In this thread, I would like to discuss: "which is more natural for a simple cosmological model: taking the long term Hubble time as a timescale (as Marcus has done), or taking the Hubble time as a timescale". I will start by paraphrasing Marcus' OP and replacing his '##\Lambda##-timescale' with the '##H_0##-timescale'.(adsbygoogle = window.adsbygoogle || []).push({});

Here is the 'paraphrased' post:

If this seems to be a valid approach, we can jump to the next level of equations developed in the original thread. The size scale of our universe (any time after year 1 million) is accurately tracked by the function

##u(y) = \sinh^{\frac{2}{3}}(\frac{3}{2}y)##

where ##y## is the cosmic time in years, scaled to a fraction of the Hubble time, ##\frac{1}{H_0}##, which is presently 14.4 billion years.

That's it. That's the model. Just that one equation. What makes it work is scaling times (and corresponding distances) down by the cosmological constant. "Dark energy" (as Lambda is sometimes excitingly called) is here treated simply as a time scale. [see note 1]

So to take an example, suppose your figure for the present is year 13.79 billion. In the scaled units, present time is

##y_{now} = \frac{13.79\ billion\ years}{14.4\ billion\ years}=0.957##

When the model gives you times and distances in terms of similar small numbers, you multiply them by 14.4 billion years, or by 14.4 billion light years, to get the answers back into familiar terms. Times and distances are here measured on the same scale so that essentially c = 1.

[Notes]

1. If we use negligible matter in this model, i.e. essentially just ##\Lambda##, the long term Hubble time would have been around 14.4 billion years. The actual long term Hubble time depends on the ratio of spacetime curvature caused by matter to the spacetime curvature caused by ##\Lambda##.

2. The basic equation is from George Jones in this post, where he wrote it more specifically for the cosmological case:

[3] My concern as expressed in Marcus' thread is that beginners may be confused by the 'new' natural timescale. I understand that in the end it might be a case of personal preference, but maybe there are more opinions about it. George Jones said: ↑For a spatially flat universe that consists of matter and dark energy (w = -1), but no radiation, the scale factor is given exactly by

##a\left(t\right) = A \sinh^{\frac{2}{3}} \left(Bt\right),##

where

##A = \left( \frac{1 - \Omega_{\Lambda 0}}{\Omega_{\Lambda 0}} \right)^{\frac{1}{3}}##

and

##B = \frac{3}{2} H_0 \sqrt{\Omega_{\Lambda 0}} .##

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# Marcus's 'no-pressure' model in different jacket

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