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| ![]() | © CAMBRIDGE UNIVERSITY PRESS 2000 |
7.2 Expansion of a homogeneous Universe
Because the cosmic background radiation is highly uniform, we infer that the Universe is isotropic - it is the same in all directions. We believe that on a large scale the cosmos is also homogeneous - it would look much the same if we lived in any other galaxy. Then, it can be shown that the length s of a path linking any two points at time t is given by integrating the expression
![]() | (7.5) |
where ,
,
are spherical polar coordinates in an
expanding curved space. Apart from their small peculiar speeds,
galaxies remain at points with fixed values of those coordinates.
The coordinate
is
dimensionless, while the distance between
galaxies expands according to the scale length
(t).
Because they follow the galaxies as the Universe expands,
,
,
are called comoving
coordinates.
The origin
= 0 looks like a
special point, but in fact it is
not. Just as at the Earth's poles where lines of longitude converge,
the curvature here is the same as everywhere else, and we
can equally well take any point to be
= 0.
The constant k specifies the curvature of space. For
k = 1, the Universe is closed, with positive curvature
and finite volume, analogous to the surface of a sphere;
(t) is
the radius of curvature.
If k = -1, we have an open Universe, a negatively curved space of
infinite volume, while k = 0 describes familiar unbounded flat space.
Near the origin, where
>> 1,
the formula for
s
is almost the same for all values of k; on a small enough scale,
curvature does not matter. If we look at a tiny region, the relationships
among angles, lengths, and volumes will be the same as they are in flat space.
Problem 7.7: In ordinary three-dimensional space, using
cylindrical polar
coordinates we can write the distance between two nearby points
(R,
where |
Further reading: For further discussion of cosmology in curved spacetime, see Chapters 6 and 7 of M.V. Berry, 1989, Principles of Gravitation and Cosmology, 2nd edition (Institute of Physics Publishing, London).
According to general relativity, Hubble's law is just one symptom of
the expansion of curved space; the distance d
between galaxies with fixed comoving coordinates
,
,
expands proportionally to
(t).
Since d
s,
Equation 7.5 tells us that the two systems are
carried away from each other at a speed
![]() | (7.7) |
here H(t) is the Hubble parameter, which presently has the value H0.
Relativity tells us that the distance between two events happening
at different times and in different places depends on the motion of
the observer. But all observers will measure the same proper
time along a path through
space and time connecting the events, given by integrating
![]() | (7.8) |
Light rays always travel along paths of zero proper time,
= 0.
If we place ourselves at the origin of coordinates,
then the light we receive from a galaxy at comoving distance
e has followed the
radial path
![]() | (7.9) |
it covers less comoving distance per unit of time as the scale of the Universe grows. We can integrate this equation for a wavecrest that sets off at time te, arriving at our position at the present time t0:
![]() | (7.10) |
Suppose that e is
the wavelength
of the emitted radiation; then the following wavecrest sets off later,
by a time
te
= c
e.
We receive it with wavelength
obs, at time
t = c
obs after the
previous crest. But the galaxy's comoving position
e, and the
integral on the right of Equation 7.10, have not changed;
so the left side also stays constant:
![]() | (7.11) |
as long as t <<
(t) /
(t).
The wavelength grows along with the scale length
(t),
while the frequency, momentum, and energy of each photon decay
proportionally to 1 /
(t).
The measured redshift z of a distant galaxy
tells us how much expansion has taken place since the time
te when
its light was emitted. This is the cosmological redshift of
Equation 1.28:
![]() |
To describe processes in the expanding Universe, we can use
redshift as a substitute for time: z(t) is the redshift of light
emitted at time t, reaching us now at time t0.
The time
corresponding to a given redshift depends on the function
(t);
once we know this, Equation 7.10 tells us the comoving
distance
e from
which the light would have started.
The rate at which the Universe expands is set by the gravitational pull
of matter and energy within it. We first use Newtonian physics to
calculate the expansion, and then discuss how general relativity
modifies the result. Consider a small sphere of radius
r, at a time t when our homogeneous Universe has density
(t);
we take r >>
(t), so that we can
neglect the curvature of space.
Everything is symmetric about the origin r = 0, so we appeal to
Newton's first theorem in Section 3.1: the gravitational force at
radius r is determined only by the mass
M (< r) within the
sphere. If our sphere is large enough that
gas pressure forces are much smaller than the pull of gravity (see
Section 7.4 below), then Equation 3.20
gives the force on a gas cloud of mass m at that radius:
![]() | (7.12) |
Our sphere of matter is expanding along with the rest of the Universe,
so its radius r(t)
(t). The mass
m of the cloud cancels out, giving
![]() | (7.13) |
the higher the density, the more strongly gravity slows the expansion.
Nothing enters or leaves our sphere, so the mass within it does not
change: (t)
3 (t) is
constant. Multiplying
by
(t) tells us
how the kinetic energy decreases as the sphere expands:
![]() | (7.14) |
where the time t0 refers to the present day. Integrating, we have
![]() | (7.15) |
where k is a constant of integration. Although we derived it using Newtonian theory, Equation 7.15 is also correct in general relativity, which tells us that the constant k is the same one as in Equation 7.15. Since the pressure p of a gas contributes to its energy and hence to its gravitational force, general relativity amends Equation 7.13 to read
![]() | (7.16) |
Equation 7.15 and 7.16 describe the Friedmann models, telling us
how the contents of the Universe determine its expansion.
For cool matter, where the sound speed cs << c,
we have
p ~
cs2, and can safely neglect the pressure term.
But for radiation, and particles moving almost at the speed of light,
pressure is important: p
c2
/ 3, where
is now the energy
density divided by c2.
For any mixture of matter and radiation, the term
+ 3p /
c2 must be
positive, so the expansion always slows down. If the Universe
enters a contracting phase, the collapse speeds up as time goes on.
The inflation theory postulates a vacuum
energy, contributing density
VAC =
/ 8
G, and
negative pressure or tension
pVAC = -
c2 / 8
G.
The vacuum energy is now very small, but there are reasons to believe
that very early, at 10-34 s
t
10-32 s,
VAC might
have been much larger than the density of matter or radiation.
During this period,
(t)
inflated, growing
exponentially by a factor ~ e100. The almost uniform cosmos that
we now observe would have resulted from the expansion of a tiny
near-homogeneous region. Because this patch was so small, the curvature
of space within it would be negligible; hence devotees
of inflation expect our present Universe to be nearly flat, with k = 0.
Problem 7.8: By substituting into Equation 7.16, show that
during inflation, expansion proceeds according to
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Since (t)
2 (t)
decreases as
(t)
grows, in a closed
Universe with k = 1 the right side of Equation 7.15 becomes negative
at large
. But
2 cannot be
negative, so the distance between galaxies does not grow forever;
(t) attains
some maximum before shrinking again. In an open Universe with k
0, there is no such limit;
expansion continues indefinitely and
(t) grows without
bound. In the borderline case k = 0,
Equation 7.16 requires that the density
is equal to the
critical value
![]() | (7.17) |
At the present day, the critical density
crit
(t0) = 3.3 x 1011 h2
M
Mpc-3: see Equation 1.24. We can measure the mass content of
the Universe
as a fraction of the critical density, defining the
density parameter
(t) as
![]() | (7.18) |
and writing 0 for
its present-day value. Equation 7.15 then becomes
![]() | (7.19) |
If the Universe is closed, with k = 1, then
(t) > 1 and
the density always exceeds the critical value, while
if k = -1, we always have
(t) < 1.
If the density is now equal to the critical value, so that
0 =
1, then
(t) = 1 at all times.
Most astronomers would agree that 0.05
0
1;
our real Universe is unlikely to be much denser than the
critical value.
In its early stage, the Universe was radiation dominated. It was
extremely hot, and its energy was almost entirely due to radiation and
relativistic particles; these are particles moving close to
light speed, so that their energy, momentum and pressure are related
in the same way as for photons. The energy density
r
c2 of a gas of photons decreases as
-4 (t): the
number per unit volume is proportional to 1 /
3 (t),
while by Equation 7.11 the
energy of each photon falls as 1 /
(t). As expansion
proceeds, the
density
m
of matter decreases more slowly, since
m (t)
-3 (t). So
at late times, its energy density
m
(t) c2 exceeds that in radiation. Since the
time teq
of matter-radiation equality, about a million years
after the Big Bang, the Universe has been matter dominated.
Problem 7.9: The cosmic background radiation is now a blackbody of
temperature T = 2.73 K: show that its energy density
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To measure the expansion of the Universe relative to
the present day, we define the dimensionless scale factor
a(t)
(t) /
(t0).
Using Equation 7.19 to rewrite
(t0) in
terms of H0 and
0, Equation 7.15 becomes
![]() | (7.20) |
Most of the structure of galaxy clusters and
voids that we see today developed after the Universe became
matter dominated. In this phase, the density falls as
a-3, and
from Equation 1.28, 1 + z = 1 / a(t); so Equation 7.20 reads
![]() | (7.21) |
If the density is exactly at the critical value,
with 0 = 1 and
k = 0,
we have
a1/2, and
a(t)
t2/3.
Since the Universe is not completely empty,
0 > 0.
Hence the first term on the right of Equation 7.21 will
be large when a(t) is small, soon after the Big Bang;
we again have a(t)
t2/3. Thus as t -> 0 in
the early Universe, space is nearly flat and
-> 1.
If the density is below critical, with
(t) < 1, then at late
times the second term of Equation 7.21 takes over, giving a(t)
t.
Expansion then proceeds almost at a constant speed,
barely slowed by the gravity of matter.
Using the redshift z to specify time indirectly,
in a matter-dominated Universe we can write
![]() | (7.22) |
We believe that 0
0.05; so the Universe
expanded with a(t)
t2/3 from the time of
matter-radiation equality until at least z ~ 18.
Problem 7.10: Use Equation 7.21 to show that if
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Problem 7.11: While blackbody radiation and relativistic particles
provide most of the energy density, so
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Problem 7.12: Even if the cosmos has infinite volume, we can
observe only a
finite portion. From Equation 7.10, light reaching us at
At early times the Universe was radiation dominated, with
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