Probing intergalactic magnetic fields with simulations of electromagnetic cascades

We determine the effect of intergalactic magnetic fields on the distribution of high-energy gamma rays by performing three-dimensional Monte Carlo simulations of the development of gamma-ray-induced electromagnetic cascades in the magnetized intergalactic medium. We employ the so-called “Large Sphere Observer” method to efficiently simulate blazar gamma ray halos. We study magnetic fields with a Batchelor spectrum and with maximal left- and right-handed helicities. We also consider the case of sources whose jets are tilted with respect to the line of sight. We verify the formation of extended gamma ray halos around the source direction, and observe spiral-like patterns if the magnetic field is helical. We apply the $Q$-statistics to the simulated halos to extract their spiral nature and also propose an alternative method, the $S$-statistics. Both methods provide a quantitative way to infer the helicity of the intervening magnetic fields from the morphology of individual blazar halos for magnetic field strengths $B\gtrsim {10}^{-15}\mathrm{G}$ and magnetic coherence lengths $L_{\mathrm{c}}\gtrsim 100\mathrm{Mpc}$. We show that the $S$-statistics has a better performance than the $Q$-statistics when assessing magnetic helicity from the simulated halos.

Figure 1

Arrival directions of photons from a monochromatic TeV blazar emitting gamma rays with energies $E_{\mathrm{TeV}}=10\mathrm{TeV}$ in a collimated jet, in the energy range of 1–100 GeV, projected onto a plane, are shown in the upper row; the color scale indicates the number of photons per bin. The deflection angles ($\theta$) of observed gamma rays as a function of the energy are presented in the bottom row. The magnetic field is stochastic with a spectrum according to Eq. (21) and a mean field strength of ${10}^{-15}\mathrm{G}$ (left column) and ${10}^{-16}\mathrm{G}$ (right column); blue dots correspond to simulation results and the black line represents the analytical prediction using Eq. (26).

Figure 2

Energy-dependent sky maps for a uniform magnetic field with $B={10}^{-15}\mathrm{G}$. We show the cases of a tightly collimated jet with magnetic field parallel (top left), perpendicular (top right), and tilted by 45 deg (bottom left) and 75 deg (bottom right) to the blazar jet direction which is taken to be along the line of sight. The different colors represent the following energy ranges: 5–10 (magenta), 10–15 (blue), 15–20 (green), 20–30 (yellow), 30–50 (orange), and 50–100 GeV (red).

Figure 3

Sky maps of arrival directions of photons from a blazar at a distance $D_{\mathrm{s}}=1\mathrm{Gpc}$ emitting photons with energy $E_{\mathrm{TeV}}=10\mathrm{TeV}$ in a jet with a half-opening angle of $\mathrm{\Psi }=5°$ directed at the observer (left column) and tilted by 5° with respect to the line of sight (right column), respectively. The magnetic field is assumed to be stochastic with rms strength of $B={10}^{-15}\mathrm{G}$, coherence length $L_{\mathrm{c}}\simeq 120\mathrm{Mpc}$, and maximal negative (upper panels, $f_H=-1$), null (central, $f_H=0$) and maximal positive (lower panels, $f_H=+1$) helicities, respectively. The colors represent the same energies as in Fig. 2.

Figure 4

Sky maps of arrival directions of photons from a blazar at a distance $D_{\mathrm{s}}=1\mathrm{Gpc}$ emitting photons with energy $E_{\mathrm{TeV}}=10\mathrm{TeV}$ in a jet with a half-opening angle of $\mathrm{\Psi }=5°$ directed at the observer. The magnetic field is assumed to be stochastic with rms strength of $B={10}^{-15}\mathrm{G}$ and a coherence length of $L_{\mathrm{c}}\simeq 50\mathrm{Mpc}$ (left), $L_{\mathrm{c}}\simeq 150\mathrm{Mpc}$ (center) and $L_{\mathrm{c}}\simeq 250\mathrm{Mpc}$ (right) for $f_H=+1$. The colors represent the same energies as in Fig. 2.

Figure 5

Different current helicity measures for the three cases shown in Fig. 3, i.e. negative current helicity ($f_{\mathrm{H}}=-1$, red), zero current helicity ($f_{\mathrm{H}}=0$, black) and positive current helicity ($f_{\mathrm{H}}=+1$, blue). The left panel shows the total distribution of “current helicity,” defined as $\mathbf{B}·(\nabla \times \mathbf{B})$ (which serves as a gauge-invariant proxy to the actual magnetic helicity), in the whole simulation box, normalized to 1. In the center panel the same measure is shown, however restricted only to the line of sight and the neighboring cells. Finally, the right panel shows the helicity values along the line of sight from the source (at $x=0\mathrm{Mpc}$) to the observer (at $x=1000\mathrm{Mpc}$).

Figure 6

$Q$-statistics for the case of a source tilted 5° with respect to the line of sight, for $D_{\mathrm{s}}=1\mathrm{Mpc}$, $E_{\mathrm{TeV}}=10\mathrm{TeV}$ and $\mathrm{\Psi }=5°$. All panels correspond to the three right-hand panels of Fig. 3, i.e. $f_{\mathrm{H}}=-1$ at the top, $f_{\mathrm{H}}=0$ in the middle $f_{\mathrm{H}}=+1$ at the bottom panel. The triplets in the legends correspond to $E_1$, $E_2$, $E_3$ in GeV, each in intervals of $[E_i,E_i+10\mathrm{GeV}]$.

Figure 7

The average polar angle $\overline{\theta }$ in dependence on the azimuthal angle $\varphi$, as calculated in Eq. (31) for the three cases $f_{\mathrm{H}}=-1$ (left), $f_{\mathrm{H}}=0$ (center) and $f_{\mathrm{H}}=+1$ (right), corresponding to the three cases in Fig. 3. Here we have chosen $n_{\mathrm{bin}}=20$. The colors correspond to the energy ranges of the arrival energies $E_{\gamma }$ in the same way as in Fig. 2, i.e. 5–10 (magenta), 10–15 (blue), 15–20 (green), 20–30 (yellow), 30–50 (orange), and 50–10 GeV (red).