Slicing COSMOS with SC4K:the evolution of typical Lya emitters and the Lya escape fraction from z~2 to z~6

We present and explore deep narrow- and medium-band data obtained with the Subaru and the Isaac Newton telescopes in the ~2 deg$^2$ COSMOS field. We use these data as an extremely wide, low-resolution (R~20-80) IFU survey to slice through the COSMOS field and obtain a large sample of ~4000 Lyman-$\alpha$ (Lya) emitters from z~2 to z~6 in 16 redshift slices (SC4K). We present new Lya luminosity functions (LFs) covering a co-moving volume of ~10$^8$Mpc$^3$. SC4K extensively complements ultra-deep surveys, jointly covering over 4 dex in Lya luminosity and revealing a global (2.5z 3.5, likely linked with the evolution of the AGN population. The Lya luminosity density rises by a factor ~2 from z~2 to z~3 but is then found to be roughly constant (~$1.1\times10^{40}$ erg s$^{-1}$ Mpc$^{-3}$) to z~6, despite the ~0.7 dex drop in UV luminosity density. The Lya/UV luminosity density ratio rises from $4\pm1$% to $30\pm6$% from z~2.2 to z~6. Our results imply a rise of a factor of ~2 in the global ionisation efficiency ($\xi_{\rm ion}$) and a factor ~$4\pm1$ in the Lya escape fraction from z~2 to z~6, hinting for evolution in both the typical burstiness/stellar populations and even more so in the typical ISM conditions allowing Ly$\alpha$ photons to escape