The hybrid scale factor, the transition from the matter dominated era to the dark energy dominated era and time evolution of the Universe
Authors: Aydiner Ekrem , Basaran-Oz Isil , Dereli Tekin , Sarisaman Mustafa 
Published: August 2021

Abstract
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.