A simple prediction of the nonlinear matter power spectrum in Brans-Dicke gravity from linear theory

Brans-Dicke (BD), one of the first proposed scalar-tensor theories of gravity, effectively makes the gravitational constant of general relativity (GR) time-dependent. Constraints on the BD parameter $\omega$ serve as a benchmark for testing GR, which is recovered in the limit $\omega \rightarrow \infty$. Current small-scale astrophysical constraints $\omega \gtrsim 10^5$ are much tighter than large-scale cosmological constraints $\omega \gtrsim 10^3$, but the two decouple if the true theory of gravity features screening. On the largest cosmological scales, BD approximates the most general second-order scalar-tensor (Horndeski) theory, so constraints here have wider implications. These constraints will improve with upcoming large-scale structure and cosmic microwave background surveys. To constrain BD with weak gravitational lensing, one needs its nonlinear matter power spectrum $P_\mathrm{BD}$. By comparing the boost $B = P_\mathrm{BD}/P_\mathrm{GR}$ from linear theory and nonlinear $N$-body simulations, we show that the nonlinear boost can simply be predicted from linear theory if the BD and GR universes are parameterized in a way that makes their early cosmological evolution and quasilinear power today similar. In particular, they need the same $H_0 / \sqrt{\smash[b]{G_{\rm eff}(a=0)}}$ and $\sigma_8$, where $G_{\rm eff}$ is the (effective) gravitational strength. Our prediction is $1\%$ accurate for $\omega \geq 100$, $z \leq 3$, and $k \leq 1\,h/\mathrm{Mpc}$; and $2\%$ up to $k \leq 5\,h/\mathrm{Mpc}$. It also holds for $G_\mathrm{BD}$ that do not match Newton's constant today, so one can study GR with different gravitational constants $G_\mathrm{GR}$ by sending $\omega \rightarrow \infty$. We provide a code that computes $B$ with the linear Einstein-Boltzmann solver hi_class and multiplies it by the nonlinear $P_\mathrm{GR}$ from EuclidEmulator2 to predict $P_\mathrm{BD}$.