The full contribution of a stochastic background of magnetic fields to CMB anisotropies

TL;DR

This paper analyzes how a stochastic background of primordial magnetic fields, modeled with a power-law spectrum $P_B(k)=A(k/k_*)^{n_B}$ and damping at $k_D$, sources scalar, vector, and tensor metric perturbations through the magnetic energy–momentum tensor. It derives exact Fourier spectra for the EMT components, implements them in a CAMB-based code, and computes the resulting CMB temperature and polarization spectra, showing that scalar perturbations dominate TT at low to intermediate $\ell$ while vector perturbations dominate at high $\ell$, with tensor contributions consistently subdominant. The vector $B$-mode signature peaks around $\ell\sim 2000$ and can exceed lensing-induced $B$ modes for certain PMF amplitudes, offering a potential observational handle on PMFs. The analysis broadens previous work by treating more $n_B$ values (including $n_B=-5/2$), providing exact convolution integrals for the EMT spectra, and quantifying the distinct imprints of PMFs on CMB anisotropies, with implications for Planck and future experiments.

Abstract

We study the contribution of a stochastic background (SB) of primordial magnetic fields (PMF) on the anisotropies in temperature and polarization of the cosmic microwave background radiation (CMB). A SB of PMF modelled as a fully inhomogeneous component induces non-gaussian scalar, vector and tensor metric linear perturbations. We give the exact expressions for the Fourier spectra of the relevant energy-momentum components of such SB, given a power-law dependence parametrized by a spectral index $n_B$ for the magnetic field power spectrum cut at a damping scale $k_D$. For all the values of $n_B$ considered here, the contribution to the CMB temperature pattern by such a SB is dominated by the scalar contribution and then by the vector one at higher multipoles. We also give an analytic estimate of the scalar contribution to the CMB temperature pattern.

Figures (5)

• Figure 1: Comparison of $k^3 |\rho_B(k)|^2$ (solid line), $k^3 |L (k)|^2$ (large dashed line) $k^3 |\Pi_{i}^{(V)} (k)|^2$ (small dashed line), $k^3 |\Pi_{ij}^{(T)} (k)|^2$ (medium dashed line) in units of $\langle B^2 \rangle^2/(1024 \pi^3)$ versus $k/k_D$. The left and right panel are for $n_B = 2$ and $n_B=-5/2$, respectively.
• Figure 2: Plot of $k^3 |\Pi^{(V)} (k)|^2$ (left panel) and $k^3 |\Pi^{(T)} (k)|^2$ (right panel) in units of $\langle B^2 \rangle^2/(1024 \pi^3)$ versus $k/k_D$ for different $n_B$ for fixed $\langle B^2 \rangle$. The different lines are for $n_B = -5/2, -3/2, -1, 0, 1, 2, 3$ ranging from the solid to the longest dashed.
• Figure 3: CMB anisotropies angular power spectrum for temperature (TT hereafter, top-left panel), temperature-E polarization cross correlation (TE hereafter, top-right panel), E polarization (EE hereafter, bottom-left panel), B polarization (BB hereafter, bottom-right panel). The solid line is the adiabatic scalar contribution in TT, TE, EE panels, whereas it is the tensor homogeneous contribution in the BB panel (for a tensor-to-scalar ratio $r =0.1$); the dotted, dot-dashed, dashed are the scalar, vector and tensor contributions of a SB of PMF respectively for $\sqrt{\langle B^2 \rangle} = 7.5 \,$ nG, $n_B = 2$ and $k_D = 2 \pi \, {\rm Mpc}^{-1}$. The dotted line in the BB panel is the lensing contribution. The cosmological parameters of the flat $\Lambda CDM$ model are $\Omega_b \, h^2 = 0.022$, $\Omega_c \, h^2 = 0.123$, $z_{\rm re}= 12$, $n_s=1$, $H_0 = 100 \, h \, {\rm km \, s}^{-1} \, {\rm Mpc}^{-1} = 72 \, {\rm km \, s}^{-1} \, {\rm Mpc}^{-1}$.
• Figure 4: CMB angular power spectrum for TT (left top panel), TE (left top panel), EE (bottom left), BB (bottom right). The solid line is the adiabatic scalar contribution in TT, TE, EE panels, whereas it is the tensor homogeneous contribution in the BB panel (for a tensor-to-scalar ratio $r =0.1$); the dotted, dot-dashed, dashed are the scalar, vector and tensor contributions of a SB of PMF respectively for $\sqrt{\langle B^2 \rangle} = 7.5 \,$ nG, $n_B = -5/2$ and $k_D = 2 \pi \, {\rm Mpc}^{-1}$. The dotted line in the BB panel is the lensing contribution. The cosmological parameters of the flat $\Lambda CDM$ model are the same as in Fig. 3.
• Figure 5: Vector contributions to the CMB angular power spectrum for BB. The solid line is the tensor homogeneous contribution for a tensor-to-scalar ratio $r =0.1$ and the dotted line is the lensing contribution with cosmological parameters as in the previous figures; the triple-dotted, long dashed, dot-dashed and dashed are the vector spectra obtained with $\sqrt{\langle B^2 \rangle} = 7.5 \,$ nG, $k_D = 2 \pi {\rm Mpc}^{-1}$, for $n_B = -2.5, -1.5, -1, 2$, respectively. Note how the spectra for $n_B=-1$ and $n_B=2$ are super-imposed since the Fourier spectra of the vector part of the PMF EMT are both white noise for $k \ll k_D$ for these spectral indexes.