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• Hyperbolic and elliptic models or regions. 

where H q and H qi follow from (38), while c ( t − t$_{i}$ ) is given by (50) and (55). 

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). 

From (9), (31), (40) and (42a) applied to A = m , con- dition (9) for standard regularity that avoids (58b) now reads 

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## VIII. THE HELLABY–LAKE CONDITIONS. 

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). 

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$_{i}$ 

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We will also need an expression for the radial gradient in terms of our initial value functions. This expression follows by differentiating both sides of (47) and (52) with respect to r : 

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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. 

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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). 

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 

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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 

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]. 

We look again at Γ in (60), but now considering that H q changes sign (from positive to negative) at some t = 

## A. Parabolic models/regions. 

## B. Hyperbolic models/regions. 

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## C. Elliptic models/regions. 

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