| Firstly, we brie?y introduce the elementary theory of the modern cosmology,and describe the evolution of the universe. Besides, we present some importantastronomy experiments that bring the enormous e?ects for the modern cosmol-ogy. In the last few years, people found that the expanding universe was speed-ing up rather than slowing down by recent astronomy observations, which defiesthe theory of standard cosmology. Thus, people present dark energy to inter-pret the accelerating universe in standard cosmology. Furthermore, we introducesome popular models interpreting the accelerating universe, such as quintessence,phantom, quintom and braneworld model. Secondly, we apply the double complexfunction method to the cosmology to explain the accelerating universe. On theone hand, by introducing the double complex scalar field, the double Lagrangianwith the potential V (Φ(J),Φ?(J)) = V (|Φ(J) |) is constructed, which not onlycan describe the evolution of quintom universe, but also can naturally give thespintessence and hessence universe. On the other hand, the U(1,J) symmetry ofthe double complex Lagrangian is verified, and the total conserved charge withinthe physical volume is derived by means of the Noe..ther theorem. Moreover, it canbe pointed out that the"imaginary motion"of the angular parameter in Ref.[71],in fact, is only a real phase displacement in our model. At last, we utilize theextra-dimensional theory to interpret the accelerating universe. In this part, basedon the five-dimensional cosmological model with a big bounce, by using the formsof the e?ective equation of state for dark energy, the expression of the interestingfunction f(z) is given in this model. Furthermore, the evolution of some cosmo-logical quantities with the redshift z is presented. In addition, it is shown thatthe evolution of the dimensionless density parameters ?dm,?de, and decelerationparameter q for the first case is the best fit one, which consists with the presentobservations.
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