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Generation of magnetic fields in the early Universe
Physics Letters A 310 (2003) 182–186 www.elsevier.com/locate/pla
Generation of magnetic fields in the early Universe P.K. Shukla 1 Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Received 17 January 2003; accepted 8 February 2003 Communicated by V.M. Agranovich
Abstract Nonlinear interactions between intense photon and neutrino beams and relativistically hot electron–positron–ion plasmas of the early Universe are considered. It is shown that a small fraction of ions produces large space charge electric fields in the presence of the plasma pressure gradient, relativistic photon ponderomotive force and weak nuclear driving force of neutrinos. Large space charge electric fields can separate pairs, generating the currents which are required for creating magnetic fields in plasmas. The present theory thus offers a possible clue to the origin of primordial magnetic fields in the early Universe. 2003 Elsevier Science B.V. All rights reserved. PACS: 13.10.-g; 52.30.-q; 98.62.En; 98.80.Cq
It is well known [1–4] that during the later stage of lepton era of the early Universe, the medium was predominantly populated with intense photons, electrons and positrons, and their associated neutrinos and antineutrinos, in addition to a small fraction of protons. In the early Universe, the neutrino plasma temperatures are between 1–2 MeV and 10 MeV and the electron–positron charge separation density (ne − np ) is close to the proton (ion) number density Zi ni 10−10ne , where ne = np + Zi ni , nσ is the number density of the particle species σ (σ equals e for electrons, p for positrons, and i for ions), and Zi is the ion charge state. At relativistic temperatures, intense neutrinos (and antineutrinos) are produced from fusion reactions in violent processes following the Big Bang leading to what we call the relic neutrinos-equivalent to the microwave background, while intense photons are produced due to annihilation of electron–positron pairs. The latter are usually created by the electron bremsstrahlung [5,6], or by accelerated electrons to relativistic speeds in a strong wakefield [7] which is created by intense photons. Computer simulations [8] and recent laboratory experiments [9] have reported the generation of high-energy electrons and positrons by ultraintense lasers. Furthermore, the early Universe plasma also contains the minority ion population, which can affect the electrodynamics of the composite system [10–12]. Intense photons interacting with pair plasmas generate many interesting non-linear effects [10], which have received a great deal of attention in laser–plasma interactions and in studies of radio pulsar emissions from neutron stars. Besides, in
E-mail address: [email protected] (P.K. Shukla). 1 Also at the Department of Plasma Physics, Umeå University, SE-90187 Umeå, Sweden, and the Center for Interdisciplinary Plasma
Science, Max-Planck Institut für Extraterrestrische Physik und Plasmaphysik, D-85740 Garching, Germany. 0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00336-0
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electron–positron–ion plasmas, we have the possibility of relativistic collisionless shock waves [12] and relativistic density pulses which have relevance to astrophysical settings. On the other hand, neutrinos interacting with matter generate charged and neutral currents [13,14] associated with the weak Fermi nuclear interaction whose strength is proportional to GF nσ , where GF = 9 × 10−30 erg cm−3 is the Fermi constant. Charged weak currents appear due to the exchange of the charged vector boson W ± (with mass mW = 80 GeV/c2 , where c is the speed of light in vacuum) associated with the processes involving leptons σ interacting with neutrinos µσ of the same flavor, while neutral weak currents are due to the exchange of the neutral vector boson Z 0 (with mass mZ ≈ 91 GeV/c2 ) associated with processes involving neutrinos of all types interacting with electrons, positrons, protons, and neutrons. Neutrino matter interactions in particle physics community are generally described in terms of discrete-particle collisional effects [15], yielding typically small scattering cross-sections that are proportional to G2F . The small cross-section implies a large mean free path and a long interaction timescale for the neutrino (or momentum) deposition into the plasma environment. On the other hand, self-consistent neutrino-plasma interactions [16,17] produce new non-linear collective phenomena, viz. the excitation of plasma waves whose growth rate is proportional to GF if electron–ion collisions are considered, 2/3 or more strongly ∝ GF . Although discrete-particle effects seem to dominate at very large temperatures [18], collective effects are supposed to play a dominant role during the later stage of the early Universe when the neutrino-plasma temperatures range 1 MeV < T < 10 MeV. The importance of collective interactions has also been emphasized in connection with gravitational waves in cosmology [19,20]. In this Letter, we consider the combined action of intense photon and neutrino beams on electrons and positrons, and show how the presence of a small fraction of ions can produce large space charge electric fields. The latter can separate charges, generating the currents which are required for the magnetic field generation. The present theory thus offers a new scenario for the early Universe primordial magnetic fields [21,22] whose origin has not been understood so far. Furthermore, our results may help to understand the origin of galactic and large scale extra-galactic magnetic fields [23] for which there is no detection yet on scales larger than megaparsec. We consider a relativistically hot electron–positron–ion plasma in the presence of intense photon and neutrino beams. We assume that at equilibrium a small fraction of ions in a pair plasma modifies the quasi-neutrality condition ne − np ≡ Le = Zi ni , while the non-zero chemical potential of the electron–neutrinos and electron– antineutrinos gives rise to a finite value for the neutrino–lepton number density Lν = nνe − nν¯ e , where nνe and nν¯ e are the number densities of the electron–neutrinos and electron–antineutrinos, respectively. Intense circularly polarized photons exert a ponderomotive force on the electrons and positrons. The low-frequency (in comparison with the photon frequency) ponderomotive force comes from the averaging of the non-linear advection, mσ (Tσ )vσ · ∇(γσ vσ ), and the non-linear Lorentz force, qσ vσ × B/c over one photon period, where mσ (Tσ ) is the particle mass [24] depending on the plasma temperature Tσ , γσ = (1 − vσ2 /c2 )−1/2 is the relativistic gamma factor, vσ is the quiver velocity in the photon fields, qσ is the charge (qe = −e, qp = e and qi = Zi e, where e is the magnitude of the electron charge), B = ∇ × A⊥ is the photon magnetic field, and A⊥ is the perpendicular (to an axial direction) component of the vector potential A. We use the Coulomb gauge ∇ · A = 0. The relativistic ponderomotive force of intense photon beams then turns out to be [10]
Fpσ = −mσ (Tσ )c2 ∇γpσ ,
(1)
where γpσ = 1 + e2 |A⊥ |2 /m2σ (Tσ )c4 − 1, mσ (Tσ ) = m0σ G(m0σ c2 /Tσ ), m0σ is the rest mass (typically, in a pair plasma we have m0e = m0p ≡ m0 ), G(ξ ) = K3 (ξ )/K1 (ξ ), and K3 (K1 ) is the McDonald function of the third (first) order. For non-relativistic temperatures (Tσ m0σ c2 ), the effective mass reduces to mσ (Tσ ) = m0σ + 5Tσ /2c2 , while for the ultrarelativistic high temperatures (Tσ m0σ c2 ), the effective mass becomes mσ = 4Tσ /c2 m0σ . On the other hand, collective neutrino-plasma interactions occur when a plasma particle interacts via weak nuclear interaction with the electron–neutrinos and electron–antineutrinos, and behaves as if it were under the
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influence of an effective force [25–27] uσ 1 Gσ ν − ∇nν + 2 ∂t ν + 2 × (∇ × ν ) , Fν = c c
(2)
ν=νe ,¯νe
with ∇ · ν + ∂t nν = 0. Here, nν and = nν vν denote the number density and particle flux of the neutrino fluxes of species ν, vν is the neutrino fluid velocity, uσ is the plasma slow fluid velocity, the effective weak interaction charge Gσ ν is assumed to posses the charge parity symmetry property, namely Geν = −Gpν = −Geν¯ = Gpν¯ , √ √ and Geν = (GF / 2 )(1 + 4 sin2 θW ) ≈ 2 GF . Here, sin2 θW = 1 − (mW /mZ )2 ≈ 0.23 and θW is the Weinberg mixing angle. The relativistic ponderomotive force of the photon beams and weak nuclear force of the neutrinos act on electrons and positrons. The direct action of these two forces on ions is much weaker. Thus, on a timescale much larger than the photon and neutrino periods [28], the equations of motion for the electrons and positrons are, respectively, ue × B − ne me (Te )c2 ∇γpe ne Dt e pe = −∇Pe − ne e E + c √ 1 ue + 2 GF ne − ∇δnν + 2 ∂t δ ν + 2 × (∇ × δ ν ) , (3) c c and
up np Dtp pp = −∇Pp + np e E + × B − np mp (Tp )c2 ∇γpp c √ up 1 − 2 GF np − ∇δnν + 2 ∂t δ ν + 2 × (∇ × δ ν ) , c c
(4)
where Dt σ = ∂t + uσ · ∇, pσ = mσ (Tσ )γσ uσ is the relativistic momentum [24] for the plasma slow motion, γσ = (1 − u2σ /c2 )−1/2 , Pσ = nσ Tσ + σSB Tσ4 /4, σSB = 4π 2 /45h¯ 3 c3 is the Stefan–Boltzmann constant, h¯ is the Planck constant divided by 2π , δnν = nνe − nν¯ e , and δ νe = νe − ν¯ e . The expression for the total pressure Pσ [2] holds when the pair plasma is in equilibrium with the radiation. The space charge electric field and spontaneously created magnetic field are denoted by E and B, respectively. The origin of various terms in Eqs. (3) and (4) is obvious. The left-hand sides are the linear and non-linear inertial acceleration of the electrons and positrons, while the first (second) term in the right-hand side represents the pressure (plasma particles plus photons) gradient (electromagnetic) force, and the photon ponderomotive force and the neutrino weak nuclear force are given by the last two terms in the right-side of Eqs. (3) and (4). We note that the photon ponderomotive force on the electrons and positrons is symmetric, while the neutrino weak nuclear forces acting on electrons and positrons are opposite to each other, thereby breaking the pair dynamics symmetry. The mean magnetic field B is determined from Faraday’s law ∂t B = −c∇ × E.
(5)
The space charge electric field E is obtained by adding Eqs. (3) and (4), yielding √ GF ∇P c2 1 ∇δnν + 2 ∂t δ ν E = − − nσ mσ (Tσ )∇γpσ + 2 eZi ni eZi ni σ =e,p e c +
nσ mσ (Tσ ) J × BP ν − Dt σ γσ uσ , ceZi ni eZi ni σ =e,p
(6)
where √ P = Pe + Pp , J = e(np up − ne ue ) is the sum of the positron and electron currents, and BP ν = B − 2 (GF /ec)∇ × δ ν . Eq. (6) reveals that large space charge electric fields in a pair plasma are excited
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only when the positive ions in the background plasma are present. The positive ions attract electrons and repel positrons, thereby separating them apart and producing currents. The resulting current density J is given by Ampère’s law c ∇ × B. 4πe Taking the curl of Eq. (6) and using Eq. (7) we obtain ∇P × ∇(Zi ni ) c2 nσ mσ (Tσ ) ∇ × E = − ∇γpσ × ∇ e σ =e,p Zi ni eZi2 n2i √ √ GF ∇ × B × (B − 2 (GF /ec)∇ × δ ν ) − 2 2 ∂t (∇ × δ ν ) + ∇ × ec 4πZi ni e2 nσ mσ (Tσ ) − ∇× Dt σ γσ uσ . eZi ni σ =e,p J =
(7)
(8)
Combining Eqs. (5) and (8) we have c ∇P × ∇(Zi ni ) c3 nσ mσ (Tσ ) + ∇γ × ∇ pσ e e σ =e,p Zi ni Zi2 n2i √ √ GF c ∇ × B × (B − 2(G/ec)∇ × δ ν ) ∂t (∇ × δ ν ) − + 2 ∇× ec 4πe2 Zi ni c nσ mσ (Tσ ) + ∇× Dt σ γσ uσ , e σ =e,p Zi ni
∂t B = −
(9)
which is the evolution equation for the mean magnetic field B in a relativistically hot electron–positron–ion plasma. The first term in the right-hand side of Eq. (9), which is the main result of our Letter, is a new Biermann battery (or a baroclinic vector driver), while the second and third terms are associated with the ponderomotive force of non-uniform photon beams and asymmetric neutrino fluxes in a non-uniform pair plasma. The fourth term in Eq. (9) is the curl of Hall forces involving the plasma and neutrino effects, while the fifth term represents the rotational inertial motions of the electrons and positrons. It turns out that the first three terms in the right-hand side of Eq. (9) are the appropriate sources generating magnetic fields, and the Hall and inertial forces are important in non-linear evolution studies. We now present some specific result for the magnetic fields at relativistic temperatures where me (Te ) = mp (Tp ) = m0 (T ) ≡ 4T /c2 . Hence, Eq. (9) for ne ≈ np simplifies to √ GF 16c 2cne σSB 4 ne T ∇γ ∂t (∇ × δ ν ), + + ∇(Z n ) × ∇ T + T × ∇ 2 ∂t B = (10) i i p 4ne e Zi ni ec eZi2 n2i where we have denoted γp = 1 + e2 |A|2 /16T 2 − 1, and retained only sources which generate the magnetic fields. Thus, for given profiles of the ion number density, the plasma temperature, the photon intensity, and neutrino fluxes, one is able to determine the magnitude of the magnetic field strength from Eq. (10). As an illustration, taking ne ∼ 1010Zi ni and T ∼ 1 MeV, we find from Eq. (10) that the Biermann battery generates the magnetic field of one µG over a second when the scale sizes of the density and temperature inhomogeneities are of the order of several milli pc. In summary, we have presented a new theory for the magnetic field generation in an electron–positron–ion plasma in the presence of intense inhomogeneous photon beams and asymmetric neutrino fluxes. Such an admixture of charged particle fluids and energetic photons and neutrinos appears in the early Universe during the later stage
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of lepton era after the Big Bang. Our results show that in a pair plasma with a small fraction of ions, there exists a powerful Biermann battery, in addition to sources involving the photon ponderomotive force and the neutrino driving force. These drivers are capable of generating the primordial magnetic fields in a non-uniform plasma of the early Universe. Our results may also help to explain the origin of magnetic fields in most spiral galaxies, galaxy clusters [21], as well as in extra-galactic environments. The present results may also have relevance to intense magnetic fields that are produced on the surface of neutron stars and during the interaction of super-intense laser beams with hot laboratory plasmas.
Acknowledgements This work was partially supported by the Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 591.
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Generation of magnetic fields in the early Universe P.K. Shukla 1 Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Received 17 January 2003; accepted 8 February 2003 Communicated by V.M. Agranovich
Abstract Nonlinear interactions between intense photon and neutrino beams and relativistically hot electron–positron–ion plasmas of the early Universe are considered. It is shown that a small fraction of ions produces large space charge electric fields in the presence of the plasma pressure gradient, relativistic photon ponderomotive force and weak nuclear driving force of neutrinos. Large space charge electric fields can separate pairs, generating the currents which are required for creating magnetic fields in plasmas. The present theory thus offers a possible clue to the origin of primordial magnetic fields in the early Universe. 2003 Elsevier Science B.V. All rights reserved. PACS: 13.10.-g; 52.30.-q; 98.62.En; 98.80.Cq
It is well known [1–4] that during the later stage of lepton era of the early Universe, the medium was predominantly populated with intense photons, electrons and positrons, and their associated neutrinos and antineutrinos, in addition to a small fraction of protons. In the early Universe, the neutrino plasma temperatures are between 1–2 MeV and 10 MeV and the electron–positron charge separation density (ne − np ) is close to the proton (ion) number density Zi ni 10−10ne , where ne = np + Zi ni , nσ is the number density of the particle species σ (σ equals e for electrons, p for positrons, and i for ions), and Zi is the ion charge state. At relativistic temperatures, intense neutrinos (and antineutrinos) are produced from fusion reactions in violent processes following the Big Bang leading to what we call the relic neutrinos-equivalent to the microwave background, while intense photons are produced due to annihilation of electron–positron pairs. The latter are usually created by the electron bremsstrahlung [5,6], or by accelerated electrons to relativistic speeds in a strong wakefield [7] which is created by intense photons. Computer simulations [8] and recent laboratory experiments [9] have reported the generation of high-energy electrons and positrons by ultraintense lasers. Furthermore, the early Universe plasma also contains the minority ion population, which can affect the electrodynamics of the composite system [10–12]. Intense photons interacting with pair plasmas generate many interesting non-linear effects [10], which have received a great deal of attention in laser–plasma interactions and in studies of radio pulsar emissions from neutron stars. Besides, in
E-mail address: [email protected] (P.K. Shukla). 1 Also at the Department of Plasma Physics, Umeå University, SE-90187 Umeå, Sweden, and the Center for Interdisciplinary Plasma
Science, Max-Planck Institut für Extraterrestrische Physik und Plasmaphysik, D-85740 Garching, Germany. 0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00336-0
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electron–positron–ion plasmas, we have the possibility of relativistic collisionless shock waves [12] and relativistic density pulses which have relevance to astrophysical settings. On the other hand, neutrinos interacting with matter generate charged and neutral currents [13,14] associated with the weak Fermi nuclear interaction whose strength is proportional to GF nσ , where GF = 9 × 10−30 erg cm−3 is the Fermi constant. Charged weak currents appear due to the exchange of the charged vector boson W ± (with mass mW = 80 GeV/c2 , where c is the speed of light in vacuum) associated with the processes involving leptons σ interacting with neutrinos µσ of the same flavor, while neutral weak currents are due to the exchange of the neutral vector boson Z 0 (with mass mZ ≈ 91 GeV/c2 ) associated with processes involving neutrinos of all types interacting with electrons, positrons, protons, and neutrons. Neutrino matter interactions in particle physics community are generally described in terms of discrete-particle collisional effects [15], yielding typically small scattering cross-sections that are proportional to G2F . The small cross-section implies a large mean free path and a long interaction timescale for the neutrino (or momentum) deposition into the plasma environment. On the other hand, self-consistent neutrino-plasma interactions [16,17] produce new non-linear collective phenomena, viz. the excitation of plasma waves whose growth rate is proportional to GF if electron–ion collisions are considered, 2/3 or more strongly ∝ GF . Although discrete-particle effects seem to dominate at very large temperatures [18], collective effects are supposed to play a dominant role during the later stage of the early Universe when the neutrino-plasma temperatures range 1 MeV < T < 10 MeV. The importance of collective interactions has also been emphasized in connection with gravitational waves in cosmology [19,20]. In this Letter, we consider the combined action of intense photon and neutrino beams on electrons and positrons, and show how the presence of a small fraction of ions can produce large space charge electric fields. The latter can separate charges, generating the currents which are required for the magnetic field generation. The present theory thus offers a new scenario for the early Universe primordial magnetic fields [21,22] whose origin has not been understood so far. Furthermore, our results may help to understand the origin of galactic and large scale extra-galactic magnetic fields [23] for which there is no detection yet on scales larger than megaparsec. We consider a relativistically hot electron–positron–ion plasma in the presence of intense photon and neutrino beams. We assume that at equilibrium a small fraction of ions in a pair plasma modifies the quasi-neutrality condition ne − np ≡ Le = Zi ni , while the non-zero chemical potential of the electron–neutrinos and electron– antineutrinos gives rise to a finite value for the neutrino–lepton number density Lν = nνe − nν¯ e , where nνe and nν¯ e are the number densities of the electron–neutrinos and electron–antineutrinos, respectively. Intense circularly polarized photons exert a ponderomotive force on the electrons and positrons. The low-frequency (in comparison with the photon frequency) ponderomotive force comes from the averaging of the non-linear advection, mσ (Tσ )vσ · ∇(γσ vσ ), and the non-linear Lorentz force, qσ vσ × B/c over one photon period, where mσ (Tσ ) is the particle mass [24] depending on the plasma temperature Tσ , γσ = (1 − vσ2 /c2 )−1/2 is the relativistic gamma factor, vσ is the quiver velocity in the photon fields, qσ is the charge (qe = −e, qp = e and qi = Zi e, where e is the magnitude of the electron charge), B = ∇ × A⊥ is the photon magnetic field, and A⊥ is the perpendicular (to an axial direction) component of the vector potential A. We use the Coulomb gauge ∇ · A = 0. The relativistic ponderomotive force of intense photon beams then turns out to be [10]
Fpσ = −mσ (Tσ )c2 ∇γpσ ,
(1)
where γpσ = 1 + e2 |A⊥ |2 /m2σ (Tσ )c4 − 1, mσ (Tσ ) = m0σ G(m0σ c2 /Tσ ), m0σ is the rest mass (typically, in a pair plasma we have m0e = m0p ≡ m0 ), G(ξ ) = K3 (ξ )/K1 (ξ ), and K3 (K1 ) is the McDonald function of the third (first) order. For non-relativistic temperatures (Tσ m0σ c2 ), the effective mass reduces to mσ (Tσ ) = m0σ + 5Tσ /2c2 , while for the ultrarelativistic high temperatures (Tσ m0σ c2 ), the effective mass becomes mσ = 4Tσ /c2 m0σ . On the other hand, collective neutrino-plasma interactions occur when a plasma particle interacts via weak nuclear interaction with the electron–neutrinos and electron–antineutrinos, and behaves as if it were under the
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influence of an effective force [25–27] uσ 1 Gσ ν − ∇nν + 2 ∂t ν + 2 × (∇ × ν ) , Fν = c c
(2)
ν=νe ,¯νe
with ∇ · ν + ∂t nν = 0. Here, nν and = nν vν denote the number density and particle flux of the neutrino fluxes of species ν, vν is the neutrino fluid velocity, uσ is the plasma slow fluid velocity, the effective weak interaction charge Gσ ν is assumed to posses the charge parity symmetry property, namely Geν = −Gpν = −Geν¯ = Gpν¯ , √ √ and Geν = (GF / 2 )(1 + 4 sin2 θW ) ≈ 2 GF . Here, sin2 θW = 1 − (mW /mZ )2 ≈ 0.23 and θW is the Weinberg mixing angle. The relativistic ponderomotive force of the photon beams and weak nuclear force of the neutrinos act on electrons and positrons. The direct action of these two forces on ions is much weaker. Thus, on a timescale much larger than the photon and neutrino periods [28], the equations of motion for the electrons and positrons are, respectively, ue × B − ne me (Te )c2 ∇γpe ne Dt e pe = −∇Pe − ne e E + c √ 1 ue + 2 GF ne − ∇δnν + 2 ∂t δ ν + 2 × (∇ × δ ν ) , (3) c c and
up np Dtp pp = −∇Pp + np e E + × B − np mp (Tp )c2 ∇γpp c √ up 1 − 2 GF np − ∇δnν + 2 ∂t δ ν + 2 × (∇ × δ ν ) , c c
(4)
where Dt σ = ∂t + uσ · ∇, pσ = mσ (Tσ )γσ uσ is the relativistic momentum [24] for the plasma slow motion, γσ = (1 − u2σ /c2 )−1/2 , Pσ = nσ Tσ + σSB Tσ4 /4, σSB = 4π 2 /45h¯ 3 c3 is the Stefan–Boltzmann constant, h¯ is the Planck constant divided by 2π , δnν = nνe − nν¯ e , and δ νe = νe − ν¯ e . The expression for the total pressure Pσ [2] holds when the pair plasma is in equilibrium with the radiation. The space charge electric field and spontaneously created magnetic field are denoted by E and B, respectively. The origin of various terms in Eqs. (3) and (4) is obvious. The left-hand sides are the linear and non-linear inertial acceleration of the electrons and positrons, while the first (second) term in the right-hand side represents the pressure (plasma particles plus photons) gradient (electromagnetic) force, and the photon ponderomotive force and the neutrino weak nuclear force are given by the last two terms in the right-side of Eqs. (3) and (4). We note that the photon ponderomotive force on the electrons and positrons is symmetric, while the neutrino weak nuclear forces acting on electrons and positrons are opposite to each other, thereby breaking the pair dynamics symmetry. The mean magnetic field B is determined from Faraday’s law ∂t B = −c∇ × E.
(5)
The space charge electric field E is obtained by adding Eqs. (3) and (4), yielding √ GF ∇P c2 1 ∇δnν + 2 ∂t δ ν E = − − nσ mσ (Tσ )∇γpσ + 2 eZi ni eZi ni σ =e,p e c +
nσ mσ (Tσ ) J × BP ν − Dt σ γσ uσ , ceZi ni eZi ni σ =e,p
(6)
where √ P = Pe + Pp , J = e(np up − ne ue ) is the sum of the positron and electron currents, and BP ν = B − 2 (GF /ec)∇ × δ ν . Eq. (6) reveals that large space charge electric fields in a pair plasma are excited
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only when the positive ions in the background plasma are present. The positive ions attract electrons and repel positrons, thereby separating them apart and producing currents. The resulting current density J is given by Ampère’s law c ∇ × B. 4πe Taking the curl of Eq. (6) and using Eq. (7) we obtain ∇P × ∇(Zi ni ) c2 nσ mσ (Tσ ) ∇ × E = − ∇γpσ × ∇ e σ =e,p Zi ni eZi2 n2i √ √ GF ∇ × B × (B − 2 (GF /ec)∇ × δ ν ) − 2 2 ∂t (∇ × δ ν ) + ∇ × ec 4πZi ni e2 nσ mσ (Tσ ) − ∇× Dt σ γσ uσ . eZi ni σ =e,p J =
(7)
(8)
Combining Eqs. (5) and (8) we have c ∇P × ∇(Zi ni ) c3 nσ mσ (Tσ ) + ∇γ × ∇ pσ e e σ =e,p Zi ni Zi2 n2i √ √ GF c ∇ × B × (B − 2(G/ec)∇ × δ ν ) ∂t (∇ × δ ν ) − + 2 ∇× ec 4πe2 Zi ni c nσ mσ (Tσ ) + ∇× Dt σ γσ uσ , e σ =e,p Zi ni
∂t B = −
(9)
which is the evolution equation for the mean magnetic field B in a relativistically hot electron–positron–ion plasma. The first term in the right-hand side of Eq. (9), which is the main result of our Letter, is a new Biermann battery (or a baroclinic vector driver), while the second and third terms are associated with the ponderomotive force of non-uniform photon beams and asymmetric neutrino fluxes in a non-uniform pair plasma. The fourth term in Eq. (9) is the curl of Hall forces involving the plasma and neutrino effects, while the fifth term represents the rotational inertial motions of the electrons and positrons. It turns out that the first three terms in the right-hand side of Eq. (9) are the appropriate sources generating magnetic fields, and the Hall and inertial forces are important in non-linear evolution studies. We now present some specific result for the magnetic fields at relativistic temperatures where me (Te ) = mp (Tp ) = m0 (T ) ≡ 4T /c2 . Hence, Eq. (9) for ne ≈ np simplifies to √ GF 16c 2cne σSB 4 ne T ∇γ ∂t (∇ × δ ν ), + + ∇(Z n ) × ∇ T + T × ∇ 2 ∂t B = (10) i i p 4ne e Zi ni ec eZi2 n2i where we have denoted γp = 1 + e2 |A|2 /16T 2 − 1, and retained only sources which generate the magnetic fields. Thus, for given profiles of the ion number density, the plasma temperature, the photon intensity, and neutrino fluxes, one is able to determine the magnitude of the magnetic field strength from Eq. (10). As an illustration, taking ne ∼ 1010Zi ni and T ∼ 1 MeV, we find from Eq. (10) that the Biermann battery generates the magnetic field of one µG over a second when the scale sizes of the density and temperature inhomogeneities are of the order of several milli pc. In summary, we have presented a new theory for the magnetic field generation in an electron–positron–ion plasma in the presence of intense inhomogeneous photon beams and asymmetric neutrino fluxes. Such an admixture of charged particle fluids and energetic photons and neutrinos appears in the early Universe during the later stage
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of lepton era after the Big Bang. Our results show that in a pair plasma with a small fraction of ions, there exists a powerful Biermann battery, in addition to sources involving the photon ponderomotive force and the neutrino driving force. These drivers are capable of generating the primordial magnetic fields in a non-uniform plasma of the early Universe. Our results may also help to explain the origin of magnetic fields in most spiral galaxies, galaxy clusters [21], as well as in extra-galactic environments. The present results may also have relevance to intense magnetic fields that are produced on the surface of neutron stars and during the interaction of super-intense laser beams with hot laboratory plasmas.
Acknowledgements This work was partially supported by the Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 591.
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