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

[kurdewiusz]

In case I'm wrong and this integration is correct, I ask for clarification. If I'm right and this integration is wrong, then the observable universe radius calculated with it is also wrong. In that case, I'd like to know, what's the proper formula for this calculation. If there is no other, then I think we have a problem and I kindly ask you to acknowledge it.

Downvotes given out of personal grudge after discussion - this is closely related to the original question in terms of physics, moderation and the astronomy/stackexchange community.

Edited October 15, 2023 by kurdewiusz

[vlaiv]

Hi and welcome to SGL.

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At any instant, the entire path traveled by the photon scales uniformly.

This is correct.

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The described integration means dividing this path into segments that expand at different rates depending on the time they've been traversed by the photon.

This is also correct.

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If different values of the scaling factor describe consecutive spacetime segments, then the photons reaching us would have the original wavelength from the time of their emission, because the scaling factor in our place and time has a value of 1 for us.

Not sure how you came up with this part though.

You should look at it this way:

For every different spacetime segment there is a small change in scale factor which corresponds to small change in spatial length. Photon being in that segment of space time will have its wavelength altered due to this change in spatial length. Wavelength of photon is distance between two "crests" of EM wave - increase this distance (while photon is in this patch of space) and wavelength gets longer - photon is red shifted.

Total wavelength red shift / increase in wavelength is sum of all small increases in wavelength that are consequence of scale factor changes and respective spatial length increases as photon travels thru each patch of spacetime

 

[kurdewiusz]

4 minutes ago, vlaiv said:

Total wavelength red shift / increase in wavelength is sum of all small increases in wavelength that are consequence of scale factor changes and respective spatial length increases as photon travels thru each patch of spacetime

This integration means that a fragment of space with an initial length equal to the original wavelength at the emission site expanded 1100 times, and subsequent ones expanded less and less, until no expansion at our place and time.

[vlaiv]

1 minute ago, kurdewiusz said:

This integration means that a fragment of space with an initial length equal to the original wavelength at the emission site expanded 1100 times, and subsequent ones expanded less and less, until no expansion at our place and time.

Integration is summation process.

It represents the fact that at each "leg" of the journey photon expands by tiny little bit and all those tiny expansions add up to x1100 when it reaches us.

[kurdewiusz]

1 minute ago, vlaiv said:

Integration is summation process.

Exactly

2 minutes ago, vlaiv said:

It represents the fact that at each "leg" of the journey photon expands by tiny little bit and all those tiny expansions add up to x1100 when it reaches us.

That' the problem: all these legs have a different expansion rates and they all end up with different lengths.

[vlaiv]

Just now, kurdewiusz said:

Exactly

That' the problem: all these legs have a different expansion rates and they all end up with different lengths.

I think I now see why we don't understand each other.

There are two different sizes we can attribute to term size of the observable universe.

One is based on light travel and other is based on "hypothetical ruler length" / or size in co-moving coordinates.

equation addresses following question - thru how much (local) space did photon need to travel from CMB to reach us? That distance is not the same distance that we would get if we had hypothetical ruler that we placed at the same spot in universe and other end at our location - such ruler would give different answer.

In each little segment of it's flight - photon had to "traverse" some universe. Universe between photon and us expanded, and universe between origin and photon expanded as well. However, in this case we are not interested in stretching between origin and photon - as photon already traversed that part and that stretching won't contribute to total length photon traversed to reach us. Only expansion between us and photon is considered.

Now, we can simply look at above formula as much more common everyday scenario. You are in the car and you want to calculate the distance you traveled from the speed function.

For constant speed we know that s = v * t (s being distance, v - velocity and t - time). However, if you have changing speed - you need to perform integral of v(t) * dt - or sum every little v*t for small t intervals where v can be considered constant.

In above integral we almost have the same thing. With one difference - velocity is constant and can be brought in front of the integral - but scale factor - which represents change in length (and hence "speed" if we stick to definition of speed as length / time) is variable in time - so we must integrate it.

If we observe following formula:

which says that if we have some distance d0 now - at Hubble time t - that distance was smaller by factor of a(t).

Now from perspective of photon - it traveled "full" distance, even if it was smaller at time t - so we need to do inverse - to divide with a(t) in order to get "true" distance photon traversed.

That is why we have 1/a(t) in integral for light travel size of universe.

[kurdewiusz]

2 hours ago, vlaiv said:

For constant speed we know that s = v * t (s being distance, v - velocity and t - time). However, if you have changing speed - you need to perform integral of v(t) * dt - or sum every little v*t for small t intervals where v can be considered constant.

When you integrate the changing speed or the constant speed with the variable time step, you add up all these small, partial sections. Space is not made of sections of different lengths (that expanded at different rates) unlike the path calculated by the numerical integration of a motion of a body moving with variable speed. Can you see the difference? It's crucial.

2 hours ago, vlaiv said:

That is why we have 1/a(t) in integral for light travel size of universe.

I introduce parameter b(t)=1/a(t) and have the exact, same problem.

It occurred to me, that this integration would be suitable for the path of gravitationally redshifted photon, because gravitationally curved spacetime retains its distributions of time dilation and length contraction factors at all times at all distances:



This is not the case with expanding, intergalactic space.

Edited July 27, 2023 by kurdewiusz

[andrew s]

You may find this paper worth studying.  While it can take some effort to get into the various diagrams I think if will illuminate the issues you are discussing. 

Regards Andrew 

[vlaiv]

9 hours ago, kurdewiusz said:

Space is not made of sections of different lengths (that expanded at different rates) unlike the path calculated by the numerical integration of a motion of a body moving with variable speed. Can you see the difference? It's crucial.

But it is.

Along the path as photon is traversing any particular patch of space - it does so at certain time and due to expansion of universe - that patch of space being traversed is at different scale factor than the rest of the path.

Ok, so here is simplest explanation that I can think of.

Imagine following scenario. There is some example universe in which expands in particular way - say we have three epochs - early, middle and current epoch. Scale factor for each epoch is 1/3, 2/3 and 1. In each epoch photon travels with speed C which is some "epoch kilometers per second".

Each epoch has its own "kilometers" which should not be mixed. We have early kilometers, middle kilometers and current kilometers. Think of these as different units of length, maybe that is better - like miles and kilometers - they measure the distance but have different numerical value for same length and there is a conversion factor (scale factor).

Now, if we say that distance to edge of observable universe is 13.7Bly - what does that mean? Well that just means that if we had static universe and we shot the photon at C - it would take it 13.7 billion years to reach said distance. So we are using measure of distance that we know from present time - light year. It is "present" light year - one that we are familiar with (not early light years - from the past).

Let's say that photon is traveling for 1/3 of the time in each epoch. How do we calculate in "present kilometers" how much it traveled?

We will have:

c_early * 1/3 + c_middle * 1/3 + c_present * 1/3

Where c_period has the same numerical value but is expressed in different kilometers - either early kilometers, middle kilometers or present kilometers. But we want answer in present kilometers to make sense to us, so what do we do? Well, it is easy - we have conversion factor, so we can convert it to modern kilometers by dividing with scale factor at given epoch.

so it will be:

c * sum ( 1 / early_scale + 1 / middle_scale + 1  / modern_scale) * delta_t (or 1/3 in our case)

Do you see anything familiar in this formula?

[kurdewiusz]

4 hours ago, andrew s said:

You may find this paper worth studying.  While it can take some effort to get into the various diagrams I think if will illuminate the issues you are discussing. 

This document was my starting point. The title of my question refers to it. It's also given as one of the references at the bottom.

Edited July 27, 2023 by kurdewiusz

[kurdewiusz]

2 hours ago, vlaiv said:

But it is.

But it's not. I understand everything that you wrote below.

Space, that has been traversed by CMB photon does no longer exist, as well as the space that has been expanding behind it, moving away its emission place, as well as the space that has been expanding in front of it, lengthening its path. Current space, that separates us from the current position of the place of its emission, this space has a current metric, other than the metric of space traversed by the photon, other than the space that was expanding in front of it and behind it during its travel. Former space no longer exists not only because of the changes of matter/energy density in the expanding space traversed by the photon, but above all because spacetime exists only for the time of its metric.

[vlaiv]

4 minutes ago, kurdewiusz said:

Space, that has been traversed by CMB photon does no longer exist, as well as the space that has been expanding behind it, moving away its emission place, as well as the space that has been expanding in front of it, lengthening its path.

You will need to explain a bit more this statement. What do you mean by does not exist?

It is the same space - but with "changed properties". Property of distance changed.

 

[iantaylor2uk]

5 minutes ago, kurdewiusz said:

But it's not. I understand everything that you wrote below.

Space, that has been traversed by CMB photon does no longer exist, as well as the space that has been expanding behind it, moving away its emission place, as well as the space that has been expanding in front of it, lengthening its path. Current space, that separates us from the current position of the place of its emission, this space has a current metric, other than the metric of space traversed by the photon, other than the space that was expanding in front of it and behind it during its travel. Former space no longer exists not only because of the changes of matter/energy density in the expanding space traversed by the photon, but above all because spacetime exists only for the time of its metric.

So by your logic if a photon is emitted by a galaxy which travels towards us, the galaxy no longer exists. I don't think so! 

[kurdewiusz]

16 minutes ago, vlaiv said:

You will need to explain a bit more this statement. What do you mean by does not exist?

It is the same space - but with "changed properties". Property of distance changed.

We characterize and describe the fabric of spacetime with its properties - values of the terms of the metric tensor, that contain information about the rate of the flow of time and the density of space. If it has different properties, it's different fabric, just like the material characterized by its properties.

Edited July 27, 2023 by kurdewiusz

[vlaiv]

2 minutes ago, kurdewiusz said:

We characterize and describe the fabric of spacetime with its properties - values of the terms of the metric tensor, that contain information about the rate of the flow of time and the density of space. If it has different properties, it's different fabric, just like the material characterized by its properties.

Yes, and these properties can change - but that does not mean that spacetime vanishes or appears - it is the same spacetime just with different properties.

One of these properties is distance - and we can integrate this distance as it changes over some other parameter - for example time

[kurdewiusz]

On 27/07/2023 at 14:25, iantaylor2uk said:

So by your logic if a photon is emitted by a galaxy which travels towards us, the galaxy no longer exists. I don't think so! 

No sir, space within the galaxy is not expanding, unlike the intergalactic space.

*EDIT: As far as I know, parameters describing the building blocks of matter - massive particles - have not changed either.

Edited July 30, 2023 by kurdewiusz

[kurdewiusz]

11 minutes ago, vlaiv said:

Yes, and these properties can change - but that does not mean that spacetime vanishes or appears - it is the same spacetime just with different properties.

That's exactly what it means. It's replaced with a new one with changed properties. If you say, that it's the same spacetime, you must tell, what is the property, that connects them and remains unchanged. Distance is not it.

Edited July 27, 2023 by kurdewiusz

[vlaiv]

2 minutes ago, kurdewiusz said:

That's exactly what it means. It's replaced with a new one with changed properties. If you say, that it's the same spacetime, you must tell, what is the property, that connects them, that remained unchanged. Distance is not it.

Content? All the stuff inside it?

There is no reason to think it is different spacetime and that previous stopped existing and now different one exists.

Even if that is the case - we can still integrate numerical values as that is simple summation of things. You can calculate how much you earn in a month by summing up all the expenses and the money you are left with at the end of the month - even if you no longer have that money with you.

In any case - distance is just comparison between two quantities of "stuff" that is inside space time. Stuff did not cease to exist nor was it created / replaced in the mean time.

[kurdewiusz]

13 minutes ago, vlaiv said:

Content? All the stuff inside it?

13 minutes ago, vlaiv said:

In any case - distance is just comparison between two quantities of "stuff" that is inside space time. Stuff did not cease to exist nor was it created / replaced in the mean time.

Unchanged since the big bang? Same density? No conversion of matter into energy and vice versa?

13 minutes ago, vlaiv said:

There is no reason to think it is different spacetime and that previous stopped existing and now different one exists.

There is one, totally sufficient reason - all the parameters by which we describe and define spacetime have changed.

13 minutes ago, vlaiv said:

Even if that is the case - we can still integrate numerical values as that is simple summation of things. You can calculate how much you earn in a month by summing up all the expenses and the money you are left with at the end of the month - even if you no longer have that money with you.

But someone else has these money. You are adding up things, that have ceased to exist and no one has them anymore.

Edited July 27, 2023 by kurdewiusz

[vlaiv]

2 minutes ago, kurdewiusz said:

But someone else has these money. You are adding up things, that heave ceased to exist and no one has them anymore.

But all the energy / matter still exists and hence relationship between their properties still exist (in one form or another) - much like money that might have changed from paper to coin or to electronic form - but nominal value remained or even changed based on inflation.

I think that we have problem with world view above all else and as such, I'm not sure I can contribute anything further

[kurdewiusz]

7 minutes ago, vlaiv said:

But all the energy / matter still exists and hence relationship between their properties still exist (in one form or another) - much like money that might have changed from paper to coin or to electronic form - but nominal value remained or even changed based on inflation.

I wanted to point this out before reaching the end of your sentence  Value is the only significant parameter that characterizes money. If it changes, it's not the same money.

7 minutes ago, vlaiv said:

I think that we have problem with world view above all else and as such, I'm not sure I can contribute anything further

Indeed. Thank you for your interest.

Edited July 27, 2023 by kurdewiusz

[andrew s]

1 hour ago, kurdewiusz said:

This document was my starting point. The title of my question refers to it. It's also given as one of the references at the bottom.

Sorry missed that.

I think you may have missunderstood the paper. Maybe you could point out which diagrams are incorrect and mathematically why. It's very difficult to get a handle on your concern via "words" when discussing mathematics. 

The integral is well defined and I belive your concept of " retaining the scale factor from the past" is mistaken or at least I don't know what it means.  For a given model, LCDM for example, the scale factor is well defined so the integration has no mysteries.  

In cosmology concepts such as distance are model dependent, and while I am sure LCDM is not the last word on the topic, there are several other predictions which are observationaly supported such as the time delay in super nove light curves which respect the distance scale it defines.

Regards Andrew 

 

[andrew s]

1 hour ago, kurdewiusz said:

No sir, space within the galaxy is not expanding, unlike the intergalactic space.

I would put it the gravitationally bound system don't expand as the force is order of magnitude greater that the tension due to metric expansion. Even more so for electromagnetically bound systems.

However,  metric expansion is still is going locally even if we can't see it.

Regards Andrew 

[andrew s]

Looking at the last few posts between you and @vlaiv it might be worth noting that in GR spacetime is about geometry and how stress energy shapes it. 

The standard model has the initial state (or at least as far as we can reliably go back) as hot and dense and it has been evolving from there. Spacetime evolves dynamically as the the stress energy has continued to dilute from it starting point as space metrical expanded with time. 

The geometry does not stop and start it just is.

Regards Andrew 

[kurdewiusz]

1 minute ago, andrew s said:

However,  metric expansion is still is going locally even if we can't see it.

I agree.

10 minutes ago, andrew s said:

Maybe you could point out which diagrams are incorrect and mathematically why.

If the observable universe radius calculation is wrong, then the formula for recession velocity is wrong and thus the Figure 2. diagram is wrong, showing the velocity as a function of redshift. Recession velocity formula contains the same integral, that gives the distance travelled by photons through the expanding space:



The scale factor is interchangeable with the redshift.

20 minutes ago, andrew s said:

It's very difficult to get a handle on your concern via "words" when discussing mathematics. 

True. Words + my diagram, showing numerical integration and the images, showing spacetime with photons.

22 minutes ago, andrew s said:

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. 

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.