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

Abstract

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 $\ddot{I}‰$ serve as a benchmark for testing GR, which is recovered in the limit $\ddot{I}‰\to \infty$. Current small-scale astrophysical constraints $\ddot{I}‰\gtrsim 10^5$ are much tighter than large-scale cosmological constraints $\ddot{I}‰\gtrsim 10^3$, 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 $H_0/\sqrt{G_{\mathrm{eff}}(a=0)}$ and $\ddot{I}{ƒ}_8$, where $G_{\mathrm{eff}}$ is the (effective) gravitational strength. Our prediction is $1\%$ accurate for $\ddot{I}‰\ge 100$, $z\le 3$, and $k\le 1h/\mathrm{Mpc}$; and $2\%$ up to $k\le 5h/\mathrm{Mpc}$. 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 $\ddot{I}‰\to \infty$. 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 $P_{\mathrm{BD}}$.

12 pages, 8 figures. Accepted for publication in A&A. A code that computes the non-linear boost can be found at https://github.com/hersle/jbd