James Bond and the true value of the cosmological curvature constant

[marcus]

Einstein discovered that general covariance allows his GR equation to have just TWO gravitational/geometric constants: Newton G and a curvature constant he called Lambda. So the symmetry of the theory requires us to put both constants into the equation and investigate empirically whether or not Lambda is zero (in which case we could say there was just one constant after all )

With the passage of some years, the cosmological curvature constant was at last finally measured and turned out not to be zero. What is its value? If you want, that is, to write it as an actual constant of nature, not as a changing percentage on some sliding scale, but as an actual constant.

Well the curvature in question, as appearing in the Einstein equation, is a reciprocal area. One very convenient way to express length in GR context is in lightseconds, or if no confusion can arise, in seconds. And it's convenient as well, if we want to use a metric unit, to express area in
square seconds. So curvature, in that context, can be an inverse square second (s^-2) quantity.

Now when Planck mission recently measured this tiny curvature constant Lambda it just happened to turn out to be
1.007 x 10^-35 seconds^-2
This, it must be confessed, immediately calls to mind the irreverent notion that the Creator likes powers of ten, perhaps He has ten fingers and wears a wristwatch with a second hand.

That would explain the number 1, and the 10^-35, in this profoundly important physical constant. But what accounts for the .007? Some secret agency, no doubt.

So how does this relate to the history of the universe that you see in one of Jorrie's tables, like this?$${\scriptsize \begin{array}{|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize \begin{array}{|c|c|} \hline S&a&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 10.900&0.092&0.479&0.720333&31.375&2.878&4.391&2.179&3.996\\ \hline 8.584&0.116&0.686&1.030150&29.384&3.423&5.344&2.041&3.323\\ \hline 6.760&0.148&0.982&1.471721&27.142&4.015&6.454&1.885&2.728\\ \hline 5.324&0.188&1.404&2.098399&24.621&4.625&7.722&1.710&2.204\\ \hline 4.192&0.239&2.005&2.980242&21.797&5.199&9.132&1.514&1.745\\ \hline 3.302&0.303&2.855&4.200217&18.648&5.648&10.642&1.295&1.345\\ \hline 2.600&0.385&4.044&5.833312&15.177&5.837&12.179&1.054&1.001\\ \hline 2.048&0.488&5.676&7.890822&11.427&5.581&13.633&0.794&0.707\\ \hline 1.612&0.620&7.837&10.231592&7.510&4.658&14.882&0.522&0.455\\ \hline 1.270&0.788&10.560&12.525003&3.620&2.851&15.835&0.251&0.228\\ \hline 1.000&1.000&13.787&14.399932&0.000&0.000&16.472&0.000&0.000\\ \hline 0.794&1.259&17.257&15.648602&3.109&3.914&16.842&0.216&0.250\\ \hline 0.631&1.585&20.956&16.410335&5.731&9.083&17.047&0.398&0.554\\ \hline 0.501&1.995&24.789&16.836447&7.890&15.743&17.153&0.548&0.935\\ \hline 0.398&2.512&28.694&17.063037&9.638&24.210&17.204&0.669&1.419\\ \hline 0.316&3.162&32.638&17.180008&11.040&34.912&17.224&0.767&2.032\\ \hline 0.251&3.981&36.601&17.239540&12.160&48.409&17.240&0.844&2.808\\ \hline 0.200&5.012&40.575&17.269607&13.051&65.411&17.270&0.906&3.788\\ \hline 0.158&6.310&44.553&17.284732&13.760&86.821&17.285&0.956&5.023\\ \hline 0.126&7.943&48.534&17.292324&14.324&113.777&17.292&0.995&6.580\\ \hline 0.100&10.000&52.516&17.296130&14.772&147.715&17.296&1.026&8.540\\ \hline \end{array}}$$

 

[marcus]

The answer to that question is that you can see in the table that both the Hubble radius R and the cosmic event horizon distance D_hor are converging to 17.3 Gly.

And that 17.3 billion lightyears is a disguised form of the cosmological curvature constant.

To reveal this, all you need to do is calculate (Λ/3)^-.5

For example, paste this into the google window:
(1.007e-35/3 s^(-2))^(-.5)
and you get
1.729622 × 10¹⁰ years
In light of the remark about using time-units for distance, this amounts to the 17.3 Gly which emerges as longterm limit of Hubble radius in the tabulated segment of universe history.

 

[fzero]

The limiting value of the Hubble parameter is $H_\Lambda = \sqrt{\Lambda/3}$. Since that isn't in the table, why wouldn't you just use the same method as in your DE density post. The math doesn't care what side of the EFE you put the CC/DE on.

 

[marcus]

Hi fzero!

In this thread's discussion I'd like (for aesthetic reasons primarily) to stick to the authentic Einstein conception (cosmological curvature constant) and avoid dubious or outright fictional concepts and quantities. They seem like unnecessary clutter--extra baggage.

You might enjoy the article by Bianchi and Rovelli "Why all these prejudices against a constant?". Have you read it? If not, just google "prejudices against a constant".

Thanks for the question :-)

 

[fzero]

Hi fzero!

In this thread's discussion I'd like (for aesthetic reasons primarily) to stick to the authentic Einstein conception (cosmological curvature constant) and avoid dubious or outright fictional concepts and quantities. They seem like unnecessary clutter--extra baggage.

You might enjoy the article by Bianchi and Rovelli "Why all these prejudices against a constant?". Have you read it? If not, just google "prejudices against a constant".

Thanks for the question :-)

I'm familiar with the concept and the last sentence of my post is actually irrelevant. For some reason I thought that what was being called $R$ in the table was the scale factor and not $cH^{-1}$ (though if I had bothered looking at the numerical value, I'd have realized my error.

 

[marcus]

I'm familiar with the concept and the last sentence of my post is actually irrelevant. For some reason I thought that what was being called $R$ in the table was the scale factor and not $cH^{-1}$ (though if I had bothered looking at the numerical value, I'd have realized my error.

Thanks for the comment. Yes it seems in about half the sources one encounters the scale factor is denoted "a" as Jorrie has it in his table, and in the other half it is denoted "R". I hope people who are used to the latter notation can get used to this style.

In the table-maker Jorrie provides us, there are two main model parameters that the user controls:
the present-day Hubble radius R₀ and the longterm eventual Hubble radius R_∞.

So the user puts in, say, 14.4 Gly and 17.3 Gly. The units are cleaner and more transparent, perhaps, than some of the alternatives. These correspond to distance growth rates of 1/144% per million years, and 1/173% per million years.
So the user specifies those two Hubble radii, or in effect those two percentage growth rates, and the calculator constructs the rest.