"Geschlossenes Universum mit ΩΛ>1"
sol=Quiet[FullSimplify[DSolve[{
a'[t]/a[t]==H0 Sqrt[ΩΛ+(1-ΩΛ)/a[t]^2],
a[0]==a0}, a[t], t], ΩΛ>1]]

"Skalenfaktor"
a[t_]:=Tanh[H0 t Sqrt[ΩΛ]+ArcTanh[(a0 ΩΛ)/Sqrt[ΩΛ (1+
(-1+a0^2) ΩΛ)]]]/Sqrt[-((ΩΛ Sech[H0 t Sqrt[ΩΛ]+
ArcTanh[(a0 ΩΛ)/Sqrt[ΩΛ (1+(-1+a0^2) ΩΛ)]]]^2)/(-1+ΩΛ))]


"Beispiel"
ΩΛ=11/10; H0=1; a0=1; tMax=2; H[t_]:=a'[t]/a[t];
Plot[a[t], {t, 0, 2tMax}, Frame->True, GridLines->{{tMax}, {}}, AspectRatio->1/2]
Plot[H[t], {t, 0, 2tMax}, Frame->True, GridLines->{{tMax}, {}}, AspectRatio->1/2]

"Minimal a"
H→0 @ a→Sqrt[1-1/ΩΛ];