Here’s why it’s possible that doubling CO2 won’t make much difference.
The carbon that’s already up in the atmosphere absorbs most of the light it can. CO2 only “soaks up” its favorite wavelengths of light, and it’s close to saturation point. It manages to grab a bit more light from wavelengths that are close to its favorite bands, but it can’t do much more, because there are not many left-over photons at the right wavelengths.
The natural greenhouse effect is real, and it does keep us warm, but it’s already reached its peak performance.
Throw more carbon up there and most of the extra gas is just “unemployed” molecules.
This graph shows the additional warming effect of each extra 20ppm of atmospheric CO2.