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 $$a(t)=a_{0} (t/t_{0})^{\alpha } e^{ht/t_{0}}$$ a ( t ) = a 0 ( t / t 0 ) α e h t / t 0 . This peculiar result reveals that the scale factor behaving as $$a (t) \propto (t/t_{0})^{\alpha }$$ a ( t ) ∝ ( t / t 0 ) α in the range $$t/t_{0}\le t_{c}$$ t / t 0 ≤ t c corresponds to the matter-dominated era while $$a(t) \propto \exp (ht/t_{0})$$ a ( t ) ∝ exp ( h t / t 0 ) in the range $$t/t_{0}>t_{c}$$ 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 $$t_{0} \approx 9.8$$ t 0 ≈ 9.8 Gyear. We attain that the presented model leads to precisely correct results so that the crossover time $$t_{0}$$ t 0 and $$\alpha $$ α are completely consistent with the exact solution of the FLRW and re-scaled Hubble parameter $$H_{0}$$ H 0 lies within the observed limits given by Planck, CMB and SNIa data (or other combinations), which lead to consistent cosmological quantities such as the dimensionless Hubble parameter h, deceleration parameter q, jerk parameter j and EoS parameter w. We also discuss time dependent behavior of the dark energy and dark matter to show their roles on the time evolution of the universe. Additionally, we observe that all main results completely depend on the structure of the interaction potential when the parameter values are tuned to satisfy the zero energy condition. Finally, we conclude that interactions in the dark sector may play an important role on the time evolution and provides a mechanism to explain the late time transition of the Universe.

The late time crossover from matter dominated era (represented power-law evolution) to the dark energy dominated era (represented exponential evolution) of the Universe evolution is the major problem in today’s physical cosmology. Unless this critical transition problem is solved, it is not possible to reach a holistic theory of cosmology. To explain this critical transition we propose a new model where the dark matter and dark energy interacting through a potential. Based on the FLRW framework we analytically solve this model and obtain the scale factor a(t). In addition, we numerically compute all cosmological quantities. We find more significant results to enlightening the physical mechanism of the critical transition. Firstly, we show that the scale factor a(t) has a hybrid form as a(t) = a0(t/t0) α e ht/t0 . This is main and important result in the presented work, which clearly indicates that the transition from the power-law to the exponential expansion of the Universe. The numerical results clearly provide that there is a time crossover tc in the scale factor a curve, which indicates the transition from the power-law to the exponential expansion of the Universe. Below t/t0 ≤ tc, matter era dominated hence time evolution of the Universe is given by a(t) ∝ (t/t0) α , on the other hand, above t/t0 > tc, the evolution is represented by a(t) ∝ exp(ht/t0). It is first time, the hybrid result for scale factor is exactly obtained from the presented model without use any approximation. Secondly, we fit the scale factor below and above tc. Surprisingly, we find that the scale factor behaves as a(t) ∝ (t/t0) 2/3 below t/t0 ≤ tc, and as a(t) ∝ exp(ht/t0) which indicates that the Hubble parameter takes the value in the interval of the around H0 = 69.5 and H0 = 73.5 km s−1Mpc−1 depend on the weak and strong interactions between dark components above t/t0 > tc, respectively. These are remarkable that α = 2/3 is completely consistent exact solution of the FLRW and re-scaled Hubble parameter H0 is the observable intervals given by Planck, CMB and SNIa data (or other combinations) for chosen interaction values are purely consistent with cosmological observations. Thirdly, we find from the model the transition point from matter dominated era to the dark energy dominated era in the cosmic time is the t0 = 9.8 Gyear which is consistent with the theoretical solution and observations. Additionally, we numerically obtain and analyse other cosmological quantities such as dimensionless Hubble parameter h, deceleration parameter q, jerk parameter j and EoS parameter w. We show that all cosmological quantities of this model are consistent observational results for the matter and dark energy dominated eras. As a result, we consider late time crossover of the Universe, we propose an interacting dark matter and dark energy model, we show that this model can explain the late time crossover phenomena of the Universe and our solutions are very good consistent with theoretical and observational results. Finally, we state that this work makes essential steps towards solving a critical outstanding problem of the cosmology, and has a potential to creates a paradigm for future studies in this field. Furthermore, the model also sheds light on the interaction mechanism of dark matter and dark energy in the Universe.