11 stitute in (50) and (55). Since ˆ Ω$_{i}$ , H qi are roughly equivalent to inhomogeneous generalization of fiducial Omega and Hubble factors, parametrizing the analytic solutions with these initial value functions could be more intuitive than doing it with the density and spatial curvature profiles m$_{qi}$, k$_{qi}$ . The solutions (50) and (55) become m$_{qi}$ = H 2 qi ˆ Ω$_{i}$ , k$_{qi}$ = H 2 qi ( ˆ Ω$_{i}$ − 1) , (81) c ( t − t$_{i}$ ) = W − W$_{i}$ H qi . (82) H $_{qi}$c ( t − t$_{i}$ ) =     W − W$_{i}$ expanding phase π ˆ Ω$_{i}$( ˆ Ω$_{i}$ − 1) − 3 / 2 − W − W$_{i}$ collapsing phase (83) where the functions W and W$_{i}$ are • Elliptic models or regions: ˆ Ω$_{i}$ − 1 ≥ 0. • Hyperbolic models or regions: ˆ Ω$_{i}$ − 1 ≤ 0. It is straightforward to parametrize the analytic solu- tions in terms of ˆ Ω$_{i}$ , H $_{qi}$. For the parabolic case we have ˆ Ω$_{i}$ = 1, and thus we just obtain (46) with √ 2 m$_{qi}$ = H $_{qi}$. For the hyperbolic and elliptic models or regions we sub- A. Analytic solutions in terms of ˆ Ω i Hyperbolic models or regions Elliptic models or regions We can also parametrize Γ, as well as the Hellaby–Lake conditions, in terms of ˆ Ω$_{i}$ , H qi and their gradients. For cosmic time t$_{i}$ ˆ Ω$_{i}$ = 2 m$_{qi}$ H 2 qi = 2 m$_{qi}$ 2 m$_{qi}$ − k$_{qi}$ , ˆ Ω$_{i}$ − 1 = k$_{qi}$ H 2 qi = k$_{qi}$ 2 m$_{qi}$ − k$_{qi}$ , (80) Notice that the sign of ˆ Ω$_{i}$ − 1 in any region of r deter- mines the sign of ˆ Ω − 1 for all times, which is the same as with k$_{qi}$ and k$_{q}$ . Hence, ˆ Ω − 1 behaves as the FLRW Omega factor for spatial curvature, so that ˆ Ω$_{i}$ − 1 deter- mines the kinematic class: parabolic if ˆ Ω$_{i}$ = 1, elliptic if ˆ Ω$_{i}$ > 1 and hyperbolic if 0 < ˆ Ω$_{i}$ < 1. Also, irrespective of the kinematic class we have for every LTB model or region: ˆ Ω → 1 as L → 0 (near the central curvature singularity). For all hyperbolic models or regions ˆ Ω is bounded between 0 and 1 for all choices of ˆ Ω$_{i}$ , H qi (or m$_{qi}$, k$_{qi}$ ), hence: ˆ Ω → 0 as L → ∞ , whereas for elliptic models or regions ˆ Ω → ∞ as L → L max because H q → 0. W = [ ˆ Ω$_{i}$ + (1 − ˆ Ω$_{i}$) L ] 1 / $^{2}$L 1 / 2 1 − ˆ Ω$_{i}$ − ˆ Ω$_{i}$ 2(1 − ˆ Ω$_{i}$) 3 / $_{2}$arccosh ( 2 L ˆ Ω$_{i}$ + 1 − 2 L ) , (84a) ˆ W$_{i}$ = 1 1 − ˆ Ω$_{i}$ − Ω$_{i}$ 2(1 − ˆ Ω$_{i}$) 3 / $_{2}$arccosh ( 2 ˆ Ω$_{i}$ − 1 ) . (84b) W = ˆ Ω$_{i}$ 2( ˆ Ω$_{i}$ − 1) 3 / $_{2}$arccos ( 2 L ˆ Ω$_{i}$ + 1 − 2 L ) − [ ˆ Ω$_{i}$ − ( ˆ Ω$_{i}$ − 1) L ] 1 / $^{2}$L 1 / 2 ˆ Ω$_{i}$ − 1 , (85a) ˆ W$_{i}$ = Ω$_{i}$ 2( ˆ Ω$_{i}$ − 1) 3 / $_{2}$arccos ( 2 ˆ Ω$_{i}$ − 1 ) − 1 ˆ Ω$_{i}$ − 1 . (85b) t max = ct bb + π ˆ Ω$_{i}$ 2 H $_{qi}$[ ˆ Ω$_{i}$ − 1] 3 / $_{2}$, (87a) t coll = ct bb + π ˆ Ω$_{i}$ H $_{qi}$[ ˆ Ω$_{i}$ − 1] 3 / $_{2}$. (87b) ct bb = ct$_{i}$ − W$_{i}$ H qi , (86) Setting t = t bb and L = 0 in (82) and in the expanding phase of (83) yields the bang time for hyperbolic and elliptic configurations while the maximal expansion and collapse times, t = t max and t = t coll , given by (57) are