Dark Matter

Dark Matter is not enough. The Universe ought to be weirder

Dr Andrew Pontzen


Department of Physics & Astronomy UCL

www.cosmocrunch.co.uk                    @apontzen





Oscar II (21 January 1829 – 8 December 1907), was King of Sweden from 1872 until his death and King of Norway from 1872 until 1905. He had a great interest in maths and science but he was particularly worried about the future of the Solar System. He was convinced that it would fall apart and that some of the planets/other solar bodies would collide with each other. In some respects he wasn’t alone in this. In fact the problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since Newton’s time. This was known originally as the three-body problem and later the n-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the 19th century. Indeed in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:

“Given a system of arbitrarily many mass points that attract each according to Newton’s law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.”

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, “This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics.”


Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science.


In other words Poincaré won the prize by stating that it was “impossible to predict the future of the Solar System”


In its traditional sense, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton’s laws of motion and of universal gravitation).



——and we still can’t predict the future of the Solar System. Some of the planets may simply fly off or the Planets could attract more “stuff” into the Solar System. Who knows?


We do know that the Solar System is pretty stable now.


The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable historically, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is stated to be chaotic, and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.

The Solar System is stable in human terms, and far beyond, given that none of the planets will collide with each other or be ejected from the system in the next few billion years, and the Earth’s orbit will be relatively stable.

Since Newton’s law of gravitation (1687), mathematicians and astronomers (such as Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold and Jürgen Moser) have searched for evidence for the stability of the planetary motions, and this quest led to many mathematical developments, and several successive ‘proofs’ of stability for the Solar System.

In fact we can be more confident in making a statement on the future of the Universe than the future of the Solar System even though this involves dark matter (which we can’t see) and dark energy.

In fact physicists investigating dark energy and dark matter have actually been able to use their work to successfully model aspects of the global economy.







The average family income is 2T. The most probable family income is T.

The article reviews progress in applications of statistical physics to probability distributions of money, income, and wealth in a society. Developing an analogy between the probability distributions of energy in physics and money in economics, they argued that the distribution of money should follow the exponential Boltzmann-Gibbs law for certain classes of models with interacting economic agents. Analysing the empirical data, they found that income distribution in the USA has a well-defined two-class structure. The majority of the population (about 97%) belongs to the lower class characterized by the exponential (“thermal”) distribution. The upper class (about 3% of the population) has a Pareto power-law (“superthermal”) distribution, whose parameters change in time with the rise and fall of financial markets. Recent studies, such as the role of debt and the distribution of energy consumption around the world, were discussed.

In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution, probability measure, or frequency distribution over various possible states of a system,[clarification needed] with the form


where E is state energy (which varies from state to state), and kT (a constant of the distribution) is the product of Boltzmann’s constant and thermodynamic temperature (temperature in kelvin).


The Boltzmann constant (kB or k), named after Ludwig Boltzmann, is a physical constant relating energy at the individual particle level with temperature.


As mentioned previously the Solar System is considered to be chaotic.



Chaos can refer to any state of confusion or disorder and chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, sociology, physics, engineering, economics, biology, and philosophy. Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect.

One of Dr Pontzen’s favourite experiments that demonstrate chaotic movement is the double rod pendulum.



In the above left image you can see the double pendulum with an LED attached. Using a special camera the motion of the pendulum can be recorded.


The above picture shows Dr. Pontzen with the special camera on his right

In physics and mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behaviour with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of coupled ordinary differential equations. For certain energies its motion is chaotic. It demonstrates unpredictable motion.



The above picture is the finished article. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that have chaotic solutions.


The double pendulum can go round and round for quite a while as energy is continually being transferred. It may be jointed but it can make lots of squiggles.



An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.

In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will come in range of Earth and other planets. Every now and again the gravitational force pulls planets together. Over time the effects build up but they are very hard to predict.

This chaotic behaviour applies to all space, but what is in space?

If you look up at a dark piece of the sky with your eyes at night you are seeing about 1000 stars. If you use a telescope you would see a lot more. You would see gas, dust and dim stars.

A galaxy contains a couple of billion stars. It is crammed with stuff.

The Andromeda Galaxy (M31) is the closest large galaxy to the Milky Way and is one only ten galaxies that can be seen unaided, as a smear of light, from the Earth. It is a spiral galaxy approximately 2.5 million light-years (2.4 × E19 km) from Earth in the Andromeda constellation. Also known as Messier 31, M31, or NGC 224, it is often referred to as the Great Andromeda Nebula in older texts. The Andromeda Galaxy may be the nearest spiral galaxy to our Milky Way galaxy, but not the nearest galaxy overall. It gets its name from the area of the sky in which it appears, the constellation of Andromeda, which was named after the mythological princess Andromeda. The Andromeda Galaxy is the largest galaxy of the Local Group, which also contains the Milky Way. It is accompanied by at least 10 satellite galaxies the most notable of which is the Triangulum Galaxy, and about 44 other smaller galaxies. It contains about 200 billion stars.

In approximately 4.5 billion years the Andromeda Galaxy and the Milky Way are expected to collide.






The Hubble Space Telescope (HST) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. With a 2.4-metre mirror, Hubble’s four main instruments observe in the near ultraviolet, visible, and near infrared spectra. The telescope is named after the astronomer Edwin Hubble.

Over ten consecutive days between December 18 and December 28, 1995 the Hubble telescope was turned towards what appeared to be a dark part of the sky. It covered an area 2.5 arcminutes across, about one 24-millionth of the whole sky, which is equivalent in angular size to a 65 mm tennis ball at a distance of 100 metres. The image was assembled from 342 separate exposures taken with the Space Telescope’s Wide Field and Planetary Camera 2.


The field is so small that only a few foreground stars in the Milky Way lie within it; thus, almost all of the 3,000 objects in the image are galaxies, some of which are among the youngest and most distant known. By revealing such large numbers of very young galaxies, the HDF has become a landmark image in the study of the early universe, with the associated scientific paper having received over 800 citations by the end of 2008.


The Hubble Deep Field is shown above left. The HDF, shown on the right, is at the centre of this image of one degree of sky. The Moon as seen from Earth would fill roughly one quarter of this image.

The final images were released at a meeting of the American Astronomical Society in January 1996, and revealed a plethora of distant, faint galaxies. About 3,000 distinct galaxies could be identified in the images, with both irregular and spiral galaxies clearly visible, although some galaxies in the field are only a few pixels across. In all, the HDF is thought to contain fewer than twenty galactic foreground stars; by far the majority of objects in the field are distant galaxies.

There are about fifty blue point-like objects in the HDF. Many seem to be associated with nearby galaxies, which together form chains and arcs: these are likely to be regions of intense star formation. Others may be distant quasars. Astronomers initially ruled out the possibility that some of the point-like objects are white dwarfs, because they are too blue to be consistent with theories of white dwarf evolution prevalent at the time. However, more recent work has found that many white dwarfs become bluer as they age, lending support to the idea that the HDF might contain white dwarfs.


Details from the HDF illustrate the wide variety of galaxy shapes, sizes and colours found in the distant universe.

There is believed to be 100 billion galaxies in the Universe each with their own billions of stars.


M81 galaxy is one of the galaxies shown in the HDF. It is a spiral galaxy about 12 million light-years away in the constellation Ursa Major. Due to its proximity to Earth, large size and active galactic nucleus (which harbours a 70 million M☉supermassive black hole), Messier 81 has been studied extensively by professional astronomers. The galaxy’s large size and relatively high brightness also make it a popular target for amateur astronomers.


In the picture above left is a Hubble Space Telescope (HST) image of Messier 81. In the picture above right is an infrared image of Messier 81, taken by the Spitzer Space Telescope. The blue colours represent stellar emission observed at 3.6 μm. The green colours represent 8 μm emissions originating primarily from polycyclic aromatic hydrocarbons in the interstellar medium. The red colours represent 24 μm emissions originating from heated dust in the interstellar medium.

To investigate it, or any other galaxy, you could look at how much light is being emitted from it. The brightness can give you some idea how much stuff and how many stars are present.

Another method of investigating how much stuff there is in a galaxy is to see how fast it spins. The spinning is due to gravity and the greater the gravity the faster the galaxy spins. The amount of gravity is due to the amount of matter in the galaxy. Gravity is a natural phenomenon by which all physical bodies attract each other.

However in analysing the spinning of galaxies it turns out there is five times more mass than we can see.

Dark matter is the term we give to the matter we can’t see.


Dark matter doesn’t just occur in galaxies. NGC 2215 is an open cluster of galaxies in the constellation Monoceros.




The gravity within such a cluster is so great that it can bend light.


The amount of bending tells you how much gravity which in turn tells you how much stuff. The answer comes out once again as five times more than we can see.

We would like to know what this dark matter is made of and we can use the real world to help us.

If we could focus on to something, getting closer and closer and seeing smaller and smaller objects we would eventually see matter particles that cannot be divided up any further. Therefore we could assume that dark matter must be made up of dark matter particles.

The next step is to investigate how dark matter particles behave. Matter particle experiences forces. We experience the force of gravity pulling us downwards and electromagnetic forces preventing us falling through the floor.

Electromagnetic forces are also the reason why we see things as light is part of the electromagnetic spectrum.

Dark matter must experience gravity or it wouldn’t be able to do the things we think it does however it doesn’t seem to experience electromagnetic forces as we can’t see it.

Where is dark matter, how do we find it? We can make predictions of how much of it is there.

Even though we believe there is five times more dark matter than normal matter the Earth is totally made up of matter. At the start of the Solar System rocks and dust bumped into each other and stuck together and the Earth was formed. We believe that dark matter can’t be sticky so where there is large amounts of matter then there is very little dark matter.

If the calculations are correct dark matter particles are arriving at about 100kms-1 into the Earth and because they don’t experience any forces there are no interactions and they are simply streaming out again.

If it were possible to freeze time you could count the dark matter particles streaming through and it could be about one million. This sounds a lot but because the particles are so small this amounts to a millionth of a millionth of a millionth of a kg.

How do we know the speed and quantity of dark matter in the room? The answer is we can produce a computer simulation.


The computer can create a virtual galaxy and calculate all the gravitational forces within it that can affect the virtual blob of dark matter (assuming that it is only affected by gravity).


The blob swings back and forth in an unpredictable chaotic manner being attracted to the centre of the galaxy. Mathematically the motion is chaotic so it is hard to predict where the blob is going.

Using different computers can give different results for the motion. This becomes more noticeable the longer you run the simulations. There are bigger and bigger differences appearing until eventually the blob on the 64-Bit computer is doing something completely different from the blob on the 32-Bit computer.


Two different answers about what dark matter is doing. The reason is that computers don’t store numbers accurately enough. Increasing the precision of the computer still further simple gives you a completely different answer. It doesn’t matter how accurately you keep track of all the numbers involved in the future of the Solar System as the calculations will still go wrong. An infinite number of digits are needed and we simply can’t account for the positions of everything within the Solar System to an infinite number of digits.

The same problem occurs when investigating dark matter as computers are simply not good enough. If we lose a tiny bit of accuracy we get different measurements each time we run the simulation.

A long exposure selfie was described earlier using a special camera and the double pendulum.


We can understand the pattern. There are not many lines at the top and the pattern never goes outside the circle. The pattern also seems to go back towards the centre a lot.

The circle is easy to understand as the length of the pendulum is set and the maximum height is to do with energy. The maximum energy at the start is dissipated over time so the pendulum can’t get above a certain point. The continuous return to the middle is due to degeneracy. There are lots of routes towards the middle when the pendulum is near it. It seems like there is an attraction to the centre but in fact it is statistical. We can understand it with calculations.



The theory is very similar to fact as the general pattern is reproduced. We can’t predict exactly what is going to happen to the pendulum. Similarly we can’t predict what is going to happen to one single blob of dark matter but we can understand the general patterns. The details may go wrong in the computer but the general pattern holds. We take dark matter seriously because we can do calculations.


Simulations can be used to show galaxy formation and over time little galaxies are seen to merge because of the presence of dark matter. We can even simulate our own galaxy. We can’t do the calculations exactly but using all the information that we know we can produce a galaxy very much like our own. We can find things out on average.


Dark energy is based on the evidence that the Universe expanding, with galaxies moving apart, at an accelerated rate. In order for this expansion to happen you need to be able to add energy from nowhere. Quantum mechanics has an answer to this problem in that in the quantum world a vacuum is not totally empty. There is a trace of energy known as vacuum energy.

In reality we see strange things and we reach out for things to explain them which is why we have dark matter and dark energy. We are using what we know to try and explain what we don’t know. However it could all be much weirder.

Despite the problems we can get models right and on average get the expected answers if our questions are broad enough. Where things go wrong is when we are interested in something specific. However even if all the work that has been done so far turns out to be wrong it still may be useful.

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