We investigate the timescale with which the IR luminosity decreases after a complete and rapid quenching of star formation using observations of local and high-redshift galaxies. From SED modelling, we derive the time since quenching of a subsample of 14 galaxies from the Herschel Reference Survey suffering from ram-pressure stripping due to the environment of the Virgo cluster and of a subsample of 7 rapidly quenched COSMOS galaxies selected through a state-of-the-art statistical method already tested on the determination of galaxies star formation history. Three out of the 7 COSMOS galaxies have an optical spectra with no emission line, confirming their quenched nature. Present physical properties of the two samples are obtained as well as the past L$_{IR}$ of these galaxies, just before their quenching, from the long-term SFH properties. This past L$_{IR}$ is shown to be consistent with the L$_{IR}$ of reference samples of normally star-forming galaxies with same $M_*$ and $z$ than each of our quenched galaxies. We put constraints on the present to past L$_{IR}$ ratio as a function of quenching time. The two samples probe different dynamical ranges in terms of quenching age with the HRS galaxies exhibiting longer timescales (0.2-3,Gyr) compared to the COSMOS one ($<100$,Myr). Assuming an exponential decrease of the L$_{IR}$ after quenching, the COSMOS quenched galaxies are consistent with short e-folding times less than a couple of hundreds of Myr while the properties of the HRS quenched galaxies are compatible with timescales of several hundreds of Myr. For the HRS sample, this result is consistent with ram pressure stripping due to the environment. For the COSMOS sample, different quenching processes are acting on short to intermediate timescales. Processes such as galaxy mergers, disk instabilities or environmental effects can produce such strong star formation variability.
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