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Time variation of G and , acceleration of the universe, coincidence problem and Mach's cosmological coincidence

AbstractWe study a gravitational model in which scale transformations play the key rolein obtaining dynamical G and Λ. We take a non-scale invariant gravitational actionwith a cosmological constant and a gravitational coupling constant. Then, by a scaletransformation, through a dilaton ﬁeld, we obtain a new action containing cosmologicaland gravitational coupling terms which are dynamically dependent on the dilaton ﬁeldwith Higgs type potential. The vacuum expectation value of this dilaton ﬁeld, throughspontaneous symmetry breaking on the basis of anthropic principle, determines the timevariations of G and Λ. The relevance of these time variations to the current acceleration ofthe universe, coincidence problem, Mach’s cosmological coincidence and those problemsof standard cosmology addressed by inﬂationary models, are discussed. The currentacceleration of the universe is shown to be a result of phase transition from radiationtoward matter dominated eras. No real coincidence problem between matter and vacuumenergy densities exists in this model and this apparent coincidence together with Mach’scosmological coincidence are shown to be simple consequences of a new kind of scale factordependence of the energy momentum density as ρ ∼ a