He investigates the effects of the background PMF on the CMB. The sound speed of the tightly coupled photon-baryon fluid is increased by the background PMF. The increased sound speed causes the odd peaks of the CMB temperature fluctuations to be suppressed, and the CMB peak positions to be shifted to larger scale. The background PMF causes the stronger decaying potential, and increases the amplitude of the CMB. These two effects of the background PMF on a smaller scale cancel out, and the overall effects of the background PMF are suppression of the CMB around the first peak and shifting peaks to large scale. We also discuss obtaining information about the PMF generation mechanisms, and examine the non-linear evolution of the PMF by the constraint on the maximum scale for the PMF distributions. Finally, we discuss degeneracies between the PMF parameters and the standard cosmological parameters.

The cosmic microwave background (CMB) provides important information about the Universe. The positions of peaks and the amplitude of the CMB temperature perturbations are reflected by the sound speed of the photon-baryon fluid and the changing potential. Since a magnetic field increases the sound speed of fluid, and affects the evolution of density perturbations, if this amplitude as background is large enough, the magnetic field gives the critical effects on the CMB.

   A power law (PL) is one of the most familiar spectra for the various physical processes including the PMF on cosmological scale (see Yamazaki, et al. Phys. Rep. 517, 141, 2012 and references therein). The main parameters of the PL-PMF are the field strength on the coherent scale: &lambda$; , B_λ and the spectral index: n_B. These parameters also have been tried constraining from the cosmological observations.

   Since an average of the strength of the PMF as the background is zero, while an average of the background PMF energy density is a finite value, previous studies have constrained the background PMF energy density (ρ_PMF) from big bang nucleosynthesis (BBN). ρ_PMF is proportional to the scale invariant field strength of the PMF (B_SI) which is as a function of λ, B_λ, n_B, and upper and lower scales of the PMF at a its generation time (Yamazaki and Kusakabe, 2012). From previous studies (Yamazaki and Kusakabe, 2012) ρ_PMF for larger n_B is comparable to the constrained PMF energy density by the BBN and this influence in the CMB is not negligible. In this paper, we investigate effects of the background PMF on the CMB as a function of the PL-PMF parameters: the field strength on the coherent scale, the spectral index, the maximum scales of the PMF. We also report degeneracies of these PL-PMF parameters and the standard cosmological parameters in the PMF influences in the CMB for the first time.

Fig. 1; The contribution of the PMF and the standard cosmological parameters on the CMB. The black curves are the theoretical result without PMF effects. The red curves are the theoretical result with the PMF effects of ( nB, Bλ, kmax,ρMF/ργ) = (-2.0, 10 nG, 1500 Mpc -1, 0.0160). The standard cosmological parameters of all curves in this figure except the red curve of Panel (b), which is displayed by moving your mouse on Fig. 1, are the WMAP 9yr best fit parameters in ΛCDM and the tensor mode.These standard cosmological parameters are (Ωb, ΩCDM, ns, 109Δ2R, H0, τ, r) = (0.0442, 0.210, 0.992, 2.26, 72.6, 0.091, 0.38). The different standard cosmological parameters of the red curve in the panel (b) is the baryon density and the CDM density. These parameter values are (Ωb, ΩCDM) = (0.0461, 0.195).

   The background PMF increases the sound speed of the tightly coupled photon-baryon fluid, and causes the stronger decaying potential. The overall effect of the background PMF changes amplitudes of the CMB around the first peak stronger than in smaller scale [Fig. 1(a)].

   We report the case that k_max is assumed to be a free parameter. The energy density of the background PMF is dependent on k_max, and k_max is dependent on a PMF generation model. Hence, if one determines the k_max by constraining the magnetic energy density from the CMB, we obtain information of the PMF generation mechanisms. We also discuss the constraint on the k_max as being an examination for the non-linear evolution of the PMF.

   Finally we discuss the possibility of degeneracies between the background PMF, the baryon, and the CDM [Fig. 1(a) and (b)]. Considering the fundamental understanding of the baryon and matter densities effects on the CMB, we expect the positive correlation between Ω_b and the background PMF, and the negative correlation between Ω_CDM and the background PMF. In fact, the affected CMB by the PMF can be adjusted by changing Ω_b and Ω_CDM as Fig. 1(a) and (b). In previous constraints on the PMF without the background effects, the degeneracy between the PMF and the standard cosmological parameters are negligible small. If n_B and k_max are sufficiently small, the ρ_MF is too small to affect the CMB, and the previous result is no problem. However, a lot remains to be established about the PMF, it is too early to discuss the PMF effects and generation mechanisms in such narrow parameter range. To understand the PMF correctly, we should constrain the background PMF and the standard cosmological parameters simultaneously.

   If we promote the effects of the background PMF on the CMB, and constrain them by the latest and future observations, it will permit development of better studies for the generation and evolution of the PMF and provide new insight into the early universe with the PMF.