Cosmology friedmann

Published on December 29th, 2010 | by Carl Mundy

0

Describing our Universe with the Friedmann Equation

This article concentrates on how we can determine the fate of our unimaginably large universe, and everything within it, using an elegant but powerful bit of maths. Yummy. We all love maths, don’t we? While not going into too much detail behind things – I’ll leave that for you and/or future articles – this should give you a decent idea of how we can describe our universe, and the possible paths it will take. Lets dive straight into the deep end.

The Friedmann Equation

GCSE physics teaches us that the total energy, U, for a system in equilibrium (stable) is given by the sum of its kinetic energy  T, that is energy from its motion, and its potential energy V. As the system is in equilibrium, this energy is constant and is conserved

    \[ U=T+V \]

Consider our system to be a spherical shell of matter, with uniform (unchanging) density \rho, centred at some point and a small particle with a mass m at some distance r from the origin. An example would be the Earth orbiting the Sun, with the Earth representing the small particle. Assuming the particle is outside this spherical shell of matter, we can obtain expressions for the kinetic and potential energies of our particle.

The kinetic energy of our particle is given by an equation we should all know and love, T=\frac{1}{2}mv^{2}, while the potential energy is just due to the gravitational field produced by the shell of matter in our system, V(r)=-\frac{GM(r)m}{r} where M(r) is all the mass enclosed within a radius r. So we can write the total energy of our particle as

    \[   U=\frac{1}{2}mv^{2}-\frac{GM(r)m}{r}=\mathrm{constant} \]

Before moving on, we know that the mass within a certain distance is related to the volume at that distance and the uniform density of the matter so that M(r) = \frac{4}{3}\pi\rho r^3 and our total energy expression becomes:

    \[   U=\frac{1}{2}mv^{2}-\frac{4 \pi G}{3}m \rho r^{2}=\mathrm{constant} \]

Now we have to introduce another way of thinking about the distance between objects in our universe to take account of any expansion (or contraction) of the universe. The physical distance between two objects (i.e. the distance we see or measure) can be given by the product of a scale factor and a coordinate that gets carried along with any expansion that might occur – this is the comoving coordinate.

    \[   r(t)=a(t)x \]

where r(t) is the physical distance at any time t, the comoving coordinate is x and the scale factor describing the expansion rate of the universe is a(t).

Our next step is to differentiate this (for a quick tutorial on differentiation, see MathWorld) expression for the physical distance. Doing so, we arrive at the expression below.

    \[   \large{\dot{r}=a\dot{x}+\dot{a}x=\dot{a}x} \]

The a(t)\dot{x} term in the middle expression is zero because x doesn’t change, it’s a constant, and so its derivative is zero. After this, we substitute in the original equation for x and realise that \dot{r} is velocity.

    \[    \dot{r}=v(t)=\frac{\dot{a}}{a} r(t)=H(t)r(t) \]

where we have defined H(t) = \dot{a}/a, called the Hubble parameter.

We can now piece it all together. We have an expression for the total energy of the system, which is dependent only on r and \dot{r}, and we have expressions for these two things as well. All we have to do is substitute in these expressions into our total energy equation and do some rearranging to get:

    \[   H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho - \frac{kc^{2}}{a^{2}} \]

which is called the Friedmann Equation, where we have defined \frac{kc^2}{a^2}=-\frac{2U}{mx^2}.

The last term on the right hand side is called the curvature term because k is a constant that describes the geometry, or shape, of space in our universe – it can tell us the fate of our universe. Whether the universe expands for ever, remains static or re-collapses and ends all depends on one letter. Isn’t that amazing!

Summary

If you just want the gist of what the past 600 words was trying to tell you, then this is the paragraph for you. The expansion of the universe can be described through the Friedmann equation which relies on the density of all the stuff in the universe as well as the curvature of the universe. From this one, elegant expression we can determine whether or not the universe will expand forever, eventually re-collapse or decelerate. Don’t forget that observations tell us that our universe is currently expanding at an accelerating rate!

Further Your Knowledge

Cosmology is one of the most exciting areas of research at the moment. It gives us the framework we need to describe our universe and how it behaves. It gives us testable predictions and allows us to look back to the very beginning of our universe. If you want to learn more, do some reading up on the following things:

  • Cosmological constant
  • Cosmic microwave background radiation
  • Dark matter
  • Fluid equation

Header image by Robert Couse-Baker.

Tags: , , , , , , , , , ,


About the Author

Astronomy PhD student from the UK with a passion for astronomy and science outreach projects. Involved with weekly science-based radio programme The Science Show on University Radio Nottingham (URN).



Back to Top ↑