"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&&H0>0&&a0>(Sqrt[1-1/ΩΛ])]]

FullSimplify[Limit[((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))])),a0->(Sqrt[1-1/ΩΛ])],ΩΛ>1&&H0>0&&t>=0]

a[t_]:=(Sqrt[((-1+ΩΛ)/(ΩΛ))]) Cosh[H0 t (Sqrt[ΩΛ])]
H[t_]:=a'[t]/a[t]

"Skalenfaktor"
"a[t]"->FullSimplify[a[t]]
"a'[t]"->FullSimplify[a'[t]]
"a''[t]"->FullSimplify[a''[t]]
"a'''[t]"->FullSimplify[a'''[t]]
"Hubbleparameter"
"H[t]"->FullSimplify[H[t]]
"H'[t]"->FullSimplify[H'[t]]
"H''[t]"->FullSimplify[H''[t]]
"H'''[t]"->FullSimplify[H'''[t]]

"Beispiel"
ΩΛ=11/10;H0=1;T=2;
Plot[a[t],{t,0,T}]
Plot[a'[t],{t,0,T}]
Plot[a''[t],{t,0,T}]
Plot[a'''[t],{t,0,T}]
Plot[H[t],{t,0,T}]
Plot[H'[t],{t,0,T}]
Plot[H''[t],{t,0,T}]
Plot[H'''[t],{t,0,T}]