We present FIR-CO luminosity relations ($log L_{rm FIR} = alpha log L_{rm CO} + beta$) for the full CO rotational ladder from J=1-0 to J=13-12 for 62 local (z < 0.1) (Ultra) Luminous Infrared Galaxies (LIRGs) using data from Herschel SPIRE-FTS and ground-based telescopes. We extend our sample to high redshifts (z > 1) by including 35 (sub)-millimeter selected dusty star forming galaxies from the literature with robust CO observations. The addition of luminous starbursts at high redshifts enlarge the range of the FIR-CO luminosity relations towards the high-IR-luminosity end while also significantly increasing the small amount of mid-/high-J CO line data available prior to Herschel. This new data-set (both in terms of IR luminosity and J-ladder) reveals linear FIR-CO luminosity relations ($alpha sim 1$) for J=1-0 up to J=5-4, with a nearly constant normalisation ($beta sim 2$). This is expected from the (also) linear FIR-(molecular line) relations found for the dense gas tracer lines (HCN and CS), as long as the dense gas mass fraction does not vary strongly within our (merger/starburst)-dominated sample. However from J=6-5 and up to J=13-12 we find an increasingly sub-linear slope and higher normalization constant with increasing J. We argue that these are caused by a warm (~100K) and dense ($>10^4{rm cm^{-3}}$) gas component whose thermal state is unlikely to be maintained by star formation powered far-UV radiation fields (and thus is no longer directly tied to the star formation rate). We suggest that mechanical heating (e.g., supernova driven turbulence and shocks), and not cosmic rays, is the more likely source of energy for this component. The global CO spectral line energy distributions (SLEDs), which remain highly excited from J=6-5 up to J=13-12, are found to be a generic feature of the (U)LIRGs in our sample, and further support the presence of this gas component.
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