Emergent Dark Energy and Cyclic Dynamics from Dual-Sector Energy Exchange: A Phenomenological Toy Model

I present a phenomenological toy model exploring whether dark energy and cyclic cosmological evolution can emerge from energy exchange between two separate Friedmann–Robertson–Walker sectors, rather than from a fundamental cosmological constant. The observable sector x and a hidden partner sector y are coupled through an interaction term Q = λHxρy, which modifies the continuity equation of sector x and produces an effective equation of state wₑff = −1 − λρy/ (3ρx). Combined with a loop-quantum-cosmology bounce mechanism, this creates a closed feedback loop between energy density, expansion rate, and energy transfer, admitting non-singular cyclic dynamics. A qualitative numerical integration using a Runge–Kutta scheme confirms alternating expansion–contraction phases and sign changes in the interaction term, consistent with the proposed mechanism. The model is intentionally minimal; the coupling term is phenomenological and no perturbation analysis is included. A heuristic higher-dimensional interpretation is discussed, in which the two sectors correspond to brane-like universes in a five-dimensional bulk and the coupling term represents energy exchange through the extra dimension. The goal is to demonstrate conceptual consistency and to outline a programme for further theoretical development.