Cosmology
First FriedMann equation: how it derives the Λ-CDM Model and the fate of universe
Cosmology is a study about the structure of universe, how does it evolve to the
presence state, and how it will evolve to be. One of the base equations in the study of
cosmology is the First Friedmann equation, derived from the Einstein’s equation of the field
of gravity, which constructs a model of the universe, how it changes and its most possible
fate. Although it sounds like a physics topic, through analyzing the equation mathematically,
we can obtain some crucial information about the universe, even deducing the model and
the fate of our cosmos.
Introducing the equation
Below is the content of the original First Friedmann equation:
𝑎̇
8𝜋𝐺
𝑘𝑐 2
( )2 −
𝜌= − 2
𝑎
3
𝑎
In the equation, 𝑎 represents the scale factor of the universe, corresponding to the
size of the universe. 𝑎̇ represents the acceleration rate of the universe, namely the time
derivative of 𝑎.
8𝜋𝐺
3
is a constant term introduced from the Einstein gravity field equation,
while 𝜌 refers to the density of matter in the whole universe. Lastly, 𝑘 refers to the shape
of universe, having three possible values (1, 0, and -1). If 𝑘 = 1, the universe is closed and
have a limited space, being like a 4D sphere where two parallel lines will meet each other
(Fig a). If 𝑘 = 0, the universe is flat and has no distinctive curvature. In this universe, two
parallel lines drawn will never meet each other, yet always having a constant distance from
each other (Fig b). If 𝑘 = −1, the universe is open, being like a 4D saddle in which two
parallel lines will become farther apart (Fig c).
Fig(a)
Fig(b)
Fig(c)
Analysing the equation: Expansion of Universe
The LHS of the equation can be separated into two parts. The first term on the LHS,
𝑎
(𝑎̇ )2, corresponds to the kinetic energy of the universe to keep expanding. The second term
on the RHS,
8𝜋𝐺
3
𝜌, corresponds to the potential energy of the universe, from the gravity by
the mass of the matter in the universe. If LHS > 0 (KE > PE), the universe will expand faster
and faster. If LHS = 0 (KE = PE), the expansion of universe will come to a halt. If LHS < 0, (KE <
PE), the universe will turn to collapse into a singularity due to the gravity that pulls it
together. By analysing LHS of the equation mathematically, we can predict the expansion of
universe.
𝑎
After measuring the Hubble parameter 𝐻, which is equal to (𝑎̇ ), and 𝜌, we find that
the LHS > 0, so the universe will expand faster and faster. It is also confirmed by other
observation. However, when considering the RHS of the equation, the astronomers notice
an abnormality. If LHS > 0, 𝑘 should be −1, which means the universe should be open. But
the calculation of the geometry of the universe (drawing triangles in the universe dimension)
reveals that the universe is flat (𝑘 = 0). The equation doesn’t balance! Logically, the
universe should also slow down its expansion if it is flat, as the energy for expansion will be
diluted throughout the increasing space of the universe. There must be an additional
element in the equation, which reduces the LHS even further to 0 to matches the RHS, as
well as balancing the diluted energy to keep the expansion. Thus, a new crucial element in
the universe, the dark energy, is deduced.
Adjusting the equation: Presence of Dark Energy
Upon facing the need to balance the First Friedmann equations, the astronomers
remember the ‘mistake’ Einstein made in his gravity field equation – the cosmological
constant Λ, which was intended to create a static, non-expanding universe model. However,
as it was proved the universe is expanding, Einstein abandoned this term in his equations.
Now, this term is needed again in the First Friedmann Equation to make it looks like the
following:
𝑎̇
8𝜋𝐺
Λ𝑐 2
𝑘𝑐 2
( )2 −
𝜌−
= − 2
𝑎
3
3
𝑎
In this adjusted equation, the term with Λ refers to the dark energy, an energy from the
empty space or vacuum, which is yet another force to pushes the universe to expand. Now
that this new term is added, the equation is balanced again! We can conclude a more
complete universe model, including the newly discovered property, the dark energy. In this
model, the universe is flat, but the energy of expansion would not be diluted as there is
energy coming from the empty space that propel the universe to expand. In fact, it even
propels the universe to expand in an accelerating rate, which we will discuss in the later
section.
Obtaining information about the universe: Hubble Parameter, Critical Density, and the 𝛬CDM Model
To identify how fast the universe is expanding, the astronomers establish a parameter –
𝑎
the Hubble Parameter 𝐻, which is equal to (𝑎̇ ). Thus, we can deduce a formula from the
first Friedmann Equation to find its relationship with the density of space and dark energy as
follow:
8𝜋𝐺
𝑘𝑐 2 Λ𝑐 2
𝜌− 2 +
3
𝑎
3
The Hubble parameter, being the accelerating rate of the universe divided by the size of the
𝐻2 =
universe, becomes greater as the universe’s expansion is accelerated. By assuming that 𝑘
and Λ are constant, and that the density of universe 𝜌 decreases as the space of the
universe increases, we can see that the parameter is in fact decreasing over time. However,
it will never reach zero due to the cosmological constant Λ (dark energy). Does that mean
the universe is decelerating in its expansion instead of accelerating? It will be discussed in
the next session.
Another property of universe we can deduce from the first Friedmann equation is the
critical density of the universe, 𝜌𝑐 . The following equation of 𝜌𝑐 is derived by assuming 𝑘
and Λ as 0, while 𝐻0 represents the Hubble parameter in the present day.
3𝐻02
8𝜋𝐺
The equation describes an ideal density of matter in a flat (𝑘 = 0) matter-dominated
universe (without dark energy). However, when considering the density of matter in our flat
universe, the astronomers discover a weird scenario -- 𝜌𝑐 is much greater than the density
of matter in the universe. Along with other evidence, the scientists propose another kind of
matter, the dark matter, which contributes to 85% of the mass of matter in the universe. This
𝜌𝑐 =
leads to the completion of the Λ-CDM Model of the universe, in which Λ refers to the dark
energy and CDM refers to cold dark matter. Through further evaluation of the model, the
scientists discover the dark matter and dark energy takes up a large part of the universe, as
shown in Fig (d).
Fig (d)
Analysing the equation: An universe with accelerating expansion
Now going back to whether our universe is expanding in an accelerating or decelerating
rate. We have mentioned that the Hubble parameter 𝐻 has been decreasing over time. But
𝑎
𝐻 refers to (𝑎̇ )2, in which 𝑎̇ represents the expansion rate of the universe. We can see
that even with 𝐻 decreasing over time, the size of universe (𝑎) is still increasing as well as
the expansion rate (𝑎̇ ). It is only that the ratio of 𝑎̇ to 𝑎 is decreasing. Moreover, as 𝑎
increases while 𝐻 remains non-zero, 𝑎̇ is increasing exponentially. The universe has
always been accelerating its expansion. In the far future, as the universe has expanded so
much that its density 𝜌 is almost 0, we can eliminate the term with 𝜌 as well as the term
with 𝑘 (as the universe is flat) to reach the equation below:
𝑎̇
Λ𝑐 2
𝐻 2 = ( )2 =
𝑎
3
We can see that at the far future, the expansion of universe is only due to the dark energy.
However, as 𝐻 never reaches 0, the expansion will keep on accelerating. It may be that any
two different particles in the universe will eventually become so far apart in space that there
will be no more reactions in the cosmos. Thus is the fate of universe that we predict based
on the first Friedmann Equation.
Conclusion: Importance of Mathematics in Cosmology and the Flaw in the above Analysis
Through the above mathematical analysis, combined with the data obtained from the
astronomers, we can obtain the model of universe as well as predicting its fate. In different
cosmological topics, such as calculating the percentage of matter in universe and modelling
the current universe or the early universe just after Big Bang, Mathematics is equally
important as the observation data and physics laws. Without the Mathematics tools such as
calculus and matrices, these investigations would have been impossible. It can be concluded
that Mathematics is essential in dealing with an every-day matter, our universe.
It may also be noticed that the above analysis about the First Friedmann Equation
𝑎
contains some flaws. For example, as the term (𝑎̇ ) is squared in the equations, we cannot
determine whether the universe is expanding or collapsing only by analysing it. Instead, the
astronomers have observed that the distant galaxies are going farther and farther away from
us. It’s due to this that we reach a conclusion that the universe is expanding. However, the
fact that Hubble parameter 𝐻 doesn’t equal to 0 does indicate that the universe is
accelerating either its expansion or collapse. There are also many other observation data
and calculations that determine the presence of dark matter, dark energy and the Λ-CDM
Model. The study of the universe is truly broad and deep that it is impossible to include all of
them in this essay.
Sources:
Applications of Mathematics (Cosmology) – Mathigon
Will the Universe Expand Forever? – YouTube
Why the Universe Needs Dark Energy - YouTube
What Does Dark Energy Really Do? - YouTube
The Lambda CDM model - 2.2.1.1 - YouTube
Is the rate of expansion of the universe (Hubble 'constant') increasing or decreasing over
time? - Quora
Friedmann equations (Density Parameter) - Wikipedia