If the output of a detector changes exponentially with time, the time required for it to change from its initial value by the fraction 1- exp(-t/τc) (for t = τ c) of the final value, is called the time constant.
Source:
PAC, 1995, 67, 1745. 'Nomenclature, symbols, units and their usage in spectrochemical analysis-XI. Detection of radiation (IUPAC Recommendations 1995)' on page 1751 (https://doi.org/10.1351/pac199567101745)