Spacial curvature and expanding space If we take the analogy that in an empty space the space is just a flat sheet then if there is a single planet or a star then the flat sheet will curve below the planet leaving a curvature shaped like a hemisphere below the object, my question is, Does this curvature change as space expands?
 A: If you take an isolated spherically symmetric object then the spacetime curvature around it is described by the Schwarzschild metric. The bending of the rubber sheet is meant to be an analogy for this curvature, but bear in mind it's just an analogy and is in many ways a poor representation of what actually happens.
Anyhow, the Schwarzschild metric only applies if:


*

*the planet/star is the only object in the universe

*the universe is time independant i.e. the planet/star has existed forever and will continue to exist forever
Obviously neither of these conditions are true for real stars in the real universe, but in most cases we expect the Schwarzschild metric to be a good approximation.
You specifically ask about the effect of the expansion of the universe. A convenient model for an expanding universe is the de Sitter universe. Like the Schwarzschild metric this is only an approximation to the real universe as the de Sitter universe contains no matter, but again it's not a bad approximation to our universe and actually over the next few billion years it will become a better and better approximation as the expansion becomes dominated by dark energy.
So your question can be rephrased as how is the Schwarzschild metric changed in a de Sitter universe? and actually we have an exact answer to that. The spacetime curvature due to a planet/star in a de Sitter universe is described by the de Sitter–Schwarzschild metric. However as long as you're close to the star planet there's little difference from the Schwarzschild metric.
All this is inevitably a bit technical, and I'd guess it's probably a deeper explanation than you wanted. But there isn't really any way to simplify it except to say that the expansion of the universe does change the curvature around an astronomical body, but not very much.
