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Current cosmological theories have the universe be made up of multiple forms of energy. The three most commonly involved are dark energy, matter, and radiation. These three energy forms exist at various densities, and through their interactions with gravity dictate the way in which the universe expands at the largest scales.

The exact way that these determine the universes' expansion is given by the Friedmann equation,

H^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{R^2}

Here, H is the Hubble parameter (the rate of expansion of the universe, by Hubbles' law), G is the gravitational constant, ρ is the density of the universe, R is the size of the universe, and k is a constant related to the curvature of the universe.

The density of the universe, however, will change over time, because the universe is expanding. Rewritten in terms of the size of the universe, we could write \rho = \sum \frac{\rho_n}{a^n}, where a is a dimensionless value related to the size of the universe and ρn is the density of the specific type of energy that drops off at a rate proportional to a-n.

For example, matter density drops off as a-3; the density of matter is the inverse of the volume. Doubling the size of the universe will divide the matter density by eight. However, different types of energy will fall off at different rates; radiation density drops off as a-4 (doubling the size will divide radiation density by sixteen) and dark energy density does not drop off at all (doubling size will leave dark energy density unchanged).

The energy density can be divided by a constant to give the dimensionless value \Omega_n = \frac{8 \pi G}{3 H^2} \rho_n, which is more convenient in equations. Putting this into the original equation gives

\frac{H^2}{H_0^2} = \sum_{n} \Omega_n a^{-n} - \left( 1 - \Omega_0 \right) a^{-2}, with the first term being due to the density and the second term being due to the curvature of space. Each Ωn in the above equation corresponds to the density of a different type of energy. This energy type drops off with the expansion of the universe as a-n.

Energy Density from Topology

Some energy types can be considered to be regions of constant energy density, unchanging with scale, but which are confined in a certain number of dimensions.

Ω0 - Cosmological constant - Volume Energy

Energy from the cosmological constant, also known as dark energy, is a constant density, but is unconfined in space. This means that, regardless of the size of the universe, it has the same density.

Ω1 - Domain wall - Sheet Energy

Energy that drops off as a-1 is known as a domain wall, and is confined to two dimensions. They can be considered to be an infinite sheet of constant energy density. As the universe expands, in two dimensions it will remain the same energy density and in one dimension it will decrease, giving a dropoff of a-2 as expected.

Ω2 - Cosmic string - Line Energy

By analogy, energy that drops of as a-2 is a cosmic string, confined to one dimension. In one dimension, energy density is the same, and in the other two dimensions, it decreases.

Ω3 - Matter - Point Energy

Using this same process, matter can be considered to be energy that is confined to zero dimensions. Since energy density is not constant in any direction, it drops off as a-3. This is the same as saying that energy from matter is constant.

Other Types

The following types of energy are theoretically possible, but do not have straightforward interpretations in terms of topology.

Ω4 - Radiation

The energy density of radiation drops off with the size of the universe as a-4, which would imply that energy was confined to minus one dimensions, which is probably impossible. This is because the energy of the radiation is based upon its wavelength, which also increases as the universe expands.

Ω<0 - Phantom Energy

Phantom energy is any energy for which the energy density increases with the size of the universe, rather than decreases. Because of this, as the universe expands phantom energy will eventually dominate the energy content, resulting in an inevitable Big Rip.

Ω>4 - Ultralight

Ultralight is energy for which the energy density drops off faster than a-4, which gives a greater attractive gravitational pressure than ordinary matter and radiation. 

Altering Topological Defects

Another way of producing energy with a specific fallof is to curve a topological defect (domain wall or cosmic string). A domain wall rolled into a tube will act as Ω2, a domain wall rolled into a sphere will act as Ω3, and a cosmic string rolled into a circle will act as Ω3.

Yet another way is to move a topological defect at a high velocity. When moving at relativistic speeds, a domain wall will act as Ω2, a cosmic string will act as Ω3, and a matter particle will act as Ω4.

Generalised

In an n-dimensional universe, Ωn corresponds to matter, Ωn+1 corresponds to radiation, Ω0 corresponds to the cosmological constant, Ωn-m corresponds to m-dimensional topological defects, and Ωn-m+1 corresponds to m-dimensional topological defects moving at relativistic speeds.

Example Solutions

If energy of only one type is considered (including assuming a flat universe, so the curvature term is zero too), then an analytic solution for a(t) is possible. Here, a0 is an arbitary constant that can be calculated if the value of a(t) at a specific time is known. For most cases, this will be a(0) = 0.

Energy Solution
Only Ω-n

Phantom energy dominated

a = 4^\frac{1}{n} \left( \frac{1}{n^2 \Omega_{-n} \left( a_0 + H_0 t \right)^2 } \right)^\frac{1}{n}
Only Ω0

Cosmological constant dominated

a = a_0 e^{H_0 \sqrt{\Omega_0} t}
Only Ω1

Domain wall dominated

a = \frac{1}{4} \Omega_1 {\left( a_0 + H_0 t \right)}^2
Only Ω2

Cosmic string dominated

a = \sqrt{\Omega_2} \left( a_0 + H_0 t \right)
Only Ω3

Matter dominated

a = \left( \frac{3}{2} \right)^\frac{2}{3} \sqrt[3]{\Omega_3} \left( a_0 + H_0 t\right)^\frac{2}{3}
Only Ω4

Radiation dominated

a = \sqrt{2} \sqrt[4]{\Omega_4} \sqrt{a_0 + H_0 t}
Only Ωn

Ultralight dominated

a = 4^{-\frac{1}{n}} \left( n^2 \Omega_n \left( a_0 + H_0 t \right)^2 \right)^\frac{1}{n}

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