twin universe cosmology Twin Universes cosmology (p 10)
13) Red shift and the Robertson-Walker metric with a variable light velocity.
...The derivation of the distance from the red shift z, with "variable constants", has already been presented. See reference [13], sections 3 to 7. The index 1 refers to the emitter and the index 2 to the receiver. For example, c2 is the current value of the velocity of light, as measured in the observatory. It is assumed that the Rydberg constant (ionization energy of the hydrogen) follows (94)
Then we find :
(95)
The value g = 1 is chosen in order to fit the classical value.
Then, expanding the function 1/R(t) into a series with respect to
(96)
we get :
(97)
Which is nothing but the Hubble's red shift law, which still applies in this variable light velocity conditions. From measurement of d2, c2 and z we can derive the so called Hubble's constant, i.e. the age of Universe.
(98)
identical to the standard value. Then the distance to the object d2 is evaluated :
(99)
...When z tends to infinity we find the cosmological horizon 3/2 c2 t2, which is twice smaller than the standard value 3 c2 t2. If we compare the present model to the EdS model, we get, for the distances, the ratio :
(100)
...They are similar for weak z values, as shown on the next figure. For weak z values, the distances, as derived from the present model, are slightly larger. h is close to unity for z = 1.5. Then h tends to 0.5 when z tends to infinity. For z < 2.5 the difference of the two distance evaluation is less than 5% .
Figure 20 : The distances for the present model and for the Einstein-de Sitter model, and the ratio h of these distances, versus the red shift.
...In reference [14], section 3 the evolution of the angular size of a distant object, versus z, was computed. For the EdS model and constant size objects, the law is :
(101)
...This function of z has a minimum for z = 1.25 and then f tends to grow linearly versus z. The figure 21 explains why it provides an overestimation of f, for large z values.
Figure 21 : Why the classical model overestimates the angular size of large red shift objects. The measure, at the reception time, corresponds to a "fossil" angular size, when the object was closer. ** **
In the present model, the situation is basically different for the objects are supposed to expand with the Universe. See figure 22
Figure 22 : Present model : The light moves along geodesics. The angular size is unchanged.
The corresponding formula is :
(102)
When z tends to infinity, f tends to be constant.
Notice that in our model :
...In the reference [14] this was used to compare the present model to the EdS model, applying to radio-QSO data (Barthel and Miley, 1988 [35]), giving a slight advantage to the first. Obviously, a single test, implying many assumptions about the nature of the observed objects, could not validate the model. See the discussion in reference [14].