Does the scale factor need to be normalized?

[hedgehug]

Let me say it this way: Before calculating the proper size of the observable universe, we don't know how many times it has expanded since - let's say - the Planck time. We know it after the calculation. Its result allows to calculate it, so it's like the output. Again, my common sense tells me, that we should know how many times the universe has expanded since the Planck time BEFORE the calculation of the particle horizon, we should use this factor to calculate it, and it shouldn't cancel out! Since we don't, I have a problem with the result of this calculation, which in my opinion outputs the data that should be its input.

 

[Ibix]

Now I know exactly why the proper distance doesn't depend on for a mathematical reason, but my common sense is asking the question Why shouldn't it depend on it?

Because it's part of the process of defining coordinates, and that's a personal decision.

My analogy with switching between meter and kilometer grid squares was not an idle one. Whether I call the distance to the shop 1 unit or 1000 units, it'll still take the same time to walk and I'll still burn the same number of calories doing it. There's no experiment I can do that will tell me that the distance is "really 1 unit" or "really 1000 units" - until I've specified my unit. Then everyone will agree how many of my units it is to the shop (even if they prefer to say it's 5/8 of a mile).

In the FLRW case, you have a similar freedom to rescale your coordinates. If you use the $a_\mathrm{now}=1$ rule and decide on a 1ly coordinate grid then you would define a galaxy currently at $10^6\mathrm{ly}$ to be at $r=10^6$. An astronomer who lived when the universe was half its present scale but used the same rule would say the same galaxy was at $r=5\times 10^5$ (assuming it's a sufficiently long-lived and co-moving galaxy). And one who lives now but uses meters to assign coordinates would say it's at $r=10^{19}$. They absorb the differences between their grid sizes into their definitions of $a$.

They can even use completely different non-uniform coordinate grids if they want. That would change the functional form of the metric as well without changing the prediction of any actual physical quantities. This is a founding principle of GR, by the way, called "general covariance". Physics is independent of coordinate choice, and it must be so precisely because it's a choice.

 

[hedgehug]

@Ibix why on earth did you delete $a(t_0)$ in my quote "Now I know exactly why the proper distance doesn't depend on $a(t_0)$ for a mathematical reason..."?

How many times the universe has expanded since - let's say - the Planck time till now IS NOT a matter of choice of distance coordinates, because it's a ratio. And I'm asking about it BEFORE calculating the particle horizon, because I think that this calculation should depend on it, but it doesn't, and that bothers me.

My first guess:
$d(t_0)=a(t_0)\int_{t_P}^{t_0}\frac{cdt}{(a(t_0)/a(t_P))a(t)}$
$d(t_0)=a(t_P)\int_{t_P}^{t_0}cdt/a(t)$
where $t_P$ is the Planck time.

My second guess:
$d(t_0)=a(t_0)\int_{t_P}^{t_0}\frac{cdt}{(a(t_P)/a(t_0))a(t)}$
$d(t_0)=\frac{a(t_0)^2}{a(t_P)}\int_{t_P}^{t_0}cdt/a(t)$

 

[Ibix]

@Ibix why on earth did you delete $a(t_0)$ in my quote "Now I know exactly why the proper distance doesn't depend on $a(t_0)$ for a mathematical reason..."?

Forum quirk. Highlight a block of text and click reply in the little popup and it quotes the text but drops quoted LaTeX. If you need to copy LaTeX you have to use the reply button at the bottom of the post and edit down the quoted text. I didn't notice this time.

In a similar vein, please try to avoid colouring your text. Dark mode doesn't alter explicitly coloured text so the added text in your last post is near illegible to anybody using that:

How many times the universe has expanded since - let's say - the Planck time till now IS NOT a matter of choice of distance coordinates, because it's a ratio.

Indeed. I thought you were still talking about the free choice of constant scale of $a$, but you seem to be asking why the scale ratio isn't simply related to the particle horizon diameter.

To see this, consider a case with almost no expansion, then a short period of rapid expansion, and then almost no expansion again. (I'm not sure this is physically plausible, but I don't have to care for this purpose.) We can model $a(t)$ as a step function with value 1 before the expansion phase and 2 after. All "how many times bigger is the universe now than some earlier time" questions are clearly always either 1 or 2, depending on whether the rapid expansion happened between the two times or not. However, the particle horizon is always bigger - maybe many times bigger - at a later time because its growth doesn't depend solely on the expansion but also on the travelling of light.

This is less obvious in a more realistic universe because the expansion never "turns off" like that, but the particle horizon radius still depends on the integral of the scale factor, not just the scale factor. So the ratio of particle horizon diameters is the wrong thing to be using to measure the ratio of scale factors.

 

[hedgehug]

I didn't color the text, and the extra ratio I used in the calculation of the particle horizon is the ratio of scale factors, not the ratio of particle horizons.

 

[Ibix]

I didn't color the text

Hm. Perhaps another forum quirk.

the extra ratio I used in the calculation of the particle horizon is the ratio of scale factors, not the ratio of particle horizons.

Apparently I don't understand what question you are asking. I thought you were asking why the ratio of particle horizon radii did not depend simply on the ratio of scale factors, but apparently that's not what you are asking. Can you state in one sentence what you want to know?

 

[hedgehug]

I think you've said more than enough in the matter, and I thank you for your crucial explanations in my threads.

 

[hedgehug]

I was wrong here: https://www.physicsforums.com/threa...or-need-to-be-normalized.1083850/post-7297819

We get $a(t)$ by solving the Friedmann equations, so we already know the ratio $a(t_0)/a(t_P)$ (where $t_P$ is the Planck time) BEFORE calculating particle horizon for the bottom integration limit of $t_P$ instead of $0$. And we already know that this ratio is irrelevant to this calculation.

Let me repeat. It's irrelevant to the proper distance calculation of particle horizon accounting for both the expansion and light travel, how many times the universe has expanded during $t_0-t_P$ period of time, which can be very well approximated by $t_0$.

It doesn't matter whether the universe has expanded $\times 10^{50}$ or $\times 10^{100}$ since the Planck time. We get the same result of the calculation of the observable universe radius, whatever its current size is.

Maybe someone else will feel bothered with it too.