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If we live in $4+n$ dimension and compactify the extra $n$ dimensions so that they are of typical size $R$, then on scales $r\gg R$ gravity is known to act in the normal manner. That is, the gravitational potential is $V(r)\sim \frac{1}{r}$ (in regimes where the gravitational potential is a reasonable quantity to consider, that is).

In the DGP model there is one extra dimension which is very large in extent and the phenomena opposite to the compactification case occurs. That is, on small scales gravity acts as though it would in four dimensions, $V(r)\sim \frac{1}{r}$, but on large scales, $r\gg R$ for some scale $R$, gravity acts acts five dimensionally, $V(r)\sim \frac{1}{r^2}$.

What's the intuition one should have about these two cases? Why is gravity modified only at small scales in the first case and only long distances in the other?

Is it crucial that in the compactification case the action consists of a single Einstein-Hilbert term whereas in the DGP case there is both a bulk and brane Einstein-Hilbert term? That is $$S_{\rm compactification}=\int d^{4+n}x\, \sqrt{-g}R[g]$$ and $$S_{\rm DGP}=\int d^5 X\, \sqrt{-G}R[G]+\int d^4 x\, \sqrt{-g}R[g].$$

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