Skip to main content

Newtonian Cosmology

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 which appears in the Friedman equationsI 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.


Popular posts from this blog

Three Things William Lane Craig Gets Wrong About the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

How Should Thatcherites Remember the '80s?

Every now and again, when I talk to people about the '80s I'm told that it was a time of unhinged selfishness, that somehow or other we learned the price of everything but the value of nothing. I can just remember that infamous line from Billy Elliot; 'Merry Christmas Maggie Thatcher. We all celebrate today because its one day closer to your death'. If it reflected the general mood of the time, one might wonder how it is she won, not one but three elections.

In an era when a woman couldn't be Prime Minister, her launch into power was accidental owing in part to Manchester liberals and the Winter of Discontent. Yet I'm convinced her election victory in '79 was the only one that ever truly mattered. Simply consider the calamity of what preceded it, the 1970s was a decade of double-digit inflation, power cuts, mass strikes, price and income controls, and the three day week. Britain was sick, it needed fundamental restructuring but no one seemed to quite under…

Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…