Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.
Let's go back and look again briefly at our derivation of the dynamics of the universe in a previous lecture. We will examine how the fate of the expansion depends on the density of matter ... and on another, hypothetical, factor.
Imagine that the universe is homogeneous and uniform, filled with matter which has the same density everywhere. Let's write the density of material as ρ(t), so at the current time, it has value ρ(t0), Consider a spherical shell of material of radius r, which is currently expanding at velocity v.
What will happen to the shell as time progresses?
Because the universe is uniform, the gravitational forces from all the material OUTSIDE the shell cancel each other. That means that the dynamics of the shell depend only on the total mass enclosed by it. We will assume that the peculiar motions of matter are so small, compared to the large-scale motion of the shell as a whole, that the interior mass doesn't change as the shell expands.
The shell itself has some mass m. We can write the total energy of the shell as a sum of its kinetic and gravitational potential energies. This total energy will remain constant as the shell expands; I'll write that total energy in one particular manner.
Now, both of the terms on the left-hand side represent energy: the first is kinetic, and the second is potential. As you may recall from first-year physics classes, every potential energy is associated with a force. In general, we can derive the force in some direction x associated with some potential energy U by taking a partial derivative like so:
Q: Derive the force in the radial direction associated with
the gravitational potential energy term in the equation above.
How does this force affect the motion of an object?
Right. The gravitational force pulls the shell back toward its center, acting to slow the expansion.
Will the gravitational force succeed in slowing the expansion so much that the shell eventually halts, then starts to contract? The answer depends on Mint, the mass of all the objects inside the shell. In our previous discussion, we found a way to re-write this energy equation in terms of slightly different variables. But the general terms are still the same:
The term inside square bracket is the key: if it is positive (the kinetic energy overwhelms gravitational forces), then the shell will expand forever; if it is negative (the gravitational forces win), the shell will eventually halt, then collapse back on itself. Astronomers can measure the Hubble constant H0 and the average density of matter in the universe ρ0; then, using this equation, they can figure out how our universe will evolve in the future.
As we saw last time, the answer is -- the actual density is not large enough to stop the expansion.
| Component | Value |
| Ordinary matter | 4.6 x 10-31 g/cm3 |
| Critical density | 9.9 x 10-30 g/cm3 |
Since it's rather annoying to keep writing numerical values for densities which are all very small -- keeping track of the exponents can be difficult -- cosmologists prefer to express values of matter density (or other densities) in relative terms, using the capital Greek letter Ω. The critical density is normalized to a value of Ω = 1, and all other densities are expressed as some fraction of this value. In these terms,
| Component | Value |
| Ordinary baryonic matter | Ωb = 0.05 |
| Critical density | Ω = 1 |
We use the term "baryonic matter" to describe all the matter made up of particles known as baryons, which include protons and neutrons; in other words, "baryonic" matter includes all the ordinary matter on Earth, or in the Solar System, or in the Milky Way Galaxy: planets, stars, clouds of cold gas, clouds of hot gas -- everything we can see. When we add up all this material, we find it falls far short of the amount required to cause the universe to stop expanding.
Q: Is there any evidence for matter that we CAN'T see?
Yes, of course! Analysis of the rotation curves of spiral galaxies, or the motions of galaxies in giant clusters, or the deflection of light from distant quasars as it flies through galaxy clusters, suggests that very large concentrations of some sort of material which
We don't know exactly what it is, but astronomers can measure the amount of this dark matter via several different techniques. We find that there is considerably more dark matter than ordinary baryonic matter in the universe ... but it's still not enough to halt the expansion.
| Component | Value |
| Ordinary baryonic matter | Ωb = 0.05 |
| Dark matter | Ωdm = 0.24 |
| Total matter | Ωm = 0.05 + 0.24 = 0.29 |
| Critical density | Ω = 1 |
So, in a universe filled only with matter, it seems that the expansion will continue forever.
Back in 1917, soon after publishing his theory of general relativity, when Einstein was going through a relativistic version of this analysis, he realized that a universe filled only with matter would yield non-stationary solutions. That is, he deduced that a universe governed by gravitational forces alone would NOT be static (just as we did). It could expand, or contract, but it couldn't remain fixed in size.
Unfortunately, this date -- 1917 -- was still a decade before Hubble and Humason and other observers began to see the evidence for expansion. As far as Einstein knew,
So, in order to cause the theory to agree with the observations (of that day), Einstein modified the equation by adding one term. We can paraphrase his work in a simplified way like so:
This new term acts like a potential energy. It must therefore have an associated force.
Q: Derive the force associated with this potential
energy.
How does this force affect the motion of an object?
Right. The cosmological constant terms adds a force which pushes radially outward. That's exactly what Einstein needed in order to counteract the force of gravity, so that the universe could remain static, balanced perfectly between the gravitational pull inward and the lambda-powered expansion outward.
Just as we speak about the density of matter in two different ways,
we can also discuss the cosmological constant in two ways, the second of which is both normalized to the "critical" value and unitless.
About 12 years after Einstein put forth this "cosmological constant" as a means to prevent the universe from expanding (or shrinking), Hubble and others showed convincingly that the universe WAS expanding. Rats! If Einstein had not inserted that constant into his equations, he could have predicted that galaxies would be moving away from us (or speeding toward us).
Actually, several years after Einstein's work, other cosmologists (Alexander Friedman and Georges Lemaitre) did suggest that the relativistic theory would be consistent with an expanding or contracting universe, rather than rejecting the possibility. So the idea was present in the literature before Hubble's observational evidence in 1929.
It seems very likely that Albert Einstein considered the introduction of Λ into his theory a very unfortunate mistake. I suspect he would be very happy to learn of later developments in the field, as we are about to see ourselves.
Okay, so we've added a new parameter to the list of cosmological quantities. What effect does it have on the evolution of the universe?
The simple answer is "it fights against the pull of gravity". If the density of matter in the universe is just a bit larger than the critical value, we would expect the universe to cease expanding and collapse on itself; but if Λ is larger than zero, it can cause the universe to keep expanding.
Let's play around a little.
Suppose we set the amount of matter in the universe to much more than the critical density -- say, ΩM = 3.5. By itself, that would cause the universe to stop expanding and contract back on itself quite quickly. But if we add even small amounts of Λ to the universe, we can delay the contraction, or even prevent it entirely.
If we instead set the amount of matter to something like its observed value, then even a small amount of Λ can lead to a much faster expansion -- which will never turn around.
What else can we say about Λ in a qualitative way?
The large-scale geometry of the universe, its curvature, if you prefer to think of it that way, depends on both the amount of matter and on Λ. In fact, the curvature of the universe depends on the sum of
"What does it mean for space to be curved?" you might ask.
There are a number of consequences, but one relatively simple
one involves drawing a triangle.
On a flat plane -- which is where we teach students about geometry --
the sum of the interior angles of any triangle is always 180 degrees.
But on a curved surface, the sum may be larger or smaller.
Consider this right triangle, drawn on the surface of the Earth.
One way to determine the value of the cosmological constant
is to find some sort of
standard candle:
a class of objects which are all identical in luminosity,
or absolute magnitude.
If one measures the apparent magnitude of such objects
at different redshifts,
one will see
So, in theory,
making precise measurements of the change in apparent magnitude
with redshift should allow astronomers to figure out the values
of both
ΩM
and
ΩΛ.
In practice, this is almost impossible.
I'll provide three (and there are more).
Nonetheless, because this puzzle is such a fascinating one,
scientists spent many years of effort trying to acquire enough
observations to make the test.
In the late 1990s, two large teams of astronomers, each finding
and monitoring type Ia supernovae,
announced their results nearly simultaneously.
Here's the abstract from the group led by Adam Riess and
Brian Schmidt ...
... and here's the abstract from the "Supernova Cosmology Group"
led by Saul Perlmutter.
Both groups claimed that
What was the evidence behind their claims?
Take a look at the diagram below,
which shows the apparent magnitude of a large number
of type Ia supernovae over a wide range of distances.
The exact shape of this curve --
how quickly objects fade as a function of redshift --
depends on the
geometry of the universe;
specifically, on the values of the
matter density and cosmological constant.
The graph shows both measurements (red, white, and yellow
circles with errorbars)
and models (dashed blue lines).
Since all the important information is crammed into that small
region near the top-right area of the graph,
it helps to display the same information in a slightly different
way.
The figure below shows a similar graph in the upper panel,
but a new view in the lower panel:
the differences in apparent magnitude of the supernovae
when compared to one particular baseline model
(in which there is no cosmological constant, and the
matter density is equal to the critical density).
It is pretty clear that the high-redshift supernova data
support a non-zero value
for the cosmological constant.
The work by these two groups was so persuasive, and so significant,
that the
2011 Nobel Prize for Physics
was awarded to the leaders of these teams.
At the time of writing,
the astronomical community has
settled on a "standard model" which
has value of roughly
Image courtesy of
NASA / WMAP Science Team
Q: What is the sum of the angles in a triangle?
Location Latitude Longitude
------------------------------------------------------------
North Pole
Quito, Ecuador
Libreville, Gabon
------------------------------------------------------------
Q: What is the size of each angle in this triangle?
Q: What is the sum of the angles in this triangle?
So, what's the value of Λ?
Q: Why is it so hard to test various models of the universe in
this way? Provide at least two reasons.
Abstract of
Riess et al., AJ 116, 1009 (1998)
Abstract of
Perlmutter et al., ApJ 517, 565 (1999)
Q: Which of the four blue models (four blue dashed lines) best fits
the measurements of type Ia supernovae?
Figure 2 slightly modified from
Perlmutter et al., ApJ, 517, 565 (1999)
Figure 5 taken from
Riess et al., AJ 116, 1009 (1998)
Q: Which of the three models shown above fits the data best?
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Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.