10 which, because of (42c), implies that δ ( m $^{)}$− (3 / 2) δ ( k ) ≥ 0 necessarily holds for all times if (61) holds. Since condi- tion (74) implies (69), then the latter (also necessary but not sufficient) follows from (72) too. It is straightforward to show from (31), (57), (62), (63b) and (73) that conditions (72) are equivalent to the Hellaby–Lake conditions (11). We have expressed these conditions in terms of m$_{qi}$, k$_{qi}$, δ ( m ) i , δ ( k ) i , but it is straightforward to rewrite them in terms of other sets initial value functions, like m$_{qi}$, H $_{qi}$, δ ( m ) i , δ ( H ) i . Because of (A6) and (40), we do not need to express (72) in terms of the ± signs for R $^{′}$, as it is understood that the signs of t ′ bb and t ′ coll must be, respectively, the same and the opposite of the sign of R ′ i (which because of (A6) will be the same as the sign of R $^{′}$). The fact that H q behaves in (38) as a Hubble scalar of a FLRW dust universe, has motivated some authors [15, 16] dealing with cosmological and astrophysical applications to define an “Omega” parameter analogous to that of a FLRW dust cosmology IX. OMEGA AND HUBBLE EXPANSION PARAMETERS. ˆ Ω ≡ κρ$_{q}$ 3 H 2 q = 2 m$_{q}$ H 2 q = 2 m$_{q}$ 2 m$_{q}$ − k$_{q}$ = 2 m$_{qi}$ 2 m$_{qi}$ − k$_{qi}$L , (75) ˆ Ω − 1 = k$_{q}$ 2 m$_{q}$ − k$_{q}$ = k$_{qi}$L 2 m$_{qi}$ − k$_{qi}$L , (76) so that where we use the symbol ˆ Ω, and not Ω$_{q}$, because this quantity is not the quasi–local dual of the scalar 2 m/ H 2 under the definition (24) (see section XIII). The following FLRW–like scaling laws hold for ˆ Ω and ˆ Ω − 1: while the Hamiltonian constraint (38) can be rewritten as an analogue of the FLRW Hubble expansion The relative fluctuation (43) of the quasi–local Hubble factor, δ ( H $^{)}$, takes the appealing form which can be given in terms of metric functions L, Γ and initial value functions ˆ Ω$_{i}$ , δ ( m ) i and δ ( H ) i by means of (42a), (42b), (43) and (77). We can identify ˆ Ω$_{i}$ in (77), (78) and (79) as analogous to an r − dependent FLRW Omega factor at a fiducial H 2 q = H 2 qi [ ˆ Ω$_{i}$ L 3 + 1 − ˆ Ω$_{i}$ L 2 ] . (78) 2 δ ( H ) = ˆ Ω δ ( m ) + ( ˆ Ω − 1) δ ( k $^{)}$, (79) ˆ Ω = ˆ Ω$_{i}$ ˆ Ω$_{i}$ − ( ˆ Ω$_{i}$ − 1) L , ˆ Ω − 1 = ( ˆ Ω$_{i}$ − 1) L ˆ Ω$_{i}$ − ( ˆ Ω$_{i}$ − 1) L , (77) Γ ≈ − 3 H q qi [ δ ( m ) i − δ ( k ) i − H qi c ( t$_{i}$ − t$_{1}$ ) ( δ ( m ) i − 3 2 δ ( k ) i )] = − 3 H q ct ′ bb 3 R ′ $_{i}$/R$_{i}$ + c ( t$_{1}$ − t bb ) ( δ ( m ) i − 3 2 δ ( k ) i )] , (70) Γ ≈ 3 |H q | H qi [ H qi c ( t$_{2}$ − t$_{i}$ ) ( δ ( m ) i − 3 2 δ ( k ) i ) + δ ( m ) i − δ ( k ) i ] = 3 |H q | ct ′ bb 3 R ′ $_{i}$/R$_{i}$ + c ( t$_{2}$ − t bb ) ( δ ( m ) i − 3 2 δ ( k ) i )] , (71) Since (70) must hold as t$_{1}$ → t bb and (71) must hold as t$_{2}$ → t coll , each expression furnishes a second and third necessary condition for (61). Since both conditions that follow from (70) and (71) must hold for all t in the range t bb ( r ) < t < t coll ( r ) for all r , these two conditions taken together are then the necessary and sufficient conditions for the fulfillment of (61), which can be written as where we used c ( t coll − t$_{i}$ ) = c ( t coll − t bb ) − c ( t$_{i}$ − t bb ), with t bb and t coll given by (57), and the equal sign holds only at a symmetry center. The fact that one of the three con- ditions in (72) to avoid shell crossings in elliptic models is basically a sign condition on the gradient t ′ coll has not been, apparently, noticed in the extensive literature that uses the conventional variables. The reader is requested to compare the simplicity and elegance of (72) with the cumbersome form (11). Conditions (72) imply the following necessary (but not sufficient) condition for (61) δ ( m ) i − 3 2 δ ( k ) i ≥ 0 , (74) ct ′ coll 3 R ′ $_{i}$/R$_{i}$ = ( δ ( m ) i − 3 2 δ ( k ) i ) c ( t coll − t bb ) + ct ′ bb 3 R ′ $_{i}$/R$_{i}$ , (73) ct ′ bb 3 R ′ $_{i}$/R$_{i}$ ≤ 0 , ct ′ coll 3 R ′ $_{i}$/R$_{i}$ ≥ 0 , δ ( m ) i ≥ − 1 , (72) with Near the initial singularity (bang) we have H q → ∞ as t → t bb , while near the collapsing singularity we have H q −∞ as t → t coll with t coll given by (57). We consider t = t$_{1}$ such that t$_{i}$ > t$_{1}$ > t bb but t$_{1}$ ≈ t bb , as well as t = t$_{2}$ such that t coll > t$_{2}$ > t$_{i}$ with t$_{2}$ ≈ t coll . For these times Γ takes the form t max where L = L$_{max}$ = 2 m$_{qi}$/k$_{qi}$ = 2 /x$_{i}$ , so that 0 < L ≤ L$_{max}$ and we have the presence of a second (collapsing) singularity as L → 0 when t → t coll , with t max and t coll given by (57). Near the maximal expansion L → L$_{max}$ and so H q → 0, while c ( t − t$_{i}$ ) → c ( t max − t$_{i}$ ). Thus, we have in this limit the following necessary (not sufficient) condition for (61): Γ = 1 + 3( δ ( m ) i − δ ( k ) i ) > 0 . (69)