Chapter 30
Newtonian Gravity and Cosmology
The Universe is mostly empty space, which might suggest that a Newtonian description of gravity (which is valid in the weak gravity limit)
is adequate for describing the large-scale structure of the Universe.
• But whether general relativity effects are important relative to a
Newtonian description may be estimated in terms of the ratio of
an actual radius for a massive object compared with its radius of
gravitational curvature.
• If we apply such a criterion to the entire Universe, reasonable estimates for the mass–energy contained in the Universe indicate
that the actual radius of the known Universe and the corresponding gravitational curvature radius could be comparable.
• Thus, a description of the large-scale structure of the Universe
(cosmology) must be built on a covariant gravitational theory,
rather than on Newtonian gravity.
• Even so, we can understand a substantial amount concerning the
expanding Universe simply by using Newtonian concepts.
949
950
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
Distant
galaxy
Density = ρ
r
Earth
Homogeneous mass
distribution
Figure 30.1: Newtonian model of the expanding Universe.
30.1 Expansion and Newtonian Gravity
Consider the test galaxy illustrated in Fig. 30.1. The gravitational
potential acting on the galaxy is
U=
−GMm
,
r
where m is the mass of the galaxy and
Total mass within sphere = M = 43 π r3ρ ,
which is constant since ρ decreases with time and r increases but the
product ρ r3 is constant. Thus
U = − 34 π Gr2ρ m.
30.1. EXPANSION AND NEWTONIAN GRAVITY
Distant
galaxy
Density = ρ
Earth
951
r
Homogeneous mass
distribution
If the motion of the galaxy is caused entirely by the Hubble expansion, its radial velocity relative to the Earth is H0 r. This implies a
kinetic energy
T = 21 mv2 = 21 mH02 r2,
where m is the inertial mass of the galaxy, assumed to be equivalent
to its gravitational mass. The total energy of the galaxy is then
E = T +U = 12 mH02 r2 − 34 π Gr2ρ m
2
2
8
1
= 2 mr H0 − 3 π Gρ .
952
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
30.2 The Critical Density
For the expansion to halt, we must have E = 0 and thus
H02 = 83 π Gρ .
Solving for ρ , the critical density that will just halt the expansion is
3H02
ρc =
≃ 1.88 × 10−29h2 g cm−3 .
8π G
The corresponding critical energy density is
εc = ρcc2 = 1.05 × 10−2h2 MeV cm−3
= 1.69 × 10−8h2 erg cm−3.
The critical density corresponds to an average concentration of only six hydrogen atoms per cubic meter of space
or about 140 M⊙ per cubic kiloparsec.
We may distinguish three qualitative regimes for the actual density ρ :
1. If ρ > ρc the Universe is said to be closed and the expansion will
stop in a finite amount of time.
2. If ρ < ρc the Universe is said to be open and the expansion will
never halt.
3. if ρ = ρc the Universe is said to be flat (or Euclidean) and the
expansion will halt, but only asymptotically as t → ∞.
30.2. THE CRITICAL DENSITY
Thus, in this simple picture the ultimate fate of the Universe is determined by its present matter density. (We
shall see that this conclusion is modified—profoundly—
by the apparent presence of vacuum energy.)
953
954
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
30.3 Baryonic and Non-Baryonic Matter
Baryonic matter is “ordinary” matter consisting of protons
and neutrons. Non-baryonic matter consists of particles
that do not undergo the strong interactions. For example,
neutrinos are one example of non-baryonic matter.
Constraints can be placed on the baryonic matter density ρB by comparing observed and predicted abundances of light isotopes such as
3 He and 7 Li that are formed in the early Universe.
1. One finds ρB ≃ 2 × 10−31 g cm−3,
2. This is far too small to close the Universe since ρc ≃ 10−29 g cm−3
3. However, this does not settle the issue because
• There is substantial evidence that most matter in the Universe is non-baryonic and in the form of dark matter that is
visible only to gravitational probes.
• There is a substantial dark energy contribution to the current
evolution of the Universe.
4. We shall discuss candidates for this non-baryonic matter and the
role of vacuum energy in subsequent chapters.
30.3. BARYONIC AND NON-BARYONIC MATTER
Extending the trend started by Copernicus: we are not the
center of the Universe, and we aren’t even made up of the
dominant matter of the Universe. Not only are we not the
center, we aren’t even made of the right stuff!
955
956
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
Radial Velocity
Observed
Predicted if mass
traces luminosity
Radius
Figure 30.2: Schematic velocity curves for spiral galaxies.
30.3.1 Evidence for Dark Matter: Galaxy Rotation Curves
In spiral galaxies, if we balance the centrifugal and gravitational forces
at a radius R, the tangential velocity v should obey the relation
r
GM
v=
R
implied by Kepler’s laws, with R the radius and M the enclosed mass.
• Well outside the main matter distribution, we expect v ≃ R−1/2 .
• The velocities can be measured using the Doppler effect, both
for visible light from the luminous matter, and from the 21 cm
hydrogen line for non-luminous hydrogen.
• For many spirals we find not v ≃ R−1/2 but almost constant velocity well outside the bulk of the luminous matter.
This is illustrated schematically in Fig. 30.2.
30.3. BARYONIC AND NON-BARYONIC MATTER
957
vr (km s-1)
300
200
100
0
0
20
40
60
80
100
120
140
160
Distance (arcminutes)
Figure 30.3: Rotation curve for the Andromeda Galaxy. White points indicate
measured velocities (open circles at large distance are RF observations).
Observational data on the rotation curve for the Andromeda Galaxy (M31) are displayed in Fig. 30.3. Converting the angular size to kpc using the distance of 778
kpc to Andromeda we see that
• The obvious visible matter lies within about 60′ ∼
14 kpc of the center.
• RF observations suggest that the rotation curve is
constant out to at least about 150′ ≃ 36 kpc.
• Direct measurements suggest constant velocities out to at least
30 kpc in many spirals, and
• Indirect means suggest that constant velocities may extend out
to 100 kpc or more in some spirals.
This indicates the presence of substantial gravitating matter distributed
in a halo beyond the visible matter.
958
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
Aug 1991
Aug 1994
1"
Figure 30.4: Gravitational lensing: the Einstein Cross.
30.3.2 Evidence for Dark Matter: Gravitational Lensing
The path of light is curved in a gravitational field.
• This can cause gravitational lensing, where intervening masses
act as “lenses” to distort the image of distant objects.
• A spectacular example of gravitational lensing is the Einstein
Cross, shown in Fig. 30.4.
• In this image, a single object appears as four objects.
• A very distant quasar is thought to be positioned behind a massive galaxy.
• The gravitational effect of the galaxy has created multiple images through gravitational lensing on the light from the quasar.
The individual stars in the foreground galaxy may also be acting
as gravitational lenses, causing the images to change their relative
brightness in these two images taken three years apart, as stars change
position in the lensing galaxy.
30.3. BARYONIC AND NON-BARYONIC MATTER
959
Quasar
images
Einstein Cross
Nucleus
Bar
Faint lensing
galaxy
Spiral arms
Figure 30.5: The Einstein Cross and the lensing galaxy. The intensity has been
displayed on a logarithmic scale so that the very bright quasar images and the extremely faint bar and arms of the lensing galaxy can be seen at the same time.
Image courtesy W. Keel, University of Alabama.
This interpretation of the Einstein Cross is bolstered by
Fig. 30.5, which shows in faint outline the foreground
lensing galaxy surrounding the bright central nucleus of
the spiral and the four quasar images.
• The lensing galaxy is a relatively nearby barred spiral.
• Both the spiral arms and the central bar of the foreground galaxy can be seen if one looks carefully (see
the annotated version of the figure in the right panel).
960
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
The strength of a gravitational lens depends on the total
mass contained within it, whether that mass is visible or
not.
• Gravitational lenses can serve as excellent indicators
of how much unseen matter is present in the region
of the lens.
• Extensive analysis of gravitational lensing by large
masses leads to conclusions similar to those suggested above by the rotation curves for spiral galaxies:
More than 90% of the mass contributing to
the strength of large gravitational lenses is
dark.
30.4. DARK ENERGY
961
30.4 Dark Energy
Dark matter may appear exotic by normal standards, since we don’t
know what it is and therefore do not know why it fails to couple
strongly through any force other than gravity.
• However, we shall see in Chapter 31 that there is growing evidence that the evolution of the present Universe is being dominated by something even more exotic: dark energy.
• Dark energy (also known as vacuum energy) behaves fundamentally differently from either normal matter and energy, or dark
matter.
• It appears to cause the force of gravitation to become repulsive.
• To understand and to deal adequately with this remarkable notion will require a covariant formulation of gravitation.
Therefore, we defer substantial discussion of the evidence for and role
played by dark energy until the following two chapters.
962
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
30.5 Cosmic Scale Factor
As we have seen, the Hubble expansion makes it convenient to introduce a cosmic scale factor a(t) that sets the global distance scale for
the Universe.
• If peculiar motion is ignored, the expansion is governed entirely
by a(t) and all distances simply scale with this factor.
• Example: if present time is t0 and present scale factor is a0 , a
wavelength of light λ emitted at time t < t0 is scaled to λ0 at
t = t0 by the universal expansion:
λ0
λ
.
=
a0 a(t)
• Likewise, if r0 and ρ0 are the present values of r and ρ ,
a0 3
r(t) a(t)
ρ (t)
=
=
,
r0
a0
ρ0
a(t)
• This permits us to express all dynamical equations in terms of
the scale factor. Example: Newtonian gravitational force acting
on the galaxy
Mm
∂U
= −G 2 = − 34 π Gρ rm,
∂r
r
and the corresponding gravitational acceleration is
GM
FG
= − 2 = − 34 π Gρ r.
r̈ =
m
r
Then from r/r0 = a/a0 and ρ = (a30 /a3)3 a,
FG = −
a30 r0
4
r0
r̈ = ä = − π Gρ0 3 a →
a0
3
a a0
(acceleration of the scale factor).
ä = − 34 π Gρ0a30
1
.
a2
30.6. DENSITY PARAMETERS
963
30.6 Density Parameters
It is convenient to introduce the total density parameter evaluated at
the present time
ρ
8π Gρ
.
Ω≡ =
ρc
3H02
where ρ is the current total density coupled to gravity.
• Thus, the closure condition implies that Ω = 1 (critical density).
• The subscript “0” is often used on Ω and ρ to indicate explicitly that they are evaluated at the present time; we suppress that
subscript to avoid notational clutter in later equations.
The acceleration of the scale factor may be expressed in terms of the
density parameter Ω,

1

3

ä = − 34 π Gρ a0 2


a
− 21 H02 a30Ω
.
→ ä =
2

a
2 4
8π Gρ
ρ

= ( 3 π Gρ ) 
Ω ≡ ρc =

2
H0
3H0
(where ρ ≡ ρ0 and Ω ≡ Ω0). Anticipating the later treatment of the
expansion using general relativity, we may expect that the density
parameter gets contributions from three major sources in the current
Universe:
1. Matter (with density denoted by ρm )
2. Radiation (with density denoted by ρr)
3. Vacuum or dark energy (with density denoted by ρv or ρΛ ).
964
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
These densities may be used to define corresponding density parameters Ωi through
ρr (a) = ρcΩr
ρm (a) = ρc Ωm
ρv (a) = ρcΩv ,
where we shall show (Chapter 31) that the total density changes with
a(t) according to
Ωr Ωm
ρ (a) = ρc 4 + 3 + Ωv
(a(t0 ) ≡ 1),
a
a
• we have assumed the standard convention of normalizing the
current value of the scale parameter a(t0 ) to unity.
• We shall make no explicit distinction between mass density ρ
and the corresponding energy density ε = ρ c2, since they are
numerically the same in c = 1 units.
• Note that the different densities scale differently with a(t), and
thus differently with time.
For baryonic matter alone, we obtain from the observed ρB ≃ 2 ×
10−31 g cm−3 that
ρB
Ω = Ωm =
≃ 0.024.
(baryonic matter).
ρc
This is well below the critical density (Ω = 1) but, as we
have previously noted, baryonic matter is not the dominant
matter in the Universe and we must include the effect of
• non-baryonic dark matter and
• dark energy
to determine the true value of Ω.
30.6. DENSITY PARAMETERS
Table 30.1: Density parameters
Source
Value (Ωi = ρi /ρc )
Total matter
Ωm = 0.3
Baryonic matter
ΩB = 0.024
Total radiation
Ωr <∼ 8 × 10−5
Total vacuum
Ωv = 0.7
Curvature
Ωc ≤ 0.01
Some estimates of the current density parameters for the
radiation, matter, baryonic portion of the matter, and the
vacuum energy are given in Table 30.1 (the curvature density entry will be explained in Chapter 31).
965
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
966
30.7 Time Dependence of the Scale Factor
1d 2
Identity: ä =
ȧ
2 da
− 21 H02 a30Ω
Earlier: ä =
a2





−→




1 d 2 − 21 H02 a30Ω
ȧ =
.
2 da
a2
Solving this for dȧ2 and integrating from the present time t0 back to
an earlier time t,
Z t
t0
dȧ
2
= −H02 a30Ω
Z a
da
a0
→
a2
ȧ
2
= ȧ20 + H02a30 Ω
1 1
−
,
a a0
and since ȧ0 = a0H0 (Exercise),
ȧ2 = a20H02 f (Ω,t),
where we define
f (Ω,t) = 1 + Ω
a0
− Ω,
a(t)
which must obey the condition
f (Ω,t) ≥ 0,
since ȧ2 can never be negative.
We may use this condition to enumerate different possibilities for the history of the Universe. NOTE: Ω ≡ Ω0 in
these equations.
30.8. EXPANSION HISTORIES FOR THE UNIVERSE
967
30.8 Expansion Histories for the Universe
Let us consider as an example, dust-filled universes (universes containing only pressureless, non-relativistic matter and negligible amounts
of radiation or vacuum energy). Three qualitatively different scenarios for such a Universe, depending on the value of Ω ≡ Ω0 = Ωm.
1. Ω < 1 (undercritical): In this case, as a(t) → ∞,
f (Ω,t) = 1 + Ω
a0
−Ω
a(t)
−→
1 − Ω > 0.
Thus ȧ never goes to zero (ȧ2 ∝ f (Ω,t) and we live in an open,
ever-expanding universe if Ω < 1.
2. Ω = 1 (critical): For this case, as a(t) → ∞,
f (Ω,t) −→ 0,
but it only reaches 0 at t = ∞. Hence, if Ω = 1, the universe
is ever-expanding (constraint: expanding now) but the rate of
expansion approaches zero asymptotically as t → ∞.
3. Ω > 1 (overcritical): Now as t increases
f (Ω,t) −→ 0,
but in a finite time tmax . Beyond this time we still must satisfy the
condition f (Ω,t) ≥ 0. Thus, if Ω > 0 the expansion turns into a
contraction at time tmax and the universe begins to shrink.
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
968
Ω<1
Scale factor a(t)
Open
Flat
Ω=1
Closed
Ω>1
Now
Time t
Figure 30.6: Behavior of the scale factor a(t) as a function of time for a dust-filled
universe.
The evolution of the corresponding scale factor is sketched
in Fig. 30.6.
30.9. THE DECELERATION PARAMETER
969
30.9 The Deceleration Parameter
The density of the Universe is clearly related to the rate at
which the Hubble expansion is changing with time.
If we expand the cosmic scale factor to second order in time,
a(t) ≃ a0 + ȧ0(t − t0 ) + 12 ä0(t − t0 )2
(where ȧ0 ≡ (da/dt)t=t0 , and so on), introduce the deceleration parameter at the present time q0 ≡ q(t0 ) through
q0 ≡ −
and utilize
ä0
ä0
= −a0 2 ,
2
a0H0
ȧ0
ȧ0
= H0 ,
a0
we obtain


a(t) = a0 1 + H0(t − t0 ) − 12 H02 q0(t − t0)2 + . . . .
|
{z
} |
{z
}
Hubble
Corrections to Hubble
The deviation from the Hubble law is quadratic in time to
leading order.
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
970
H0 = 72 km s-1 Mpc-1
0
-0
-1
1.8
.5
2.0
0.
1
1.6
1.4
1.2
Now
1.0
0
0.8
0.6
0.4
Ωm=0, Ωr=0
Ωv=1 q0=-1.0
Ωm=1, Ωr=0.5
Ωv=0 q0=1.0
Ωm=0, Ωr=0
Ωv=0.5 q0=-0.5
1
Ωm=1, Ωr=0
Ωv=0 q0=0.5
0
-24
3
5
10
Ωm=0, Ωr=0
Ωv=0 q0=0
0.2
-20
-16
-12
-8
Time
-4
(109
0
Redshift
Scale factor relative to today
5
4
8
12
16
years)
Figure 30.7: Quadratic deviations from the Hubble expansion. The different
curves correspond to different assumed values of the density parameters and the
corresponding deceleration parameter q0 . Each curve has the same linear term but
a different quadratic (acceleration) term. Positive values of the deceleration parameter correspond to a slowing of the expansion and negative values to an increase in
the rate of expansion with time.
Quadratic deviations from the Hubble law are illustrated
in Fig. 30.7.
30.9. THE DECELERATION PARAMETER
971
30.9.1 Deceleration and Density Parameters
Generally, the deceleration parameter q0 is related to the density parameters Ωi through (Exercise, Ch. 31)
q0 =
Ωm
+ Ωr − Ωv .
2
The parameters of Table 30.1 suggest that the deceleration parameter
for the present Universe is negative,
q0 ≃
Ωm
+ Ωv = 12 (0.3) − 0.7 ≃ −0.55,
2
and that the expansion is currently accelerating.
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
972
2.0
1.8
H0 = 72 km s-1 Mpc-1
q0 = 0
1.4
1.2
Now
1.0
0
0.8
0.6
0.4
Ωm=0, Ωr=1
Ωv=1 q0=0
Ωm=0, Ωr=0
Ωv=0 q0=0
Ωm=1, Ωr=0
Ωv=0.5 q0=0
0.2
0
-24
1
-20
-16
-12
-8
Time
-4
(109
0
Redshift
Scale factor relative to today
1.6
3
5
10
4
8
12
16
years)
Figure 30.8: Different choices of matter, radiation, and vacuum energy densities
that give the same deceleration parameter. The curves all agree near the present
time to second order, but have very different long-time behaviors.
30.9.2 Deceleration and Cosmology
Figure 30.8 illustrates that H0 and q0 determine the behavior of the
Universe only near the present time.
• The three curves have the same H0 and q0 = 0, but very different
mixtures of matter, radiation, and vacuum energy densities.
• Within the gray box the curves are essentially indistinguishable
but at redshifts of 1 or larger they are very different.
• For example, these three curves predict ages of the Universe (intercepts with the lower axis) that differ by almost a factor of 2.
30.9. THE DECELERATION PARAMETER
Until very recently, the primary quest in cosmology was
to determine the Hubble constant H0 and the deceleration
parameter q0 . Acquisition of precision cosmology data
through
• The study of high-redshift Type Ia supernovae
• The detailed analysis of the cosmic microwave background
mean that the cosmological data now are beginning to constrain a broader range of parameters than just these two.
We shall discuss this in more detail in Chapter 31.
973
974
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
30.10 Lookback Times
Telescopes are time machines:
• Lookback time: tL how far back in time we are looking when we view an object having a redshift z,
tL = t(0) − t(z),
where t(z = 0) is the present age of the Universe and
t(z) is the age when light observed today with redshift z was emitted.
• Example: in a flat universe (Exercise)
t(z) 2
= (1 + z)−3/2
τH
3
t(0) 2
=
τH
3
and the lookback time is
tL 2 2
= − (1 + z)−3/2
τH 3 3
2
1
=
,
1−
3
(1 + z)3/2
where τH = 1/H0 is the Hubble time.
• Thus light from an object that we observe with a redshift z ∼ 5 was emitted when
1. The Universe was only ∼7% of its present age
2. The cosmic scale factor a(t) was six times
smaller than it is today.
30.10. LOOKBACK TIMES
975
H0 = 72 km/s/Mpc
Ω=1
Ω=0.5
z=5
Ω=0.1
8
tL
Figure 30.9: Geometrical interpretation of the lookback time tL for a dust Universe
with three different values of the density parameter Ω = Ωm .
The lookback time as a function of redshift is interpreted
geometrically in Fig. 30.9.
CHAPTER 30. NEWTONIAN GRAVITY AND COSMOLOGY
976
12
Lookback time tL (109 years)
Ω = 0.1
10
Ω = 0.5
8
Ω = 1.0
6
4
2
0
0
1
2
3
4
5
Redshift z
Figure 30.10: Lookback time as a function of redshift for three different assumed
values of the density parameter in a dust model with H0 = 72 km s−1 Mpc−1 .
The lookback time is plotted for various assumed values
of the density parameter Ω in Fig. 30.10 for a dust model.
• For small redshifts tL ≃ zτH , as we would expect from
the Hubble law.
• For larger redshifts the curves in Fig. 30.10 differ
substantially from this approximation.
30.11. PROBLEMS WITH NEWTONIAN COSMOLOGY
977
30.11 Problems with Newtonian Cosmology
As promised, we have been able to make considerable headway in understanding the expanding Universe simply by using Newtonian gravitational concepts. However, the purely Newtonian approach leads to
some problems and inconsistencies. For example,
1. At large distances the expansion leads to recessional velocities
that can exceed the speed of light. How are we to interpret this?
2. Newtonian gravitation is assumed to act instantaneously, but because light speed is the limit for signal propagation, there should
be a delay in the action of gravitation.
3. In the Newtonian picture we had a uniform isotropic sphere expanding into nothing, which causes conceptual problems in interpreting the expansion. Alternatively, if the sphere is assumed
to be of infinite extent, there are formal difficulties with even
defining a potential.
These and other difficulties suggest that we need a better
theory of gravitation to adequately describe cosmologies
built on expanding universes. In Chapter 31 we shall develop an understanding of the expanding Universe based
of general relativity that will deal with these problems.