Cosmology 101- Friedmann equation

The Friedmann Equation is a very important equation describing the expansion of the universe.

H^{2} = \frac{8 π G}{3} (ρ_{m} + ρ_{r}) - \frac{k c^{2}}{a^{2} {R_{0}}^{2}} + \frac{Λ c^{2}}{3}

$H = \frac{\dot{a}}{a} = \frac{d a}{a d t}$ is called the Hubble parameter, it is a function of time, the present day value of Hubble parameter is the Hubble constant. $a$ is called scale factor, it is a dimensionless quantity that describe the size of the universe, it is also a function of time, the present day value of the scale factor is $1$ . $R_{0}$ is the radius of curvature of the Universe.

The Friedmann Equation indicates that the expansion (or contraction) of our universe is influenced by 4 parts:

Matter component ( $ρ_{m}$ term, matter refers to both ordinary matter and dark matter)
Radiation component ( $ρ_{r}$ term)
Curvature component ( $k$ term)
Dark energy (cosmological constant) component ( $Λ$ term)

To get a better understanding of what are dark matter and dark energy is, watch this video made by Kurzgesagt.

With the Friedmann Equation, we can define the critical density of universe. (to see how it is defined and derived, click here).

The expression for critical density is (this does have a density dimension, check it by your self!):
$ρ_{cr} (t) = \frac{{3H(t)}^{2}}{8 π G}$

Critical density is also a function of time

However only $Ω_{m}$ and $Ω_{r}$ have been defined, since there is only matter density and radiation density in the Friedmann Equation, but if we define density parameter for both curvature and cosmological constant, we can apply the definition of dnesity parameter and simplify the Friedmann Equation, let:
$Ω_{k} = \frac{k c^{2}}{{R_{0}}^{2} a^{2} {H(t)}^{2}}$ and
$Ω_{Λ} = \frac{Λ c^{2}}{3 {H(t)}^{2}}$

With the density parameter defined, we can alter the Friedmann Equation into:

Ω_{m} (t) + Ω_{r} (t) + Ω_{k} (t) + Ω_{Λ} (t) = 1

Before we actually start to analyze the equation, we need to state some restrcitions on what value each parameter can take. It is obvious that $Ω_{m}$ and $Ω_{r}$ can only accept positive value, since there is no such thing as negative density. However, both $Ω_{k}$ and $Ω_{Λ}$ can take both postive and negative value (they can also take 0 as its value):

if $Ω_{k}$ has a positive value, it is a positively curved universe, with a shape of a sphere-like structure.
If $Ω_{k}$ has a negative value, it is a negatively curved universe, with a shape of a hyperbola-like structure.
If $Ω_{k}$ equals to 0, it is a flat universe, with a shape of a flat-surface. Currently, we believe the universe is flat, i.e. it has no curvature.

click here

The situation for

Ω_{Λ}

is a little different since it has nothing to do with the shape of the universe. Cosmological constant acts like a repulsive force against gravity and drives the accelarated expansion of our universe.

If the value of $Ω_{Λ}$ is positive, it means that there exist a repulsive force stronger than gravity, it will accelarate the expansion of the universe.
If the value of $Ω_{Λ}$ is negative, it means that there exist a contractive force (basically gravity but does not follow Newton's universal gravitation law) that will deaccelarate the expansion of the universe.
If the value of $Ω_{Λ}$ is 0, it means that there is no such thing as cosmological constant.

According to data from the Planck probe, the modern value for all the density parameters are:

$Ω_{m} = 0.308$
$Ω_{r} = 8.4 × 10^{-5}$
$Ω_{Λ} = 0.692$
$Ω_{k} = 0$

One can understand this result as cosmological constant makes up 69.2% of our universe, while matter (both dark and ordinary matter) makes up 30.8% of our universe. If we want to split matter into dark and ordinary matter, then dark matter accounts for 25.9% and ordinary matter accounts for 4.9%. We will discuss how to use these data to calculate the fate of universes with different density parameter.