I do not have the solution referenced in the title, but I seriously ask if one could possibly exist. Given the observed redshift data, can there be a solution to any mathematical model which DOES NOT result in an expanding universe? Credit to user Chronos sig line: If a theory appears to be the only one possible, assume you don't understand the problem. - Popper Two possible alternative theories to consider in reconciling the data with a non-expanding universe: Redshift due to changing propagation of light over the duration of the cosmos. The spectral redshift of distant cosmic light sources is presumed to be due to rarefied light carriers (photons); analagous to rarefied sound carriers (air molecules). In lieu of the usual receding train analogy, consider the Doppler effect as it manifests in various materials. http://arstechnica.com/science/2011/03/inverse-doppler-effect/ Hypothesize that the composition of the early universal was such that the average speed of light through "empty" space was slower than it is through the vacuous "empty" space of the current universe. Assume a smooth slow transition from then to now. Note: "empty" in this context does not necessarily imply vacuum. It is just a label for the transparent column between a distant object and an earthbased observer. Given that hypothesis and a non-expanding universe; it is reasonable to conclude that the light from extremely distant light sources would take longer to arrive at earth than would be expected if the average speed of light through "empty" space were constant. An expected earth based observation based on this hypothesis when looking at extremely distant light sources; would be a shift of characteristic spectral lines in the direction of the red end of the visible spectrum; due to the rarefied arrival of photons traveling at a lower average speed over the entire course of their journey. Redshift due to simultaneous contraction of all matter in the universe. Hypothesize that all matter everywhere is slowly shrinking at uniform rates. This means that subatomic particles are shrinking but the relative distance between them in an atom remains proportionally consistent. The fundamental forces continue to operate in a consistent manner. Gravity and atomic forces still pull stuff together. All of our tools for measuring would be shrinking too, thus almost everything locally would appear unchanged. One way the shrinking may be observable would be to look deep into the (past)cosmos at the spectra of light generated from the fusion of atoms. In the distant past, the spectra of a given element would result from fusion of larger atoms than the spectra of the same element in the present. Given that everything else is proportional, it might be assumed that the spectra generated from larger identically configured atoms would result in 'slower' but still characteristic spectra. QED redshift.