Relation between dark energy and universe expansion

Question

I'm trying to understand the role of dark matter/energy in universal evolution. What I know is they occupy a large percentage of our unverse's energy density, and to observe them we use type Ia supernova as a "candle", but I still confused with two point

1. what's the exaclty method that we used (with type Ia supernova observation) to determine the relative contributions of dark energy

2. how does the dark energy affect the universe (why we usually related them with unversal expansion)

Answer 1

how does the dark energy affect the universe (why we usually related them with universal expansion)

To answer your second question, you need to look at the acceleration equation.

$$\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}\sum_i(\varepsilon_i + 3w_i\varepsilon_i)$$

$$\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}\sum_i\varepsilon_i(1 + 3w_i)$$

Where $\varepsilon_i$ represents the energy density and $w_i$ equation of state parameter where $i = m, r, \Lambda$

For simplicity let me denote the sum as $K$ such that $K \equiv \sum_i\varepsilon_i(1 + 3w_i)$. In this case the acceleration equation reduces to

$$\frac{\ddot a}{a}=-\frac{4\pi G}{3c^2}K$$

Now by looking at this equation, it's very clear that if $K>0$ the universe decelerates, and if $K<0$, the universe accelerates.

Let me calculate the $K$ for two types of universe.

$1$ - Universe with $\varepsilon_{\Lambda} = 0$

In this universe model, you'll see that

$$K = \sum_i\varepsilon_i(1 + 3w_i) =2\varepsilon_r+\varepsilon_m > 0$$ which is clearly positive, since $\varepsilon_r > 0$ and $\varepsilon_m > 0$

This implies that, this type of universe decelerates.

$2$- Universe with $\varepsilon_{\Lambda} \ne 0$

At this point we should ask, what do we need to have an accelerated universe ?

Well, we need to satisfy $K<0$ as I have mentioned before. To do that, let me set a simple inequality.

$$K < 0 \rightarrow 2\varepsilon_r+\varepsilon_m + \varepsilon_{\Lambda}(1+3w_{\Lambda}) <0$$

If we assume that $\varepsilon_{\Lambda}$ is positive, to have an accelerating universe the only possible solution becomes $$1+3w_{\Lambda}<0$$

Or $$w_{\Lambda} < -1/3$$

So, to have an accelerated universe we need some sort of material where its equation of state parameter is less than $-1/3$. Dark energy is some sort of "energy" which satisfies this condition and allows an expanding universe model.

Current favoured dark energy model sets $w_{\Lambda} = -1 < -1/3$.

In this case we have,

$$K = 2\varepsilon_r+\varepsilon_m -2 \varepsilon_{\Lambda}$$

As the universe expands $\varepsilon_r$ and $\varepsilon_m$ decreases but $\varepsilon_{\Lambda}$ remains the same. This allows $K$ to be negative at some point in the history of the universe.