It works for photons. A spiral has a "radius" that is changing at a constant rate, which would follow from a constant speed of light. (The speed of the photon is still c and would be pointed along the direction tangent to the spiral. It will work, though the calculation of the radius would have to be done in SR. The red shift won't be a factor as the speed of a red photon is the same as the speed of a violet photon. (If we have to we can use GR instead of SR.)
I can't really speculate on a general problem where dark energy or dark mattter exist. All I can say is that it would look similar to a spiral but the radius wouldn't be likely to have a constant rate of change.