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.
