Geometrical Diagnostics For The Modified Theory Of Gravity With The Different Formalisms

In the first section of this thesis,we briefly introduce the accelerating universe and the normal interpretations for this acceleration.In the second section of this thesis,several popular models of dark energy are described,and the state-finder geometrical diagnostic for these models are exhibited.Chapter 3 and chapter 4 are the main work for me.We apply the geometrical diagnostic methods to distinguish the modified theory of gravity(including the f(R)and the GBD)with the different formalisms.It is well known that,several observational and theoretical motivations require us to investigate the modified theories of GR.In modified gravitational theory,there exist the metric formalism and the Palatini formalism.In these two formalisms,the dynamical variables are considered to be different.Geometrical-diagnostic methods were often applied to distinguish the gravitational models.But it is scarce to investigate the differences for the different formalisms(e.g.the metric formalism and the Palatini formalism)in the modified theories of gravity.In this thesis,we discriminate the different formalisms of the modified gravity by using the diagnostic methods.For considered modified theories of gravity(including the f(R)theory and the GBD theory),we can see that the difference between the two formalisms are notable according to the geometrical diagnostics.And relative to the ?CDM model,there are the more deviation in metric formalism than those in Palatini formalism,according to the {r,s} diagnostic.Given that the GBD is a time-variable Newton gravitational constant theory,the differences between the variable-G theory and the constant-G theory are studied by using the diagnostic methods.According to the observational limits on G,we plot some pictures on the geometrical quantities.For the influence of variable G,the values of r(or s)in metricformalism GBD are smaller than those in f(R)theory,which indicates that the variation of Newton gravitational constant can induce remarkable effects on geometrical quantities for the existence of BD scalar field.In the Palatini formalism,the shapes of r-s curves between the GBD theory and the f(R)theory could be obviously.