RAUM: I was thinking.
ZEIT: Uhuh. “Question,” you said.
RAUM: Yeah, well, would you happen to know what the exact number for the Cosmological Constant is?
ZEIT [slightly puzzled]: No. No, I don’t. I don’t. But – as you youngsters tend to say: “Google is your friend,” right. Nothing needs to be remembered these days. So, no I don’t. Let’s see…
Here you go, UniverseToday.com says: 7.3 x 10^−27 kg/m3 (1). They (even) call it the “best estimate today” [smiling].
But why exactly do you want to know?
RAUM: Yeah, well, the “exact number” of the observed value for the Cosmological Constant seems to be increasing over time. Not that it’s actually increasing itself, in fact it’s expected to be very constant, right – throughout space and possibly even throughout time. No – what I mean is, the more exact we are able to measure it, the higher the value seems to become. (Observed) values are really pretty much all over the place, ranging from “at least” 10^-28 kg/m3 to 8 x 10^27 kg/m3. But they, as said, dó seem to increase as we’re able to observe and compute better ((2) to (5)).
ZEIT: And still you’re saying that the Constant is actually “constant”. “Throughout space, and probably time”?
RAUM: Yeah. Yes. And there’s more. Bear with me.
ZEIT: Bearing with you.
RAUM: Well, to tell you Why I want to know – I was thinking. About E = m c^2. You know, like all normal people do [laughing]. And the sort of strange importance in this equation the sheer value of the speed of light has. Squared even.
And very elegant as well. I know – the result of the combination and reduction of a number of far more complex equations. It’s more or less a discovery like – well, like any other, really. But still.
ZEIT: Uhuh. It seems arbitrary and fundamental at the same time. Thus magical..
RAUM [enthused]: Yes! That’s exactly it!
Well, then, dear friend, forward to the “there’s more part”. Imagine now a sphere of vacuum with a radius of one light-second: 2.997924580 x 10^8 meters. Remember: c is important. Why not sweat this asset, right?
ZEIT [smiling]: Imaginging.. Ehhrr, radius: 2.99 something x 10^8 meters. Yes. Go on.
RAUM [smiling]: Well, then this sphere of vacuum would have a volume of 4/3 x π x (2.997924580×10^8)^3, or 112,862,773,403,637,000,000,000,000 cubic meters (7)
ZEIT: I think I know where this is going..
RAUM [smiling, again]: Well, my dearest friend, where is this – where am I headed you think..
ZEIT: It’s obvious, isn’t it. You’re going to hand me the exact mass for this volume, which will be have a – not so – surprising value.
RAUM [clearing his throat]: Yeah, well that would be the easy way – You’re right of course – but there’s more. It all has even more elegance to it.
ZEIT: Hmmm. Go on.
RAUM: If this light-second sphere volume would result in a meaningful value for it’s weight, then c would once again be fundamental –
ZEIT: and arbitrary!
RAUM: Well, yes. Seemingly arbitrary, more leaning towards fundamental, I would prefer.
[RAUM takes a proper sip of his single malt – Coughs]
RAUM: The Cosmological Constant is the number for vacuum density, right.
RAUM: And the Hubble Constant says what the critical density is for the universe to expand the way it does, right?
ZEIT: Uhuh. Pretty accurate. Right.
RAUM: Well, the current most accurate number for the Hubble Constant – the critical density is 9.02 x 10^-27 kg/m3 (8) or 10 hydrogen atoms per cubic meter (9).
ZEIT: Hmmm. Pretty close to the value of the Cosmological Constant.. [meaningful smile]
RAUM: Exactly. In fact, there’s repeated mention of the Cosmological Constant – vacuum density – being ~70% of the Hubble Constant – critical density (10).
But. With the Cosmological Constant growing over time – due to measurement and computational power as I mentioned earlier – why wouldn’t they ultimately be proven to have the exact same value? After all, Einstein needed the Cosmological Constant to make the expanding universe a static one. Only logical he ultimately needed the exact same value to be able to completely compensate for the value that is describing the actual expansion, right?
ZEIT: quite some assumption, but I follow.
Where’s your elegant mass in all of this?
RAUM: I got to that right now. But my first conclusion would be that the Cosmological Constant and the Hubble Constant have the exact same value.
ZEIT: Yes, but what is its value? And I still don’t see where the light-second comes in.
RAUM: For a Cosmological Constant of 7.3 x 10^−27 kg/m3, the mass of a sphere with a radius of one light-second is 7.3 x 10^−27 x 112,862,773,403,637,000,000,000,000 = 0.8238982458 kilogram. In itself not spectacularly elegant, but quite close to 1 kilogram, wouldn’t you say so?
ZEIT: Sure. Yes, I would.
RAUM: And with the Cosmological Constant growing, it might very well turn out to be exactly one kilogram – if only to gain its deserved elegance..
ZEIT: Ehhrr, yes. But why assume –
RAUM: This is again both arbitrary and elegant, right. And it’s again the importance of c, right..?
ZEIT: Right. Right.
RAUM: And now the Hubble Constant arrives on the scene. 9.02 x 10^-27 kg/m3 gives us a mass for this volume of 1.0180222161! Quite elegant wouldn’t you say?
ZEIT [actually impressed]: I would indeed.
RAUM: But I’m going for the magic you introduced: For a mass of the mentioned volume of exactly 1 kilogram, I derive a Hubble Constant, and thus a Cosmological Constant of 8.86031744429714 x 10^027 kg/m3.
ZEIT [speechless]: – –
RAUM [very pleased smile]: Thank you. Allow me to summarise, my dear friend:
1. c is fundamental. It invites to be used as a starting point for further investigation;
2. Prediction: The Cosmological Constant and the Hubble Constant have the exact same value;
3. Due to the elegance of the internal consistency a kilogram can be used as a fundamental starting point;
4. Assumption / prediction: A sphere with a radius of one light-second has a weight of exactly one kilogram;
5. Prediction: The exact value of the Hubble Constant and the Cosmological Constant is 8.86031744429714 x 10^27 kg/m3.
ZEIT: My young friend. I need a drink.
[ZEIT pours himself a stiff one as het tops up his young friend’s glass]
(7) 4/3 x π x (2.997924580×10^8)^3 ->
1.3333333333 x 3.14159265359 x (2.997924580×10^8)^3 ->
4.1887902048 x (2.997924580×10^8)^3 ->
4.1887902048 x 26,944,002,417,374,000,000,000,000 ->
(10) Instead of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the critical density of the universe, the tipping point for a sufficient density to stop the universe from expanding forever. This ratio is usually denoted ΩΛ, and is estimated to be 0.6911 ± 0.0062, according to the recent Planck results released in 2015. In a flat universe ΩΛ is the fraction of the energy of the universe due to the cosmological constant, i.e., what we would intuitively call the fraction of the universe that is made up of dark energy. Note that this value changes over time: the critical density changes with cosmological time, but the energy density due to the cosmological constant remains unchanged throughout the history of the universe: the amount of dark energy increases as the universe grows, while the amount of matter does not. https://en.wikipedia.org/wiki/Cosmological_constant