Unifying Cosmic Epochs and Effective Dynamical Brane Tension
We present a single analytic scale factor \(a(t)=e^{H(t)t}\bigl(1-e^{-k(t)t}\bigr)^{b(t)}\) that unifies the cosmic expansion history from inflation to the present in a smooth and differentiable manner. By eliminating conventional piecewise definitions, the model provides a compact global description of cosmic evolution while remaining consistent with SNIa, CMB, and BAO observations. The model introduces a time-dependent scale-modulation parameter $k(t)$, with the dimensionless combination $x(t)$ governing the asymptotic evolution across all epochs. This construction finds a compelling realization as a brane-inspired effective framework: interpreted as the ratio $x(t)\simρ(t)/λ(t)$ of the energy density to a dynamical brane tension, the model necessitates a time-varying $λ(t)$, thereby driving acceleration without a cosmological constant. This single parameter bridges the observational successes of the Randall-Sundrum II and chrono-brane paradigms within a coherent, effective framework. Although phenomenological in origin, the construction offers an analytically tractable and observationally testable alternative to $Λ$CDM that requires neither a fixed vacuum energy nor abrupt phase transitions.