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.
