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Normalizing the Einstein-Hilbert action by the volume functional makes the theory invariant under constant shifts in the Lagrangian. The field equations then resemble unimodular gravity whose otherwise arbitrary cosmological constant is now determined as a Machian universal average. We first prove that an empty space-time is necessarily flat, and then demonstrate, by invoking the scalar field paradigm, that the cosmological constant is actually zero. Normalized general relativistic cosmology is then discussed at the mini-superspace level, where we confirm the generic zero cosmological constant result, and furthermore prove that if matter is attractive then the Universe cannot be closed. The only non-vanishing cosmological constant solutions are associated with static Einstein-like closed universes.

The Hamiltonian for the quantized gravitational and matter fields contains both positive and negative energy particle contributions, which leads through a positive and negative energy symmetry of the vacuum to a cancellation of the zero-point vacuum energy and a vanishing cosmological constant in the presence of a gravitational field. The positive and negative energy particles interact only weakly through gravity. As in the case of antimatter, the negative energy matter is not found naturally on Earth or in galaxies in the universe. We introduce a graviton momentum cutoff ΛG ≤ 2 x10-3 eV that leads to a gravitational stability of the Minkowski spacetime vacuum with a lifetime greater than the age of the universe. A positive energy spectrum and a consistent unitary field theory for a pseudo- Hermitian Hamiltonian is obtained by demanding that the pseudo-Hermitian Hamiltonian is PT symmetric. The quadratic divergences in the two-point vacuum fluctuations and the self- energy of a scalar field are removed. By adopting a Higgsless model of electroweak theory, we remove the fine-tuning associated with a spontaneous symmetry breaking vacuum density. We also postulate that there are no phase transitions associated with QCD and at higher particle physics energy scales, removing all theorized quark-gluon vacuum density condensates and their fine-tuned vacuum densities and cosmological constant contributions.

Theories of gravity having connection as an independent dynamical variable can have a massive spin -2 particle together with the massless graviton in their spectrum. We study the potentiality of such models as a consistent infrared modification of gravity. It will be shown that these models are free from ghost like Boulware-Deser mode in the background of arbitrary torsionless Einstein manifolds. At least for weak for weak enough curvature the dangerous spin zero mode has a healthy kinetic energy in our backgrounds. vDVZ singularity seems to be a generic feature which may be cured by Vainshtein mechanism as in other infrared modified theories of gravity.