We use the EAGLE galaxy formation simulation to study the effects of baryons on the power spectrum of the total matter and dark matter distributions and on the velocity fields of dark matter and galaxies. On scales $k{stackrel{>}{{}_sim}} 4{h,{rm Mpc}^{-1}}$ the effect of baryons on the amplitude of the total matter power spectrum is greater than $1%$. The back-reaction of baryons affects the density field of the dark matter at the level of $sim3%$ on scales of $1leq k/({h,{rm Mpc}^{-1}})leq 5$. The dark matter velocity divergence power spectrum at $k{stackrel{<}{{}_sim}}0.5{h,{rm Mpc}^{-1}}$ is changed by less than $1%$. The 2D redshift-space power spectrum is affected at the level of $sim6%$ at $|vec{k}|{stackrel{>}{{}_sim}} 1{h,{rm Mpc}^{-1}}$ (for $mu>0.5$), but for $|vec{k}|leq 0.4{h,{rm Mpc}^{-1}}$ it differs by less than $1%$. We report vanishingly small baryonic velocity bias for haloes: the peculiar velocities of haloes with $M_{200}>3times10^{11}{{rm M}_{odot}}$ (hosting galaxies with $M_{*}>10^9{{rm M}_{odot}}$) are affected at the level of at most $1~$km/s, which is negligible for $1%$-precision cosmology. We caution that since EAGLE overestimates cluster gas fractions it may also underestimate the impact of baryons, particularly for the total matter power spectrum. Nevertheless, our findings suggest that for theoretical modelling of redshift space distortions and galaxy velocity-based statistics, baryons and their back-reaction can be safely ignored at the current level of observational accuracy. However, we confirm that the modelling of the total matter power spectrum in weak lensing studies needs to include realistic galaxy formation physics in order to achieve the accuracy required in the precision cosmology era.
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