Magnetoelastic dynamics of the "spin Jahn-Teller" transition in CoTi$_{2}$O$_{5}$

CoTi$_{2}$O$_{5}$ has the paradox that low temperature static magnetic order is incompatible with the crystal structure owing to a mirror plane that exactly frustrates magnetic interactions. Despite no observable structural distortion with diffraction, CoTi$_{2}$O$_{5}$ does magnetically order below $T_{\rm N}$ $\sim$ 25 K with the breaking of spin ground state degeneracy proposed to be a realization of the spin Jahn-Teller effect in analogy to the celebrated orbital Jahn-Teller transition. We apply neutron and Raman spectroscopy to study the dynamics of this transition in CoTi$_{2}$O$_{5}$. We find anomalous acoustics associated with a symmetry breaking strain that characterizes the spin Jahn-Teller transition. Crucially, the energy of this phonon coincides with the energy scale of the magnetic excitations, and has the same symmetry of an optic mode, observed with Raman spectroscopy, which atypically softens in energy with decreasing temperature. Taken together, we propose that the energetics of the spin Jahn-Teller effect in CoTi$_{2}$O$_{5}$ are related to cooperative magnetoelastic fluctuations as opposed to conventional soft critical dynamics which typically drive large measurable static displacements.