and Cosmology

4.2 An Expanding Universe
in which v is the radial velocity of a source at distance D
from us. Therefore, setting t = t 0 and H 0 ≡ H(t 0 ),(4.8)
is simply the Hubble law, in other words, (4.8) is a generalization
of (4.9) for arbitrary time. It expresses the
fact that any observer expanding with the sphere will
observe an isotropic velocity field that follows the Hubble
law. Since we are observing an expansion today –
sources are moving away from us – we have H 0 > 0,
and ȧ(t 0 )>0.
4.2.3 Dynamics of the Expansion
The above discussion describes the kinematics of the
expansion. However, to obtain the behavior of the function
a(t) in time, and thus also the motion of comoving
observers and the time evolution of the density of the
sphere, it is necessary to consider the dynamics. The
evolution of the expansion rate is determined by selfgravity
of the sphere, from which it is expected that it
will cause a deceleration of the expansion.
Equation of Motion. We therefore consider a spherical
surface of radius x at time t 0 and, accordingly, a radius
r(t) = a(t) x at arbitrary time t. The mass M(x)
enclosed in this comoving surface is constant in time,
and is given by
M(x) = 4π 3 ρ 0 x 3 = 4π 3 ρ(t) r3 (t)
= 4π 3 ρ(t) a3 (t) x 3 , (4.10)
where ρ 0 must be identified with the mass density of
the Universe today (t = t 0 ). The density is a function
of time and, due to mass conservation, it is inversely
proportional to the volume of the sphere,
ρ(t) = ρ 0 a −3 (t). (4.11)
The gravitational acceleration of a particle on the spherical
surface is GM(x)/r 2 , directed towards the center.
This then yields the equation of motion of the particle,
¨r(t) ≡ d2 r
dt 2 =−GM(x) r 2 =− 4πG ρ 0 x 3
3 r 2 , (4.12)
or, after substituting r(t) = xa(t), an equation for a,
ä(t) = ¨r(t)
x
=−4πG 3
ρ 0
a 2 (t) =−4πG ρ(t) a(t).
3
(4.13)
It is important to note that this equation of motion does
not dependent on x. The dynamics of the expansion,
described by a(t), is determined solely by the matter
density.
“Conservation of Energy”. Another way to describe
the dynamics of the expanding shell is based on the
law of energy conservation: the sum of kinetic and potential
energy is constant in time. This conservation of
energy is derived directly from (4.13). To do this, (4.13)
is multiplied by 2ȧ, and the resulting equation can be
integrated with respect to time since d(ȧ 2 )/dt = 2ȧä,
and d(−1/a)/dt = ȧ/a 2 :
ȧ 2 = 8πG
3 ρ 1
0
a − Kc2 = 8πG
3 ρ(t) a2 (t) − Kc 2 ;
(4.14)
here, Kc 2 is a constant of integration that will be interpreted
later. After multiplication with x 2 /2, (4.14) can
be written as
v 2 (t)
2 − GM x2
=−Kc2
r(t) 2 ,
which is interpreted such that the kinetic + potential
energy (per unit mass) of a particle is a constant
on the spherical surface. Thus (4.14) in fact describes
the conservation of energy. The latter equation also
immediately suggests an interpretation of the integration
constant: K is proportional to the total energy of
a comoving particle, and thus the history of the expansion
depends on K. The sign of K characterizes the
qualitative behavior of the cosmic expansion history.
• If K < 0, the right-hand side of (4.14) is always positive.
Since da/dt > 0 today, da/dt remains positive
for all times or, in other words, the Universe will
expand forever.
• If K = 0, the right-hand side of (4.14) is always positive,
i.e., da/dt > 0 for all times, and the Universe
will also expand forever, but in a way that da/dt → 0
for t →∞ – the limiting expansion velocity for
t →∞is zero.
• If K > 0, the right-hand side of (4.14) vanishes if
a = a max = (8πGρ 0 )/(3Kc 2 ). For this value of a,
da/dt = 0, and the expansion will come to a halt.
After that, the expansion will turn into a contraction,
and the Universe will re-collapse.
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