<p>The cooling of the stratosphere in response to increasing carbon dioxide concentration is a fingerprint of human effects on climate. However, the mechanisms that control the magnitude and vertical structure of this cooling have not been clear. Here we use idealized models of spectroscopy and radiative transfer to explain the sensitivity of stratospheric temperature to carbon dioxide concentration. We find that stratospheric cooling is mainly driven by the distribution of mass absorption coefficients in the primary carbon dioxide band and modulated by the longwave cooling of water vapour and ozone in other parts of the spectrum. These spectral mechanisms explain why the stratosphere cools more aloft than it does below, why each doubling of carbon dioxide yields roughly 0 to 8 degrees Kelvin of cooling across the depth of the stratosphere and why stratospheric cooling increases the top of atmosphere radiative forcing of carbon dioxide by about 50%. This theory implies that stratospheric cooling is not a fundamental consequence of increasing the optical thickness of a greenhouse gas but rather the unique result of the spectroscopy of that gas.</p>