5.1.5. The = 1 solution

As emphasized in Chapter 4, this model is predicted by inflation and hence remains the preferred one. In chapter 1, we have established these relations for this model

which we can now substitute into equation 31 to yield

The solution to this partial differential equation is a power law of the form

Substituting this in yields

which has two solutions, = 2/3 and = -1. These two solutions correspond to the growing mode (2/3) and the decaying mode (-1). As the decaying mode has much faster time evolution it quickly becomes unimportant leading to the result that the growing mode amplifies at the rate of

In the static medium we derived an exponential growth rate but in the expanding medium the growth rate is very much slower. In fact, such a slow growth rate caused early cosmologists (Lifshitz 1946, Bonnor 1957) to conclude that gravitational instability could not have produced the structure that we observe today because the growth rate is to slow relative to the age of the Universe. Indeed, this remains a problem for the very largest scales features (e.g., the Great Wall) that we see in the galaxy distribution.