Equivalence between formulations in Cosmological Perturbation Theory: The primordial magnetic fields as an example

Nowdays, Cosmological Perturbation Theory is a standard and useful tool in theoretical cosmology. In this work, we compare the 1+3 covariant formalism in perturbation theory (Ellis et al.) to the gauge invariant approach (Bruni et al.), and we show the equivalence of these formalisms to fix the choice of the perturbed variables (gauge choice) in magnetogenesis. We analyze the evolution of primordial magnetic fields through perturbation theory and we discuss the similarities and differences between these two approaches. We get the Maxwell's equations and show a cosmic dynamo like equation written in Poisson gauge, computing the evolution of primordial magnetic fields. Finally, prospects around these formalisms in the study of magnetogenesis are discussed.