CMB constraints on DHOST theories

We put constraints on the degenerate higher-order scalar-tensor (DHOST) theories using the Planck 2018 likelihoods. In our previous paper, we developed a Boltzmann solver incorporating the effective field theory parameterised by the six time-dependent functions, $\alpha_i$ $(i={\rm B},{\rm K},{\rm T},{\rm M},{\rm H})$ and $\beta_1$, which can describe the DHOST theories. Using the Markov-Chain Monte-Carlo method with our Boltzmann solver, we find the viable parameter region of the model parameters characterising the DHOST theories and the other standard cosmological parameters. First, we consider a simple model with $\alpha_{\rm K} = \Omega_{\rm DE}(t)/\Omega_{\rm DE}(t_0)$, $\alpha_{\rm B}=\alpha_{\rm T}=\alpha_{\rm M}=\alpha_{\rm H}=0$ and $\beta_1=\beta_{1,0}\Omega_{\rm DE}(t)/\Omega_{\rm DE}(t_0)$ in the $\Lambda$CDM background where $t_0$ is the present time and obtain $\beta_{1,0}=0.032_{-0.016}^{+0.013}$ (68\% c.l.). Next, we focus on another theory given by $\mathcal{L}_{\rm DHOST} = X + c_3X\Box\phi/\Lambda^3+ (M_{\rm pl}^2/2+c_4X^2/\Lambda^6)R + 48c_4^2X^2/(M_{\rm pl}^2\Lambda^{12}+2c_4\Lambda^6X^2)\phi^\mu\phi_{\mu\rho}\phi^{\rho\nu}\phi_\nu$ with $X:=\partial_\mu\phi\partial^{\mu}\phi$ and two positive constant parameters, $c_3$ and $c_4$. In this model, we consistently treat the background and the perturbations, and obtain $c_3 = 1.59^{+0.26}_{-0.28}$ and the upper bound on $c_4$, $c_4<0.0088$ (68\% c.l.).