# What are the proper parameters?

So, there Big Bang model explains several observed features of the universe. But it contains a number of parameters -- knobs and buttons, you might consider them -- which control the precise behavior of the universe as it expands. Some of the most important of these are:

• Hubble constant: the current rate of expansion
• matter density: how much matter is there?
• cosmological constant: a "repulsive" property of space?

Check out the cast of characters for a summary of these and the major players in the Big Bang theory.

We need to figure out the values of these parameters in order to predict the past (and future) evolution of the universe.

#### Hubble Constant

In order to measure the Hubble Constant -- the rate at which the universe is currently expanding -- one must measure

• distance to a set of objects
• radial velocity to a set of objects

The Hubble Constant is then simply the ratio of the two:

```                   radial velocity    km/sec
H  =   ------------------   ------
distance          Mpc
```

One complication is that the velocities of some galaxies are modified by gravitational forces from other galaxies. For example, a galaxy in a cluster may have a peculiar velocity of up to 1,000 km/sec, positive or negative, due to its motion within the cluster. If the "natural" radial velocity of a galaxy would be only 2,000 km/sec, its motion within a cluster could modify its apparent radial velocity greatly.

In order to make accurate measurements of H, it helps to look at very distant galaxies, because their radial velocities are so large -- say, 20,000 km/sec -- that gravitationally-induced peculiar velocities are negligible. Unfortunately, it is very difficult to measure distances to such galaxies. It's a bit of a catch-22:

1. distances to nearby galaxies are easier to measure, but their radial velocities are usually perturbed significantly by motions due to neighbors
2. distances to far-away galaxies are extremely hard to measure, but their radial velocities are not significantly modified by neighbors

The best current estimates for the Hubble Constant are somewhere around

```           H  =    60 - 80     km/sec/Mpc
```

What does the value of H imply?

``` If H is large         the universe is expanding     age since the
rapidly, so it didn't         starting point
take long to reach its        is "small"
current size

If H is small         the universe is expanding     age since the
slowly, so it has taken       starting point
a long time to reach its      is "large"
current size
```

In fact, to a very rough approximation, the value of H gives the age of the universe; just figure out its numerical value after cancelling all the units you can, and then take the reciprocal.

```               km/sec        m / sec         1
H   =   ------   =   --------   =   -----
Mpc            m            sec

1
---  =   sec
H
```

Now, this isn't the true age of the universe, just a rough approximation that may be accurate to within a factor of two or so. Why isn't it the true age? Because it assumes that the universe has been expanding at the current rate ever since the starting point. But that's not true, because gravitational forces have been slowing down the expansion ... so the true age depends on how much matter there is in the universe.

And, if the cosmological constant is not zero, it has also been changing the expansion rate, causing it to increase.

Is there any way to check this age estimate? Yes. One way to is use the known rates at which radioactive elements decay:

• if one knows how long it takes element X to decay into element Y,
• and one can measure how much of element X there is in a star,
• and one can measure how much of element Y there is in a star,
• and one knows the initial ratio of element X to element Y
then one can calculate the age of the star. In Febrary, 2001, a team of astronomers announced the results of such a calculation. They used one of the 8.2-m Very Large Telescopes in Chile to take the spectrum of a star, CS 31082-001.

In the star, they found lines of uranium (the parent element) and thorium (the daughter element). Notice just how tricky it is to measure the strength of the uranium line!

Based on their measurements, they determine an age of 12.5 +/- 3 billion years.

#### Matter Density

The more matter there is, the more its gravitational force opposes the expansion of the universe. If there is enough matter, more than the critical density, then the universe may eventually stop expanding and begin to contract.

How can we measure the amount of matter in the universe? There are several ways:

1. Add up the visible matter: When we look in the sky, we see stars, galaxies, clouds of gas and dust. We can estimate the mass of all this material and add it up. If we know the distance to each bit of material (oops, there's the need for distance measurements again!), we can calculate the density of visible matter.

Typical values for visible matter density is

```    visible matter density  =  0.001 to 0.01 * (critical density)
```

2. Add up the dark matter, too: There are several different kinds of evidence for some sort of dark matter: for example, galaxies in some clusters move more quickly than they would under the influence of the visible matter in those clusters. If we calculate the mass of material required to produce the observed motions via gravitational forces, we typically find about ten times as much dark matter as visible matter:
```    dark matter density  =  0.05 to 0.2 * (critical density)
```

3. Use nucleosynthesis: We believe that, in the first few minutes after the starting point of the Big Bang, when the temperatures were in the millions of degrees and the density was very high, nuclear reactions produced a set of light elements:
• lots of hydrogen
• some helium
• tiny amounts of deuterium (heavy hydrogen)
• tiny amounts of lithium

It turns out that the relative amounts of these light elements depends very sensitively on the density of the universe during those early times. The higher the density, the larger the amount of helium produced, and the smaller the amount of deuterium (because the deuterium fuses into helium more easily when density is high).

Our best measurements of the abundances of these light elements lead us to conclude that the density of baryonic matter (ordinary matter, which makes up the ordinary elements) is

```   baryonic matter density  =  0.015 to 0.05 * (critical density)
```

No matter how you slice it, it appears that the amount of matter (visible, dark, or baryonic) is far below the critical amount needed to halt the expansion of the universe. So, if the only factor in the evolution of the universe was the gravitational attraction of matter, the universe would never stop expanding....

#### The Cosmological Constant

The behavior of space is governed by the theory of general relativity (GR). GR describes space as (possibly) being "curved". What does that mean?

Consider an analogy to our three-dimensional universe: a two-dimensional world, a Flatland upon which creatures wriggle. If the creatures have taken high-school geometry, they can test their 2-D universe for curvature by drawing a triangle and measuring the angles. In a Euclidean, flat space, the angles will add up to exactly 180 degrees:

But if the creatures happen to live on a sphere, a positively curved space, they will notice that the sum of the angles of a triangle is more than 180 degrees:

Drawing big triangles isn't the only way to test for curvature. Some other tests are:

• measure the brightness of a source of light as it moves farther and farther away
• measure the apparent size of an object as it moves farther and farther away

There are three possibilities for the geometry of a 3-D universe

```curvature      angles          brightness             size
--------------------------------------------------------------
flat          sum = 180      dims as the         shrinks as the
inverse square       inverse of
of distance          distance

positive      sum > 180      dims more slowly    shrinks more
(sphere)                      than inverse         slowly than
square               the inverse

negative      sum < 180      dims more quickly   shrinks more
(saddle)                      than inverse         quickly than
square               the inverse
```

Solutions to Einstein's equations of general relativity indicate that a completely empty universe would have negative curvature, and would expand rapidly and forever.

But GR also indicates that space can be "warped" by the presence of matter. Matter has two effects on the space in its vicinity:

1. it applies a positive curvature
2. it slows the expansion
We usually think of the second item as due to gravitational forces between bits of matter.

When Einstein solved the equations of general relativity, he found that there was another factor, besides matter, which might "warp" space. He called this factor the cosmological constant, usually denoted by the capital Greek letter lambda. Now, the cosmological constant is wierd:

1. it applies a positive curvature to space (like matter)

but

2. it accelerates the expansion (unlike matter)

So, is there any evidence for a non-zero cosmological constant? Yes -- but it's a bit indirect. The evidence goes something like this:

• If we know the matter density already, we can calculate how much it curves space ...
• and if we then observe the actual curvature of space ...
• then we can figure out how much extra (if any) curvature must be due to the cosmological constant
Two of the current methods astronomers currently use to observe the curvature of space are
1. Observations of distant Type Ia supernova: When we compare the distance of a nearby galaxy to its radial velocity, we find a simple linear relationship. But if we do the same for very, very distant objects, this relationship can change:
• if the matter density is exactly equal to the critical density, and cosmological constant is zero, then the linear trend continues
• if space is positively curved, objects appear to be closer (and brighter) than their radial velocity (redshift) would predict
• if space is negatively curved, objects appear to be farther away (fainter) than their radial velocity (redshift) would predict

If we believe we know the true luminosity of Type Ia supernovae, and we can observe them at great enough distances, we might be able to check for any deviation from the linear relationship -- which would indicate the overall curvature of the universe.

The current observations indicate that there is probably some negative curvature to space; and, if given the very small matter density, this indicates a relatively large value for the cosmological constant. The basic result from the supernova observations is that the cosmological constant is probably between 0.5 and 1.0.

2. Observations of the cosmic microwave background: When we look at the distribution of galaxies in the nearby universe, we see some clumpiness, both in the real world:

and in simulations of the universe:

These "clumps" are clusters of galaxies, and groups of clusters, and concentrations of groups of clusters, and so on. Because material at the center of clump exerts a strong gravitational pull on its surroundings, it attracts nearby material, and grows with time. A big clump at the current time must have been a smaller clump at an earlier time.

In fact, we can run simulations of the universe, in which we take the current distribution of galaxies and follow them backwards in time. The big, obvious clusters and superclusters turn into smaller, very subtle blobs in the early universe. Our simulations tells us how big those blobs ought to be when the universe became neutral and transparent for the first time.

If we compare the actual size of the blobs in the microwave background with the size predicted by simulations, we can check for the curvature of space:

• if space is flat, the blobs will be close to the predicted size (about 1 degree wide)
• if space is negatively curved, the blobs will smaller than predicted
• if space is positively curved, the blobs will be larger than predicted

The size of the blobs in the BOOMERANG maps are pretty darn close to what we'd expect if the universe is geometrically flat. Again, since we can estimate how much curvature is caused by the relatively small matter density, we can figure out how much of the remainder must be due to the cosmological constant. The basic result from BOOMERANG is that the cosmological constant is probably between 0.5 and 1.0.

Another balloon-borne experiment to measure the CMB, called MAXIMA, has reached a similar conclusion. It looked at a somewhat smaller area of the sky (and in a different direction), but with higher resolution. The maps made by the two experiments are very similar. The pictures below show a square roughly 10 degrees on a side from BOOMERANG on the left, MAXIMA on the right:

If we combine information from the supernova observations and the CMB observations, we can place reasonably strong constraints on two important cosmological parameters:

• the density of matter (both ordinary baryonic matter and the mysterious dark matter)
• the "density" contributed by the cosmological constant

At the present time (Feb 13, 2001), it looks like the matter density is only about 30 percent of the critical density (with only 5 percent or so due to ordinary baryonic matter). Because the overall geometry of the universe appears to be flat, that means that the cosmological constant must contribute about 70 percent of the critical density.