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Flat FLRW Einstein-tensor divergence

For

d s^{2} = - d t^{2} + a (t)^{2} d x^{2}

, the nonzero connection coefficients are

Γ_{ij}^{t} = a \overset{a}{˙} δ_{ij}, Γ_{t j}^{i} = Γ_{j t}^{i} = H δ_{j}^{i} .

With

G_{t}^{t} = - 3 H^{2}

and

G_{i}^{j} = - (2 \overset{a}{¨} / a + H^{2}) δ_{i}^{j}

, direct substitution into the mixed-tensor covariant derivative gives zero.

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