8 t bb ( r )). For every comoving observer r = const., the time evolution is contained in the range t bb ( r ) < t < t coll ( r ). Rewriting the analytic solutions (5)–(7) as in (46), and as the implicit forms (50) and (55) is very useful, as it al- lows us to relate (by implicit radial derivation) the gradi- ents of L (and thus of R ) with gradients of the intial value functions m$_{iq}$, k$_{iq}$ (see section 11). Notice that the time splitting associated with u a and the scaling of R with R$_{i}$ in L imposes on LTB models a constraint between t bb and m$_{qi}$, k$_{qi}$ , since (47) and (52) imply that any choice of these functions uniquely determines the “age” t$_{i}$ − t bb of the initial slice t = t$_{i}$ for any comoving observer. Also, the initial slice t = t$_{i}$ in elliptic models always occurs in the expanding phase [25]. VII. CURVATURE SINGULARITIES. where Setting t = t bb and L = 0 in (50) and using (49) yields the bang time as a function of m$_{qi}$ and | k$_{qi}$ | : x$_{i}$ = | k$_{qi}$ | m$_{qi}$ , y$_{i}$ = | k$_{qi}$ | 3 / 2 m$_{qi}$ . (51) ct bb = ct$_{i}$ − Z$_{h}$ ( x$_{i}$ ) y$_{i}$ . (52) In this case, the implicit solution t = t ( R ) follows from eliminating η from the first equation in (7) and substi- tuting in the second one. The resulting implicit solution has two branches, an “expanding” one (0 < η < π with ˙ R > 0) and a “collapsing” one ( π < η < 2 π with ˙ R < 0): The new variables allow us to express the coordinate locus of curvature singularities exclusively in terms of L and Γ by means of (41), (42) and (43). Notice that for reasonable initial value functions (bounded and contin- uous), all scalars A$_{q}$ = m$_{q}$, k$_{q}$, H q diverge as L → 0, whereas local scalars A = m, k, can also diverge if Γ → 0 (even if L > 0). Following the standard criteria that define a curvature singularity and considering (A4), we can identify two known possible singular surfaces Notice that if Γ > 0, then all scalars A and A$_{q}$ only diverge at the central singularity L = 0, but if Γ → 0, then all the relative fluctuations δ ( A ) diverge (with A$_{q}$ = 0), so that local scalars A diverge while their quasi– local duals A$_{q}$ remain bounded. This is an obviously unphysical effect of shell crossings that must be avoided. It is important to remark that characterizing the cur- vature singularities as in (58a) and (58b) represents an improvement in clarity over the conventional variables. In these latter variables “ R = 0” can be either a regular symmetry center, or a curvature singularity. Likewise, “ R ′ = 0” can be either a shell crossing singularity or a regular “turning value” r = r$_{tv}$ associated with a closed model. There is no such ambiguity in our parametriza- tion, since L > 0 holds at a symmetry center and Γ > 0 holds at the turning value. In order to test the avoidance of (58b) we need to compute Γ. For this purpose, we derive both sides of the solutions (46), (50) and (55), then we use (31) and L $^{′}$/L = (1 − Γ) R ′ $_{i}$/R$_{i}$ to eliminate m ′ $_{qi}$, k ′ qi and L ′ in terms of δ ( m ) i , δ ( k ) i and Γ. The result is Γ = 1 + δ ( m ) i − δ ( m ) i L 3 / $_{2}$. (59) ct bb = ct$_{i}$ − Z$_{e}$ ( x$_{i}$ ) y$_{i}$ , (57a) ct max = ct bb + π y$_{i}$ = ct$_{i}$ + π − Z$_{e}$ ( x$_{i}$ ) y$_{i}$ , (57b) ct coll = ct bb + 2 π y$_{i}$ = ct$_{i}$ + 2 π − Z$_{e}$ ( x$_{i}$ ) y$_{i}$ . (57c) Notice that (in general) t max = t max ( r ) and t coll = t coll ( r ), so neither one coincides with a $^{3}$T [ t ] hypersurface (like Setting t = t bb and L = 0 in the expanding phase of (55) yields a bang time function, while t = t coll and L = 0 in the collapsing phase yields the “crunch” time associated with the collapsing singularity. The maximal expansion time follows by substituting t = t max and L = L max in either branch of (55). These times are given by It follows from (55)–(54) that L is restricted by 0 < L ≤ L max , where the maximal expansion is L max = 2 /x$_{i}$ = 2 m$_{qi}$/k$_{qi}$ , characterized by ˙ L = 0 and H q = 0. L ( t, r ) = 0 , central singularity (58a) Γ( t, r ) = 0 shell crossing singularity . (58b) x$_{i}$ = k$_{qi}$ m$_{qi}$ , y$_{i}$ = k 3 / 2 qi m$_{qi}$ , (56) | E | 3 / 2 M c ( t − t bb ) = { Z$_{e}$ ( ¯ R ) expanding phase 2 π − Z$_{e}$ ( ¯ R ) collapsing phase , (53) where R = ( | E | /M ) R and Z$_{e}$ is given by u ↦→ Z$_{e}$ ( u ) = arccos(1 − u ) − u 1 / $^{2}$(2 − u ) 1 / $^{2}$. (54) Proceeding as in the hyperbolic case, we transform (53) into y$_{i}$ c ( t − t$_{i}$ )+ Z$_{e}$ ( x$_{i}$ ) =    Z$_{e}$ ( x$_{i}$L ) expanding phase 2 π − Z$_{e}$ ( x$_{i}$L ) collapsing phase (55) where C. Elliptic models or regions: k qi > 0 • Parabolic models or regions.