Explore nonlinear effects in the growth of perturbations in the gravitational age— when radiation and the cosmological constant can be ignored—by considering the evolution of a sphere in which the matter density is uniform and exceeds the external density by a small quantity.

(a) Use the Friedmann equations (28.16), (28.18), and (28.19) to show that the sphere behaves like a universe with density greater than the critical density and stops expanding when its density exceeds the external value by a factor of 9π²/16.

(b) Assume that the perturbation remains strictly spherical, and determine by what additional scale factor the external universe will have expanded when the perturbation collapses to a point.

(c) Argue that realistic perturbations behave quite differently, that nonspherical perturbations grow during the collapse, and that the in fall kinetic energy effectively randomizes during the collapse. Show that the collapse stops when the radius of the sphere is roughly half its maximum value and that this occurs when the average density exceeds that in the still-expanding external universe by a factor of ∼150.

Equations.

Transcribed Image Text:

ಠ8π 3 -pa². - k R²2 (28.16)