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

[kurdewiusz]

On 27/07/2023 at 15:10, andrew s said:

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.

All the parameters by which we describe and define spacetime have changed. This is geometry with different parameters, thus different geometry. The equations describing this geometry have not changed (probably).
In the same way you can say that mathematics has not ceased to exist - it just is.

[andrew s]

But the functional form is not wrong. You are missing an equation the relates the scale factor to the material in the Universe and an equation of state for the stress energy. For example taking the  Universe as an expanding isotropic gas you get the Friedman equation and the fluid equations which can be solved analytically. 

For a matter dominated state you get a(t)  = (t/t0)^2/3 , and an equation for density. 

For a radiation dominated state you get a(t) = (t/t0)^1/2 a d a different equation for density.

Similarly for the full LCDM model.

There are no issues integrating these equations and the standard results apply.

I can recommend "An introduction to Modern Cosmology" by Andrew Liddle that covers all this in detail.

Regards Andrew 

 

[andrew s]

On 27/07/2023 at 15:36, kurdewiusz said:

All the parameters by which we describe and define spacetime have changed. This is geometry with different parameters, thus different geometry. The equations describing this geometry have not changed (probably).
In the same way you can say that mathematics has not ceased to exist - it just is.

This is not a standard view. Regards Andrew 

[kurdewiusz]

1 minute ago, andrew s said:

You are missing an equation the relates the scale factor to the material in the Universe and an equation of state for the stress energy.

For me, metric tensor is the property of the material or fabric of spacetime, and the values of its terms changed along with the scale factor. Are we talking about Einstein equations?

[kurdewiusz]

9 minutes ago, andrew s said:

There are no issues integrating these equations and the standard results apply.

Except the one we're talking about right now, but you say, that it's not an issue, because there are no issues.

[andrew s]

6 minutes ago, kurdewiusz said:

For me, metric tensor is the property of the material or fabric of spacetime, and the values of its terms changed along with the scale factor. Are we talking about Einstein equations?

I was talking about a simple solution to them. The Friedman equations and an equation of state.

3 minutes ago, kurdewiusz said:

Except the one we're talking about right now, but you say, that it's not an issue, because there are no issues

So you feel integration of t^-x where X =2/3 or 1/2  with respect to t difficult?

As I say I think you would find Liddle's book answers your questions in detail.

I don't think I can help you further. 

Regards Andrew 

[kurdewiusz]

21 hours ago, andrew s said:

I was talking about a simple solution to them. The Friedman equations and an equation of state.

You wrote, that I was "missing an equation the relates the scale factor to the material in the Universe and an equation of state for the stress energy." That's why I asked about Einstein equations.

21 hours ago, andrew s said:

So you feel integration of t^-x where X =2/3 or 1/2  with respect to t difficult?

Now I totally feel that you are deliberately missing my point of integrating the expanding space - whether numerically or analytically - it makes no difference. Their results may be infinitesimally close. Numerical integration simply shows the problem.

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." and I gave you the answer, what it means, but you didn't comment.

I also think you can't help me anymore. Thank you for discussion.

Edited July 28, 2023 by kurdewiusz

[kurdewiusz]

@andrew s Sorry, you can still help me with the clarification of the relation between Friedmann equation and the equation for the distance calculation in the expanding space - the one with the problematic integration. I know, that this integration uses the explicit form of the scale factor, calculated from Friedmann equation (by another integration - numerical or not). In my understanding, this is the only element, that connects them. Choosing the integration-based method to calculate the distance in the expanding space has nothing more to do with Friedmann equation, except the scale factor, calculated from it. Am I right?

[vlaiv]

23 hours ago, kurdewiusz said:

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 think that cosmological principle plays a part here.

Scale factor is the same for every point with same proper time from the big bang. Proper times are also "synchronized" so to say as general curvature of space time is the same in every point due to cosmological principle and expansion of space does not imply that points are moving away from each other in local sense / special relativity sense.

In simple terms - time evolved in the same way in origin of the photon, our location and every point along the trajectory. Also - all of these points of interest do not move in relativistic sense - they are all on the same "now" slice with respect to us (or close enough so we can assert that scale factor is the same).

if we have non linear time - we can simply introduce a parameter that is linear and instead of having a(t(p)) - which is a * t(p) we can simply replace that with a'(p) in linear parameter.

There is no reason why integral in our proper time would not work for scale factor along the whole trajectory.

[kurdewiusz]

@vlaiv Sorry, but you write too many things that are clear. You quoted my condition for this integration to be correct and you gave your explanation, why this condition is not correct. The reason, why this integration is not correct, remains the same, no  matter how many times you'll repeat, that there is no reason for it to be incorrect.

I can only repeat my reasoning, the same as you: 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 and unlike the path of gravitationally redshifted photon - I pasted the picture with the description.

You claim, that it is. That's obviously false to me. At the same time, you also write, that the scale factor "is the same for every point with same proper time from the big bang" - that's obviously true. It can't be both. The scale factor can't be the same everywhere if the spacetime consists of sections expanding at different rates, depending on the time they've been traversed by the photon.

Edited July 28, 2023 by kurdewiusz

[vlaiv]

20 minutes ago, kurdewiusz said:

That's obviously false to me.

Can you explain why, for my benefit.

Try to explain it in simple terms so I can understand.

[kurdewiusz]

@vlaiv I'll try to draw it, because I would have to use the same words and that makes no sense.

[kurdewiusz]

On 27/07/2023 at 11:26, vlaiv said:

so it will be:

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

First of all, can we agree, that the scale factor increases with time, so its reciprocal decreases with time to a0 = 1 in our place and time?

[vlaiv]

1 hour ago, kurdewiusz said:

First of all, can we agree, that the scale factor increases with time, so its reciprocal decreases with time to a0 = 1 in our place and time?

Yes

[kurdewiusz]

@vlaiv I'm sorry, I thought I could come up with something new, but I keep drawing the same thing all over again.

Your summation, which is simplified integration, is in my opinion qualitatively suitable for calculating the distance in gravitationally curved spacetime, because early_scale, middle_scale and modern_scale are valid simultaneosly at different distances from  a star or a black hole, they just need renaming to close_scale, middle_scale, far_scale. They are unsuitable for the expanding, intergalactic space, because it's uniform at all times and only one value of the scale is valid for the current space, thus we can't use the other ones to calculate the distance in current space. I feel like I'm stuck with this explanation, sorry.

I used the word "simultaneously"  It may be problematic, if we start to talk about simultaneity of events. That's why my first choice of words was "at all times at all distances" - this in turn implies untrue assumption of constant mass, but I make such assumption for simplification and the sake of argument.

Edited July 28, 2023 by kurdewiusz

[kurdewiusz]

@vlaiv @andrew s 

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?

[andrew s]

Your equation (z+1) = 1/a only holds for the current now. In general the equation is

(z+1) = a(tr)/a(te) where a(tr) is the scale factor at the the time the light is observed and a(te) the scale factor at the time of emission.

In an  expanding Universe a(tr) > a(te) and so no problems arise. 

Figure 1 of the Expanding Confusions paper show a plot with a -> infinity. Its equation 23 is the one I quote but with R in place of a.

Hope this helps.

Regards Andrew 

Edited August 12, 2023 by andrew s

[kurdewiusz]

Thx @andrew s but i'm still confused. If (z+1)=a(tr)/a(te), does it mean, that a(tr)=1100 and a(te)= 1 or the other way around? If a(tr) is in the numerator and equal to 1100, why do we have (z+1)=a0/a, where a0=1?

Edited August 12, 2023 by kurdewiusz

[andrew s]

Ok. In the past t ~ 0 Gyr a ~ 0 (see attached image from the paper). By about 1 Gyr a~ 0.2. At 13.6 Gyr (now) a = 1 in the future at about 25 Gly a = 2 . It will not get to 1100 until 60 or more Gyr.

So you have it in reverse so its the other way round.

Regards Andrew 

PS It's confusing the a is sometimes labelled a0 for now. In all the above t=0 is at the start of the current expansion/  big bang and a0 refers to t ~ 13.6 Gyr

Edited August 12, 2023 by andrew s

[Bloom]

On 27/07/2023 at 17:50, andrew s said:

 

I can recommend "An introduction to Modern Cosmology" by Andrew Liddle that covers all this in detail.

 

 

It is the best cosmology book I ever read. When I finished it (I was not just reading it, like a novel, or a popular science book; I was closely following the equations - but they are not that many, and not extremely advanced), I felt illuminated.

[kurdewiusz]

@andrew s

2 hours ago, andrew s said:

in the future at about 25 Gly a = 2 

So in the future, when a = 2, will a0 = 2 or still 1? I have a problem with both cases, that remains the same, so we're back at the start point:

 

Edited August 12, 2023 by kurdewiusz

[andrew s]

1 minute ago, kurdewiusz said:

@andrew s

So in the future, when a = 2, will a0 = 2 or still 1?

 a0 = 1 always that is the scale is fixed so that the  scale factor is normalised to one now. At future epochs a will be greater than 1 as the universe expands and was less than 1 in the past but a0 I.e. a at t = 13.6Gyr will remain 1. That's why you need the full equation for (1+z) = a(tr)/a(te)  when doing sums about redshifts seen in the past or future not the one only valid for now.

The key is that "a" is a normalised scale. Cosmologists chose to normalise it to 1 at the current time to make their sum with data colleted now simpler.

Regards Andrew 

[kurdewiusz]

@andrew s Thank U!
So z+1 is equal to:
normalized a0/a for a <= 1,
unnormalized a(tr)/a(te) for a >= 1.
So for a = 1, we have a(tr)=a(te)=1, right? How is that possible?

Edited August 12, 2023 by kurdewiusz

[andrew s]

56 minutes ago, kurdewiusz said:

@andrew s Thank U!
So z+1 is equal to:
normalized a0/a for a < 1,
unnormalized a(tr)/a(te) for a > 1.
So for a = 1, we have a(tr)=a(te)=1, right? How is that possible?

No.

a(te) is the scale factor for when the light was emitted I.e. in our past. a(tr) is the scale factor at the time it is received. 

Consider a light ray emitted when t ~ 1Gly I.e. te = 1 Gly (about 40 Glyrs away) a(te) ~ 0.01. It is received today tr ~ 13.6 Gly so a(tr) = 1 so (1+z) = 1/0.01 = 100

Now consider a light ray emitted today te = 13.6 Gyr  about 10 Glys away it will reach us at tr ~ 25 Gyr and a(tr) ~ 2 so (1+z) = 2/1 = 2

I have taken the numbers very approximately from the diagram.

Regards Andrew 

Edited August 12, 2023 by andrew s

[kurdewiusz]

@andrew s The scale factor function is continuous on the interval 1-2 and on every other interval. I am asking you about the point (Now, 1) and its proximity, where normalized a0/1 becomes unnormalized a(tr)/a(te). Consider the point that lies on the curve and its coordinates are (Now+dt, 1+da), where  dt and da are infinitesimal and positive. Does this point has a physical meaning? Normalized and unnormalized values need to be equal for a=1. How would you explain the fact, that for a=1.00000001 the values of a(tr) and a(te) are practically equal, since we're already in unnormalized range for a > 1.