Emergent universe scenario in the modified Chaplygin gas: towards an exact solution and observational constraints

The modified Chaplygin gas (MCG) model is revisited to examine its capability in describing the complete cosmic evolution within a unified framework. The model is found to be consistent with observational constraints, including the cosmic microwave background (CMB) shift parameter, while also supporting standard structure formation alongside late-time cosmic acceleration. Due to the highly nonlinear nature of the governing field equations, exact analytical solutions in cosmic time are generally not attainable. To overcome this, we adopt a first-order approximation, which enables us to obtain an analytical expression for the scale factor and determine the corresponding flip time. The effective equation-of-state parameter evolves from a positive value in the early matter-dominated era to weff ≈ -1 w eff ≈ - 1 at late times, in agreement with observations. In addition, a solution in the phantom regime exhibits an emergent, non-singular behavior, where the universe originates from a finite minimum scale factor and smoothly evolves toward a late-time de Sitter-like phase, approaching the Λ Λ CDM limit. The emergent solution is shown to be geodesically complete and free from curvature singularities, as confirmed by the regular behavior of the Kretschmann scalar. We also perform a CMB analysis for all the considered cases, and in each case the obtained values of Rₜₕ R th and χ ²CMB χ CMB 2 remain consistent with observations within the 1σ 1 σ confidence level. Furthermore, the combined Hubble-57 and CMB analysis leads to significantly improved constraints on the model parameters. Moreover, the obtained solutions are independently verified using the Raychaudhuri equation, providing a consistency check that further reinforces the robustness of the methodology. Overall, the results demonstrate that the MCG model provides a self-consistent and observationally viable description of cosmic evolution without requiring modifications to general relativity.