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2004/11/07 | F-L宇宙模型[F-L Cosmological Model]
类别(Ω〖物理〗)
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发表于 13:53
A cosmological model formulated by
Friedmann
in 1922 and independently by
Lemaître
(1927). It assumes a homogeneous and isotropic
universe
with positive
cosmological constant
and
expansion parameter
a
governed by the equations of motion
(1)
and
(2)
where
is the total mass-energy density,
is the mass density, and
c
is the
speed of light
. These equations were originally rigorously derived from
general relativity
, but can also be rigorously derived from Newtonian mechanics and thermodynamics for small length scales, i.e.,
r
such that
(3)
where
M
is the mass
The unintegrated form of (1) is
(4)
(5)
so
(6)
Adding a term taking the
cosmological constant
into account,
(7)
The integrated form of (1) is
(8)
(9)
(10)
(11)
(12)
(13)
Equation (2) can be derived by assuming the expansion of the
universe
is
adiabatic
,
(14)
where
is the heat change (an inexact differential),
dE
is the energy change,
P
is the pressure, and
V
is the volume. Then
(15)
(16)
and
(17)
so
(18)
(19)
(20)
But, since
(21)
(22)
The time evolution for a large
cosmological constant
(23)
is
(24)
(25)
(26)
(27)
(28)
For a matter dominated
universe
at initial time with
expansion parameter
a
,
(29)
The redshift at a time
with
expansion parameter
is defined as
(30)
Plugging in the curvature
(31)
yields
(32)
Density goes as
(33)
so plugging (33) into (32)
(34)
(35)
(36)
For a matter dominated
universe
,
where
q
is the deceleration parameter, so (36) simplifies to
(37)
The static solution is
(38)
(39)
(40)
(41)
If
P
= 0, then
(42)
where
H
is the
Hubble constant
. Also
(43)
where
q
is the
deceleration parameter
.
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