Perturbations on Large and Small Scales

Nonetheless, we can get a feel for the evolution of perturbations by considering the smallest scales, where Newtonian gravity is sufficient. We start in physical (non-expanding) coordinates and simply write down the equations of motion  

Although these are the equations that we would right down from first principles in a Newtonian analysis, they are also the small-scale and small-velocity limit of General Relativity. To account for the expansion of the Universe, we change to comoving coordinates, r and peculiar velocity, u, defined from physical coordinates, x, and velocities, v as 

Although we can t write down a general solution to this equation, we can examine it in various special cases and limits. 

In a flat Universe, dominated by pressureless matter like Cold Dark Matter, we have a, oc2/3 and, by definition, p = cs = 0. The solution to the second-order equation is = A t2/3 + The first term, proportional to f2// 3 oc 

In a Universe dominated instead by curvature or an accelerating component such as a cosmological constant, the expansion of the universe is more rapid than in a matter dominated universe. In both of these cases, there is no longer a growing mode neither solution to Equation 10.10 grows with time. 

---