CMB constraints on a stochastic background of primordial magnetic fields

We constrain a stochastic background of a primordial magnetic field (PMF) by its contribution to the angular power spectrum of cosmic microwave background anisotropies. We parametrize such a stochastic background by a power-law spectrum with index $n_B$ and by its Gaussian smoothed amplitude $B_{\lambda }$ on a comoving length $\lambda$. We give an approximation for the spectra of the relevant correlators of the energy-momentum of the stochastic background of PMF for any $n_B$. By using the WMAP 7 yr data in combination with ACBAR, BICEP, and QUaD, we obtain the constraint $B_{1\mathrm{Mpc}}<5.0\mathrm{nG}$ at the 95% confidence level for a stochastic background of a nonhelical PMF. We discuss the relative importance of the scalar and vector contributions to CMB anisotropies in obtaining these constraints. We then forecast Planck capabilities in constraining $B_{1\mathrm{Mpc}}$.

Figure 1

Comparison of the spectral fit (dashed line) and exact spectrum (solid line) for the rescaled magnetic energy density $|{\rho }_B(k){|}^2/U$ (top panels), scalar Lorentz force $|L_B(k){|}^2/U$ (middle panels), and the vector part of the anisotropic stress $|{\Pi }^{(V)}(k){|}^2/(2U)$ (bottom panels) as a function of $k/k_D$, with $U=(A^2{k}_D^{2n_B+3})/(512{\pi }^4)$. In the left (right) columns $n_B=2.3$ ($n_B=-2.7$) is displayed.

Figure 2

The cross correlator $k^3⟨{\rho }_B(k)L_B(k)⟩/U$ (dotted line) for $n_B=2$ and $n_B=-5/2$ is plotted versus $k/k_D$ in comparison with $k^3|{\rho }_B(k){|}^2/U$ (solid line) and $k^3|L_B(k){|}^2/U$ (dashed line) with $U=(A^2{k}_D^{2n_B+3})/(512{\pi }^4)$.

Figure 3

We computed the scalar power spectrum with the contribution of the Lorentz force-energy density cross correlation for $n_B=2$ (left panel) and $n_B=-2.5$ (right panel). The solid line represents the correct assumption on the cross correlation, the dotted line represents the result considering the Lorentz force and the energy density fully anticorrelated, the triple dotted-dashed line represents the uncorrelated sum, and the dashed line represents the result assuming full correlation.

Figure 4

Results of the MCMC constrained with WMAP 7 yr, BICEP, QUaD, and ACBAR data. Curves are the 68% and 95% confidence levels. Note that $B_{\lambda }$ (with $\lambda =1\mathrm{Mpc}$) is in 10 nG units.

Figure 5

Results of the MCMC constrained with WMAP 7 yr, BICEP, QUaD, and ACBAR data. Curves are the 68% and 95% confidence levels. Solid lines are the results considering both scalar and vector magnetic modes, while dashed lines are the results taking into account only the vector contribution. Note that $B_{\lambda }$ (with $\lambda =1\mathrm{Mpc}$) is in 10 nG units.

Figure 6

Results of the MCMC for the PMF spectral index constrained with WMAP 7 yr, BICEP, QUaD, and ACBAR data, and a logarithmic prior for the PMF amplitude. Curves are the 95% and 68% confidence levels.

Figure 7

Comparison of the results of the MCMC with the real data used in Figs. 5, 6 (solid line) and simulated Planck data (dashed line). Curves are the 95% and 68% confidence levels. Note that $B_{\lambda }$ (with $\lambda =1\mathrm{Mpc}$) is in 10 nG units.