The Diversity of Diffuse Ly$\alpha$ Nebulae around Star-Forming Galaxies
  at High Redshift
  We report the detection of diffuse Ly$\alpha$ emission, or Ly$\alpha$ halos
(LAHs), around star-forming galaxies at $z\approx3.78$ and $2.66$ in the NOAO
Deep Wide-Field Survey Bo\"otes field. Our samples consist of a total of
$\sim$1400 galaxies, within two separate regions containing spectroscopically
confirmed galaxy overdensities. They provide a unique opportunity to
investigate how the LAH characteristics vary with host galaxy large-scale
environment and physical properties. We stack Ly$\alpha$ images of different
samples defined by these properties and measure their median LAH sizes by
decomposing the stacked Ly$\alpha$ radial profile into a compact galaxy-like
and an extended halo-like component. We find that the exponential scale-length
of LAHs depends on UV continuum and Ly$\alpha$ luminosities, but not on
Ly$\alpha$ equivalent widths or galaxy overdensity parameters. The full
samples, which are dominated by low UV-continuum luminosity Ly$\alpha$ emitters
($M_{\rm UV} \gtrsim -21$), exhibit LAH sizes of 5$\,-\,6\,$kpc. However, the
most UV- or Ly$\alpha$-luminous galaxies have more extended halos with
scale-lengths of 7$\,-\,9\,$kpc. The stacked Ly$\alpha$ radial profiles decline
more steeply than recent theoretical predictions that include the contributions
from gravitational cooling of infalling gas and from low-level star formation
in satellites. On the other hand, the LAH extent matches what one would expect
for photons produced in the galaxy and then resonantly scattered by gas in an
outflowing envelope. The observed trends of LAH sizes with host galaxy
properties suggest that the physical conditions of the circumgalactic medium
(covering fraction, HI column density, and outflow velocity) change with halo
mass and/or star-formation rates.
