Cosmography of chameleon gravity

The paper discusses a particular model of chameleon gravity where the scalar field has an exponential as $V=M^{4}\exp(\frac{\alpha\phi}{M_{pl}})$ and an exponential coupling to the matter density component of the Universe as $\rho_{m}\exp(\frac{\beta\phi}{M_{pl}})$. We finds the relations between the four (in principle free) parameters of the model and the first five cosmographic parameters (which can be written in terms of the Hubble parameter and its first four derivatives). If such cosmographic parameter are known then the we present the algebraic relations which can be used to derive the free parameters in terms of the cosmographic ones. We show that, determining the third derivative of scale factor $Q$ is of great importance if the chameleon parameter $\beta$ and coupling constant $\alpha$ are known. On the other hand, if nothing can be proposed about $(\alpha,\beta)$, reconstructing the free parameters of the model can be done by measuring the first five derivatives of the scale factor and testing the validity of theory is only possible by its first six derivatives, so that the parameters $(\alpha,\beta)$ will also be determined automatically. We also find relation between cosmographic parameters and parameters of the model when it dynamically approaches its critical points.