Relic Gravitational Waves in the Noncommutative Foliated Riemannian Quantum Gravity

We present a study of relic gravitational waves based on a foliated gauge field theory defined over a spacetime endowed with a noncommutative algebraic-geometric structure. As an ontological extension of general relativity concerning manifolds, metrics and fiber bundles, the space and time coordinates, which conventionally correspond to classical numbers, are replaced in this formulation by complementary quantum dual fields. In this framework, consistently complying within the Bekenstein criterion, combined with the Hawking-Hertog multiverse conception, singularities merge, portrayed into a new cosmic helix-like scale factor, analytically continued to the complex plane. This scale-factor captures the essence of an intricate topological quantum-leap transition between two phases of the branching universe: a contraction phase preceding the surpassed conventional concept of a primordial singularity and a subsequent expansion phase whose transition region between the two phases is characterized by a Riemannian topological foliated structure. The present linearized formulation, underlying a slightly gravitational field perturbation reveals, during the universe phase transition, high sensitivity of the relic gravitational waves amplitudes to the primordial matter and energy content. The present formulation reveals a stochastic homogeneous distributions of gravitational-wave intensities arising from the capture of short and long spacetime scales within the noncommutative algebraic framework. These results are consistent with the expected anticipated future observations of relic gravitational waves, which are expected to fill the universe as a stochastic homogeneous background.