Quite generally, cosmological constant defines itself a length scale R=1/Λ^{1/2}. r= (8π)^{1/4}(Rl_{P})^{1/2} - essentially the geometric mean of cosmological and Planck length - defines second much shorter length scale r. The density of dark energy assignable to flux tubes in TGD framework is given as ρ=1/r^{4}.

In TGD framework these scales corresponds two p-adic length scales coming as half octaves. This predicts a discrete spectrum for the length scale dependent cosmological constant Λ . For instance, one can assign to ..., galaxies, stars, planets, etc... a value of cosmological constant. This makes sense in many-sheeted space-time but not in standard cosmology.

Cosmic expansion is replaced with a sequence of fast jerks reducing the value of cosmological constant by some power of 2 so that the size of the system increases correspondingly. The jerk involves a phase transition reducing Λ by some negative power of 2 inducing an accelerating period during which flux tube thickness increases and magnetic energy transforms to ordinary matter. Thickening however increases volume energy so that the expansion eventually halts. Also the opposite process could occur and could correspond to a "big" state function reduction (BSFR) in which the arrow of time changes.

An interesting question is whether the formation of neutron stars and super-novas could involve BSFR so that these collapse phenomena would be kind of local Big Bangs but in opposite time direction. One can also ask whether blackhole evaporation could have as TGD analog BSFR meaning return to original time direction by a local Big Bang. TGD analogs of blackholes are discussed here.

Consider now some representative examples to see whether this picture can be connected to empirical reality.

- Cosmological constant in the length scale of recent cosmology corresponds to R∼ 10
^{26}m (see this). The corresponding shorter scale r= (8π)^{1/4}(Rl_{P})^{1/2}is identified essentially as the geometric mean of R and Planck length l_{P}and equals to r∼ 4× 10^{-4}m: the size scale of large neuron. This is very probably not an accident: this scale would correspond to the thickness of monopole flux tubes.

- If the large scale R is solar radius about 7× 10
^{8}m, the short scale r∼ 10^{12}m is about electron Compton length, which corresponds to p-adic length scale L(127) assignable to Mersenne prime M_{127}=2^{127}-1. This is also the size of dark proton explaining dark fusion deduced from Holmlid's findings (see this): this requires h_{eff}∼ 2^{12}!

**Remark**: Dark proton sequences could be neutralized by a sequence of ordinary electrons locally. This could give rise to analogs of atoms with electrons being very densely packed along the flux tube.

The prediction of the TGD based model explaining the 10 year old puzzle related to the fact that nuclear abundances in solar interior are larger than outside (see this) assumes that nuclear reactions in Sun occur through intermediate states which are dark nuclei. Hot fusion in the Sun would thus involve the same mechanism as "cold fusion". The view about cosmological constant and TGD view about nuclear fusion lead to the same prediction.

- If the short scale is p-adic length L(113) assignable to Gaussian Mersenne M
_{G,113}=(1+i)^{113}-1 defining nuclear size scale of r∼ 10^{-14}m, one has R∼ 10 km, the radius of a typical neutron star (see this) having a typical mass of 1.4 solar masses.

A possible interpretation is as a minimum length of a flux tube containing sequence of nucleons or nuclei and giving rise to a tangle. Neutron would take volume of about nuclear size - size of the magnetic body of neutron? Could supernova explosions be regarded as phase transitions scaling the stellar Λ by a power of 2 by making it larger and reducing dramatically the radius of the star?

- Short scale r ∼ 10
^{-15}m corresponding to proton Compton length gives R about 100 m. Could this scale correspond to quark star (see this)? The known candidates for quark stars are smaller than neutron stars but have considerably larger radius measured in few kilometers. Weak length scale would give large radius of about 1 cm. The thickness of flux tube would be electroweak length scale.

For a summary of earlier postings see Latest progress in TGD.

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