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The $x$ Euler-Lagrange equation has the first integralwhere the constant follows from the initial data. Combining this with proper-time normalization gives the Timelike geodesic in the expanding flat slicing of de Sitter spacetime:ConsequentlyIntegrating from $t=0$, where $\tau =0$, givesAs $t\to -\infty$, the first ratio tends to $1$, soThus $t$ tends to $-\infty$ at the finite proper time ${\tau }_{-}$.