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Planned cosmic microwave background (CMB) experiments can dramaticallyimprove what we know about neutrino physics, inflation, and dark energy. Thelow level of noise, together with improved angular resolution, will increasethe signal to noise of the CMB polarized signal as well as the reconstructedlensing potential of high redshift large scale structure. Projected constraintson cosmological parameters are extremely tight, but these can be improved evenfurther with information from external experiments. Here, we examinequantitatively the extent to which external priors can lead to improvement inprojected constraints from a CMB-Stage IV (S4) experiment on neutrino and darkenergy properties. We find that CMB S4 constraints on neutrino mass could bestrongly enhanced by external constraints on the cold dark matter density$\Omega_{c}h^{2}$ and the Hubble constant $H_{0}$. If polarization on thelargest scales ($\ell<50$) will not be measured, an external prior on theprimordial amplitude $A_{s}$ or the optical depth $\tau$ will also beimportant. A CMB constraint on the number of relativistic degrees of freedom,$N_{\rm eff}$, will benefit from an external prior on the spectral index$n_{s}$ and the baryon energy density $\Omega_{b}h^{2}$. Finally, an externalprior on $H_{0}$ will help constrain the dark energy equation of state ($w$).

External priors for the next generation of CMB experiments