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The Smooth Universe
Adrian Down
September 01, 2006
1 Statics
1.1 The Cosmological Principle
The smooth universe model is a first approximation that will allow us to
understand the large-scale structure and evolution of the universe. Of course,
the existence of stars and planets indicates that the smooth universe model is
not complete accurate. However, we will include perturbations to the model
later.
A basic assumption of the smooth universe model is the cosmological
principle, which states that on large scales, the universe is spatially isotropic
and homogeneous.
Note. • Isotropic means the universe looks the same in all directions.
Homogeneous means that the density of the universe is constant.
• It is possible to have a homogeneous universe that is not isotropic; The
two conditions are not, in general, equivalent. Isotropy about one point
does not guarantee homogeneity. Isotropy about more than one point,
however, does imply homogeneity.
• The cosmological principle is only true on large scales, meaning that
the mean density over large enough patches of the universe should be
approximately the same at any point in the universe. This result is
supported by sky surveys containing many galaxies. Isotropy is very
well confirmed by the cosmic microwave background.
The perfect cosmological principle expands upon the cosmological princi-
ple to include time as well, and basically postulates a quasi-static universe.
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This view of cosmology is outdated, and all current evidence disfavors a
steady state universe.
1.2 Hubble parameter
1.2.1 Hubble law
Definition (Hubble constant). The Hubble law states that the recession
velocity of a galaxy is proportional to the distance to that galaxy. The
constant of proportionality is called the Hubble constant H,
v = Hr
Note. • The Hubble was discovered in 1928 and based on empirical data.
• The velocity v can be determined from the redshift of a galaxy. The
velocity v is easy to determine compared to the distance to the galaxy,
r. There are various ways to measure distance to an object, many of
which use standard candles, which are objects for which we assume that
the luminosity as a function of distance is known exactly.
• To measure the hubble law, it is necessary to look beyond the effects of
local gravity. For example, the andromeda galaxy is actually moving
towards us due to gravitational attraction.
• The linear relationship must also break down as v approaches c, indi-
cating that this law is only the lowest order approximation of the true
distance-velocity relationship.
• Although H is called the Hubble constant, it is not strictly constant
over galactic time. We will later investigate how H changes with the
evolution of the universe.
This linear proportionality of the Hubble law is very important. The
simplest explanation for the linear relationship is expansion of the universe.
1.2.2 Units and notation
From the Hubble law, the Hubble constant must have units of inverse time.
However, it is usually quoted in terms of a dimensionless quantity h and in
2
units that are convenient for astronomical observations,
H = 100h km s−1 km−1
where h is an unknown parameter (that is indicative of our ignorance).
Note. • 1 Megaparsec = 106 parsecs. A parsec = 3.26 light-years, which
is about 3 × 1018 cm. The parsec is based on measurements of parallax.
As the Earth rotates about the Sun, the line of sight from the earth
to a particular astronomical object changes. Half of the maximum
angle subtended by these lines of sight is defined as the parallax. The
parsec is defined so that an object at 1 parsec will have a parallax of 1
arc-second.
• The most recent measurement by the Hubble space telescope, presented
in the Astrophysical Journal (553, 47, 2001), predict H0 = 72 ± 8 km
s−1 Mpc−1 . For this and all other quantities, the naught subscript is
used to denote the present value.
• Hubble measured h to be about 5 from H0 = 500 km s−1 Mpc−1 .
In units of time,
h h
H= ≈ 10
9.78810 yr 10 yr
This number is useful to know. Since h is on the order of 1, the Hubble time
defines a galactic time scale.
1.3 Scale Factor
1.3.1 Motivation
The scale factor, denoted by a(t), is a useful way of describing the expansion
of the universe. It relates physical coordinates to coordinates in a frame that
is expanding with the universe, called a comoving frame. We will use r to
denote the physical distance between two objets, and x to denote comoving
coordinates. Comoving coordinates are useful because they are independent
of the expansion rate of the universe, which changes with time and can vary
between cosmological models.
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1.3.2 Definition
A physical distance r is related to a distance x in comoving coordinates via
the scale factor a(t),
r = a(t) · x
Taking the time derivative,
˙ ˙ ˙
r = v = ax + ax
Replacing a by x ,
r
a˙
˙
r = r + ax
˙
a
Comparing with Hubble’s law, this is the true expression for the relationship
between velocity and distance in an expanding universe. By comparison with
the Hubble law, we can identify the Hubble parameter,
˙
a
H=
a
Note. This basic equation is used extensively in cosmology.
The second term of the expression above for velocity is called the peculiar
velocity, and describes motion relative to comoving frame that is independent
of expansion.
Note. The peculiar velocity makes the measurement of H very difficult. At
large distances, peculiar velocities decrease. However, large distances become
harder to measure.
1.3.3 Conventions
There are different conventions for the units of the scale factor . The a that
we use is dimensionless. Since the scale factor is as relative measure, we can
choose to normalize expansion to the present universe by setting a0 = 1. In
other words, the comoving frame is defined such that comoving distance is
equal to physical distance in the present universe.
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It is also possible to define a scale factor r with units of length and let
comoving distances be dimensionless. The relation between this scale factor
and our convention is,
R(t)
≡ a(t)
R0
where R0 is related to the curvature of space.
2 Dynamics
2.1 The Friedman equation
2.1.1 Motivation
The Friedman equation describes how the scale factor a evolves with time,
and thus is of critical importance to astronomy. The full derivation uses
general relativity (GR) and the Einstein equations. However, we will obtain
the Friedman equation using a quasi-Newtonian argument that will give the
right answer, although with a bit of hand-waving.
2.1.2 Derivation
Using the cosmological principle, assume the universe is the same in all di-
rections. Thus it is only necessary to consider radial distance r since there
is no angular dependence.
Homogeneity, also from the cosmological principle, indicates that the den-
sity of the universe ρ is constant in space, although it could vary in time.
Note. Later we will let ρ refer to mass and energy density. However, to
simplify this Newtonian argument, we consider only mass density.
Assume the universe is an adiabatically expanding homogeneous media.
The adiabatic approximation implies that there is no net change in entropy,
and thus no transfer of heat.
Consider the motion of an expanding thin spherical shell of matter of
radius a. From Newton’s law of gravitation, the behavior of the shell depends
only on the matter enclosed by that shell. Since the mass density of the
universe is assumed to be constant, the matter contained in the shell is equal
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to the mass density times the volume of the shell,
4π 3
M(< a) = aρ
3
We can define the energy per unit mass of the shell,
E 1 GM
= a2 −
˙
m 2 a
1 2 4π
= a −˙ Gρa2 (1)
2 3
The first term is due to the kinetic energy of expansion, and the second term
is due to the gravitational potential of the matter in the sphere.
We now have a differential equation defining the evolution of the sphere
of radius a. The behavior of the solution to this equation will depend on
the value of the total energy in the sphere. If the total energy of the sphere
is too small, the gravitational energy of the sphere will eventually cause it
to collapse, and the value of the radius of the sphere will be bound. With
enough energy, the sphere could expand forever, and the radius would be
unbound. We can encapsulate this asymptotic behavior in a parameter k,
which is usefully related to E,
> 0 ⇒ E < 0 bound
2
kc ≡ −2E = 0 ⇒ E = 0 critical
< 0 ⇒ E > 0 unbound
2
Substituting k into (1), multiplying by a , and rearranging produces the Fried-
man equation,
2
˙
a 8π kc2
= Gρ − 2
a 3 a
Derivation using GR The Einstein equation in GR is,
Gµν = 8πT µν
The left hand side of this equation represents the properties of space, and the
right hand side represents the properties of the energy within that space. The
Freedman equation comes from a combination of time-time and space-space
parts of this equation.
Note. This formulation is analogous to the four-vector formulation of elec-
tromagnetism.
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2.1.3 Evolution of ρ
To determine the evolution of the density ρ with time using classical physics,
we can use the first law of thermodynamics. The derivation of the Friedman
equation assumed that the expansion of the universe is adiabatic, so that the
charge in entropy of the universe, ∆S is equal to 0. In this case, the first law
of thermodynamics can be used to relate the energy of the universe to the
density,
dU = −P dV
where U is the internal energy, P is the pressure, and V is the volume.
From dimensionality, the energy must be equal to the mass times volume.
The volume is equal to the cube of the length, and thus is proportional to
the cube of the scale factor,
d(ρc2 · a3 ) = −P · d(a3 )
Note. We will often speak of mass in units of energy, with the implicit conver-
sion factor of c2 . Also, we will henceforth take c = 1 unless noted otherwise,
so that mass and energy are in explicitly equivalent units.
Performing the differentiation,
ρa3 + 3ρa2 a = −3P a2 a
˙ ˙ ˙
Thus the rate of change of density is related to both the density and the
pressure,
˙
a
ρ = −3 (ρ + P )
˙
a
Note. In GR, this equation can be obtained from the energy conservation
equation, Tνµν = 0.
2.1.4 Acceleration of expansion
We can combine the two previous results to obtain an expression for the
second derivative of a. Take the time derivative of the Freedman equation
and multiply by a2 to get,
8π d
2a¨ =
˙a G (ρa2 )
3 dt
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k is a constant and hence its derivative is equal to 0. Performing the deriva-
tive on the right and factoring out the a2 term,
8π 2 ˙
a
˙a
2a¨ = ˙
Ga (ρ + 2 ρ)
3 a
˙
Using the equation above for ρ,
8π 2 a˙ ˙
a
˙a
2a¨ = Ga − ρ − 3 P
3 a a
8π
˙
= − Gaa(ρ + 3P )
3
Thus,
¨
a 4π
= − G(ρ + 3P )
a 3
Note. • This is an important equation, sometimes also called a Freedman
equation. These two Friedman equations are the only two independent
equations that can be obtained from the smooth universe.
¨
• If ρ + 3p is positive, then a is negative, and the expansion of the uni-
verse is decelerating. We would expect the expansion of the universe
to be slowed by gravitational attraction. However, if the universe is
accelerating, as present data seem to indicate, it must be possible to
obtain a negative pressure somehow, since ρ can never be negative.
• In GR, this result can be obtained from the time-time part of the
Einstein equation, Gµν = 8πT µν .
2.1.5 Comparison to Newtonian physics
For a self-gravitating body in one dimension, Newton’s law of gravitation
gives,
GM
¨
x=−
x2
Writing the mass in terms of the density, which we assume to be constant,
4π
¨
x= Gρ · x
3
¨
x 4π
⇒ = − Gρx
x 3
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Thus the pressure does not come into play in the classical argument. Cosmo-
logically, the pressure represents an effective mass that is important during
the inflation of the universe.
2.2 Equation of state
2.2.1 Motivation
Thus far, we have obtained the Friedman equations, which describe the evo-
lution of the scale factor a based on a model of expansion of a thin spherical
shell. To solve these equations and determine a(t), we must specify the den-
sity ρ, the curvature k, and have some understanding of the relation between
ρ and P . The equation of state provides this relation between the pressure
and the density.
2.2.2 Derivation
Because we assume that the universe is homogeneous, ρ and P are both
constants, respectively, at a given time. However, both ρ and p can change
with time. The equation of state describes the relationship between ρ and P
at any point in time,
P = wρ(c2 )
where w is a parameter defined by this equation.
It is possible to use this equation of state to rephrase the equation for the
evolution of the density in terms of the scale factor,
˙
a
˙
ρ = −3 (ρ + P )
a
˙
a
˙
⇒ ρ = −3 (1 + w)ρ
a
˙
ρ ˙
a
⇒ = −3 (1 + w)
ρ a
Integrating and solving for ρ,
ρ ∝ a−3(1+w)
Note. The final integration in this equation assumes that w is constant in
time. Many cosmological models use this assumption, although there are
theories that allow w to a time-dependent.
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We can use this relation between the scale factor and the density to
investigate the evolution of the universe due to certain types of matter.
2.2.3 Pressure-less matter
This is the case of regular (boring) Newtonian matter. An example would
be cold dark matter.
Because the matter is pressureless, P ≡ 0. Since ρ = 0, it must be that
w = 0. Hence,
ρ ∝ a−3
This result makes sense based on mass conservation, as density is propor-
tional to the inverse of volume.
2.2.4 Radiation
This is the extreme opposite to Newtonian matter. An example would be a
relativistic gas composed of photons or very light neutrinos.
Using kinetic theory, it can be shown that w = 1 for a highly-relativistic
3
substance. Thus,
ρ ∝ a−4
The energy density goes down by another factor of a relative to the Newto-
nian matter density because not only does the density of the matter decrease
as the inverse of the volume, but the energy of the particles is also decreased
due to redshifting, with the energy loss proportional to a−1 .
2.2.5 Cosmological constant Λ
This scenario refers to the case of w = −1. The hypothesized matter with this
equation of state is sometimes referred to as the “vacuum energy”. Taking
˙
w = −1, p = −ρ and ρ = 0. This scenario represents a universe that is
continually being filled with matter as it expands.
2.2.6 Dark energy
Dark energy refers to a more general form of negative pressure. As long as
1
¨
w ≤ − 3 , then a is positive, and the universe is accelerating.
10
An accelerating universe seems to be favored by the present data, al-
though the value of w varies between theories. Most data indicate w is near
−1, if w is assumed to be constant in time.
2.2.7 Graphical interpretation
15
10
matter
radiation
cosmological constant
10
10
density ρ
5
10
0
10
−5
10
−5 −4 −3 −2 −1 0
10 10 10 10 10 10
scale factor a(t)
Figure 1: Evolution of the density with the scale factor. Radiation dominates
in the early universe. Matter soon comes to dominate the universe. The
evolution of the present universe seems to be governed by a cosmological
constant.
A plot of the evolution of the density as a function of scale factor is shown
in 1.
Note. The scale factor a is proportional to time, so increasing a corresponds
to increasing time along the horizontal axis, with the present at the far right
of the graph.
11
Both matter and radiation appear as downward sloping lines, and the
cosmological constant appears as a horizontal line. The slopes of each of
these lines is determined by the respective equation of state.
This graph indicates that according to this model of the smooth universe,
radiation is dominant in the early universe. Because the energy density
of radiation declines more quickly with a than that of matter, matter will
eventually come to dominate. The point where matter and radiation cross is
called matter-radiation equality. A significant indication of this plot is that
at large time, the cosmological constant will dominate. Also, energy density
from radiation is negligible in the present universe.
The main point is that the universe evolves from a radiation dominated
era to a matter dominated era and finally to a Λ dominated era. The energy
density evolves differently in each of these eras depending on the dominant
type of matter.
2.3 Density parameter Ω
2.3.1 Definition
Definition (Density parameter). The density parameter Ω is a dimensionless
measure of the mass density of the universe,
ρ(t)
Ω(t) ≡
ρc (t)
where ρc is the critical density.
Note. The density parameter has time dependence through both ρ and ρc .
Definition (Critical density). The critical density ρc is defined to be the
density necessary to asymptotically halt the expansion of the universe.
2.3.2 Derivation
Before deriving the critical density, it is instructive to consider a classical
analog. The escape velocity of a particle is a the velocity required so that
the particle asymptotically approaches rest at large distances. The total
energy of the particle traveling the escape velocity is equal to 0, the critical
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value between a bound and unbound system,
1 GM
E = 0 = v2 −
2 R
2GM 8π
⇒ vesc = = GρR3
R 3
The critical density can be obtained from the Friedman equation. The
expansion of the universe will stop in the asymptotic limit of the kinetic
energy of expansion is exactly match by the gravitational potential, thus
when the total energy E is 0. From the definition of κ = −2E, E = 0 ⇔
κ = 0. Hence taking κ = 0 in the Friedman equation,
2
˙
a 8π
= Gρc
a 3
3H 2
⇒ ρc =
8πG
where we have used the definition of the Hubble parameter,
˙
a
H=
a
Note. • The critical density has time dependence since the Hubble pa-
rameter H is a function of time.
• This expression for the critical density is identical to the expression for
the escape velocity after some algebra.
2.3.3 Matter density of the present universe
Given H0 , it is possible to determine ρc , which indicates the amount of matter
necessary to create an asymptotically closed universe. The present value of
ρc is,
3H0
ρ0,c = = 2.78 · 1011 h2 M⊙ Mpc−3
8πG
= 1.88 × 10−29 h2 g cm−3
where M⊙ is the mass of the sun.
13
Because h is of order 1, the present critical density is quite small. If the
universe were filled only with protons, for example, it would take only 10−5 h2
protons per cubic centimeter to achieve a closed universe.
There are some useful numbers to keep in mind when considering the
density of the universe. A rule of thumb is that the density of a thumb is
approximately equal to that of water, 1 gram per cubic centimeter, which is
equal to about 1024 protons per cubic centimeter. Another rule of thumb is
that the density of the interstellar medium within a galaxy is also 1, although
in units of 1 proton (hydrogen atom) per cubic centimeter. This is about
10−24 grams per cubic centimeter. The density outside of galaxies, however,
is much lower than 1 proton per cubic centimeter, and as we will see later,
matter alone is insufficient to create a closed universe.
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