twin universe cosmology Astrophysics of matter and ghost matter.
1. The geometric framework. The matter era and the Newtonian approximation. (p3)
(33-a)
(33-b)
(33-c)
(33-d)
… From (33-b) and (33-d), the curvature indices k and k* must be negative; we therefore obtain, with k = k* = –1*. The initial evolution laws are simply linear: R = R* » r*. However, as will be shown later, the matter densities may become different. We then obtain the following system:
(34-a)
(34-b)
(34-c)
(34-d)
from which we immediately deduce:
(35-a)
(35-b)
By introducing mass conservation in both folds:
(36)
w R³ = constant w* R³ = constant
The system then becomes:
(37-a)
(37-b)
… Notice that R = R** implies R¨ = R¨* = 0*. On the other hand, if the two universes were "fully coupled," that is, if R/R = constant*, they would then correspond to Friedmann models, with "parallel evolutions." However, we consider that they are coupled by the gravitational field, via (37-a) and (37-b), which show that the linear expansion is unstable. For example, if R > R**, then R¨ > 0 and R¨* < 0*. The system can be solved numerically; the typical solution corresponds to Figure 1.
Fig. 1: Evolution of the scale parameters of the universe and the ghost universe.
There is a "common history," described through the common system of coordinates:
{ t, u, q, j }
… Using equations (13) to (16), we can return to the systems { t, r, q, j } and { t* , r* , q, j }. Note that the light velocities c and c**, as well as the characteristic times T and T**, can be different. If c = c** and T = T** = 1, we simply get (t = t ; t* = – t*).
Why can't we simply set r* = r*?
Because the length scales R and R* are found to be different. Consider two sets of conjugated points (A, A** ) and (B, B** ). Assume (q_A = q_B ; j_A = j_B ). The two sets correspond to radial markers u_A and u_B. Since they are conjugated, A and A** refer to the same radial marker u_A. The same applies to the conjugated points B and B**, corresponding to the value u_B. The distance AB is R (u_B – u_A), while the distance AB is R* (u_B – u_A). They are different, since R* ≠ R.
Fig. 2: Different distances between the conjugated points (A, B) and (A, B).**
… If we assume that the coordinates (t, x, y, z) and (t*, x*, y*, z*) describe two observers located in the folds F and F**, they are like two spectators watching the same movie in two different rooms, but:
- the screens have different sizes (R and R**);
- the order of events is opposite (t and t* have opposite signs);
- what is "right" on one screen is "left" on the other (enantiomorphy).
This is an extension of Sakharov's initial idea ([5], [6], [7] and [8]), with different spatial scales.