Jacek

Sigh @Mordred indeed. There is a scale factor in the metric tensor in the Einstein field equations for the expanding universe. And the Ricci tensor and the Ricci scalar are non-zero because of it. Go ahead, look it up.

The FLRW metric is the description of the expanding spacetime governed by GR, and there is a scale factor in this metric. What's more, Friedmann equations are a solution to the Einstein field equations for this metric, and they also contain the different powers of the scale factor corresponding to the changing density parameters. Taking it all into account, how can you say that there is no scale factor in GR?

So you're talking to yourself by quoting yourself... Where did I say that the proper time IS the conformal time... Nowhere @Mordred As you see, I also can talk to myself and quote myself. What a delightful discussion, right? Deal with what I wrote.

I guess that makes me a rat.

And another rat terrier on this fourm.

My patience also has its limits @Mordred Not 380 million years, but 380 thousand years after the BB. Quite a significant difference and something to neglect in comparison to 13.8 billion years. Your tendency to throw in the bits of information that are actually irrelevant to the discussion, like 2.73K, is just repulsive, and comes from your need to show how smart you are. We, on Earth, observe the CMB dipole, because we have a peculiar velocity and we're not at rest with respect to the CMB. Your credentials closed your mind so tightly, that you probably can't think for yourself anymore. The FLRW metric doesn't work without the CMB reference frame. And where did I say that the proper time IS the conformal time... Nowhere @Mordred

If your observer hasn't been at rest in the CMB reference frame since its emission, then his proper time is not equal to the proper age of the universe, neglecting the fact that the emission happened some time after the BB. You talk too much.

Enough is enough. You also need your observer to rest in the CMB reference frame so that his proper time can be equal to the proper age of the universe, so you also need the CMB reference frame.

That's why my physical definition of the conformal time requires the point at which the CMB is isotropic, so you can assign its position to a comoving observer. It requires the CMB rest frame, and the fact that proper time is coordinate independent chages nothing in that matter. Your answers are one of the most evasive in the world, @Mordred and you're very, very tiring.

Why on earth are you talking to me about the Lorentz transformation, when I'm asking you whether the age of these comoving observers is equal to the universe age?

Are you saying that comoving observers separated by the distance greater than Hubble Radius and receding with v>c are not the same proper age at the same cosmic time in the spatially flat universe, assuming they've been existing since the emission of the CMB? Are you saying that universe age is not equal to the proper age of these comoving observers, neglecting the fact that the CMB was emitted some time after the BB?

Yes it is, in spatially flat universe. I restrict myself to it and leave you with all the versatility.

No, I didn't miss your edit. You made it after my comment. I live in spatially flat universe assuming large scale homogeneity. You don't?

But conformal time of our universe is 47 Gy and it directly corresponds to the proper distance 47 GLy. You just multiply this conformal time by c to get the proper distance.

What is incomprehensible for you in my previous comment, where I state, that I use all the contributions in the Friedmann equations? How many times are you going to repeat that I need all of them?

I use a(t) calculated from the Friedmann equations with all the contributions, and I use it in the conformal time calculation ∫dt/a(t)=-∫dz/H(z), so all the history of expansion with its changing rate is factored in.

Nothing I wrote contradicts your statements.

I don't want to use a scale factor of the expanding universe without matter. I'm using the scale factor from the Friedmann equations with all our density parameters. You're talking about varying expansion rates and I'm repeating that expansion rate is totally based on the scale factor and its time derivative. Answering your problematic question: Conformal time is better for the universe because universe is mostly spacetime filled with radiation in terms of its volume. In terms of energy contribution the radiation is negligible, but it expands along with the universe and it's filling it.

Oh gee, like I didn't write that the universe itself would be aging along with the decreasing energy density and temperature of its radiation. It seems that we can agree about it. Now the question is How would you calculate its age?

Do me a favor, forget about 47 and 13.8 billion years in case of a universe without matter, and answer my question. What would be aging and how?

Yes, I do. I also know a useful solution of the Friedmann equations for the radiation dominated universe. It would be other scale factor function of time, but it would still be explicit. So @Mordred can you finally answer my preliminary question?

Your question was answered in my preliminary question, because I was asking about the EXPANDING universe without matter. Expansion rate is (da/dt)/a, so it's based on the scale factor as a function of time, so the conformal time calculation ∫dt/a(t) definitely takes it into account. If you want, we can change it to ∫dz/H(z) and the result will be the same.

Universe from my preliminary question is filled with radiation. If you want, you can also have dark energy, but no dark matter, no baryonic matter and no other massive particles.

@Mordred as far as I can tell, conformal time η=∫dη=∫dt/a(t)=47Gy is in billion years, so it's not dimensionless. I totally agree that a tick of material clock remains constant, while a tick of radiation extends with its period due to the expansion. Before I answer your question why would it be a better representation for the age of a universe, I'm asking you to answer my preliminary question: What would be aging and how in the expanding universe without matter?

So the more exclamation marks, the more right you are and the more errors in my thinking? I don't argue when I'm wrong, like in case of recombination not happening all at once. That's what I call cancel culture: https://www.scienceforums.net/topic/140238-cancel-culture-in-major-astronomy-communities/ Again, I don't argue when I'm wrong, and your criticism based on my assumed misunderstanding of simple phase transition is one of the nastiest I've ever read. Congratulations. Please, find someone else to impress you, ok? Or impress yourself by this calculation. No wonder you have 40 pages of banned/suspended users. Think for a second about your own attitude and your own choice of words.