https://insu.hal.science/insu-03645648Vitenti, S. D. P.S. D. P.VitentiIAP - Institut d'Astrophysique de Paris - INSU - CNRS - Institut national des sciences de l'Univers - SU - Sorbonne Université - CNRS - Centre National de la Recherche ScientifiqueFalciano, F. T.F. T.FalcianoPinto-Neto, N.N.Pinto-NetoCovariant Bardeen perturbation formalismHAL CCSD201498.80.-k04.20.Cv04.25.-g98.80.JkCosmologyFundamental problems and general formalismApproximation methodsequations of motionMathematical and relativistic aspects of cosmologyAstrophysics - Cosmology and Nongalactic AstrophysicsGeneral Relativity and Quantum Cosmology[SDU] Sciences of the Universe [physics]Sorbonne Université, Gestionnaire HAL 42022-04-25 15:12:012023-05-01 03:58:032022-04-25 15:12:02enJournal articleshttps://insu.hal.science/insu-03645648/document10.1103/PhysRevD.89.103538application/pdf1In a previous work we obtained a set of necessary conditions for the linear approximation in cosmology. Here we discuss the relations of this approach with the so-called covariant perturbations. It is often argued in the literature that one of the main advantages of the covariant approach to describe cosmological perturbations is that the Bardeen formalism is coordinate dependent. In this paper we will reformulate the Bardeen approach in a completely covariant manner. For that, we introduce the notion of pure and mixed tensors, which yields an adequate language to treat both perturbative approaches in a common framework. We then stress that in the referred covariant approach, one necessarily introduces an additional hypersurface choice to the problem. Using our mixed and pure tensors approach, we are able to construct a one-to-one map relating the usual gauge dependence of the Bardeen formalism with the hypersurface dependence inherent to the covariant approach. Finally, through the use of this map, we define full nonlinear tensors that at first order correspond to the three known gauge invariant variables Φ, Ψ and Ξ, which are simultaneously foliation and gauge invariant. We then stress that the use of the proposed mixed tensors allows one to construct simultaneously gauge and hypersurface invariant variables at any order.