Friedmann Cosmology from Newtonian Gravity

I thought it would be useful to write a short series of posts explaining key and well established concepts in physics starting with cosmology, in order to have something to refer back to, when I write on more speculative topics like inflation. I'll start off then with the most important equation in all of Cosmology, the first Friedman equation, then I hope to discuss the fluid equation, the acceleration equation (the second Frideman equation), the Robertson-Walker metric and physical meaning and value of k which appears in the Friedman equations. I may add to this list later on, but currently I plan to keep it at five posts.

I will however also discuss at some point, Einstein's field equations for now I won't, as these involve tensors and I don't right now feel like explaining these, my derivation here is entirely Newtonian but one should be aware that the first Friedman equation can also be derived by taking the relevant features our universe into account, i.e., it's homogeneity, isotropy and their expression in the Robertson-Walker metric and using these to solve the field equations by taking each point in time this metric and allowing the curvature and scale factor to vary with time. The second Friedmann equation can then be derived using both the first and the fluid equation which describes the evolution of the universe's density as a function of time.

Big Bang cosmology is based on four sets of equations. Friedmann's equations discovered in 1922 and 1924 are two important pieces. First though we need to go back and discuss Newton's shell theorem because this becomes very important for classical cosmology. Newton proved two very important conclusions, first that inside a spherically semetric distribution of matter a particle feels no gravitational force from the shell itself, only from objects at small radii, no matter where it is in that shell and second the object acts as though all its mass were concentrated at its centre. We'll call this, the centre of mass. Further more we also will need to remember Newton's famous law of gravitation, the inverse square law:

Given that W = f x d and f = m x a, then W or energy (in this case gavitational potential energy), E = m x g x d so therefore, the gravitational potential energy in a radial gravitation field is equal to:

Both these equations and the theorem will be important later on. Suppose we have a uniformly expanding medium, whose density (mass per unit volume) is equal to the Greek symbol rho. Then we can take the mass of that system to be equal to:

Putting this back into equation (1) you can derive gravitational potential (this step requires Newton's theorem) that is equal to: 

This is only the gravitational potential energy, under energy conservation a particle has both kinetic and potential energy: 

Where T is equal to the Kinetic energy:

I'm only using derivatives with respect to time but with both T and V defined we can now define U as:

This equation describes the evolution of two particles separations and thus is important in its own right, but we can do better than this if we make some further assumption. We assume large scale homogeneity and isotropy, that is to say that the Universe looks the same at each point and the Universe looks the same in all directions. This means the above equation for U applies to any two particles in the Universe and thus we can change the coordinate system. In your mind you should picture a square grid which is expanding, the points of intersection each represent a galaxy, but notice that the points themselves are at rest, they are in a sense, carried along with the expansion of the grid. We call these coomoving coordinates and in fact, it's just like real cosmology. A galaxy is at rest with respect to space, rather it is space time itself that expands. The expansion itself is uniform. Our assumption of homogeneity allows us to assume that the scale factor a(t) is a function of time alone. Thus we relate the initial distance of a galaxy to the coomoving distance

Remember that x is a fixed coordinate. It's just like a stationary point on a two dimensional graph and so its derivitive is equal to zero. We can now plug this equation back into our value for U and we get: 

Simply to clean up the equation I'm going to multiply each term by

This is the Friedmann equation but it's not very elegant in its current form. I'll take

In its final form as presented in text books, I substitute kc squared into the Friedmann equation and rearrange it to get: 

This is the final form of the equation with the term for the cosmological constant being omitted. Add Lambda/3 to the right hand side for the correction. Einstein famously added the term to counterbalance the density so that H(t) = 0 and restore a static universe but he later cited it as the "greater blunder" of his carrier. Such a delicate balance across the universe is unstable but this parameter becomes far more important for the acceleration equation that we'll discuss in the future. One further detail is that the left hand side is identical to the square of the Hubble paramter, so one may see it presented in a slightly different form. One may also see in some text books the c term representing the speed of light has been dropped entirely because the author is using so called natural units in which c = 1. 

Friedmann's equation tells us that the distance between galaxies gets bigger over time. As the time increases the density of any homogenous distribution of galaxies decreases. In the above derivation it's clear that I derived this equation as a measure of energy, but it's proper interpretation is understood in the context of general relativity as a measure of the curvature of space. I'll derive Friedmann's equation from general relativity in a later post. For now I'll leave it at that.