^{0})

^{2}-r

_{M}

^{2})

^{1/2}, and from the assumption that (quantum) criticality corresponds to a vanishing 3-curvature meaning that 3-space is Euclidian.

The condition that the induced metric of the a= constant section is Euclidian, fixes the critical cosmology apart from its duration a_{0} from the existence of its vacuum extremal imbedding to M^{4}× S^{2},

where S^{2} homologically trivial geodesic sphere:

ds^{2} = g_{aa}da^{2} -a^{2} (dr^{2} +r^{2}dΩ^{2}) ,

g_{aa}= (dt/da)^{2}=1- ε^{2} /(1-u^{2}) , u=a/a_{0} , ε=R/a_{0} .

sin(Θ)= +/- u , Φ= f(r) ,

1/(1+r^{2}) -ε^{2}(df/dr))^{2}=1 .

From the expression for dt/da one learns that for the small values of a it is essentially constant equal to dt/da=(1 ε^{2})^{1/2}. When a/a_{0} approaches to (1-ε^{2})^{1/2}, dt/da approaches to zero so that the rate of expansion becomes infinite. Therefore critical cosmology is analogous to inflationary cosmology with exponential expansion rate. Note that the solution is defined only inside future or past light-cone of M^{4} in accordance with zero energy ontology.

After this a transition to Euclidian signature of metric happens (also a transition to radiation dominated cosmology is possible): this is something completely new as compared to the general relativistic model. The expansion begins to slow down now since dt/da approaches infinity at a/a_{0}=1. In TGD framework the regions with Euclidian signature of the induced metric are good candidates for blackhole like objects. This kind of space-time sheets could however accompany all physical systems in all scales as analogs for the lines of generalized Feynman diagrams. For sin(Θ)=1 at a/a_{0}=1 the imbedding ceases to exist. One could consider gluing together of two copies of this cosmology together with sin(Θ)= sin(π-Θ)= a/a_{0} to get a closed space-time surface. The first guess is that the energy momentum tensor for the particles defined by wormhole contacts connecting the two space-time sheets satisfies Einstein's equations with cosmological constant.

Quantum criticality would be associated with the phase transitions leading to the increase of the length and thickness of magnetic flux tubes carrying Kähler magnetic monopole fluxes and explaining the presence of magnetic fields in all length scales. Kähler magnetic energy density would be reduced in this process, which is analogous to the reduction of vacuum expectation value of the inflation field transforming inflaton vacuum energy to ordinary and dark matter.

At the microscopic level one can consider two phase transitions. These phase transitions are related to the hierarchy of Planck constants and to the hierarchy of p-adic length scales corresponding to p-adic primes near powers of 2.

- The first phase transition increases Planck constant h
_{eff}=nh in a step-wise manner and increases the length and width of the magnetic flux tubes accordingly but conserves the total magnetic energy so that no magnetic energy is dissipated and one has adiabaticity. This sequence of phase transitions would be analogous to slow roll inflation in which the vacuum expectation of inflation field is preserved in good approximation so that vacuum energy is not liberated. The flux tubes contain dark matter.

- Second phase transition increases the p-adic length scale by a power of 2
^{1/2}and increases the length and width of magnetic flux tubes so that the value of the magnetic field is reduced by flux conservation (magnetic flux tubes carry monopole fluxes made possible by CP_{2}homology). This phase transition reduces zero point kinetic energy and in the case of magnetic fields magnetic energy transforming to ordinary and dark matter.

- The latter phase transition can be accompanied by a phase transition reducing Planck constant so that the length of the flux tubes is preserved. In this transition magnetic energy is liberated and dark matter is produced and possibly transformed to ordinary matter. This kind of phase transitions could take place after the inflationary adiabatic expansion and produce ordinary matter. As a matter fact, I

have originally proposed this kind of phase transition to be the basic phase transition involved with the metabolism in living matter (see

this), which suggests that the creation of ordinary matter from dark magnetic energy could be seen as kind of metabolism in cosmological scales.

In zero energy ontology one can ask whether one could assign to the Minkowskian and Euclidian periods a sequence of phase transitions increasing Planck constants but proceeding in opposite time directions.

- During the inflationary period the size scale of the Universe should increase by a factor of order 10
^{26}at least. This corresponds to 2^{87}- that is 87 2-foldings, which is a more natural notion than e-folding now. If the size of the sub-Universe is characterized by a p-adic length scale, this would correspond in the final state to p∼ 2^{174}at least: this p-adic length scale is about 4× 10^{-5}meters roughly and thus of order cell size. If the foldings correspond to increase of secondary p-adic length scale characterizing causal diamond, 89 foldings would correspond to Mersenne prime assignable to weak bosons.

- How the transition to radiation dominated cosmology takes place is an interesting question. The decay of the magnetic energy to ordinary matter should take place during the Euclidian period initiating therefore the radiation dominated period. For the radiation dominated cosmology the scale factor behaves as t∝ a
^{2}so that dt/da approaches zero. Since this occurs also when the Euclidian period starts, the guess is that space-time sheets with radiation dominated sub-cosmologies assignable to sub-CDs (CD is shorthand for causal diamond) begin to be created.

Although this picture is only an artist's vision and although one can imagine many alternatives, I have the feeling that the picture might contain the basic seeds of truth.

For details see the chapter TGD and Cosmology or the article BICEP2 might have detected gravitational waves.

## 2 comments:

Matti, what can be considered Minkowski content in TGD? In the sense of

https://en.wikipedia.org/wiki/Minkowski_content. also offtopic , are you familiar with the urantia book?

The Minkowski content of a set (named after Hermann Minkowski), or the boundary measure, is a basic concept in geometry and measure theory which generalizes to arbitrary measurable sets the notions of length of a smooth curve in the plane and area of a smooth surface in the space. It is typically applied to fractal boundaries of domains in the Euclidean space, but makes sense in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure.

Definition

Let be a metric measure space, where d is a metric on X and μ is a Borel measure. For a subset A of X and real ε > 0, let

be the ε-extension of A. The lower Minkowski content of A is given by

and the upper Minkowski content of A is

If M*(A) = M*(A), then the common value is called the Minkowski content of A associated with the measure μ, and is denoted by M(A).

Minkowski content in Rn

Let A be a subset of Rn. Then the m-dimensional Minkowski content of A is defined as follows. The lower content is

where αn−m is the volume of the unit (n−m)-ball and is -dimensional Lebesgue measure. The upper content is

As before, if the upper and lower m-dimensional Minkowski content of A agree, then the Minkowski content of A, Mm(A), is defined to be this common value.

To crow:

I already answered this message but as it sometimes

happens the reply disappeared mysteriously although

it was certainly at the page (I always do the check).

Minkowksi content is new notion to me, in no manner related to Minkowski metric. In TGD the existence of first derivatives of imbedding space coordinates is essential since otherwise induced field quantities would be ill-defined.

One can have many-valued partial derivatives as in case of branching of space-time sheet occurring in the vertex representing particle decay.

In any case, the geometric measures defined by the induced metric are enough.

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