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Extensive measurements of the X-ray background (XRB) yield a reasonably reliable characterization of its basic properties. Having resolved most of the cosmic XRB into discrete sources, the levels and spectral shapes of its main components can be used to probe both the source populations and also alternative cosmological and large-scale structure models. Recent observations of clusters seem to provide evidence that clusters formed earlier and are more abundant than predicted in the standard Λ cold dark matter (ΛCDM) model. This motivates interest in alternative models that predict enhanced power on cluster scales. We calculate predicted levels and spectra of the superposed emission from groups and clusters of galaxies in ΛCDM and in two viable alternative non-Gaussian and early dark energy models. The predicted levels of the contribution of clusters to the XRB in the non-Gaussian models exceed the measured level at low energies and levels of the residual XRB in the 2–8 keV band; these particular models are essentially ruled out. Our work demonstrates the diagnostic value of the integrated X-ray emission from clusters, by considering also its dependences on different metallicities, gas and temperature profiles, Galactic absorption, merger scenarios and on a non-thermal pressure component. We also show that the XRB can be used for an upper limit for the concentration parameter value. [ABSTRACT FROM AUTHOR]