ODIN: A New Lyman Alpha Blob Selection Method, Sample, and Statistical Analysis at $z\sim3.1$

Ly$\alpha$ blobs (LABs) are large, spatially extended Ly$\alpha$-emitting objects whose nature remains unclear. Their statistical properties such as number densities and luminosity functions are still uncertain because of small sample sizes and large cosmic variance. The One-hundred-deg$^2$ DECam Imaging in Narrowbands (ODIN) survey, with its large volume, offers an opportunity to overcome these limitations. We describe our LAB selection method and present 112 new LABs in the 9 deg$^2$ E-COSMOS field. We begin with the conventional LAB selection approach, cross-matching LAEs with extended Ly$\alpha$ sources, yielding 89 LAB candidates. To obtain a more complete LAB sample, we introduce a new selection pipeline that models all galaxies detected in deep broadband imaging, subtracts them from the narrowband image, and then directly detects extended Ly$\alpha$ emission. This method successfully identifies 23 additional low-surface-brightness LABs which could otherwise be missed by the conventional method. The number density of ODIN LABs near an ODIN protocluster ($n=7.5\times10^{-5}$ cMpc$^{-3}$) is comparable to that found in the SSA22 proto-cluster and is four times higher than the average across the field. The cumulative Ly$\alpha$ luminosity function within the protocluster regions is similar to that measured for the LABs in the SSA22 proto-cluster, suggesting a large excess of luminous LABs relative to the average field. These findings suggest the Ly$\alpha$ luminosities and number densities of LABs are environment-dependent. ODIN will provide an expansive LAB and protocluster samples across six additional fields and two more redshifts, allowing us to investigate the nature of LABs in relation to their environments.