From Aeon to Zeon to Zeit, simplifying the standard cosmic model

[RyanH42]

So If I try to calculate that objects position 0.1 billion years later position.Then the same thing 5sinh^2/3(3/2(13.9/17.3)) and that's equal(I guess) e^Ht=e^(0.07.0.1)

 

[RyanH42]

I go too fast I guess.

 

[marcus]

Let's try to talk only in zeits (for time) and lightzeits( for distance). Talking the time time about "billions of lightyears" is a bother, I think.
Let's use easy numbers.

The present is 0.8 zeit.
How much did distances expand between the time the Earth was forming (around 0.54 zeit) and now? By what factor did they expand?

That's easy. You just have to calculate the ratio a(.8)/a(.54)

Here is something you can paste into google:

sinh(1.5*0.8)^(2/3)/sinh(1.5*0.54)^(2/3)

Can you think of an even simpler example to work? Simple examples are good.

 

[marcus]

By what factor will distances expand between NOW and ONE ZEIT FROM NOW?
That is another very easy exercise. It is good to do several for practice. now = 0.8 and one zeit from now in the future is 1.8
sinh(1.5*1.8)^(2/3)/sinh(1.5*0.8)^(2/3)

Distances will be almost 3 times what they are at present. You can find the more precise figure.

 

[marcus]

Our galaxy disk formed around time t = 0.29 zeit and the Earth formed much later at t = 0.54 zeit.

By what factor did distances expand in the time between those two events?

 

[RyanH42]

sinh(1.5*0.54)^(2/3)/sinh(1.5*0.29)^(2/3)

I found 1.591.This number means If we call scale factor 1 at 0.29 zeit in 0.54 zeit scale factor will be 1.591.So distance R in 0.29 zeit will be R*1.591in 0.54 zeit.

Discovering universe is the greatest thing.
(I hope my idea is true )

 

[RyanH42]

perfect! except
a'=sinh^-1/3(3/2x)cosh(3/2x)
and except for the coth at the very end
a'/a=coth(3/2x)=H

There is something you learn in differential calculus called (in English) "the chain rule" that enables you to take the derivative of NESTED functions like f(g(x)) where you first do g(x) and then put the result of that into f( . )

a=sinh^2/3(3/2x) involves doing sinh and then doing X --> X^2/3
so the functions are nested, one inside the other
taking the derivative involves the chain rule
the derivative of f(g(x)) is f'(g(x)) g'(x)
the derivative of the first multiplied by the derivative of the second.

I know derivatives,simple partial derivatives(very simple ones) and integral(Not too much but enough to understand many applications)
If I made a mistake here probably that's reason is I calculate something wrong,The reason cannot be knowladge. I am curious person and I want to everything about cosmology.Problem I am learning too fast and that causes sometimes wrong results.

 

[marcus]

sinh(1.5*0.54)^(2/3)/sinh(1.5*0.29)^(2/3)

I found 1.591.This number means If we call scale factor 1 at 0.29 zeit in 0.54 zeit scale factor will be 1.591.So distance R in 0.29 zeit will be R*1.591in 0.54 zeit.

Discovering universe is the greatest thing.
(I hope my idea is true )

YES!
I fell asleep early last night around 10 pm pacific time and did not see your posts. I just woke up and came downstairs, it is around 6 am pacific. I am very happy to see several Ryan posts! 1.59 is exactly right. The galaxy disk formed at around 0.29 zeit (we think) and then later when the Earth formed, large-scale distances (not in solar system or within our local group of galaxies which is held together by gravity but REALLY large-scale distances) had grown to about 1.6 times their earlier size.

 

[marcus]

...

I am curious person and I want to everything about cosmology.Problem I am learning too fast and that causes sometimes wrong results.

Yes, I understand that. It is good to be curious and learn fast.

Let f(x) = x^2/3
then f'(x) = (2/3) x^-1/3

this is just an application of the general rule that the derivative of xⁿ is nx^n-1

now the chain rule says f(g(x)) derivative is f'(g(x))g'(x)

so the derivative of (g(x))^2/3 is (2/3)g(x) g'(x)

there is one other detail. In this case (the distance growth in universe) the function g(t) is sinh(1.5t) so again by chain rule we have
g'(t) = 1.5 cosh(1.5t)
a factor of 1.5 comes out when we take the derivative
so that 1.5 cancels the 2/3 that appeared earlier.
there are really two applications of the chain rule here

 

[RyanH42]

I am very happy know.

 

[marcus]

I also am happy. It was nice to find these posts when I woke up this morning. Now I will go get some coffee.

 

[RyanH42]

Thats great.Have a nice day

Finally I learned and understand the idea.

Thank you.

 

[marcus]

I got some coffee and am back. I would like to discuss something else, and proceed slowly. You said you had some integral calculus.
Please take a look at the numberempire.com web page where they have a "definite integral calculator" and see if you understand how to use it
http://www.numberempire.com/definiteintegralcalculator.php

 

[marcus]

It's good to start with very simple examples
http://www.numberempire.com/definiteintegralcalculator.php

When you go there, if you scroll down the page to where it says EXAMPLES there is a box you can click on that says "Example 1"
If you click on this it will show the first simple example, how to calculate the definite integral from 0 to 4 of the function x² using the computer. Actually that is such a simple problem that you don't need to use the computer! you would simply evaluate (1/3)x³ at x=4
but integrating more complicated functions sometimes requires using computer, so it is good to know how to do this

 

[RyanH42]

Yeah,I lookd

 

[RyanH42]

It's good to start with very simple examples
http://www.numberempire.com/definiteintegralcalculator.php

When you go there, if you scroll down the page to where it says EXAMPLES there is a box you can click on that says "Example 1"
If you click on this it will show the first simple example, how to calculate the definite integral from 0 to 4 of the function x²

I know that x^3/3 then 4^3/3-0

 

[RyanH42]

It's good to start with very simple examples
http://www.numberempire.com/definiteintegralcalculator.php

When you go there, if you scroll down the page to where it says EXAMPLES there is a box you can click on that says "Example 1"
If you click on this it will show the first simple example, how to calculate the definite integral from 0 to 4 of the function x²

I know that x^3/3 then 4^3/3-0

 

[RyanH42]

I am busy right know can we start 1 hour later.

 

[marcus]

I know that x^3/3 then 4^3/3-0

Good! It is a case where the function is so simple we do not need "numerical integration"---that is we do not need the computer.

but when you do need the computer in more complicated cases it is good to know how.

You type the function to be integrated into the box, at the top of the page. And you type the limits (like 0 and 4). And you press "calculate".

 

[marcus]

I am busy right know can we start 1 hour later.

Sure.

 

[RyanH42]

I am here now

 

[marcus]

Oh, there you are! BTW there is no rush. No need to hurry on to a new topic.
I only wanted to LAY OUT the next topic for when you might want to proceed. It is distance. Calculating the distance that some light is now, from its source galaxy, (how far it has come,with the help of expansion and its own speed)---this requires getting numberempire or some other tool to calculate a definite integral.

For example, you remember the Earth formed about .54 zeit and the present day is .8 zeit. How far would some light travel in that time?

By itself, without help by expansion it could only go .26 lightzeit.

But now that we have the sinh^2/3 function we can also factor in how much each little step the light takes will be enlarged by expansion.

You should not feel any pressure to proceed. Only when you feel curious about this and are ready, and are not too busy with other work. But I only want to set this topic out, so we know what the next thing is.

 

[marcus]

$$D(0.54) = \int_.54^.8 \frac{cdt}{a(t)}$$

here t is some moment in time in the interval between 0.54 and 0.8
and cdt is a little step that the light takes at time t, a little distance.
and a(t) is how much smaller the distances were then, at time t, than they are now.

So that dividing by a(t) scales the little step up to its size now. IOW 1/a is the factor by which the little step at time t is magnified, between time t and the present.

a(t) = sinh^2/3(1.5t)/sinh^2/3(1.5*0.8)

And D(0.54) is how far the light (that was emitted by its source galaxy at time .54) has traveled. How far away it is from home now.

 

[RyanH42]

0.26 lightzeit come from (0.8zeit-0.54zeit)*17.3=0.26 lightzeit.

I do the integral and I found ln(a(0.8))-ln(a(0.54)) will be the answer.I can do that just a second

 

[RyanH42]

0.234 I forget c.c*0.234 lightzeit.But c unit must be in zeit isn't it ?

 

[marcus]

c=1
light travels 1 lightzeit per zeit
that is, by itself without counting the help of expansion.

 

[marcus]

Isn't it true that sinh^2/3(1.5*0.8) is about equal to 1.3?
So we could make it easy for ourselves and just take a(t) to be sinh^2/3(1.5t)/1.3

That is approximately right. Is that OK?

So the distance the light has come, how far it is now from home, measured in lightzeits, is

$$D(0.54) = 1.3\int_.54^.8 \frac{dt}{\sinh^{2/3}(1.5t)}$$

we need to see how to put that into numberempire (I have dropped the c, it is just 1)

 

[RyanH42]

I didnt see your edit.So I made wrong.Now I can see

 

[RyanH42]

0.3042 lightzeit ?

 

[marcus]

actually I never did that integral at numberempire. I don't know if that website has the sinh function. : ^)
We will see. Before now I always used a different version of the integral that involved a calculus trick called "change of variable".
But we should try this version.

What should go in the box? You see the box I mean? It is where you type the function to be integrated. Like in Example 1 the function was x*x

1.3*sinh(1.5*t)^(-2/3)

Let's try that, and make the limits .54 and .8, and in the variable box put t. The default is x but we are using t, either is OK,it does not matter.

I am slow. you already went ahead. Now I want to try

I get
==quote==
Integral of 1.3*sinh(1.5*t)^(-2/3) by t on the interval from .54 to .8:
.3046035045325962
==endquote==

YES! I see that is the same as what you got.