<<Units`
Ωm0=.3089;ΩΛ0=.6911;Ωr0=4 10^-5;
H0=Convert[67.74 Kilo Meter/Second/(Mega Parsec),1/(Giga Year)];
solution=NDSolve[{a'[t]== H0 *(Giga Year)Sqrt[Ωm0/a[t]+Ωr0/a[t]^2+ΩΛ0 a[t]^2],a[10^-10]==2.95 10^-7},a[t],{t,0,100}]

(a[t]/.solution[[1]])/.t->0

Show[Plot[a[t]/.solution[[1]],{t,0,15},AxesLabel->{"t in Gigayears","((a (t))/(Subscript[a, 0]))"}],Graphics[{Dashed,Line[{{0,1},{13.8,1}}]}],Graphics[{Dashed,Line[{{13.8,0},{13.8,1}}]}]]

NSolve[(a[t]/.solution[[1]])==1,t]

Show[Plot[{a[t]/.solution[[1]],(Sqrt[t])/32.6},{t,0,10^-6},(* 10^-6 Giga Years = 1000 years *)AxesLabel->{"t in Gigayears","((a (t))/(Subscript[a, 0]))"}],Graphics[{Dashed,Line[{{0,1},{13.8,1}}]}],Graphics[{Dashed,Line[{{13.8,0},{13.8,1}}]}]]

Show[Plot[{a[t]/.solution[[1]],t^(2/3)/6.58},{t,0,5},AxesLabel->{"t in Gigayears","((a (t))/(Subscript[a, 0]))"}],Graphics[{Dashed,Line[{{0,1},{13.8,1}}]}],Graphics[{Dashed,Line[{{13.8,0},{13.8,1}}]}]]

Show[Plot[{a[t]/.solution[[1]],.495Exp[.057 t]},{t,0,50},AxesLabel->{"t in Gigayears","((a (t))/(Subscript[a, 0]))"}],Graphics[{Dashed,Line[{{0,1},{13.8,1}}]}],Graphics[{Dashed,Line[{{13.8,0},{13.8,1}}]}]]

H[t_]=D[a[t]/.solution[[1]],t]/a[t]/.solution[[1]];

Show[Plot[H[t]/H[13.8],{t,10,50},AxesLabel->{"t in Gigayears","((H (t))/(Subscript[H, 0])) = (((OverDot[a]) (t))/(a (t) (Subscript[H, 0])))"},PlotRange->{0,1.1}],Graphics[{Dashed,Line[{{10,.831},{50,.831}}]}],Graphics[{Dashed,Line[{{10,1},{13.8,1}}]}],Graphics[{Dashed,Line[{{13.8,1},{13.8,0}}]}]]

H[100]/H[13.8]