Mathematical Principles of Theoretical Physics

460 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

whereR,ρ,pare the unknown functions.
Equations (7.5.15)-(7.5.17) are called the Friedmann cosmological model, from which we
can derive the Newtonian cosmology equations (7.5.14). To see this, by (7.5.15) and (7.5.16),
we have

(7.5.18)

( ̇

R

R

) 2

=−

kc^2

R^2

+

8 πG

3

ρ.

By the approximatep/c^2 ≃0, (7.5.12) follows from (7.5.17). Then we deduce (7.5.14) from
(7.5.18) and (7.5.12).

From the equation (7.5.18), the densityρccorresponding to the casek=0 is

(7.5.19) ρc=

3

8 πG

( ̇

R

R

) 2

=

3

8 πG

H^2 ,

whereH=R ̇/Ris the Hubble constant, and by (7.5.2) we have

(7.5.20) ρc= 10 −^26 kg/m^3.

Thus, by the Friedmann model we can deduce the following conclusions.

Conclusions of Friedmann Cosmology 7.23.1) By (7.5.18) we can see that

(7.5.21)

ρ>ρc ⇔k= 1 the Universe is closed:M=S^3 ,

ρ=ρc ⇔k= 0 the Universe is open:M=R^3 ,

ρ<ρc ⇔k=− 1 the Universe is open:M=L^3.

2) Let E 0 be the total kinetic energy of the Universe, M is the mass, then we have

(7.5.22) E 0 =











3

5

GM^2

R

for k= 0 ,

2

3 π

GM^2

R

−

1

2

Mc^2 for k= 1 ,

where the first term represents the total gravitational bound potential energy, and the second
term is the energy resisting curvature tensor.
3) By (7.5.18),R ̇6≡ 0 , and consequently the universe is dynamic.
4) By (7.5.15),R ̈< 0 , the dynamic universe is decelerating.

9.The Lemaˆıtre cosmology. Consider the Einstein gravitational field equations with the
cosmological constantΛterm:

(7.5.23) Rμ ν=−

8 πG

c^4

(Tμ ν−

1

2

gμ νT)+Λgμ ν, Λ> 0 ,