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tokens: appropriate choice of the arbitrary function t 0 ( R ). Let us now find the general expressions for t sc ( R ) and t hor ( R ) for each type of solution ( 14 )–( 16 ). For further purposes, it is convenient to rewrite the solution in dimensionless unitsIn the rest of our paper, we shall use these dimensionless units and thus prefer to omit the tilde over the letters for brevity. The solution ( 14 )–( 16 ) in dimensionless units ( 26 ) reads: 1. The case of flat space: 2. The case of closed space: 3. The case of open space: By solving Equations ( 23 ) and ( 24 ) successively for each case ( 27 )–( 29 ), we obtain: 1. For k = 0 2. For k = 1 3. For k = −1 Here, in our calculations, we have used the fact that β is very large for realistic objects (estimative size of the universe is about m, and r g of a typical black hole in the galaxy center is about ∼10 10 m, i.e., one has β ∼ 10 16 ). Thus, we have employed the Taylor expansion for large β to get rid of transcendental equations for the open and closed spaces. Analyzing the expressions ( 30 )–( 32 ) and applying the condition ( 25 ), we come to the following form of the arbitrary functions t 0 ( R ): It is easy to see that these functions have the same limit at small R and the result ( 17 ) remains valid. To give an example of the behavior of the obtained model, we present in Figures 2 – 3 the profiles of the energy density and the Hubble parameter for the parabolic case of our solution ( 14 ). The energy density is given by ( 19 ). The Hubble parameter describing the rate of expansion of the inhomogeneous universe is generally defined according to Ellis ( 2009 ) as followswhere l is some characteristic length that corresponds to the scale factor a ( t ) in Friedmann models, and τ is proper time given in the standard way:Θ is the volume expansion given bywhere u α is usual 4-velocity of the dust particles. Figure 2. Energy density profile for the parabolic case of our solution given in dimensionless units. (a) At the initial (Big Bang) singularity there is an infinite value of the energy density that is decreasing with time, while in the center of symmetry there always holds ε ( R = 0, t ) = 0. (b) The structure of the profile at the initial singularity in the vicinity of the central point . There is a punctured infinite energy density cusp, which is rapidly decreasing with time. (c) The character of the energy density dependence on time for different values of R . The dashed curve corresponds to the parabolic Friedmann solution. The bigger is R , the closer is ε ( R , t ) to the Friedmannian line. (d) The picture at the epoch of the Milky Way birth. The number of each curve corresponds to the index of the value t (or R ) listed in the legend. Figure 3. The Hubble parameter profile for the parabolic case of our solution given in dimensionless units. (a) The line t = t +0( R ) marks the line of the initial singularity. (b) The structure of the profile at the initial singularity in the vicinity