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Write $H=\dot{a}/a$. The nonzero Christoffel symbols for the flat FLRW metric areThe mixed Einstein tensor isFor a mixed tensor,For $\mu =t$, direct substitution givesFor each spatial $\mu$, homogeneity makes the partial derivatives vanish and the remaining diagonal connection terms cancel pairwise. This verifies the flat FLRW Einstein-tensor divergence directly.

In vacuum,The $tt$ equation isChoosing the positive root gives the expanding de Sitter scale factor in flat slicingand the spatial equation is then automatically satisfied.