content_image/399494.jpg content_image/399495.jpg content_image/399493.jpg content_image/399507.jpg content_image/399506.jpg content_image/399510.jpg content_image/399485.jpg content_image/399491.jpg content_image/399492.jpg content_image/399503.jpg content_image/399501.jpg content_image/399486.jpg content_image/399505.jpg
the parabolic case, we substitute into (59) and (64) 

while for hyperbolic and elliptic models or regions we substitute into (60), (68) and (72) equation (81) plus 

where R$_{i}$ can be specified as a choice of radial coordinate. The initial value functions ˆ Ω$_{i}$ , H qi can also be used in the analytic solutions given in the forms (5)–(7), in terms of the parameter η . For this purpose, we eliminate M and E by means of 

where R$_{i}$ can be prescribed as a radial coordinate gauge. The bang time is then given by (86). 

## X. MAPPING OF DENSITY TO DENSITY OR VELOCITY TO VELOCITY PROFILES. 

Krasi´ nski and Hellaby [7, 10] have examined an inter- esting proposal: the possibility that arbitrary radial pro- files of density (or velocity or density and velocity) given at two different cosmic times for a finite radial range can be fit (or “mapped”) by a unique LTB model. The initial value parametrization of the analytic solutions is ideally suited to examine this issue, which can also be under- stood in terms of the equivalence of boundary and initial conditions to solve the evolution equations of sections XI and XIII. Our aim here is to illustrate how the ini- tial value parametrization can be helpful, hence we only examine in detail the mapping between the quasi–local density profiles by a hyperbolic model, since all other similar mappings can be handled in the same manner. 

Consider two radial profiles of m$_{q}$ . Since t = t$_{i}$ is arbitrary, then m$_{qi}$ can be one of the profiles, while we take m$_{qj}$ = m$_{q}$ ( ct$_{j}$, r ) for some t$_{j}$ > t$_{i}$ as the second profile. If m$_{qi}$ and m$_{qj}$ are known, then 

follows from (37) (remember that L$_{i}$ = 1). For a parabolic model, we have from (46) 

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a constraint that will not be satisfied unless m$_{qi}$ and m$_{qj}$ have very restricted forms. For a hyperbolic model we rewrite (50) as 

<img src='content_image/399485.jpg'>

where Z$_{h}$ is given by (49) and 

Since a unique LTB model is determined by prescribing m$_{qi}$ and | k$_{qi}$ | , and we assume m$_{qi}$ and m$_{qj}$ known, then (93) becomes a constraint to find the missing initial value function | k$_{qi}$ | that determines the LTB model. Solving this constraint for known m$_{qi}$, m$_{qj}$ and c ( t$_{j}$ − t$_{i}$ ) can only be done numerically, but by looking at the properties of the function F ( | k$_{qi}$ | ) we can find the conditions for the existence of this solution, as it was done in [7, 10]. 

Considering that in an expanding model we have m$_{qi}$ > m$_{qj}$ for t$_{j}$ > t$_{i}$ , we have α > β in general, and since Z$_{h}$ is monotonously increasing, then the left hand side of (93) is positive (like the right hand side). We also have 

<img src='content_image/399491.jpg'>

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which, together with similar expansions of dF/d | k$_{qi}$ | , show that F ( | k$_{qi}$ | ) is monotonously decreasing and bounded by 

<img src='content_image/399503.jpg'>

Hence, considering (91), a unique solution of (93) must exist if 

<img src='content_image/399501.jpg'>

which is the same result found in [7]. Comparison with (46) shows that the density-density mapping requires the LTB model that maps m$_{qi}$ to m$_{qj}$ to expand faster than a parabolic model. For the elliptic case we follow the same method: using the solution (55) (in the expanding and collapsing phases) as a constraint to find the missing initial value function k$_{qi}$ . If the local density profiles m$_{i}$ and m$_{j}$ are known (instead of m$_{qi}$ and m$_{qj}$ ), then we have to use the scaling law (41a) with Γ given by (40) 

<img src='content_image/399486.jpg'>

<img src='content_image/399505.jpg'>

