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Dimensional analysis suggests that $\Lambda R^2 \sim O(1)$, where $\Lambda$ is the cosmological constant and $R$ is the radius of the universe. $\Lambda$ is measured to be around $10^{-52}$ m$^{-2}$, which implies a radius of curvature of $R \sim 10^{26} m \sim 10$ billion ly. Of course this scale is of the same order scale as the observable universe, which suggests that we should be able to detect appreciable deviations from flatness at the scales that we can observe cosmologically. Yet to my knowledge no experimental probe has found any deviations from global flatness, even at fairly enormous scales. How does one reconcile these? Is this a problem that can be waved away as a consequence of insufficient experimental sensitivity?

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Dimensional analysis suggests that $\Lambda R^2 \sim O(1)$, where $\Lambda$ is the cosmological constant and $R$ is the radius [of curvature] of the universe.

The flaw in this inference is that spatial curvature is not linked to the cosmological constant. In a universe dominated by a cosmological constant, the comoving synchronous spatial surfaces are flat, i.e. there is no curvature scale. To the extent that these spatial surfaces might be curved, that curvature is set by the stuff that isn't the cosmological constant.

A universe with a cosmological constant has a cosmological event horizon, though, and dimensional analysis correctly predicts that its scale is $\sim \Lambda^{-1/2}$.

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