In this contribution, motivated by the quest to understand cosmic acceleration, based on the theory of Hořava–Lifshitz and on the branch-cut gravitation, we investigate the effects of non-commutativity of a mini-superspace of variables obeying the Poisson algebra on the structure of the branch-cut scale factor and on the acceleration of the Universe. We follow the guiding lines of a previous approach, which we complement to allow a symmetrical treatment of the Poisson algebraic variables and eliminate ambiguities in the ordering of quantum operators. On this line of investigation, we propose a phase-space transformation that generates a super-Hamiltonian, expressed in terms of new variables, which describes the behavior of a Wheeler–DeWitt wave function of the Universe within a non-commutative algebraic quantum gravity formulation. The formal structure of the super-Hamiltonian allows us to identify one of the new variables with a modified branch-cut quantum scale factor, which incorporates, as a result of the imposed variable transformations, in an underlying way, elements of the non-commutative algebra. Due to its structural character, this algebraic structure allows the identification of the other variable as the dual quantum counterpart of the modified branch-cut scale factor, with both quantities scanning reciprocal spaces. Using the iterative Range–Kutta–Fehlberg numerical analysis for solving differential equations, without resorting to computational approximations, we obtained numerical solutions, with the boundary conditions of the wave function of the Universe based on the Bekenstein criterion, which provides an upper limit for entropy. Our results indicate the acceleration of the early Universe in the context of the non-commutative branch-cut gravity formulation. These results have implications when confronted with information theory; so to accommodate gravitational effects close to the Planck scale, a formulation à la Heisenberg’s Generalized Uncertainty Principle in Quantum Mechanics involving the energy and entropy of the primordial Universe is proposed.

Based on an analytically continued Riemannian foliated quantum gravity super-Hamiltonian, known as branch cut quantum gravity (BCQG) we propose a novel approach to investigating the effects of noncommutative geometry on a minisuperspace of variables, influencing the acceleration behavior of the Universe’s wave function and the cosmic scale factor. Noncommutativity is introduced through a deformation of the conventional Poisson algebra, enhanced with a symplectic metric. The resulting symplectic manifold provides a natural setting that enables an isomorphism between canonically conjugate dual vector spaces, spanning the BCQG cosmic scale factor and its complementary quantum counterpart. Using this formulation, we describe the dynamic evolution of the Universe’s wave function, the cosmic scale factor, and its complementary quantum image. Our results strongly suggest that the noncommutative algebra induces late-time accelerated growth of the wave function, the Universe’s scale factor, and its complementary quantum counterpart, offering a new perspective on explaining the accelerating cosmic expansion rate and the inflationary period. In contrast to the inflationary model, where inflation requires a remarkably fine-tuned set of initial conditions in a patch of the Universe, analytically continued non-commutative foliated quantum gravity captures short and long scales, driving the evolutionary dynamics of the Universe through a reconfiguration of the primordial cosmic content of matter and energy. This reconfiguration is encapsulated into a quantum field potential, which leads to the generation of relic gravitational waves, a topic for future investigation. Graphical representations and contour plots indicate a characteristic torsion (or twist) deformation of spacetime geometry. This result introduces new speculative elements regarding the reconfiguration of matter and energy as a driver of spacetime torsion deformation, generating relic gravitational waves and serving as an alternative topological mechanism for the Universe’s acceleration. However, these assumptions require further investigation.

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 conventional space and time coordinates, typically treated as classical numbers, are replaced by complementary quantum dual fields. Within this framework, consistent with the Bekenstein criterion and the Hawking–Hertog multiverse conception, singularities merge into a helix-like cosmic scale factor that encodes the topological transition between the contraction and expansion phases of the universe analytically continued into 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 now-surpassed conventional concept of a primordial singularity and a subsequent expansion phase, whose transition region is characterized by a Riemannian topological foliated structure. The present linearized formulation, based on a slight gravitational field perturbation, also reveals a high sensitivity of relic gravitational wave amplitudes to the primordial matter and energy content during the universe’s phase transition. It further predicts stochastic homogeneous distributions of gravitational wave intensities arising from the interplay of short- and long-spacetime effects within the non-commutative algebraic framework. These results align with the anticipated future observations of relic gravitational waves, expected to pervade the universe as a stochastic, homogeneous background.

In the present contribution, a preliminary analysis of the effects of the Generalized Uncertainty Principle (GUP) with a minimum length, in the context of compact stars, is performed. On basis of a deformed Poisson canonical algebra with a parametrized minimum length scale that induces deviations from conventional Quantum Mechanics, fundamental questions involving the consistence, evidences and proofs of this approach as a possible cure for unbounded energy divergence are outlined. The incorporation of GUP effects into semiclassical 2N-dimensional systems is made by means of a time-invariant distortion transformation applied to their non-deformed counterparts. Assuming the quantum hadrodynamics $\sigma -\omega$ approach as a toy-model, due to its simplicity and structured description of neutron stars, we perform a preliminary analysis of GUP effects with a minimum spacetime length on these compact objects. The corresponding results for the equation of state and the mass-radius relation for neutron stars are in tune with recent observations with a maximum mass around 2.5 M_⊙ and radius close to 12 km. Our results also indicate the smallness of the noncommutative scale.

We propose a novel extension to the recently developed non-commutative Riemannian foliated branch-cut quantum gravity (BCQG). On basis in an extended Faddeev–Jackiw symplectic deformation of the conventional Poisson algebra, we investigate non-commutativity effects on a symplectic topological manifold that provides a natural isomorphic setting composed by a triad of canonically conjugate scalar complex fields which comprise quantum complementary dualities. The resulting extended manifold spans the canonical BCQG cosmic scale factor and, in addition to its complementary quantum counterpart, outlined in the perfect fluid domain of Hermann Weyl, a scalar-complex inflaton-inspired field. By constructing a triad of quantized fields underlying a noncommutative spacetime, this extension allows the elaboration of a scenario where the original branch-cut domain coexists with a complex scalar inflaton-type field, settled on a plateau of a strong gravity interaction structure. Alongside shaping this structure settled on a background composed by the primordial energy, radiation-, quintessence-, stiff-, baryon- and dark matter, combined with the spacetime components of the scalar Ricci curvature at different orders, cosmic inflation is driven. The presence of the inflaton field in the present formulation transcends the character of a simple reference field, to the extent that its original motivation has been to driven inflation, a role played by the original formulation of the BCQG. The resulting symplectic algebraic extension allows additionally a liaison with primordial cosmic density perturbation predictions based on heuristic derivations of the scale invariance of the primordial spectrum of the Universe, as well as with eternal inflation and primordial chaotic cosmic mechanisms proposals. Given the peculiar characteristic of BCQG in overcoming the primordial singularity, the non-commutative symplectic extension proposal, due to the complex character of the constituent fields, additionally outlines the evolutionary process of the mirror Universe that encompasses analytically continued complex conjugate Friedmann's-type equations. This brings unique characteristics to the main scenario for the BCQG cosmic evolution, with a mirrored parallel evolutionary Universe, adjacent to ours, nested in the structure of space and time. The evolutionary process of the mirror Universe occurs in the negative sector of cosmological thermal time, and is characterized by a continuous cosmic contraction, with a systematic and continuous increase in temperature and a decrease in entropy before reaching the transition region to the current stage of the Universe. In contrast, the subsequent present evolutionary stage of cosmic expansion is characterized by a positive complex cosmological-time sector, accompanied by a systematic decrease in temperature and increase in entropy. On basis on complementary analytically continued Friedmann’s-type equation, combined with a quantum approach based on the Ho\v{r}awa-Lifshitz quantum gravity, we describe the dynamic evolution of the Universe’s wave function, unfolding unprecedented predictions for the cosmic evolution and inflation. This approach offers a new perspective to explain the accelerated cosmic expansion of the Universe, strongly suggesting that non-commutative algebra induces the late accelerated growth of the wave function of the Universe as well as of the corresponding scale factor and its complementary quantum counterparts. In contrast to the conventional inflationary model, where inflation requires a remarkably fine-tuned set of initial conditions in a patch of the Universe, non-commutative foliated quantum gravity, analytically continued to the complex plane, captures short and long scales of spacetime, leading to an evolutionary cosmic dynamic through a topological reconfiguration of the primordial cosmic matter and energy content. Still, we consider in this work the interplay between naturalness and a fine-tuning set constraining of the cosmological initial conditions. This result introduces new speculative framework elements regarding the reconfiguration of matter and energy due to an underlying non-commutative spatio-temporal structure as a driver of spacetime cosmic acceleration.