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We present the generalized Friedmann equations describing the cosmological evolution of a finite thick brane immeresed in a five-dimensional Schwarzschild Anti-de Sitter spacetime. A linear term in the density in addition to a quatratic one arises in the Friedmann equation, leading to the standard cosmological evolution at late times without introducing an ad hoc tension term for the brane. The effective four-dimensional cosmological constant is then uplifted similar to the KKLT effect and vanishes for a brane thickness equal to the AdS curvature size, up to the third order of the thickness. The four-dimensional gravitational constant is then equal to the five-dimensional one divided by the AdS curvature radius, similar to that derived by dimensional compactification. An accelerating brane cosmology may emerge at late times provided there is either a negative transverse pressure component in the brane energy-momentum tensor or the effective brane cosmological constant is positive.
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