Generation of Large-Scale Magnetic Fields from Inflation in Teleparallelism

Transcription

1 VIIIth Rencontres du Vietnam - Beyond the Standard Model Generation of Large-Scale Magnetic Fields from Inflation in Teleparallelism Reference: JCAP 10 (2012) 058 Ling-Wei Luo Department of Physics, National Tsing Hua University, Hsinchu, Taiwan (NTHU ) Collobrators: K. Bamba (KMI, Nagoya University, Japan), C.Q. Geng (NCTS/NTHU, Taiwan) Quy Nhon, Dec. 21, 2012 Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 0/ 18

2 Outline 1 Geometrical Background 2 Large-Scale Magnetic Fields 3 Magnetic Fields in Teleparallelism 4 Summary and Conclusions Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 1/ 18

3 Outline 1 Geometrical Background 2 Large-Scale Magnetic Fields 3 Magnetic Fields in Teleparallelism 4 Summary and Conclusions Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 1/ 18

4 Riemann-Cartan Geometry U 4 Gravitational theory described on Riemannian geometry V 4 with metric g µν and metric-compatible Levi-Civita connection { ρ µν} = 1 2 gρλ (g νλ,µ + g λν,µ g µν,λ ) is Einstein s general relativity established in The metric-compatible affine connection in U 4 is Γ ρ µν = { ρ µν} + K ρ µν, where the second term is contorsion tensor K ρ µν = 1 2 (T ρ µν T µ ρ ν T ν ρ µ ) with the torsion tensor is the antisymmetric part of affine connection T ρ µν = Γ ρ µν Γ ρ νµ. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 2/ 18

5 Introducing torsion: in order to couple spin angular momentum in gravity (Élie Cartan 1922.) Gravity as a gauge theory Einstein-Cartan-Sciama-Kibble theory (ECSK) S ECSK = d 4 x [ ] 1 g 2κ 2 R(g, T ). Include massless Rarita-Schwinger field (gravitino, Rarita & Schwinger 1941) Supergravity (gauge SUSY) Include torsion part in the Lagrangian Poincaré gauge theory Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 3/ 18

6 What is Teleparallelism? A special case of Poincaré gauge theory (only quadratic torsion part) with absolute parallelism. Introduce the orthonormal frame (veirbein) g µν = η AB e A µ e B ν = e µb e B ν g = e. Weitzenböck geometry W 4 is define by the metric-compatible Weitzenböck connection W ρ µν = e ρ A µe A ν with Riemann tensor R σ ρµν(w ) = 0. The gravitaty theory on Weitzenböck geometry is so-called new general relativity (NGR) (Hayashi & Shirafuji 1979.) The torsion tensor in terms of the vierbein is T ρ µν e ρ A ( µ e A ν ν e A µ ). Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 4/ 18

7 NGR Lagrangian L NGR = 1 [ ] 2κ 2 e a 1 T ρ µνt µν ρ + a 2 T ρ µνt νµ ρ + a 3 T ρ µρt νµ ν. The vector part of torsion tensor is v µ T ν νµ and the axial-vector part is a µ 1 6 ɛ µνρσt νρσ. The NGR Lagrangian can be rewritten to L NGR = 1 [b 2κ 2 e 1 t ρµνt ] µν ρ + b 2 v µ v µ + b 3 a µ a µ. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 5/ 18

8 Teleparallel Equivalent to GR: R(W ) = R({}) + T 2 µ v µ 0. }{{} boundary term TEGR Lagrangian L T EGR = 1 [ 1 2κ 2 e Sometimes written the form 4 T ρ µνt ρ µν T ρ µνt νµ ρ T ρ µρt νµ ν L T EGR 1 2κ 2 e T and T S ρ µν T ρ µν with S ρ µν 1 2 ]. ( ) K µν ρ + δ ρ µ T σν σ δρ ν T σµ σ. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 6/ 18

9 [arxiv:gr-qc/ ] Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 7/ 18

10 Outline 1 Geometrical Background 2 Large-Scale Magnetic Fields 3 Magnetic Fields in Teleparallelism 4 Summary and Conclusions Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 7/ 18

11 Mystery The origin of the magnetic fields observed in the galaxies and galaxy clusters. Magnectic fields with strength 10 6 G at coherence scale 1 10kpc G Large-scale magnetic fields at coherence scale 10kpc 1Mpc. Future experiments on CMB radiation PLANCK QUIET B-Pol LiteBIRD Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 8/ 18

12 Dynamo mechanism: the kinetic energy associated with the differential rotation is converted into magnetic field energy. Galactic dynamo: to amplify the energy of seed magnetic field of spiral galaxies into the 10 6 G strength fields observed in the galaxies. Some generation mechanisms (not natural) Plasma instability Magnetohydrodynamics (MHD): the interaction between magnectic fields and plasmas. Cosmological phase transition Magnetogensis: need out-of-thermal equilibrium and macroscopic parity violation. Matter perturbation before or after the recombination Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 9/ 18

13 The most natural mechanism EM quantum fluctuation during inflation (Turner & Widrow 1988) Break conformal symmetry by non-minimal coupling. From quantum electrodynamics in curved space-time the non-minimal coupling is owing to the one-loop vacuum-polarization effect (Drummond & Hathrell 1980.) Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 10/ 18

14 Outline 1 Geometrical Background 2 Large-Scale Magnetic Fields 3 Magnetic Fields in Teleparallelism 4 Summary and Conclusions Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 10/ 18

15 Model From Einstein equation two classes to explain the accelerating expansion. R µν 1 2 g µνr = 8πGT µν. }{{}}{{} modified matter modified gravity The action of inflation model in teleparallelism is [ ] f(t ) S = d 4 x e 2κ 2 + L M. (Ferraro & Fiorini 2007) Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 11/ 18

16 The electromagnetic sector is described by a non-minimal I(T )-Maxwell theory S em = d 4 x e ( 14 ) I(T ) F µνf µν, with second quantization A i (t, x) = d 3 k(2π) 3/2 [ ˆb(k, σ)ɛ i (k, σ)a(t, k)e ik x σ=1,2 +ˆb (k, σ)ɛ i (k, σ)a (t, k)e ik x ] The EoM by using conformal time η is A (k, η) + (I /I) A (k, η) + k 2 A(k, η) = 0. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 12/ 18

17 Analytical Solution for Magnetic Field By solving the EoM lead to A(k, η) 2 at the late times with A(k) 1 ( ) 1 ηr ] I [1 1/2 1 2k 2 I + iki η I ( η) d η e ikη η=η k By using the comoving magnetic field B i (t, x), the proper magnetic field is expressed as B i proper (t, x) = a 1 B i (t, x) = a 2 ɛ ijk j A k (t, x). The energy density of the generated magnetic field per unit logarithmic interval of k is ρ B (k, η) 1 4πk 3 2 (2π) 3 Bproper (k, η) 2 I(η) = k A(k) 2 k 4 2π 2 a 4 I(η). Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 13/ 18

18 the density parameter and spectral index are given by Ω B (k, η) = ρ B(k, η R ) ρ γ (η R ) n B d ln Ω B(k) d ln k where ρ γ (η R ) = N eff ( π 2 /30 ) T 4 R. I(η) I(η R ) = = 4 + k4 T 4 R a4 R d ln k A(k) 2 d ln k Set the specific form of non-minimal coupling 15k A(k) 2 N eff π 4 I(η),, I(η) = I ( η η ) β, with I = I(η ) and we obtain k A 1 2 = 1 2I(η k ) 1 (β + 2i) 2(β + 1) 2 A 2I(η k ), where A(= O(1)) is a constant of the order of unity. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 14/ 18

19 From the relation B(η 0, k) 2 = 2ρ B (η 0, k) = 2Ω B (η 0, k) ρ γ (η 0 ) with ρ γ (η 0 ) GeV 4 and 1 G = GeV 2 The current amplitude of the magnetic field [ 7.2 B(η 0, L) = 2.7 (5.1) 4 π A I(η 0) I(η R ) ] β/ β/4 N (β 4)/8 eff ( HR M Pl ) β/4 ( ) β/2 2 L G. [Mpc] Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 15/ 18

20 Numerical Results Suppose the power-law inflation (a = a 0 (t/t 0 ) p ), the relation between t and η is t t 0 = [a 0 t 0 (p 1) ( η)] 1/(p 1). The case of a power-law type coupling I(T ) = (T/T 0 ) n I(η) = ( 6/T 0 ) n (p/t 0 ) 2n [a 0 t 0 (p 1)] 2n/(p 1) ( η) 2n/(p 1). N eff = 100, H R = GeV (T R = GeV ), L = 1Mpc, A = 1, I(η R ) = I(η 0 ), and β = 4.2 B(η 0, L = 1 Mpc) = G. Similarly, H R = GeV (T R = GeV ) and β = 4.6, met for p = 10 and n = 19.7 B(η 0, L = 1 Mpc) = G. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 16/ 18

21 Outline 1 Geometrical Background 2 Large-Scale Magnetic Fields 3 Magnetic Fields in Teleparallelism 4 Summary and Conclusions Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 16/ 18

22 Summary and Conclusions Current observation of the large-scale magectic field strength is 10 6 G. Large-scale magectic field can be naturally generated through the conformal symmetry broken by non-minimal coupling of torsion and electromagnetic fields in the teleparallesim during inflation. The magnetic field with its strength of 10 9 G and the coherence scale of 1Mpc at the present time can be generated. Resultant field strength is compatible with the upper limit of G obtained from CMB radiation as well as that of being smaller than G from CMB radiation on the present strength with scales larger than the present horizon. The constraint on the current strength of the magnetic fields on the BBN horizon scale h 1 Mpc (h = 0.7), is smaller than 10 6 G. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 17/ 18

23 End Thank you! Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 18/ 18

24 Backup Slides Outline 5 Backup Slides Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 18/ 18

25 Backup Slides Observational Methods Synchrotron emission: from pulsar to superclusters. Faraday rotation: the change of the polarization angle. Zeeman splitting (energy splitting): eg. 21-cm hydrogen line. Polarization of optical starlight: can reveal the presence of large-scale magnetic fields in our Galaxy and nearby. Ling-Wei Luo VIIIth Rencontres du Vietnam - Beyond the Standard Nhon 19/ 18