Rearranging the Friedmann Equation

[TRB8985]

Homework Statement

Assume that the universe today is flat with both matter and a cosmological constant, the latter with energy density which remains constant with time. Integrate Eq. (1.2) to find the present age of the universe. That is, rewrite Eq. (1.2) as:

dt = H₀^-1 da / a [Ω_Λ + (1 - Ω_Λ ) / a³ ]^-1/2

where Ω_Λ is the ratio of energy density in the cosmological constant to the critical density.

Homework Equations

[/B]
Equation 1.2:

H²(t) = 8πG/3 [ρ(t) + (ρ_cr - ρ₀) / a²(t)]

ρ_cr ≡ 3H₀² / 8πG

The Attempt at a Solution

I'm having a little difficulty getting equation 1.2 into the format desired above. My idea was to factor out a ρ_cr from the right hand side, which leads me to the following:

H²(t) = 8πG/3 * ρ_cr [ρ(t)/ρ_cr + (ρ_cr - ρ₀) / a²(t)*ρ_cr]

= 8πG/3 * (3H₀²/8πG) [ρ(t)/ρ_cr + (ρ_cr - ρ₀) / a²(t)*ρ_cr]

= H₀² [ρ(t)/ρ_cr + (ρ_cr - ρ₀) / a²(t)*ρ_cr]

At this point, I run into a little difficulty switching things up inside the brackets like I need to.

Using some things I've gleaned from various sources online, I've come to the following idea:

ρ(t)/ρ_cr = Ω

Where Ω = Ω_matter + Ω_Λ

I'm not really quite sure what the difference ρ_cr - ρ₀ is supposed to represent, so my last step looks like the following:

H²(t) = H₀² [Ω + 1 / a² - ρ₀ / (ρ_cr * a²)]


Without actually telling me what to do, could someone provide a hint as to what I may be missing? Thank you very much.

 

[nrqed]

Homework Statement

Assume that the universe today is flat with both matter and a cosmological constant, the latter with energy density which remains constant with time. Integrate Eq. (1.2) to find the present age of the universe. That is, rewrite Eq. (1.2) as:

dt = H₀^-1 da / a [Ω_Λ + (1 - Ω_Λ ) / a³ ]^-1/2

where Ω_Λ is the ratio of energy density in the cosmological constant to the critical density.

Homework Equations

[/B]
Equation 1.2:

H²(t) = 8πG/3 [ρ(t) + (ρ_cr - ρ₀) / a²(t)]

ρ_cr ≡ 3H₀² / 8πG

The Attempt at a Solution

I'm having a little difficulty getting equation 1.2 into the format desired above. My idea was to factor out a ρ_cr from the right hand side, which leads me to the following:

H²(t) = 8πG/3 * ρ_cr [ρ(t)/ρ_cr + (ρ_cr - ρ₀) / a²(t)*ρ_cr]

= 8πG/3 * (3H₀²/8πG) [ρ(t)/ρ_cr + (ρ_cr - ρ₀) / a²(t)*ρ_cr]

= H₀² [ρ(t)/ρ_cr + (ρ_cr - ρ₀) / a²(t)*ρ_cr]

At this point, I run into a little difficulty switching things up inside the brackets like I need to.

Using some things I've gleaned from various sources online, I've come to the following idea:

ρ(t)/ρ_cr = Ω

Where Ω = Ω_matter + Ω_Λ

I'm not really quite sure what the difference ρ_cr - ρ₀ is supposed to represent, so my last step looks like the following:

H²(t) = H₀² [Ω + 1 / a² - ρ₀ / (ρ_cr * a²)]


Without actually telling me what to do, could someone provide a hint as to what I may be missing? Thank you very much.

Do you know the definition of H(t)? How does it relate to the scale factor a(t) ?

 

[George Jones]

Homework Statement

Assume that the universe today is flat with both matter and a cosmological constant, the latter with energy density which remains constant with time.

$\rho_0$ is $\rho\left( t \right)$ evaluated at $t = t_0 = t_{\mathrm{now}}$. Since $\rho\left( t \right) = \rho_m \left( t \right) + \rho_\Lambda \left( t \right)$, $\rho_0 = \rho_m \left( t_0 \right) + \rho_\Lambda \left( t_0 \right)$, and $\rho_\Lambda \left( t_0 \right) = \rho_\Lambda \left( t \right)$, since $\rho_\Lambda$ is constant.

Adding to what @nrqed asked, how is "the universe today is flat" expressed mathematically?

 

[TRB8985]

Good evening nrqed and George, thank you for your hints.

I think I've come to the answer to this problem with your help, but I'm not sure if I'm skipping some justifications or not.

Given what you said George, "the universe today is flat" ⇒ ρ₀ / ρ_cr = Ω₀ = 1.

Continuing from where I left off in my first post, and using the definition of H(t) as you suggested nrqed, here's what I have:

[(da/dt)/a]^2 = H₀² [Ω(t) + 1 / a²(t) - 1/ a²(t)]

= H₀² [Ω(t)]

= H₀² [Ω_m(t) + Ω_Λ]

= H₀² [(1 - Ω_Λ)/a³ + Ω_Λ]

On page 4 of the Modern Cosmology textbook, Dodelson writes ".. the energy density of matter scales as a^-3". So hence, I figured slapping that a^-3 below (1 - Ω_Λ) was justified by those words. (That, and I don't see any other way to get an a^-3 in the expression to match what I need.)

If this process is correct, I'll have absolutely no problem separating the variables and integrating for the next part. I just don't know if that last step is okay.