From my understanding, we use the CMB data to measure the spatial curvature of the universe today. Why is it the value for today if the CMB data reflects the universe at recombination (380K years after the big bang). Shouldn't the CMB data give the spatial curvature of the universe at 380K years? Part 2, what are the measured values of spatial curvature at recombination and today?
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We do not use CMB data to directly measure the curvature of the universe. Instead, we use CMB data to estimate the average density (as well as the size of density fluctuations) in the early universe. Plugging this value into a cosmological model (the Lambda-CDM model is a common choice) and running time forward to the present day allows us to place an upper bound on the current curvature of the universe.
This Wikipedia article contains a summary of current knowledge about the local and global geometry of the universe.
