Near-extremal hydrodynamics and the holographic product formula

A bstract The holographic product formula is used to determine the general form taken by holographic spectral functions in the near-extremal hydrodynamic regime, with energy ω , momentum k and temperature T much smaller than a hard scale μ . The resulting expressions simplify in the extremal limit T ≪ ω , k ≪ μ , 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 ω , k , T ≪ μ at leading order in T / μ . 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.