twin universe cosmology Astrophysics of ghost-matter.3 : The radiative era: the problem of the "origin" of the universe.
The problem of the homogeneity of the early universe. (p3)
...The characteristic Schwarzschild length Rs varies like the space scale factor R. The characteristic Jeans length is : (43)
write : (44)
then : (45)
...The characteristic Jeans length varies like the space scale factor R.
Combining (35) and (42) we get :
(46)
...The Compton length varies like the space scale factor R. (47)
...The Planck length varies like the space scale factor R. Combining (17) and (42) we get : (48)
m » R
and : (49)
...The Kepler law asserts that the square of the revolution period To2 varies like the third power Ro3 of the orbit radius. Assume this is unchanged during the process : (50)
R3 » T2 or : (51)
R » T2/3
...This is a simple relation linking the space scale R and the time scale T. Combining with (40) and (48) we get immediately : (52)
(53)
(54)
and : (55)
(56)
(57)
The energies are constant (but not the masses).
Note: as we needed one more equation to define the set of constants, the space scale R and the time scale variations, instead of the hypothesis (50), we could have assumed that mc2 is conserved: the two are equivalent. We find that all the characteristic times vary like the time scale factor T. For example, the Jeans and Planck times: (58)
The Poisson equation does not pose a specific problem : (59)
(60)
becomes : (61)
It is normal, because the Poisson equation comes from the field equation. Now let us return to the Maxwell equations (25) to (29). Using (35), we obtain : (62)
(26) gives : (63)
(25) transforms into : (64)
and (28) into (65)
The invariance of these equations is ensured if : (66)
Assuming that the electric and magnetic energy are conserved : (67)
and combining with (63), we find E = c B.
In order to be consistent with the rest, assume :
- the fine structure constant a is an absolute constant
- the Bohr radius Rb varies like the space scale factor R
- the cross section Q varies like R2.
(68)
we find : (69)
gauge electromagnetic laws.
...We can check that the Rydberg energy is an absolute constant, while the Debye length varies like R. In this model, where we define a space scale factor R, a time scale factor T, so-called constants of physics are treated as variables, invariance of all physical equations is required, and energies are conserved :
...- All the characteristic lengths vary like the space scale factor R
...- All the characteristic times vary like the time scale factor T
...As a consequence, we can specify the evolution law, returning to x° = ct and introducing (51). The evolution law becomes : (70)
R = R* » t2/3
...As all the parameters are linked, we can choose any as a leading parameter. If we choose the time t, the general evolution scheme becomes : (71)
R » t 2/3 G » t - 2/3 m » m e » t 2/3 h » t c » t - 1/3 r » t - 4/3 v » t - 1/3 e » t 1/3 E » t B » t - 2/3 m o » t 2/3
...And these quantities are linked to this generalized gauge process. We can choose any as a leading parameter (here : t).
...We could have chosen, during that radiative era, the density r » rr as a leading parameter : (72)
