We have conducted two-component, non-LTE modeling of the CO lines from J = 1-0 through J = 13-12 in 87 galaxies observed by the Herschel SPIRE Fourier Transform Spectrometer (FTS). We find the average pressure of the cold molecular gas, traced especially by CO J = 1-0, is $sim 10^{5.0 pm 0.5}$ K cm$^{-3}$. The mid- to high-J lines of CO trace higher-pressure gas at $10^{6.5 pm 0.6}$ K cm$^{-3}$; this pressure is slightly correlated with LFIR. Two components are often necessary to accurately fit the Spectral Line Energy Distributions (SLEDs); a one-component fit often underestimates the flux of CO J = 1-0 and the mass. If low-J lines are not included, mass is underestimated by an order of magnitude. Even when modeling the low-J lines alone or using an $alpha_{CO}$ conversion factor, the mass should be considered to be uncertain to a factor of at least 0.4 dex, and the vast majority of the CO luminosity will be missed (median, 65%). We find a very large spread in our derived values of $alpha_{CO}$, though they do not have a discernible trend with LFIR; the best fit is a constant 0.7 M$_{odot}$/ (K kms$^{-1}$ pc$^2$), with a standard deviation of 0.36 dex, and a range of 0.3-1.6 M$_{odot}$/ (K kms$^{-1}$ pc$^2$). We find average molecular gas depletion times ($tau_{dep}$) of $10^8$ yr that decrease with increasing SFR. Finally, we note that the J = 11-10/J = 1-0 line flux ratio is diagnostic of the warm component pressure, and discuss the implications of this comprehensive study of SPIRE FTS extragalactic spectra for future study post-Herschel.
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