Age of Universe (Liddles Modern Cosmology)

[hawker3]

Hi,
I am currently reading Liddles Introduction to Modern Cosmology (2nd Ed) and having trouble with problem 8.4, about the age of the universe with a cosmological constant.
The question asks to derive the formula for the age by first writing the Fridemann equation in such a model as
[itex]\frac{da}{dt}[/itex] = [itex]H^{2}_{0}[/itex] [itex][Ω_{0}a^{-1}[/itex] + [itex]\left(1-Ω_{0})a^{-2}\right][/itex]

But I'm not sure how this step is found, so can't proceed any further. I would appreciate any help to get me started.

 

[cepheid]

Welcome to PF hawker3!

Hi,
I am currently reading Liddles Introduction to Modern Cosmology (2nd Ed) and having trouble with problem 8.4, about the age of the universe with a cosmological constant.
The question asks to derive the formula for the age by first writing the Fridemann equation in such a model as
[itex]\frac{da}{dt}[/itex] = [itex]H^{2}_{0}[/itex] [itex][Ω_{0}a^{-1}[/itex] + [itex]\left(1-Ω_{0})a^{-2}\right][/itex]

But I'm not sure how this step is found, so can't proceed any further. I would appreciate any help to get me started.

I'd be happy to help you, but in order to do so, you need to tell me what form of the Friedmann equation you are starting with. Then I can give you suggestions on what manipulations to perform in order to get it into the form given above.

 

[BillSaltLake]

I'm having trouble finding a set of conditions in which the formula is correct. Look at the matter-only case for example: a [itex]\propto[/itex]t^2/3. Then the left side is [itex]\propto[/itex] t^-1/3 whereas the 2 right side terms are [itex]\propto[/itex] t^-2/3 and t^-4/3.

 

[cepheid]

I'm having trouble finding a set of conditions in which the formula is correct. Look at the matter-only case for example: a [itex]\propto[/itex]t^2/3. Then the left side is [itex]\propto[/itex] t^-1/3 whereas the 2 right side terms are [itex]\propto[/itex] t^-2/3 and t^-4/3.

I think that the relation a ~ t^2/3 is specific to the "critical" or "Einstein-de Sitter" universe which is not merely matter-dominated, but also happens to have a critical density of matter so that Ω₀ = Ω_m = 1. Hence (1-Ω₀) = 0.

Even so, the equation is also a bit off. At the risk of giving too much away, it should be (da/dt)² on the left-hand side. You can see that this makes things work out for the critical case.

The a^-2 dependence on the (1-Ω₀) term also seems wrong to me. If you assume that there is no spatial curvature, and that the matter shortfall (1-Ω₀) is made up of "something" that has a constant energy density (e.g. dark energy), then this term should have an a² dependence. If you assume that there is nothing else aside from matter, then (1 - Ω₀) is the curvature term, and it should have an a⁰ dependence (i.e. no dependence on a at all).

 

[sardar]

Hello,
I am a scientist with a background in cosmology and I would be happy to help you with this problem. The first step to solving this problem is to understand the components of the Friedmann equation. The Friedmann equation is a key equation in modern cosmology that describes the expansion of the universe. It relates the rate of expansion (represented by the derivative of the scale factor, da/dt) to the density and curvature of the universe. In its simplest form, the Friedmann equation is written as:

\frac{da}{dt} = H(a)

Where H(a) is the Hubble parameter, which is a function of the scale factor. In order to incorporate a cosmological constant into this equation, we can rewrite the Hubble parameter as:

H(a) = H_{0} \sqrt{Ω_{0}a^{-3} + Λa^{-2} + (1-Ω_{0}-Λ)a^{-1}}

Where H_{0} is the present-day value of the Hubble parameter, Ω_{0} is the present-day matter density parameter, Λ is the cosmological constant, and (1-Ω_{0}-Λ) is the present-day dark energy density parameter.

Now, we can substitute this expression for H(a) into the Friedmann equation and simplify to get:

\frac{da}{dt} = H_{0} \sqrt{Ω_{0}a^{-3} + Λa^{-2} + (1-Ω_{0}-Λ)a^{-1}}

Next, we can rearrange this equation to get:

\frac{da}{dt} = H_{0} \sqrt{Ω_{0}a^{-1} + \left(1-Ω_{0})a^{-2}\right) + Λ}

Finally, we can substitute this expression into the original Friedmann equation and solve for the age of the universe, which is given by:

t = \frac{1}{H_{0}} \int_{0}^{a} \frac{da}{\sqrt{Ω_{0}a^{-1} + \left(1-Ω_{0})a^{-2}\right) + Λ}}

I hope this helps you get started on solving problem 8.4. Remember to carefully consider the components of the Friedmann equation and how they relate to the expansion