9 A. Parabolic models/regions. From (59), if L → 0 then Γ → − δ ( m ) i /L 3 / $^{2}$, while Γ → 1 + δ ( m ) i for L → ∞ . Considering (42a), the necessary and sufficient condition for (61) to hold is simply which from (31) implies m ′ qi ≤ 0. It is evident from (62) and (63a) that (64) implies t ′ $_{bb}$/R ′ i ≤ 0 and M ′ ≥ 0, and so it is equivalent to the Hellaby–Lake conditions (10). − 1 ≤ δ ( m ) i ≤ 0 , (64) B. Hyperbolic models/regions. For a fixed r all initial value functions are finite, con- sider then t$_{i}$ > t$_{1}$ > t bb but t$_{1}$ ≈ t bb , so that L ≈ 0 and H q → ∞ , and thus (60) becomes 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 )] , (65) where we used (63b). Now consider t → ∞ so that L → ∞ for all r . From (38) and (50) we have in this limit H q → 0 and H $_{q}$c ( t − t$_{i}$ ) ≈ 1+ Z$_{h}$ ( x$_{i}$ ) + ln(2 x$_{i}$L ) − 2 x$_{i}$L + O ( L − $^{2}$) , (66) hence Γ in (60) becomes Γ ≈ 1 + 3 2 δ ( k ) i + O ( L − $^{2}$) . (67) Bearing in mind, from (9) and (A6), that R ′ i > 0 for hyperbolic models/regions and that (61) must hold as t$_{1}$ → t bb in (65), and since the scaling law (42a) and (65)–(67) must hold for all t and r , the conditions for (61) are [47] where t ′ bb is given in terms of our initial value functions by (63b). Notice, from (42b), that all regular hyperbolic models comply with δ ( k ) ≥ − 2 / 3 for all their time evolu- tion. From (34) and (62), it is evident that (68) are com- pletely equivalent to the Hellaby–Lake conditions (10). Since these conditions are necessary and sufficient, (68) are also necessary and sufficient [24, 25, 39]. C. Elliptic models/regions. We look again at Γ in (60), but now considering that H q changes sign (from positive to negative) at some t = ct ′ bb ≤ 0 , δ ( k ) i ≥ − 2 3 , δ ( m ) i ≥ − 1 . (68) where t bb in (63b) is given by (52) with Z$_{h}$ ( x$_{i}$ ) or Z$_{e}$ ( x$_{i}$ ), respectively, for hyperbolic and elliptic models. As men- tioned before, R$_{i}$ can be always be prescribed as a radial coordinate gauge. ct ′ bb 3 R ′ $_{i}$/R$_{i}$ = c ( t$_{i}$ − t bb ) δ ( m ) i = 2 δ ( m ) i 3 √ 2 m$_{qi}$ , parabolic (63a) ct ′ bb 3 R ′ $_{i}$/R$_{i}$ = δ ( m ) i − δ ( k ) i H qi − c ( t$_{i}$ − t bb ) ( δ ( m ) i − 3 2 δ ( k ) i ) , hyperbolic and elliptic (63b) M ′ M = 3 R ′ i R$_{i}$ [ 1 + δ ( m ) i ] , E ′ E = 3 R ′ i R$_{i}$ [ 2 3 + δ ( k ) i ] . (62) We will also need an expression for the radial gradient t ′ bb in terms of our initial value functions. This expression follows by differentiating both sides of (47) and (52) with respect to r : In order to examine the Hellaby–Lake conditions in the initial value parametrization, we need to relate the gra- dients M ′ and E ′ appearing in (10)–(11) with the initial fluctuations by specializing (34) to t = t$_{i}$ We obtain in this section the Hellaby–Lake conditions (10)–(11) that fulfill the regularity condition (61) [24, 25, 39], but now as restrictions on the initial value functions. These conditions guarantee absence of shell crossings for all t . Notice from (9) and (40) that avoidance of this singularity already excludes a surface layer singularity associated with (123). VIII. THE HELLABY–LAKE CONDITIONS. Γ > 0 ∀ ( ct, r ) such that L > 0 (61) The fulfillment of this condition is examined in the fol- lowing section. From (9), (31), (40) and (42a) applied to A = m , con- dition (9) for standard regularity that avoids (58b) now reads Inserting (59) or (60) (depending on the kinematic class) into (41), (42) and (43) leads to closed analytic expres- sions for all these scaling laws in terms of L and ini- tial value functions in the context of the initial value approach described in previous sections. The functional forms of Γ and H q in (38) and (59) allow us to obtain analytic expressions for all the scaling laws of covariant scalars (as functions of L and initial value functions). where H q and H qi follow from (38), while c ( t − t$_{i}$ ) is given by (50) and (55). Γ = 1 + 3( δ ( m ) i − δ ( k ) i ) ( 1 − H q qi ) − 3 H q c ( t − t$_{i}$ ) ( δ ( m ) i − 3 2 δ ( k ) i , (60) • Hyperbolic and elliptic models or regions.