Generation and evolution of cosmological magnetic fields at the electroweak epoch Temperature of primordial plasma [GeV] Oleg 100 50 10 5 1 0.5 0.1 0.05 105 Short RUCHAYSKIY 4 wavelength ψ Aλ(k2) 10 modes 1000 Transfer of magnetic helicity ψ 100 Bν (p) 10 Long wavelength Magnetic Helicity modes ψ Aµ(k1) 1 Electroweak QCD epoch COSMO 2015 epoch Time from Big Bang September 10, 2015 ¡ Phys. Rev. Lett. 108 (2012) 031301 [arXiv:1109.3350] (with A. Boyarsky and J. Frohlich)¨ Phys. Rev. Lett. 109 (2012) 111602 [arXiv:1204.3604] (with A. Boyarsky and M. Shaposhnikov) Phys. Rev. D (2015) [arXiv:1504.04854] (with A. Boyarsky and J. Frohlich)¨ Magnetic fields in the Universe Our Universe today is magnetized: (earth, stars, galaxies, clusters of galaxies) Magnetic fields in the spiral (i.e. rotating) galaxies can be generated by turbulent effect from tiny primordial seeds (“dynamo mechanism”) Magnetic fields in elliptical galaxies? Magnetic fields in clusters of galaxies = sum of magnetic fields of galactic magnetic fields 6 Even intergalactic medium seems to be filled with magnetic fields Neronov & Vovk’10; Dolag et al.’10; Are we observing the evidence of process in the very early Tavecchio et Universe? al.’11 Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 1 Magnetic fields in the early Universe Magnetic fields affect every important process in the early Universe: Many, many works. – Change the nature of electroweak phase transition Recent reviews: – Affect baryogenesis Widrow et – Leave its imprints in production of gravity waves al.’11; – Affect BBN Kandus et al.’11; – Leave its imprints in CMB Yamazaki et – Affect structure formation al.’12; – Could have played a role of seeds of galactic magnetic fields Durrer& Neronov’13 Many mechanisms of generation of primordial magnetic fields exist, usually associated with violent events (at inflation, electroweak transition, QCD transition, etc.) It is commonly believed that noticeable magnetic fields are not generated in the Universe filled with Standard Model particles Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 2 Main message Primordial plasma (MeV-GeV temperatures) behaves very different from a laboratory plasmas In fact this is a rare situation when quantum effects affect dynamics at macroscopic (arbitrarily large) scales What I discuss in this talk does not rely on any new physics and is true fully within the Standard Model of particle physics Who should care about this? – People who study cosmological magnetic fields – People who work on leptogenesis and related themes Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 3 Axial symmetry Massless fermions can be left and right-chiral (left and right moving): ¨* 0 ! ¨m¨ i(@ + ~σ ~ ) µ ¨H ¨ t L (iγ @µ ¨mH) = − ·0r = 0 ¨¨* − i(@ ~σ ~ ) ¨m R t − · r ¨− where γ = and γ = iγ γ γ γ . 5 R;L ± R;L 5 0 1 2 3 R 3 y R 3 y Number of left NL = d x L L and right NR = d x R R particles is conserved independently Electric charge: Q = NL + NR – gauge symmetry Axial charge: Q = N N – global symmetry 5 L − R Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 4 Axial anomaly Gauge interactions respects chirality (Dµ = @µ + eAµ). 0 ! ¨¨* ~ ¨¨m i(Dt + ~σ D) L − ¨* 0· = 0 i(D ~σ D~ ) ¨m¨ R t − · ¨− . but the difference of left and right-movers is not conserved once the quantum corrections are taken into account — axial anomaly dQ d(N N ) Z α Z 5 = L − R = d3~x @ j5 = d3 ~xE~ B~ dt dt µ µ π · e2 α ≡ 4π – fine-structure constant Aν jµ h 5 i Aµ Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 5 Chiral anomaly Right particle with electric charge eR in magnetic field B Landau energy levels Asymmetric branch on n=0 Landau level QM interpretation: spectralE(pz flow)= cpz 2 2 Ε ~ Landau levels: E = pz + e B (2n + 1) + 2e(pBz)~s j j · G.E. Volovik, n=2cond- Only particles with (B~ ~s< 0) have mat/9802091 · massless branches: n=1 −p left branch (~p · ~s) > 0 E = z pz right branch (~p · ~s) < 0 n=0 pz e R Electric field E~ = Ez^ creates right particle( because pz(t) = pz(0) + eEt) Electric field destroys left particles Total number does not change n=-1 Difference of left minus right n=-2 appears – chiral anomaly! Particles creation from vacuum Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 6 left particles can In applied electric field E : compensate the effect • if pz = eRE left-right symmetry is exact Particles from negative energy levels of Dirac vacuum are pushed into positive energy levels of matter • n π2 2 2 = (1 / 4 ) [(eR) - (eL) ] E•B Axial anomaly change of helicity $ If we introduce magnetic helicity Z = d3x A~ B~ H · . then Z 1d (t) d3x E~ B~ = H · −2 dt Axial anomaly couples change of left-right asymmetry to the change of magnetic helicity: d α (N N ) + = 0 dt L − R πH Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 7 Axial anomaly at finite densities – Fermi level for left B =0 µL 6 fermions µL µR µR – Fermi level for right fermions Change helicity of B~ -field due to axial anomaly of N changes: ) L;R Z tf Z α Z δNL;R = dt N_ L;R(t) = dt dV E B ti ∓ π · Nielsen & Ninomiya α(µ µ ) Z The energy: δ = δN µ + δN µ = L − R dV A B (1983); E L L R R 2π · Rubakov (1986) Free energy of magnetic fields in plasma: δ = R dV α(µL−µR)A B F 2π · Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 8 Chern-Simons term In coordinate space Π = 0 leads to a Chern-Simons term: 2 6 1 Z α [A~] = d3x B~ 2 + µ A~ B~ F 2 π 5 · Consider configuration A~ = A0 cos(kz); sin(kz); 0 The magnetic field B~ = kA~ (maximally helical configuration) − 1 Z α The effective action: [A] = d3x k2 k µ A2 < 0 F 2 − π 5 0 α Increasing amplitude for k < µ we lower energy Instability π 5 ) Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 9 Chiral Magnetic effect In coordinate space Π = 0 leads to a Chern-Simons term: Vilenkin’78; 2 6 Redlich & 1 Z α [A~] = d3x B~ 2 + µ A~ B~ Wijewardhana F 2 π 5 · (1985); Rubakov’86; Chiral magnetic effect: Cheianov, Frohlich,¨ Alexeev’98; δ [A~] 2α ~| = F = µ5B~ Frohlich¨ & δA~ π Pedrini’00,’02; Fukushima, Kharzeev, Warringa’08 Son & Surowka’09 Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 10 Maxwell equations The presence of difference of chemical potential of left and right fermions leads to additional terms in the effective Lagrangian for electromagnetic fields – Chern-Simons term As a result Maxwell equations contain current, proportional to µ5 Kharzeev’11 — MHD turns into chiral MHD: Vilenkin (1978) @B~ curl E~ = Frohlich¨ & − @t Pedrini 2α (2000–2001) curl B~ = σE~ + µ5B~ Chiral magnetic effect π Joyce & Shaposhnikov (1997) In addition, µ5 should be allowed to become dynamical: d(NL − NR) dµ5 α Z / / d3x E~ · B~ dt dt π Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 11 Dynamics of µ5 Boyarsky, Frohlich,¨ O.R., PRL (2012) @B~ curl E~ = − @t 2α curl B~ = σE~ + µ B~ Chiral magnetic effect π 5 Z @µ5 2α d3x E~ B~ Γ µ chirality-flipping due to finite mass of fermions @t / π · − flip 5 Without B chirality flipping reactions drive µ 0 (µ = µ e−Γflipt) 5 ! 5 0 2 −k t Without µ5 finite conductivity drives B 0 (Bk = B0e σ ). This is called magnetic diffusion ! Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 12 Instability Maxwell equations with µ5 are unstable: @B 1 αµ = 2B + 5 curl B @t σr π magnetic diffusion instability magnetic diffusion α Exponential growth for k < µ (for one of the circular polarizations π 5 depending on the sign of µ5)— generation of helical magnetic fields 2 k αkµ5 B± = B exp t t 0 − σ ± π σ Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 13 Attractor solution Consider sharply peaked at k0 maximally helical field ρB(k) dµ5 XXX = ρ µ µ¯ ΓXµX dt − B 5 − 5 − flip X5 dρ ρ µ B = B 5 1 dt tσ (¯µ5) − where 2σ and tσ = 2 ρB Γflip k k0 0 2πk0 Large ρB drives µ5 to an attractor solution µ¯5 = O.R. with α A. Boyarsky, J. Frohlich¨ PRL 2012 Electric conductivity of the plasma is finite but magnetic diffusion [1109.3350] is compensated by the presence of µ5 Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 14 Evolution of magnetic energy density T@GeVD PRL (2012) [1109.3350] 100 50 10 5 1 0.5 0.1 1 0.1 B G 0.01 Change of 0.001 - 10 4 16.0 16.5 17.0 17.5 18.0 18.5 Conformal time lgHM*TL See also Tashiro et al. [1206.5549] X-axis – log of time Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 15 Two modes ρB(k) T@GeVD 100 50 10 5 1 0.5 0.1 105 104 3 T k 1000 H 100 Helicity 10 1 16.0 16.5 17.0 17.5 18.0 18.5 k k T@GeVD ÐÓÒg ×hÓÖØ 100 50 10 5 1 0.5 0.1 0.05 5105 In case of two modes the helicity gets transferred from the shorter 2105 T one to the longer one 1105 Μ 5106 Chemical potential follows the wave-number of the mode with 2106 2πk 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 higher helicity µ = α Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 16 Evolution of chemical potential T@GeVD 100 50 10 5 1 0.5 0.1 0.05 5105 2105 T Μ 1105 5106 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 Conformal time lgHMTL Process continues while ρB Γflip Oleg RuchayskiyGENERATION AND EVOLUTION OF COSMOLOGICAL MAGNETIC FIELDS 17 Evolution of chemical potential 105 106 T 107 Μ 108 109 1010 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 Conformal time, lgHMTL Continuous initial spectrum with Hk / k and fraction of magnetic energy density 5 × 10−5 (blue) or 5 × 10−4 (green).