On almost commutative Friedmann–Lemaître–Robertson–Walker geometries

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We analyze the leading terms of the spectral action for a model of noncommutative geometry, which is a product of 4D Riemannian manifold with a two-point space exploring the previously neglected case when the metrics over each sheet are different. Assuming the Friedmann–Lemaître–Robertson–Walker type of the metric for both sheets we obtain the action, which in addition to the the usual cosmological constant terms and the Einstein–Hilbert term involves a nonlinear interaction term. We study qualitative picture of potential consequences of such term in the basic cosmological models.