General non-singular accelerating cosmological solutions for an initialcosmic period of pure vacuum birth era are derived. This vacuum era isdescribed by a varying cosmological "constant" suggested by the RenormalisationGroup flow of Asymptotic Safety scenario near the ultraviolet fixed point. Inthis scenario, natural exit from inflation to the standard deceleratingcosmology occurs when the energy scale lowers and the cosmological "constant"becomes insignificant. In the following period where matter is also present,cosmological solutions with characteristics similar to the vacuum case aregenerated. Remarkably the set of equations allow for particle production andentropy generation. Alternatively, in the case of non-zero bulk viscosity,entropy production and reheating is found. As for the equations of motion, theymodify Einstein equations by adding covariant kinetic terms of the cosmological"constant" which respect the Bianchi identities. An advance of the proposedframework is that it ensures a consistent description of both a quantum vacuumbirth of the universe and a subsequent cosmic era in the presence of matter.
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