Testing consistency of general relativity with kinematic and dynamical probes

In this work, we test consistency relations between a kinematic probe, the observational Hubble data, and a dynamical probe, the growth rates for cosmic large scale structure, which should hold if general relativity is the correct theory of gravity on cosmological scales. Moreover, we summarize the development history of parametrization in testings and make an improvement of it. Taking advantage of the Hubble parameter given from both parametric and non-parametric methods, we propose three equations and test two of them performed by means of two-dimensional parameterizations, including one using trigonometric functions we propose. As a result, it is found that the consistency relations satisfies well at $1\sigma$ CL and trigonometric functions turn out to be efficient tools in parameterizations. Furthermore, in order to confirm the validity of our test, we introduce a model of modified gravity, DGP model and compare the testing results in the cases of $\Lambda$CDM, "DGP in GR" and DGP model with mock data. It can be seen that it is the establishing of consistency relations which dominates the results of the testing. Overall, the present observational Hubble data and growth rate data favor convincingly that the general relativity is the correct theory of gravity on cosmological scales.