Slow electron-phonon relaxation controls the dynamics of the superconducting resistive transition

We investigate the temporal and spatial scales of resistance fluctuations ($R$-fluctuations) at the superconducting resistive transition accessed through voltage fluctuations measurements in thin epitaxial TiN films. This material is characterized by a slow electron-phonon relaxation, which puts it far beyond the applicability range of the textbook scenario of superconducting fluctuations. The measured Lorentzian spectrum of the $R$-fluctuations identifies their correlation time, which is nearly constant across the transition region and has no relation to the conventional Ginzburg-Landau time scale. Instead, the correlation time coincides with the energy relaxation time determined by a combination of the electron-phonon relaxation and the relaxation via diffusion into reservoirs. Our data is quantitatively consistent with the model of spontaneous temperature fluctuations and highlight the lack of understanding of the resistive transition in materials with slow electron-phonon relaxation.