  We used a combination of polarized Raman spectroscopy and spin wave
calculations to study magnetic excitations in the strong spin-orbit-coupled
(SOC) bilayer perovskite antiferromagnet $Sr_3Ir_2O_7$. We observed two broad
Raman features at ~ 800 $cm^{-1}$ and ~ 1400 $cm^{-1}$ arising from magnetic
excitations. Unconventionally, the ~ 800 $cm^{-1}$ feature is fully symmetric
($A_{1g}$) with respect to the underlying tetragonal ($D_{4h}$) crystal lattice
which, together with its broad line shape, definitively rules out the
possibility of a single magnon excitation as its origin. In contrast, the ~
1400 $cm^{-1}$ feature shows up in both the $A_{1g}$ and $B_{2g}$ channels.
From spin wave and two-magnon scattering cross-section calculations of a
tetragonal bilayer antiferromagnet, we identified the ~ 800 $cm^{-1}$ (~ 1400
$cm^{-1}$) feature as two-magnon excitations with pairs of magnons from the
zone-center $\Gamma$ point (zone-boundary van Hove singularity X point). We
further found that this zone-center two-magnon scattering is unique to bilayer
perovskite magnets which host an optical branch in addition to the acoustic
branch, as compared to their single layer counterparts. This zone-center
two-magnon mode is distinct in symmetry from the time-reversal symmetry broken
spin wave gap and phase mode proposed to explain the ~ 92 meV (742 $cm^{-1}$)
gap in RIXS magnetic excitation spectra of $Sr_3Ir_2O_7$.
