1. and The Cosmic Microwave Background (Yamasaki etal, ApJL 625:L1=astro-ph/0410142 & astro-ph/0509xxx) National Astronomical Observatory of Japan The Primordial Magnetic Field The University of Tokyo & D. G. Yamazaki, K. Ichiki, T. Kajino, & G. Mathews COSMO 05 CMB Session

2. cmbast : U. Seljak, et al., 1997, CBI: B. S. Mason et al., 2003, WMAP: Bennett, et al., 2003, ACBAR: Kuo et al., 2004. Background and Motivation Background and Motivation For higher l, the temperature anisotropy of CMB is not enough There is the gap between observations and theoretical calculations for higher l WMAP best fit cmbfast

3. We need some new physical process for higher l. Several semi-analytic studies point out that the effect of the primordial magnetic field (PMF) is very important in CMB for higher l. (Jedamzik et al. 2000: Durrer et al. 2000, Mack et al. 2002 Subramanian and Barrow, 1998, 2002) The PMF is one of the new physical process for higher l For higher l, our understanding of the temperature anisotropy of CMB is not enough. Introduction 1

4. Those semi-analytic studies develop the CMB analysis. Their approximations are appropriate for lower l, however, their accuracy is not enough to compare theoretical CMB for higher l with observations. We want to estimate the effect of the PMF on CMB accurately, So we construct new computation program which can calculate scalar and vector mode effects of magnetic fields on CMB.

5. We can solve these problems simultaneously by studying the effect of the PMF on the CMB. A cluster of galaxies have magnetic field of 0.1-1  ( T. E. Clarke et. al. 2000). But the origin and evolution of magnetic field in the cluster of galaxies are not clearly understood. The study of the PMF at the last scattering surface of photons will provide important information to solve this problem. Another interesting subject The attractive point of our study Introduction 2

6. 1. We construct a new computation program which can calculate scalar and vector-mode effects of the PMF on CMB. 2. We estimate the PMF at 1Mpc by likelihood analysis with the Markov Chain Monte Carlo (MCMC) method, in order to solve the discrepancy between the theoretical primary CMB and observational data (WMAP: Verde et al. 2003, ACBAR: Kuo, C.L., et al., 2004, and CBI: Mason, B. S., et al., 2003) for higher l. 3. We then discuss the evolution of the PMF. Purpose Purpose

7. Effect of PMF The Lorentz force changes only vectors of baryons The magnetic field increases the fluid pressure MF baryons Lorentz force Thomson scattering Photon Lines of magnetic force MF Repulsion between lines of magnetic force Lines of magnetic force magnetic pressure Vector of photons is changed by Thomson scattering. (photons and baryons are tight-coupled before the last scattering surface ). E all = E PMF + E fluid All energy is the sum of the PMF and the fluid

8. Primordial Magnetic Field n B n B : power spectral index of the magnetic field B B : magnetic comoving mean-field amplitude (at 1Mpc) We discard MHD back reaction onto the field itself within the linear approximation ( Durrer et al., 2000). We consider the primordial stochastic magnetic field. The conductivity of the primordial plasma is very large, and it is “frozen-in” (Mack et al. 2002). So, Electric field is neglected A time evolution of a magnetic field decouple from its spatial structure on sufficiently large scales: B (τ, x)=B (x)/a 2, the power law: Our cosmological magnetic field model on the early universe is a statistically homogeneous and isotropic random Our purpose is to constraint these two parameters.

9. 1. Combining Einstein equations with the fluid equations (Ma and Bertschinger 1995, Hu and White 1997), we obtain evolution equations of scalar and vector perturbations. 2. We evaluated the likelihood functions of WMAP, ACBAR, and CBI data sets in a wide range of the magnetic field strength B and power spectral index of the primordial magnetic field n B, with other cosmological parameters, h,  b h 2,  c h 2, n s,A s, and  in flat Universe models. To explore the parameter space, we make use of the Markov chain technique (Lewis 2002). 3. We also take account of the SZ effect in our analysis. For that, we follow an estimate of Komatsu and Seljak, with  8 = 0.9 (Spergel et at. 2003; Komatsu and Seljak 2002). Estimation of Primordial magnetic field strength

10. B=8nG B=6nG 500 100015002500 2000 l l(l+1)C l [   ] Result and Discussion I Numerical estimations For higher l, the effect of a primordial magnetic field is much more important The magnetic effect to CMB perturbation becomes strong for higher l

11. Excluded and allowed regions at 1 and 2  on two parameter plane |B | vs. n B, where |B | is the primordial magnetic field strength and n B is the power-law spectral index. Result and Discussion I Numerical estimations The upper limit of the magnetic field strength is |B | < 5.5 nG (1  )   WMAP+ACBAR+CBI:  %) WMAP+ACBAR+CBI :1  (68%)

12. Excluded and allowed regions at 1 and 2  on two parameter plane |B | vs. n B, where |B | is the primordial magnetic field strength and n B is the power-law spectral index. Result and Discussion I Numerical estimations The upper limit of the magnetic field strength is |B | < 5.5 nG (1  )   WMAP+ACBAR+CBI:  %) WMAP+ACBAR+CBI :1  (68%)

13. The multiple constraints on generation scenario of PMF ③ Limit from gravity wave (Caprini & Durrer 2002) ② Limit from the cluster of galaxies B λ > 1.0 nG ① Our limit from WMAP + ACBAR + CBI date sets B λ < 5.5 nG(1Mpc) 1.0 nG < B < 5.5 nG -3.0 < n B < -2.3 31 2 + + BBN: QCD: 1 2 + inflation: 1 WMAP+ACBAR+CBI:  %) BBN QCD inflation WMAP+ACBAR+CBI :1  (68%) Lower limit from Cluster of galaxies BBN limits on B from the PMF generation epoch Allowed region of the PMF from the multiple constraints categorized by the generation epochs 1 2 3

14. The multiple constraints on generation scenario of PMF ⅠⅡ ③ Limit from gravity wave (Caprini & Durrer 2002) ② Limit from the cluster of galaxies B λ > 1.0 nG ① Our limit from WMAP + ACBAR + CBI date sets B λ < 5.5 nG(1Mpc) 1.0 nG < B < 5.5 nG -3.0 < n B < -2.3 Ⅲ ⅢⅠ Ⅱ + + BBN: QCD: Ⅰ Ⅱ + inflation: Ⅰ WMAP+ACBAR+CBI:  %) BBN QCD inflation WMAP+ACBAR+CBI :1  (68%) Lower limit from Cluster of galaxies BBN limits on B from the PMF generation epoch Allowed region of the PMF from the multiple constraints categorized by the generation epochs

15. 2. Likelihood analysis of WMAP data with MCMC method gives constraint on a primordial magnetic field, B < 5.5nG Summary 3. All constrains from the constraint of PMF by gravity wave and recent magnetic field strength in clusters of galaxies, 1 nG< B < 5.5 nG, -3.0 < n B < -2.3 in our estimated allowed parameter region. 1. We confirmed numerically (without approximation) that potential discrepancy of CMB at higher l between theory and observation is explained by the primordial magnetic field.

16. We considered only the isotropic collapse effect without other evolutions of the PMF after the LSS (the last scattering of photons). If we include new effective evolution processes; cluster merger → shock driven Weidel instability, AGN origin of magnetic field, the upper limit of the PMF may decrease from the present estimate. We should research others effective evolutions of the cosmological primordial magnetic field after the last scattering of photons. Discussion

17. Thank you very much for your attention

18. The energy momentum tensor of the electromagnetic field The energy momentum tensor of the electromagnetic field We consider a energy momentum tensor of the electromagnetic field. Because of the “frozen in” We replace the energy momentum tensor in the Einstein and Euler equation with normal, :Lorentz force energy density of magnetic field

19. (3) We considered only the isotropic collapse effect without other evolutions of the PMF after the LSS. ⇒ If we include new effective evolution processes; ・ cluster merger → shock driven Weidel instability, ・ AGN origin of magnetic field, the upper limit of the PMF may decrease from the present estimate.

20. The generation story of CPMF ●Inflation: n B < -2, B < nG In the case of Simple and rough scenario, The strength of the CPMF : 1 nG< B < 2.5 nG The power low index of the CPMF : -2.3 < n B < -2.0 ● 1 ～ 10nG by the magnetic field strength in clusters of galaxies ●lower limit on LSS: B = 10 -7 nG, at least B = 10 -16 nG The new scenarios of the evolution of the CPMF after the LSS

21. Thank you very much for your attention When you ask me, I am very glad if you talk slowly.

22. (1)We did not use data on the most effective regime such as ACBAR and CBI ⇒ Now Our PCs in Japan are calculating those regime more precisely by including ACBAR and CBI data. Coming soon new result! Future works (2) We did not consider the second order effect (i.e. back reaction, etc.). ⇒ We should carefully calculated the second order effect which may change the effect of the CPMF.

23. Effect of PMF 磁場 CMB 全エネルギー = + 磁場 バリオン ( 電 子 ) ローレンツ 力 トムソン散 乱 光子 図 2, 磁場エネル ギー 図 3, ローレンツ力。磁場によるローレン ツ力を受けたバリオン ( 電子 ) が、トムソ ン散乱によって光子と反応し、間接的に 磁場が CMB 光子に影響する。 図 4, 磁気圧。磁力線同士 の反発する力によって生 じた磁気圧が、流体全体 の圧力に影響し、音速を 変化させる。 磁力線同士の反発 磁場圧 磁力線 磁力線

24. For higher l, the effect of a primordial magnetic field is much more important the magnetic effect to CMB perturbation becomes strong for higher l Result and Discussion I Numerical estimations

25. WMAP only 宇宙論パラメータの決定に際して、 初期磁場の影響は見られない

26. The energy momentum tensor of the electromagnetic field The energy momentum tensor of the electromagnetic field We consider a energy momentum tensor of the electromagnetic field. Because of the “frozen in” :Lorentz force energy density of magnetic field

27. The Lorentz force changes only vectors of baryons Vector of Photons is changed by Thomson scattering. (photons and electron are tightly coupled before the last scattering ). Magnetic Field & CMB Baryons transfer the vector element of a magnetic field to photons by Thomson scattering Barrow et al. 1997, Subramanian et al. 1988, Jedamzik et al. 2000, Mack et al. 2002 Thomson scattering through electron interactions

28. wave number optical depth momentum density L (1) L (1) : Lorentz force quadrupole moments gravity constant a : a : scalar factor pressure of neutrino, photon, and baryon energy density of photon and baryon shear stress of neutrino and photon x e x e : ionization ratio Hu & White 1997, Mack et al. 2002 vector potential : neutrino : photon : baryon : Equations Thomson scattering Lorentz force

29. ○the vertical axis ： the magnetic field strength at 1Mpc ○the horizontal axis ： the power spectral index of the magnetic field ○contours ：⊿ χ 2, ⊿ χ 2 <2.3 ⇒ 68.3 ％（ 1σ ）、 ⊿ χ 2 <6.17 ⇒ 95.4 ％ （ 2σ ） 、 ⊿ χ 2 <18.4 ⇒ 99.99 ％ ○critical line: the limit of primordial magnetic field from that in the cluster of galaxies at present: 10nG ⇒ isotropic collapse ⇒ 1μG ●-2 ≦ n ≦ <-0.4, B ≫ 10nG We need the physical process in which the primordial magnetic field is strongly damped. ●n ≦ -2, -0.4 ≦ n, B ≪ 10nG, strongly increased. ●the upper limit of strength: B<20nG Result and Discussion Ⅱ Result and Discussion Ⅱ Likelihood with WMAP

30. Result and Discussion Ⅲ : Likelihood with CBI+ACBAR ｎ Log(B[G]) critical line ●Our results support that the evolution of a cosmological magnetic field is enhancing process. ●the upper limit of its strength: B<9nG

31. Result and Discussion Ⅳ ｎ Log(B[G]) critical line For higher l, ACBAR and CBI have more data points than WMAP, and the effect of a magnetic field is stronger for higher l. The difference of “CBI + ACBAR” and “WMAP”

32. Result and Discussion Ⅳ Likelihood analysis of ACBAR+CBI data gives stronger constraint on a primordial magnetic field ｎ Log(B[G]) critical line The difference of “CBI + ACBAR” and “WMAP”

33. the cosmological magnetic field on the early universe A conductivity of a inter galactic medium is effectively infinite. So we assume that the magnetic field on the cosmological scale is frozen in the spatial structure. Our cosmological magnetic field model on the early universe is a statistically homogeneous and isotropic random

34. Lorentz force (Mack et al. 2002) : Primordial Magnetic Field n n : power spectral index of the magnetic field B λ B λ : magnetic comoving mean-field amplitude (at 1Mpc) One of our purpose is constraint these parameters Our cosmological magnetic field model on the early universe is a statistically homogeneous and isotropic random

35. Lorentz force (Mack et al. 2002) : Primordial Magnetic Field n n : power spectral index of the magnetic field B λ B λ : magnetic comoving mean-field amplitude (at 1Mpc) We discard MHD back reaction onto the field itself within the linear approximation ( Durrer et al., 2000). We consider the primordial stochastic magnetic field. The conductivity of the primordial magnetic field is very large, and it is “frozen-in” (Mack et al. 2002). Electric field is neglected, and a time evolution of a magnetic field decouple from its spatial structure on sufficiently large scales: B(τ, x)=B(x)/a 2, the power law: P B (x) ∝ k n One of our purpose is constraint these parameters

36. (2) Likelihood analysis of ACBAR+CBI data gives stronger constraint on a primordial magnetic field, B < 9nG, than the analysis of WMAP data, B < 20 nG. Summary (1) We confirmed numerically that potential discrepancy of CMB at higher l between theory and observation is explained by the primordial magnetic field. (3) In the same way as Summary (2), Our results support that the evolution of a cosmological magnetic field is enhancing process.

37. Future Work WMAP 2nd year data and PLANCK(2007) data will provide us with more accurate data of polarization. This will help solve the problems concerning CMB at higher l and cosmological magnetic field through our accurate theoretical calculations.

38. Result and Discussion Ⅱ Likelihood with WMAP ○the vertical axis ： the magnetic field strength at 1Mpc ○the horizontal axis ： the power spectral index of ○contours ：⊿ χ 2, ⊿ χ 2 <2.3 ⇒ 68.3 ％（ 1σ ）、 ⊿ χ 2 <6.17 ⇒ 95.4 ％ （ 2σ ） 、 ⊿ χ 2 <18.4 ⇒ 99.99 ％ ○critical line: the limit of magnetic field from the present magnetic field in the cluster of galaxy. ・ -2 ≦ n ≦ <-0.4, B ≫ 10nG We need the physical process in which the primordial magnetic field is strongly damped. ・ n ≦ -2, -0.4 ≦ n, B ≪ 10nG We need the physical process in which the primordial magnetic field is strongly increased.

39. (1) We confirmed numerically (without approximation) that potential discrepancy of CMB at higher l between theory and observation is explained by the primordial magnetic field.Summary

40. Future Work ○ Likelihood analysis including WMAP TE mode (our PCs are now estimating, already finished) ○ Jointing the new Boltzmann code to estimate accurately the epoch of neutrino decoupling. ○ Likelihood with WMAP 2 nd year data ○ Likelihood with PLANCK (2007) we will solve the problems of CMB and the cosmological magnetic field simultaneously.

41. For B ≫ 10nG (much more than the critical line): we need the physical process in which the primordial magnetic field is strongly damped. ⇒ -2 ≦ n ≦ <-0.4 For B ≪ 10nG (much less than the critical line): we need the physical process in which the primordial magnetic field is strongly increased. ⇒ n ≦ -2, -0.4 ≦ n, (2) WMAP ⇒ B<20nG, ACBAR+CBI ⇒ B < 9nG Likelihood using ACBAR+CBI data gives us stronger constrains of a primordial magnetic field, because they have more higher l data. Final constraint is B < 9nG at 1MpcSummary

42. Hu & White 1997, Mack et al. 2002 wave number optical depth gravity constant momentum density Lorenz force quadrupole moments a : a : scalar factor pressure of neutrino, photon, and baryon energy density of photon and baryon shear stress of neutrino and photon background : neutrino : photon : baryon : Equations

43. The Lorentz force changes only vectors of baryons Photons are change it’s vector by Thomson scattering. (photons and baryons tight coupled before the last scattering surface ). Magnetic Field & CMB Baryons transmit the vector element of a magnetic field to photons by Thomson scattering Barrow et al. 1997, Subramanian et al. 1988, Jedamzik et al. 2000, Mack et al. 2002 Thomson scattering

44. Hu & White 1997, Mack et al. 2002 background : neutrino : photon : baryon : Equations Lorentz force Thomson scattering wave number optical depth momentum density L (1) L (1) : Lorentz force quadrupole moments gravity constant a : a : scalar factor pressure of neutrino, photon, and baryon energy density of photon and baryon shear stress of neutrino and photon x e x e : ionization ratio

45. Result I : Likelihood by WMAP ○ critical line : Now the magnetic field strength in a cluster of galaxy is μG, so, if the isotropic collapse is just only process which is rising a magnetic field strength, for the last scattering surface, about 10μG is the critical magnetic field in the cluster of galaxy. a magnteic field strength B is proportional to a length squared and a (energy) density ρ is is proportional to legnth cubed so magnetic field strength B is proportional to a density ρ to the power two thirds (2/3)

46. 初期磁場計算モデル Cutoff scale Alfven 波を考慮すると このスケール以下の磁場の影響は０とおけ る （ K. Subramanian, et. al, 1998 ） ∝ B -2/(n+5)

47. 磁場の相関関数のベクトルモード 磁場のストレステンソル （ K. Jedamzik, et al., 2000 ） 初期磁場計算モデル ∝ k 2n+3 ∝ B 4 ∝ B 4-2(2n+3)/(n+5) n< -3/2 n> -3/2