12 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 the parabolic case, we substitute into (59) and (64) F ( | k$_{qi}$ | ) ≡ Z$_{h}$ ( α | k$_{qi}$ | ) − Z$_{h}$ ( β | k$_{qi}$ | ) β k$_{qi}$ | 3 / 2 = c ( t$_{j}$ − t$_{i}$ ) , (93) α = 1 m 2 / 3 qi m 1 / 3 qj , β = 1 m$_{qi}$ . (94) where Z$_{h}$ is given by (49) and while for hyperbolic and elliptic models or regions we substitute into (60), (68) and (72) equation (81) plus 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 F ( | k$_{qi}$ | ) ≈ √$_{2}$ 3 β ( α 3 / 2 − β 3 / 2 ) + O ( | k$_{qi}$ | ) , | k$_{qi}$ | ≈ 0 , (95a) F ( | k$_{qi}$ | ) ≈ α − β β | k$_{qi}$ | + O ( | k$_{qi}$ | − 3 / $^{2}$) , | k$_{qi}$ | → ∞ , (95b) ˆ Ω$_{i}$ = 2 M 2 M + E R$_{i}$ , H 2 qi = 2 M + E R$_{i}$ R 3 i , (90) δ ( m ) i = R$_{i}$ 3 R ′ i [ ˆ Ω ′ i ˆ Ω$_{i}$ + 2 H ′ qi qi ] , (89a) δ ( k ) i = R$_{i}$ 3 R ′ i [ ˆ Ω ′ i ˆ Ω$_{i}$ − 1 + 2 H ′ qi qi ] , (89b) 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). 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 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) 1 + 3 2 √ 2 m$_{qi}$c ( t$_{j}$ − t$_{i}$ ) < ( m$_{qi}$ m$_{qj}$ ) 1 / 2 = L 3 / 2 j , (97) Hence, considering (91), a unique solution of (93) must exist if 0 < F ( | k$_{qi}$ | ) ≤ √$_{2}$ 3 β ( α 3 / 2 − β 3 / 2 ) = √$_{2}$ 3 [ m − 1 / 2 qj − m − 1 / 2 qi ] (96) which, together with similar expansions of dF/d | k$_{qi}$ | , show that F ( | k$_{qi}$ | ) is monotonously decreasing and bounded by 3 m$_{i}$ m$_{j}$ R 2 $_{i}$R ′ i = 3 L$_{j}$ Γ$_{j}$ R 2 $_{i}$R ′ i = ( R 3 i L 3 $_{j}$) $^{′}$, (98) c ( t$_{j}$ − t$_{i}$ ) = 2 3 √ 2 m$_{qi}$ [ ( m$_{qi}$ m$_{qj}$ ) 1 / 2 − 1 ] , (92) L$_{j}$ ( r ) = L ( ct$_{j}$, r ) = ( m$_{qi}$ m$_{qj}$ ) 1 / 3 (91) follows from (37) (remember that L$_{i}$ = 1). For a parabolic model, we have from (46) X. MAPPING OF DENSITY TO DENSITY OR VELOCITY TO VELOCITY PROFILES. δ ( m ) i = 2 H ′ $_{qi}$/ H qi 3 R ′ $_{i}$/R$_{i}$ , (88)