Rotation Curve of Galaxy:

Dynamical studies of the Universe began in the late 1950's. This meant that instead of just looking and classifying galaxies, astronomers began to study their internal motions (rotation for disk galaxies) and their interactions with each other, as in clusters. The question was soon developed of whether we were observing the mass or the light in the Universe. Most of what we see in galaxies is starlight. So clearly, the brighter the galaxy, the more stars, therefore the more massive the galaxy. By the early 1960's, there were indications that this was not always true, called the missing mass problem.

The first indications that there is a significant fraction of missing matter in the Universe was from studies of the rotation of our own Galaxy, the Milky Way. The orbital period of the Sun around the Galaxy gives us a mean mass for the amount of material inside the Sun's orbit. But, a detailed plot of the orbital speed of the Galaxy as a function of radius reveals the distribution of mass within the Galaxy. The simplest type of rotation is wheel rotation shown below.

Rotation following Kepler's 3rd law is shown above as planet-like or differential rotation. Notice that the orbital speeds falls off as you go to greater radii within the Galaxy. This is called a Keplerian rotation curve.

To determine the rotation curve of the Galaxy, stars are not used due to interstellar extinction. Instead, 21-cm maps of neutral hydrogen are used. When this is done, one finds that the rotation curve of the Galaxy stays flat out to large distances, instead of falling off as in the figure above. This means that the mass of the Galaxy increases with increasing distance from the center.

The surprising thing is there is very little visible matter beyond the Sun's orbital distance from the center of the Galaxy. So, the rotation curve of the Galaxy indicates a great deal of mass, but there is no light out there. In other words, the halo of our Galaxy is filled with a mysterious dark matter of unknown composition and type.

Cluster Masses:

Most galaxies occupy groups or clusters with membership ranging from 10 to hundreds of galaxies. Each cluster is held together by the gravity from each galaxy. The more mass, the higher the velocities of the members, and this fact can be used to test for the presence of unseen matter.

When these measurements were performed, it was found that up to 95% of the mass in clusters is not seen, i.e. dark. Since the physics of the motions of galaxies is so basic (pure Newtonian physics), there is no escaping the conclusion that a majority of the matter in the Universe has not been identified, and that the matter around us that we call `normal' is special. The question that remains is whether dark matter is baryonic (normal) or a new substance, non-baryonic.

Mass-to-Luminosity Ratios:

Exactly how much of the Universe is in the form of dark matter is a mystery and difficult to determine, obviously because its not visible. It has to be inferred by its gravitational effects on the luminous matter in the Universe (stars and gas) and is usually expressed as the mass-to-luminosity ratio (M/L). A high M/L indicates lots of dark matter, a low M/L indicates that most of the matter is in the form of baryonic matter, stars and stellar remnants plus gas.

A important point to the study of dark matter is how it is distributed. If it is distributed like the luminous matter in the Universe, that most of it is in galaxies. However, studies of M/L for a range of scales shows that dark matter becomes more dominate on larger scales.

Most importantly, on very large scales of 100 Mpc's (Mpc = megaparsec, one million parsecs and kpc = 1000 parsecs) the amount of dark matter inferred is near the value needed to close the Universe. Thus, it is for two reasons that the dark matter problem is important, one to determine what is the nature of dark matter, is it a new form of undiscovered matter? The second is the determine if the amount of dark matter is sufficient to close the Universe.

Baryonic Dark Matter:

We know of the presence of dark matter from dynamical studies. But we also know from the abundance of light elements that there is also a problem in our understanding of the fraction of the mass of the Universe that is in normal matter or baryons. The fraction of light elements (hydrogen, helium, lithium, boron) indicates that the density of the Universe in baryons is only 2 to 4% what we measure as the observed density.

It is not too surprising to find that at least some of the matter in the Universe is dark since it requires energy to observe an object, and most of space is cold and low in energy. Can dark matter be some form of normal matter that is cold and does not radiate any energy? For example, dead stars?

Once a normal star has used up its hydrogen fuel, it usually ends its life as a white dwarf star, slowly cooling to become a black dwarf. However, the timescale to cool to a black dwarf is thousands of times longer than the age of the Universe. High mass stars will explode and their cores will form neutron stars or black holes. However, this is rare and we would need 90% of all stars to go supernova to explain all of the dark matter.

Another avenue of thought is to consider low mass objects. Stars that are very low in mass fail to produce their own light by thermonuclear fusion. Thus, many, many brown dwarf stars could make up the dark matter population. Or, even smaller, numerous Jupiter-sized planets, or even plain rocks, would be completely dark outside the illumination of a star. The problem here is that to make-up the mass of all the dark matter requires huge numbers of brown dwarfs, and even more Jupiter's or rocks. We do not find many of these objects nearby, so to presume they exist in the dark matter halos is unsupported.

Non-Baryonic Dark Matter:

An alternative idea is to consider forms of dark matter not composed of quarks or leptons, rather made from some exotic material. If the neutrino has mass, then it would make a good dark matter candidate since it interacts weakly with matter and, therefore, is very hard to detect. However, neutrinos formed in the early Universe would also have mass, and that mass would have a predictable effect on the cluster of galaxies, which is not seen.

Another suggestion is that some new particle exists similar to the neutrino, but more massive and, therefore, more rare. This Weakly Interacting Massive Particle (WIMP) would escape detection in our modern particle accelerators, but no other evidence of its existence has been found.

The more bizarre proposed solutions to the dark matter problem require the use of little understood relics or defects from the early Universe. One school of thought believes that topological defects may have appears during the phase transition at the end of the GUT era. These defects would have had a string-like form and, thus, are called cosmic strings. Cosmic strings would contain the trapped remnants of the earlier dense phase of the Universe. Being high density, they would also be high in mass but are only detectable by their gravitational radiation.

Lastly, the dark matter problem may be an illusion. Rather than missing matter, gravity may operate differently on scales the size of galaxies. This would cause us to overestimate the amount of mass, when it is the weaker gravity to blame. This is no evidence of modified gravity in our laboratory experiments to date.

Current View of Dark Matter:

The current observations and estimates of dark matter is that 20% of dark matter is probably in the form of massive neutrinos, even though that mass is uncertain. Another 5 to 10% is in the form of stellar remnants and low mass, brown dwarfs. However, the combination of both these mixtures only makes about 30% the amount mass necessary to close the Universe.

The rest of dark matter is called CDM (cold dark matter) of unknown origin, but probably cold and heavy. The combination of all these mixtures only makes 20 to 30% the amount mass necessary to close the Universe. Thus, the Universe appears to be open, i.e. _M is 0.3.

Dark Energy:

With the convergence of our measurement of Hubble's constant and _M, the end appeared in site for the determination of the geometry and age of our Universe. However, all was throw into turmoil recently with the discovery of dark energy. Dark energy is implied by the fact that the Universe appears to be accelerating, rather than decelerating, as measured by distant supernovae.

This new observation implies that something else is missing from our understanding of the dynamics of the Universe, in math terms this means that something is missing from Friedmann's equation. That missing something is the cosmological constant, .

In modern cosmology, the different classes of Universes (open, flat or closed) are known as Friedmann universes and described by a simple equation:

In this equation, `R' represents the scale factor of the Universe (think of it as the radius of the Universe in 4D spacetime), and H is Hubble's constant, how fast the Universe is expanding. Everything in this equation is a constant, i.e. to be determined from observations. These observables can be broken down into three parts gravity (matter density), curvature and pressure or negative energy given by the cosmological constant.

Historically, we assumed that gravity was the only important force in the Universe, and that the cosmological constant was zero. Thus, if we measure the density of matter, then we could extract the curvature of the Universe (and its future history) as a solution to the equation. New data has indicated that a negative pressure, or dark energy, does exist and we no longer assume that the cosmological constant is zero.

Each of these parameters can close the Universe in terms of turn-around and collapse. Instead of thinking about the various constants in real numbers, we perfer to consider the ratio of the parameter to the value that matches the critical value between open and closed Universes. For example, the density of matter exceeds the critical value, the Universe is closed. We refer to these ratios as (subscript M for matter, k for curvature, for the cosmological constant). For various reasons due to the physics of the Big Bang, the sum of the various must equal one. And for reasons we will see in a later lecture, the curvature is expected to be zero, allowing the rest to be shared between matter and the cosmological constant.

The search for the value of matter density is a much more difficult undertaking. The luminous mass of the Universe is tied up in stars. Stars are what we see when we look at a galaxy and it fairly easy to estimate the amount of mass tied up in stars, gas, planets and assorted rocks. This is contains an estimate of what is called the baryonic mass of the Universe, i.e. all the stuff made of baryons = protons and neutrons. When these numbers are caluclated it is found that for baryons is only 0.02, a very open Universe. However, when we examine motion of objects in the Universe, we quickly realize that most of the mass of the Universe is not seen, i.e. dark matter, which makes this estimate of to be much too low. So we must account for this dark matter in our estimate.

Einstein first introduced to produce a static Universe in his original equations. However, until the supernova data, there was no data to support its existence in other than a mathematical way.

The implication here is that there is some sort of pressure in the fabric of the Universe that is pushing the expansion faster. A pressure is usually associated with some sort of energy, we have named dark energy. Like dark matter, we do not know its origin or characteristics. Only that is produces a contribution of 0.7 to , called , so that matter plus dark energy equals an of 1, a flat Universe.

With a cosmological constant, the possible types of Universes are numerous. Almost any kind of massive or light, open or closed curvature, open or closed history is possible. Also, with high 's, the Universe could race away. Above are a series of models for the three values of k (positive, zero, negative) and a range of values. Note that _E is the value proposed by Einstein to keep the Universe static and not expanding (which it is not).

Fortunately, observations, such as the SN data and measurements of allow us to constraint the possible models for the Universe. In terms of for k (curvature), M (mass) and (where the critical values are =1), the new cosmology is given by the following diagram.

SN data gives =0.7 and _M=0.3. This results in _k=0, or a flat curvature. This is sometimes referred to as the Benchmark Model which gives an age of the Universe of 13.5 billion years.