 Solution

Pressureless matter obeys the cosmological perfect-fluid continuity equation, ${\dot{\rho }}_m+3H{\rho }_m=0$. Integrating gives ${\rho }_m={\rho }_{m,0}(a_0/a{)}^3$. Define the present critical density and cosmological density parameters byHere $c=1$ and the curvature contribution is an effective density parameter, not an additional matter fluid. Substitution into the Friedmann equation givesUsing the Hubble parameter $H=\dot{a}/a$ therefore yieldsAt $a=a_0$ the same equation implies ${\Omega }_{m,0}+{\Omega }_{k,0}+{\Omega }_{\Lambda ,0}=1$. The power in the matter term is $a_0^3$; the converted TeX's $a_0^2$ is an error not present in the original PDF.