^{4}is an important piece of story and restricts the set the parameters of imbeddable cosmologies dramatically.

One can try to understand the situation microscopically in terms of the cosmic strings which gradually develop higher than 2-D M^{4} projection during cosmic evolution and become magnetic flux tubes carrying magnetic monopole fluxes explaining the presence of magnetic fields in cosmology.

At microscopic level magnetic flux tubes are the key structural elements. The phase transitions increasing Planck constant for the matter associated with flux tubes and thus also the lengths of magnetic flux tubes should be important as also the phase transitions increasing p-adic prime and reducing Planck constant originally emerged in the modelling of TGD inspired quantum biology are highly suggestive. First transitions would mean adiabatic expansion with no heat generation and latter transitions would liberate magnetic field energy since flux conservation forces field strength to be reduced and leads to liberation of magnetic energy producing ordinary matter and dark matter. Dark energy in turn is identifiable as magnetic energy.

The key question concerns the mechanism causing the isotropy and homogeny of the cosmology. There are two possible identifications.

- According to two decades old TGD proposal primordial cosmology before the emergence of space-time sheets could be regarded as string gas in M
^{4}_{+}× CP_{2}at Hagedorn temperature determined by CP_{2}radius: T_{H}∼ hbar/R(CP_{2}). This phase could be present also after the transition to radiation dominated cosmology and consist of strings, whose thickness is gradually increasing and which contain carry dark energy and dark matter. The horizon radius is infinite for this cosmology thus providing at least partial explanation for the homogeny and isotropy and visible matter would represent deviations from it.

- The accelerating expansion period towards the end of the critical period could smooth out inhomogenities and thus provide an additional mechanism leading to homogenous and isotropic Big Bang. This for given space-time sheet representing R-W cosmology: in many-sheeted cosmology one can imagine distribution of parameters for the cosmology. The rapid expansion period could however also develop large fluctuations! Indeed, the time a
_{F}<a_{1}(density would be infinite for a_{1}) for its end - and therefore local mass density - must have a distribution after the rapid expansion ends. This expansion would generate separate smoothed out radiation dominated space-time sheets with slightly different mass densities and cosmic temperatures. A splitting to smooth radiation dominated sub-cosmologies would take place.

The formulas used to make back of the envelope (see this) calculations in inflation theory discussed in a guest posting in Lubos's blog given some idea about TGD counterpart for the generation of gravitons. Inflationary period is replaced with essentially unique critical cosmology containing only its duration as a free parameter. The fluctuations in the duration of this parameter explain scalar temperature fluctuations assoiated with CMB.

**How the local polarization of CMB is generated?**

There is a nice discussion about the mechanism leading to the generation of CMB polarization (see this). The polarization is generated after the decoupling of CMB photons from thermal equilibrium and is due to the scattering of photons on free electrons during decoupling. This scattering is known as Thomson scattering. The page in question contains schematic illustrations for how the polarization is generated. The scattering from electrons polarizes the photons in direction orthogonal to the scattering plane. In thermal equilibrium the net polarization of scattered radiation vanishes. If however the scattered photons from two perpendicular directions have different intensities a net polarization develops.

Polarized photons could be produced only during a short period during recombination scattering from free electrons was still possible and photons could diffuse between regions with different temperature. Polarized photons were generated when electrons from hot and cold regions where scattering on same electrons. CMB polarization indeed varies over sky but not in long length scales since photons could not diffuse for long lengths.

So called quadrupole anisotropy of CMB temperature contains information about the polarization. There are three contributions: scalar, vector, and tensor.

- Scalar contributions is due to density fluctuations reflecting themselves as temperature fluctuations and does not distinguish between polarizations: this is what has been studied mostly hitherto. A natural TGD mechanism for their generation would be different time for the end of the critical period leading to splitting of critical cosmology to radiation dominated cosmologies with slightly different temperatures.

- There is also so called vorticity distribution due to the flow which has vorticity and would due to defects/string like objects present also in TGD. The simplified situation corresponds to are region in which one has two flows in opposite direction locally. Depending on whether the scattering photons are upstream or down stream they are blue-shifted or red-shifter so that the temperatures are slightly different in up-stream and down.The flows in opposite direction give rise to a situation in which photons with different temperatures scatter and produce polarization. The effects of vorticity are expected to disappear during the fast expansion period. Probably because the gradients of velocity giving rise to vorticity are smoothed out.

- The third contribution is tensor contribution and due to gravitons generating stretching and squeezing of space in two orthogonal directions defining polarization tensor. Stretching increases wavelengths and decreases temperature. Squeezing does the opposite. Therefore temperature differences distinguishing between the two directions are generated and the outcome is polarization of the CMB background much later. This corresponds to the so called E and B modes.

One can decompose polarization as vector field to two parts: the first one - the E-mode - is gradient and thus irrotational and second is curl and thus rotational and with vanishing divergence (incompressible liquid flow is a good concrete example).

**How the polarization anisotropies could be generated in TGD Universe?**

One can try to understand microscopically how the polarization anisotropies are generated in TGD framework using poor man's arguments.

- One can introduce a vision vision about fractal 3-D network of cosmic strings forming a kinds of grids with nodes in various scales. These grids would be associated with different levels of the hierarchy of space-time sheets associated with many-sheeted space-time. Coordinate grid is of course an idealization since three coordinate lines would meet in single node. A weaker form of grid would involve meeting of two coordinate lines at given node. There is data about our own galactic nucleus understood if it correspond to the node at which two magnetic flux tubes meet. Ordinary visible matter would be generated in nodes.

One might say that galaxies are due to traffic accidents in which dark matter arriving along two cosmic strings collides in the crossing of the roads. Flux tubes would be attracted together by gravitational attraction so from the crossing.

- Amusingly, the notion grid emerged also TGD inspired quantum biology as a proposal for how living system codes morphogenetic position information. Flux tubes carry dark matter and ordinary matter is associated with the nodes at which coordinate lines meet each other. This web can give rise to a generalization of topological quantum computation using 2-braids. Coordinate lines define strings which can be knotted in 3-dimensions and define braids making possible topological quantum computation using macroscopic quantum phases defined by the dark matter. The time evolutions of coordinate lines defines string world sheets and in 4-D space-time the string world sheets can be knotted and braided so that also higher level TQC becomes possible with string reconnection and going above or below the other define two bits in each node.

- The presence of grid could also explain the honeycomb like structure of Universe with the recent typical size of honeycomb about 10
^{8}ly.

- In this framework the illustrations for how the gravitational waves induce the polarization of CMB. The radiation beams entering from opposite directions can be assigned with two magnetic flux tubes meeting at the node and in slightly different temperatures due to the interaction with gravitons much earlier. The gravitons can be regarded as larger space-time sheets at which the two flux tubes had contacts so that space associated with the flux tubes was forced to stretch or squeeze. This in turn increased of reduced photon wavelength so that photon temperature at flux tubes was different and the difference were preserved during subsequent evolution.

**Back on the envelope calculations in TGD framework**

One can modify the back on the envelope calculations of John Preskill (see this) in Lubos's blog to see what could happen in TGD framework. Now one however starts from the critical cosmology fixed apart from its duration and looks what it gives rather than starting from Higgs potential for inflaton field. The obvious counterpart for inflaton scalar field would be magnetic field intensity having same dimension but one should avoid too concrete correspondences.

The key question is whether the critical period generates the rapid expansion smoothing out inhomogenities or whether it generates them. The original guess that it smooths them out turns out be wrong in closer examination.

- The basic equation in inflationary model is given by

(da/dt)

^{2}= V/m_{p}^{2}

If V is small this has as solution a(t)= a(0)exp(Ht) if H= V

^{1/2}/m_{p}is constant. De Sitter cosmology allows partial imbedding in TGD but the imbedding is naturally static and has interpretation as black-hole interior with constant mass density. One can find coordinates in which the solution looks like expanding cosmology without Big Bang but these coordinates are not natural from the view of imbedding space.

- In TGD the expression for da/dt for critical cosmology is

da/dt= [a

_{0}^{2}-a^{2}]^{1/2}/[a_{0}^{2}-R^{2}-a^{2}]^{1/2}.

a

_{0}is roughly the duration of cosmology and R is CP_{2}radius of order 10^{3.5}Planck lengths. The almost uniqueness follows from the condition that the imbedding is such that the induced metric at the 3-surfaces defined by intersections with hyperboloids of M^{4}_{+}is flat rather than hyperbolic. This cosmology differs from de-Sitter cosmology.

- For a→ 0 one has

da/dt ≈ a

_{0}^{2}/[a_{0}^{2}-R^{2}]≈ 1 .

so that one has da/dt ≈ 1 and a≈ t for small values of a in accordance with the replacement of Big Bang with a "silent whisper amplified to a Big Bang" (density of matter goes as 1/a

^{2}) Hubble constant goes like H∝ 1/a so that Hubble radius divergence. This does not guarantee that horizon radius becomes infinite. Rather, the horizon is finite and given in good accuracy by the duration a_{1}=[a_{0}^{2}R^{2}]^{1/2}of the period. One can however explain the isotropy and homogenity of the string gas in light-cone M^{4}_{+}carrying flux tubes carrying dark matter and energy in terms of the infinite horizon of M^{4}.

There is no exponential time evolution at this period since one has a≈ t in good approximation for a/a

_{0}<<1. The TGD counterpart of V would behave like 1/a^{2}, which conforms with the idea that V corresponds to energy density.

- As the limit a→ a
_{1}=[ta_{0}^{2}-R^{2}]^{1/2}is approached, the expansion rate approaches infinite and for a>a_{1}at the latest one expects radiation dominated cosmology: otherwise a region of Euclidian signature of the induced metric results. The expectation is that a transition to radiation dominated cosmology takes place before a=a_{1}at which also energy density would diverge. The question is whether this period means smoothing out of inhomogenities or generation of them or both.

- Although it is not clear whether rapidly accelerating expansion is needed to to smooth out homogenities, one can just find what conditions this would give on the parameters. For a
_{i}= kR at which phase transition began the condition that a was increased at least by factor e^{50}∼ 5× 10^{21}(50 e-folds) this would give a_{1}≈ a_{0}>e^{50}kR. For k∼ 1 this gives something like 10^{-18}seconds, which happens to correspond atomic length scale. Below it will be found that this period more naturally corresponds to the period during which large fluctuations in density distribution and metric are generated.

- The earlier estimate for the emergence of radiation dominated cosmology assumed that the transition to radiation dominated cosmology takes place at CP
_{2}temperature defining Hagedorn temperature at which temperature of the string gas cannot be raised anymore since all the energy goes to the generation of string excitations rather than to kinetic energy, gives a_{F}∼ 10^{-10}seconds, which is by factor 10^{8}larger. If this were true, the fast expansion period a_{F}would increase the scale factor to about 68 e-folds equivalent to 98 2-folds. p-Adic prime p≈ 2^{196}would correspond to p-adic length scale about L(196) ∼ .1 meters. The crucial assumption would be that the the time a_{F}at which the expansion ends is same everywhere. There is no reason to assume this and this would mean that the period in question generates inhomogenities and isotropies of mass distribution and temperature distribution.

Note that if the distribution of the time a

_{F}<a_{1}at which the critical period ends is responsible for the CMB fluctuations then the number of foldings characterizes the smoothness of given local radiation dominated cosmology and could be rather large.

- The rapid accelerating expansion occurs as g
_{aa}approaches zero. Indeed, for

a→ a

_{1}= [a_{0}^{2}-R^{2}]^{1/2}

a very rapid expansion occurs and da/dt approaches infinite value. Near to a

_{1}one can write a/a_{1}=1-δ and solve δ approximately as function of t as

δ =[3R

^{2}/4a_{1}^{2}]^{2/3}[t-t_{1}/a_{1}]^{2/3},

t_{1}= ∫_{0}^{a1}[1-a^{2}/a_{1}^{2}1/2/[1-a^{2}/a_{1}^{2}]^{1/2}.

Hubble constant behaves as

H= (da/dt)/a = (R

^{2}/2a_{1}^{3})δ^{-1/2}.

- What is interesting is that applying the naive dimensional estimate for the amplitude of gravitational fluctuations to be δ h
_{T}^{2}∼ H^{2}/m_{P}^{4}. This would mean that at the limit a→ a_{F}< a_{1}gravitational fluctuations become very strong and generate the strong graviton background. Same applies to fluctuations in mass density.

**Summary**

The possibility of very rapid expansion near a=a_{F}<a_{1} leading to radiation dominated cosmology should have some deep meaning. The following tries to catch this meaning.

- The explosive period could lead to a radiation dominated cosmologies from string dominated cosmology with Hagedorn temperature. It could involve h
_{eff}increasing phase transitions for string gas during the initial period and liberation of magnetic energy during the end period as massless particles: this would explain why the mass density of the space-time sheet increases dramatically. The critical cosmology could correspond to a phase transition from a phase with Hagedorn temperature identified as T_{H}∝ hbar/R_{H}to radiation dominated cosmology.

- The cooling of string gas would lead to the generation of hierarchy of Planck constants and liberation of the magnetic energy of strings as massless particles during the end of critical period topologically condensing to space-time sheets such as massless extremals. This process could correspond to the rapid increase of energy density towards the end of the critical period.

- Isotropy and homogenity appear both at the level of imbedding space and space-time sheets. The infinite horizon of M
^{4}_{+}would explain the isotropy and homogenity of string gas in H both before and after the emergence of space-time sheets at Hagedorn temperature around a∼ R(CP_{2}). In particular, the smoothness of the cosmology of dark matter and dark energy would find explanation. The rapid expansion would in turn smooth out inhomogenities of individual space-time sheets.

- The Hubble scale 1/H approaches to zero as a=a
_{F}<a_{1}is approached. The rapid expansion destroys anisotropies and inhomogenities of radiation dominated space-time sheet corresponding to particular value of a_{F}. The distribution for values of a_{F}in turn explains CMB scalar fluctuations since the energy density in final state is highly sensitive to the precise value of a_{F}. This distribution would be Gaussian in the first approximation. One can say that the fluctuation spectrum for inflaton field is replaced with that for a_{F}.

- Also the generation of gravitational radiation and its decoupling from matter could take place during the same end period. After this gravitational fields would be essentially classical and assignable to space-time sheets. Essentially formation of gravitationally bound states would be in question analogous to what happens photons decouple from matter much later. The reduction of the temperature of string gas below Hagedorn temperature could generate also the massless graviton phase decoupling from matter and inducing the temperature fluctuations and polarization during decoupling.

Gravitons and also other particles would topological condense at "massless extremals" (MEs,topological light rays) and particles - in particular photons - would interact with gravitons by generating wormhole contacts to gravitonic MEs. The interaction between MEs assignable to gravitational radiation and photons would have caused the fluctuations of CMB temperature.

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

## 7 comments:

Dear Matti,

What is physical meaning of coupling of the spinors to a half odd multiple of the Kahler potential?

(i am studying the appendix of CP2 geometry and standard model symmetries)

Dear Hamed,

I already answered once. For some mysterious reason the response did not appear at the blog. Not the first time. Maybe I am once again a victim of terror (should not say this aloud since it give justification for labelling me a paranoid). This is

really terrible.

In any case, I try again.

Mathematical reason for the coupling of quark and lepton H-chiralities to n=1 and n=3 multiples

of Kahler potential is that otherwise one does not obtain respectable spinor structure.

The physical reason is that one obtains in this manner correct em charges. Kahler coupling corresponds to coupling to weak hyper charge (U(1) part of photon field).

Dear Matti,

Thanks,

Why covariant derivatives are defined in this form by summing veilbein connections and kahler potential:

A=V+B/2*(n+ 1+ + n- 1- )

This post was so excellently written it almost brought liquid to an eye or two of mine. I vaguely remember something about Penrose and "fortuitous geometric alignment with past light cones" ... anyway. Fascinating that the free parameter in TGD cosmology is duration which is so closely related to the theory of point processes which I am fond of and understand well, even though generally considered very different.

The main reason for my comment though, is to ask the question, what role does viscosity play in TGD Cosmology?

To crow:

Interesting question. Viscosity is certainly there. I have not even tried to consider it.

The point is however that in TGD one can get very powerful results just from imbeddability. Usually one starts from complex kinetic equations for particle densities and deduces the time evolution of cosmology from them using numerical calculations.

Now the evolution of particle densities is necessary in the background determined by imbeddability condition and must be consistent with it.

This is one of the strengths of sub-manifold gravity.

One must remember that critical period is only

period in cosmology. As in ordinary cosmology one has radiation and matter dominated phases and also the phase of accelerating expansion which is also critical phase. By fractality criticality appears in many length scales.

The understanding of the analogs of inflationary cosmology and cosmology with accelerating expansion as same thing but in different time periods and scales is also characteristic for TGD.

As also the notion of string gas now understood as gas of cosmic strings or magnetic flux tubes. One can say that the cosmology follows more or less

from general principles. Kinetic equations are not needed for this.

To Hamed:

Connection is linear object. Linear combination of parts corresponding to different Cartesian factors of the vielbein group now extended to contain U(1) factor to obtain consistency. Vielbein connection represents SU(2)_L and Kahler potential represents U(1). Multiplication with A.dx represents infinitesimal gauge rotation in parallel translation by dx^mu and parallel translation along curve corresponds to ordered exponential integral exp(iA.dx).

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