<<xAct`xPert`
<<xAct`xpand`
<<xAct`xTensor`
<<xAct`xCoba`;

$PrePrint=ScreenDollarIndices;(*needed for nice printouts*)
$CVVerbose=False;

DefManifold[M, 4, {α, β, γ, μ, ν, λ, σ}]

(*$ConformalTime=False;*)
DefMetric[-1, g[-α,-β], CD,{";","D"}, PrintAs->"g"];

SetSlicing[g,n,H,cd,{"|","D"},"FLFlat"];

DefConformalMetric[g,a];

DefMetricPerturbation[g,h,ε]

(*VisualizeTensor[h[LI[1],μ,ν]/.SplitMetric[g,dg,H,"SynchronousGauge"],H]*)

$FirstOrderVectorPerturbations=True;
$FirstOrderTensorPerturbations=True;

Lag = Conformal[g,ga2][RicciCD[-α,-β]]

MyResult = ExpandPerturbation@Perturbed[Lag, 1]

ExtractComponents[ExtractOrder[MyResult,1],H,{"Time","Time"}]