# Rearranging the Friedmann Equation

• TRB8985
In summary, The problem involves finding the present age of the universe by integrating Eq. (1.2), which is rewritten in the form dt = H0-1 da / a [ΩΛ + (1 - ΩΛ ) / a3]-1/2, where ΩΛ is the ratio of energy density in the cosmological constant to the critical density. To do this, we must first factor out a ρcr from the right hand side of Eq. (1.2) and use the definition of ρcr to simplify. From there, we can use the definition of H(t) and the fact that the universe is assumed to be flat to arrive at the desired form and proceed with the
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 [ΩΛ + (1 - ΩΛ ) / a3 ]-1/2

where ΩΛ 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) + (ρcr - ρ0) / a2(t)]

ρcr ≡ 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 ρcr from the right hand side, which leads me to the following:

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

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

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

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)/ρcr = Ω

Where Ω = Ωmatter + ΩΛ

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

H2(t) = H02 [Ω + 1 / a2 - ρ0 / (ρcr * a2)]Without actually telling me what to do, could someone provide a hint as to what I may be missing? Thank you very much.

TRB8985 said:

## 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 [ΩΛ + (1 - ΩΛ ) / a3 ]-1/2

where ΩΛ 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) + (ρcr - ρ0) / a2(t)]

ρcr ≡ 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 ρcr from the right hand side, which leads me to the following:

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

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

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

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)/ρcr = Ω

Where Ω = Ωmatter + ΩΛ

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

H2(t) = H02 [Ω + 1 / a2 - ρ0 / (ρcr * 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) ?

TRB8985 said:

## 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?

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 / ρcr = 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)]

= H02m(t) + ΩΛ]

= H02 [(1 - ΩΛ)/a3 + ΩΛ]

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 - ΩΛ) 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.

## 1. What is the Friedmann Equation and why is it important in cosmology?

The Friedmann Equation is a mathematical equation that describes the expansion of the universe in the context of general relativity. It is important in cosmology because it allows us to understand the dynamics of the universe and make predictions about its future behavior.

## 2. How does the Friedmann Equation relate to the concept of dark energy?

The Friedmann Equation includes a term for the energy density of the universe, which can be divided into two parts: matter (including dark matter) and dark energy. Dark energy is thought to be responsible for the accelerating expansion of the universe and is an important factor in the Friedmann Equation.

## 3. Can the Friedmann Equation be rearranged to solve for different variables?

Yes, the Friedmann Equation can be rearranged to solve for different variables, such as the Hubble parameter or the critical density of the universe. This allows us to make different calculations and predictions about the universe based on the available data.

## 4. How does the Friedmann Equation account for the effects of gravity on the expansion of the universe?

The Friedmann Equation takes into account the curvature of spacetime caused by the mass and energy in the universe, which is described by Einstein's theory of general relativity. This curvature affects the rate of expansion of the universe and is incorporated into the equation through the term for the energy density.

## 5. Are there any limitations to the Friedmann Equation?

While the Friedmann Equation is a useful tool for understanding the expansion of the universe, it does have some limitations. It assumes a homogeneous and isotropic universe, which may not accurately reflect the actual distribution of matter and energy. Additionally, it does not account for the effects of dark matter or other unknown factors, which may impact the expansion of the universe.

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