Are $^{12}$CO lines good indicators of the star formation rate in
  galaxies?
  In this paper, we investigate the relevance of using the $^{12}$CO line
emissions as indicators of star formation rates (SFR). For the first time, we
present this study for a relatively large number of $^{12}$CO transitions (12)
as well as over a large interval in redshift (from z$\sim$0 to z$\sim$6). For
the nearby sources (D$\leq$10 Mpc), we have used homogeneous sample of
$^{12}$CO data provided by Bayet et al. (2004, 2006), mixing observational and
modelled line intensities. For higher-z sources (z $\geq$ 1), we have collected
$^{12}$CO observations from various papers and have completed the data set of
line intensities with model predictions which we also present in this paper.
Finally, for increasing the statistics, we have included recent $^{12}$CO(1-0)
and $^{12}$CO(3-2) observations of intermediate-z sources. Linear regressions
have been calculated for identifying the tightest SFR-$^{12}$CO line luminosity
relationships. We show that the \emph{total} $^{12}$CO, the $^{12}$CO(5-4), the
$^{12}$CO(6-5) and the $^{12}$CO(7-6) luminosities are the best indicators of
SFR (as measured by the far-infrared luminosity). Comparisons with theoretical
approaches from Krumholz and Thompson (2007) and Narayanan et al. (2008) are
also performed in this paper. Although in general agreement, the predictions
made by these authors and the observational results we present here show small
and interesting discrepancies. In particular, the slope of the linear
regressions, for J$_{upper}\geq$ 4 $^{12}$CO lines are not similar between
theoretical studies and observations. On one hand, a larger high-J $^{12}$CO
data set of observations might help to better agree with models, increasing the
statistics. On the other hand, theoretical studies extended to high redshift
sources might also reduce such discrepancies.
