Cosmological bulk-to-boundary propagator

For a free mode function $f(k,\eta )$ and a correlator evaluated at ${\eta }_0$, the two cosmological bulk-to-boundary propagators are $K_R=f(k,\eta )f^{\ast }(k,{\eta }_0)$ and $K_L=f^{\ast }(k,\eta )f(k,{\eta }_0)$, up to the chosen contour-label convention.