15 XII. GAUGE INVARIANT AND COVARIANT SPHERICAL PERTURBATIONS ON A FLRW BACKGROUND. these authors, a perturbation scheme on FLRW cosmolo- gies is covariant if the “lumpy” model S is described by variables defined in the framework of the 1+3 fluid flow variables associated with the system (18). Although our description of LTB spacetimes is not based on these scalars (the representation (17)) but on the quasi–local representation (35), it is still a covariant description be- cause m$_{q}$ and H $_{q}$, are covariant scalars by virtue of their connection with the invariants M, R and their deriva- tives in (28). Hence, the formalism based on m$_{q}$ and H q would also be covariant. As commented by Ellis and Bruni [32], by virtue of the Stewart–Walker gauge invariance lemma [35], all covari- ant objects in S that would vanish in the background ¯ S (a FLRW cosmology in this case) are gauge invariant (GI), to all orders, and also in the usual sense (as in [37]). The tensorial quantities in LTB models that vanish for a dust FLRW cosmology in the 1+3 formalism are the shear and electric Weyl tensors σ ab and E $^{ab}$, given by (15) and (16) in terms of the scalar functions Σ and E , which from (30), can be written in terms of the fluctuations m − m$_{q}$ and H − H $_{q}$. From (31), it is evident that these fluctuations and δ ( A $^{)}$, as well as the radial gradients m ′ q and H ′ q are all “first order” quantities (in the perturbation scheme) that are GI to all orders, though m$_{q}$, H q are not (which is expected because these are “zero order” background variables). As a consequence, the fluid flow dynamics of LTB models in the quasi–local scalar representation (35) also provides a rigorous characterization of LTB models as spherical, non–linear GIC perturbations on a FLRW background. We note that spatial curvature k = $^{3}$R / 6 and its quasi– local dual k$_{q}$ = $^{3}$R $_{q}$/ 6 do not appear in the dynamic equations (102a)–(102d), though they can be used when specifying initial conditions as in (108) (see also the ap- pendices of [31]). Spatial curvature is GI only if the FLRW cosmology ¯ S is spatially flat, but its associated variables δ ( k ) and k ′ q are GI. In fact, from (33), (37) and (38) we can always eliminate either one of m$_{q}$, H q or δ ( m $^{)}$, δ( H ) in terms of k$_{q}$ and δ ( k $^{)}$, and construct a sys- tem of evolution equations equivalent to (102a)–(102d), but describing the dynamics in terms of spatial curvature k$_{q}$ and its perturbation δ ( k $^{)}$. Given the correspondence between (102) and the dy- namics of non–linear perturbations on a FLRW back- ground, it is important to examine the connection with linear perturbation theory of dust sources. For this pur- pose we derive a second order equation for the density perturbation δ ( m ) by differentiating both sides of (102a) and use the remaining equations (102b)–(102d) to elimi- nate all derivatives except ¨ δ ( µ ) and ˙ δ ( m $^{)}$. We obtain which is an exact non–linear equation for δ ( m ) (a sim- ilar equation was obtained in [41]). For near homoge- ¨ δ ( m ) − [ ˙ δ ( m $^{)}$]2 1 + δ ( m ) + 2 H q ˙ δ ( m ) − κ 2 ρ$_{q}$ δ ( m ) ( 1 + δ ( m ) ) = 0 , (112) associated with (110), LTB models characterized by the metric (1) and source (2) can be considered then as spher- ical non–linear “perturbed” dust FLRW cosmologies in the “comoving” gauge. The perturbation scheme that we described fits nat- urally to the gauge invariant and covariant (GIC) ap- proach of Dunsbury, Ellis and Bruni [32, 33]. Following δ ( A ) = A − Φ( ¯ A ) Φ( ¯ A ) (111) defines a “background model” (associated to a FLRW cosmology) in LTB models through the quasi–local scalars A$_{q}$ (which are LTB objects satisfying FLRW dy- namics). Given their common normal geodesic represen- tation and considering the perturbations The map Φ can be defined rigorously as follows. Let ¯ X and X be, respectively, the sets of smooth integrable scalar functions in ¯ S and S , then for all covariant FLRW scalars ¯ A ∈ ¯ X (we denote FLRW objects with an over– bar) the map A perturbation formalism between an idealized space- time, ¯ S (a dust FLRW cosmology), and an “lumpy” uni- verse model, S (a dust LTB model), follows by defining a “background model” in S , constructed by objects (in S ) that are the images of a suitable map Φ between ob- jects in ¯ S to objects in S [32]. Though, perturbations in general are not uniquely defined by such an abstract map Φ, because of the ambiguity of the gauge freedom associated with choices of coordinates and hypersurfaces. This fact requires carefully defining perturbed variables that are “gauge–invariant” [37]. However, in the specific case that concerns us we face a simpler task, as both the idealized FLRW and the perturbed LTB spacetimes are LRS (locally rotationally symmetric) and can be com- pletely described by scalar functions [36]. Hence, Φ can be a map of scalars to scalars, while the fact that both spacetimes are spherically symmetric and are given in the same normal geodesic coordinate (or frame) representa- tion, greatly suppresses most of the gauge freedom that we would find for a general S .[48] The connection between the system (102) and a per- turbation formalism on a FLRW background is clearly suggested by the fact that the evolution equations (102a) and (102b) for the quasi–local scalars m$_{q}$, H q are for- mally identical to the energy balance and Raychaudhuri equations of a dust FLRW cosmology. Also, from (28), (29), (37) and (38), we have ˙ k$_{q}$ = − 2 k$_{q}$ H $_{q}$, which is the same evolution equation for $^{3}$R in a FLRW spacetime, while the δ ( A ) defined by (31) and its relation with the scalars in (32) clearly highlights a sort of perturbation definition. We provide in this section a rigorous charac- terization of this intuitive resemblance. Φ : ¯ X → X, ¯ A ↦→ Φ( ¯ A ) = A$_{q}$ ∈ X, (110)