PITSZI: Probing Intra-cluster medium Turbulence with Sunyaev-Zel'dovich Imaging -- Application to the triple merging cluster MACS J0717.5+3745

Turbulent gas motions are expected to dominate the non-thermal energy budget of the intracluster medium (ICM). The measurement of pressure fluctuations from high angular resolution Sunyaev-Zel'dovich imaging opens a new avenue to study ICM turbulence, complementary to X-ray density fluctuation measures. We develop a methodological framework designed to optimally extract information on the ICM pressure fluctuation power spectrum statistics, and publicly release the associated software named PITSZI. We apply this tool to the NIKA data of the merging cluster MACSJ0717 to measure its pressure fluctuation power spectrum at high significance, and to investigate the implications for its nonthermal content. Depending on the choice of the radial pressure model and the details of the applied methodology, we measure an energy injection scale $L_{inj} \sim 800$ kpc. The power spectrum normalization corresponds to a characteristic amplitude reaching $A(k_{peak}) \sim 0.4$. These results are are obtained assuming that MACSJ0717 can be described as pressure fluctuations on top of a single (smooth) halo, and are dominated by systematics due to the choice of the radial pressure model. Using simulations, we estimate that fitting a radial model to the data can suppress the observed fluctuations by up to 50\%, while a poorly representative radial model can induce spurious fluctuations, which we also quantify. Assuming standard scaling relations between the pressure fluctuations and turbulence, we find that MACSJ0717 presents a turbulent velocity dispersion $\sigma_v \sim 1200$ km/s, a kinetic to kinetic plus thermal pressure fraction $P_{k} / P_{k+th} \sim 20\%$, and we estimate the hydrostatic mass bias to $b_{HSE} \sim 0.3-0.4$. Our results are in excellent agreement with alternative measurements from X-ray surface brightness fluctuations, and in agreement with the fluctuations being adiabatic in nature.