content_image/397454.jpg content_image/397455.jpg content_image/397468.jpg content_image/397453.jpg content_image/397448.jpg content_image/397456.jpg content_image/397460.jpg content_image/397461.jpg content_image/397463.jpg
neous conditions, as assumed in linear perturbations with “small” perturbations | δ ( m $^{)}$|  ≪ 1, (112) reduces to 

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. 

## XIII. A THEORETICAL CONTEXT FOR THE OMEGA PARAMETER. 

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: 

<img src='content_image/397454.jpg'>

<img src='content_image/397455.jpg'>

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 

<img src='content_image/397468.jpg'>

so that an Omega parameter follows as Ω$_{eff}$ = κρ/ (3 H 2 $_{eff}$). 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. 

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 

<img src='content_image/397453.jpg'>

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 

<img src='content_image/397448.jpg'>

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 . 

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). 

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 

<img src='content_image/397456.jpg'>

<img src='content_image/397460.jpg'>

<img src='content_image/397461.jpg'>

where we have introduced the dimensionless variables 

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. 

<img src='content_image/397463.jpg'>

