content_image/431949.jpg content_image/431948.jpg content_image/431947.jpg content_image/431969.jpg content_image/431967.jpg content_image/431968.jpg content_image/431966.jpg content_image/431942.jpg content_image/431943.jpg content_image/431957.jpg content_image/431959.jpg content_image/431945.jpg content_image/431946.jpg content_image/431955.jpg
constant, leads to 

<img src='content_image/431949.jpg'>

where H q follows from (38) and 

<img src='content_image/431948.jpg'>

with x$_{i}$, y$_{i}$, Z$_{h}$ and Z$_{e}$ given by (51), (49), (56) and (54). Since x$_{i}$ and y$_{i}$ depend only on m$_{qi}$ and k$_{qi}$ , a simulta- neous big–bang t ′ bb = 0 implies the following constraint on these initial value functions 

<img src='content_image/431947.jpg'>

which, in turn, implies (by differentiation) the following constraint between the δ ( m ) i and δ ( k ) i (hence, between the gradients m ′ qi and k ′ $_{qi}$) 

<img src='content_image/431969.jpg'>

As a consequence of (126), a simultaneous big–bang al- lows us to prescribe only one of the two functions m$_{qi}$ and k$_{qi}$ , the other one must be found by solving the al- gebraic constraint (126) or solving (127) as a differential equation for d k$_{qi}$/ d m$_{qi}$ (the function R$_{i}$ ( r ) remains free and can be fixed as a radial coordinate gauge). Since we need both functions m$_{qi}$ and k$_{qi}$ to solve (102) or (103), finding previously a numeric solution of the alge- braic constraint (126) is not a problem (it is part of the work needed to provide appropriate initial conditions). However, assuming that m$_{qi}$ is selected and bearing in mind that 

it is evident that the hyperbolic branch of (126) does not appear to be restrictive at all to find | k$_{qi}$ | . For the elliptic branch, a necessary condition for a solution of (126) that yields k$_{qi}$ is 

which, again, does not seem to be restrictive (see [10] for comparison). Still, since we cannot solve (126) analyti- cally, nor invert (125) to find L , the case of a simultane- ous big–bang may be easier to handle analytically with the conventional variables M, E and R by means of (48) or (53), since we only need to fix t bb = t (0) bb and then pre- scribe the functions M and E taking care to fulfill the Hellaby–Lake conditions (10) and (11). Nevertheless, we can still use m$_{qi}$ and k$_{qi}$ to examine qualitatively these regularity conditions in these models through (124). 

<img src='content_image/431967.jpg'>

<img src='content_image/431968.jpg'>

<img src='content_image/431966.jpg'>

• Hyperbolic models 

We examine (124) in the limits L ≈ 0 and L → ∞ along constant but arbitrary r (so that the time dependence is concentrated on L ). Consider the following series expansions of H $_{q}$c ( t − t (0) bb ): 

<img src='content_image/431942.jpg'>

<img src='content_image/431943.jpg'>

where we used (49), (38) and (125). Taking the leading terms in these expansions, Γ in (124) be- comes 

leading to the following conditions for the fulfill- ment of (61) 

which are the same as (68) with t ′ bb = 0. 

• Elliptic models 

The behavior of Γ in (124) around L ≈ 0 for t ≈ t (0) bb follows from an expansion similar to (130b), but using Z$_{e}$ instead of Z$_{h}$ . This yields also H $_{q}$c ( t − t (0) bb ) ≈ 2 / 3 + O ( L ), and lead to Γ ≈ 1 + δ ( m ) i . Considering now L ≈ 0 but in the collapsing phase, so that t ≈ t coll , we have H $_{q}$c ( t − t (0) bb ) → −∞ . Hence, Γ in this limit takes the form 

<img src='content_image/431957.jpg'>

while Γ for t ≈ t max takes the form (69). Therefore, the conditions for the fulfillment of (61) are simply 

<img src='content_image/431959.jpg'>

Notice, by comparing with (73), that the second condition above is equivalent to ct ′ coll ≥ 0, hence (133) are simply (72) with t ′ bb = 0. It is important to notice that a simultaneous big–bang t ′ bb = 0 does not imply a simultaneous collapsing singularity or maximal expansion ( t ′ coll and t ′ max are not zero). 

## C. Simultaneous maximal expansion and simultaneous collapsing time. 

These elliptic LTB configurations follow by setting the maximal expansion and collapse times, t max , t coll , in (57) 

<img src='content_image/431945.jpg'>

<img src='content_image/431946.jpg'>

<img src='content_image/431955.jpg'>

