2024A&A...689A..39S

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 ω serve as a benchmark for testing GR, which is recovered in the limit ω → ∞. Current small-scale astrophysical constraints ω ≳ 10⁵ are much tighter than large-scale cosmological constraints ω ≳ 10³, 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_BD. By comparing the boost B = P_BD/P_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 σ₈, where G_eff is the (effective) gravitational strength. Our prediction is 1% accurate for ω ≥ 100, z ≤ 3, and k ≤ 1 h/Mpc; and 2% up to k ≤ 5 h/Mpc. It also holds for G_BD that do not match Newton's constant today, so one can study GR with different gravitational constants G_GR by sending ω → ∞. We provide a code that computes B with the linear Einstein-Boltzmann solver HI_CLASS and multiplies it by the nonlinear P_GR from EUCLIDEMULATOR2 to predict P_BD.