476 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
Here, we remark that in the classical Einstein field equations whereφ=0, the relation
(7.6.9) still holds true, by which we can deduce thatp=0. It implies that the Friedmann
equations has no stationary solution fork 6 =0, and has only stationaryR=0 fork=0.
Therefore, the PID gravitational theory is essential for establishing a static cosmological
model.
Theory of dark energy
In the static cosmology, dark energy is defined in the following manner. LetEobbe the
observed energy, andRbe the cosmic radius. We define the observable mass and the total
mass as follows:
Mob=Eob
c^2(7.6.18) ,
MT=
Rc^2
2 G(7.6.19).
IfMT>Mob, then the difference
(7.6.20) ∆E=ET−Eob
is called the dark energy.
The CMB measurement and the WMAP analysis indicate that the difference∆Ein (7.6.20)
is positive,
∆E> 0 ,
which is considered as another evidence for the presence of dark energy.
From the PID cosmological model (7.6.15)-(7.6.17), we see that the dark energy∆Ein
(7.6.20) is essentially due to the dual gravitational potentialφ. In fact, we infer from (7.6.15)
that
(7.6.21)
φ= 0 ⇔ R= 2 MobG/c^2 (i.e.∆E= 0 ),
φ> 0 ⇔ ∆E> 0.Hence, dark energy is generated by the dual gravitational field. This fact is reflected in the
PID gravitational force formula derived in subsections hereafter.
If we can measure precisely, with astronomical observations, the energy (7.6.18) and the
cosmic radiusR(i.e.MTof (7.6.19)), then we can obtain a relation between the parameters
α 1 andα 2 in (7.6.16). In fact, we deduce from (7.6.15) and (7.6.16) that
(7.6.22) ρ+
βp
c^2= 0 , β=1 + 24 π α 1
α 2 + 8 π α 1.
As we get
(7.6.23)
∆M
Mob=
MT−Mob
Mob=k (k> 0 ).Then by (7.6.22) and
ρ=3 Mob
4 πR^3, p=−c^4
8 πGR^2