M Theory Lesson 299
Recall that in the charged lepton and neutrino mass matrices, there is a peculiar phase given by the fraction
$\phi = \frac{1}{9 \pi}$,
in contrast to the cubed root of unity, which corresponds to the rational fraction $\omega = 1/3$. We have seen these numbers before. Observe that
$\frac{1}{3} \omega - 3 \phi = \frac{1}{9} \frac{\pi - 3}{\pi} = \frac{1}{9} \Omega_{B}$,
where $\Omega_{B} = 0.0450703$ is the baryonic matter fraction, first computed by Louise Riofrio in her varying speed of light cosmology. In other words, to put it more simply,
$27 \phi = \Omega_{DM}$,
the dark matter fraction. One suspects that the factor of $27 = 3^3$ is due to the dimensionality of the information space required to describe the dark matter quantum numbers.
$\phi = \frac{1}{9 \pi}$,
in contrast to the cubed root of unity, which corresponds to the rational fraction $\omega = 1/3$. We have seen these numbers before. Observe that
$\frac{1}{3} \omega - 3 \phi = \frac{1}{9} \frac{\pi - 3}{\pi} = \frac{1}{9} \Omega_{B}$,
where $\Omega_{B} = 0.0450703$ is the baryonic matter fraction, first computed by Louise Riofrio in her varying speed of light cosmology. In other words, to put it more simply,
$27 \phi = \Omega_{DM}$,
the dark matter fraction. One suspects that the factor of $27 = 3^3$ is due to the dimensionality of the information space required to describe the dark matter quantum numbers.
posted by Kea | 8:14 AM
15 Comments:
Do you know exactly what calculation Louise's omega_(Baryonic) = 0.0450703 comes from? I'm wondering why the 6 significant figures? Is it something like 1/(8*Pi) (not exactly that obviously, that is too small), i.e. just from fundamental constants, or does it include experimental data on the density of the universe? I can't find Louise's own derivation of that ratio anywhere.
(By fundamental constants, I mean dimensionless mathematical ratios like Pi, not dimensionful physical presumed "constants" which will have units.)
Nigel, it comes from pi - 3 divided by pi.
P.S. To be more precise, what this says is that the neutrino space is 27 times the dimension of the DM space, because the smaller angle (phi) determines the larger dimension. Eg. think of the three qutrit space.
3/Pi = 0.9549... This seems to differ slightly from 0.0450703, but thanks for helping, Marni!
(Pi - 3) / Pi
Thanks Anonymous, for bothering to include brackets, which switches the gesalt and allows my brain to stop seeing Marni's "-" as a dash and to see it instead as a minus sign. (Sigh. If only mathematicians generally would be logical and distinguish between reciprocals and inverses, superscripts in vector and indices, etc., etc., instead of relying on the poor number-dyslexic reader's intelligence to interpret the symbols.) Now I know that it is an exact mathematical constant that doesn't contain experimental data, do you happen to know where Louise's derivation of that fraction can be found? Or do you have the brainpower to work out the derivation from first principles? Thank you, Nige
Nigel, it is in Louise's main short paper, linked at the top of her blog ... not explicitly, but close enough.
I'm surprised that kneemo has not reminded us yet that 27 is the dimension of the octonionic Jordan algebra.
Thanks Kea,
If you mean http://www-conf.slac.stanford.edu/einstein/talks/aspauthor2004_3.pdf
then as I've commented before, it gives a different result.
Equation 4 states the Friedmann result for a critical density with zero cosmological constant; (v/R)^2 = (8/3)*Pi*G*Rho, where v = HR. Hence, critical density, Rho = (3/8)(H^2)/(Pi*G).
Louise's equation 1 is: GM = tc^3. Here, t = 1/H for flat spacetime, and M = Rho*(4/3)*Pi*R^3, giving density Rho = (3/4)(H^2)/(Pi*G). If this is baryonic matter and the critical density is the entire matter, then the ratio is 2 not (Pi-3)/Pi.
Louise in directly indicates one assumption in her equation 6, where she uses the volume of curved spacetime, 2*(Pi^2)*R^3, not the volume of the observed flat spacetime, (4/3)*Pi*R^3.
Women are used to being constantly corrected by guys ... even when they are right.
"constantly corrected by guys", Marni, you are so wrong about that!
(giggles)
I've given in to the concept that I'm going to spend 24 hours on airplanes and will meet you in Novemenber. Now instead of dreading the trip I'm looking forward to it. Heck, I always manage to do nice physics on planes (reduced oxygen levels maybe).
Carl, I'm guessing it won't be as bad as Oxford-Christchurch direct. Anyway, see you soon.
Hi Marni,
I'm not correcting anyone, just trying to find out what assumptions are used in the theory that 4.50703% of the universe is baryonic matter and the rest dark matter. I think this is important to quesstion. As everyone knows, science isn't about believing theories without going into the details, just to be friendly.
We know the earliest stage of the universe was extremely flat from the small size of the density fluctuations in the CBR, and we know the universe is now extremely flat because the universe isn't observably decelerating. I.e., gravitational curvature due to inward directed attraction is offset over cosmological distances by acceleration (dark energy which is easiest to explain as spin-1 gravitons, nothwithstanding the ignorant prejudice in favour of unpredictive, unobservable spin-2 gravitons).
Thanx for the linx! Whether in the UK or NZ, Kea is truly ahead of the curve.
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