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Friedmann inflation in Horava-Lifshitz gravity with a scalar field

We study Friedmann inflation in general Horava-Lifshitz (HL) gravity with detailed and non-detailed but also without the projectability conditions. Accordingly, we derive the modifications in the Friedmann equations due to single scalar field potentials describing power-law and minimal-supersymmetrically extended inflation. By implementing four types of the equations-of-state charactering the cosmic background geometry, the dependence of the tensorial and spectral density fluctuations and their ratio on the inflation field is determined. The latter characterizes the time evolution of the inflation field relative to the Hubble parameter. Furthermore, the ratio of tensorial-to-spectral density fluctuations is calculated in dependence on the spectral index. The resulting slow-roll parameters apparently differ from the ones deduced from the standard General Relativity (Friedmann gravity). We also observe that the tensorial-to-spectral density fluctuations continuously decrease when moving from non-detailed HL gravity, to Friedmann gravity, to HL gravity without the projectibility, and to detailed HL gravity. This regular patter is valid for three types of cosmic equations-of-state and different inflation potential models. The results fit well with the recent PLANCK observations.