 Solution

Without radiation, the Friedmann acceleration equation changes sign when ${\rho }_m=2{\rho }_{\Lambda }$. The cosmological redshift satisfies $1+z=a_0/a$, and conservation of pressureless matter gives ${\rho }_m(z)={\rho }_{m,0}(1+z{)}^3$. HenceThe expanding spatially flat model is decelerating above this redshift and accelerating below it. Positive ${\rho }_{\Lambda }$ is required for such a transition; a negative value gives no accelerated expanding epoch. If the displayed redshift is negative, the transition lies in our future.