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

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the primordial magnetic field


CMB Session

The Primordial Magnetic Field

and The Cosmic Microwave Background(Yamasaki etal, ApJL 625:L1=astro-ph/0410142 & astro-ph/0509xxx)

The University of Tokyo &

D. G. Yamazaki ,

K . Ichiki , T. Kajino, & G. Mathews

National Astronomical Observatory of Japan


Background and Motivation

There isthe gap between observations and theoretical calculations for higher l

WMAP best fit cmbfast

For higher l, the temperature anisotropy of CMB is not enough

cmbast : U. Seljak, et al., 1997, CBI: B. S. Mason et al., 2003,

WMAP: Bennett, et al., 2003, ACBAR: Kuo et al., 2004.


Introduction 1

For higher l, our understanding of the temperature anisotropy of CMB is not enough.

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


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.


Introduction 2

Another interesting subject

A cluster of galaxies have magnetic field of 0.1-1m (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.

The attractive point of our study

We can solve these problems simultaneously by studying the effect of the PMF on the CMB.



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.


All energy is the sum of the PMF and the fluid

Eall = EPMF + Efluid



Effect of PMF

Lines of magnetic force

Lines of magnetic force

Lorentz force


magnetic pressure


Repulsion between lines of magnetic force

Thomson scattering

Vector of photons is changed by

Thomson scattering. (photons and baryons are tight-coupled before the last scattering surface ).

The Lorentz force changes only vectors of baryons

The magnetic field increases the fluid pressure


Primordial Magnetic Field

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:Bl(τ, x)=Bl (x)/a2,

the power law:

Our cosmological magnetic field model on the early universe is a statistically homogeneous and isotropic random

Bl: magnetic comoving mean-field amplitude (at 1Mpc)

nB:power spectral index of the magnetic field

Our purpose is to constraint these two parameters.


Estimation of Primordial magnetic field strength

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 Bl and power spectral index of the primordial magnetic field nB, with other cosmological parameters, h, Wbh2, Wch2, ns,As, and t 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 s8 = 0.9 (Spergel et at. 2003; Komatsu and Seljak 2002).




l(l+1)Cl[m K2]







Result and Discussion INumerical estimations

The magnetic effect to CMB perturbation becomes strong for higher l

For higher l, the effect of a primordial magnetic field is much more important


Result and Discussion INumerical estimations

WMAP+ACBAR+CBI :1 s(68%)

WMAP+ACBAR+CBI: 2 s (95%)


Excluded and allowed regions at 1 and 2s on two parameter plane |Bl| vs.nB , where |Bl| is the primordial magnetic field strength and nB is the power-law spectralindex.


The upper limit of the magnetic field strength is |Bl| < 5.5 nG (1s)




Lower limit from Cluster of galaxies


①Our limit from WMAP + ACBAR + CBI date sets

Bλ < 5.5 nG(1Mpc)




②Limit from the cluster of galaxies

Bλ > 1.0 nG

③Limit from gravity wave (Caprini & Durrer 2002)



1.0 nG < Bl < 5.5 nG








-3.0 < nB < -2.3




The multiple constraints on generation scenario of PMF

WMAP+ACBAR+CBI :1 s(68%)

BBN limits on B from the PMF generation epoch

WMAP+ACBAR+CBI: 2 s (95%)

Allowed region of the PMF from the multiple constraints categorized by the generation epochs



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.

2.Likelihood analysis of WMAP data with MCMC method gives constraint on a primordial magnetic field, B < 5.5nG

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 < nB < -2.3

in our estimated allowed parameter region.



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,

AGNorigin 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.