A Single-Field Model of Cosmic Expansion with a Geometric Disformal Coupling

We propose a mechanism that unifies the cosmic expansion history using a single scalar field coupled to a disformal lapse function. In the Jordan frame, where matter is minimally coupled, the lapse modifies the Hubble rate measured by matter observers. As a result, matter observers perceive three successive expansion epochs, including early inflation, intermediate deceleration, and late time acceleration, even though the underlying Einstein-frame dynamics exhibits no distinct phases. The lapse function is constrained to satisfy the horizon entropy condition and to approach unity asymptotically, ensuring agreement with standard Einstein-frame dynamics at late times. In this framework, a small early-time value of the lapse compresses the physical time experienced by matter observers, producing the appearance of a rapid inflationary expansion occurring within a very short interval of their proper time. The time dilation effect induced by the lapse is alleviated at late times as the lapse approaches unity. A Noether symmetry approach is implemented to derive the scalar potential, which is exponential and leads to smooth power-law evolution in the Einstein frame. Although such solutions are known to have phenomenological limitations, the disformal lapse modifies the effective expansion dynamics experienced by matter observers while preserving the simplicity of power-law evolution.