What is the price of abandoning dark matter? Cosmological constraints on alternative gravity theories

Any successful alternative gravity theory that obviates the need for dark matter must fit our cosmological observations. Measurements of microwave background polarization trace the large-scale baryon velocity field at recombination and show very strong, $O(1)$, baryon acoustic oscillations. Measurements of the large-scale structure of galaxies at low redshift show much weaker features in the spectrum. If the alternative gravity theory's dynamical equations for the growth rate of structure are linear, then the density field growth can be described by a Green's function: $\delta(\vec x,t) = \delta(\vec x,t')G(x,t,t')$. We show that the Green function, $G(x,t,t')$, must have dramatic features that erase the initial baryon oscillations. This implies an acceleration law that changes sign on the $\sim 150$ Mpc scale. On the other hand, if the alternative gravity theory has a large nonlinear term that couples modes on different scales, then the theory would predict large-scale non-Gaussian features in large-scale structure. These are not seen in the distribution of galaxies nor in the distribution of quasars. No proposed alternative gravity theory for dark matter seems to satisfy these constraints.