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, $\alpha$, the electron-proton mass ratio, $\mu$, and the Newtonian gravitational constant, $G$. Large-scale spatial fluctuations in the fine structure constant are bounded by $\delta \alpha /\alpha \lesssim 2\times {10}^{-9}$ and $\delta \alpha /\alpha \lesssim 1.2\times {10}^{-8}$ in the Bekenstein-Sandvik-Barrow-Magueijo and varying-speed-of-light theories, respectively, fluctuations in the electron-proton mass ratio by $\delta \mu /\mu \lesssim 9\times {10}^{-5}$ in the Barrow-Magueijo theory and fluctuations in $G$ by $\delta G/G\lesssim 3.6\times {10}^{-10}$ in the Brans-Dicke theory. These derived bounds are significantly stronger than any obtainable by direct observations of astrophysical objects at the present time.