7 VI. THE ANALYTIC SOLUTIONS IN TERMS OF AN INITIAL VALUE APPROACH. We rephrase in this section the analytic solutions (5)– (7) in terms of { m$_{qi}$, k$_{qi}$ } , which are related to the free parameters M and E in (5)–(7) by Inserting (43) into (30) allows us to find scaling laws sim- ilar to the ones above for H = 3Θ and the scalars Σ , E , respectively associated by (15) and (16) to the shear and electric Weyl tensor. Using (31), we can express the scal- ing laws (42) as relations between the gradients H ′ $_{q}$, m ′ q and k ′ q and L $^{′}$. Notice that it is always possible to use (38) and (43) to eliminate any one of the three pairs H $_{q}$, δ ( H ) or m$_{q}$, δ ( m ) or k$_{q}$ and δ ( k ) in terms of the other two. The scaling laws (41), (42), and (43) depend on both L and Γ. While the effects of their dependence on L is easy to grasp because the qualitative time dependence of L (for a fixed r ) can be appreciated directly from the Friedman–like equation (38), there is no simple way to guess the qualitative behavior of Γ (either in the t or r directions). As a consequence, there is not much we can do with these scaling laws as long as we lack an expression for Γ in terms of L and the initial value functions (we obtain this expression in section VII). B. A radial coordinate gauge. A. Parabolic models or regions: k qi = 0 Of course, other combinations of initial value functions, such as { m$_{qi}$, H qi } or { k$_{qi}$, H qi } , can also be considered, as (38) relates the three available functions. Since the fluctuations follow from the gradients of these functions from (31), any two of { m$_{qi}$, k$_{qi}$, H qi } forms an irreducible set of initial value functions. As we show in section XI, these initial value functions provide also the basic initial conditions for the evolution equations under a numeric approach. A third free function (the “bang time”, t bb ) necessarily appears as an “integration constant” in both, the solu- tions of ˙ L in (38) and those of ˙ R in (3). However, as we show below, if we rewrite (5)–(7) in the context of an initial value problem associated with m$_{qi}$, k$_{qi}$ and L (con- sidering that L$_{i}$ = 1), the bang time t bb can be obtained as a function of m$_{qi}$, k$_{qi}$ . M = m$_{qi}$ R 3 $_{i}$, E = − k$_{qi}$ R 2 $_{i}$. (45) We express M and R in terms of m$_{qi}$ and L from (36) and (45) in (5), after re–arranging terms we get the fol- lowing closed analytic expression for L where we are only considering expanding configurations ( L increases for t > t$_{i}$ ). The bang time follows by setting L = 0 and t = t bb in (46) L = [ 1 + 3 2 √ 2 m$_{qi}$ c ( t − t$_{i}$ ) ] 2 / 3 , (46) ct bb = ct$_{i}$ − 2 3 √ 2 m$_{qi}$ = ct$_{i}$ − 2 3 H qi . (47) B. Hyperbolic models or regions: k qi < 0 We obtain the following implicit solution of the form t = t ( R ) by eliminating the parameter η from the equa- tion for R in (6) and substituting in the equation for t : E 3 / 2 M c ( t − t bb ) = Z$_{h}$ ( ¯ R ) , (48) where ¯ R = ( E/M ) R and Z$_{h}$ is the function u ↦→ Z$_{h}$ ( u ) = u 1 / $^{2}$(2 + u ) 1 / 2 − arccosh(1 + u ) . (49) We express then M, E and R in (48) in terms of m$_{qi}$, k$_{qi}$ and L from (36) and (45). The result is y$_{i}$ c ( t − t$_{i}$ ) = Z$_{h}$ ( x$_{i}$L ) − Z$_{h}$ ( x$_{i}$ ) , (50) However, other radial coordinate gauges are often used (for example setting M as radial coordinate, as in [7– 10]). And, as pointed before, (44) cannot be used in cases where a regular zero of R ′ occurs. For closed models, R$_{i}$ must be selected so that it vanishes at both centers of symmetry and R ′ i has a common same order zero with F = √$_{1 +}$$_{E}$ (to avoid the surface layer singularity asso- ciated with (123), see [40]). Moreover, even if we use the parametrization associated with (39), we are not forced to use the radial coordinate gauge for fixing R$_{i}$ , it is still possible to keep this initial value function unspecified and use the gauge freedom to fix any one of the other initial value functions. where R$_{0}$ is an arbitrary constant characteristic length scale. The gauge (44) is a popular choice in the literature [14–17], not only due to its simplicity, but because setting r = R$_{i}$/R$_{0}$ allows us to regard radial dependence as a dependence on an fiducial value of an invariant quantity ( R ) that has a clear physical and geometric meaning. Also, the choice of R$_{0}$ provides a physical length scale for the radial coordinate. R$_{i}$ = R$_{0}$ r, (44) The metric of LTB models in the form (39) is also in- variant under an arbitrary rescaling r = r (¯ r ). Since L and Γ are time dependent, it is natural when using (39) to identify this radial coordinate gauge freedom with the freedom to choose the initial value function R$_{i}$ ( r ). How- ever, because of (A6) and (A7), the choice for a function R$_{i}$ is not completely arbitrary: it depends on the topol- ogy of the space slices $^{3}$T [ t ] (see [25] and Appendix A 3). For open models (or LTB regions in which (9) allows for R ′ > 0 to hold everywhere) R$_{i}$ can be any monotonously increasing function complying with R$_{i}$ (0) = 0 and R ′ i > 0 for all r . Evidently, the simplest choice in these cases is