Cosmology 101 Part 2

We are bounded in a nutshell of Infinite space: Blog Post #29, Worksheet # 9.1, Problem #2: Cosmology 101 Part 2

2. GR modification to Newtonian Friedmann Equation:

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, \(\rho\) 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. \(\Lambda\) 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.

\[\dot{\rho} c^2 = -3 \frac{\dot{a}}{a}(\rho c^2 + P). \]   To derive this equation, first multiply \(a^2\)  on both sides of  the first equation and then take time derivative on both sides; plug the second equation into your expression to eliminate \(\ddot{a}\).

Now we can use these equations to derive some fun consequences of different kinds of universe, some of which describe our own universe. For simplicity, in the exercise below, let us always set \(k = 0 \) , namely consider a flat universe. Luckily for us, state-of-the-art observations suggest that our universe is likely flat.

(b) Cold matter dominated universe. If the matter is cold, its pressure P = 0, and the cosmological constant \(\Lambda\). 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 \(a_0\) , where \(\rho_0\)  and \(a_0\)  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. Solve this differentiation equation to show that \(a(t) \propto t^{2/3}\)  . This is the characteristic expansion history of the universe if it is dominated by matter. (Hint: When solving this final differential equation, recall that at time t = 0, a = 0 (the Big Bang). At time \(t = t_0 , a = a_0 = 1\)  (present day).)

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

The result also has a simple interpretation. Imagine the radiation being a collection of photons. Similar to the matter case, the number density of the photon is diluted, inversely proportional to the volume. Now the difference is that, in contrast to the dust particle, each photon can be thought of as wave. As you learned last week, the wavelength of the photon is also stretched as the universe expands, proportional to the scale factor of the universe. According to quantum mechanics, the energy of each photon is inversely proportional to its wavelength: \(E = h\nu\). Unlike the dust case where each particle has a fixed energy. So in an expanding universe, the energy of each photon is decreasing inversely proportional to the scale factor. Check that this understanding is consistent with the result you got. Again using the relation between \(\rho\) and a and the first Friedmann equation to show that \(a(t) \propto t^{1/2}\)  for the radiation only universe.

(d) Cosmological constant/dark energy dominated universe. Imagine a universe dominated by the cosmological-constant-like term. Namely in the Friedmann equation, we can set \(\rho = 0\)  and P ­= 0 and only keep \(\Lambda\) nonzero. As a digression, notice that we said “cosmological-constant-like” term. This is because the effect of the cosmological constant may be mimicked by a special content of the universe which has a negative pressure \( P = - \rho c^2\)  . Check that the effect of this content on the right-hand-side of third Friedmann equation is exactly like that of the cosmological constant. To be general we call this content the Dark Energy. How does the energy density of the dark energy change in time? Show that the scale factor of the cosmological-constant-dominated universe expands exponentially in time. What is the Hubble parameter (constant) of this universe?

Hint: While calculating the scale factor as a function of time, you will find that setting \(a(0) = 0\) leads to a negative infinity. Feel free to ignore this term to show the dependence.

(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 accelerated 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?

(a) Following the above instructions in order to derive the third Friedmann equation, we first take the first equation: \[ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi}{3}G\rho - \frac{kc^2}{a^2} + \frac{\Lambda}{3}, \] and multiply both sides by the scale factor a squared. \[  \dot{a}^2 = \frac{8\pi}{3}G\rho a^2 - kc^2 + \frac{\Lambda}{3} a^2, \] and now taking the time derivative, with respect to the density \(\rho\) and the scale factor a: \[2\dot{a} \ddot{a} = \frac{8\pi}{3}G(a^2 \dot{\rho} + 2\rho a \dot{a}) + \frac{\Lambda}{3} \cdot 2 a \dot{a}, \] and solving for \(\ddot{a}\) : \[\ddot{a} = \frac{\frac{4\pi}{3}G}{\dot{a}}(a^2 \dot{\rho} + 2\rho a \dot{a}) + \frac{\Lambda}{3}\cdot a,\] \[\ddot{a} = \frac{ 4\pi G a^2 \dot{\rho}}{3 \dot{a}} + \frac{8 \pi G \rho a}{3} + \frac{\Lambda}{3}\cdot a,\] and now return one scale factor a to the denominator of the left side of the equation,  \[\frac{\ddot{a}}{a} = \frac{ 4\pi G a \dot{\rho}}{3 \dot{a}} + \frac{8 \pi G \rho }{3} + \frac{\Lambda}{3},\] and now this can be equated to the Second Friedmann equation, seeing as they both have \(\frac{\ddot{a}}{a}\) as one side of their equation. So:

\[\frac{\ddot{a}}{a} = \frac{ 4\pi G a \dot{\rho}}{3 \dot{a}} + \frac{8 \pi G \rho }{3} + \frac{\Lambda}{3},\]

\[ \frac{\ddot{a}}{a} = -\frac{4\pi G }{3c^2} (\rho c^2 + 3P) + \frac{\Lambda}{3}.\]

\[\frac{ 4\pi G a \dot{\rho}}{3 \dot{a}} + \frac{8 \pi G \rho }{3} + \frac{\Lambda}{3} =  -\frac{4\pi G }{3c^2} (\rho c^2 + 3P) + \frac{\Lambda}{3},\] which allows us to do some quick algebra to cancel out al lot of terms, \[\frac{ 4\pi G a \dot{\rho}}{3 \dot{a}} + \frac{8 \pi G \rho }{3} =  -\frac{4\pi G }{3c^2} (\rho c^2 + 3P) ,\] \[\frac{ \pi G a \dot{\rho}}{ \dot{a}} + 2 \pi G \rho  =  -\frac{\pi G }{c^2} (\rho c^2 + 3P) ,\]  \[\frac{ a \dot{\rho}}{ \dot{a}} + 2 \rho  = -\rho - \frac{3P}{c^2} ,\] \[\dot{\rho}\frac{ a }{ \dot{a}} = -3\rho - \frac{3P}{c^2} ,\] \[\dot{\rho}\frac{ a }{ \dot{a}} c^2 = -3\rho c^2 - 3P ,\] \[\dot{\rho} c^2 = -3\frac{\dot{a}}{ a }(\rho c^2 + P),\] which is the third Friedmann equation we have been trying to solve for.

(b) In the case of a cold matter dominated universe, there are several considerations that can and should be taken advantage of, which are all outlined by the instructions. These conditions (\(\Lambda = 0 \), P = 0 and k = 0 ), turn the third equation into: \[\dot{\rho} c^2 = -3\frac{\dot{a}}{ a }(\rho c^2),\] \[\frac{\dot{\rho}}{\rho} = -3\frac{\dot{a}}{ a },\] and we take the antiderivative with respect to time on both sides: \[ \int_{\rho_0} ^\rho \frac{\frac{d\rho}{dt}}{\rho} = -3 \int_{a_0} ^a \frac{\frac{da}{dt}}{ a },\] which turns into: \[\ln(\frac{\rho}{\rho_0} )= -3 \ln(\frac{a}{a_0}) , \] and can be simplified into: \[\frac{\rho}{\rho_0} = \frac{a_0 ^3}{a^3}\ ,\] \[\rho = \frac{\rho_0 a_0 ^3}{a^3} .\] And since the question also states that  \(t = t_0 , a = a_0 = 1\)  and \(\rho_0 = 1\), so the relationship becomes: \[\rho = \frac{1}{a^3} .\]

We now take this equation and combine it with the First Friedmann equation, and get: \[ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi}{3}G\rho ,\] \[ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi}{3}G \frac{1}{a^3} ,\]  \[ \dot{a}^2 = \frac{8\pi}{3}G \frac{1}{a} ,\] \[ \dot{a} = a^{-1/2}\sqrt{\frac{8\pi}{3}G } ,\] and since we are only looking for a proportion, the constants can be ignored and considered c: \[ \dot{a} = a^{-1/2} c ,\] now we take an integral on both sides: \[ \frac{da}{dt} = a^{-1/2} c ,\] \[ a^{1/2}da = c \cdot dt ,\] \[ \int_0 ^a a^{1/2}da =\int_0 ^t c \cdot dt ,\]  \[\frac{2}{3} a^{3/2} = t c,\] \[ a = t^{2/3} \frac{3}{2}c,\] therefore: \[ a(t) \propto t^{2/3} . \] This is the characteristic expansion history of the universe if it is dominated by matter.

(c) Another case for the development of the universe is the way it might be radiation dominated, meaning the entirety of the electromagnetic spectrum. Here, we follow a very similar procedure to the last problem, except now we take \(P = \frac{1}{3} \rho c^2\), which turns the equation into: \[\dot{\rho} c^2 = -3\frac{\dot{a}}{ a }(\rho c^2 + \frac{1}{3} \rho c^2),\] \[\dot{\rho} c^2 = -4\frac{\dot{a}}{ a }(\rho c^2),\] which makes a small but meaningful difference once we follow the same process of integration we saw in the previous problem. The integration leaves us with: \[\rho = \frac{1}{a^4} \], which is now plugged into the First Friedmann equation, and we have: \[ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi}{3}G \frac{1}{a^4} ,\] and once again following the exact same process of separation of variables, we have: \[ \frac{da}{dt} = a^{-1} c ,\] which can be evaluated by integrating, and we get: \[ a(t) \propto t^{1/2} ,\] which describes the development of a radiation only universe.

(d) For this problem, different considerations have to be taken into account, mainly the factor we have been eliminating in the other problems, \(\Lambda\). Here, all the terms with \(\rho\) and P will cancel out and we will be left with, from the First Friedmann equation: \[ \left(\frac{\dot{a}}{a}\right)^2 =  \frac{\Lambda}{3}, \] \[ \frac{\dot{a}}{a} =  \sqrt{\frac{\Lambda}{3}}, \]which can be expanded to find a general equation relating the development rate to the new Hubble Parameter. So we have: \[ \frac{\frac{da}{dt}}{a} =  \sqrt{\frac{\Lambda}{3}}, \] \[ \frac{da}{a} = \sqrt{\frac{\Lambda}{3}} dt , \] \[\int_{a_0} ^a \frac{da}{a} = \int_{t_0} ^t \sqrt{\frac{\Lambda}{3}} dt , \] and keeping the definitions we had for initial time and expansion rate, we get:  \[ \ln(a) =  \sqrt{\frac{\Lambda}{3}} t , \]  \[ a =  e^{\sqrt{\frac{\Lambda}{3}} t} = exp\left(\sqrt{\frac{\Lambda}{3}} t \right) .\] So we now know the Hubble parameter of the universe is \(\sqrt{\frac{\Lambda}{3}}\), and that the expansion of this dark Energy dominated Universe is exponential.

(e) As the universe continues expanding, matter would become the dominating component since its rate of expansion is larger and its slope of separation (-3 vs. -4 for radiation) is lower so it will continue to be present and dominate after the radiation has dissipated.

(f) Dark energy will become the dominant component in the universe, since its expansion is the greatest of all, since it depends on a true exponential increase at all times. Eventually, the universe would be so expanded, that gravity no longer would exert enough force to pull matter back in on itself, disproving the “Big Crunch” Theory, and the dark energy would continue propagating as far as possible, turning the universe into a cold and seemingly barren place, with hundreds of orders of magnitudes of space for every speck of matter.