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We present the first detection of the cross-correlation angular powerspectrum between the thermal Sunyaev-Zel'dovich (tSZ) effect and the X-rayemission over the full sky. The tSZ effect and X-rays are produced by the samehot gas within groups and clusters of galaxies, which creates a naturallystrong correlation between them that can be used to boost the joint signal andderive cosmological parameters. We computed the correlation between the ROSATAll Sky Survey in the 0.5-2 keV energy band and the tSZ effect reconstructedfrom six Planck all-sky frequency maps between 70 and 545 GHz. We detect asignificant correlation over a wide range of angular scales. In the range$50<\ell < 2000$, the cross-correlation of X-rays to tSZ is detected at anoverall significance of 28 $\sigma$. As part of our systematic study, weperformed a multi-frequency modelling of the AGN contamination and thecorrelation between cosmic infra-red background and X-rays. Taking advantage ofthe strong dependence of the cross-correlation signal on the amplitude of thepower spectrum, we constrained $\sigma_8 = 0.804 \pm 0.037$, where modellinguncertainties dominate statistical and systematic uncertainties. We alsoderived constraints on the mass indices of scaling relations between the halomass and X-ray luminosity, L$_{500}$-M$_{500}$, and SZ signal,Y$_{500}$-M$_{500}$, $a_{\rm sz} + a_{\rm x} = 3.37 \pm 0.09$, and on theindices of the extra-redshift evolution, $\beta_{\rm sz} + \beta_{\rm x} =0.4^{+0.4}_{-0.5}$.

Cosmological constraints from the observed angular cross-power spectrum between Sunyaev-Zel'dovich and X-ray surveys