Constraining Quasar Relativistic Reflection Regions and Spins with
  Microlensing
  We present an analysis of Chandra spectra of five gravitationally lensed
active galactic nuclei. We confirm the previous detections of FeK$\alpha$
emission lines in most images of these objects with high significance. The line
energies range from 5.8 to 6.8 keV with widths from unresolved to 0.6 keV,
consistent with emission close to spinning black holes viewed at different
inclination angles. We also confirm the positive offset from the
Iwasawa-Taniguchi effect, the inverse correlation between the FeK$\alpha$
equivalent width and the X-ray luminosity in AGN, where our measured equivalent
widths are larger in lensed quasars. We attribute this effect to microlensing,
and perform a microlensing likelihood analysis to constrain the emission size
of the relativistic reflection region and the spin of supermassive black holes,
assuming that the X-ray corona and the reflection region, responsible for the
iron emission line, both follow power-law emissivity profiles. The microlensing
analysis yields strong constraints on the spin and emissivity index of the
reflection component for Q2237+0305, with $a > 0.92$ and $n > 5.4$. For the
remaining four targets, we jointly constrain the two parameters, yielding
$a=0.8\pm0.16$ and an emissivity index of $n=4.0\pm 0.8$, suggesting that the
relativistic X-ray reflection region is ultra-compact and very close to the
innermost stable circular orbits of black holes, which are spinning at close to
the maximal value. We successfully constrain the half light radius of the
emission region to $< 2.4$ $r_g$ ($r_g = GM/c^2$) for Q2237+0305 and in the
range 5.9--7.4 $r_g$ for the joint sample.
