Conjoined constraints on modified gravity from the expansion history and cosmic growth

In this paper we present conjoined constraints on several cosmological models from the expansion history $H(z)$ and cosmic growth $f\sigma_8(z)$. The models we study include the CPL $w_0w_a$ parametrization, the Holographic Dark Energy (HDE) model, the Time varying vacuum ($\Lambda_t$CDM) model, the Dvali, Gabadadze and Porrati (DGP) and Finsler-Randers (FRDE) models, a power law $f(T)$ model and finally the Hu-Sawicki $f(R)$ model. In all cases we perform a simultaneous fit to the SnIa, CMB, BAO, $H(z)$ and growth data, while also following the conjoined visualization of $H(z)$ and $f\sigma_8(z)$ as in Linder (2017). Furthermore, we introduce the Figure of Merit (FoM) in the $H(z)-f\sigma_8(z)$ parameter space as a way to constrain models that jointly fit both probes well. We use both the latest $H(z)$ and $f\sigma_8(z)$ data, but also LSST-like mocks with $1\%$ measurements and we find that the conjoined method of constraining the expansion history and cosmic growth simultaneously is able not only to place stringent constraints on these parameters but also to provide an easy visual way to discriminate cosmological models. Finally, we confirm the existence of a tension between the growth rate and Planck CMB data and we find that the FoM in the conjoined parameter space of $H(z)-f\sigma_8(z)$ can be used to discriminate between the $\Lambda$CDM model and certain classes of modified gravity models, namely the DGP and $f(T)$.