Near-extremal hydrodynamics and the holographic product formula

The holographic product formula is used to determine the general form taken by holographic spectral functions in the near-extremal hydrodynamic regime, with energy $\omega$, momentum $k$ and temperature $T$ much smaller than a hard scale $\mu$. The resulting expressions simplify in the extremal limit $T \ll \omega,k\ll \mu$, for which the low-temperature gapless modes and the IR conformal behavior factorize. In some cases, this factorization extends to the general near-extremal regime $\omega,k,T\ll\mu$ at leading order in $T/\mu$. Several examples are discussed with different types of gapless modes and IR CFTs, including new numerical results for low temperature quasi-normal modes. We end with a concrete application that shows how the inclusion of the IR conformal behavior improves the description of the spectral function at low energies.