LCDM cosmological coincidences

[mikehelland]

Here's 1 / Hubble's parameter, H(t), times 1 pc/yr and 1 ly/yr, the dotted lines. And the age and radius also there.

I thought it was interesting that the radius of the observable universe, 46 Gly, is 14 Gpc.

That kind of looks like very fundamental aspects of the universe, line up exactly with the pc / ly conversion rate, right as present time, as opposed to the next few billion or trillion years.

[HanDeBruijn]

Let $H=$ Hubble parameter, $G=$ gravitational constant, $\rho_0=$ mass density here and now, OI = from Origin of Inertia, the book by Amitabha Ghosh, Mach = another value from the same book, VMT = version of the Variable Mass Theory (and ΛCMD = oh well). Then we have

$$ H = \sqrt{\chi G \rho_0} \quad \mbox{with} \quad \chi = 8\pi \times \begin{cases} 1/8 & (: \mbox{OI}) \\ 1/6 & (: \mbox{Mach}) \\ 1/3 & (: \Lambda\mbox{CDM}) \\ 0.3883945571 & (: \mbox{VMT}) \end{cases} $$

But there are other theories about the Hubble parameter saying that

$$ H = \begin{cases} 2/\mbox{age} & (: \mbox{VMT}) \\ 1/\mbox{age} & (: \mbox{other}) \end{cases} $$

Let us assume that the first set of formulas is correct.
According to the Variable Mass Theory, mass is increasing with time. Consequently, in a Static Euclidean Universe, the mass density must be increasing with time as well. And so the Hubble parameter must be increasing with time.
However, according to the second set of formulas, the Hubble parameter is inversely proportional with time and so, because time is always increasing, the Hubble parameter must be decreasing with time.
Contradiction. What is it?

[mikehelland]

Maybe it's the case in ΛCDM, but that theory is rejected by many

Sure. The point of the thread was just to talk about some of the coincidences in that theory.

FWIW, I (personally) would not claim that the universe is infinite or eternal. One reason is those are bold claims and there's no empirical evidence of that. Another is that my model actually says nothing about the age of the universe. Here is what Edwin Hubble said:

The familiar interpretation of red-shifts as velocity-shifts very seriously restricts not only the time scale, the age of the universe, but the spatial dimensions as well. On the other hand, the alternative possible interpretation, that red-shifts are not velocity-shifts, avoids both difficulties, and presents the observable region as an insignificant sample of a universe that extends indefinitely in space and in time.

That's how I see it. The universe is indefinitely old and large. That's more scientific to my ears than eternal and infinite.

[HanDeBruijn]

Another is that my model actually says nothing about the age of the universe.

Perhaps not explicitly, but implicitly .. ?? Please answer this question: what is the name of the value $A=1/H$ of Minkowski time that is equivalent to $t\to -\infty$ as in

$$ 1+H\tau = e^{-\infty} = 0 \quad \Longleftrightarrow \quad \tau = -1/H = -A $$

That's how I see it. The universe is indefinitely old and large. That's more scientific [ .. ]

I could not agree more, Mike !! What's in a word is the summary of our misunderstanding, it seems. Here is what Gauss said:

[ Was nun aber Ihren Beweis für (1) betrifft, ]
so protestire ich zuvörderst gegen den Gebrauch einer unendlichen Größe
als einer Vollendeten, welcher in der Mathematik niemals erlaubt ist. Das
Unendliche ist nur eine façon de parler, indem man eigentlich von Grenzen
spricht, denen gewisse Verhältnisse so nahe kommen als man will, während
andern ohne Einschränkung zu wachsen verstattet ist.

: C.F. Gauss in a letter to Schumacher, 12 July 1831. Translation:

I protest against the use of infinite magnitude as something completed, which
is never permissible in mathematics. Infinity is merely a way of speaking, the
true meaning being a limit which certain ratios approach indefinitely close,
while others are permitted to increase without restriction.

[mikehelland]

I would call 1/H the Hubble limit, as that's it's name. At that point, time is dilated to infinity, and all information before that is lost.

[HanDeBruijn]

At that point, time is dilated to infinity

More precise: "measured" time $t$ is dilated to infinity.
But there is also time $\tau$ in the Minskowski frame which is not dilated to infinity.

and all information before that is lost.

Information? Before that? I am lost here.

[mikehelland]

Minkowski time isn't a real thing, it's what you'd have, hypothetically, without time dilation.

The idea is that the universe goes back in time indefinitely, it's just everything before 14 billion years ago is on the wrong side of the Hubble limit, meaning it is in accessible to us.