Dark Matter

 

VISHVA MEHTA (Y13)

This article was submitted as part of the WBGS Fuller Research Prize Competition 2022.

Dark matter is a component of the universe which makes up approximately 30.1% of the matter-energy composition of the universe. Its existence was determined from its gravitational attraction rather than its luminosity that visible matter (makes up 0.5% of the Universe) has. 

During the 1920s, the very first astronomer to imply the existence of dark matter was Jacobus Kaptelyn. Later on, Jacobus and an astronomy pioneer hypothesised the existence of dark matter in 1932. A year later, an astrophysicist called Fritz Zwicky who was studying  a galaxy cluster called the Coma cluster (a galactic supercluster which contains more than 1,000 galaxies) made a similar conclusion by using the virial theorem to obtain evidence of unseen matter which he referred to as ‘dunkle materie’. The virial theorem relates the total kinetic energy of a stable system of discrete particles to the total potential energy of said system. This theorem is applicable to all stable systems, classical and quantum. From a mathematical perspective the theorem proceeds to state the following:  < T > = -½ N:Σ(k=1) <Fₖ ⋅ r ₖ> where the angled brac
kets represent the average of quantity over time. Fₖ represents the force on the kth particle and r ₖ

represents the position vector of the kth particle. This theorem does not depend upon temperature and sustains itself for systems that are not in a state of thermal equilibrium (no net flow of thermal energy between two physical systems). Additionally, it allows the average total kinetic energy to be calculated for very complicated systems that defy an exact solution. However not everyone accepted this evidence as the virial theorem assumes that the clusters are gravitationally bound but in reality they might be flying away from each other. Furthermore, the 800 galaxies that Fritz Zwicky has studied should have only had a velocity dispersion (this is the statistical dispersion (an extent to which data is squashed or stretched) of velocities about the mean velocity for a group of astronomical objects) of 80 kilometres per second, however the real value was found to be much larger than that - 1000 kilometres per second. At such a high speed, the stars should have been able to escape their mutual gravitational pull.

Unlike normal visible matter, dark matter does not interact with electromagnetic forces which means that it does not absorb or reflect light - this means that dark matter must be dark. Instead, it bends space as when

astronomers proceed to observe distant galaxies from the Earth they appear distorted (appear stretched and oddly shaped). This distortion is caused by a phenomenon called gravitational lensing which is caused by the gravitational force of dark matter. In fact, this gravitational force is so huge and profound that it physically bends light around galaxies which results in a distorted appearance when we try to observe said galaxies. The more massive an object that you are trying to observe, the more such lensing will also be observed. This distortion occurs at large-range zooms. Distortion essentially describes how the magnification of the image changes across the field of view at a fixed working distance. For example, the image of a square object formed by a lens is not a square as it is distorted. This happens due to the fact that the magnification produced by the lens for different parts of the object are different, as different parts of the lens are having different axial distances from the lens. Hence different parts of the object are magnified differently.

Furthermore, the observation of spiral galaxies (a type of galaxy which consists of a flat disk which is rotating that contains stars, gas and dust.) shows that rotational velocities do not decrease with distance from 

the centre. However, if we use Kepler’s second law of planetary motion then we expect the rotation velocities to decrease as you move away from the centre as there will be more potential energy and less kinetic energy. The law states that the line between the Sun and the planet sweeps equal area at equal times. This means that the torque (τ) on the planet will be zero as the angle θ between the position vector r and the force vector F is zero. Since the torque is zero, this would mean that angular momentum must be some constant value. This is because torque is defined as ||r|| ||F||sinθ as well the rate of change of angular momentum: d=dL/dt. The orbital radius and angular velocity of a planet in an elliptical orbit will vary however the areal velocity will remain constant.

However, the galaxy rotation curve remains flat as the distance from the centre increases. This would essentially mean that the mass distribution in spiral galaxies are not similar to our solar system and there might perhaps be an abundance of dark matter on the outskirts of the galaxy which is undetectable but bends light.

The work of galaxy rotation curves by Vera Rubin, Kent Ford and Ken Freeman’s work in the 1960s and 1970s by using spectrographs concluded with the fact that most galaxies contain about six times as much dark as visible mass. 

Much later, the new contender to explain the concept of dark matter was massive compact halo objects (MACHOs) which are a type of astronomical bodies that emit little to no radiation. However, this explanation fell out of favour as the EROS project in the late 1990s showed that the MACHOs were not numerous enough to account for the mass that was needed. Now, the leading contender are weakly interacting massive particles which is a new elementary particle that interacts via gravity and any

forces, which is as weak or weaker than the weak nuclear force. This is because if you ask what ratio of particle and antiparticle interaction (annihilation) is required to give the ratio of dark matter we see today then the answer would be that they need to interact through the weak nuclear force. This is often called the WIMP miracle.  This is also because of the Bullet Cluster, where two clusters of galaxies collide. This collision shows that dark matter remains unchanged therefore it must weakly interact with itself. Additionally, it must be cold as it is not moving at very high speeds, otherwise, after the Big Bang it would continue to keep moving away. Furthermore, it must be stable because if it was not stable then it would decay into other particles and it would not make up such a large amount of our Universe.

Supersymmetry predicts the existence of the WIMP particle and it comes about naturally in string theory. Supersymmetry is essentially the symmetry between Bosons (force particles) and fermions (matter particles). Supersymmetry states that for every boson particle there is a corresponding fermion particle and vice versa. However, supersymmetry has not been proven yet.

Another candidate is the axion which was a theorised particle to solve the charge-conjugation parity symmetry problem in quantum chromodynamics. It essentially states that the laws of physics should be the same if a particle is replaced by its antiparticle. The problem can be simplified slightly further: a neutron has a spin in an electric field as it is made up of up to three quarks that have spin; however the neutron has no spin in an electric field. For there to be no spin observed whatsoever, the value of theta in the quantum chromodynamics lagrangian equation would have to be zero or close to zero. However, this problem was solved by particle physicists Roberto Peccei and Helen Quinn who made theta into a field. Most fields tend to maintain the least possible tendency of zero and this is exactly what the field theta was made to do. Like a quark field or an electromagnetic field, a field excitation would create a new particle called an axion. However, due to the nature of the theta field it would make the mass of the axion extremely small. Although, it is considered as a favourable candidate for a dark matter particle among many scientists as it does not rely on supersymmetry and it can be put in a standard model next to the Higgs Boson. Also, its very low mass would make up for very large numbers of the particle as if you had 1cm³ of space, it would fit one WIMP but 10¹⁶ axions.