CMB anisotropies in the presence of a stochastic magnetic field

Primordial magnetic fields present since before the epoch of matter-radiation equality have an effect on the anisotropies of the cosmic microwave background (CMB). The CMB anisotropies due to scalar perturbations are calculated in the gauge-invariant formalism for magnetized adiabatic initial conditions. Furthermore, the linear matter power spectrum is calculated. Numerical solutions are complemented by a qualitative analysis.

Figure 1

The spectra determining the autocorrelation functions of the magnetic anisotropic stress, energy density contrast and the Lorentz term are shown. For $B=10\mathrm{nG}$ and spectral index $n_B=-2.9$ in each graph, the numerical solution of the double integral, together with the numerical interpolation (spline) used in the code to calculate the CMB anisotropies, and the analytical approximation in terms of incomplete Gamma functions is shown. The lower horizontal axis shows $k$, the upper one shows the ratio $k/k_m$. In this example, the magnetic damping wave number is given by $k_m=20{\mathrm{Mpc}}^{-1}$ (cf., Eq. (3.15)).

Figure 2

The spectra determining the autocorrelation function of the magnetic anisotropic stress ${\mathcal{P}}_{{\pi }_B}$ are shown for different values of the magnetic field strength $B$ and magnetic spectral index $n_B$. In this case, the numerical interpolation of the double integral appearing in Eq. (3.24) was used. The magnetic damping wave number $k_m$ is given by Eq. (3.15).

Figure 3

The angular power spectra determining the autocorrelation functions of the temperature anisotropies of the CMB ($C_{\ell }^{TT}$), the polarization of the CMB ($C_{\ell }^{EE}$) and the temperature-polarization cross correlation function ($C_{\ell }^{TE}$) for different values of the magnetic field strength while keeping fixed the magnetic field spectral index $n_B=-2.9$. These have been calculated using the best fit values of the 6-parameter $\Lambda \mathrm{CDM}$ fit of WMAP7. For comparison, the pure $\Lambda \mathrm{CDM}$ model corresponding to $B=0$ has also been shown.

Figure 4

The angular power spectra determining the autocorrelation functions of the temperature anisotropies of the CMB ($C_{\ell }^{TT}$) for the magnetic field strength $B=5\mathrm{nG}$ while varying the spectral index. These have been calculated using the best fit values of the 6-parameter $\Lambda \mathrm{CDM}$ fit of WMAP7. For comparison, the $\Lambda \mathrm{CDM}$ model with $B=0$ has also been shown. Left: The $TT$ angular power spectrum for the whole range of multipoles. Middle: The first acoustic peak. Right: The second acoustic peak.

Figure 5

The different contributions to the total temperature angular power spectrum due to the Sachs-Wolfe term (SW), the integrated Sachs-Wolfe effect (ISW) and the Doppler term. The light solid line shows always the case in the presence of a magnetic field with $B=10\mathrm{nG}$ and $n_B=-2.9$.

Figure 6

The numerical solution of the evolution of the effective temperature perturbation is shown for different wave numbers. The WMAP7 best fit solution is shown in comparison with the solution in presence of a stochastic magnetic field with $B=20\mathrm{nG}$ and magnetic spectral index $n_B=-2.9$. Indicated is the time of last scattering, which in this case is ${\tau }_{\mathrm{LS}}=285\mathrm{Mpc}$. Left: The solutions for the wave numbers $k=0.0140{\mathrm{Mpc}}^{-1}$ and $k=0.0172{\mathrm{Mpc}}^{-1}$ have a maximum at the time of last scattering. These wave numbers correspond to the region of the first acoustic peak, whereas the former corresponds to $\ell =200$, the latter corresponds to $\ell =244$. Right: The solutions for the wave numbers $k=0.0364{\mathrm{Mpc}}^{-1}$ and $k=0.0396{\mathrm{Mpc}}^{-1}$ have a minimum at the time of last scattering. These wave numbers correspond to the region of the second acoustic peak, whereas the former corresponds to $\ell =523$, the latter corresponds to $\ell =569$.

Figure 7

The total matter power spectrum for different spectral index and amplitude of the magnetic field. The data points included show the estimated power spectrum from the Automatic Plate Measurement angular galaxy catalogue [42] (see also 43).

Figure 8

The spectrum determining the cross correlation function of the magnetic energy density and anisotropic stress is shown for $B=10\mathrm{nG}$ and spectral index $n_B=-2.9$. The numerical solution of the double integral, together with the numerical interpolation (spline) used in the code to calculate the CMB anisotropies, is plotted. The lower horizontal axis shows $k$, the upper one shows the ratio $k/k_m$. In this example, the magnetic damping wave number is given by $k_m=20{\mathrm{Mpc}}^{-1}$ (cf., Eq. (3.15)).

Figure 9

The angular power spectra determining the autocorrelation functions of the temperature anisotropies of the CMB ($C_{\ell }^{TT}$) (cf., Eq. (4.17)), the polarization of the CMB ($C_{\ell }^{EE}$) and the temperature-polarization cross correlation function ($C_{\ell }^{TE}$) (cf. Eq. (4.18)) for the compensated magnetic mode for $B=10\mathrm{nG}$ and $n_B=-2.9$. These have been calculated using the best fit values of the 6-parameter $\Lambda \mathrm{CDM}$ fit of WMAP7.

Figure 10

Left: The total linear matter power spectrum due to the compensated magnetic mode. For comparison, the total linear matter power spectrum for the WMAP7 best fit model is included, which corresponds to a pure adiabatic mode. Right: The total linear matter power spectrum due to the adiabatic mode and the compensated magnetic mode assuming that these are uncorrelated. Both graphs have been calculated using the best fit values of the 6-parameter $\Lambda \mathrm{CDM}$ fit of WMAP7.