It would be a miracle if we came to mutual understanding after all we wrote, but you gave me a tiny spark of hope.

You're asking me "is commoving coordinate independent of its geometry ? Or is that now a coordinate time ?"

I assume that geometry you mention is given by the metric, so coordinate time differential dt is independent of this metric and proper time differential dτ is not. For null geodesic used to calculate the proper distance equal to the radius of the observable universe we have:

dτ = 0

∫dr = ∫cdt/a(t)

dt is the differential of cosmological time t. Now I'm asking you: Is t the comoving, coordinate time of the comoving observer? I'm asking, because in integration over this time we use limits corresponding both to the universe age, as well as the proper time of the observer resting in the CMB reference frame since its emission, if we neglect the fact that it was emmited some time after the BB, because his proper time is equal to the universe age (13.8 billion years).

Ps. Not that the Christoffel symbols were needed and you said it yourself, but you couldn't help yourself... Now I get it - you're promoting your own thread.

Edited December 30, 2025Dec 30 by Jacek

Commoving time and conformal time including the formula for age of the Universe is all forms of coordinate time.

When you apply a scale factor to a metric in the case of the FLRW you are specifying you are a commoving observer and the radius for that scale factor is also a form of coordinate. So the change in radius is a commoving event..

The proper time being along the worldline or null geodesic between emitter and receiver. The age of the Universe ie for example 10^-43 sec after the BB occurs prior to the formation of the CMB. For that matter it would extend beyond the Cosmic neutrino background assuming we can ever eventually measure it.

The time periods prior to nucleosynthesis is needed as it provides a timeline for inflation and electroweak symmetry breaking both which occur prior to nucleosynthesis which forms the CMB.

Now consider the following argument as to why conformal time is preferred ? Take a redshift value you can establish a distance as well as time the signal is emitted but that depends on coordinates between observer and emitter and not some clock following the null geodesic.

As to an observer at CMB ? Well see above I already explained that the calculation for the age of the Universe is not the proper time age. The wiki link I posted yesterday specifically stated it's conformal time. The Peebles article further highlighted that detail.

Look back time is the formula used for age of universe and it accounts for expansion which entails commoving coordinates

Edited December 30, 2025Dec 30 by Mordred

3 minutes ago, Mordred said:

As to an observer at CMB ? Well see above I already explained that the calculation for the age of the Universe is not the proper time age. The wiki link I posted yesterday specifically stated it's conformal time. The Peebles article further highlighted that detail.

Don't you have a problem with calling 13.8 billion years a conformal time?

Not really, once you delve deep enough you start to learn how useful conformal time is with regards to measurements via luminosity distance, angular diameter distance or redshift.

Unbelievable. If the universe age equal to 13.8 billion years is already in the conformal time, then I can die happily. Thank you @Mordred

Edited December 30, 2025Dec 30 by Jacek

Yw

https://en.wikipedia.org/wiki/Particle_horizon

And now I think I really need to read about Marzke-Wheeler "light clock".

Marzke, R. F.; Wheeler, J. A. (1964). Chiu, H. Y. (ed.). Gravitation and relativity. Benjamin. pp. 40–64.

Edited December 30, 2025Dec 30 by Jacek

Glad to see your studying, a couple of points on conformal distance vs proper distance the latter being the actual physical distance while the conformal distance is rescaled, with the addition of the scale factor. Through this rescaling this simplifies redshift relations, angular diameter distance etc to a fundamental observer.

Just a side note the calc in my signature for example initially uses conformal distance then converts to proper distance. There is a link for the tutorial on how to use it including which formulas are used

This article may help understand how it simplifies some key distance relations.

Though careful one detail this uses a rescaled proper distance in essence the equivalent of a commoving distance as opposed to the actual physical (proper distance). It specifies that detail and discusses it

https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture05.pdf

Edited December 30, 2025Dec 30 by Mordred

@Mordred you shouldn't try to teach anyone anything anymore, because 13.8 billion years is not the age of the universe in the conformal time.

Edited December 30, 2025Dec 30 by Jacek

Are you sure about that ? Better supply a reference of where I am in error.

Lookback time

The lookback time tL to an object is the difference between the age to of the Universe now (at observation) and the age te of the Universe at the time the photons were emitted (according to the object). It is used to predict properties of high-redshift objects with evolutionary

models, such as passive stellar evolution for galaxies. Recall that E(z) is the time derivative

of the logarithm of the scale factor a(t); the scale factor is proportional to (1 + z), so the

product (1 +z) E(z) is proportional to the derivative of z with respect to the lookback time

https://ned.ipac.caltech.edu/level5/Hogg/paper.pdf

@Mordred yes, I am. Read the underlined text and text in the red frame.

https://www.scienceforums.net/topic/140204-physical-conformal-age-of-the-universe/page/5/#findComment-1305391

And find the same text in the wikipedia - there is a link.

47 gly is the conformal time of the universe today correct ? As that's what you have on you opening post.

That scale factor you used is the universe today. Did you apply the conformal time to age integral ? At BB n=0

Dt=a(n)dn

T(n)=int_0^ n a(\prime{n})d\prime{n}

Edited December 30, 2025Dec 30 by Mordred

@Mordred ...

I asked you whether 13.8 billion years is the age of the universe in the conformal time, and you confirmed. I asked you whether you don't have a problem with calling 13.8 billion years a conformal time, and you denied. And you insinuated yourself earlier multiple times, that it's the conformal time, so now your evasiveness reached its peak and you're practically a liar at this point.

More than that. At the very end you asked me if I'm sure that you're wrong. Yes @Mordred you were wrong, and now you're trying to act like you weren't.

That's very, very low, and I totally disrespect you for it.

Edited December 30, 2025Dec 30 by Jacek

@swansont @Phi for All Can you see now what I'm dealing with on your forum regarding the fact that @Mordred is Resident Expert, Super Nerd and Glorious Leader?

52 minutes ago, Jacek said:

@swansont @Phi for All Can you see now what I'm dealing with on your forum regarding the fact that @Mordred is Resident Expert, Super Nerd and Glorious Leader?

But it was you that got banned from the astronomy forum, was it not? Perhaps a bit of self-reflection?............just a thought...........

Do you not understand the difference of conformal time today as opposed to conformal time during expansion history ?

The equation you had during opening post only applied scale factor today not the scale factor of earlier times from BB forward hence the integral.

I lost count the number of times I posted lookback time with E(z) including the evolutionary history of matter and radiation as terms. The formula you used in the opening post ignored the expansion history and you had already that the expansion history must be taken into account.

Just calculating today's moment of the expansion radius is not including the expansion history.

In essence the formula you used in the opening post was the equivalent of start of signal from an emitter today sending a signal to observer today how long would the signal take to arrive with no further expansion. In other words you did apply any form of look back time. Which I posted references to numerous times.

Edited December 30, 2025Dec 30 by Mordred

@Mordred ...

Conformal time η=∫dη=∫dt/a(t) includes the whole expansion history, because the scale factor function of time a(t) IS the expansion history.

You're worse than a liar at this point. Much worse.

@Mordred EVERYTHING you wrote indicated that COSMIC or COSMOLOGICAL time IS THE CONFORMAL TIME.

Try to deny it, please, make my day.

---

2 hours ago, exchemist said:

But it was you that got banned from the astronomy forum, was it not? Perhaps a bit of self-reflection?............just a thought...........

You know exactly what I was banned for.

https://www.scienceforums.net/topic/140238-cancel-culture-in-major-astronomy-communities/

You shouldn't defend a liar, you know? Just a thought.

Edited December 30, 2025Dec 30 by Jacek

43 minutes ago, Jacek said:

@Mordred ...

Conformal time η=∫dη=∫dt/a(t) includes the whole expansion history, because the scale factor function of time a(t) IS the expansion history.

You're worse than a liar at this point. Much worse.

@Mordred EVERYTHING you wrote indicated that COSMIC or COSMOLOGICAL time IS the conformal time.

Try to deny it, please, make my day.

---

You know exactly what I was banned for.

https://www.scienceforums.net/topic/140238-cancel-culture-in-major-astronomy-communities/

You shouldn't defend a liar, you know? Just a thought.

Yes, for posting what they considered pseudoscience. At least here nobody has accused you of that.

1 minute ago, exchemist said:

Yes, for posting what they considered pseudoscience. At least here nobody has accused you of that.

OMG, you're all so kind to me. Thank You!

46 minutes ago, Jacek said:

You know exactly what I was banned for.

Was it for spending more time on insulting the people who are discussing science with you than you spent on the science itself? That's probably what you'll get banned for here.

If your arguments are strong, why do you need all the toxicity? Can't you explain yourself without derision and condescension? This is a pattern for you. You join science discussion forums, break their rules, get banned, and then sign up again from a different account and start the whole shitshow all over again.

Please focus on the science you're trying to persuade others to accept, lose the snark, and try to make your points without being a complete dick about it. It would go a LONG way towards helping everyone understand what you so desperately are trying to convey.

@Phi for All I wasn't banned on Reddit's r/Astronomy for evading a ban, arguing with them or insulting anyone. I was banned for telling them they created a cancel culture, and I gave the explanation. They couldn't handle it. I couldn't even argue with them afterwards, because they muted me right away.

You know exactly why I'm calling @Mordred a liar. Go ahead, ban me, you're great at it giving 40 pages of banned users. I don't care.

I'm insulting you now because you actually are a nasty, indecent bunch of people. And @Mordred is the worst of you.

You told me not to be a COMPLETE DICK about my post and my points. Go ahead, be a COMPLETE CUNT about me and ban me.

Edited December 30, 2025Dec 30 by Jacek

1 hour ago, Jacek said:

@Mordred ...

Conformal time η=∫dη=∫dt/a(t) includes the whole expansion history, because the scale factor function of time a(t) IS the expansion history.

You're worse than a liar at this point. Much worse.

@Mordred EVERYTHING you wrote indicated that COSMIC or COSMOLOGICAL time IS THE CONFORMAL TIME.

Try to deny it, please, make my day.

---

You know exactly what I was banned for.

https://www.scienceforums.net/topic/140238-cancel-culture-in-major-astronomy-communities/

You shouldn't defend a liar, you know? Just a thought.

No that formula does not today scale factor a=1 there is a difference between conformal time now and conformal time then. This will be my last post this thread be well.

Thank god for that @Mordred .

Thank you for admitting you are a sickpuppet account

That's a good one :) I really like it, seriously.

In a sick world you have be a psycho to be normal.

Edited December 30, 2025Dec 30 by Jacek