Cosmological models

These applets allow you to investigate the solutions of the Friedmann equation for an homogeneous universe
 

(dR/dt)2=Wo Ho2/R + l R2 Ho2-(Wo+ l-1) Ho2

(1)

where R is the universe scale factor at a time t  and its current value ,R_o, is equal to 1; W_o is the ratio of the density of the universe to the critical density (for a universe with a zero cosmological constant, the critical density divides the case of a closed universe which will eventually stop expanding and recollapse from an open universe which will instead expand for ever), H_o is the Hubble constant and l is the reduced cosmological constant defined as l=L/ 3 H_o². You can explore the evolution of R for different values of the Hubble constant, of  the density of the universe and of the cosmological constant.
 

Explore the evolution of the Universe in the following cases:

cosmological constant=0; W_o<1 and W_o>1; for W_o<1 derive, analytically, the maximum value of R  and compare it with the value obtained by the applet.
cosmological constant<0; is there any value of W_o such that the universe is open?
cosmological constant>0; the evolution depends on the sign of (W_o+l-1). For (W_o+l-1)>0, there is a critical value of l which separates expanding models from oscillating/bouncing models. Derive the critical value analytically and verify the evolution of R for  all the cases.

In the first applet the cosmological constant is set equal to 0; in the second applet you can choose the value of the cosmological constant; in the third applet the cosmological constant is set equal to zero,  H_o is set equal to
65 km/(s Mpc) but you can set the maximum time for the integration (this feature allows you to explore in detail the solution at small times or the asymptotic evolution of R at large times). 

Go to the applet 2

Go to the applet 3