Temporal Continuum Cosmology with a Quadratic Density Potential: Background Evolution, Stability, and Early-Universe Signatures

This work presents a four-dimensional covariant cosmological model based on a conserved temporal current coupled to general relativity.The theory is formulated from a primitive continuity law for a temporal density and flow vector, from which a variational action and stress tensor are derived. The temporal sector modifies the Einstein equations without introducing additional dimensions or independent scalar inflation fields, and should be interpreted as an effective continuum description valid in the high-density regime. To obtain a predictive model, the temporal potential is fixed to a quadratic form in the density. When the equations are reduced to a homogeneous and isotropic spacetime, the continuity law determines the evolution of the temporal density and leads to modified Friedmann equations. In the high-density limit the quadratic term dominates, producing a stiff pre-radiation phase in which the energy density scales faster than radiation. Linear perturbations are analysed on the resulting background, and stability conditions are derived that constrain the temporal parameters and ensure recovery of general relativity at low density. The model predicts a modified expansion history in the earliest cosmological epoch, which may influence the initial conditions for standard cosmology while remaining consistent with late-time observations. This paper provides a predictive formulation of temporal continuum cosmology in four dimensions and presents the first explicit background evolution, perturbation analysis, and parameter constraints for a quadratic density potential.