# Proving an Equation for a Single Component Flat Universe

• Arman777
In summary: The question asks you to first show the relation and then to find particular values of w. The OP's issue is related to the first part, not the second.The question is whether you are trying to show that ##dz/dt_0## is related to ##H_0## in the way that the OP has written, or whether you are trying to derive the OP's equation from some other set of assumptions. The two are different things. To prove that ##dz/dt_0## is related to ##H_0## in the way that the OP has written, you don't need to know what ##w## is, you just need to know that ##w < -1/3##.That's what I thought
Arman777
Gold Member
I am trying to prove that for a single component flat universe

$$\frac{dz}{dt_0} = H_0(1 + z) - H_0(1 + z)^{\frac{3 + 3w}{2}}$$

For a single component flat universe,
##q = \frac{2}{3 + 3w}##
##a(t) = (t/t_0)^q##
##t_0 = qH_0^{-1}##
##1 + z = (t_0/t_e)^q##

Now here is my approach,

$$\frac{dz}{dt_0} = qt_0^{q-1}t_e^{-q} = q (t_0/t_e)^qt_0^{-1}$$
$$\frac{dz}{dt_0} = q ( 1 + z) q^{-1}H_0 = H_0 (1 + z)$$

Can someone point it out what am I missing. Like I am trying over 3 days now. Either I suck at math or there's something that I am missing

Edit: This identity can be derived from above identities but I wanted to write
$$t_e = \frac{t_0}{(1 + z)^{3(1+ w)/2}} = \frac{2}{3(1 + w)H_0} \frac{1}{(1+z)^{3(1+ w)/2}}$$

Last edited by a moderator:
Arman777 said:
I am trying to prove that for a single component flat universe

Your first equation doesn't make sense; ##t_0## is a constant, so ##dz / dt_0## is meaningless.

PeterDonis said:
Your first equation doesn't make sense; ##t_0## is a constant, so ##dz / dt_0## is meaningless.
I know I am also confused too. This is a question in the Cosmology book

I tried to think it as ##dz/dt## and then make ##t = t_0## substitution but that also does not work.

Arman777 said:
This is a question in the Cosmology book

What book?

PeterDonis said:
What book?
Barbara Ryden 2nd Ed

This is is not a cosmology problem.
This is not a calculus problem.
It is not even an Algebra 2 problem.
It is an Algebra 1 problem.

Moving to homework forum since this is a problem from a textbook.

This is is not a cosmology problem.
This is not a calculus problem.
It is not even an Algebra 2 problem.
It is an Algebra 1 problem.
Well that's not much helpful. Can you guide us ?

Arman777 said:

Well, that's not much effort.

Since past experience indicates that you're never going to put the effort in, and this thread is going to around and around in circles, the answer is:

3(1+w)/2 < 1
w < -1/3

Arman777 said:
I am trying to prove that for a single component flat universe

$$\frac{dz}{dt_0} = H_0(1 + z) - H_0(1 + z)^{\frac{3 + 3w}{2}}$$

For a single component flat universe,
##q = \frac{2}{3 + 3w}##
##a(t) = (t/t_0)^q##
##t_0 = qH_0^{-1}##
##1 + z = (t_0/t_e)^q##

Now here is my approach,

$$\frac{dz}{dt_0} = qt_0^{q-1}t_e^{-q} = q (t_0/t_e)^qt_0^{-1}$$
$$\frac{dz}{dt_0} = q ( 1 + z) q^{-1}H_0 = H_0 (1 + z)$$

Can someone point it out what am I missing. Like I am trying over 3 days now. Either I suck at math or there's something that I am missing

You are missing something: ##t_e## is a function of ##t_0##, i.e., ##t_e = t_e \left( t_0 \right)##. Consequently, when ##z## is differentiated with respect to ##t_0##, because of the quotient (or product) rule, you should get a second term.

Imagine that, over millions of years, you watch a distant galaxy through a telescope. As you watch, you see the galaxy progress in time. But what you see in the eyepiece is the galaxy as it was at the time of emission, i.e., the emission time ##t_e## changes as the observer's time ##t_0## changes.

I have derived the required result using this method, but I doubt that what I have done is what Ryden expects. I will look into this.

Arman777
Honestly, the question is not very well defined as what is being observed is not well defined. If you consider observing a comoving object, then you need to take the change in ##t_e## with ##t_0## into account. However, if you observe something like a spacelike surface of constant cosmological time (such as the CMB), then clearly ##t_e## would be fixed.

It should be better specified what is intended by ”light source”.

Gentlemen, it is not nearly this complicated. At t0, every element on the right hand side has a value, so dz/dt0 does too. If it has a value, it has a sign.

3(1+w)/2 < 1
w < -1/3
It does not make any sense. We don't know the value of w.

What does the question ask for?

George Jones said:
You are missing something: tetet_e is a function of t0t0t_0, i.e., te=te(t0)te=te(t0)t_e = t_e \left( t_0 \right).
Orodruin said:
If you consider observing a comoving object, then you need to take the change in tetet_e with t0t0t_0 into account.
What kind of dependence are we talking about ? Can you show by using equations ?
George Jones said:
but I doubt that what I have done is what Ryden expects
Why so ?

I am kind of lost. Is this dependence can be derived from the equations in the first post or is it something else ?

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Orodruin said:
It should be better specified what is intended by ”light source”.
I guess yeah. And as Peter Donis mentioned, derivative of ##z## w.r.t ##t_0## seems really odd.

What does the question ask for?
We are trying to prove the identity. The question says "Show that ..".

What does the question ask for?
The question asks you to first show the relation and then to find particular values of w. The OP’s issue is related to the first part, not the second.

Arman777 said:
I guess yeah. And as Peter Donis mentioned, derivative of ##z## w.r.t ##t_0## seems really odd.

The idea is to let the observation time ##t_0## progress into the future. I do not like this notation, because, for example, it would mean that the the expression ##a \left( t \right) = \left( t/t_0 \right)^q## (valid for fixed ##t_0##) would have to change to something like ##a \left( t \right) = A t^q##. (Why?) Better to drop the subscipt ##0## from the observation time, i.e., just write the observation time as ##t##. Then, ##a \left( t \right) = \left( t/t_0 \right)^q## for a fixed ##t_0## is valid. The subscipt ##0## can be inserted at the end

Take a comoving galaxy as the light source. Taking into account that ##t_e## is a function of observation time ##t##, find ##dz/dt##. To differentiate ##a \left( t_e \right)##, use the chain rule.

Arman777
I am trying but I am kind of stuck again.

##\frac{dz}{dt_0} = \frac{d}{dt_0}(t_0^q t_e^{-q} - 1)##
So we can write
##\frac{dz}{dt_0} = qt_0^{q-1}t_e^{-q} - qt_0^qt_e^{-q-1} \frac{dt_e}{dt_0}##
So first term is ##H_0(1+z)##
##\frac{dz}{dt_0} = H_0(1+z) - qt_0^qt_e^{-q-1} \frac{dt_e}{dt_0}##
In the other side we have again ##H_0(1+z)##
$$\frac{dz}{dt_0} = H_0(1+z)[1 - t_e^{-1}t_0 \frac{dt_e}{dt_0}]$$

Is it correct up to here ?

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Arman777 said:
$$\frac{dz}{dt_0} = H_0(1+z)[1 - t_e^{-1} \frac{dt_e}{dt_0}]$$

Is it correct up to here ?

Almost, but I think that you are missing a factor of ##t_0## in the second term in the square brrackets on the right.

Arman777
Arman777 said:
q−10
George Jones said:
Almost, but I think that you are missing a factor of ##t_0## in the second term in the square brrackets on the right.
Yes you are right.
so we have

$$\frac{dz}{dt_0} = H_0(1+z)[1 - t_e^{-1}t_0 \frac{dt_e}{dt_0}]$$

So it means that ##dt_e/dt_0 = 1/(1 + z) ## ?

## What is a Single Component Flat Universe?

A Single Component Flat Universe refers to a theoretical model of the universe where there is only one type of matter or energy present and the overall curvature of the universe is flat. This means that the universe is neither expanding nor contracting, but instead maintains a constant size over time.

## How is an Equation for a Single Component Flat Universe proven?

An equation for a Single Component Flat Universe can be proven through mathematical and observational evidence. The equation must take into account the properties of the matter or energy present, as well as the overall geometry of the universe. Observational evidence, such as measurements of the cosmic microwave background radiation, can also support the validity of the equation.

## Why is proving an Equation for a Single Component Flat Universe important?

Proving an equation for a Single Component Flat Universe is important because it helps us better understand the fundamental nature of the universe and its evolution. It also allows us to make accurate predictions and calculations about the behavior of the universe, such as its expansion rate and the formation of galaxies and other structures.

## What are some challenges in proving an Equation for a Single Component Flat Universe?

One of the main challenges in proving an equation for a Single Component Flat Universe is the limited understanding of the properties of dark matter and dark energy, which are thought to make up the majority of the matter and energy in the universe. Additionally, obtaining accurate observational data and accounting for potential errors and biases can also be challenging.

## What implications does a Single Component Flat Universe have on our understanding of the universe?

A Single Component Flat Universe has significant implications on our understanding of the universe. It suggests that the universe is homogeneous and isotropic on a large scale, and that the expansion of the universe is driven by a single dominant source of energy. This can also inform our understanding of the origins and fate of the universe.

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