Linear wave spectrum associated with collective neutrino-plasma interactions in the early universe

The effects of collective neutrino-plasma interactions on the linear wave spectrum supported by a magnetized electron-positron plasma in the presence of a neutrino-antineutrino medium are investigated. When a pair-symmetric background neutrino-plasma medium is perturbed by space-charge waves (electrostatic waves associated with electron-positron charge separation), our analysis shows that the neutrino and antineutrino fluids also separate and the pair symmetry of the background medium is broken. The cosmological implications of this pair-symmetry breaking mechanism are briefly discussed.


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associated with processes involving neutrinos of all types interacting with arbitrary charged and neutral particles.
A measure of the strength of weak interactions is expressed in terms of the neutrino meanfree-path ν 1.4 × 10 11 T −5 MeV cm [7], where T MeV denotes a characteristic temperature expressed in MeV units. Although neutrino-matter interactions are very weak under normal conditions (T 1 MeV), we note that the ratio of the neutrino-matter collision rate (c/ ν 0.21 T 5 MeV s −1 ) to the Hubble expansion rate (cH 0.67 T 2 MeV s −1 ) is larger than unity for neutrino-matter temperatures above 1-2 MeV. Such high temperatures occurred within the first second after the Big Bang [7] and, consequently, the neutrino medium is strongly coupled to the matter medium during the early universe (T > 1 MeV).
Neutrino-matter interactions are described either in terms of discrete-particle collisional effects, with typical scattering cross-sections which scale with the strength of weak interactions as G 2 F (e.g., ν ∝ G −2 F ), or in terms of collective (self-consistent) effects, which scale as G F (see below). Although discrete-particle effects tend to dominate over collective effects at very large temperatures [8], collective effects may be important during the later eras of the early universe. Indeed, the characteristic frequency associated with collective plasma oscillations (ω p 2.7 × 10 20 T

3/2
MeV rad s −1 ) is larger than the neutrino-matter collision rate c/ ν by a factor ω p ν /c 1.3 × 10 21 T −7/2 MeV , which is larger than unity when the neutrino-matter temperatures are smaller than 10 6 MeV. In addition, the ratio of the plasma frequency to the photon-matter collision rate (c/ γ 4.8 × 10 17 T 3 MeV s −1 ) is expressed as ω p γ /c 560 T

−3/2
MeV , which exceeds unity for temperatures less than 70 MeV. Hence, the short-time evolution of the primordial neutrino-plasma medium (in the temperature range 1 MeV < T < 10 MeV) appears to be dominated by collective (collisionless) effects.
In this paper, therefore, we investigate collective neutrino-plasma interactions in the early universe when the neutrino-plasma temperatures are between 1-2 MeV and 10 MeV. This temperature range corresponds to the later stage of the lepton era [7], when the universe is predominantly populated with photons, electrons, positrons and their associated neutrinos and antineutrinos.

Effective weak-interaction charge G σν
As a neutrino propagates through a stationary (unpolarized) matter medium, it experiences a force derived from an effective potential [9,10] of the form σ G σν n σ , where n σ denotes the matter density associated with particle species σ and G σν denotes the effective weak-interaction charge associated with σ-ν interactions. For example, the effective potential experienced by electron neutrinos (or antineutrinos) propagating in a stationary electron-positron medium is expressed in terms of the effective weak-interaction charge [9,10] where the first term combines the lowest-order (∝ G F ) contributions of charged and neutral weak currents (with sin 2 θ W = 1 − (m W /m Z ) 2 0.22), while the second term involves higherorder (∝ G F /m 2 W ) contributions, where E ν denotes the neutrino (antineutrino) energy and E σ denotes the average matter-particle energy for species σ. For an electron-positron plasma, we 97.3 note that the effective potential becomes where ∆n e = n e − n e denotes the electron-positron charge-separation density and the (±) signs in the first term refer to ν e neutrinos (+) or ν e antineutrinos (−). Note that the first term in equation (2) is antisymmetric to the charge-parity (CP) interchange (e ↔ e and ν ↔ ν) while the second term is CP-symmetric (see [9] for further discussion). The charge neutrality of the early universe during the lepton era, however, dictates that the charge-separation density ∆n e be very small (i.e., the charge separation density ∆n e is equal, by quasi-neutrality, to the proton density n p 10 −10 n e ). The ratio of the CP-asymmetric term to the CP-symmetric term in equation (2) is expressed approximately as (∆n e /10 −10 n e )/4 T 2 MeV and, hence, the CP-symmetric term dominates over the CP-asymmetric term for neutrino-plasma energies above 1 MeV, i.e., for equilibrium conditions of primordial electron-positron plasmas typically found during the lepton era of the early universe.
If the primordial electron-positron plasma is perturbed by space-charge waves (electrostatic waves associated with electron-positron charge separation), however, neutrinos can feel the influence of the effective CP-asymmetric potential. We henceforth ignore higher-order corrections in equation (1) and investigate collective neutrino-plasma interactions in an electronpositron plasma perturbed by space-charge waves (for which ∆n e 10 −10 n e ). In this space-charge scenario, the effective weak-interaction charge G σν is assumed to possess the CP-symmetry property where σ = e (σ = e) and ν = ν e (ν = ν e ), and G eν √ 2G F . In the present work, we also ignore neutrino collective self-interactions [10] since we are interested in collective neutrino-plasma effects only.

Collective neutrino-plasma interactions
Our discussion of collective neutrino-plasma effects begins with the problem of a single neutrino propagating in an unpolarized matter medium composed of charged and/or neutral particles. When a neutrino (of species ν) propagates (with velocity v ν ) in a moving matter medium, it interacts by weak interaction with matter particles (of species σ) and behaves as if it were under the influence of an effective force [11] where n σ and Γ σ = n σ v σ /c denote the density and normalized particle flux of the matter fluid of species σ. Similarly, when a matter particle (of species σ) propagates (with velocity v σ ) in a moving neutrino medium, it interacts by weak interaction with neutrinos and behaves as if it were under the influence of an effective force [11] ν G σν − ∇n ν + 1 c 97.4 where n ν and Γ ν = n ν v ν /c denote the density and normalized particle flux of the neutrino fluid of species ν. The ν-σ symmetry of the two effective forces (4) and (5) arises from the variational formulation of collective neutrino-plasma interactions introduced in [11]. The existence of effective forces (4) and (5) on a single neutrino-matter particle implies that neutrino-matter fluids are self-consistenly coupled through collective weak-interaction effects. Hence, for example, electromagnetic fields can influence neutrino-fluid dynamics in the presence of a charged-plasma medium just as if neutrinos carried an effective electrical charge [12,13,14]. The purpose of the present work is to investigate the effects of collective neutrino-plasma interactions on the linear wave spectrum supported by a magnetized electron-positron plasma in the presence of a neutrino-antineutrino medium.

Organization
The remainder of this paper is organized as follows. In section 2, the basic model for collective neutrino-plasma interactions in the presence of an electromagnetic field is derived on the basis of nonlinear fluid equations derived by variational principle by Brizard, Murayama, and Wurtele [11]. In section 3, the nonlinear fluid equations derived in section 2 are linearized about a stationary uniform neutrino-plasma medium. We also assume the background medium to be pair symmetric, i.e., the electron lepton number density L e = n e − n e and the neutrino lepton number density L ν = n ν − n ν are both assumed to be zero at equilibrium (i.e., the chemical potential for each fluid species is zero). Since the primordial plasma is thought to be magnetized (for recent reviews on primordial magnetic fields, see [15]), our analysis also includes the presence of a uniform magnetic field. The normal-mode analysis of the linearized fluid equations leads to a dispersion relation whose branches include the standard plasma branches associated with electromagnetic and electrostatic waves propagating in a magnetized electron-positron plasma (see [16]- [19]), now modified by collective neutrino-plasma effects and a pure neutrino-sound branch associated with bulk neutrino-density propagating waves.
In section 4, we investigate space-charge waves [20] associated with electrostatic modes driven by electron-positron charge separation. In summarizing our work in section 5, we discuss the potential implications of neutrino-modified space-charge waves for early universe cosmology, since they provide an effective CP-symmetry breaking mechanism leading to leptogenesis.

Fluid model for collective neutrino-plasma interactions
The equations for the relativistic fluid dynamics associated with collective neutrino-plasma interactions in the presence of an electromagnetic field were recently derived from a variational principle by Brizard, Murayama, and Wurtele in [11]. Each fluid is represented by its proper density γ −1 s n s , its fluid velocity v s and its normalized particle-flux density Γ s = n s v s /c, its fluid pressure p s and its normalized fluid enthalpy w s = h s /m s c 2 (where enthalpy h s = µ s + S s T s is defined in terms of the chemical potential µ s , entropy S s , and temperature T s ). Here, the fluid label s is either s = e and e for electrons and positrons, respectively (with mass m e = m e , electric charge q e = e = − q e , and effective weak charge G eν = − G eν ), or s = ν and ν for electron-neutrinos and electron-antineutrinos, respectively (with mass m ν = m ν = 0 and effective weak charge G eν = − G eν ).
Each fluid species s satisfies a continuity equation The relativistic force equations for the electron-positron plasma fluids are [11] where d s /dt = (∂ t + v s ·∇) denotes a total convective time derivative for fluid species s and the asymmetric neutrino-flux four-vector The first two sets of terms on the right-hand side of equations (7) and (8) represent the classical pressure-gradient and electromagnetic forces. The remaining terms are associated with CPasymmetric neutrino-antineutrino forces, which arise as a result of collective neutrino-plasma interactions.
Next, the relativistic force equations for neutrino-antineutrino fluids are [11] where the asymmetric particle-flux four-vector ∆Γ α e = (∆n e , ∆Γ e ) is defined as ∆Γ α e ≡ Γ α e − Γ α e . The driving forces appearing on the right-hand side of equations (9) and (10) include not only the classical pressure-gradient forces but also the asymmetric electron-positron driving forces.
Lastly, the evolution of the electromagnetic field is expressed in terms of the Maxwell which couple the evolution of the electromagnetic field to plasma charge densities and currents through the asymmetric electron-positron particle-flux four-vector. Note that using equation (11) we find where 2 = ∇ 2 −c −2 ∂ 2 t is the D'Alembertian operator. Thus, substituting these expressions into equations (9) and (10), we find that collective neutrino-plasma interactions allow electromagnetic fields to influence neutrino dynamics just as if neutrinos carried an effective electric charge [12,13,14].
In summary, the coupled equations (7)-(11) provide a self-consistent model for the collective neutrino-plasma interactions in the presence of an electromagnetic field. This self-consistency implies, for example, that electromagnetic fields can generate asymmetric neutrino flows in the presence of a plasma medium or that collective neutrino-plasma interactions can assist in generating electromagnetic fields [11].

Normal-mode representation
We now consider the linear wave spectrum supported by a warm electron-positron plasma (with densities n e0 = n e0 and temperatures T e0 = T e0 ) interacting with a warm neutrino-antineutrino fluid (with densities n ν0 = n ν0 and temperatures T ν0 = T ν0 ) in the presence of a uniform magnetic field B 0 = B 0 z.
We first introduce the following representation for the electromagnetic field where we take the background magnetic field to be B 0 = B 0 z, and the neutrino-plasma medium n s = n s0 + δn s Γ s = δΓ s = n s0 δv s /c p s = p s0 + (dp s0 /dn s0 )δn s where we assume (for simplicity) that the scalar pressures p e0 and p ν0 satisfy the equation of state p s0 = p s0 (n s0 ). We adopt a semi-relativistic approach based on the expansion of the relativistic factors γ s to second order in |v s | 2 /c 2 while retaining the enthalpy correction to the particle's mass M s = m s + h s0 /c 2 . We also introduce the dimensionless quantity K s = 1 M s c 2 dp s0 dn s0 defined as the square of the ratio of the speed of sound in fluid species s to the speed of light.
Next, we introduce the normal-mode representation for the perturbation fields given in equations (13) and (14), where δ(· · ·) → δ(· · ·) exp(ik·x − iωt) and k = k (cos θ z + sin θ x) ≡ k k. We also introduce the bulk energy density and (normalized) bulk energy density flux and the CP-asymmetry density and (normalized) current δQ s = −(4πiec/ω) (δn s − δn s ) The normalization (16) is chosen so that the perturbed densities δQ s and δJ s have the same units as the perturbed electromagnetic fields. 97.7

Linearized coupled wave equations
The normal-mode analysis involves 22 perturbation fields associated with the electronpositron fluid (δ e , δp e ; δQ e , δJ e ), the neutrino-antineutrino fluid (δ ν , δp ν ; δQ ν , δJ ν ), and the electromagnetic field (δE, δB). The continuity equations (6) where s = e or ν. The perturbed electromagnetic field (δE, δB) satisfies the linearized Maxwell equations kc·δE = − ωδQ e kc·δB = 0 We can combine the last two Maxwell equations to eliminate the perturbed magnetic field δB in favour of the perturbed electric field δE to obtain where N = N k k and N = k 2 c 2 /ω 2 denotes the square of the refractive index. Next, the linearized versions of the electron and positron fluid equations (7) and (8) yield where we have substituted (kc/ω)δ e = N·δp e and we have defined X = ω 2 c /ω 2 and Y = ω 2 p /ω 2 . Here, we have introduced the following enthalpy-modified neutrino-plasma parameters: the electron-positron plasma frequency the electron-positron gyrofrequency ω c ≡ eB 0 /M e c, and the dimensionless neutrino-plasma interaction coefficient so that α e 1 for T e 10 GeV. Similarly, the electron-neutrino and electron-antineutrino fluid equations (9) and (10) become where α ν ≡ 2n ν0 G eν /M ν c 2 1. The corresponding equations for δ ν and δQ ν can be obtained from equations (22) and (23), respectively, by taking their dot product with (kc/ω) and using the continuity equations (17) to obtain

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We note that the neutrino-bulk equation (24) yields the linear dispersion relation 1 = K ν N or ω 2 = K ν k 2 c 2 , which describes neutrino sound waves in a neutrino-antineutrino medium when δ ν = 0 (i.e., when δn νe = −δn νe ). Under the model considered here (which retains only a CP-asymmetric weak-interaction charge), however, neutrino sound waves are decoupled from collective neutrino-plasma effects.

Linear dispersion relation for collective neutrino-plasma interactions
We now show that collective neutrino-plasma interactions couple density perturbations δn νe and δn νe in such a way that, if δn νe = 0, then δ ν = 0 and δQ ν = 0, i.e., collective neutrino-plasma interactions generate CP asymmetries in the neutrino-antineutrino medium. First, equation (20) can be inverted to give where ∆ e = 1 − K e N = det(I − K e N). Next, equation (23) can be inverted to yield where ∆ ν = 1 − K ν N = det(I − K ν N) = 0. Note that, according to equation (25), the CPasymmetry of the neutrino-antineutrino medium (δQ ν = 0) and charge separation δQ e = 0 imply that 1 − K ν N = 0 and thus the neutrino-antineutrino bulk density must vanish δ ν = 0 (or δn νe = −δn νe ). Lastly, the second term on the right-hand side of equation (21) can be rewritten using equation (26) as while the third term on the right-hand side of equation (21) can be rewritten using equation (27) as where β = α e α ν 1. By combining these results and introducing the dielectric tensor we find the following non-vanishing components for the dielectric tensor N cos 2 θ) ,

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where R = K e + β (K ν + N − 2)/∆ ν and By substituting equation (28) into equation (19), the Maxwell equations become where the dispersion tensor D (ω, N, θ) has the non-vanishing components Nontrivial solutions of equation (29) exist for δE only if which holds if D yy = 0 or D xx D zz = D xz D zx . Note that the non-diagonal components D xz and D zx vanish at θ = 0 (parallel propagation, k B 0 ) and θ = π/2 (perpendicular propagation, k⊥B 0 ). Since the charge neutrality of the electron-positron plasma invalidates the assumptions of our space-charge model, we turn our attention to electrostatic modes.

Electrostatic modes
Electrostatic modes satisfy the condition k·δE = 0 and δB = 0; these modes are also referred to as space-charge waves since δQ e = −(kc/ω)·δE = 0 and δJ e = − δE, according to equation (18). We also note that space-charge waves can induced bulk motion in the electron-positron medium, with δp e = − δE × B 0 and δ e = 0. For parallel propagation (θ = 0 and k = z), the wave polarization is δE = δE z and the dispersion relation is the dispersion relation Here, collective neutrino-plasma effects (proportional to β) modify the plasma dispersion relation K e N = 1 − Y ≡ P or ω 2 = ω 2 p + k 2 (K e c 2 ), where K e c 2 represents the square of the speed of sound in the electron-positron medium (which includes effects due to finite enthalpy).
For perpendicular propagation (θ = π/2 and k = x), the wave polarization is δE = δE x and the dispersion relation is

Space-charge waves
We now consider how space-charge (electrostatic) waves in a warm electron-positron plasma are affected by collective neutrino-plasma interactions (see [16,18] for further deatils). Since δQ e = 0 for space-charge waves, we find from (25). There are two branches associated with space-charge waves: the Langmuir branch D zz = 0 (for θ = 0 and δE B 0 ) and the upper-hybrid branch D xx = 0 (for θ = π/2 and δE⊥B 0 ).

Langmuir branch
The dispersion relation (32) for the Langmuir branch yields the quadratic equation where P = 1 − Y . There are two solutions N ± to this equation: Expanding the solution N + up to first order in β (for K s = 0) yields so that Substituting these expressions into equation (34) yields Since the neutrino-sound waves are recovered from the solution for N + in the absence of neutrinoplasma coupling (β = 0), we call this solution the neutrino solution of the Langmuir branch. Similarly, expanding the solution N − up to first order in β (for K s = 0) yields so that

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Substituting these expressions into equation (34) now yields Since the standard Langmuir plasma waves are recovered from the solution for N − in the absence of neutrino-plasma coupling (β = 0), we call this solution the plasma solution of the Langmuir branch.

Upper-hybrid branch
A similar treatment of the upper-hybrid branch is facilitated by the fact that the formulae presented in section 1.1 can be used with the substitution P → S = 1 − X − Y . Hence, the coupled neutrino-plasma upper-hybrid branch has a neutrino solution N + and a plasma solution N − , with similar expressions for δQ (±) ν . The neutrino solution of the upper-hybrid branch is and while the plasma solution of the upper-hybrid branch is and It is clear that upper-hybrid waves retain their relationship with Langmuir waves as their magnetized analogues [16] even in the presence of collective neutrino-plasma interactions.

Discussion
We now comment on the relations between the perturbed CP-asymmetric densities δQ (±) ν and δQ e associated with Langmuir and upper-hybrid waves in a magnetized neutrino-plasma medium. The neutrino solution δQ (+) ν associated with the Langmuir and upper-hybrid waves is proportional to α −1 e δQ e δQ e . Hence, even a small electron-positron charge separation can lead to a large CP-asymmetric neutrino-antineutrino density δQ (+) ν . The plasma solution δQ (−) ν , on the other hand, is proportional to α ν δQ e δQ e . Hence, a small electron-positron charge separation leads to a very small CP-asymmetric neutrinoantineutrino density δQ (−) ν . We can reverse the argument, however, and say that a CP-asymmetric neutrino-antineutrino density δQ ν can induce a large electron-positron charge separation δQ (−) e proportional to α −1 ν δQ ν δQ ν . The present work has thus revealed that collective neutrino-plasma interactions may be quite efficient at breaking the leptonic CP-symmetry of the early universe when it is perturbed by spacecharge waves (whether a primordial magnetic field exists or not). The role of leptonic asymmetry 97.12 in cosmology has received a lot of attention recently within the context of the formation of largescale structure and the cosmic microwave background anisotropy [21,22,23], and big-bang nucleosynthesis [24]. Hence, it is possible that in the last era of the early universe (just before neutrinos and antineutrinos decoupled from matter), the electron and electron-neutrino lepton number densities L e = n e − n e and L ν = n νe − n νe both experienced large fluctuations as a result of CP-symmetry breaking collective neutrino-plasma interactions. It is perhaps possible that such large fluctuations could be imprinted in the cosmic neutrino background itself [3,25].