Providing stringent star formation rate limits of z$\sim$2 QSO host galaxies at high angular resolution

We present integral field spectrograph (IFS) with laser guide star adaptive optics (LGS-AO) observations of z=2 quasi-stellar objects (QSOs) designed to resolve extended nebular line emission from the host galaxy. Our data was obtained with W. M. Keck and Gemini-North Observatories using OSIRIS and NIFS coupled with the LGS-AO systems. We have conducted a pilot survey of five QSOs, three observed with NIFS+AO and two observed with OSIRIS+AO at an average redshift of z=2.15. We demonstrate that the combination of AO and IFS provides the necessary spatial and spectral resolutions required to separate QSO emission from its host. We present our technique for generating a PSF from the broad-line region of the QSO and performing PSF subtraction of the QSO emission to detect the host galaxy. We detect H$\alpha$ and [NII] for two sources, SDSS J1029+6510 and SDSS J0925+06 that have both star formation and extended narrow-line emission. Assuming that the majority of narrow-line H$\alpha$ is from star formation, we infer a star formation rate for SDSS J1029+6510 of 78.4 M$_\odot$yr$^{-1}$ originating from a compact region that is kinematically offset by 290 - 350 km/s. For SDSS J0925+06 we infer a star formation rate of 29 M$_\odot$yr$^{-1}$ distributed over three clumps that are spatially offset by $\sim$ 7 kpc. The null detections on three of the QSOs are used to infer surface brightness limits and we find that at 1.4 kpc distance from the QSO that the un-reddened star formation limit is $<$ 0.3 M$_\odot$yr$^{-1}$kpc$^{-2}$. If we assume a typical extinction values for z=2 type-1 QSOs, the dereddened star formation rate for our null detections would be $<$ 0.6 M$_\odot$yr$^{-1}$kpc$^{-2}$. These IFS observations indicate that if star formation is present in the host it would have to occur diffusely with significant extinction and not in compact, clumpy regions.