Blog #29, Worksheet #9.1, Question 2, GR modifications to Friedmann Equations

2. In Question 1, you have derived the Friedmann Equation in a matter-only universe in the New- tonian approach. That is, you now have an equation that describes the rate of change of the size of the universe, should the universe be made of matter (this includes stars, gas, and dark matter) and nothing else. Of course, the universe is not quite so simple. In this question we’ll introduce the full Friedmann equation which describes a universe that contains matter, radiation and/or dark energy. We will also see some correction terms to the Newtonian derivation.

(a) The full Friedmann equations follow from Einstein’s GR, which we will not go through in this course. Analogous to the equations that we derived in Question 1, the full Friedmann equations express the expansion/contraction rate of the scale factor of the universe in terms of the properties of the content in the universe, such as the density, pressure and cosmological constant. We will directly quote the equations below and study some important consequences.

Starting from these two equations, derive the third Friedmann equation, which governs the way average density in the universe changes with time.

As suggested in the assignment, much of the math has been shortened or skip due to tediousness but the processes will be described for each step. We begin by multiplying equation (1) on the sheet by a and taking the time derivative then plugging in equation 2 to cancel \(\ddot{a}\). We end up with \[\frac{4 \pi G a \dot{a}}{3c^2} (\rho c^2 + 3P) = \frac{8}{3} \pi a \dot{a} G \rho + \frac{4}{3} \pi a^2 G \dot{\rho}\]We then cancel and simplify both sides to get \[- \frac{3P \dot{a}}{c^2} = 3 \dot{a} \rho + a \dot{\rho}\]Finally we multiply both sides by \(c^2\) and divide by a to get our desired final answer of \[\dot{\rho} c^2 = -3 \frac{\dot{a}}{a} (\rho c^2 + P) \]
 

(b) Cold matter dominated universe. If the matter is cold, its pressure P = 0, and the cosmological constant Λ = 0. Use the third Friedmann equation to derive the evolution of the density of the matter \(\rho\) as a function of the scale factor of the universe a. You can leave this equation in terms of (\rho\) , (\rho\)0,a and a0, where (\rho\)0 and a0 are current values of the mass density and scale factor. 

The result you got has the following simple interpretation. The cold matter behaves like “cosmological dust” and it is pressureless (not to be confused with warm/hot dust in the interstellar medium!). As the universe expands, the mass of each dust particle is fixed, but the number density of the dust is diluted - inversely proportional to the volume.

Using the relation between (\rho\) and a that you just derived and the first Friedmann equation, derive the differential equation for the scale factor a for the matter dominated universe. 

We can derive from the third Friedmann equation that \[\frac{\dot{\rho}}{\rho} \propto \frac{\dot{a}}{a} \rightarrow \rho \propto a^{-3} \]We then solve the first Friedmann equation with \(\Lambda = 0\) and \(k = 0 \) giving us \[\dot{a}^2 = \frac{8 \pi}{3} G \rho a^2 \propto \frac{8 \pi}{3} G a^{-1} \]Which means \[\dot{a} \propto a^{- \frac{1}{2}} \propto \frac{da}{dt} \]We then separate and integrate both sides \[\int{a^{\frac{1}{2}} da} = \int{dt} \] \[a^{\frac{3}{2}} \propto t \rightarrow a \propto t^{\frac{2}{3}}\]

c) Let us repeat the above exercise for a universe filled with radiation only. For radiation, \(P = \frac{1}{3} \rho c^2 \) and Λ = 0. Again, use the third Friedmann equation to see how the density of the radiation changes as a function of scale factor.

As explained in the assignment, parts b, c, and d are very similar in their mathematical process. We follow the same steps as in part b to show \[\rho \propto a^{-4}\]Which, using the same reasoning would give us \[\dot{a} \propto a^{-1} \propto \frac{da}{dt} \] and when we integrate we would get \[a^2 \propto t \rightarrow a \propto t^{\frac{1}{2}}\]

d) Imagine a universe dominated by the cosmological-constant-like term. Namely in the Friedmann equation, we can set ρ = 0 and P = 0 and only keep Λ nonzero. 

To avoid repetition, we will skip the same steps seen in part b and c and give the answer as \[a(t) \propto e^t\]Meaning that in a dark energy dominated universe, the scale factor increases exponentially as time progresses. 

(e)  Suppose the energy density of a universe at its very early time is dominated by half matter and half radiation. (This is in fact the case for our universe 13.7 billion years ago and only 60 thousand years after the Big Bang.) As the universe keeps expanding, which content, radiation or matter, will become the dominant component? Why? 

Simply put, if we look at our answers for b) and c) we see that \[a_{matter} \propto t^{\frac{2}{3}} \qquad and \qquad a_{radiation} \propto t^{\frac{1}{2}}\]Thus we can say that the portion of the universe dominated by matter will expand much faster than that of radiation so over a long period of time, the matter will become dominant because it is constantly expanding faster. 

(f) Suppose the energy density of a universe is dominated by similar amount of matter and dark energy. (This is the case for our universe today. Today our universe is roughly 68% in dark energy and 32% in matter, including 28% dark matter and 5% usual matter, which is why it is acceleratedly expanding today.) As the universe keeps expanding, which content, matter or the dark energy, will become the dominant component? Why? What is the fate of our universe?
In a similar manner \[a_{matter} \propto t^{\frac{2}{3}} \qquad and \qquad a_{energy} \propto e^t\] \(e^t\) expands much much faster than \(t^{\frac{2}{3}}\) so our universe will soon be far dominated by dark matter. We have a very dark future to look forward to.