How can we distinguish between the metric expansion of space and the speed of light slowing over time?
I understand that the universe is expanding. But it seems that our main measurement techniques involve assuming the speed of light is constant over time. I know that we define length in terms of a constant speed of light and our unit of time, the second. Does this mean the two are indistinguishable - that is, the expansion and the slowing of the speed of light are the same thing?
Type 1a supernovae explode with a fixed luminosity, then fade over a fixed time.
If the speed of light varies over cosmic time, the light will be redshifted for more distant supernovae, but the duration of the fading will on average be the same. Cosmic expansion, on the other hand, will increase the wavelength of the light but also the time between photons to be emitted and in effect also make them appear to fade more slowly.
So, if supernovae at higher redshift on average fade more slowly, it's Cosmic expansion. If not, it's a variable speed of light. Turns out they do indeed fade more slowly when seen at higher redshift, decisively ruling in favor of Cosmic expansion.
If the speed of light were variable, it would affect plenomena at all scales from microscopic to cosmological. However the expansion of space has no effect on matter bound by forces, such as on anything within a galaxy. The expansion of space is observed (based on the FLRW metric) only on the intergalactic cosmological scale, so it cannot be explained by a variable speed of light.
Cosmological expansion has observable effects, such as redshifts.
A change in a dimensionful constant is unobservable. See Is it possible to speak about changes in a physical constant which is not dimensionless?
Therefore there is no link between these two ideas.
I would just like to point out that Type 1a supernovae can not be used to differentiate between cosmic expansion and variable speed of light.
Since the the speed of light is a dimensionful constant, local changes are not observable. Locally a supernova will not have a redshift nor fade more slowly. However when observed from a long distance away (where the speed of light is slower), it will appear that it is both redshifted and fade more slowly.
One can see this by timing the rate of fade based on the number of wavelengths of light, which will be the same measured both locally and far away.
