Reparametrization Invariance and the Phenomena of Dark Energy and Dark Matter

The nature of dark energy (DE) challenges our comprehension of the cosmos, appearing as the source responsible for the Universe’s accelerating expansion. This work considers the Einstein Cosmological Constant (Λ_E) as a manifestation of DE, interpreted through the lens of Reparametrization Invariant Scaling Symmetry (RISS). Within this paradigm, Λ_E emerges as a “kinetic energy” term, derived from relative temporal motion, setting it apart from the conventional kinetic energy based on spatial relative motion.

Central to this exploration is the scale factor λ(t), representing a reparametrization capable of rendering Λ_E dynamically within the extended equations of Einstein’s General Relativity (EGR) as Λ = Λ_E λ² . Through meticulous derivations, the governing equations of λ(t) and its interplay with Λ_E are articulated. Imposing reparametrization symmetry on the equations of motion reveals a new avenue for addressing the missing mass problem evident at galactic and extragalactic scales. Here, improper/non-affine (non-co-moving) temporal parameterizations introduce fictitious forces, whose presence is reconciled through the symmetry framework.

This symmetry-based approach naturally yields the MOND-like relationship g² ∼ (a₀g_N), where g denotes gravitational acceleration, a₀ represents the fundamental MOND acceleration, and g_N is the Newtonian acceleration. The theoretical predictions for Λ_E and a₀ demonstrate remarkable alignment with their observed magnitudes, lending credence to this interpretation. This synthesis underscores a potential unifying principle in our understanding of dark energy and dark matter phenomena.