Magnetic field plays a very important role in many astronomical phenomena at various scales of the universe. Before recombination epoch for the photon last scattering (PLS) by the electron, because photons and electrons interact with each other through Thomson scattering, the primordial magnetic field (PMF) affects the photon dynamics. Thus the radiation of cosmic microwave background (CMB) is influenced by the PMF through electron-photon dynamics. In addition, because the energy density, pressure, and tension of the PMF affect gravitational collapse of the plasma, the fluctuation of matter density, which is an origin of the large scale structures (LSS), should be influenced by the PMF. Therefore, precise observations of the CMB anisotropies can place a stringent constraint on the PMF parameters. Several theoretical models of the origin and amplification mechanism of the PMF in the scale of galaxy cluster have recently been proposed and studied intensively in a number of papers. However, there is not satisfactory understanding of the origin and evolution of the PMF yet. Especially, there still remains missing link between the PMF and the observed magnetic field in galaxy clusters.

　In order to compare the CMB anisotropy induced by the PMF with observations precisely, we need to perform numerical calculations of the fully linearized MHD equations along with a realistic recombination history of the universe. In particular, we need to develop a numerical method to predict the theoretical spectrum for intermediate angular scales, in which the previous analytic approximation becomes inappropriate. The approximate PMF power spectrum adopted in the previous works applies only to and around the PMF power spectral index nB = -1. In this thesis we take an assumption that the PMF is expressed as B ∝ Bλk(nB), where Bλ is the strength and nB is the power spectral index of the PMF. This approximation is useful for studying PMF effect around nB = -1. However, it is dangerous to use such approximation for wider range of nB values because the origin of PMF is still open to debate and becuse the PMF parameters including n are not well constrained. Therefore we need to calculate the PMF power spectrum more generally for all available nB values more generally. In this thesis we developed the new method to estimate PMF power spectrum without approximations.

　Using our new formula, we can numerically investigate the effects of stochastic PMF on the fluctuation of the matter density and CMB with high accuracy. We found that the PMF is one of the most important cosmological physical processes on a "middle" cosmological scale. In this thesis we denote that the middle cosmological scale means less than the Silk damping scale which corresponds to relatively larger mutilpoles 1000 〓 l in the CMB power spectrum. For such scale there is a potential discrepancy between the best-fit cosmological model and the CMB anisotropy data from the Wilkinson Microwave Anisotropy Probe (WMAP), the Arcminute Cosmology Bolometer Array Receiver(ACBAR), and the Cosmic Background Imager (CBI) in the liner perturbation theory. We found that the PMF is one of the most predominant physical processes for resolving such a discrepancy.

　We confirmed numerically without approximation that the excess power in the CMB at higher l was able to be explained by the existence of PMF. Also, we confirmed that the vector mode effect of the PMF dominates the TT mode of CMB fluctuation from the PMF. A likelihood analysis utilizing the WMAP, ACBAR and CBI data with the Markov Chain Monte Carlo(MCMC) method was applied for the first time to constrain the upper limit on the strength of the PMF, which turned out to be

We also considered three conditions on the generation and evolution of the cosmological PMF: 1) the constraint on the PMF from the CMB (our result); 2) the lower limit of the PMF from the magnetic field of galaxy clusters; and 3) the constraint on the PMF from gravity waves. Combining these three conditions, we found the following concordance region for the PMF parameters;

We will be able to constrain the PMF more accurately when we combine our study and future data of observed CMB anisotropies and polarizations for higher multipoles l via the Planck Surveyor.

　We also investigated the effect of the PMF on the energy density fields by numerically studying the stochastic PMF that depends on scales, and we quantitatively discussed the effect of the PMF on the seeds of LSS in the early universe. We considered not only the magnetic field tension but also the increases of pressure and energy density perturbations from the field. Furthermore, by considering the correlation between the PMF and the fluctuation of the matter density, and taking the mathematically exact stochastic PMF power spectrum, we obtained reasonable and accurate evolutions of baryon, cold dark matter (CDM), photon, and therefore the LSS, too. We showed that the PMF can play very different roles in the evolution of density perturbations according to the correlation between the PMF and the primordial density fluctuations. After decoupling, CDM is also influenced indirectly by the PMF through gravitational interaction. We have estimated the effects numerically and found that the amplitude ratio between the density spectra with and without PMF, |P(k)/P0(k)| at k > 0.2 Mpc(-1), lies between 75% and 130% at present for the range of PMF parameters nB = -2.001 and -1.0, 0.5 nG <|Bλ| < 1.0 nG, and -1 < s < 1, where s is the correlation parameter between the PMF and the primordial density fluctuations.

　It is reported that the magnetic field around |Bλ| 〜 nG at large scales (λ = 1Mpc) provides a new interpretation for the excess of CMB anisotropies at smaller angular scales for the reason below. If the PMF had such strength it is very likely, as shown in the present thesis, that the PMF has affected the formations of LSS. Yoshida, Sugiyama and Hernquist (2003) suggested that in order to avoid false coupling of the baryon and CDM at small scales, it is preferable to use independent transfer functions for the baryon and CDM. The PMF could be another source of this difference in the transfer function for baryon and CDM. The LSS with the PMF should have evolved differently from that without the PMF, since the density perturbations in the early universe have evolved to the LSS at present epoch. We have shown that the perturbations of baryon and the CDM energy density follow very different evolutions in the presence of the PMF. The evolution of large scale structure should become more complicated and interesting with the PMF being taken into consideration.

　本論文は5章からなる。第1章はイントロダクションであり、宇宙の構造形成における初期磁場の重要性、具体的には、初期磁場の形成、増幅のメカニズムと、初期磁場がCMB(マイクロ波背景輻射)非等方性や大規模構造にどのように影響するかについての従来の理解がまとめられ、本論文でどのように初期磁場を研究していくかの方針が示される。本論文ではCMBについては、その高次のモーメントに対する、再結合前に生成された初期磁場の影響が考察される。また大規模構造については、密度ゆらぎと磁場の間の相関を考慮した時にどのような影響があるかについて検討がなされる。

　第2章では、初期磁場のスカラーモード、ベクターモード、テンサーモードについての定式化が与えられる。特に、従来用いられていた近似は初期磁場のパワースペクトルのインデックスnBが-1にごく近い時にのみ有効であったのに対して、本論文での定式化は任意のnBの値について正確な値を与えるものであることが述べられる。さらに、密度ゆらぎの発展に対する磁場の影響の定式化が与えられる。

　第3章では、CMB非等方性に対する初期磁場の影響が議論される。現在の標準的な宇宙モデルと、CMB非等方性の観測値の間には、角度スケール0.1度程度の高次の領域で観測値がモデルの予測値の数倍大きいというずれがあることが知られている。この領域では観測値の不定性が大きいためにこのずれは決定的なものではないが、初期磁場としてある程度の強さのものを考えることでこのずれをなくすことができることが示される。具体的には、磁場のパワースペクトルのインデックスnBを-1.5とするなら、共動座標で1Mpcスケールでの磁場強度βλが4ないし8nG程度であると観測と良く一致する。またこの強度が4nGであるなら、nBが-0.5前後であるなら観測と良く一致する。この結果から、観測と矛盾しないnBとBλの組み合わせの範囲に制限をつけることができた。nB<-2では、この制限はおよそBλ<8nGとなる。これは、初期磁場の強度について、銀河団磁場の観測とは別に上限を与える重要な結果であるといえる。

　第4章では、構造形成に対する初期磁場の影響が議論される。Tashiro and Sugiyama (2006)が初期磁場の、非常に初期の構造形成、特に種族III星の形成時期への影響について議論しているが、彼らは初期磁場が密度ゆらぎと無相関な場合だけを扱っている。初期磁場の形成メカニズムによっては無視できない相関があるケースも考えられるため、本論文では密度の初期ゆらぎと初期磁場の相関係数をパラメータとして扱うことで、従来考えられていなかった相関があるケースについて調べている。観測的な情報が現段階では不足しているため実際に初期磁場がどの程度影響をするのかについての定量的な議論は十分行えない。しかしここで行われた議論や手法は、今後初期磁場の形成過程と構造形成への影響を考える上で重要な第一歩になると考えられる。第5章は結論にあてられている。

　以上の結果の中で特に重要なことは、スペクトルインデックスnBの広い範囲で適用可能な初期磁場のパワースペクトルの計算式を初めて定式化したこと、その結果を用いてCMBの観測結果から初期磁場の強度に対して制限を与えることに成功したことである。さらに、構造形成の観測自体から初期磁場の強度、形成メカニズムに対して制限を与える可能性も示唆された。これらの結果は磁場の起源の理解に対して重要な意味をもつものであり、高く評価できる。

　本論文の第2章から第4章までについては、市來浄與、梅津健一、G.J. Mathews、梶野敏貴、及び花山秀和との共同研究であるが、論文提出者が主体となって定式化、数値計算を行ったもので、論文提出者の寄与が十分であると判断する。

従って、博士(理学)の学位を授与できると認める。