An upper limit to differential magnification effects in strongly gravitationally lensed galaxies

Differential magnification is now well-known to distort the spectral energy distributions of strongly gravitationally lensed galaxies. However, that does not mean that any distortions are possible. Here I prove an analytic upper bound to differential magnification effects. For example, a thermal or sub-thermal CO ladder cannot be made to appear super-thermal just from gravitational lensing, and the Balmer decrement emission line ratio H$\alpha$:H$\beta$ cannot reduce below the case B prediction just from differential magnification. In general, if a physical model of a galaxy predicts upper and/or lower bounds to an emission line ratio, then those bounds also apply to the differentially magnified strongly gravitationally lensed case. This applies not just for velocity-integrated emission lines, but also for the line emission in any rest-frame velocity interval.