Résumé : This thesis aims to study the freeze-in mechanism of dark matter production in an early universe. Specifically, we focus on the production of a Feebly Interacting Massive Particles (FIMP) through the decay of a mother particle that is in thermal and chemical equilibrium with the primordial bath. We assume that this freeze-in occurs during a matter-dominated epoch, just after inflation, when the radiation bath has not yet formed. In this case, the universe could be dominated by a scalar field, called the inflaton, which drives the inflation. When the age of the universe becomes comparable to its lifetime, it decays, forming the radiation bath. For a period after inflation, the universe is dominated by the inflaton rolling at the bottom of its potential, that could potentially be described with an equation of state typical of matter. In this work, we are interested in the consequences of this decay on the dark matter momentum distribution function. In particular, we use this distribution to calculate the mean dispersion velocity of a FIMP. Finally, we derive a constraint on the FIMP mass using observations of the Lyman-α forest. As an example, we illustrate this new constraint for a leptophilic dark matter model. We show that this new constraint enlarges the phase space to a new region initially excluded when we considered a dark matter freeze-in during a radiation-dominated universe.