Chapter 29
The Hubble Expansion
The observational characteristics of the Universe coupled
with theoretical interpretation to be discussed further in
subsequent chapters, allow us to formulate a standard picture of the nature of our Universe.
29.1 The Standard Picture
The standard picture rests on but a few ideas, but they have profound
significance for the nature of the Universe.
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CHAPTER 29. THE HUBBLE EXPANSION
29.1.1 Mass Distribution on Large Scales
Observations indicate that the Universe is homogeneous
(no preferred place) and isotropic (no preferred direction),
when considered on sufficiently large scales.
• When averaged over distances of order 50 Mpc, the
fluctuation in mass distribution is of order unity,
δ M/M ≃ 1
• When averaged over a distance of 4000 Mpc (comparable to the present horizon), δ M/M ≤ 10−4
• Thus, averaged over a large enough volume, no part
of the Universe looks any different from any other
part.
• The idea that the Universe is homogeneous and
isotropic on large scales is called the cosmological
principle.
The cosmological principle as implemented
in general relativity is the fundamental theoretical underpinning of modern cosmology.
29.1. THE STANDARD PICTURE
The cosmological principle should not be confused with
the perfect cosmological principle, which was the underlying idea of the steady state theory of the Universe.
• In the perfect cosmological principle, the Universe is
not only homogeneous in space but also in time.
• Thus it looks the same not only from any place, but
from any time.
• This idea once had a large influence on cosmology
but is no longer considered viable because it is inconsistent with modern observations that show a Universe evolving in time.
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CHAPTER 29. THE HUBBLE EXPANSION
29.1.2 The Universe Is Expanding
Observations indicate that the Universe is expanding; the
interpretation of general relativity is that this is because
space itself is expanding
• The distance ℓ between conserved particles is changing according to the Hubble law
v≡
dℓ
= H0 ℓ,
dt
deduced from redshift of light from distant galaxies.
• H0 is the Hubble parameter (or Hubble “constant”,
but it changes with time; the subscript zero indicates
that this is the value at the present time).
• The Hubble parameter can be determined by fitting
the above equation to the radial velocities of galaxies
at known distances.
• The uncertainty in H0 is sometimes absorbed into a
dimensionless parameter h by quoting
H0 = 100h km s−1 Mpc−1 = 3.24 × 10−18h s−1,
where h is a scaling parameter of order one.
• The currently accepted value of the Hubble constant
is H0 = 72 ± 8 km s−1 Mpc−1 , corresponding to h =
0.72.
• Note: units of H0 are actually (time)−1.
29.1. THE STANDARD PICTURE
925
• We may define a Hubble length LH through
LH =
c
≃ 4000 Mpc.
H0
• Thus, for a galaxy lying a Hubble length away from us,
v=
c
dℓ
= H0
= c,
dt
H0
• This implies that the recessional velocity of a galaxy further
away than LH exceeds the speed of light, if the observed redshifts
are interpreted as Doppler shifts.
• We shall find that the redshift of the receding galaxies is not a
Doppler shift caused by velocities in spacetime, but is a consequence of the expansion of space itself, which stretches the
wavelengths of all light.
• The light speed limit of special relativity applies to velocities in
spacetime; it does not apply to spacetime itself.
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CHAPTER 29. THE HUBBLE EXPANSION
It is important to understand that local objects are not partaking of the general Hubble expansion.
• The Hubble expansion is is not caused by a force. It
only occurs when forces between objects are negligible.
• Smaller objects, such as our bodies, are held together
by chemical (electrical) forces. They do not expand.
• Larger objects like planets, solar systems, and galaxies are also held together by forces, in this case gravitational in origin. They generally do not expand with
the Universe either.
• It is only on much larger scales (beyond superclusters of galaxies) that gravitational forces among local objects are sufficiently weak to cause negligible
perturbation on the overall expansion.
29.1. THE STANDARD PICTURE
927
29.1.3 The Expansion is Governed by General Relativity
• It is possible to understand much of the expanding Universe using only Newtonian physics and insights borrowed from relativity.
• However, in the final analysis there are serious technical and
philosophical difficulties that eventually arise and that require
replacement of Newtonian gravitation with the Einstein’s general theory of relativity for their resolution.
• Central to these issues is the understanding of space and time in
relativity compared with that in classical Newtonian gravitation.
– In relativity, space and time are not separate but enter as a
unified spacetime continuum.
– Even more fundamentally, space and time in relativity are
not a passive background upon which events happen.
– Relativistic space and time are not “things” but rather are
abstractions expressing a relationship between events.
Thus, in this view, space and time do not have a separate existence from events involving matter and energy.
• On fundamental grounds, the gravitational curvature radius of
the Universe could be (it actually isn’t) comparable to the radius
of the visible Universe → general relativity.
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CHAPTER 29. THE HUBBLE EXPANSION
29.1.4 There Is a Big Bang in Our Past
Evidence suggests that the Universe expanded from an initial condition of very high density and temperature.
• This emergence of the Universe from a hot, dense
initial state is called the big bang.
• The popular (mis)conception that the big bang was a
gigantic explosion is in error because it conveys the
idea that it happened in space and time, and that the
resulting expansion of the Universe is a consequence
of forces generated by this explosion.
• The general relativistic interpretation of the big bang
is that it did not happen in spacetime but that space
and time themselves are created in the big bang.
• Thus, questions of “what happened before the big
bang?” and “what is the Universe expanding into?”
become meaningless because these questions presuppose the existence of a space and time background
upon which events happen.
• The big bang should be viewed not as an explosion
but as an initial condition for the Universe.
• (Very) loosely we may view the big bang as an “explosion” because of the hot, dense nature of the initial state. But then we should view the explosion as
happening at all points in space, so that it makes no
sense to talk about a “center” for the big bang.
29.1. THE STANDARD PICTURE
General relativity implies that the initial state was a spacetime singularity.
• Whether general relativity is correct on this issue will
have to await a full theory of quantum gravitation,
since general relativity cannot be applied too close
to the initial singularity (on scales below the Planck
scale) without incorporating the principles of quantum mechanics.
• However, for most (but not all!) issues in cosmology
the question of whether there was an initial spacetime
singularity is not relevant.
• For those issues, all that is important is that once the
Universe expanded beyond the Planck scale it was
very hot and very dense.
• This hot and dense initial state is what we shall mean
in simplest form when we refer to the big bang.
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CHAPTER 29. THE HUBBLE EXPANSION
29.1.5 Particle Content Influences Evolution of the Universe
The Universe contains a variety of particles and their associated fields that influence strongly its evolution.
• The “ordinary” matter composed of things that we
find around us is generally termed baryonic matter.
• Baryons are the strongly interacting particles of halfinteger spin such as protons and neutrons.
• Although baryonic matter is the most obvious matter
to us, data indicate that only a small fraction of the
total mass in the Universe is baryonic.
• The bulk of the mass in the Universe appears to be
in the form of dark matter, which is easily detected
only through its gravitational influence.
• We don’t know what dark matter is, but there is good
reason to believe that it is primarily composed of
as-yet undiscovered elementary particles that interact only weakly with other matter and radiation.
• There is also growing evidence that the evolution of
the Universe is strongly influenced by dark energy,
which permeates even empty space and causes gravity to effectively become repulsive.
• We do not know the origin of dark energy but a reasonable guess is that it is associated with quantummechanical fluctuations of the vacuum, or possibly
with an as-yet undiscovered elementary particle field.
29.1. THE STANDARD PICTURE
931
A fundamental distinction for particles and associated fields is whether
they are massless or massive.
• Lorentz-invariant quantum field theories require that massless
particles must move at light speed and that particles with finite
mass must move at speeds less than that of light.
• Therefore, massless particles like photons, gravitons, and gluons, and nearly massless particles like neutrinos, are highly relativistic. Roughly, particles with rest mass m are non-relativistic
at those temperatures T where kT << mc2
– Electrons have a rest mass of 511 keV and they are nonrelativistic at temperatures below about 6 × 109 K.
– Protons have a rest mass of 931 MeV and they remain nonrelativistic up to temperatures of about 1013 K.
– Conversely (massless) photons, gluons, gravitons, and (nearly
massless) neutrinos are always relativistic
• In cosmology, it is common to refer to such massless or nearly
massless particles as radiation.
• Conversely, particles with significant mass have velocities well
below that of light (unless temperatures are extremely high) and
are non-relativistic. In cosmology non-relativistic particles are
termed matter (or dust).
• Non-relativistic matter has low velocity and exerts little pressure
compared with relativistic matter.
The present energy density of the Universe is dominated by nonrelativistic matter and by vacuum energy density (dark energy). However, this was not always the case: the early Universe was dominated
by radiation.
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CHAPTER 29. THE HUBBLE EXPANSION
29.1.6 The Universe is Permeated by a Microwave Background
As noted in the preceding section, radiation plays a role in the evolution of the Universe.
The most important feature of the standard picture, other
than the Hubble expansion, is that the Universe is filled
with a photon background lying in the microwave region
of the spectrum that is extremely smooth and isotropic.
• Any theoretical attempt to understand the standard
picture must as a minimal starting point account for
the expansion of the Universe and for this cosmic microwave background (CMB) that fills all of space.
• Conversely, precision measurements of tiny fluctuations in the otherwise smooth CMB are presently
turning cosmology into a highly quantitative science.
• Although the CMB currently peaks in the microwave
region of the spectrum, its wavelength has been
steadily redshifting since the big bang and it was
originally much higher energy radiation.
• For example, near the time when the temperature
dropped low enough for electrons to combine with
protons the spectrum of what is now the microwave
background peaked in the near-infrared region.
• The CMB accounts for more than 90% of the photon
energy density (less than 10% in starlight).
29.1. THE STANDARD PICTURE
933
Time (Billion Years)
72
80
Scale Factor Relative to Today
60
Now
0
1
60
72
3
5
10
80
12.2
13.5
16.2
Age (109 years)
Figure 29.1: Expansion of the Universe for three values of the Hubble constant
( km s−1 Mpc−1 ). The corresponding Hubble times estimating the age of the Universe are indicated below the lower axis. Redshift is indicated on the right axis.
Hubble Law :
v≡
dℓ
= H0 ℓ
dt
H0 ≃ 3.24 × 10−18h s−1,
The Hubble expansion is most consistently interpreted in terms of an
expansion of space itself.
• Convenient to introduce a scale factor a(t) that describes how
distances scale because of the expansion of the Universe. Hubble’s law for evolution of the scale factor is illustrated in Fig. 29.1.
• The slopes of the straight lines plotted there define the Hubble
constant H0 .
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CHAPTER 29. THE HUBBLE EXPANSION
We shall interpret the Hubble constant as being characteristic of a
space (thus constant for the Universe at a given time) but having possible time dependence as the Universe evolves.
• The subscript zero on H0 is used to denote that this is the value
of the Hubble constant today, in anticipation that the coefficient
governing the rate of expansion changes with time.
• One often refers to the Hubble parameter H = H(t), meaning an
H that varies with time, and to the Hubble constant H0 to mean
the value of H(t) today.
• It is also common to use the term Hubble constant loosely to
mean a parameter that is constant in space but that may change
with time.
Hubble’s original value was H0 = 550 km s−1 Mpc−1.
This is approximately an order of magnitude larger than
the presently accepted value of about 72 km s−1 Mpc−1.
The large revision (which implies a corresponding shift in
the perceived distance scale of the Universe) was because
• Hubble’s original sample was a poorly-determined
one based on relatively nearby galaxies.
• There was confusion over the extra-galactic distance
scale at the time because of issues like misinterpreting types of variable stars and failing to account for
the effect of dust on light propagation.
29.1. THE STANDARD PICTURE
935
Table 29.1: Some peculiar velocities in the Virgo Cluster
Galaxy
Redshift (z)
vr (km s−1 )
IC 3258
−0.001454
−436
M86 (NGC 4406)
−0.000901
−270
NGC 4419
−0.000854
−256
M90 (NGC 4569)
−0.000720
−216
M98 (NGC 4192)
−0.000467
−140
NGC 4318
+0.004086
+1226
NGC 4388
+0.008426
+2528
IC 3453
+0.008526
+2558
NGC 4607
+0.007412
+2224
NGC 4168
+0.007689
+2307
M99 (NGC 4254)
+0.008036
+2411
NGC 4354
+0.007700
+2310
Source: SIMBAD
29.1.7 Redshifts
If a galaxy has a spectral line normally at wavelength λemit that is
shifted to a wavelength λobs when we observe it, the redshift z is
z≡
λobs − λemit
.
λemit
• A negative value of z corresponds to a blueshift.
• A positive value of z corresponds to a redshift.
• Since the Universe is observed to be expanding, the Hubble law
gives rise only to redshifts.
• Thus, any blueshifts correspond to peculiar motion of objects
with respect to the general Hubble flow (Table 29.1).
• “Peculiar” = “a property specific to an object” (6= “strange”).
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CHAPTER 29. THE HUBBLE EXPANSION
The few galaxies observed to have blueshifts are nearby,
in the Local Group or the Virgo Cluster, where peculiar
motion is large enough to partially counteract the overall
Hubble expansion.
• The Andromeda Galaxy (M31), which is part of our
Local Group of galaxies, is moving toward us with
a velocity of about 300 km s−1 and will probably
collide with the Milky Way in several billion years.
• The most extreme blueshifts (negative radial velocities) found in the Virgo Cluster are the largest
blueshifts known with respect to our galaxy.
29.1. THE STANDARD PICTURE
29.1.8 Expansion Interpretation of Redshifts
The redshifts associated with the Hubble law may be approximately viewed as Doppler shifts for small redshifts.
• This interpretation is problematic for large redshifts.
• The Hubble redshifts (large and small) are most consistently interpreted in terms of the expansion of
space, which may be parameterized by the cosmic
scale factor a(t).
• If all peculiar motion is ignored the time dependence
of the expansion is lodged entirely in the time dependence of a(t), and all distances simply scale with this
factor.
• A simple analogy on a 2-dimensional surface will be
exploited in later discussion: distances between dots
placed on the surface of a balloon all scale with the
radius of the balloon as it expands.
• In the general case a(t) may be interpreted as setting
a scale for all cosmological distances.
• In the special case of a closed universe, we may think
of a(t) loosely as a radius for the universe.
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CHAPTER 29. THE HUBBLE EXPANSION
938
As we shall show, light traveling between two galaxies separated by
cosmological distances follows the null curve
c2 dt 2 = a2(t)dr2
where the scale factor a(t) sets the overall scale for distances in the
Universe at time t and r is the coordinate distance. Therefore,
cdt
= dr.
a(t)
Consider a wavecrest of monochromatic light with wavelength λ ′ that
is emitted at time t ′ from one galaxy and detected with wavelength λ0
at time t0 in the other galaxy. Integrating both sides of the above
equation gives
Z t0
Z r
dt
c
=
dr = r.
t ′ a(t)
0
The next wavecrest is emitted from the first galaxy at t = t ′ + λ ′ /c and
is detected in the second galaxy at time t = t0 + λ0/c. For the second
wave crest, integrating as above assuming that the interval between
wavecrests is negligible compared with the timescale for expansion
of the Universe gives
c
Z t0 +λ0 /c
dt
t ′ +λ ′ /c
a(t)
=
Z r
dr = r.
0
Since from above
c
Z t0
dt
t′
a(t)
=r
c
Z t0 +λ0 /c
dt
t ′ +λ ′ /c
a(t)
we may equate the left sides to obtain
Z t0
dt
t′
a(t)
=
Z t0 +λ0 /c
dt
t ′ +λ ′ /c
a(t)
,
=r
29.1. THE STANDARD PICTURE
939
which may be rewritten as
Z t ′ +λ ′ /c
dt
Z
Z t0
t0
dt
dt
=
+
+
a(t)
t ′ +λ ′ /c a(t)
t ′ +λ ′ /c a(t)
{z
} |
{z
}
|
t′
Cancel
t0
a(t)
Cancel
Z t ′ +λ ′ /c
dt
a(t)
t′
Z t0 +λ0 /c
dt
=
Z t0 +λ0 /c
dt
a(t)
t0
.
Because the interval between wave crests is negligible compared with
the characteristic timescale for expansion, we may bring the factor
1/a(t) outside the integral to obtain
1
a(t ′ )
Z t ′ +λ ′ /c
t′
1
dt =
a(t0 )
Z t0 +λ0 /c
dt
−→
t0
λ ′ a(t ′ )
,
=
λ0 a(t0 )
This demonstrates explicitly
• the stretching of wavelengths caused by the expansion of the
Universe
• that the cosmological redshift is not a Doppler shift (no velocities appear in the formula).
From the definition for the redshift z,
λ0 − λ ′ λ0 a(t0 )
= ′=
1+z = 1+
λ′
λ
a(t ′ )
−→
z=
a(t0 )
− 1.
a(t ′ )
It is conventional to normalize the scale parameter so that its value in
the present Universe is unity, a(t0 ) ≡ 1, in which case
z=
1
− 1.
a(t ′ )
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CHAPTER 29. THE HUBBLE EXPANSION
Thus the redshift that enters the Hubble law
• depends only on the ratio of the scale parameters at the time of
emission and detection for the light,
1+z =
a(t0 )
.
a(t ′ )
• It is independent of the details of how the scale parameter changed
between the two times.
• The ratio of the scale parameters at two different times is determined by the cosmological model in use.
• Measuring the redshift of a distant object is then equivalent to
specifying the scale parameter of the expanding Universe at the
time when the light was emitted from the distant object, relative
to the scale parameter today.
• Thus, measuring redshifts tests cosmological models.
EXAMPLE: If from the spectrum for a distant quasar one
determines that z = ∆λ /λ = 5, the scale factor of the Universe at the time that light was emitted from the quasar
was equal to 16 of the scale factor for the current Universe:
1
1
a(t ′ )
=
= .
a(t0 ) z + 1 6
29.1. THE STANDARD PICTURE
941
Time (Billion Years)
72
80
Scale Factor Relative to Today
60
Now
0
1
60
72
3
5
10
80
12.2
13.5
16.2
Age (109 years)
The preceding discussion indicates that we may use the
scale factor a(t) or the redshift z interchangeably as time
variables for a universe in which the scale parameter
changes monotonically (compare the right and left axes
of the above figure).
CHAPTER 29. THE HUBBLE EXPANSION
942
Time (Billion Years)
72
80
Scale Factor Relative to Today
60
Now
0
1
60
72
3
5
10
80
12.2
13.5
16.2
Age (109 years)
29.1.9 The Hubble Time
The Hubble parameter has the dimensions of inverse time:
[H0 ] = [km s−1 Mpc−1] = time−1.
Thus, 1/H0 defines a time called the Hubble time τH ,
τH ≡
1
= 9.8h−1 × 109 y.
H0
• If the Hubble law is obeyed with a constant value of H0, the
intercept of the curve with the time axis gives the time when the
scale factor was zero.
• Hence, the value of τH = 1/H0 is sometimes quoted as the age of
the Universe.
29.1. THE STANDARD PICTURE
943
• This is simply a statement that if the expansion rate today is the
same as the expansion rate since the big bang, the time for the
Universe to evolve from the big bang to today is the inverse of
the Hubble constant.
• In general the Hubble time is not a correct age for the Universe
because the Hubble parameter can remain constant only in a Universe devoid of matter, fields, and energy.
• The realistic Universe contains all of these and the expansion of
the Universe is accelerated (positively or negatively, depending
on the details) because of gravitational interactions.
• In later cosmological models we shall see that the age of the Universe may be substantially longer or shorter than τH , depending
on the details of the matter, fields, and energy contained in the
Universe.
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CHAPTER 29. THE HUBBLE EXPANSION
29.1.10 A Two-Dimensional Hubble Expansion Model
Figure 29.2 illustrates a two-dimensional Hubble expansion as viewed
from two different vantage points (Show interactive animation of this).
See: http://csep10.phys.utk.edu/guidry/darkEnergy/universeExpander.html
29.1. THE STANDARD PICTURE
Figure 29.2: The same two-dimensional Hubble expansion as viewed from two
different vantage points.
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CHAPTER 29. THE HUBBLE EXPANSION
946
Figure 29.3: Hubble parameter extracted from observations.
29.1.11 Measuring the Hubble Constant
The Hubble constant may be determined observationally
by measuring the redshift for spectral lines and comparing
that with the distance to objects at a range of distances sufficiently large that peculiar motion caused by local gravitational attraction is small compared with the motion associated with the Hubble expansion.
Figure 29.3 illustrates the determination of the Hubble constant from
a variety of observations. The adopted value is
H0 = 72 ± 8 km s−1 Mpc−1,
corresponding to h = 0.72.
29.2. LIMITATIONS OF THE STANDARD WORLD PICTURE
947
29.2 Limitations of the Standard World Picture
We shall demonstrate that the standard picture has been remarkably
successful in describing many features of our Universe. However,
there are two aspects of this picture suggesting that it is (at best) incomplete:
1. In order to get the big bang to produce the present universe, certain assumptions about initial conditions must be taken as given.
While not necessarily wrong, some of these assumptions seem
unnatural by various standards.
2. As the expansion is extrapolated backwards, eventually one would
reach a state of sufficient temperature and density that a fully
quantum mechanical theory of gravitation would be required.
• This is the Planck era, and the corresponding scales of distance, energy, and time are called the Planck scale.
• Since we do not yet have a consistent theory of quantum
gravity, the presently understood laws of physics may be
expected to break down on the Planck scale.
• Thus the standard picture says nothing about the Universe
at those very early times.
In later chapters we shall address these issues, to consider whether
modifications of the standard picture can alleviate some of these problems.