  Prospects for discovery of the standard model Higgs boson are examined at
center of mass energies of $7$ and $10$ TeV at the CERN Large Hadron Collider.
We perform a simulation of the signal and principal backgrounds for Higgs boson
production and decay in the $W^+ W^-$ dilepton mode, finding good agreement
with the ATLAS and CMS collaboration estimates of signal significance at 14 TeV
for Higgs boson masses near $m_H = 160$~GeV. At the lower energy of $7$~TeV,
using the same analysis cuts as these collaborations, we compute expected
signal sensitivities of about $2$ standard deviations ($\sigma$'s) at $m_H =
160$~GeV in the ATLAS case, and about 3.6~$\sigma$ in the CMS case for
$1$~fb$^{-1}$ of integrated luminosity. Integrated luminosities of
8~$\rm{fb}^{-1}$ and 3~$\rm{fb}^{-1}$ are needed in the ATLAS case at $7$ and
$10$~TeV, respectively, for $5~\sigma$ level discovery. In the CMS case, the
numbers are 2~$\rm{fb}^{-1}$ and 1~$\rm{fb}^{-1}$ at $7$ and $10$~TeV. Our
different stated expectations for the two experiments arise from the more
restrictive analysis cuts in the CMS case. Recast as exclusion limits, our
results show that with $1~{\rm fb}^{-1}$ of integrated luminosity at 7~TeV, the
LHC may be able to exclude $m_H$ values in the range 160 to 180~GeV provided no
signal is seen.
