### Correlated Polygons in Standard Cosmology and in TGD

Peter Woit had an interesting This Week's Hype . The inspiration came from a popular article in Quanta Magazine telling about the proposal of Maldacena and Nima Arkani-Hamed that the temperature fluctuations of cosmic microwave background (CMB) could exhibit deviation from Gaussianity in the sense that there would be measurable maxima of n-point correlations in CMB spectrum as function of spherical angles. These effects would relate to the large scale structure of CMB. Lubos Motl wrote about the article in different and rather aggressive tone.

The article in Quanta Magazine does not go into technical details but the original article of Maldacena and Arkani-Hamed contains detailed calculations for various n-point functions of inflaton field and other fields in turn determining the correlation functions for CMB temperature. The article is technically very elegant but the assumptions behind the calculations are questionable. In TGD Universe they would be simply wrong and some habitants of TGD Universe could see the approach as a demonstration for how misleading the refined mathematics can be if the assumptions behind it are wrong.

It must be emphasized that already now it is known and stressed also in the articl that the deviations of the CMB from Gaussianity are below recent measurement resolution and the testing of the proposed non-Gaussianities requires new experimental technology such as 21 cm tomography mapping the redshift distribution of 21 cm hydrogen line to deduce information about fine details of CMB now n-point correlations.

Inflaton vacuum energy is in TGD framework replaced by Kähler magnetic energy and the model of Maldacena and Arkani-Hamed does not apply. The elegant work of Maldacena and Arkani-Hamed however inspired a TGD based consideration of the situation but with very different motivations. In TGD inflaton fields do not play any role since inflaton vacuum energy is replaced with the energy of magnetic flux tubes. The polygons also appear in totally different manner and are associated with symplectic invariants identified as Kähler fluxes, and might relate closely to quantum physical correlates of arithmetic cognition. These considerations lead to a proposal that integers (3,4,5) define what one might called *additive primes* for integers n≥ 3 allowing geometric representation as non-degenerate polygons - prime polygons. On should dig the enormous mathematical literature to find whether mathematicians have proposed this notion - probably so. Partitions would correspond to splicings of polygons to smaller polygons.

These splicings could be dynamical quantum processes behind arithmetic conscious processes involving addition. I have already earlier considered a possible counterpart for conscious prime factorization in the adelic framework. This will not be discussed in this section since this topic is definitely too far from primordial cosmology. The purpose of this article is only to give an example how a good work in theoretical physics - even when it need not be relevant for physics - can stimulate new ideas in completely different context.

For details see the chapter More About TGD Inspired Cosmology or the article Correlated Triangles and Polygons in Standard Cosmology and in TGD .

For a summary of earlier postings see Latest progress in TGD.
## 2 Comments:

Dark Energy and the Schwarzian Derivative

http://arxiv.org/abs/1403.5431

6 Conclusion

In this paper I have shown how the Schwarzian derivative enters the formula for the change of the density of dark energy under temporal re-parameterisations and how the Schwarzian tensor enters when considering conformal rescalings of the metric. I have illustrated this by considering a ΛCDM cosmology. It is striking that a similar behavour crops up in the change of the stress tensor of a two-dimensional CFT under conformal transformations. This seems to hint at a deeper connection between dark energy and CFT’s in 3+1 spacetime dimensions. In this connection it would be interesting to see whether or how the Schwarzian derivative enters the transformation formulae for the stress tensor.

Interesting finding. It seems that the idea is to represent the effect of time reparametrization of time coordinate as a modification of G and Lambda. Since the metric is scaled by a conformal factor in conformal transformation this brings in Schwarzian derivative in the transformation of cosmological term. The re-defined cosmological term emerges from the change of time coordinate and contaisn third time derivative although transformation formula for tensor involves only first derivatives of coordinate variables.

This is tricky game: one replaces G and Lambda with time dependent constants: I do not believe that this helps much. The appearence of Schwartzian derivative suggests that there might be however some deeper involved

The cosmological constant term would be analogous to energy momentum tensor in conformal field theories. One interpretation in TGD could be that string world sheets and magnetic flux tubes accompany each other and magnetic energy gives rise to analog of dark energy characterized by cosmological constant. Second interpreration could be that the appearance of small four-volume term in twistor lift of Kaehler action corresponds to this term. Volume term generalizes bosonic string action (area).

In TGD framework this relates to the extension of 2-D conformal symmetries to their 4-D analogs. One has supersymplectic symmetries with structure of conformal algebra and extened conformal symmetries at light-cone boundary and at light-like orbits of partonic 2-surfaces and also ordinary conformal symmetries at string world sheets.

Somehow all this should extend to a 4-D analog of conformal symmetry. Most naturally to quaternionic structure and quaternonic analyticity, which can be formulated in terms of Cauchy-Riemann-Fueter conditions (http://tgdtheory.fi/public_html/tgdquantum/tgdquantum.html#twistorstory ). Quaternionic analyticity would be a property of preferred extremals.

I wish I could make this intuition more explicit. The generalization of conformal field theory should be algebraic continuation from complex plane to quaternions.

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