In Question 1, you have derived the Friedmann Equation in a matter-only universe in the Newtonian 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.
The first Friedmann equation:
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi}{3}G\rho - \frac{kc^2}{a^2}+\frac{\Lambda}{3}\]
The second Friedmann equation:
\[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3c^2}(\rho c^2+3P)+\frac{\Lambda}{3}\]
In these equations, ρ and P are the density and pressure of the content, respectively. k is the curvature parameter; k = -1, 0, 1 for open, flat and closed universe, respectively. Λ is the cosmological constant. Note that in GR, not only density but also pressure are the sources of energy.
Starting from these two equations, derive the third Friedmann equation, which governs the way average density in the universe changes with time.
WARNING: MATH AHEAD
First let's multiply both sides of the first equation by \(a^2\) to get:
\[\dot{a}^2=\frac{8\pi}{3}G\rho a^2 - {kc^2}+\frac{\Lambda}{3}a^2\]
Now let's take the derivative with respect to t.
\[2\dot{a}\ddot{a}=\frac{8\pi}{3}G(\dot{\rho}a^2 +2\rho a\dot{a}) - {kc^2}+\frac{2\Lambda a \dot{a}}{3}\]
The second equation solved for \(\ddot{a}\) can be expressed as:
\[\ddot{a}=-\frac{4\pi G}{3c^2}(\rho c^2+3P)a+\frac{\Lambda a}{3}\]
Plugging this in for \(\ddot{a}\) and simplifying gives us:
\[2\dot{a}(-\frac{4\pi G}{3c^2}(\rho c^2+3P)a+\frac{\Lambda a}{3})=\frac{8\pi}{3}G(\dot{\rho}a^2 +2\rho a\dot{a}) - {kc^2}+\frac{2\Lambda a \dot{a}}{3}\]
\[-\frac{8\pi G}{3c^2}(\rho c^2+3P)a\dot{a}+\frac{2\Lambda a\dot{a}}{3}=\frac{8\pi}{3}G(\dot{\rho}a^2 +2\rho a\dot{a}) - {kc^2}+\frac{2\Lambda a \dot{a}}{3}\]
\[-\frac{8\pi G}{3c^2}(\rho c^2+3P)a\dot{a}=\frac{8\pi}{3}G(\dot{\rho}a^2 +2\rho a\dot{a}) - {kc^2}\]
Since our universe is flat, k=0, allowing further simplification.
\[-\frac{1}{c^2}(\rho c^2+3P)a\dot{a}=\dot{\rho}a^2 +2\rho a\dot{a}\]
\[-\rho c^2 a\dot{a} -3Pa\dot{a}=\dot{\rho}a^2 c^2 +2\rho a\dot{a}c^2\]
\[-3Pa\dot{a}=\dot{\rho}a^2 c^2 +3\rho a\dot{a}c^2\]
\[\dot{\rho}a^2 c^2= -3Pa\dot{a}-3\rho a\dot{a}c^2\]
\[\dot{\rho}a^2 c^2= -3a\dot{a}(P+\rho c^2)\]
\[\boxed{\dot{\rho}c^2= -3\frac{\dot{a}}{a}(\rho c^2+P)}\]
This is the Third Friedmann Equation which described the density of the universe as a function of time.
(b) What is the evolution of a cold matter dominated universe? (P=0, \(\Lambda=0\))
Then, solve for a in terms of t using the 1st equation.
Let's start with the Third Friedmann Equation.
\[\dot{\rho}c^2= -3\frac{\dot{a}}{a}(\rho c^2+P)\]
We are told that \(P=0\) and \(\Lambda=0\), so:
\[\dot{\rho}= -3\frac{\dot{a}}{a}\rho \]
We now need to solve this differential equation, so:
\[\frac{\dot{\rho}}{\rho}= -3\frac{\dot{a}}{a}\]
\[\int^{\rho}_{\rho_0}\frac{d\rho}{\rho}= -3\int^a_{a_0}\frac{da}{a}\]
\[ln\left(\frac{\rho}{\rho_0}\right)=-3ln\left(\frac{a}{a_0}\right)\]
\[\boxed{\frac{\rho}{\rho_0}=\left(\frac{a}{a_0}\right)^{-3}}\]
Now we can use this result along with the 1st equation to find how a and t scale. This means we can drop all constants.
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi}{3}G\rho - \frac{kc^2}{a^2}+\frac{\Lambda}{3}\]
Earlier we were told that \(P=0\), \(\Lambda=0\), and \(k=0\) so:
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi}{3}G\rho\]
Let's drop all constants.
\[\left(\frac{\dot{a}}{a}\right)^2 \propto \rho\]
Let's apply our previous result and integrate:
\[\left(\frac{\dot{a}}{a}\right)^2 \propto a^{-3}\]
\[\dot{a}^2 \propto a^{-1}\]
\[\dot{a} \propto a^{-1/2}\]
\[\frac{da}{dt} \propto a^{-1/2}\]
\[a^{1/2}da \propto dt\]
\[\int a^{1/2}da \propto \int dt\]
\[a^{3/2} \propto t\]
\[\boxed{a \propto t^{2/3}}\]
(c) Radiation dominated universe: \(P=\rho c^2 /3\), \(\Lambda =0\).
I'll omit the math from here on out and just give answers.
\[\boxed{\frac{\rho}{\rho_0}=\left(\frac{a}{a_0}\right)^{-4}}\]
\[\boxed{a \propto t^{1/2}}\]
(d) Dark Energy dominated universe \(P=0\) \(\rho=0\)
\[\boxed{a \propto e^t}\]
(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?
(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?
(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?
So far we have determined that radiation disperses faster than matter. Dark Energy on the other hand, disperses slower than matter, making it the dominant component as time passes. Thus the fate of our universe, is for matter and radiation to be spread thin, while dark energy becomes the primary component of the universe.
I worked on this problem with B. Brzycki, G. Grell, and N. James.
Nice job
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