"Geschlossenes Universum mit ΩM>1"
sol=Quiet[FullSimplify[DSolve[{
a'[t]/a[t]==H0 Sqrt[ΩM/a[t]^3+(1-ΩM)/a[t]^2],
a[0]==a0}, a[t], t], ΩM>1&&t>0]]

"Funktion"
f[x_]:=(ΩM Log[-Sqrt[1-ΩM] Sqrt[x]+Sqrt[ΩM+x-ΩM x]]
Sqrt[ΩM+x-ΩM x]+Sqrt[1-ΩM] Sqrt[x] (ΩM+x-ΩM x))/((1-ΩM)^(3/2)
x^(3/2) Sqrt[(ΩM+x-ΩM x)/x^3])

"Argument"
x[t_]:=(-Sqrt[a0 (1-ΩM)] (-ΩM+a0 (-1+ΩM) (1+H0 t
Sqrt[(a0+ΩM-a0 ΩM)/a0^3]))+ΩM Sqrt[a0+ΩM-a0 ΩM]
Log[-Sqrt[a0 (1-ΩM)]+Sqrt[a0+ΩM-a0 ΩM]])/(a0^(3/2)
(1-ΩM)^(3/2) Sqrt[(a0+ΩM-a0 ΩM)/a0^3])

"Skalenfaktor"
a[t_]:=InverseFunction[f[#1]&][x[t]]

"Beispiel"
ΩM=11/10; H0=1; a0=1/1000;
tMax=FindMaximum[a[t], {t, 1}, WorkingPrecision->32][[2, 1, 2]];
T:=If[t<tMax, t, 2tMax-t]
h[t_]:=a'[t]/a[t]; H[t_]:=If[t<tMax, h[t], -h[2tMax-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]