A Method for Obtaining Cosmological Models Consistency Relations and Gaussian Processes Testing

In the present work, we apply consistency relation tests to several cosmological models, including the flat and non-flat $\Lambda$CDM models, as well as the flat XCDM model. The analysis uses a non-parametric Gaussian Processes method to reconstruct various cosmological quantities of interest, such as the Hubble parameter $H(z)$ and its derivatives from $H(z)$ data, as well as the comoving distance and its derivatives from SNe Ia data. We construct consistency relations from these quantities which should be valid only in the context of each model and test them with the current data. We were able to find a general method of constructing such consistency relations in the context of $H(z)$ reconstruction. In the case of comoving distance reconstruction, there were not a general method of constructing such relations and this work had to write an specific consistency relation for each model. From $H(z)$ data, we have analyzed consistency relations for all the three above mentioned models, while for SNe Ia data we have analyzed consistency relations only for flat and non-flat $\Lambda$CDM models. Concerning the flat $\Lambda$CDM model, some inconsistency was found, at more than $2\sigma$ c.l., with the $H(z)$ data in the interval $1.8\lesssim z\lesssim2.4$, while the other models were all consistent at this c.l. Concerning the SNe Ia data, the flat $\Lambda$CDM model was consistent in the $0<z<2.5$ interval, at $1\sigma$ c.l., while the nonflat $\Lambda$CDM model was consistent in the same interval, at 2$\sigma$ c.l.