Late time transition of Universe and the hybrid scale factor

In this study, we propose an interacting model to explain the physical mechanism of the late time transition from matter-dominated era to the dark energy-dominated era of the Universe evolution and to obtain a scale factor a ( t ) representing two eras together. In the present model, we consider a minimal coupling of two scalar fields which correspond to the dark matter and dark energy interacting through a potential based on the FLRW framework. Analytical solution of this model leads to a new scale factor a ( t ) in the hybrid form a(t)=a₀ (t/t₀)α e^ht/t₀ a ( t ) = a 0 ( t / t 0 ) α e h t / t 0 . This peculiar result reveals that the scale factor behaving as a (t) ∝ (t/t₀)α a ( t ) ∝ ( t / t 0 ) α in the range t/t₀≤ tc t / t 0 ≤ t c corresponds to the matter-dominated era while a(t) ∝ exp (ht/t₀) a ( t ) ∝ exp ( h t / t 0 ) in the range t/t₀>tc t / t 0 > t c accounts for the dark energy-dominated era, respectively. Surprisingly, we explore that the transition from the power-law to the exponential expansion appears at the crossover time t₀ ≈ 9.8 t 0 ≈ 9.8 Gyear. We attain that the presented model leads to precisely correct results so that the crossover time t₀ t 0 and α α are completely consistent with the exact solution of the FLRW and re-scaled Hubble parameter