Dark matter and the value of

Introduction

As stated on our main page, the aim of this research-project is to find the answer to the question: How sure are we that there exists dark matter. To do this, we have decided to examine the assumptions which are involved in the different kinds of evidence, which are also stated on our main page. One of these involves the value of and this one will be discussed here to answer the question: How strong is the evidence for the existence of dark matter following from the value of ? is defined to be the ratio of the average mass density to the critical mass density of the universe. I will explain this in more detail in the following section, in which I will also explain how this parameter is related to dark matter. At that point, the following subquestions follow in a natural way:

  1. How can we measure the value of ?
  2. How can we estimate a value for based on the mass, which we can detect?
If there exist dark matter, these two values for will differ from each other, since the first includes dark matter, while the second is purely based on the matter we are actually able to detect.

What is ?

There are three possible scenario's for the future of the universe:

  1. The universe is open. In this case, the matter of the universe has gained velocities during the Big Bang, which exceeded the escape velocity, so the universe will expand forever. In other words, the gravitational pull acting on all the mass in the universe, is too small to bring the expansion to a halt.
  2. The universe is closed. In this case, the velocity, which the matter of the universe has gained during the Big Bang, is smaller than the escape velocity, so the universe will eventually stop expanding and collapse again. In other words, the gravitational pull acting on all the mass in the universe is strong enough to bring the expansion to a hold and let the universe contract again.
  3. The universe is flat. In this case, the velocity, which the matter of the universe has gained during the Big Bang, is equal to the escape velocity, so the universe will expand slower and slower until the expansion stops when age of the universe is equal to infinity; the gravitational pull acting on the different parts of the universe is just to small to let the universe contract again.
All these scenario's are represented in the following graph (Zeilik, Gregory and Smith, 1992). R(t) is the scaling factor of the universe, which is directly proportional to the radius of the universe, which can be regarded as a sphere, as a function of time.

An important quantity in cosmology is the critical mass density. This is defined to be the mass density, which is required for a flat universe. is closely related to these quantities, since it is defined in the following way:

How is related to dark matter? The problem is, that when is measured, values of about 0.3 to 0.45 (Eke e.a. 1998, De Laix 1998) are obtained, but when is estimated based on the mass we can detect, values of about 0.1 to 0.2 (Evrard 1987, Huchra and Geller 1982) are obtained. The difference between these values has to be dark matter, since that is the only matter, which is not taken into account in the estimation, but is in the measurements.

For this research-project, the question to be answered is: how strong is this piece of evidence for the existence of dark matter. To answer this question, the following questions automatically arise:

  1. How can we measure ?
  2. How can we estimate a value for based on the mass, which we can detect?
These questions have to be answered, since they are the only way to find the assumptions, which lead to the values mentioned above, which in turn suggest the existence of dark matter, which is an important fact in view of our main question.

How can we measure ?

I've found four ways to measure :

  1. A method involving redshifts of galaxies. De Laix e.a. (1998) have made an error-analysis on this method, which was found by Alcock and Paczynski. Although they intended this method to be used for measuring Lambda (another cosmological parameter), it can and is used for measuring . This seemed the most promising method, because it offered a way to measure directly instead of measuring other parameters first and calculating from those results and it is based on redshifts, which we can measure quite accurately, so I've chosen to concentrate on this method.
  2. A method involving the cosmological background radiation.
  3. A method involving the evolution of clusters (Eke e.a. 1998).
  4. A method involving gravitational lensing. I heard of the existence of such a method from Steven Louise , who examined the evidence for dark matter following from gravitational lensing, but unfortunately, he hasn't put a reference about this sunject on his page.

The theory behind the method involving redshifts consists of one formula, in which the decelleration-parameter, q0 and are the only constants, One can find values for these constants by fitting one's measurements with this formula. The best way to find the assumptions which are involved in this technique is following the path, that leads to this formula. Since the mathematics are quite cumbersome, I have decided to give a sketch of what the method does and list of assumptions which are involved. For a more thorough discussion, on will have to go through the mathematics.

The description, which follows, is not an accurate description, but it does give an idea of what this method does in a way, which is a lot easier to imagine than the real line of reasoning. Imagine, that you are looking at a sample of stars (or other objects, but I will keep calling them stars) on the sky, which form a group. These stars will have a certain redshift. This redshift can be explained by the Doppler-effect, since the stars will be moving with respect to the earth. But let's assume for the moment, that those stars don't move with respect to us, so they're moving with us, but that we still measure the same redshift and that the equation for the Doppler-shift is still valid. The only way we can accomplish this construction is to accept, that the speed of light is NOT constant!

Since we can only measure the redshift of stars in our line of sight, the speed of light only changes in the direction of the line of sight and not in the direction perpendicular to it, so we have mathematically created a new reality in which the speed of light within a group of stars has two different values in two perpendicular directions. This has important consequences.

Now, imagine, that you lived in that group of stars in that mathematically created world. As mentioned above, the speed of light would have two different values in two perpendicular directions. Imagine, that you would apply the normal Doppler-equation in your situation and would measure the redshift of two stars: one in the direction in which the speed of light hasn't changed and one in the direction in which the speed of light has. Imagine you measured the same redshift for both stars. In that case, the outcome of your experiment would be, that these stars wouldn't have the same velocity.

The consequence of this is, that one of the stars will move quicker away from you than the other. Now imagine that you do this with a group of stars, which are more or less spherically distributed around you and which have the same redshift. In one direction the speed of light will differ from the other two (we're now entering the third dimension, which is also unaffected, since there is only one direction in which the value of the speed of light changes). The result is, that if you would live long enough, you would see how this sphere deformed into an ellipse, since some stars are moving out quikker than other stars. This rate of deformation from a sphere into an ellipse can be measured and expressed in terms of and q0 (Alcock and Pazcynski 1979)

To get to the final result, one has to run through a lot of mathematics, during which one has to make the following assumptions:

  1. the universe is isotropic, which means, that it doesn't matter in which direction one looks: one should see the same everywhere;
  2. the universe is homogeneous, which means, that it doesn't matter where one is, since the universe is physically the same everywhere;
  3. the physically measurable pressure in the universe is equal to zero;
  4. the internal energy of the universe is equal to the density time the square of the speed of light;
  5. the expansion of the universe is uniform, which means, that the universe expands at the same speed at every position in it;
  6. an assumption about the dynamical evolution of the universe, but one can wonder of this is really another assumption, since it follows from the general theory of relativity with the assumptions 3 and 4 mentioned above.
The final result is:

This expression is only valid if the curvature of the universe, determined by the value of a constant k, follows from k=1. There are also formulae for other curvatures (Alcock and Paczynski 1979), but I'll only use this one, since all measurements and estimations indicate, that <1, so k seems to be equal to 1, so I only mention this one here. Since this is a very complicated expression and because results are obtained by fitting data to it, I've made some graphs, which relate the ratio /z to z for different values of 0.

Before I could start plotting those graphs, I needed to make some choices:

  1. A choice for the curvature of the universe. As mentioned above, this choice was already made by chosing a certain formula.
  2. A choice for Since measurements using this technique result in values of ca. 0.4, I have decided to make three graphs: one for each of the values 0.35, 0.4 and 0.45. I have chosen to do so, since by doing this, it might be possible to see, how sensitive this technique is: if the graphs differ a lot, the technique will be more accurate than in the case, in which they are quite similar.
  3. A choice for the value of q0. Geometrically, this parameter is given by:

    which is equivalent to:

    which should look very familiar. Since I'm assuming an open universe, I don't want the universe to be in simple harmonic motion, so I shouldn't take the value of q0 positive. If I choose q0=0, then there wouldn't be any relation between the decelleration of the universe and the size of the universe. Personally, I regard this to be very unlikely, since the distances between all the objects in the universe is a very important factor for the gravitational pull acting on all those objects. Since the decelleration of "the outer rim" of the universe is hard to determine (we can only measure the recessional speed), I will put q0=-0.5 q0=-1 and q0=-1.5, so that it is possible to get an idea of the impact of this parameter on the resulting graph.
By doing this, I get the following graphs (all graphs display /z versus z).



The upper-left graph shows the result for q0=-0.5, the upper-right one for q0=-1.0 and the lower-left graph for q0=-1.5, all for equal to 0.35, 0.4 and 0.45. Since the curves are quite similar for q0=-0.5 and q0=-1.0, it will be hard to get an accurate value for by fitting your data on this graph. But the only thing, that is relevant for this research project, is the difference between the values for following from measurement by, for example, this method and from estimation by, for example, the method involving Mass-to-Light-ratios described below. Since estimates give a value of about 0.1 to 0.2, I have made another graph showing this relationship for equal to 0.4, 0.2 and 0.1 respectively for q0=-1.0. This is the lower-right graph. As you can see, these graphs differ enough from each other to decide which value of fits best to your data, so we can't deny this difference by pointing out, that it is just the result of inaccuracies in the measurements.

I've made these graphs using Mathematica and by entering the following commands (the result is the upper-right graph):

Although there are a lot of assumptions needed to arrive at the final equation, I still think, that this is a very good method, since the variables which have to be measured can be measured with a high degree of accuracy and this method offers a way to find a value for , which does not depend on one single measurements. but a value, which gets better by doing more measurements, since the measurements can be chosen to be totally independent by spreading them out over a wide range of different redshifts. The only problem is, that the value one finds for

De Laix (1998) has made a thorough study of the errors in this method and has come up with = 0.3+/-0.18, while Eke e.a. (1998), who have used a method involving cluster evolution, have found = 0.45+/-0.25. I think that the latter value is somewhat too high since this was found by means of cluster evolution and I think that the average density of the universe is somewhat lower than that of a cluster.

How can we estimate a value for based on the mass, which we can detect?

One way to estimate the value of is by means of the Mass-Luminosity-Relationship of groups of galaxies. This is based on the simple relation (A.E.Evrard, 1987):

If this equation is used, the following questions can be raised:

  1. How is the Mass-Luminosity-Relationship calibrated?
  2. How is the critical Mass-Luminosity-Relationship estimated or determined?

One way in which the Mass-Luminosity-Relationship is calibrated is by using the virial theorem (Huchra and Geller, 1982). This is done by measuring the luminosity and determining the mass by means of the virial theorem and bringing those data together. For the purpose of this research-project, this isn't the best way to do it, I think, since, if there is dark matter, is will be counted as visible mass, since the use of the virial theorem doesn't take into account the difference between detectable matter and dark matter. On the other hand, even with this possible piece of dark matter included, the value for found this way is still lower than the value found by means of the redshift-method described above, so one could also regard it as a very strong piece of evidence for the existence of dark matter.

Another way to calibrate the Mass-Luminosity-Relationship is based on the Initial Mass Function. This is explained by Douwe, but he concludes, that this isn't a useful way to calibrate a Mass-Luminosity-Relationship.

Conclusions

According to direct measurements has a value between 0.3+/-0.18 (De Laix 1998) and 0.45+/-0.25 (Eke e.a. 1998), of which the latter may be somewhat too high. Values for between 0.2 and 0.48 are within on standard deviation of both measurements, so a value of more or less directly.

According to the estimations based on the detectable mass has a value of about 0.1 to 0.2 (Evrard 1987, Huchra and Geller 1982). Since this involves dynamics, there will probably some dark matter be taken as visible matter, so the value we are looking for, will, I think, be about 0.05 to 0.1.

The conclusion, that follows from these measurements is, that 50% to 90% of all the mass in the universe is dark matter.

Suggestions for further study

Since I had only two full weeks of work and needed to learn the basics of cosmology first, I didn't manage to take up all the subjects and possibilities, so there's still a lot left. One could for example study the other two methods to measure or dig further into the calibration of the Mass-Luminosity-Relationship. At the end of their article, Huchra and Geller refer to some other articles, which deal with other methods to calibrate the Mass-Luminosity-Relationship. This might be a good starting point.

Literature