twin universe cosmology, ghost matter–matter, astrophysics.
- The geometrical framework. The matter era and the Newtonian approximation.
(p1)
Commentary:
This work is based on the system of two field equations: (1)
(2)
... At the time this text was written, a model describing the radiation era, "with variable constants," already existed. However, since the referee of A & A did not comment on this part, which is the subject of paper 6, we preferred to revert to the simpler version (1) + (2). This version obviously allows matching with the standard model when radiation is present, resulting in a model that becomes "twice the standard model." However, the model then suffers from a sign reversal. Not only does it lose some of its elegance, but it presents the following peculiarity: when photons transform into matter and vice versa, or ghost photons transform into a pair of ghost matter and anti-ghost matter, their contribution to the field changes sign. The model with variable constants, applied to the radiation era, allows us to return to the original system.
(6)
(7)
... However, without this sophistication, this system of equations cannot describe the radiation era. Indeed, with variable constants, it produces, for R = R*, the trivial solution R » R* » t. An expansion then far too slow, for example, to interrupt primordial nucleosynthesis producing helium from primitive hydrogen, and ghost helium from primitive ghost hydrogen. Thus, all matter in our universe would be converted into helium.
... The analysis of the solution reveals an instability between the two expansions R(t) and R*(t) (here using the same time variable). The ghost universe somehow "propels" our universe forward, behaving, incidentally, like a kind of "cosmological constant." It is not then "the repulsive power of the vacuum," but rather "the repulsive power of the ghost universe."
... The shape of the curves in Figure 1, particularly the ratio R/R*, at a time assumed to be our present, depends on arbitrary choices of initial conditions. Different choices of initial conditions would lead to different ratios R/R*, and thus to different ratios r*/r. This is an ad hoc ratio, allowing us to match the result obtained in 1994 regarding the Hubble constant. Our model, like the one relying on the Hubble constant, is also "of variable geometry," with suitably chosen initial conditions leading to R(t) profiles yielding a larger universe age. Thus, in the work mentioned, we can multiply the universe's age by a factor of 1.6 and, starting from a Hubble constant equal to 50, arrive at an age of 15 billion years. But today, this seems less urgent. Indeed, the analysis of data collected by the Hipparcos satellite appears to have increased the calibration of Cepheid distances—the gold standard for distance measurement. Conversely, theorists have done their best to shorten the age of the oldest stars in our galaxy, based on the analysis of globular clusters and their relaxation state. Thus, "everything has now fallen into place." A sigh of relief: "the alarm was hot."
... Is the matter settled? It is perhaps too early to tell. Nevertheless, should the need arise, the ghost matter–matter model is available to extend the universe's age at will, just like the cosmological constant...
Ghost matter–matter astrophysics.
1. The geometrical framework. The matter era and the Newtonian approximation. (p1)
Ghost matter–matter astrophysics.
- The geometrical framework. The matter era and the Newtonian approximation. ** Jean-Pierre Petit and P. Midy** Observatory of Marseille, France
** ** ** **** **** **** **** ** ** **** --- **
... We study a system of massive particles involving both attractive and repulsive forces, corresponding to a two-fold geometry. The geometrical framework is specified, as well as a cosmological model for the matter-dominated era. Under conditions of small curvature and low velocities, Newton's law and the Poisson equation are derived (Newtonian approximation), justifying the chosen interaction law.
1) Geometrical framework.
** ...** In the previous paper, we explored the phenomenological aspects of a two-population system whose dynamics involves both attractive and repulsive forces. The geometric framework was briefly presented. Let us return to this question.
... We assume that the geometry of the Universe corresponds to a two-fold covering of a four-dimensional manifold M4. We call these adjacent sheets F and F*. M4 is a set of points. We can describe these points in an arbitrary coordinate system {z i}. M and M* being the corresponding points of the sheets F and F*, they are described by the same set of coordinates and are linked by this involutive mapping. We assume that sheet F, filled with ordinary matter and ordinary photons, is ours, and call sheet F* the ghost sheet, assumed to be filled with ghost matter and ghost photons (in the previous paper, we called it "repulsive dark matter," but this name no longer seems suitable for ghost matter, which attracts ghost matter). The manifold M4 can be considered a "skeleton manifold," as we use it to construct the involutive mapping linking M and M*. We will say these points are adjacent or conjugate. We introduce two metrics g and g* and assume they describe the geometries of the two sheets. We assume they are both Riemannian, with the same signature (+ - - -). The physics in the two sheets is identical, and Special Relativity holds in both. We assume that light follows null geodesics in each sheet. However, due to geometric constraints, light cannot pass from one sheet to the other.
The set of coupled field equations governing the system is a free choice. In the previous paper, we chose: (1)
(2)
which led to a sign-flip problem when matter was converted into radiation and vice versa in the two sheets. Here, we prefer to choose: (3)
(4)
S and S* are two geometric tensors constructed from the two Riemannian metrics g and g*. In the right-hand sides, they are tensors describing the energy-matter content. The subscript r refers to radiation (and ghost radiation), and the subscript m to matter (and ghost matter). With: (5)
we obtain simply: (6)
(7)
which means that: (8)
S* = - S
Consequently, the Riemann curvatures are opposite: (9)
R* = - R
and we call this conjugated geometries. Obviously, (8) does not imply g* = - g, due to the nonlinearity of the equations. In classical General Relativity, local curvature is positive or zero. Here, we allow curvature to be positive, zero, or negative in both sheets. The immediate question is: does the system (6) + (7) possess nontrivial solutions? In the following, we will develop a conjugated Robertson-Walker solution, but we will show in a subsequent paper that it also admits exact nonhomogeneous solutions.
... The system (6) + (7) corresponds to references [1] and [2]. In reference [2], we presented a cosmological model with "variable constants." We now believe, as will be developed in a future paper, that such conditions refer to the radiation era. During this era, the constants of physics—the masses, Planck's constant h, the speed of light c, the gravitational constant G, and the electromagnetic constants—vary over time. In this subsequent paper, we assume these constants depend on the electromagnetic energy density. When the radiation era ends and matter dominates, these constants become absolute constants, and that will be the subject of the present paper, devoted to describing the matter era.
We have a common coordinate system applicable to both sheets:
(10)
{ z ° , z 1 , z 2 , z 3 } = { t , u , q , j }
Left: Cartesian coordinates; right: polar coordinates.
{z 1 , z 2 , z 3 } and { u , q , j } are spatial markers. z ° = t is the time marker. We take it as a dimensionless quantity. From this set, we define dimensional coordinates applicable to both sheets. Introduce two characteristic times T and T* (positive absolute constants) and (a priori distinct) speeds of light c and c* (here considered as absolute constants). We associate the following coordinate set: (11)
{ t , x 1 , x2 , x 3 } = { t , r , q , j }
to sheet F, and the following set: (12)
{ t* , x* 1 , x* 2 , x* 3 } = { t* , r* , q , j }
to sheet F*. Both are linked to (10) via: (13)
t = T t
t* = - T* t
(14)
i¹0 xi = cT z i
xi = - cT* z i
(13) means that the time arrows are opposite; (14) means that the two sheets are considered enantiomorphic. (14) s = cT s
s* = - cT s (16)
R = cT R
R* = cT R*