Cosmological bounds on spatial variations of physical constants

We derive strong bounds on any possible large-scale spatial variation in the values of physical constants whose space-time evolution is driven by a scalar field. These limits are imposed by the isotropy of the microwave background on large angular scales in theories which describe space and time variations in the fine structure constant, , the electron-proton mass ratio, , and the Newtonian gravitational constant, G. Large-scale spatial fluctuations in the fine structure constant are bounded by /210^-9 and /1. 210^-8 in the Bekenstein-Sandvik-Barrow-Magueijo and varying-speed-of-light theories, respectively, fluctuations in the electron-proton mass ratio by /910^-5 in the Barrow-Magueijo theory and fluctuations in G by /G3. 610^-10 in the Brans-Dicke theory. These derived bounds are significantly stronger than any obtainable by direct observations of astrophysical objects at the present time.