Lecture 25 - Cosmology (4/20/99)
Quasars --- | ---
Chapter 25 (ZG4)
The Coma cluster of galaxies.
(Courtesy Omar Lopez-Cruz and Ian Shelton/
||What is the fate of the Universe?
||The cosmological energy equation
||Calculate the critical density of
- The Great Attractor
- How do massive clusters induce peculiar velocities in the
- What is the Virgocentric infall velocity of the Local Group?
- What is our expected velocity towards the Coma cluster?
- What is the Great Attractor?
- What is the magnitude of our motion towards the Great Attractor?
- Why are bulk flows coherent over the sky?
- How does gravitational collapse battle with Hubble
expansion to determine clusters and void regions?
- Newtonian Cosmology
- How do we get the cosmic energy equation
v2 - 8
R2 / 3
= -k c2
from the energy equation for a spherical distribution of matter
in a uniform background?
- What are the 3 cases for a universe (or patch thereof) corresponding
to k = 0, < 0, and > 0?
- What is the fate of a critical density Universe?
- What is a "bound" universe like?
- What is the critical density for our universe, for H_0 =
- How does the density evolve with the scale factor R?
- How is the scale factor R related to the redshift z?
- What is the deceleration parameter?
- Evolution of A "Critical" Universe
- What is the dynamical equation for a universe with the
- What is the time element dt in terms of dR of
the scale factor?
- How does the time t scale with R?
- What is the current expansion age t_0 in the k=0 universe?
- What is t_0 for H_0=60 km/s/Mpc? How does this compare
with globular cluster ages?
- How does the Hubble constant H(z) evolve with scale factor?
- What is the lookback time to z = 5 (the most distant quasar)?
- More on FRW Cosmologies
- What do we mean by Friedmann Robertson-Walker (FRW) cosmologies?
- How does the matter density scale with R?
- How does the energy density u of radiation scale with R?
- How does the blackbody temperature T of radiation vary with z?
- What do the solutions for k < 0 and k > 0 look like?
- Why does a k > 0 universe recollapse?
- What are the two major assumptions in the
- What besides homogeneity and isotropy might one invoke
for the Universe, but we haven't?
- What is a steady-state universe model?
- General Relativity and Cosmology
- What is the invariant space-time interval ds^2 in special
relativity? What is the role of the speed-of-light c?
- What is the metric?
- How does the presence of mass affect spacetime in general relativity?
- How is gravity related to curvature?
- What is the Robertson-Walker metric for homogeneous and
- How is k related to the curvature?
- How do flat, postive and negative curvatures manifest themselves in
2-D? Which are spheres, saddles, and planes?
- How does the circumference of a circle measure curvature?
- What are non-Euclidean or Riemannian geometries?
- What do we mean by flat, open, and closed
- Which universes have a finite volume, and which infinite?
- What is the curvature scale R of an FRW universe with H_0?
- How does the density parameter Omega vary with scale factor?
- Why can we consider only a flat universe when calculating things
at large z?
Large Scale Simulations of the Universe
Astrophysicists are also engaging in large numerical simulations of
how we think the universe would evolve under different assumptions about
the matter therein. Some of the faculty in this department are involved
in these endeavours. Check out some of the links to our work, and other
A simulation of a box 85 h-1 Mpc on a side,
using a model with a cold+hot dark matter. Courtesy
Greg Bryan and
Mike Norman (U.Illinois/NCSA/GC3)
The Cosmic Energy Equation
Because the laws of physics are mathematical, two systems that follow
similar laws will behave similarly. We used this when we computed the orbits
of electrons about the nucles, noting that the electric force law was similar
to the gravitational force law. If you remember when we calculated the
energy levels of the atom and when we
calculated the escape velocity from a
gravitating body, we used an energy equation. Can we make an energy equation
for the Universe?
Consider a sphere of radius R comoving (that is, expanding along with the
Universe) enclosing a mass M. The gravitational energy equation would
E/m = v2/2 - GM/R
Note we have divided through by the test mass m, the mass of the particle
that is feeling the force from the total mass M at the edge of the sphere
at radius R. If you remember the gravitational case, if E=0, then at
R of infinity v=0 and the system was just unbound. If E<0, then the
system was bound and the velocity v would reach zero at a maximum radius
Rmax = -GM(m/E)
and the particle would then fall back. If, on the other hand, E>0, then v
would be positive even for R of infinity, and the system is unbound. If
we had a fixed measure of the velocity, then we could think of an escape mass
Mesc, for which with E=0, v would be zero at R of infinity:
Mesc = v2 R / 2 G
If we want to apply this to the Universe, we should use the mass density
in the sphere (3M/4Pi R^3) instead of the mass
3Mesc/4Pi R3 = 3 v2 /
8 Pi G R2
In an expanding universe, v = H R on the comoving sphere, and so
3Mesc/4Pi R3 = 3 H2 /
8 Pi G
which no longer depends on R! This density, 3H^2 /8Pi G, is called
the critical density of the Universe. If the average mass density
in the Universe is equal to or less than this, then the Universe behaves
as if it is unbound and it will expand forever. If the average mass density
of the Universe is greater than the critical density, then it will expand
to a maximum scale length, then recollapse, behaving like it is a bound
The case with a density greater than critical gives as a closed
universe with positive curvature, finite volume, which will expand
for some time, then begin to recollapse. The case with a sub-critical
density corresponds to an open universe with negative curvature
which will expand forever, and is infinite in volume. A universe
with the critical density is flat, infinite, and will expand forever though
slowing down toward zero at infinite time in the future.
The question of whether we live in an open, flat, or closed universe is
a matter of what the mass density of the Universe is relative to the
critical density 3H^2 /8Pi G. For a Hubble constant of H = 82 km/s per
Mpc as measured by HST, the critical density is 1.26 x 10^-26 kg/m^3.
This seems really tiny, but space is really big. (Q: What is this
critical density in units of solar masses per cubic Megaparsec?)
In fact, it appears that our Universe may have only about 30% of the
critical density, and we might be living in an open universe. On the
other hand, this is a hard measurement to make, and there are some indications
(as well as some theoretical prejudices) that we live in a flat (or very
nearly flat) Universe.
Cosmological Scale Factors
In an expanding universe, the scale factor changes with time, and
thus with apparent distance. The radius R of an imaginary sphere, of radius
R_0 (say 1 Mpc) now, will change as the universe expands. Similarly, the
wavelength of light moving through the Universe will also expand (red-shift).
The ratio R/R_0 is thus related to the redshift:
R(z) = R0 / ( 1 + z )
Note that R_0 = R(z)( 1 + z ) is the redshift relation
for light of true wavelength R(z) emitted at redshift z
and observed here as an increased wavelength R_0.
For small redshifts, the Doppler shift is approximately z = v/c,
while the Hubble velocity is approximately v = H d. Thus, the
distance - redshift relation is approximatley
d = c z / H
This is true for any curvature Universe on small enough scales, since
it just involves the local expansion rate.
Introduction to General Relativistic Cosmology
What can we "see" about the Universe? It is certainly apparently very
large, as the most distant quasars (at z = 5) are over 4000 Mpc away.
It appears to be in the process of uniform expansion, with the Hubble
relation v = H d. Finally, on the largest scales it appears to be
approximately homogeneous and isotropic as per the cosmological principle.
It turns out from these considerations alone, and using Einstein's General
Theory of Relativity, one is led to some concrete predictions for the
evolution and topology of the Universe as a whole. This is subject of
cosmology, the study of the Universe as an entity.
Soon after the publication of his General Theory of Relativity in 1916,
Einstein applied his equations of gravity to the Universe as a whole.
However, when he did so he ran into a problem: the equations predicted a
Universe that was in the state of expansion or contraction, but not stasis.
Remember, this was in 1917, before the discovery of the expansion of the
universe by Hubble in 1929, and the philosophical prejudice was that the
Universe was unchanging, static, and perhaps everlasting. (Q: Compare this
prejudice to those of pre-Copernican cosmology and of pre-Darwinian
biohistory.) Einstein found he had to add a new term to his equations, the
so-called cosmological constant, to end up with a static
universe neither expanded nor contracted.
When Hubble discovered the expansion of the Universe in 1929, Einstein
found that his equations without the cosmological constant made sense. He
then declared that the cosmological constant was his "greatest blunder". He
felt that he should have had faith in the predictions of his theory, and
disregarded the unfounded (that is, not based upon observation) prejudice for
a static universe. However, in the last decade, a number of observations of
the Universe have been found to be at odds with some theoretical predictions
using simple cosmologies. Some solutions to these problems involve the
introduction of a cosmological constant term into Einstein's equations, though
in a somewhat different manner than for a purely static universe. Thus,
this may not have been a blunder after all!
Just like spacetime is curved by the presence of mass like a black hole,
the spacetime of the Universe as a whole is curved by the presence of mass
(like ourselves). There are three generic topologies for a homogeneous
and isotropic Universe with 3 spatial dimensions like ours. They are
distinguished by the curvature. In one dimension, like a line, there
is only one sense of curvature. However, in two or three dimensions,
curvature has a sign, and is positive, negative or zero. For a homogeneous
and isotropic, both the sense and magnitude of the curvature must be uniform,
that is the same everywhere.
It is easiest to use a two-dimensional analogue to begin with. A
sphere has positive curvature. In two orthogonal
(perpendicular) directions, the curvature is in the same direction, so a
positively curved space closes back on itself, like a sphere. A positively
curved space is unbounded, but finite (like the area of a sphere). A
plane has zero curvature, and is thus flat. Flat space
is unbounded and infinite. A saddle has negative curvature.
Orthogonal directions show curvature in opposite directions (upward or
downward). A space with negative curvature is unbounded and infinite.
Note that geometery as we are first taught it is only good in flat space.
This is Euclid's geometry, and thus flat (zero curvature) space is also called
Euclidean. (Note: flat space is Euclidean, but flat spacetime
is Lorentzian, because time behaves differently than space. See the Special
Lecture on the speed of light for
There are several things that are different in non-Euclidean spaces, and
can be used to tell if you are in curved or flat space. For example, one of
Euclid's postulates is that parallel lines never meet (if they are not
identical). Note that in positively curved space this is not true! Two
parallel lines on a sphere are great circles, and two distinct great circles
meet on opposite sides of the sphere at two points. The circumference and
area of a circle, and the sum of the vertex angles of a triangle are also
different in differently curved spaces.
In positively curved space, the circumference of a circle is less than
Pi x Diameter, and the area less than Pi x Radius^2. In 3-D, the volume
of a sphere is thus less than 4/3 x Pi x Radius^3! Furthermore, the
sum of the three vertex angles of a triange are greater than 180 degrees!
(Q: Test these statements using a globe (sphere). For example,
cut a circle and triangle out of paper. Now try and paste it onto the
sphere. What do you have to do to these figures to make them conform
to the sphere?)
In negatively curved space, the circumference of a circle is greater
than Pi x Diameter, the area greater than Pi x Radius^2, and the volume
of a sphere thus greater than 4/3 x Pi x Radius^3. The sum of the vertex
angles of a triangle are less than 180 degrees. (Q: Try and demonstate
these properties of negatively curved space. Though you cannot easily make
a 2-dimesional negatively curved uniform surface, you can use a saddle-shaped
surface that has negative curvature at its "saddle point". Try pasting
a circle and triangle there.)
Of course, the Universe is very large, and its curvature slight,
so you would have to use very large circles or triangles to do
The Age of the Universe
If we live in a Universe that is expanding with mass in it, the
expansion is slowing down. Thus, the Hubble constant H was higher in
the past when the Universe was expanding faster. We denote the current value,
which we measure around us locally, as H0. H0 as measured by HST is 82
km/s/Mpc, though different measurements give values in the range 40 to 100. A
universe with higher density is younger than a universe with lower
density, given the same expansion rate H0 now, since it will have slowed down
more from a higher expansion rate earlier on.
If there were no mass in the universe, then H0=H always and it would expand
at the same rate. The age of such an empty universe would thus be
t0 = 1 / H0
This represents the maximum age of an expanding universe
with a given current value of H0 (though we will find we can change this by
changing the energy equation). For H0=50 km/s/Mpc, the age of an empty
universe would be just under 20 billion years, and thus for H0=82
it would be around 12 billion years old. Since the oldest globular clusters
seem to have ages of 15 to 18 billion years, then we might expect that
H0 would be near 50. Thus, the HST value of H0=82 is incompatible with
old globular clusters even for an empty universe. Perhaps astronomers are
mistaken about the globular clusters ages (new calculations are being done
for stellar evolution). Another possible solution, using a cosmological
constant, will be discussed in the next lecture.
If we live in a flat universe with the critical density, things are even
worse. The age of a flat universe is given by
t0 = 2 / 3 H0
so for H0 = 50 km/s/Mpc the age t0 is 13 billion years, and for H0 = 82 is
8 billion years! This is the so-called age problem in cosmology, and
we are working very hard to see what the solution is. To summarize, the
oldest globular clusters appear to be 15 to 18 billion years old, while the
Sun is around 4.5 billion years old. If H0 is really as high as 82 km/s/Mpc,
and the Universe were flat, then the Universe would be younger than its oldest
stars, which is nonsensical. Either we have gotten the globular cluster ages
wrong, or H0 is really 50 km/s/Mpc or smaller, or there is some other term
in the energy equation that is making the Universe expand faster now than
in the past.
Although the relationship between the redshift and time is complicated,
there are two simple cases. For an empty universe with constant expansion
t = t0 / (1 + z)
For a flat universe
t = t0 / (1 + z)3/2
These give the age of the universe t at redshift z.
The Density of the Universe
We have from the previous lecture the scaling of lengths with redshift
R(z) = R0 / ( 1 + z )
The density of matter in a sphere of radius R is given by 3M/4Pi R^3, so
( 1 + z )3
The matter density in the Universe was higher in the past. Note however
that the critical density 3H^2/8PiG was also higher in the past, since
H was higher (expanding faster). It turns out that for a flat universe
at the critical density, the density always remains at the critical density
(since it must always be flat). Therefore, it must be that
H^2 = H0^2 ( 1 + z )3
to keep up with the matter density, and thus
H(z) = H0 ( 1 + z )3/2
Radiation (all the light in the Universe) also has a density that will
increase in number of photons per unit volume as (1+z)^3 just
like the matter density. However, the energy of the radiation is being
redshifted away, and must have been higher in the past by the factor
E(z) = E0 ( 1 + z )
where E = h f = h c / Lambda the wavelength. Thus, the
energy density of the radiation evolves as
urad(z) = urad(0) *
( 1 + z )4
Since the energy of matter goes as mc^2, the energy density of matter is
umat(z) = umat(0) *
( 1 + z )3
Thus, at some point in the past the two were equal, Umat=Urad. This
occurs at 1 + z = u_mat(0)/u_rad(0). Currently, the energy of the
Universe is matter dominated with Umat/Urad around 10^4.
Thus, at a redshift z of around 10^4, the energy density in matter
and in radiation were equal. Before this, at higher redshifts, the
Universe was radiation dominated with most of the energy in
The Stefan-Boltzmann law is essentially a statement about the
energy density of radiation. We can rephrase it as
urad = a T4
(Note: the Stefan-Boltzmann constant "sigma" is equal to ac/4.) Thus,
the temperature of the radiation must also be higher in the past by
T(z) = T0 ( 1 + z )
The Universe was hotter and denser in the past. This statement turns
out to have profound consequences, and follows from the expansion of the
Universe and the theory of relativity. In fact, if we carry it to its
extreme, there must have been a point (as it turns out not infinitely far back
in time) when the universe was infinitely dense and infinitely hot - this is
the point called the "Big Bang". This fiducial beginning is the point from
which the age of the Universe is measured. Q: What does the current
temperature of the radiation in the Universe T0 represent? How could we
Cosmology in Outline
- Clustering in the Universe
- Galaxies don't like to be alone, they seem to come in
- The Local Group of which we are a member contains
around 20 galaxies (mostly dwarf ellipticals) with 3 main
spirals (Milky Way, M31 Andromeda, and M33 Triangulum) all
within a diameter of around 1 Mpc.
- The nearest large cluster is the Virgo Cluster at
a distance of 17 Mpc. It contains more than 2500 galaxies
(again mostly dwarfs), and has a total mass of around
10^14 Msun within a radius of 1 Mpc or so.
- The Virgo cluster exerts a strong gravitational force - the
Local Group is moving toward Virgo at around 300 km/s.
- Like most larger clusters, there is a massive giant elliptical
galaxy (M87) at the center of the Virgo cluster. It may have
grown large by "cannibalizing" numerous dwarf galaxies that
happened to fall into the center of the cluster.
- M87 is also a strong radio source (Virgo A) and has an
active nucleus. It probably contains a supermassive (10^8 Msun)
- The Coma cluster at a distance of 100 Mpc has a mass of
10^15 Msun within a radius of over 1.5 Mpc. It contains
a giant elliptical and a large S0 at the center. It is
made up of mostly E's and S0's with few spirals.
- Large clusters are mostly devoid of spirals, probably due
the the high frequency of close encounters with other
- Our local group and the Virgo cluster seem to be part
of a cluster of clusters in the shape of a roughly
flattened pancake called the Local Supercluster.
- Surveys of galaxy redshifts out to 200 Mpc show a network of
filaments and walls of galaxies surrounding empty regions
called voids. The scale of these structures is
approximately 100 Mpc.
- There may be larger structures still (supervoids? superwalls?)
but current redshift survey are not "deep" enough to see
more than about 200 Mpc. There are some indications that
100 Mpc is the scale of the largest structures, but we will
not really know until we look!
Sloan Digital Sky Survey will give us a look on
larger scales, measuring redshifts for 100 million galaxies!
- There appears to be a hierarchy of clustering:
galaxies -> groups -> clusters -> superclusters -> walls?
filaments? voids? -> ?????
- Current popular theories predict a large-scale structure formation
scheme from the smallest to the largest scales, or bottom up.
The smallest scales collapse first, then organize themselves through
gravity into larger clusters and superclusters.
- Alternative models predict that the largest scale structures form
first, then fragment into smaller subunits (like star formation in
molecular clouds), or a top down scheme. This is out of
- Introduction to Cosmology
- The distribution of galaxies, on scales greater than 200 Mpc,
appears to be the same everywhere in space. Thus, we believe
the Universe is homogeneous on large scales.
- Furthermore, it appears the same in every direction we look,
- These two statements are called the cosmological principle:
the Universe is homogeneous and isotropic on large scales.
- This means the all "observers" anywhere in the Universe at the
same time will see the same general properties. There is no
special place in the Universe (the ultimate Copernicanism)!
- The laws of physics are universal, they apply the same
everywhere in the Universe and at all times.
- The laws of physics we discover based on experiments on the Earth
can be applied to the Universe at large!
- It turns out, using universal laws of physics (Einstein's theory
of Relativity in particular), the assumption of homogeneity and
isotropy of the Universe restricts the possible cosmologies
to a particular set. Requiring the solution to be expanding
(as we observe) narrows the choices down further to the
Freidmann - Robertson - Walker (FRW) solutions.
- These solutions of the Einstein equations for the Universe
results in an unbounded Universe (no edges), either finite in
volume (like the surface of a sphere) or infinite (like an
- Since the Universe is expanding, any length scale, like the average
distance between galaxies, must have been smaller in the past.
Extrapolation back in time of the equations gets us to a point where
the distance is zero - the Universe is infinitely dense. This point
is called the Big Bang, but should not be thought of as an
explosion thought it was indeed very hot then.
- Is the Sky Dark at Night?
- Does the Universe extend forever?
- The finite speed of light c: distance = time. When you
look out into the Universe, you look back in time.
- The expansion of the Universe: distance = redshift. The wavelength
of light emitted at large distances (and early times) is increased on
the way to us.
- Because the Universe is unbounded, and thus paths can be essentially
infinite in length, you would expect if it were filled uniformly
with galaxies like we see, that the sky would be bright as a
star at night, because every line of sight would end at the
surface of a star in the infinite distance somewhere.
- This like being in an infinite forest, you could never see anywhere
that you didnt see a tree!
- This is called Olber's Paradox, because the sky appears
to be pretty darn dark at night to us!
- So why is the sky dark at night if we think the Univese is unbounded?
- Well, though if we look in a particular direction, we look back in
time. If the Universe had a beginning, it would be unbounded in
space but bounded in time by its beginning!
- Thus, the sky is dark at night mostly because galaxies were formed
about 15 billion years ago (about a billion years after the "beginning"
of the Universe). Thus, the edge of the "forest" of galaxies is
at 15 billion light-years distance!
- There is another effect due to the Hubble redshift from the expansion.
- Light emitted at great distances will be redshifted to longer
wavelengths. The peak of the stellar blackbody in the visible will
be redshifted to the IR or radio waves, and thus the sky could
appear dark in the visible, but bright in the infrared or radio.
- This is a part of the solution to Olber's paradox, and as we will
discover the sky is indeed bright in the microwaves!
- The Expansion of the Universe revisited
- In 1929, Edwin Hubble at Mount Wilson Observatory near Los Angeles
discovered the Universe was expanding uniformly.
- He found that every galaxy had a Doppler redshift that implied
a velocity proportional to the distance: v = H d
- This is true at small distances where the Doppler formula holds:
wavelength obs/true = 1 + v/c
- The redshift z is defined: wavelength obs/true = 1 + z
- Thus, for small distances: d = c z / H
- This expansion of the Universe, with velocity proportional to distance,
is characteristic of a universe with uniform expansion, where the
distance between any two points is increasing at the same fractional
- This fractional rate is the Hubble constant H, in units of
distance / distance per unit time (and thus velocity / distance).
The customary unit is km/s per Mpc (or km/s/Mpc).
- The solutions to Einsteins equations give this uniform expansion due
to the creation of new space between all points in the Universe!
- There is no necessary real velocity, it is space that is expanding.
There is no explosion hurling galaxies outward from a center (there
is no center from the cosmological principle).
- Furthermore, General Relativity stated that space-time is curved
in the presence of mass (like a black hole). Thus, the space-time
of the Universe is curved due to the presence of mass (like us) in
- General Relativistic Cosmologies
- The Universe appears to us to be very large, expanding
uniformly, and on large scales approximately homogeneous
- From these observations, and using Einstein's Theory of
Relativity, it turns out that there are very concrete
predictions for the way the Universe evolves.
- Einstein, in 1917, applied his equations for gravity
to the Universe.
- Einstein found that his equations predicted an expanding
or contracting Universe, but not an unchanging one. So
he added a cosmological constant to his equations
so that the Universe could remain static.
- After Hubble discovered the expansion of the universe in
1929, Einstein called the cosmological constant his
"greatest blunder". He felt he should have believed the
predictions of his equations rather than succumb to
prejudice based not on any solid observations.
- As we will see in the final lecture, we have returned
to Einstein's idea of a cosmological constant in order
to explain some problems in cosmology. Perhaps this
wasn't a blunder after all!
- An expanding homogeneous and isotropic universe can
be characterized by its topology, or the
fundamental nature of spacetime in that universe.
- Relatively simple equations govern the way the Universe
as a whole evolves in time.
- Spacetime Curvature
- The spacetime of the Universe as a whole is
curved by the presence of mass.
- There are three generic topologies for a homogeneous
and isotropic Universe with 3 spatial dimensions, distinguished
by the curvature.
- In two dimemsions, a sphere has positive curvature,
a plane has zero curvature, and a saddle has
- Flat (zero curvature) space is also called Euclidean
(actually, it is Lorentzian in spacetime, but Euclidean in
- Two parallel lines in a postitively curved space meet at two
points (like two great circles on a sphere).
- The area of a circle is less than Pi R^2 (positive curvature),
equal to Pi R^2 (flat), or greater than Pi R^2 (negative curvature).
- The sum of the three vertex angles of a triangle are greater than
180 degrees (positive curvature), equal to 180 degrees (flat), or
less than 180 degrees (negative curvature).
- You would have to use very large circles or triangles to do
- Cosmological Scale Factors
- The radius R of a comoving sphere of radius R0 now
will change as the universe expands, just like the wavelength
of light moving through the universe.
- The ratio R/R0 can be considered as the scale factor of the
R(z) = R0 / ( 1 + z )
- Note that R0 = R(z)( 1 + z ) would be the wavelength
seen now, given the true emitted wavelength R(z) then,
thus the redshift.
- The scale factor of the Universe was higher in the past (since
redshifts are positive), thus things were closer together.
- For small redshifts, the distance redshift relation becomes:
d = v / H = c z / H
since z is approximately v/c.
- >This is true for any curvature Universe on small enough scales, since
it just involves the local expansion rate.
- The Energy Equation
- Because the laws of physics are mathematical, two systems that
follow similar laws will behave similarly.
- For a comoving sphere of radius R containing mass M, the
energy equation is:
E/m = v^2/2 - GM/R
- Setting E=0 gives us a universe that will just slow down to
v=0 at R=infinity.
- Subsituting v=HR, this corresponds to a universe with the
3M/4Pi R^3 = 3H^2/8Pi G
- If the universe has an average mass density equal to the critical
density, it is a flat universe with zero curvature, infinite
volume, and expands forever though slowing down to zero infinitely
far in the future.
- If the universe has less than the critical density, it is an
open universe with negative
curvature, is infinite, and will expand forever.
- If the universe has greater than the critical density, it is
a closed universe with positive curvature, has finite
volume, and will expand to a maximum scale factor at some time
in the future, then will recollapse.
- For the HST value of H=82 km/s/Mpc, the critical density is
1.26 x 10^-26 kg/m^3.
- Astronomical measurements give an average mass density of the
Universe somewhere in the range between 10% of to just over
the critical density.
- The Age of the Universe
- If we live in a Universe that is expanding with mass in
it, the expansion is slowing down.
- Thus, the Hubble constant H was higher in the past. We
denote the current value, which we measure, as H0.
H0 as measured by HST is 82 km/s/Mpc.
- A universe with higher density is younger than
a universe with lower density, given the same expansion
rate H0 now.
- An empty universe expands at a constant rate H0=H.
The age of an empty universe is
t0 = 1 / H0
- A flat universe with the critical density would have a
current age of
t0 = 2 / 3 H0
- For H0=50 km/s/Mpc, the age of an empty universe would
be about 20 billion years. For a flat universe, the
age would be 13 billion years.
- The oldest globular clusters appear to be 15 to 18 billion
years old, while the Sun is around 4.5 billion years old.
If H0 is really as high as 82 km/s/Mpc, and the Universe
were flat, then the Universe would be younger than its oldest
stars, which is nonsensical.
- A value of H0 larger than 50 km/s/Mpc implies an open
Universe (or does it)?
- The time is related to the redshift. For an empty Universe
t = t0 / (1 + z)
while for a flat universe
t = t0 / (1 + z)^(3/2)
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Steven T. Myers