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.

 

[closing the circle]

Redefining the scale factor from its current value of 1 to 1 at the time of the CMB emission, and from 1/(1100+1) at the time of the CMB emission to 1100+1 at the present time, shouldn't change the result of the integral for calculating the radius of the observable universe, right?

What exactly would the current integral $\int_{0}^{t_0}cdt/a(t)$ look like after this change? Integration limits must remain the same: 0 and $t_0=13.8$ billion years, and the integral must obviously remain consistent with the FLRW metric for nul

Redefining the scale factor from its current value of 1 to 1 at the time of the CMB emission, and from 1/(1100+1) at the time of the CMB emission to 1100+1 at the present time, shouldn't change the result of the integral for calculating the radius of the observable universe, right?

What exactly would the current integral $\int_{0}^{t_0}cdt/a(t)$ look like after this change? Integration limits must remain the same: 0 and $t_0=13.8$ billion years, and the integral must obviously remain consistent with the FLRW metric for null geodesic.

It will certainly change the numerical value of the result because redefining the scale factor means redefining the units of distance.

It won't change the physical meaning of the result.

just learning how to post comments. Einstein proposed action at a distance , I propose action at a point. Using Euclid 's definition of a point.

 

[Dale]

just learning how to post comments. Einstein proposed action at a distance , I propose action at a point. Using Euclid 's definition of a point.

In this context, this is clearly false. Newton proposed action at a distance, but Einstein’s proposal was a local theory.

Please be aware that personal speculation is not permitted here. All posts must be consistent with the professional scientific literature.

 

[hedgehug]

@closing the circle due to your learning to post comments in my thread, I'm learning to quote comments that were crucial to this thread after you quoted my original post with the formula for the comoving distance, which wasn't the proper distance.

Proper distance is $d(t)=a(t)\chi$ where the comoving distance is $\chi=\int_{0}^{t_0}cdt/a(t)$, so the proper distance equal to the observable universe radius should be $d(t_0)=a(t_0)\int_{0}^{t_0}cdt/a(t)$, and in my case $a(t_0)=1100+1$. Is $d(t_0)=47\text{ GLy}$? If not, what should be changed to get this value?

So if you replace the conventional $a$ with your version that is $1101a$, the 1101 outside the integral cancels the one inside and you get the same proper distance as you do with the usual definition - as you must.

 

[Jaime Rudas]

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.

I'm not sure I understand your point correctly, but it seems you might be confusing the expansion rate with the particle horizon growth rate. They are closely related concepts, but they are not the same. For example, due to expansion, since the scale factor was 0.5, distances have doubled, but the particle horizon has grown approximately 2.35 times.

 

[hedgehug]

@Jaime Rudas By how many times the universe has expanded, I meant the ratio of scale factor values at different times, not the ratio of particle horizons at different times.

 

[Jaime Rudas]

@Jaime Rudas By how many times the universe has expanded, I meant the ratio of scale factor values at different times, not the ratio of particle horizons at different times.

Since the scale factor was 0.5, it has doubled; since it was 0.1, it has expanded tenfold; since it was 0.01, it has expanded 100 times; and since it was 0.00001, it has expanded 100,000 times. I really don't see your point or what it has to do with the particle horizon you keep mentioning.

 

[hedgehug]

@Jaime Rudas Particle horizon accounts for both the expansion and light travel, and its calculation is practically a topic of this thread in context of the scale factor.

 

[Jaime Rudas]

@Jaime Rudas Particle horizon accounts for both the expansion and light travel,

I agree.

and its calculation is practically a topic of this thread in context of the scale factor.

Thanks for clarifying.

 

[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.

What bothers me most, I guess, is why do we assume zero scale factor at the time of the BB. There wasn't even such time, because there was no time, nor space at that event.

I also guess that we don't define it to be 1 instead of 0 at the BB, because we would have no idea what $a(t_0)$ should be in this case.

 

[Jaime Rudas]

What bothers me most, I guess, is why do we assume zero scale factor at the time of the BB.

No, the model doesn't assume that.

There wasn't even such time, because there was no time, nor space at that event.

Where did you get that from?

 

[Ibix]

What bothers me most, I guess, is why do we assume zero scale factor at the time of the B

We don't assume it. It follows from the Friedmann equations for the flat and open cases that $\dot{a}^2>0$ for all time, so $a$ is either always increasing or always decreasing. (Closed universes can have turning points, but they're necessarily maxima, so they have a singularity in the past and may also have one in the future.) Since $a$ is currently increasing there must be a zero in the past. The Big Bang singularity is derived, not assumed.

 

[hedgehug]

No, the model doesn't assume that.

Where did you get that from?

We don't assume it. It follows from the Friedmann equations for the flat and open cases that $\dot{a}^2>0$ for all time, so $a$ is either always increasing or always decreasing. (Closed universes can have turning points, but they're necessarily maxima, so they have a singularity in the past and may also have one in the future.) Since $a$ is currently increasing there must be a zero in the past. The Big Bang singularity is derived, not assumed.

If there was time inseparable from space at the BB, then there was space at the BB, and the universe already had its inital size.

 

[Jaime Rudas]

If there was time inseparable from space at the BB, then there was space at the BB, and the universe already had its inital size.

¿And?

 

[hedgehug]

¿And?

And you can't tell how many times it has expanded since the BB.

 

[Jaime Rudas]

If there was time inseparable from space at the BB, then there was space at the BB, and the universe already had its inital size.

I will try to clarify my previous answer: the model does not include the moment t=0 because a singularity occurs there, that is, an undefined state. The fact that there is an undefined state does not imply that, in reality, neither space nor time existed at that moment. It simply means that we do not know the characteristics of the universe at that moment.

 

[Jaime Rudas]

And you can't tell how many times it has expanded since the BB.

Yes, we can't.

 

[hedgehug]

Yes, we can't.

If we can't, why is our calculation of particle horizon correct, since it accounts for both the expansion and light travel, but at the same time we can't really tell how many times the universe has expanded since the BB if it had initial size?

Isn't this calculation correct only for zero initial size?

 

[Jaime Rudas]

If we can't, why is our calculation of particle horizon correct, since it accounts for both the expansion and light travel, but at the same time we can't really tell how many times the universe has expanded since the BB if it had initial size?

Isn't this calculation correct only for zero initial size?

Because for calculating the particle horizon, it's not relevant to know how many times the universe has expanded since the Big Bang. What's relevant is knowing the value of the scale factor for any moment t>0.

 

[hedgehug]

Because for calculating the particle horizon, it's not relevant to know how many times the universe has expanded since the Big Bang. What's relevant is knowing the value of the scale factor for any moment t>0.

But knowing $a(t\rightarrow 0)$ is just like knowing $a(t=0)$, and you already have this value from Friedmann equations. It's zero.

 

[Ibix]

If we can't, why is our calculation of particle horizon correct, since it accounts for both the expansion and light travel, but at the same time we can't really tell how many times the universe has expanded since the BB if it had initial size?

Formally, evaluate the integral with a lower limit of $\delta t$, some small positive value, and then consider what happens as $\delta t\rightarrow 0$. You can do this analytically for pure matter and pure radiation cases, for example, and show that the integral has a finite limit. On the other hand, $\lim_{\delta t\rightarrow 0} a(t_0)/a(\delta t)$ is not defined.

Intuitively, it works because the particle horizon radius is the distance between two lines that cross at the singularity while the scale factor comes from two lines that don't cross there. That means that the limits as $\delta t\rightarrow 0$ behave differently.

 

[hedgehug]

@ibis Is the particle horizon calculation correct only for zero initial size of the universe, or any initial size?

 

[Jaime Rudas]

But knowing $a(t\rightarrow 0)$ is just like knowing $a(t=0)$, and you already have this value from Friedmann equations. It's zero.

Friedman's equations don't describe reality at t=0.