To start this derivation, consider the rate of expansion of the Universe in Phase II of its evolution, exemplified by the recession velocity of a random galaxy, a distance of s_i from the centre, as given by [2], Eq.(3.19)

(B1)

In [2] this was simplified to

(B2)

where H₀ is the value of Hubble's "constant" at s_i, and in which u_i had been approximated by unity. In (B2) the latter approximation can be avoided by using the current empirical value for H₀, which automatically includes the correct effect for u_i. However, this value applies only at the cosmologically applicable time of t_C as fully explained in [2], Section 3.5.

Now if (B2) is to apply to the boundary of the Universe then it becomes

(B3)

and in using the above value of H₀ in (B3), the spatial gradient effect between s_i and s_u, both at t_C, has been ignored. This is effectively once again assuming u_i=1 over this distance. Thus (B3) includes the same degree of approximation as was used in [2] to determine the theoretical value of H_0. Therefore, treating H₀ in (B3) as a constant, (B3) can be integrated to give

(B4)

where Ln represents Naperian logarithms.

In (B4) the cosmologically applicable time t_C is the time the Universe has been expanding from the point of inflexion, and the constant of integration is therefore the Naperian logarithm of the radius of the point of inflexion from the centre. In a next paper it will be shown that this radius is equal to the gravitational radius. Thus in (B4) when t_C = 0

(B5)

which when substituted into (B4) gives

(B6)

Eq.(B6) is a simplified relationship between the radius of the Universe and the time at t_C in Phase II of its evolution from the point of inflexion. It is however, sufficiently correct in form to justify the statement made in [2] Section 2.1.1(iv) concerning the nature of the expansion of the Universe such that Hubble's law for the rate of recession of the distant galaxies was satisfied.

To eliminate a_u in (B6) substitute

(B7)

Eq.(B6) then becomes

(B8)

At the time t_C in (B8), H₀ is known to be 22.6 Km/sec/10⁶ L.Y. for which the corresponding value of r_u from [2] Eq.(3.21) is 2.1 x 10 ^-29 gm/cm³ and all other terms are constants. Thus if t_C were known, a value of s_u at t_C could be calculated. An estimate of t_C is constructed as follows.

In surveying the recession velocities of the distant galaxies and the density of the Universe, if the range of observations was between say 4 and 8 billion L.Y., and assuming a uniform distribution of observations between that range, then the value of t_C would be at the median point of the range and can accordingly be estimated to be approximately 6 billion years ago. In [2] it was stated that the evolution of the Universe was probably some 15 billion years into Phase II from the point of inflexion. The cosmologically applicable time in (B8), t_C, is then estimated as the difference between the above two numbers, i.e. 9 billion years.

Inserting this value, and those for H₀ and r_u above, together with the known values for c and g as previously stated, into (B8) then yields

(B9)

as the radius of the Universe at t_C. The mass of the Universe can now be calculated using the standard formula to be

(B10)

The values in (B9) and (B10) can now be inserted into Table 2.

Finally, as a matter of interest, a_u the gravitational radius of the Universe can now be established as

(B11)

It should be noted that all the numbers generated here, are very approximate estimates resulting from approximations in the expressions used and the uncertainties in such parameters as r_u, H₀ and t_C. However, these numbers are so large that errors of even one or two magnitudes are insignificant when used in Table 2 and FIG.1 for the purpose intended.

C2 Version 1.1.1
Ó; P.G.Bass March 2006

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