The expansion of the universe can be modeled by using the Friedmann equation. There are many forms of the Friedmann equation. In this problem, we are going to use the Fridmann equation for a universe that is matter and curvature dominated. For a universe that contains matter and curvature, the Friedmann equation for the such universe is given by, dR dt R-1+k Here, R(t) is the radius of the universe at a given time t. 2, depends on the density of the matter in the universe and 2 depends on how strong the curvature of the universe is. We take the initial condition of the differential equation as R(0) = 0, which is obvious, as at the beginning of time the radius of the universe is zero. (a) Solve the differential equation for m = 1 and 2 = 0 and plot the solution. = (b) Solve the differential equation for m =4 and 9-3 and plot the solution. (c) From the two plots we can see two different ultimate fate of the universe. For one case The universe expands continuously and the spacetime itself would get ripped apart, this is called the 'Big Rip'. In another case, the universe collapses inside itself due to gravitational force between the matters and becomes a blackhole at the end of time and this is called the 'Big Crunch'. Can you tell from the plots, Which plot suggest which scenario?