Theoretical Study On Cosmological Evolution And Problem Of Stability For K-essence Dark Energy Model

The modern astronomical observations and theoretical researches show that there are two accelerated expansion periods in the evolution history of the universe,one is the inflation in the early universe,and the other is the accelerated expansion nowadays.Scientists tried a lot of theories to make explanations,such as supposing a new matter which has a negative pressure,called dark energy,or considering the loop quantum effects,or modifying the general relativity of Einstein equations to establish various modified gravity theories.There are many dark energy models,and a kind of them is the dark energy model of scalar field.Then,after quintessence and phantom models,the k-essence dark energy model appears.It comes from k-inflation,which was originally used to explain the inflation era by the domination of kinetic energy,instead of the potential dominated one which suffers by the fine-tuning problem and coincidence problem for the sensitivity of initial values.And the equation of the state changes from 1 to-1 after the big bang during the radiation dominated period and matter dominated period,which helps explaining the cosmological evolution in the early time and the accelerating expansion nowadays.While,for the cosmological dynamical system formed by the dark energy models,the important stability properties are concerned for years.Hence,this dissertation is to deeply research the autonomous system of k-essence dark energy model in three different cosmological frames,and the concrete contents is as follows:First,in the frame work of standard cosmology,it discussed a k-essence dark energy model with kinetic energy in the form of-(?)+X,and the interaction between dark energy and dark matter which effects the critical points and stability,even the evolution of cosmological quantities.Based on the Friedmann equation and equation of motion,the phase variables are set,and the autonomous dynamical system is derived to show the evolution of the universe.By calculation,6 critical points are solved,also the existence and stability of them are analysed.Then the phase space evolutions of the four stable ones are made to analysis the future of the universe with the evolution of the cosmological quantities.Meanwhile,the effects of coupling parameter and potential parameter on the evolution of the phase space are discussed.Besides that,by using the statefinder hierarchy geometrical diagnostic,the comparison between k-essence dark energy model andΛCDM model is analyzed to show the effect of coupling parameter and potential parameter.Next,in recent years,there are many researches on scalar field dark energy in the framework of f(R)modified gravity.In this thesis,a kind of k-essence dark energy model with potential and dark matter is studied in the framework of f(R)modified gravity.The model with constant potential is firstly researched,then the model with non-constant potential model is discussed again.For both models,the phase space and autonomous system are established,unless the potential one causes some new phase variables,which makes the dimension of phase space higher.As a result,eigenvalues of zero appears,and the stability can not satisfied for whole phase space.It has to separate the phase space into stable manifolds,unstable manifolds and centre manifolds for further discussion.Finally,we get a lot of critical points,and analysis the stable points of them.Unlike the former work that only has stable sub-manifold,in our work,it has global stable points for two cases in 5-dim and 6-dim phase space.The reason is for the new kinetic energy form in our k-essence dark energy model makes the equations of the dynamical system simpler.It is worth mentioning that in the case of f(R)=R,the results of our work are in accordance with ones in the standard cosmology,which indicates that our work in f(R)modified gravity theory could be viewed as a generalization of the related results in the standard cosmology.Again,in the framework of loop quantum cosmology(LQC),two kinds of kessence dark energy models with interaction between dark energy and dark matter are discussed for their dynamical stabilities and cosmological evolution.The research shows that,in the late-time of the universe,as quantum effect vanishes,the results in LQC are in accordance with ones in standard cosmology;however,the quantum effects becomes obvious in the early universe,which performs on the evolutions of phase space and cosmological quantities.Different from the standard cosmology,in LQC the stability of some critical points is changed by the interaction.It follows that,the results obtained in LQC could be viewed as a generalization of standard cosmology.Besides,one feature of LQC is solving the singularity problem.In this work,by choosing suitable initial values,cyclic bouncing solutions are get from both k-essence models,i.e.,in future Hubble parameter H,H and energy density p will oscillate periodically,which is also called cyclic universe,and helps diminishing the future singularity and curvature singularity.Finally,DBI model(Dirac-Born-Infeld)as a sub-model of k-essence dark energy model with the interaction between dark energy and dark matter is researched in 5-dim phase space instead of the 3-dim discussion before,as the critical points,attractor and the evolution of the cosmological quantities,which shows the effect by the higher dimensions.As a result,for the critical point in 5-dim phase space,the projection of the evolution trajectories onto 2-dim phase plane and 3-dim phase space are not converged to a point,but a certain area,which shows the difference from the former research in 2-dim and 3-dim phase space.Besides that,the evolution of the cosmological quantities in DBI model are drawn with the observational data of nowadays,which shows the accordance.And the evolution are sensitive to the interaction parameter which affect the data nowadays.