Transition to Zero Cosmological Constant and Phantom Dark Energy as Solutions Involving Change of Orientation of Space-Time Manifold

Solutions with degenerate metric (\(det(g_{\mu\nu})=0\) or \(g_{\mu\nu}=0\)) in the first order formalism (FOF) are physically acceptable: they may describe topology changes (Horowitz) and reduction of "metrical dimension" (Tseytlin) of space-time. The latter implies disappearance of the volume element \(\sqrt{-g}d^4x\) of 4-D space-time. We pay attention that besides \(\sqrt{-g}\), the 4-D space-time differentiable manifold possesses also a "manifold volume measure" (MVM)described by a 4-form which is sign indefinite and generically independent of the metric. The FOF proceeds with originally independent connection and metric structures of the space-time manifold. We bring up the question whether the FOF should be supplemented with degrees of freedom of MVM. Adding such manifold degrees of freedom to the action principle in the FOF we realize very interesting dynamics. Such Two Measures Theory enables radically new approaches to resolution of the cosmological constant problem. We show that fine tuning free solutions describing a transition to \(\Lambda =0\) state involve oscillations of \(g_{\mu\nu}\) and MVM around zero. The latter can be treated as a dynamics involving changes of orientation of the space-time manifold. As we have shown earlier, in realistic scale invariant models (SIM), solutions formulated in the Einstein frame satisfy all existing tests of General Relativity (GR). Here we reveal surprisingly that in SIM, all ground state solutions with \(\Lambda\neq 0\) appear to be degenerate either in \(g_{00}\) or in MVM. Sign indefiniteness of MVM in a natural way yields a dynamical realization of a phantom cosmology (\(w