Friedmann acceleration equation

For energy density $\rho$, pressure $P$, and cosmological constant $\Lambda$,

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

For positive spatial curvature, zero cosmological constant, and $\rho ,P\ge 0$, continuity makes $\rho a^3$ nonincreasing during expansion. The negative curvature term then prevents unbounded growth of $a$, while the acceleration equation forces $\dot{a}$ to reach zero in finite time.

With zero cosmological constant, the condition $\rho +3P\ge 0$ implies $\ddot{a}\le 0$ andwherever $H\ne 0$.

If $H>0$ at one time, the inequality $d(H^{-1})/dt\ge 1$ forces $H\to +\infty$ and $a\to 0$ within finite time to the past. If $H<0$, it forces $H\to -\infty$ and $a\to 0$ within finite time to the future.

In a flat radiation-plus-$\Lambda$ universe,The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at