Polarization fraction of Planck Galactic cold clumps and forecasts for the Simons Observatory

We measure the polarization fraction of a sample of $6282$ Galactic cold clumps at $353 \, \mathrm{GHz} $, consisting of $Planck$ Galactic cold clump (PGCC) catalogue category 1 objects (flux densities measured with signal-to-noise ratio $(\mathrm{S/N}) > 4$). We find the mean-squared polarization fraction at $353 \, \mathrm{GHz} $ to be $ \langle \Pi ^ 2 \rangle = [ 4.79 \pm 0.44 ] \times 10 ^ {-4} $ equating to an $ 11 \, \sigma $ detection of polarization. We test if the polarization fraction depends on the clumps' physical properties, including flux density, luminosity, Galactic latitude and physical distance. We see a trend towards increasing polarization fraction with increasing Galactic latitude, but find no evidence that polarization depends on the other tested properties. The Simons Observatory, with an angular resolution of order $1 \, \mathrm{arcmin } $ and noise levels between $22$ and $54$ $ \mu \mathrm{ K-arcmin } $ at high frequencies, will substantially enhance our ability to determine the magnetic field structure in Galactic cold clumps. At $\ge5\,\sigma$ significance, we predict the Simons Observatory will detect at least $\sim12,000$ cold clumps in intensity and $\sim430$ cold clumps in polarization. This number of polarization detections would represent a two orders of magnitude increase over the current $Planck$ results. We also release software that can be used to mask these Galactic cold clumps in other analyses.