Today we will talk about dark matter.

We will talk about dark matter because I constantly see popular science articles overstating how

much we really know about it.

The truth is, we’re still not sure that it actually exists.

Before I say anything else, I have to tell you what we mean when we say dark matter.

Dark matter is characterized by three properties.

First, you may think it’s dark, but actually it’s transparent.

It does not emit light, but also it does not absorb light, not at any frequency.

Second, it’s matter.

If you have dark matter in a bag and you double the volume of the bag, then the energy-density

goes down to one half.

Now of course we don’t have dark matter in bags, but the same holds for the expansion

of the universe: The energy-density of dark matter falls inversely with the volume.

This is not in general the case.

For example for radiation the energy-density falls faster than the inverse volume because

the frequencies also redshift.

Third, dark matter interacts weakly, both with itself and with other matter.

When physicists say “weakly interacting” they sometimes mean as weak as the weak nuclear

force.

But this is not what’s meant here.

Dark matter interacts at least as weakly as the weak interaction but maybe weaker than

that.

And that’s what we mean when we talk about dark matter.

So why do physicists think dark matter exists?

That’s because it’s a good explanation for a lot of observations that have been made

starting in the 1930s.

First, there are the galaxy clusters.

Galaxy clusters are collections of some hundreds of galaxies that are held together by their

own gravitational pull.

The higher the total mass in the cluster, the higher the average velocities of the galaxies

in the cluster.

Now it turns out that the observed mass is not high enough to explain the observed velocities.

The easiest way to fix this is to postulate that clusters contain large amounts of dark

matter.

Second, galaxies.

The story for the galaxies is similar to the story for the galaxy clusters.

If you plot the velocity as a function of the distance from the galactic center, then

– going by the known, normal mass only – you would expect the velocities to drop.

But this is not what we observe.

Instead the velocities become constant for stars far away from the galactic center.

This is what’s known as a “flat rotation curve” and again dark matter is a simple

way to explain the observations.

Third, there is gravitational lensing.

According to Einstein’s theory of general relativity, the force we call gravity is caused

by the curvature of space and time.

So mass curves space and light is deviated by the curvature.

This means that masses act like lenses and can focus light.

Just how strongly they will focus light depends on the amount of mass.

And gravitational lenses too tell us that normal mass alone is not sufficient to explain

the observations.

Then there is the cosmic microwave background.

That’s radiation at an average temperature of about 3 Kelvin that comes from all directions.

It has its origin in the hot plasma in the early universe.

Around the average temperature of three Kelvin there are small temperature fluctuations that

correspond to spots that once where a little hotter or a little colder.

The number of spots of a certain size can be counted and you plot the number against

the size, you get what is known as the power spectrum of the cosmic microwave background.

The spectrum has a few peaks which correspond to spots of sizes that are a little more frequent

than others.

The important thing is that the relative height of the peaks depends on the amount of dark

matter.

Again the observations strongly speak for the presence of dark matter.

Fifth, there is structure formation.

As the universe expands, matter cools and it clumps and it begins to forms structures.

But when normal matter, when it clumps, builds up radiation pressure and that acts against

the clumping.

Dark matter, on the other hand interacts only weakly, and so it will not build up this pressure.

Dark matter therefore clumps more efficiently.

And Indeed it will then help the normal matter clump.

Without dark matter, the universe today wouldn’t have sufficient structures.

These five different types of evidence for dark matter have the same origin.

As I said, general relativity tells us that matter curves space, and that curvature in

return tells the matter how to move.

This relation is captured by Einstein’s field equations.

In these equations, on the one side you have the geometric quantities that describe the

curvature of space and time, on the other side you have all types of energy and mass.

And please notice that it’s “Einstein’s equations.”

It’s a plural.

It’s not merely one equation, but actually ten different ones.

And if you solve those equations, then you know how the matter moves due to the curvature

of space and time that it itself causes.

What astrophysicists now do with their observations is to go and measure what matter there is

and how it moves and then they check whether it fulfils the equations.

And what all these observations tell us is that, no, it does not work.

It does not fulfill the equations.

And dark matter is a simple fix to make the two sides of the equations match again.

Now this may look like a pretty dumb thing to do because there is always some term that

you can add to make the two sides of the equations match again.

But the point is that dark matter is a particularly simple way to make it work.

Recall that I told you that Einstein’s equations are actually ten different equations.

So correspondingly you need in general ten different components necessary to make two

sides of the equations match.

But for dark matter, you only need one.

And that one component you can define by one constant, for example the density of dark

matter today.

So, dark matter is a parametrically simple explanation.

It would be even simpler if it was a kind of particle that we know already, but unfortunately

that does not work.

From the known particles none really fit the bill.

They either interact with light, so they are not dark, or they do not clump enough in the

early universe.

We also know that dark matter cannot be some kind of dark stellar object like for example

black holes or brown dwarfs, because these would give rise to frequent gravitational

lensing and we have not seen this.

So we need something new, and most physicists think that dark matter is some new kind of

particle.

So, yes, dark matter is a simple explanation.

Or at least it used to be.

But in the last decades it has become clear that there are some observations which dark

matter can not so simply explain.

Dark matter, for example, turns out to result in density peaks in the center of galaxies,

which does not fit well with observations.

Structure formation with dark matter also predicts a large number of small galaxies

– the so-called dwarf galaxies – and those haven’t been found.

The biggest problem with dark matter however is something it does not predict.

If you look at the stars in galaxies, you can use the observed velocity to calculate

the acceleration that must acting on the star to keep it in a stable orbit.

You can also go and calculate the acceleration that comes from the pull of the normal matter

only.

So you have two types of accelerations.

The one is the total acceleration, the other is the acceleration from the normal matter

only.

Now you can go and plot the data for those two types of accellerations against each other,

and you will find that they are strongly correlated.

you see this very nicely in this wonderful plot by McGaugh and his collaborators.

Now what this means is that if you tell me how much normal matter there is in a galaxy,

I can tell you how much dark matter there is.

If you think that is an independent component of the universe this does not make any sense.

You would expect a much larger variety in the data.

But There is not.

So that’s a big mystery.

There is a pattern in the data and dark matter does not explain it.

At least it doesn’t explain it in any simple way.

In the past 20 years, astrophysicists have tried to come up with computer simulations

that would reproduce galaxies that look like the ones we actually see.

These simulation are based on the dark matter hypothesis and for a long time it worked pretty

badly.

The galaxies that they got did not look like what we actually saw.

But in the past two decades they have improved the computer simulations a lot and they have

introduced new parameters, and now they can actually produce galaxies that look pretty

much like what we see.

Now that’s not a prediction – because these simulations were made to reproduce what

we had already observed – but this removes the tension between the data and the hypothesis

of dark matter.

So everything is fine, or is it?

Now that’s where it gets interesting because there is a an alternative explanation for

the pattern in the data, and that is modified gravity.

What is modified gravity?

You can have some fun asking physicists what is modified gravity.

The most common answer I get is, believe that or not, is that it’s dark matter if you

add a term to this side of the equation and that it’s modified gravity if you add a

term to that side of the equation.

Now that, of course, does not make any sense because you can always move the term from

this side of the equation to that side of the equation.

Now the somewhat more sophisticated answer is that it’s modified gravity if you change

the function on this side of the equation.

But that’s just not how modified gravity works.

Yes, there are some theories of modified gravity which work this way but these are not the

theories that we usually use to explain the observations that are commonly attributed

to dark matter.

Instead, modified gravity changes the force of gravity and it normally does so by introducing

additional forces or, to be more specific, by introducing new fields.

You have to do something of that sort because it’s really hard to change anything about

Einstein’s theory of general relativity.

Now quantum mechanics tells us that forces are mediated by particles so whenever you

introduce a new force you also introduce new particles.

Which brings up the question well, then what is the difference between particle dark matter

and modified gravity?

The difference between modified gravity and dark matter is that for dark matter the observations

are explained only by the gravitational pull of the new type of matter.

For modified gravity, on the other hand, the gravitational pull is not the same as Einstein

taught us or there is an additional force acting on the particles.

It’s this additional force that helps to explain the observed patterns in the data.

This has been known since the 1980s, when Milgrom proposed his theory of Modified Newtonian

Dynamics which is also known as MOND.

MOND relates to modified gravity the same way that Newtonian Gravity relates to Einsteinian

gravity.

This means MOND is an approximation that works in some cases, but not in others.

Modified gravity is the complete theory that reproduces MOND in certain situations.

There are presently a few known such theories of modified gravity.

Modified gravity works well for the galaxies but not so well for the galaxy clusters and

the cosmic microwave background.

I know there are some people who want you to believe that modified gravity has been

ruled out by the Bullet Cluster but that’s just wrong.

The bullet cluster is actually a collision of two galaxy clusters.

It’s an interesting case because for this cluster we have different kinds of data.

We have on the one hand the emission of light at various frequency ranges which is shown

in red and this tells you where the normal mass is located.

On the other hand we have data from gravitational lensing which shown in blue, and this tells

you where the gravitational pull goes.

As you can see the two blobs are not in the same places.

The story now goes that this demonstrates that modified gravity must be wrong because

with modified gravity you would not be able to get the gravitational pull to go to a different

place than the center of the normal mass.

But that story is just not correct.

As I said, in modified gravity you introduce additional forces and you also introduce additional

particles.

And there is no reason that the gravitational pull, especially for a dynamical system like

the bullet cluster, should go to the same place as the normal matter.

On a more general level, the Bullet Cluster is a statistical outlier.

There are only two or three known clusters which collide at such high relative velocities.

Time-dependent systems like this are generally hard to model both with modified gravity and

dark matter.

Using the Bullet Cluster to rule out an explanation for a pattern in the data just does not make

any sense.

So the situation is that dark matter works better in some cases, and modified gravity

works better in other cases.

The big question is now what is overall the best explanation?

And we currently don’t have a good answer to this.

The reason is that no one makes direct quantitative comparisons between two types of theories.

Indeed the right answer indeed may be a combination of the two, dark matter and modified gravity.

And this is what we will talk about the next time.