2.3.3  The Velocity of Recession of a Distant Galaxy.

The recessional velocity that is to be derived here is for a galaxy far from the centre of the Universe and also at a time when the second phase of evolution is well established. This scenario has been chosen so that the resulting relationship can be approximated for comparison with empirical results determined via observations on the Earth at the present day.

The velocity of recession of such a galaxy is determined from the internal temporal rate of the Universe in phase II, which from Fig. 3, for the general point s_i, is given by

The velocity of any mass subject to an Acceleration Potential in a Relativistic Domain is given by [2], Eq. (3.2), repeated here for convenience

s_i is the current radial position of the distant galaxy from the centre of the Universe.

s₀ is the initial radial position of the galaxy from the centre of the Universe, i.e. the radial position where its inward motion from Phase I was halted in Phase II.

u₀ is the temporal rate at s₀.

s_u0 is the radius of the Universe when s_i=s₀.

If the point of inflexion, the point where the inward motion of the galaxy from phase I is halted in phase II, is close to the centre, then by comparison with s_i and s_u, s₀ may be approximated to zero, and therefore u₀ may be approximated to unity. Eq. (2.20) may then be restated as

Eq. (2.22) shows that the velocity of recession for a distant galaxy is directly proportional to its distance from the centre of the Universe. This agrees with the empirical results of Edwin Hubble in 1929 and others since. The coefficient of (2.22) can therefore be compared with the Hubble constant. This will form the subject of the next Section.

2.3.4  Derivation of a Theoretical Expression for the Hubble Constant.

With the results obtained in Section 2.3.3, the derivation of this relationship has already been effected and from (2.22) can be stated directly as

Accordingly (2.20) is the full theoretical version of Hubble's law of motion for receding galaxies in phase II.

It should be noted that, because s_u varies with time, thereby causing a variation in r_u the matter density of the Universe, from (2.23) and (2.24) below, it is therefore evident that H₀ is in fact not a constant but will also vary with time. However, in view of the magnitude of the parameters involved, mass, radius and density of the Universe, such variation will only be significant over cosmological periods. Furthermore, the presence of u_i in (2.23) shows that H₀ also varies with distance from the centre of the Universe. This variation is again however, due to the nature of u_i, a small one. These matters will be discussed in more detail in Section 3.

The relationship expressed by (2.23) can of course only be verified by comparing the numerical value it produces with those derived empirically. From (2.23) it can be seen that this would require knowledge of the energy mass of all matter in the Universe, together with its radius. Neither of these parameters are known. In addition, knowledge of u_i is required which effectively means that the location in question, i.e. the point of measurement, (the Earth), in relation to the centre of the Universe is also needed. Again, an unknown parameter. Although u_i is a variable as is evident from (2.18), its value will always be just below unity. Therefore to enable an estimate of H₀ to be made, u_i will be taken as unity which will accordingly give a slightly high value for H₀. Therefore, with u_i = 1, rewriting (2.23), as

r_u is the average density of all matter in the Universe.

enables a value for H₀ to be computed using current estimates for r_u. The average density of all matter in the Universe has been estimated from such exercises as counts of stellar objects per unit stellar volume and the density then calculated via estimates of the masses of the objects counted. Two such estimated densities are shown in the following table.

1:- By comparison with the second estimate, some allowance for the mass of dark matter may have been made here, but has not been stated.

2:- This figure, calculated from the proton mass given in [10], is quoted as the equivalent proton mass per cubic metre, (0.4), and incorporates the assumption that the mass of proposed dark matter in the Universe is 10 times that of luminous matter.

Using a figure of 6.67 x 10^{- 8} cm³/g.sec² for g, [7], insertion of the numbers in Table 1 into (2.24) gives a range of values for H₀ as

However, Hubble's constant is normally quoted in units of Km/sec/10 ⁶ L.Y., and the corresponding numbers from (2.25) in these units are

To compare this figure with empirically derived estimates, reference is made to Table 2 below where, from the "Big Bang" model of the Universe's origin, most figures have been quoted in years for the age of the Universe.

Thus the comparison between the theoretical, Eq.(2.27), and the empirical, Eq.(2.28), values for H₀ yields agreement to within some 18%. Considering the difficulty in determining such parameters as galactic distance, velocities via the Doppler shift in spectral wavelength, and galactic masses etc, this degree of agreement is considered very good. Even so, despite the improvement that could be realised by the inclusion of the correct value for u_i, it is seen that the larger of the two theoretical figures is considerably larger than the largest empirical figure. As noted earlier, the estimate for the average density of matter in the Universe, upon which this figure is based, has, via the mass to light ratio comparison method been inflated by a factor of 10. However, if it exists at all, that method of assessment of the density of dark matter, may be flawed due to the effect of the solar wind on free particles within the Solar System, dark or otherwise. Hence the higher figure for density used in the theoretical computation for H₀ may be too high. If this were the case, the agreement between the theoretically and empirically derived figures could be closer.

Finally, it is necessary to further clarify this derivation, and the comparison of H₀ with empirical results, for their cosmological time of applicability. The values of average density in Table 1 involve observations over wide parts of the cosmos from the astronomically nearby to the cosmologically distant. In doing so, the observations are of the Universe as it was in the past. The greater the distances the further in the past are the objects being observed. Because the density of the Universe is changing with time, the particular value of density that is eventually arrived at is therefore a composite made up from observed objects from relatively recent cosmological periods to extremely remote ones. This density will therefore correspond to some particular cosmological time somewhere in between the two extreme times associated with the applicable observations. Call this time t_C. Consequently, the theoretical value of Hubble's constant will likewise correspond to this particular past cosmological time t_C.

In a similar manner, the empirical determinations of the Hubble constant will suffer from exactly the same problem. The further away an observed galaxy is, the further back in time will its measured parameters correspond to. As a result the empirical values of the Hubble constant shown in Table 2 above will represent parameters over a past cosmological period. The average will therefore, as in the case of the density, correspond to some past cosmological time. The fact that the theoretical and empirical figures of H₀ agree relatively well is because their cosmological times of applicability are very close. This coincidence occurs simply because the objects observed in the determination of both parameters, density of the Universe and the empirical values of H₀, were from the same statistical population.

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