Physical insights from the spectrum of the radio halo in MACS J0717.5+3745

We present new LOFAR observations of the massive merging galaxy cluster MACS J0717.5+3745. The cluster hosts the most powerful radio halo known to date. These new observations, in combination with published uGMRT (300$-$850 MHz) and VLA (1$-$6.5 GHz) data, reveal that the halo is more extended than previously thought, with a largest linear size of $\sim2.2 \rm Mpc$. The halo shows a steep spectrum ($α_{144\,\text{MHz}}^{1.5\,\text{GHz}}\sim-1.4$) and a steepening ($α_{1.5 \text{GHz}}^{5.5 \text{GHz}}\sim-1.9$) above 1.5 GHz. We find a strong scattering in spectral index maps on scales of 50$-$100 kpc. We suggest that such a strong scattering may be a consequence of the regime where inverse Compton dominate the energy losses of electrons. The spectral index becomes steeper and shows an increased curvature in the outermost regions of the halo. We combined the radio data with \textit{Chandra} observations to investigate the connection between the thermal and non-thermal components of the intracluster medium (ICM). Despite a significant substructure in the halo emission, the radio brightness correlates strongly with the X-ray brightness at all observed frequencies. The radio-versus-X-ray brightness correlation slope steepens at a higher radio frequency (from $b_{144 \text{MHz}}=0.67\pm0.05$ to $b_{3.0 \text{GHz}}=0.98\pm0.09$) and the spectral index shows a significant anti correlation with the X-ray brightness. Both pieces of evidence further support a spectral steepening in the external regions. The compelling evidence for a steep spectral index, the existence of a spectral break above 1.5 GHz, and the dependence of radio and X-ray surface brightness correlation on frequency are interpreted in the context of turbulent reacceleration models. Under this scenario, our results allowed us to constrain that the turbulent kinetic pressure of the ICM is up to 10%.