Expanding confusion: Time integral of the reciprocal of the scaling factor from Hubble parameter equation

[kurdewiusz]

@andrew s Did you edit your comment with your claim, that te is a variable?

Edited August 13, 2023 by kurdewiusz

[andrew s]

5 minutes ago, kurdewiusz said:

@andrew s Did you edit your comment with your claim, that te is a variable?

No. I pointed out that te was the time of emission.

It is a variable as it depends on what you are observing. For the CMB its zero but for Andromeda it will Now - 2.4 Myrs

For GN -z11 it will be 430 Myrs

Regards Andrew 

[kurdewiusz]

@andrew s Sorry, false suspicion. I repeat:

You wrote:
"It is a variable as it depends on what you are observing. For the CMB its zero but for Andromeda it will Now - 2.4 Myrs"

I wrote earlier: You say, that te is variable. In the observable universe size calculation, that is the integration of the reciprocal of the scale factor function a(t), te is the time of the emission of CMB and tr is the variable, that is Now.
You wrote: Correct

Is te variable or a constant IN CASE OF THE CALCULATION OF THE OBSERVABLE UNIVERSE SIZE? I remind you, that you need tr=te for a(tr)=a(te) for a=1.

Edited August 13, 2023 by kurdewiusz

[andrew s]

7 minutes ago, kurdewiusz said:

Is te variable or a constant IN CASE OF THE CALCULATION OF THE OBSERVABLE UNIVERSE SIZE? I remind you, that you need tr=te for a(tr)=a(te) for a=1.

As I said above for the calculation of the size of the Universe 

tr = Now (that's the time we receive the light) and by convention a(Now) = 1

te = 0 ( that is the time of emission of the CMB we observe now) a(0) ~ 0.001

"I remind you, that you need tr=te for a(tr)=a(te) for a=1."

This is your error tr can't equal te. The CMB we see now was emitted some 13.6 Gyr ago and we observe it Now.

We observe (1+z) = 1101 so using (1+z) = a(tr)/a(te) which for observations Now gives (1+z) = 1/a(te). With a(te) ~ 0.001

I have been consistent on this throughout. 

Unless you accept your error there is no point continuing this discussion. 

Regards Andrew 

[kurdewiusz]

13 minutes ago, andrew s said:

This is your error tr can't equal te. The CMB we see now was emitted some 13.6 Gyr ago and we observe it Now.

(...)

I have been consistent on this throughout. 

Unless you accept your error there is no point continuing this discussion. 

That's not my error, that's Your error! That's what I'm trying to show You!

Constancy of te excludes the possibility, that te=tr for a=1, so the formula z+1 = a0/a = a(tr)/a(te) = 1 is no longer true, because a(tr) and a(te) can't be equal in this case, because te and tr can't be equal in this case for a=1.

So your formula a(tr)/a(te) = 1 is no longer true.

Edited August 13, 2023 by kurdewiusz

[andrew s]

2 minutes ago, kurdewiusz said:

So your formula a(tr)/a(te) = 1 is no longer true.

That is not my formula. I have never posted that. It's yours.

Regards Andrew 

[kurdewiusz]

9 minutes ago, andrew s said:

That is not my formula. I have never posted that. It's yours.

a0/a = a(tr)/a(te) = z+1 (we both agree on that)
a0/a = 1 for a =1
a(tr)/a(te) = 1 for a = 1 (can't be true in this case, so this formula a0/a = a(tr)/a(te) is wrong for a=1)

[andrew s]

3 minutes ago, kurdewiusz said:

a0/a = a(tr)/a(te) = z+1 (we both agree on that)
a0/a = 1 for a =1
a(tr)/a(te) = 1 for a = 1 (can't be true in this case, so this formula a0/a = a(tr)/a(te) is wrong for a=1)

No it means there is zero time and distance between the time of emission and reception. That is the source and defector are next to each other. Totally irrelevant to the detection of the CMB. 

Regards Andrew 

[kurdewiusz]

@andrew s
a0/a = a(tr)/a(te) = z+1 (we both agree on that)
a0/a = 1 for a =1
THEREFORE
a(tr)/a(te) = 1 for a = 1 (can't be true in this case, so this formula a0/a = a(tr)/a(te) is wrong for a=1)

1 hour ago, andrew s said:

No it means there is zero time and distance between the time of emission and reception. That is the source and defector are next to each other. Totally irrelevant to the detection of the CMB. 

So we agree, that the case of the calculation of the observable universe size is not the case, when the emitter and the receiver are next to each other. In case of the calculation of the observable universe size, te is the time of CMB emission, so it can't be equal to the reception time tr = Now, but that's what follows from the formula a0/a = a(tr)/a(te) = z+1 for a =1, that implies a(tr)/a(te) = 1 for a=1. 

Edited August 13, 2023 by kurdewiusz

[andrew s]

That's not how understand it. 

The red shift only depends on the scale factor at the times of emission and reception I.e  at te and tr.

However the size calculation depends on the scale factor for all  times that the light is moving between its emission and reception I.e  such that te =< t <= tr .

I will expand on this tomorrow as its late now. 

Regards Andrew 

[kurdewiusz]

@andrew s please, don't, there's nothing more to argue about, unless you want us both end up in a lunatic asylum. Nevertheless, I am ready for it if necessary.

[andrew s]

This post is mainly for my benefit so @kurdewiusz don't feel obliged to respond.

The size of the observable Universe is the furthest distance a light ray can have got to us from at the current time I.e. Now. It is also called  the particle horizon.

This is the light of the CMB arriving with a red shift of z= 1100.

One issue is to be clear what coordinate system we are using in the discussion

Cosmological distances are most simply discussed in terms of comoving distance and time. That is a coordinate system which moves with the Hubble flow.

Using the rubber band analogy consider the band have a set of tick marks at 1mm intervals at time t = 0 when the CMB was released.

 As the band expands objects on the band stay at the same "comoving" locations but move relative to a metre ruler layed next to the band.

The ruler measures the local distance as in special relativity. A set of rulers layed out along the band measures what we call proper distance   if done Now.

As the band expands to Now the ticks expand from 1mm to 1m as measured by our ruler.
 
To convert from rubber band coordinates to metres we need the number of meters per tick. This is the scale factor "a(t)".

At t = 0 a(0) = 0.001 corresponding to a red shift of z ~ 1100. At t = Now a(now) = 1 (confusingly a(Now) is labled a0 and and t(Now) as t0.

So how far can a light ray go in a radial direction from t = 0 to t = Now ?

The standard metric in the LCDM model in the radial direction is just

ds^2 = -c^2 dt^2 + a(t)^2 dr^2

Note r is the comoving coordinate distance

For a light ray ds = 0 so we get

cdt = a(t)dr or dr/dt = c/a(t)

But dr/dr is just the speed of light in the comoving coordinates.  SR tell us it is constant in local coordinates (our rulers) but it varies in comoving coordinates.  Not very surprising as when a is small there are more ticks per metre than when it has stretched.

To get the required distance we just integrate the equation dr/dt = c/a(t) from t = 0 to t = Now

r = int(c/a(t))dt from t = 0 to t = Now (sorry can't do an integration symbol)

Note a(t) changes smoothly as the Universe expands so the integration is not an issue.

To get the proper distance D you need to use

D = a(Now).r which as we have chosen a(Now) = 1 gives D = r.

However, this only applies to Now

Regards Andrew
 

Edited August 14, 2023 by andrew s

[kurdewiusz]

1. Indeed, it was for your benefit, but also for my detriment. You wrote it all, to cover the problem with a0/a = a(tr)/a(te) = z+1 for a =1, that implies a(tr)/a(te) = 1 for a=1, that can't be true for te equal to the time of emission of CMB in case of the calculation of the particle horizon, that is the observable universe size.  If there is one case, when the formula is wrong, it's no longer a valid formula, period.

2. Almost everything you wrote is true and your reasoning was also my reasoning. Everything, except one thing, and I wrote about it:

On 12/08/2023 at 16:54, kurdewiusz said:

In my opinion, the source of inconsistency between general relativity (which is supposedly a basis of calculations giving 46.5 GLy and 3.2 c values) and the Doppler is in calling the integration-based calculated recession velocity (as well as the proper size of the universe) a part of GR. Friedmann equations, as a solution of Einstein's equations, are a part of GR. Friedmann–Lemaitre–Robertson–Walker metric is a part of GR. Explicit form of the scale factor, derived from the Friedmann equations, is a part of GR. In my opinion, what is not a part of GR, is the integration of the general metric that gives the current, proper distance. This metric equation uses the explicit form of the scale factor derived from GR and may be valid for every single spacetime frame, but if you integrate it, you get this:  *[Back to my first message with my images and the description.]

You wrote again, that this integration is not an issue, (implicitly, because there are no issues), so I'll repeat myself too. No matter how many times you'll repeat, that this integration has no issues, you will not get rid of the issue with the integration described by me, unless you refute my argumentation regarding the integration itself and also the problem with a0/a = a(tr)/a(te) = z+1 for a =1, from which you ran away.

[kurdewiusz]

I also described two cases, where this integration is indeed applicable.

[andrew s]

Let's take a simpler  model than LCDM. 

A flat matter dominated Universe then a(t) = (t/tn) where tn is the time since the BB I.e. now. (Normally written as t0)

The r = ctn^2/3 Int( 1/t^2/3) from t = 0 to tn doing the integration gives  r = 3ctn

 An approximation to the full calculation using LCDM is given here which you may find accessible. It uses a simple polynomial approximation to the values of a(t).

3 hours ago, kurdewiusz said:

also the problem with a0/a = a(tr)/a(te) = z+1 for a =1, from which you ran away.

This is as I have repeated is nonsense unless the tr= te. If you solve it you get z= 0. No red shift the source and detector are next to each other.

Regards Andrew 

PS for a radiation dominated universe you get a(t) = (t/tn)^1/2 and the integration is again trivial as I said.

PPS  I am happy now I full understand the standard approach is correct so will waste no more time on it. Thanks for prompting me to get to grips with it.

[kurdewiusz]

Just now, andrew s said:

This is as I have repeated is nonsense unless the tr= te.

Andrew, it's your nonsense. It follows from your formula.

[kurdewiusz]

It doesn't matter how simple is the equation that we're integrating over time. Numerical integration only shows the problem with it.

[andrew s]

I'll let others judge who is right. Regards Andrew 

Edited August 14, 2023 by andrew s

[kurdewiusz]

Thank U, this is the best thing you could write right now. Nevertheless, physics and maths will never be a popularity contest for me, unless a voter gives the explanation of his vote.

[kurdewiusz]

btw, that was my update before they deleted my post, so you know, that my reasoning was your reasoning with the metric:

[andrew s]

I have read and reread your concerns about the integration and I don't understand it. It seems metaphysical.

All the integration is doing is integrating the instantaneous comoving speed of light c/a(t)  over a time interval. 

It's just a integral version of distance = speed x time.

That different approximations and approaches lead to the same order of distance so that makes me secure the integration is fine.

If you look at my analytic solution for the matter dominated Universe you get 13.6 x 3 = 40.8 Glyrs

The paper I linked to got 42.6 Glyrs using a numerical integration

The pros get 46.5 Glyrs

So given the approximations that looks solid to me.

Regards Andrew 

[kurdewiusz]

1 hour ago, andrew s said:

I have read and reread your concerns about the integration and I don't understand it. It seems metaphysical.

Long ago, at the beginning of this conversation, you wrote: "The integral is well defined and I believe your concept of "retaining the scale factor from the past" is mistaken or at least I don't know what it means."
I answered: "It means, that the current time flow in the place of emission of background photons is much greater than ours. If the scale factor is retained, time dilation is retained as well."
You didn't reply to my answer.
I want to clarify, that this is the condition for this integration to be correct. This condition is not correct, thus the integration is not correct.
This integration requires the different time flow (that is also defined as a value of the scale factor a(t) or its reciprocal) in the place of the emission of CMB that reach us today in our place.
This integration requires the different time flow in these two distant places CURRENTLY, but the current flow of time in these two distant places is currently the same.
This integration is applicable for the path of the gravitationally redshifted photon. Gravitationally curved spacetime retains its distributions of time-scale factor (time dilation) and space-scale factor (length contraction) at all times at all distances, so we can integrate it over time or the redshift. That is not the case with expanding, intergalactic space.

As to your calculations: I dare to claim, they're are wrong for the same reasons I'm giving you and repeating.

And don't think I will forget about a0/a = a(tr)/a(te) = 1 for a = 1 in case of the particle horizon calculation. I will keep reminding you this error, until we're done.

[andrew s]

16 minutes ago, kurdewiusz said:

I answered: "It means, that the current time flow in the place of emission of background photons is much greater than ours. If the scale factor is retained, time dilation is retained as well."

I suspect I did not understand that either. In comoving coordinates there is no time dilation. 

Any way we go in circles.  

20 minutes ago, kurdewiusz said:

And don't think I will forget about a0/a = a(tr)/a(te) = 1 for a = 1 in case of the particle horizon calculation. I will keep reminding you this error, until we're done.

We are done.

Regards Andrew 

[kurdewiusz]

We were done a few times already

21 hours ago, andrew s said:

I suspect I did not understand that either. In comoving coordinates there is no time dilation. 

Exactly! But this integration requires it.

We wouldn't go in circles, if you didn't run away from this error.

Moreover, this numerical integration, which you linked, was my start point. My plot is their plot, and I've also added this link in sources.

Edited August 15, 2023 by kurdewiusz

[kurdewiusz]

For null geodesics, the integral of the Schwarzschild metric can be reduced to the same form as the integral of the FLRW metric, which gives the time integral of the reciprocal of the scale factor. The only difference will be its equation as a function of time. We have a black hole or a star and a gravitational change in the wavelength of a photon falling on the event horizon of the hole or emitted by the star. We can calculate its path by integrating over time the inverse of the scale factor calculated from the Schwarzschild metric as a function of time. Along with passing time, the radial distance between the photon and the hole or star decreases or increases, and the value of the scale factor and the photon's wavelength change with the distance. The problem is that we use the same integration to calculate the photon's path through the expanding space. Expanding, intergalactic, time-scaling space differs fundamentally from gravitationally curved space, scaled with radial distance, because in the former, there is no stage of the background photon's travel, when there are simultaneous sections of its path with different spatial and temporal scales, as there are in the latter. The entire expanding space always scales uniformly. At all times, the scale factor has the same value in the entire expanding space, while in gravity-curved space, the scale factor has different values in different places at all times. For this reason, when using the same integration to calculate the path of a photon moving through the expanding space, we calculate its path in non-existent space, which is why we get a wrong result.

Retaining the scale factor value distribution from the past along the path traversed by CMB photon not only implies the different expansion rates at different distances at the present moment. This integration requires the different time flow (that is also defined as a value of the scale factor a(t) or its reciprocal) in the place of the emission of CMB that reach us today, and in our place. This integration requires the different time flow in these two distant places Now, but the current flow of time in these two distant places is currently the same.

Yet another, similar but different objection: By definition, the scale factor is always equal to 1 at the present moment. Somehow, nobody takes into account the fact that it is always Now (except maybe Eckhart Tolle). Everyone treats the past as if it was carved in stone, in which the function of the scale factor has been carved. However, a plain fact, that the point (Now, 1) moves in time with us, implies that this function is changing its shape. If we calculate the current size of 13.8 billion years universe now and in 1 billion years (by calculating its size 1 billion years backwards) using the same integration, we will get two different results, because the function's shape will change, and we're calculating the area under the curve of its reciprocal.

If you try to argue, that the scale factor (a) is defined to be equal to 1 just for the present age of the universe, I will ask you, what about (z+1)=1/a? What happens with the redshift (z), if the scale factor (a) exceeds 1?

Edited September 18, 2023 by kurdewiusz