Mathematical Principles of Theoretical Physics

7.6. THEORY OF DARK MATTER AND DARK ENERGY 475

then, from (7.6.5), (7.6.7) and (7.6.9)-(7.6.11) we can deduce that

(7.6.12) (R′)^2 φ′= 0.

Denoteφ=φ′′, by (7.6.12), the equations (7.6.5), (7.6.7) and (7.6.10) can be rewritten in
the form

(7.6.13)

R′′=−

4 πG

3

(

ρ+

3 p

c^2

+

φ

8 πG

)

R,

(R′)^2 =

1

3

( 8 πGρ+φ)R^2 −c^2 ,

φ′+

3 R′

R

φ=−

24 πG

c^2

R′

R

p.

Only two equations in (7.6.13) are independent. However, there are three unknown func-
tionsR,φ,pin (7.6.13). Hence, we need to add an additional equation, the equationof state,
as follows:

(7.6.14) p=f(ρ,φ).

Based on Theorem7.27, the model describing the static Universe is the equation (7.6.14)
together with the stationary equations of (7.6.13), which are equivalent to the form

(7.6.15)

φ=− 8 πG

(

ρ+

3 p

c^2

)

,

p=−

c^4

8 πGR^2

.

The equations (7.6.14) and (7.6.15) provide a theoretic basis for the static Universe, in-
cluding the dark energy.
Now, we need to determine the explicit expression for the equation (7.6.14) of state. It is
natural to postulate that the equation of state is linear. Hence, (7.6.14) can be written as

(7.6.16) p=

c^2

G

(α 1 φ−α 2 Gρ),

whereα 1 andα 2 are nondimensional parameter, which will be determined by the observed
data.
The equations (7.6.15) and (7.6.16) are the PID cosmological model, where the cosmo-
logical significants ofR,p,φ,ρare as follows:

(7.6.17)

R the cosmic radius (of the 3D spherical universe),

p the negative pressure, generated by the repulsive aspect ofgravity,

φ represents the dual gravitational potential,

ρ the cosmic density, given by

3 M

4 πR^3

=

Mtotal

π^2 R^3

,

whereMandMtotalare as in Remark7.26.