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Do expanding universe affect the curvature of spacetime? If so, does Einstein's field equation account for the change in curvature (gravitational field) around massive object in the expanding universe? What would be the mathematical equation for that?

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Yes, the expanding universe affects the curvature of spacetime.

Yes, the Einstein field equations can account for the gravitational field of a massive object in an expanding universe.

The field equations are

$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}.$$

They have analytic solutions for black holes in some simple expanding spacetimes, such as the de Sitter-Schwarzschild metric.

For our own universe, which is more complicated than de Sitter spacetime, you could solve the equations numerically. It would not be particularly illuminating, as the curvature due to the rest of the universe is small compared to the curvature of the black hole.

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