Effect of primordial magnetic field on seeds for large scale structure

Magnetic field plays a very important role in many astronomical phenomena at various scales of the universe. It is no exception in the early universe. Since the energy density, pressure, and tension of the primordial magnetic field affect gravitational collapses of plasma, the formation of seeds for large-scale structures should be influenced by them. Here we numerically investigate the effects of stochastic primordial magnetic field on the seeds of large-scale structures in the universe in detail. We found that the amplitude ratio between the density spectra with and without PMF ($|P(k)/P_0(k)|$ at $k>0.2{\mathrm{Mpc}}^{-1}$) lies between 75% and 130% at present for the range of PMF strengths $0.5\mathrm{nG}<B_{\lambda }<1.0\mathrm{nG}$, depending on the spectral index of PMF and the correlation between the matter density and the PMF distributions.

Figure 1

Ratio of stochastic PMF pressure and tension sources, $\sqrt{⟨T_{[\mathrm{EM}:\mathrm{S}1]}(\mathit{k})T_{[\mathrm{EM}:\mathrm{S}1]}^{*}(\mathit{k})⟩/⟨T_{[\mathrm{EM}:\mathrm{S}2]}(\mathit{k})T_{[\mathrm{EM}:\mathrm{S}2]}^{*}(\mathit{k})⟩}$, as a function of $n_{\mathit{B}}$. For illustration, a cut off scale $k_C$ is fixed to $k_c=10{\mathrm{Mpc}}^{-1}$. The pressure dominates for $n_B<-1.5$, and the tension dominates for $n_B>-1.5$.

Figure 2

Differences of transfer functions of CDM and baryon with ($P(k)$) and without ($P_0(k)$) the PMF, normalized by $P_0(k)$, with various strength of magnetic fields as functions of scale factor $a$ at each wave number $k$ as indicated. The curves in right and left panels correspond to the differences in CDM and baryon, respectively. The red thick, light-blue thin, orange normal, blue dotted bold, black dotted thin, and green dotted middle curves in each both side panels show the differences for parameters $(B_{\lambda },s)=(1.0\mathrm{nG},1)$, (1.0nG,0), ($1.0\mathrm{nG},-1$), (0.5nG,1), (0.5nG,0), and ($0.5\mathrm{nG},-1$), respectively. In all figures, the power spectral index of the PMF is fixed to $n=-2.001$. The curves with $s=0$ in all panels are slightly above unity, although it is difficult to read off from the figure.

Figure 3

Same as Fig. 2, but for a different power spectral index of the PMF $n=-1.0$.

Figure 4

Differences of transfer functions of CDM and baryon with ($P(k)$) and without ($P_0(k)$) the PMF, normalized by $P_0(k)$ as a function of wave number $k$ at each redshift as indicated. The curves in right and left panels correspond to the differences in CDM and baryon, respectively. The red thick, light-blue thin, orange normal, blue dotted bold, black dotted thin, and green dotted middle curves in each both side panels show $(B_{\lambda },s)=(1.0\mathrm{nG},1)$, (1.0nG,0), ($1.0\mathrm{nG},-1$), (0.5nG,1), (0.5nG,0), and ($0.5\mathrm{nG},-1$), respectively. In all figures, the power spectral index of the PMF is fixed to $n=-2.001$. The curves $s=0$ in all panels look like unit, actually, those is more than unit.

Figure 5

Same as Fig. 4, but for a different power spectral index of the PMF $n_{\mathit{B}}=-1.0$.