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

Brans-Dicke (BD) was one of the first proposed scalar-tensor theories of gravity, and effectively turns 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 these 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 will improve with upcoming large-scale structure and CMB surveys. To constrain BD with weak gravitational lensing, one needs its non-linear matter power spectrum PBD. By comparing the boost B = PBD/PGR from linear theory and non-linear N-body simulations, we show that the non-linear boost can simply be predicted from linear theory if the BD and GR universes are parametrized in a way that makes their early cosmological evolution and quasi-linear power today similar. In particular, they need the same H₀ / √[b]Geff(a=0) and σ₈, where Geff are their (effective) gravitational strengths. Our prediction is $1%$ accurate for ω ≥ 100, z ≤ 3 and k ≤ 1\,h/Mpc, and $2%$ further up to k ≤ 5\,h/Mpc. It also holds for GBD that do not match Newton's constant today, so one can study GR with different gravitational constants GGR by sending ω → ∞. We provide a code that computes B with the linear Einstein-Boltzmann solver hi_class and multiplies it by the non-linear PGR from EuclidEmulator2 to predict PBD.