# Theoretical cosmology

This is a "short" introduction some aspects of what I work on: with an introduction to the cosmological standard model and field theory. The projects on field theory have some associated movies.

Most of this is currently active research

### Contents

- Modified gravity talk
- The cosmological model
- The standard model
- Dark energy
- Modified geometry
- Modified gravity
- Is dark energy the modern day planet vulcan?
- Field theory

## Modified gravity talk

I gave a talk on modified gravity to the University of Manchester's PhysSoc lecture series (brilliantly organised by Sina Salek).

**Abstract** One of the major scientific discoveries of the past century is that the Universe is accelerating in its expansion. This discovery recently won a Nobel Prize. The problem is: we have no idea what causes the Universe to accelerate. The current way of thinking requires us to invent 75% of the Universe (dark energy).

An alternative idea, which is what I will talk about, is to assume Einstein's theory of General Relativity is wrong, in which case we need some alternatives.

In my talk I will address and discuss lots of basic questions: what is gravity? what is general relativity? what is modified gravity?

The talk will be at a level which is accessible for any undergraduate.

**YouTube**Part 1, part 2, part 2, part 4**Slides**Modified gravity

## The cosmological model

### The standard model

There is a standard model used to describe the Universe. Its ingredients are (1) a gravitational theory and (2) a description of the shape, or geometry, of the Universe. In the standard model these are (1) Einstein's theory of General Relativity and (2) the Friedmann-Robertson-Walker (FRW) geometry. The FRW geometry assumes that the Universe looks the same everywhere and in every direction (i.e. is homogeneous and isotropic).

The FRW model states that the content of the Universe should be homogeneous and isotropic on cosmological scales. Different types of "content" have different gravitational properties. For example, baryonic (and dark) matter does not have a pressure and produces an attractive gravitational force. Radiation (i.e. light) has a pressure, and also has an attractive gravitational field. Crucially, all the "known" and "understood" matter particles have attractive gravitational fields.

### Dark energy

One of the major scientific results in the past decade is that the Universe appears to be accelerating in its expansion. What this means is that whatever the substance is which dominates the content of Universe, its gravitational field is repulsive. There is no "known" particle which does this.

Using the framework of the standard model, the immediate consequence is that there is some form of "anti-gravity" which gravitationally repels objects. Approximately 74% of the Universe must be this "anti-gravity" to cause the Universe to accelerate. If one thinks about it, "normal" matter (the stuff you, I & the stars are made of) has an attractive gravitational field, which would cause the Universe to decelerate in its expansion: therefore to make the Universe accelerate the attractive force due to "normal" matter must be overcome by the repulsive "anti-gravity".

Cosmologists have begun to call the substance which has this anti-gravitational property *dark energy*, but what it actually *is* is a complete mystery. At the moment the name "dark energy" is a name for the ignorance of what 74% of the Universe is made of. There is a plethora of models of what the dark energy could be:

- Cosmological constant -- the anti-gravity energy associated with vacuum. Einstein called it his greatest blunder, but has the required properties to accelerate the Universe. Although there is an issue of it being a factor of 10^{120} too big!
- Quintessence -- a homogeneous scalar field. Scalar fields play an important role in particle physics (the Higgs field is a scalar field, currently being searched for at CERN), and so they should have an impact in the Universe.
- K-essence -- a more complicated (but more general) version of Quintessence.
- Elastic dark energy -- an inhomogeneous scalar field, where domain walls form an elastic "mesh". See below for a quick discussion on domain walls.

Infact, *dark matter* is another invented component to the content of the Universe. All in all, cosmologists "invent" about 96% of the content of the Universe. The fact that we must invent so much of the Universe perhaps signals that ** the standard model is incorrect**.

### Modified geometry

In this case one questions the validity of the assumption that the Universe is homogeneous and/or isotropic. The fact is that the Universe *is not* homogeneous on small scales, and so understanding how these small-scale inhomogeneities affect the evolution of the Universe and how they affect the propagation of photons in the Universe is a very important (and complicated) task. There are many methods that begin to understand how to tackle this problem, but none that fully use GR's complicated non-linear equations.

### Modified gravity

One can go a step further and question whether General Relativity (GR) is correct. Whilst GR has been shown to hold on "small" scales, it is not clear whether is is correct up to the biggest possible scales. Perhaps the fact that cosmologists invent 74% of the Universe to make a GR model work means that GR is wrong. To make headway on this problem one must create new gravitational models -- this is a difficult task. Especially given that GR is infact one of the simplest gravitational models (and it is incredibly complicated).

The way that new gravitational models are constructed is by modifying the gravitational "action", usually by adding in functions of the Ricci scalar.

### Is dark energy the modern day planet Vulcan?

It is usually important and instructive to look to history, so that we can perhaps learn from our past mistakes. Over 100 years ago (before the development of General Relativity) scientists had Newton's theory of gravitation. It stood up to experimental tests when applied to the earth. When Newtonian gravity was applied to the Sun (specifically, to the orbit of Mercury, which is very close to the Sun) it was found that "anomalies" were observed. Basically, their current understanding of (a) Newton's gravity and (b) the known planets *could not account for what was observed*. The result was that an **extra planet** was proposed - Planet Vulcan - whose role was to provide a "fix" to Mercury's orbit.

Eventually, when Einstein's theory of gravity was applied to the Sun and Mercury's orbit, planet Vulcan was no longer required (and indeed, does not exist).

The lesson here is that if we attempt to apply a theory to a situation for which the theory has not been tested, we should not nessecarily expect that the theory should hold. Just as Newton's theory was/is an approximation to Einstein's theory (the former held in low curvature regimes), it may be the case that Einstein's theory is only an approximation to the theory which should be applied to Cosmological scales. An artifact of applying Einstein's theory to Cosmology is that we must invent 96% of the content of the Universe. And so we must consider whether dark energy is the modern day planet Vulcan. This is an open question.

## Field theory

Studying field theory is an extraordinary task. It allows a single model to be written down that can be used to describe a variety of vastly different physical systems: liquid helium, the interior of a neutron star, bar magnets, models of atomic nuclei, particle physics and the dynamics of the Universe as a whole (to name but a few).

### Solitons

A soliton is a localized configuration of energy. A real-life example is the "Bore" phenomenon on the River Severn: a solitary wave that flows down the river. The domain walls, cosmic strings and vortons discussed below are all examples of solitons.

### Topological defects

When systems cool down phase transitions naturally occur. A simple example is the behavior of H2O as its temperature is dropped: it turns from a highly disordered system in its gaseous phase (steam) into a more ordered state when a liquid (water) and into a very ordered state when solid (ice). These states are formed as the temperature drops below critical points (the boiling or freezing points). Importantly, as a critical point is passed through (on cooling) some freedom the system had is removed (as gas will fill a volume, whereas a solid will not). This generally has the effect of creating *defects*

As a simple example, if a system is only allowed to be in one of two states at a given point in space *after* a phase transition, then there will be regions of space where the system is entirely in one state. These will be sat next to other regions where the system is in the other state. There will be an interface region: this is the defect.

To properly classify the different types of defects one needs to consider topology, which we will not do. However, we will quickly go through a few of the types of topological defect.

**Domain walls** are "sheet"-like objects in 3D which separate regions of space. The regions of space are called "domains", and thus the separating regions are domain "walls". The reason the regions of space are different is due to some fundamental property of the space which is different (being in different Higgs vacua or magnetic domains, for example). The domain walls themselves can be very thin, and cary a huge amount of energy. Most of the time domain walls collapse -- they are like stretched rubber sheets, carrying an internal tension. So if a system forms domain walls it tends to evolve until there are no walls left (they have all collapsed); this means that the "fundamental property" of the space changes in certain regions so that the property is the same over the entire space. If domain walls formed in the early universe, then we should be able to see their signatures by the way that the energy density associated with the wall gravitationally lenses light that passes through/next to the wall.

The set of images below are some domain walls collapsing in 2D. The Red and blue are the two type of domain, and the walls are the lines separating the domains. To understand these images properly, please read one of my papers.

There are many different theories which produce domain walls. I've also been studying a theory called the "Wess-Zumino" model. This has three possible places the field ends up: and so junctions can form in addition to walls. I've done something slightly different with these walls: the region where domain walls are allowed to form is confined to a circular region in the middle of a big box. At the moment, this is just a novelty. In the movie below you see the circular region with a lot of domain walls: these collapse. The two panels are energy density (on the left) and potential energy density (on the right). As the walls collapse, energy gets injected into the "outside" region.

A **cosmic string** is a line in 3D, and is formed when something like electromagnetism is present. Again, they can be produced in the universe and carry a lot of energy -- enough to affect our observation of the CMB.

It is worth stressing that cosmic strings and domain walls are not "material objects": they are configurations of energy. They have a temperature and gravitational field, but there is no "stuff" inside them.

Cosmic strings can form loops. If it is just a cosmic string in a loop, then it will collapse (like a domain wall does) due to its internal tension. However, if there is something inside the string, the loop can be prevented from collapsing. One can imagine water inside a hose-pipe (where the hose-pipe has its ends attached to make it into a loop). If there is no water the hose-pipe is limp, and can be stretched and compressed and easily deformed. However, if one now puts high pressure water inside the hose-pipe there is an internal force which makes it difficult to deform the hose-pipe. The analogy meets reality as follows: the looped hose-pipe is the cosmic string and the high pressure water is a superconducting current. A cosmic string with superconducting current is called a **cosmic vorton**. The actual term is a superconducting condensate. Just like electromagnetism one can have superconducting charge and current. So a cosmic vorton can carry both charge and current.

A looped domain wall with superconducting condensate is called a **kinky vorton**.

These vorton objects carry a lot of energy, and so if they are found to be stable they could have a substantial impact on our understanding of the evolution and history of the universe. Vortons can also be used to model the **interior of a neutron star**, where a few different superfluid interact in complicated ways.

### Ferromagnets

Another project I've been involved in was to study the stability of objects made from magnetic fields. Going back a bit: think about air. One can blow smoke rings in air: these are vortex rings which are very stable. If you just try to blow a chunk of air, that chunk will dissipate very quickly. However, if you create a proper smoke ring, that ring will travel a long way. Smoke rings are very stable objects; the reason they are stable is because the smoke rotates and has some angular momentum.

So, the question arose: can one create loops (vortex rings) in any other fluids? Another way to ask it is: can vortex rings be supported by any other media? If one wants to study objects in a medium, one needs a mathematical description of that medium and some understanding of the objects. We looked at ferromagnets. A ferromagnet is an object with a magnetic field (like a bar magnet). There is a mathematical description of a ferromagnetic material: basically, the field is constructed as the solution to a differential equation known as a Landau-Lifshitz equation. The magnetic field is a vector field (i.e. it is an entity which has three numbers at every point in the universe) of unit length (which means that the squares of these three numbers must add up to one).

Once we have this mathematical description of a magnetic medium, we then need to insert objects and see if they are stable as time increments. It is known that rings with and without twists (known as a Hopf charge) are stable objects in magnetic fields. It turns out that a ring moves up the field; large rings move slower, and rings with higher number of twists also move slower. One can then look at multiple-rings: what happens if two rings sit one above the other in a ferromagnet? The naive answer -- and indeed, the answer in a lot of cases -- is that the two rings just move upwards. However, for certain radii of the rings and certain separations, the rings leapfrog around each other: the upper ring expands, whilst the lower ring shrinks and passes through the gap to swap changes. This leapfrogging process carries on. How about, then, three rings? Can one form a trio of leapfrogging rings? Yes: I've made a movie of three leapfrogging rings in a ferromagnet.