16 neous conditions, as assumed in linear perturbations with “small” perturbations | δ ( m $^{)}$|  ≪ 1, (112) reduces to parameters of a FRLW dust spacetime. As a consequence ˆ Ω is not a background variable in the perturbation for- malism, though its gradient defined as Δ ≡ ˆ Ω $^{′}$/ ˆ Ω 3 R $^{′}$/R = δ ( m $^{)}$− 2 δ ( H ) = (1 − ˆ Ω) ( δ ( m $^{)}$− δ ( k $^{)}$) , (116) is a GIC perturbation in this formalism (since δ ( m ) and δ ( H ) are), though we have Δ = (Ω − ˆ Ω) / ˆ Ω in general (this relation only holds in the linear limit). On the other hand, it is hard to find a theoretical context for H$_{eff}$, Ω$_{eff}$ besides their resemblance to FLRW quantities and their utility in computations. A. Evolution equations in terms of ˆ Ω and Δ . XIII. A THEORETICAL CONTEXT FOR THE OMEGA PARAMETER. an equation that is formally identical to the evolution equation for linear density perturbations of a dust source around a FLRW background (characterized by ρ$_{q}$, H $_{q}$) in the comoving gauge [38], which for dust is a synchronous gauge as well. ¨ δ ( m ) + 2 H q ˙ δ ( m ) − κ 2 ρ$_{q}$ δ ( m ) = 0 , (113) H$_{⊥}$ ≡ ˙ R R = ˙ L L = H $_{q}$, (114a) H$_{‖}$ ≡ ˙ R ′ R ′ = ˙ L L + ˙ Γ Γ = H q − 3Σ = H $_{q}$(1 + 3 δ ( H $^{)}$) , (114b) ∂ H ∂τ = − H 2 ( 1 + 1 2 ˆ Ω ) , (117a) The parameter ˆ Ω that we discussed in section IX has been introduced in the literature [15, 16] (together with H $_{q}$) as useful ansatzes justified by their resemblance to the corresponding FLRW Hubble and Omega parame- ters. Another generalization of the FLRW Hubble factor, suggested by Moffat and Tartarsky [42] and used in var- ious articles [17], follows by defining two expansion fac- tors, a radial and an azimuthal one, which can be given in terms of H q and δ ( H ) as: However, given the practical utility of ˆ Ω and the con- nection of its gradient Δ to the perturbation formalism of section XII, it is still useful to construct a system of evolution equations that include evolution laws for these quantities. Rewriting the system (102) in terms of ˆ Ω and Δ yields the following dimensionless system Since ˆ Ω and H q in (75) and (78) are constructed from m$_{q}$ and k$_{q}$ , both quantities can easily be computed from a numeric solution of the systems (102) or (103). The initial value functions m$_{qi}$ and k$_{qi}$ in (107), (108) and (109) can be given in terms of ˆ Ω$_{i}$ and H qi from (81), while δ ( m ) i and δ ( k ) i follow from (89). ∂ ˆ Ω ∂τ = H ˆ Ω ( ˆ Ω − 1 ) , (117b) ∂ Δ ∂τ = H [( δ ( H ) + Δ ) ˆ Ω − ( 1 + 3Δ + 4 δ ( H ) ) δ ( H ) ] , (117c) ∂δ ( H ) ∂τ = − H [ ( 1 + δ ( H ) ) δ ( H ) + 1 2 ( δ ( H ) + Δ ) ˆ Ω ] , (117d) where we have introduced the dimensionless variables H ≡ H q H$_{0}$ , τ ≡ H$_{0}$ c ( t − t$_{i}$ ) , (118) with H$_{0}$ an inverse length scale (cm − $^{1}$), which can be identified as the cosmological Hubble scale at a specific fiducial cosmic time (we can also identify R$_{0}$ = H − 1 0 when specifying the radial coordinate gauge through R$_{i}$ ). The initial conditions for this system are basically H $_{qi}$, ˆ Ω$_{i}$ and their radial gradients, which makes their specifica- tion very intuitive and practical, as these initial value functions could tend at a given asymptotic limit to the Hubble and Omega factors in a FLRW background at a fiducial cosmic time. The system (117) can also be sup- plemented by the differential equations (104) for L and Γ. While both H q and H$_{eff}$ are covariant quantities, the theoretical context for H q may be easier to justify, as this scalar is the quasi–local dual of the fluid flow expansion scalar H = Θ / 3, and thus it is a GIC background variable in the perturbation formalism discussed in section XII (it is the image of the FLRW Hubble parameter under the map (110)). However, H 2 q is not the quasi–local dual of H $^{2}$, since the definition (24) implies that ( H $^{2}$)$_{q}$ = ( H $_{q}$) $^{2}$, hence H 2 q (and thus) ˆ Ω are not images under the per- turbation map (110) of the Omega and squared Hubble so that an Omega parameter follows as Ω$_{eff}$ = κρ/ (3 H $_{eff}$). However, both pairs H $_{q}$, ˆ Ω and H$_{eff}$, Ω$_{eff}$ are basically useful quantities in the qualitative or numeric application of LTB models, since the proper local covariant general- ization of the FLRW Hubble parameter to LTB models is neither H $_{q}$, nor H$_{eff}$ , but the expression given by equa- tion (41) of [43]. Nevertheless, it is still interesting to discuss the theoretical assumptions underlying these ex- pressions. Evidently, H$_{⊥}$ and H$_{‖}$ correspond, respectively, to ex- pansion factors of proper lengths in the direction orthog- onal and parallel to radial rays orthogonal to the orbits of SO(3). Moffat and Tartarsky define from these expan- sion factors an “effective” Hubble parameter as H 2 eff = H 2 ⊥ + 2 H$_{⊥}$H$_{‖}$ = H q H = H 2 $_{q}$(1 + δ ( H $^{)}$) , (115)