The Planck Lambda (Baez question)

1. Oct 28, 2004

marcus

Over in SPR John Baez asked what is the value of the cosmological constant (Lambda) expressed in Planck units.

I posted a reply earlier today, but it contained a mistake

---what I posted on SPR around 10AM today---
I think the current estimate of Lambda in Planck units is
4.0 E-123

The estimated dark energy density would be one third of that, expressed in Planck units, namely
1.3 E-123

....
...
----end quote---

Turns out I was right about the dark energy density----rho_Lambda---which is 1.3 E-123.
But I was wrong about the relation of Lambda itself to this.
I gather from Ted Bunn post that Lamda is 8pi times rho_Lambda. Making it (according to him) about 3.2 E-122. this is all in Planck terms.

I will correct my calculation accordingly to agree with what Ted Bunn says:
First let's calculate the dark energy density from the value of the Hubble parameter 71 km/s per Megaparsec.
This is an inverse time, and in MKS units it would be 2.3 E-18 per second,
or 2.30 E-18, if we postpone rounding off till later.
This MKS value is equivalent to 1.24 E-61 reciprocal Planck time units.

Putting G=c=1,
the critical density, rho_crit, is (3/8pi)H^2
The current estimate of the dark energy density is 73 percent of that.
So in Planck terms the dark energy density is
0.73 (3/8pi)H^2 = 0.73 (3/8pi)(1.24 E-61)^2 = 1.3 E-123

But according to the conventions about Lamda you still have to multiply by 8pi.

Lambda = 8pi x 0.73 (3/8pi)H^2 = 0.73 (3)(1.24 E-61)^2 = 3.4 E-122

Hope this is right. Anyone have suggestions, corrections?

This was Baez original post

Last edited: Oct 28, 2004
2. Nov 6, 2004

john baez

cosmological constant in Planck units

I didn't get the benefit of Ted's wisdom in time for my talk at the
Perimeter Institute, so like you I said it was about 10^{-123}.

Like you, I forgot the 8 pi. :yuck:

Good, I do that pretty often myself...

Luckily the exact value didn't matter at all for my talk. The point is just that Artem Starodubtsev and Laurent Freidel have a new spin foam model of quantum gravity where it's described using a manifestly diffeomorphism-invariant perturbation theory whose coupling constant is the cosmological constant in Planck units! Physicists always like their coupling constants small, but this time it's REALLY small!

http://math.ucr.edu/home/baez/dynamics/

3. Nov 6, 2004

Staff Emeritus
Wow! Thank you for the study program, Dr. Baez. May I ask a couple of questions.

You say "with a time slice, preferred or not, we can do Wick rotation". I understood from some discussion on s.p.r that Wick rotation in GR contexts is not understood (or maybe that is a euphemism for not well defined?). Could you comment?

And we amateurs here at PF have been looking into the Gambini-Pullen consistent discretization/relative time approach. Do you see any role for this in quantum gravity?

Thanks again

4. Nov 6, 2004

marcus

Thanks, masked man...now where did he go to?

Hey selfAdjoint, we lucked out again.
One of Doctor John's rare visits to the Fans-of-Baez club.

It will be great to have a report from the recent
Loop-Foam Americas conference at Perimeter.
I've been looking forward, and checking Baez site daily to see
if it's posted.

Over the past couple of years I've seen a bunch of Freidel papers---back to when he was collaborating with Etera Livine. I really respect the guy
(and Etera too).
If he has a perturbation series it is certainly worth listening up.

does that mean there is a foam "Vacuum State"?
around which one can make little foamy changes.
answer my dumb questions, it's the rule here!

Lambda must be one of the universe's Beautiful Numbers

we should all remember

$$\Lambda = 3.4 \times 10^{-122}$$

just like we remember

$$\alpha = \frac{1}{137}$$

and

$$\pi = 3.14$$

5. Nov 6, 2004

Staff Emeritus
WHere'd you get them last three equal signs, buddy mine? Don't pull a book of Kings on me now.

Yes I do think Dr. B is hoping for a power expansion in the coupling coefficient around some spin-foam vacuum state(to be defined). This is good strategy. The LQG people used to brag about being non-perturbative, and they charged off into that thorn bush and got stuck, just like everybody else. Non-perturbative isn't as cool, but it gets an awful lot of valid results.

6. Nov 7, 2004

marcus

:rofl:

Don't worry, I just didnt know how to spell "\approx"
Now I will fix it:

... the universe's Beautiful Numbers

we should all remember

$$\Lambda \approx 3.4 \times 10^{-122}$$

just like we remember

$$\alpha \approx \frac{1}{137}$$

and

$$\pi \approx 3.14$$

there. I hope that makes it better. Lots of books of the Bible could have been improved (not just the Book of Kings) if they had used the "approximately equal" sign. Right near the Beginning was the Word and the Word was approximately equal to God.

And suddenly there was with the angel, the multitude of the fundamental physical constants, all precise to twelve decimal places and saying:

7. Nov 25, 2004

marcus

Well what do you know! The post about the planckwise value of the Cosmological Constant, which I submitted to SPR on 28 October, finally appeared on SPR dated 25 November!

Cheered and encouraged by this good fortune, I shall soon think of something else to post there.

In the meantime we at PF should be mindful of the Universe's Beautiful Numbers. why dont we make a habit of talking about them every now and then.

these numbers are what its all about
the longterm aim of physical theory is to explain them, why they are what they are

because they, in turn, explain so much else, like why our universe contains the chemical elements necessary to produce a chocolate eclair, or a croissant.

and "landscape" thinking, or anthropery, is the urge to give up on the basic quest and say that the numbers just are what they are by accident.
It represents Susskind et al. sulking because his favorite ideas led to a negative cosmo. const. and Nature likes Lambda to be small positive. So after enduring the frustration for a while people decided there would never be an explanation (nobody is smarter than us, so who would think of it?)

Anyway the tempest in the anthropic teapot and pretty much any basic controversy I can think of is all about the Beautiful Numbers and why are what they are.

So let's think of them from time to time. Run them through your fingers like you might do a halfdozen worn amber beads on a string. these 6 numbers, or 26 numbers, howevermany, are the mystery of the universe

and I must always remember to use this elegant "\approx" wavy equal sign, or selfAdjoint will mention the Book of Kings (according to the which Pi is exactly 3)

$$\alpha \approx \frac{1}{137}$$

$$\rho_\Lambda \approx 1.3 \times 10^{-123}$$

$$mass_{proton} \approx \frac {1}{13 quintillion} = \frac {1}{13 \times 10^{18}}$$

$$\frac{mass_{proton}}{mass_{electron}} \approx 1836$$

anybody have any other that occur to them to post?

when people say go "beyond" the particle Standard Model
they mean to get it down to less than 26 fundamental numbers
(SM is still a bit clunky because 26 inputs is a lot whereas 5 or 6 would be
be more elegant----the game is explain more with less)

Oh yeah. what John Baez was talking about. I gave the dark energy density just now---the rho-(for density)-sub-Lambda. But if you want Lambda itself, the Beautiful number in the einstein equation, then you have to multiply by 8 pi, and it is

$$\Lambda \approx 3.4 \times 10^{-122}$$

8. Jan 20, 2005

marcus

I noticed something related to this today. If you express rho_Lambda the dark energy density in conventional Planck terms (using the Planck units you find at the NIST website, i.e. standard definitions)
then in fact you get 1.34 x 10-123

but if you express the same energy density in terms of the variant Planck units that Baez once proposed on SPR (which put 8piG = 1, instead of G = 1) then you get both Lambda itself, and rho_sub Lambda, to come out the same number and that number is essentially what Baez says he has heard people BANDY ABOUT, namely 10-120

if you want more precision it comes out 0.8460 x 10-120
but they way people talk they are mostly only saying orders magnitude and for all practical purposes that is 10-120

Another neat thing I noticed is that expressed in conventional Planck the numbers for rho and Lambda differ by a factor of 8pi. But in the improved Planck system they are the same number. Of course I could be mistaken.

the way you get from 1.34 to 0.846 is multiplying by the square of 8pi,which happens in unit conversion and that is essentially the square of 25, or 625, and that is where the 3 orders of magnitude come in.
10-123 is dead, long live 10-120

Last edited: Jan 20, 2005
9. Jan 20, 2005

marcus

I double checked and in the variant of Planck units I've been using (which has |8piG| = 1) the values of Lambda and rho-sub-Lambda really are the same. If you are interested you could look at this post and following.

By contrast in conventional Planck the two physical quantities differ in value by 8pi, a factor of about 25.

without making a fuss, a lot of Quantum Gravity papers do set 8piG = 1,
sometimes they call 8piG "the gravitational constant" and denote it by its own symbol, greek kappa. And to distinguish they call G the "Newtonian gravitational constant" and denote it GN.

So there are researchers out there who are using this variant (8piG = 1) planck system for their own convenience and linguistically things may be gradually shifting.

If I want to use this "Force" natural units system, variant of conventional planck, then I have to get used to Lambda having a different value in the units----it works out to 0.846 E-120. Speaking casually, the cosmological constant is E-120, and the dark energy density is E-120

$$\Lambda = \rho_{Lambda} = 0.846 \times 10^{-120} \approx 10^{-120}$$

Just to get very naive and direct with this quantity, suppose we picture taking a STEP and defining a 32 inch, or 81 centimeter, distance. There is a traditional idea of a "pace" that is right about this length (pacing off distance means counting steps often assumed to be 30 inches)

so we make a pace-size cubical BOX (very anthropo, so some people will dissaprove, which is fine)

and this box is E34 x E34 x E34 = E102 natural volume units.

And we ask HOW MUCH DARK ENERGY IS IN THIS BOX?

we obviously it has to be one quintillionth of a natural unit of energy

If there is dark energy, constant thru all space and time whose density is E-120, then that density can be written
E-18/E102

so in a E102 box there has to be E-18,
which is a quintillionth of a unit.

this unit, we calculated in another thread, was enough to heat John Baez hot tub after he has been away on a trip and let it get down to outdoors temperature.

if you want to measure this energy unit in PIZZA then it is 100 thousand food Calories. A thousand slices of pizza each of which is 100 Calories.
Or, getting more and more scientific, 100 million gram-calories or "lab" calories.

so we can say that the fabled "dark energy" if it exists constant and evenly distributed thru space and time would have E-10 of a calorie in the pace-size BOX.

except that i was using the rough estimate of E-120 instead of the more correct 0.85 E-120

so everything is really only 85 percent of what i said but I dont care about this and hope you do not either.