Testing gravity theories with cosmic microwave background in the degenerate higher-order scalar-tensor theory

We study the cosmic microwave background (CMB) in the framework of the degenerate higher-order scalar-tensor (DHOST) theory to test gravity theories. This theoretical framework includes the wide class of dark energy models such as the Horndeski theory and its extensions as certain limits, and the general relativity can be also recovered. In this study, to test gravity theories with CMB, we formulate the linear perturbations of gravity and matter in the theory and their effective description parameterised by time-dependent effective field theory (EFT) parameters, $\alpha_i$ $(i=B,K,T,M,H,L)$ and $\beta_i$ $(i=1,2,3)$. Based on the resultant DHOST framework, we develop a numerical code to solve Boltzmann equations consistently. We then show that the angular power spectra of the CMB temperature anisotropies, E-mode and lensing potential as a demonstration and find that the parameter characterising the DHOST theory, $\beta_1$, provides the larger modifications of the spectra, compared with other EFT parameters. We also show the results in the case of a specific model in which the cosmic expansion as well as the EFT parameters are consistently determined.