering. The bulkvelocity is defined as the center of mass velocity of the positive and negative charge carriers presentin the globally neutral plasma . The customary assumptions of hydromagnetic turbulence imply asolenoidal bulk velocity field with P r magn ≃ O (1) [1, 2]. Conversely, prior to photon decoupling, the If the charge carriers coincide with electrons and ions, denoting with m i and m e the masses of the electrons andions the bulk velocity of the plasma is defined as ~v b = ( m e ~v e + m i ~v i ) / ( m e + m i ). The center of mass velocity of theelectron-ion system is often called baryon velocity. ~ ∇ · ~v b = 0) and P r magn ≫
1. This means that theplasma is compressible and the divergence of the bulk velocity is directly affected by the large-scalecurvature fluctuations. Using Eq. (6) and assuming v rms = 1 the kinetic and the magnetic Reynoldsnumbers can be estimated as R kin ≃ .
03 and R magn ≃ . × for a typical last-scattering redshift z ∼ R kin by assuming that v rms coincides with the speed of lightcan be made more stringent since the large-scale flow, prior to last scattering, can be determined fromthe evolution equations of the baryon-photon system.For T <
MeV and around last-scattering the differences between the velocities of the baryonsand of the photons are quickly washed out because of the tight-coupling between ions, electrons andphotons. Recalling that H = ∂ τ ln a = aH , to lowest order in the tight-coupling approximation thetruncated set of hydromagnetic equations reads [14] ∂ τ ~v γ b + H R b R b + 1 ~v γ b = R b R b + 1 ~J × ~Bρ b a − ~ ∇ δ γ R b + 1) − ~ ∇ φ + ν th ∇ ~v γ b , (9) ∂ τ δ b = 3 ∂ τ ψ − ~ ∇ · ~v γ b + ~J · ~Eρ b a , ∂ τ δ γ = 4 ∂ τ ψ − ~ ∇ · ~v γ b ,∂ τ ~B = ~ ∇ × ( ~v γ b × ~B ) + ν magn ∇ ~B + ~ ∇ × (cid:18) ~ ∇ p e en e (cid:19) − πen e ~ ∇ × [( ~ ∇ × ~B ) × ~B ] , (10)where δ b and δ γ denote the density contrasts of the baryons and of the photons in the longitudinalgauge; φ and ψ denote, respectively, the (00) and ( ii ) fluctuations of the conformally flat backgroundgeometry adopted in the present discussion. The total Ohmic current ~J obeys an evolution equationwhich can be reduced to a consistency condition as in the Eckart approach to relativistic thermody-namics; such a relation is given by: ~J = σ (cid:18) ~E + ~v γ b × ~B + ~ ∇ p e en e − ~J × ~Ben e (cid:19) , (11)where n e = a ˜ n e . The system of Eqs. (9)–(11) is supplemented by the evolution equations of thecurvature perturbations which have been written elsewhere in the longitudinal gauge and even in fullgauge-invariant terms (see [14] and references therein).In previous studies [3, 14] the emphasis has been to see which are the effects of the large-scalemagnetic fields on the scalar modes of the geometry. The hierarchy between R kin and R magn afterelectron-positron annihilation suggests the possibility of addressing also the complementary part ofthe problem, i.e. the effect of the large-scale flow on the evolution of the magnetic field. Neglectingthe terms which are quadratic in the magnetic field as well as the thermoelectric term (i.e. the last twocontributions in Eq. (11)) the large-scale flow can be determined by using a standard WKB analysisgiving, in Fourier space, ~v γ b ( k, τ ) = i ˆ k M R ( k, τ ) sin [ kr s ( τ )] e − k /k , ˆ k = ~kk , s ( τ ) = Z τ c sb ( τ ′ ) dτ ′ , k ( τ ) = 12 Z τ ν th ( τ ′ ) dτ ′ , (12)where r s ( τ ) and k d ( τ ) denote, respectively, the sound horizon and the typical scale of diffusive damping; M R ( k, τ ) encodes the normalization inherited from the (adiabatic) curvature perturbations which arethe only source of large-scale inhomogeneities in the vanilla ΛCDM scenario: M R ( k, τ ) = 3 / c / ( τ ) (cid:20) c ( τ ) − (cid:21) T R ( τ ) R ∗ ( ~k ) , T R ( τ ) = 1 − H a Z τ a ( τ ′ ) dτ ′ . (13)As a result of Eqs. (9)–(10), the velocity field (12) is not solenoidal. This is situation differs fromstandard hydromagnetic turbulence and it is closer to the situation of acoustic turbulence (see e.g.[16], first paper). Following the standard conventions [13], the correlation function of curvature per-turbations in Fourier space is: hR ∗ ( ~p ) R ∗ ( ~q ) i = 2 π q δ (3) ( ~q + ~p ) P R ( q ) , P R ( q ) = A R (cid:18) qq p (cid:19) n s − , (14)where, according to Eq. (7), n s = 0 .
963 and A R = (2 . ± . × − ; q p = 0 .
002 Mpc − denotesthe pivot scale at which the power spectrum of curvature perturbations is conventionally normalized.Given the intrinsic inhomogeneity of the large-scale velocity flow, it is natural to generalize the kineticand the magnetic Reynolds numbers of Eq. (1) to Fourier space by keeping the dependence onthe wavenumbers in the velocity, the dependence on the redshift in the diffusion coefficients and bychoosing L v ≃ L B = 1 /k phys ( τ ) where k phys = k ( z + 1): R kin ( k, z ) = N kin F ( k, z ) , P r magn ( k, z ) = N magn G ( k, z ) N kin = 15 π / α m H η b n γ p A R , N magn = 9 π ζ (3) α em σ ln Λ C ( T ) η b (cid:18) m e T (cid:19) / ,F ( k, z ) = H [1 − c ( z )] T R ( z, z eq ) (cid:18) kk p (cid:19) n s − sin [ k r s ( z )] k c / ( z ) , G ( k, z ) = c ( z )( z + 1) / , (15)where ζ (3) = 1 . y ( z, z eq ) = [ q z eq + 1) / ( z + 1) −
1] the function T R ( z, z eq ) can be expressed as: T R ( z, z eq ) = ( z + 1) (cid:26) −
215 [ y ( z, z eq ) + 1][3 y ( z, z eq ) + 15 y ( z, z eq ) + 20][2 + y ( z, z eq )] (cid:27) . (16)In Fig. 1 the kinetic and the magnetic Reynolds numbers are illustrated for the set of cosmologicalparameters of Eq. (7). The three curves in each plot correspond to three different wavenumbers. Foreven larger wavenumbers both quantities are oscillating as it can be argued from Eq. (15).The hierarchy between the kinetic and the magnetic Reynolds number defines naturally a pertur-bative scheme where the evolution equations of the magnetic field can be averaged over the large-scale5
00 1000 1500 2000 - - - - - - - z l og R k i n
500 1000 1500 200015.015.516.0 z l og R m a gn Figure 1: The kinetic Reynolds number (left plot) and the magnetic Reynolds number (right plot)for k = 0 . − (short dashed line), k = 0 .
002 Mpc − (long dashed line) and k = 0 .
004 Mpc − (full line). In both plots on the vertical axis the common logarithm of the corresponding quantity isillustrated.flow. Consider the magnetic diffusivity equation (10) and neglect all the terms which are of higherorder in the magnetic field intensity. An iterative solution of Eq. (10) can then be constructed as B i ( ~k, τ ) = ∞ X n =0 B ( n ) i ( ~k, τ ) , G k ( y ) = e − k ν magn y (17) B ( n +1) i ( ~k, τ ) = ( − i )(2 π ) / Z τ G k ( τ − τ ) dτ Z d q Z d p δ (3) ( ~k − ~q − ~p ) × ǫ m n i ǫ a b n ( q m + p m ) v a ( ~q, τ ) B ( n ) b ( ~p, τ ) , (18)where, for simplicity, ν magn is assumed to be constant in time. From Eq. (18) the first few terms ofthe recursion are B (0) i ( ~k, τ ), B (1) i ( ~k, τ ) and B (2) i ( ~k, τ ). The term B (0) i ( ~k, τ ) = G k ( τ ) B i ( ~k ) where B i ( ~k )parametrizes the initial stochastic magnetic field obeying flux conservation (see Eq. (2)): h B i ( ~k ) B j ( ~k ′ ) i = 2 π k P ij (ˆ k ) P B ( k ) δ (3) ( ~k + ~k ′ ) , P ij (ˆ k ) = δ ij − ˆ k i ˆ k j . (19)Since the curvature perturbations are distributed as in Eq. (14), the correlation function of the velocityfor unequal times can be written as h v i ( ~q, τ ) v j ( ~p, τ ′ ) i = q i q j q U ( q, | τ − τ ′ | ) δ (3) ( ~q + ~p ) , U ( q, | τ − τ ′ | ) = v ( q ) τ c δ ( τ − τ ′ ) , (20)where, to avoid confusions with vector indices, the subscript γ b has been suppressed. The function v ( q ) appearing in Eq. (20) is v ( q ) = τ c V ( q ) , V ( q ) = M R ( q, τ ∗ ) 2 π q P R ( q ) sin [ qr s ( τ ∗ )] e − q /q , (21)6here τ ∗ denotes the last-scattering time and the correlation time τ c is the smallest time-scale whencompared with other characteristic times arising in the problem. Because of the exponential sup-pression of the velocity correlation function for τ > τ d (where τ d denotes the Silk time [15]), τ c approximately coincides with τ d . The form of the correlator given in Eq. (20) is characteristic ofMarkovian conducting fluids [16, 17].Denoting with H i ( ~k, τ ) the magnetic field averaged over the fluid flow, the terms containing anodd number of velocities will be zero while the correlators containing an even number of velocities donot vanish i.e. h B (2 n +1) i i = H (2 n +1) i = 0 and h B (2 n +2) i i = H (2 n +2) i = 0. So, for instance, h B (1) i i = 0while h B (2) i i = H (2) i is H (2) i ( ~k, τ ) = ( − i ) (2 π ) Z d q Z d p Z d q ′ Z d p ′ δ (3) ( ~k − ~q − ~p ) δ (3) ( ~p − ~q ′ − ~p ′ ) × Z τ dτ G k ( τ − τ ) Z τ dτ G p ( τ − τ ) ( q m + p m ) ( q ′ m ′ + p ′ m ′ ) ǫ b m ′ n ′ ǫ a ′ b ′ n ′ ǫ m n i ǫ a b n × h v a ′ ( ~q ′ , τ ) v a ( ~q, τ ) i B b ′ ( ~p ′ ) . (22)After averaging the whole series of Eq. (17) term by term the obtained result can be resummed andwritten as: H i ( ~k, τ ) = h B (0) i ( ~k, τ ) i + h B (2) i ( ~k, τ ) i + h B (4) i ( ~k, τ ) i + ... = e − k ν magn τ B i ( ~k ) . (23)where the magnetic diffusivity coefficient ν magn = 1 / (4 πσ ) has been renormalized as ν magn = ν magn + v , v = τ c Z dkk M R ( k, τ ∗ ) P R ( k ) sin ( k/k ∗ ) e − k /k , (24)and k ∗ = 1 /r s ( τ ∗ ). The averaging suggested here has been explored long ago in the related contextof acoustic turbulence by Vainshtein and Zeldovich [16] (see also [17]). Prior to decoupling, however,both h v i ∝ A R ≪ R kin ≪
1. The resum indicated in Eq. (23) seems then to be more plausiblein the present case than in the one of a kinetically turbulent plasma with strong inhomogeneities. Ifthe Markovian approximation is relaxed the velocity correlator becomes h v i ( ~q, τ ) v j ( ~p, τ ) i = ˆ q i ˆ q j ˜ v ( q ) Γ( q, τ , τ ) δ (3) ( ~q + ~p ) , ˜ v ( q ) = 2 π q P R ( q ) . (25)Assuming for sake of simplicity, R b ( τ ∗ ) ≪ c sb ( τ ∗ ) ≃ / √ q, τ , τ ) = 350 (cid:26) cos [ qc sb ( τ − τ )] − cos [ qc sb ( τ + τ )] (cid:27) e − q ν th ( τ + τ ) . (26)The results (24) and (26) lead to physically equivalent estimates and support the conclusion that thediffusivity wavenumber is smaller than expected from the usual arguments of the magnetic diffusivityscale at last scattering. In fact, ignoring the contribution of the bulk velocity in Eq. (10) the diffusivity7avenumber can be roughly estimated as k σ ∼ √H ∗ σ . The standard estimate of k σ must be comparedwith the diffusivity scale arising from the effect of the large-scale flow: k σ ≃ . × (cid:18) d A (cid:19) − / Mpc − ,k v ∼ (cid:18) A R . × − (cid:19) − n s+1 (cid:18) d A (cid:19) − n s+1 Mpc − , (27)where d A denotes the (comoving) angular diameter distance to last-scattering for the typical set offiducial parameters of Eq. (7). The standard analysis based on the magnetic diffusivity scale wouldimply that all the modes k > k σ are diffused. The presence of large-scale flow implies that diffusionoperates already for k ≥ k v . This means that the correct diffusion scale to be considered prior todecoupling is not k σ ∼ O (10 )Mpc − but, at most, k v ∼ O (50) Mpc − which is closer to the Silkdamping scale but qualitatively and quantitatively different.In summary, prior to electron-positron annihilation the large-scale (turbulent) flow can only bedetermined indirectly from the features of the various phase transitions. After electron-positron an-nihilation the flow can be inferred directly from the evolution of large-scale curvature perturbationsimprinted in the CMB anisotropies. The hierarchy between the kinetic and magnetic Reynolds num-bers prior to last scattering suggests an effective description of the evolution of pre-decoupling magneticwhich encompasses the conventional approach solely based on the conservation of the magnetic flux.On a more speculative ground the present considerations suggest that the Reynolds numbers aresomehow related to the largeness of the Hubble entropy during the early stages of the evolution of theplasma. References [1] D. Biskamp,
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