A novel survey for young substellar objects with the W-band filter VI: Spectroscopic census of sub-stellar members and the IMF of $\sigma$ Orionis cluster

Low-mass stars and sub-stellar objects are essential in tracing the initial mass function (IMF). We study the nearby young $\sigma$ Orionis cluster (d$\sim$408 pc; age$\sim$1.8 Myr) using deep NIR photometric data in J, W and H-bands from WIRCam on the Canada-France-Hawaii Telescope. We use the water absorption feature to photometrically select the brown dwarfs and confirm their nature spectroscopically with the IRTF-SpeX. Additionally we select candidate low-mass stars for spectroscopy and analyze their membership and that of literature sources using astrometry from Gaia DR3. We obtain the near-IR spectra for 28 very low-mass stars and brown dwarfs and estimate their spectral type between M3-M8.5 (mass ranging between 0.3-0.01 M$_{\odot}$). Apart from these, we also identify 5 new planetary mass candidates which require further spectroscopic confirmation of youth. We compile the comprehensive catalog of 170 spectroscopically confirmed members in the central region of the cluster, for a wide mass range of $\sim$19-0.004 M$_{\odot}$. We estimate the star/BD ratio to be $\sim$4, within the range reported for other nearby star forming regions. With the updated catalog of members we trace the IMF down to 4 M$_\mathrm{Jup}$ and we find that a two-segment power-law fits the sub-stellar IMF better than the log-normal distribution.