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We report the detection of thermal Sunyaev-Zeldovich effect (SZ) fluctuationsin the intracluster medium (ICM) of Coma cluster observed with ${\it Planck}$.The SZ data links the maximum observable X-ray scale to the large Mpc scale,extending our knowledge of the power spectrum of ICM fluctuations. Deprojectingthe 2-d SZ perturbations into 3-d pressure fluctuations, we find an amplitudespectrum which peaks at $\delta P/P = 33\pm 12\%$ and $74\pm19\%$ in the $15'$and $40'$ radius region, respectively. We perform tests to ensure fluctuationsare intrinsic to the cluster and not due to noise contamination. By usinghigh-resolution hydrodynamical models, we improve the ICM turbulenceconstraints in Coma, finding 3-d Mach number ${\rm Ma_{3d}}= 0.8\pm0.3$ (15'region), increasing to supersonic values at larger radii (40'), and aninjection scale $L_{\rm inj}\approx 500$ kpc. Such properties are consistentwith driving due to mergers, in particular tied to internal galaxy groups. Thelarge pressure fluctuations show that Coma is in adiabatic mode (mediated bysound waves), rather than isobaric mode (mediated by buoyancy waves). Aspredicted by turbulence models, the distribution of SZ fluctuations islog-normal with mild non-Gaussianities (heavy tails). The substantialnon-thermal pressure support implies hydrostatic mass bias $b_M=-15\%$ to$-45\%$ from the core to the outskirt region, respectively. While total SZpower probes the thermal energy content, the SZ fluctuations constrain thenon-thermal deviations important for precision cosmology. The proposed, novelapproach can be exploited by multifrequency observations using ground basedinterferometers and future space CMB missions.

Thermal SZ fluctuations in the ICM: probing turbulence and thermodynamics in Coma cluster with ${\it Planck}$