Astrophysics/Cosmology question - redshift

[tourjete]

Homework Statement

Our universe is observed to be flat, with density parameters Ω_m,0 = 0.3 in non-relativistic matter and Ω_$\Lambda$,0 = 0.7 in dark energy at the present time. Neglect the contribution from relativistic matter.

At what redshift did the expansion of the universe start to change from deceleration from acceleration?

Homework Equations

a(t=change) = 0
a(t=now) = 1 (? is this convention?)

((da/dt) * (1/a))² = H₀² E²(z) where E²(z) = Ω_$\Lambda$,0 + (1-Ω₀)(1+z)² + Ω_m,0(1+z)³

z = a⁰/a(t) -1 where a₀ denotes the present time

The Attempt at a Solution

First I thought about setting (da/dt)*(1/a) equal to zero and solving the equation for z. However, this gives a redshift of -2.32 which doesn't really make sense as the negative value implies that there is a blueshift (right?)

Upon further thought, I realized that a = 0 means that the lefthand side of the equation should be undefined / go to infinity... so could z be infinity? This doesn't really make sense to me either but it's all I've got.

I can manipulate the other equation I found to also give z=infinity by setting a(t) equal to zero.

I'm sure I'm forgetting some important concept but that's what I've got so far.

 

[mark726]

Thank you for your post. It is an interesting question to consider the redshift at which the expansion of the universe changed from deceleration to acceleration. Let me start by clarifying a few things. The convention for the scale factor, a(t), is indeed to set it equal to 1 at the present time, so a(t=now) = 1. As for your attempt at a solution, you are on the right track. Setting (da/dt)*(1/a) equal to zero is the correct approach. However, as you mentioned, this would result in a negative redshift, which does not make sense physically. This is because the equation you are using assumes a flat universe, and in a flat universe, the scale factor, a(t), can never be equal to zero. Therefore, the expansion of the universe cannot start from a(t=change) = 0.

To find the redshift at which the expansion changed from deceleration to acceleration, we need to look at the behavior of the scale factor at large values of z. As z approaches infinity, the second term in the equation for E2(z) becomes negligible compared to the first two terms. This means that at large z, the equation can be simplified to E2(z) = Ω\Lambda,0 + (1-Ω0). In order for the expansion to change from deceleration to acceleration, we need E2(z) to be equal to zero. This can only happen if (1-Ω0) is equal to -Ω\Lambda,0, which implies that Ω0 = 1. This is known as the critical density, and it is the density at which the universe is exactly flat.

Therefore, the redshift at which the expansion changed from deceleration to acceleration is z = infinity, which corresponds to the critical density of Ω0 = 1. This means that at the present time, the universe is flat and the expansion is accelerating. I hope this clarifies your doubts. Let me know if you have any further questions.