In a recent work, it has been proposed that the recent cosmic passage to a cosmic acceleration era is the result of the existence of small anti-gravity sources in each galaxy and clusters of galaxies. In particular, a swiss-cheese cosmology model which relativistically integrates the contribution of all these anti-gravity sources on galactic scale has been constructed assuming the presence of an infrared fixed point for a scale dependent cosmological constant. The derived cosmological expansion provides explanation for both the fine tuning and the coincidence problem. The present work relaxes the previous assumption on the running of the cosmological constant and allows for a generic scaling around the infrared fixed point. Our analysis reveals in order to produce a cosmic evolution consistent with the best $Lambda$CDM model, the IR-running of the cosmological constant is consistent with the presence of an IR-fixed point.
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