- Einstein proposed non-vanishing value of Λ in Einstein action as a volume term at his time in order to get what could be regarded as a static Universe. It turned out that Universe expanded and Einstein concluded that this proposal was the greatest blunder of his life. For two decades ago it was observed that the expansion of the Universe acclerates and the cosmological constant emerged again. Λ must be extremely small and have correct sign in order to give accelerating rather decelerating expansion in Robertson-Walker cooordinate. Here one must however notice that the time slicing used by Einstein was different and fr this slicing the Universe looked static.

- Λ can be however understood in an alternative sense as characterizing the dynamics in the matter sector. Λ could characterize the vacuum energy density of some scalar field, call it quintessense, proportional to 3- volume in quintessence scenario. This Λ would have sign opposite to that in the first scenario since it would appear at opposite side of Einstein's equations.

**Remark**: This is an updated version of an earlier posting, which represented a slightly wrong view about the interpretation and evolution of cosmological constant in TGD framework. This was due to my laziness to check the details of the earlier version, which is quite near to the version represented here.

**Cosmological constant in string models and in TGD**

It has turned out that Λ could be the final nail to the coffin of superstring theory.

- The most natural prediction of M-theory and superstring models is Λ in Einsteinian sense but with wrong sign and huge value: for instance, in AdS/CFT correspondence this would be the case. There has been however a complex argument suggesting that one could have a cosmological constant with a correct sign and even small enough size.

This option however predicts landscape and a loss of predictivity, which has led to a total turn of the philosophical coat: the original joy about discovering the unique theory of everything has changed to that for the discovery that there are no laws of physics. Cynic would say that this is a lottery win for theoreticians since theory building reduces to mere artistic activity.

- Now however Cumrun Vafa - one of the leading superstring theorists - has proposed that the landscape actually does not exist at all (see this). Λ would have wrong sign in Einsteinian sense but the hope is that quintessence scenario might save the day. Λ should also decrease with time, which as such is not a catastrophe in quintessence scenario.

- Theorist D. Wrase et al has in turn published an article (see this) claiming that also the Vafa's quintessential scenario fails. It would not be consistent with Higgs Higgs mechanism. The conclusion suggesting itself is that according to the no-laws-of-physics vision something catastrophic has happened: string theory has made a prediction! Even worse, it is wrong.

**Remark**: In TGD framework Higgs is present as a particle but p-adic thermodynamics rather than Higgs mechanism describes at least fermion massivation. The couplings of Higgs to fermions are naturally proportional their masses and fermionic part of Higgs mechanism is seen only as a manner to reproduce the masses at QFT limit.

- This has led to a new kind of string war: now inside superstring hegemony and dividing it into two camps. Optimistic outsider dares to hope that this leads to a kind of auto-biopsy and the gloomy period of superstring hegemony in theoretical physics lasted now for 34 years would be finally over.

The picture in which Λ in Einsteinian sense parametrizes the total action as dimensionally reduced 6-D twistor lift of Kähler action could be indeed interpreted formally as sum of genuine cosmological term identified as volume action. This picture has additional bonus: it leads to the understanding of coupling constant evolution giving rise to discrete coupling constant evolution as sub-evolution in adelic physics. This picture is summarized below.

**The picture emerging from the twistor lift of TGD**

Consider first the picture emerging from the twistor lift of TGD.

- Twistor lift of TGD leads via the analog of dimensional reduction necessary for the induction of 8-D generalization of twistor structure in M
^{4}× CP_{2}to a 4-D action determining space-time surfaces as its preferred extremals. Space-time surface as a preferred extremal defines a unique section of the induced twistor bundle. The dimensionally reduced Kähler action is sum of two terms. Kähler action proportional to the inverse of Kähler coupling strength and volume term proportional to the cosmological constant Λ.

**Remark:**The sign of the volume action is negative as the analog of the magnetic part of Maxwell action and*opposite*to the sign of the area action in string models.

Kähler and volume actions should have opposite signs. At M

^{4}limit Kähler action is proportional to E^{2}-B^{2}In Minkowskian regions and to -E^{2}-B^{2}in Euclidian regions.

- Twistor lift forces the introduction of also M
^{4}Kähler form so that the twistor lift of Kähler action contains M^{4}contribution and gives in dimensional reduction rise to M^{4}contributions to 4-D Kähler action and volume term.

It is of crucial importance that the Cartesian decomposition H=M^{4}× CP_{2}allows the scale of M^{4}contribution to 6-D Kähler action to be different from CP_{2}contribution. The size of M^{4}contribution as compared to CP_{2}contribution must be very small from the smallness of CP breaking.

For canonically imbedded M

^{4}the action density vanishes. For string like objects the electric part of this action dominates and corresponding contribution to 4-D Kähler action of flux tube extremals is positive unlike the standard contribution so that an almost cancellation of the action is in principle possible.

- What about energy? One must consider both Minkowskian and Euclidian space-time regions and be very careful with the signs. Assume that Minkowskian and Eucidian regions have
*same time orientation*.

- Since a dimensionally reduced 6-D Kähler action is in question, the sign of energy density is positive Minkowskian space-time regions and of form (E
^{2}+B^{2})/2. Volume energy density proportional to Λ is positive.

- In Euclidian regions the sign of g
^{00}is negative and energy density is of form (E^{2}-B^{2})/2 and is negative when magnetic field dominates. For string like objects the M^{4}contribution to Kähler action however gives a contribution in which the electric part of Kähler action dominates so that M^{4}and CP_{2}contributions to energy have opposite signs.

- 4-D volume energy corresponds to the magnetic energy for twistor sphere S
^{2}and is therefore positive. For some time I thought that the sign must be negative. My blunder was that I erratically deduced the volume contribution to the energy from 4-D dimensionally reduced action, which is sum of Kähler action and volume term rather than deducing it for 6-D Kähler action and then dimensionally reducing the outcome. A good example about consequences of non-commutativity!

- Since a dimensionally reduced 6-D Kähler action is in question, the sign of energy density is positive Minkowskian space-time regions and of form (E

The identification of the observed value of cosmological constant is not straightforward and I have considered several options without making explicit their differences even to myself. For Einsteinian option cosmological constant could correspond to the coefficient Λ of the volume term in analogy with Einstein's action. For what I call quintessence option cosmological constant Λ

_{eff}would approximately parameterize the total action density or energy density.

- Cosmological constant - irrespective of whether it is identified as Λ or Λ
_{eff}- is extremely small in the recent cosmology. The natural looking assumption would be that as a coupling parameter Λ or Λ_{eff}depends on p-adic length scale like 1/L_{p}^{2}and therefore decreases in average sense as 1/a^{2}, where a is cosmic time identified as light-cone proper time assignable to either tip of CD. This suggests the following rough vision.

The increase of the thickness of magnetic flux tubes carrying monopole flux liberates energy and this energy can make possible increase of the volume so that one obtains cosmic expansion. The expansion of flux tubes stops as the string tension achieves minimum and the further increase of the volume would increase string tension. For the cosmological constant in cosmological scales the maximum radius of flux tube is about 1 mm, which is biological length scale. Further expansion becomes possible if a phase transition increasing the p-adic length scale and reducing the value of cosmological constant is reduced. This phase transition liberates volume energy and leads to an accelerated expansion. The space-time surface would expand by jerks in stepwise manner. This process would replace continuous cosmic expansion of GRT. One application is TGD variant of Expanding Earth model explaining Cambrian Explosion, which is really weird event.

One can however raise a serious objection: since the volume term is part of 6-D Kähler action, the length scale evolution of Λ should be dictated by that for 1/α

_{K}and be very slow: therefore cosmological constant identified as Einsteinian Λ seems to be excluded.

- It however turns that it possible to have a large number of imbedding of the twistor sphere into the product of twistor spheres of M
^{4}and CP_{2}defining dimensional reductions. This set is parameterized by rotations sphere. The S^{2}part of 6-D Kähler action determining Λ can be arbitrarily small. This mechanism is discussed in detail in here leads also to the understanding of coupling constant evolution. The cutoff scale in QFT description of coupling constant evolution is replaced with the length scale defined by cosmological constant.

**Second manner to increase 3-volume**

Besides the increase of 3-volume of M^{4} projection, there is also a second manner to increase volume energy: many-sheetedness. The phase transition reducing the value of Λ could in fact force many-sheetedness.

- In TGD the volume energy associated with Λ is analogous to the surface energy in superconductors of type I. The thin 3-surfaces in superconductors could have similar 3-surface analogs in TGD since their volume is proportional to surface area - note that TGD Universe can be said to be quantum critical.

This is not the only possibility. The sheets of many-sheeted space-time having overlapping M

^{4}projections provide second mechanism. The emergence of many-sheetedness could also be caused by the increase of n=h_{eff}/h_{0}as a number of sheets of Galois covering.

- Could the 3-volume increase during deterministic classical time evolution? If the minimal surface property assumed for the preferred extremals as a realization of quantum criticality is true everywhere, the conservation of volume energy prevents the increase of the volume. Minimal surface property is however assumed to fail at discrete set of points due to the transfer of conserved charged between Kähler and volume degrees of freedom. Could this make possible the increase of volume during classical time evolution so that volume and Kähler energy could increase?

- ZEO allows the increase of average 3-volume by quantum jumps. There is no reason why each "big state function reduction changing the roles of the light-like boundaries of CD could not decrease the average volume energy of space-time surface for the time evolutions in the superposition. This can occur in all scales, and could be achieved also by the increase of h
_{eff}/h_{0}=n.

- The geometry of CD suggests strongly an analogy with Big Bang followed by Big Crunch. The increase of the volume as increase of the volume of M
^{4}projection does not however seem to be consistent with Big Crunch. One must be very cautious here. The point is that the size of CD itself increases during the sequence of small state function reductions leaving the members of state pairs at passive boundary of CD unaffected. The size of 3-surface at the active boundary of CD therefore increases as also its 3-volume.

The increase of the volume during the Big Crunch period could be also due to the emergence of the many-sheetedness, in particular due to the increase of the value of n for space-time sheets for sub-CDs. In this case, this period could be seen as a transition to quantum criticality accompanied by an emergence of complexity.

**Is the cosmological constant really understood?**

The interpretation of the coefficient of the volume term as cosmological constant has been a longstanding interpretational issue and caused many moments of despair during years. The intuitive picture has been that cosmological constant obeys p-adic length scale scale evolution meaning that Λ would behave like 1/L_{p}^{2}= 1/p≈ 1/2^{k}.

This would solve the problems due to the huge value of Λ predicted in GRT approach: the smoothed out behavior of Λ would be Λ∝ 1/a^{2}, a light-cone proper time defining cosmic time, and the recent value of Λ - or rather, its value in length scale corresponding to the size scale of the observed Universe - would be extremely small. In the very early Universe - in very short length scales - Λ would be large.

A simple solution of the problem would be the p-adic length scale evolution of Λ as Λ ∝ 1/p, p≈ 2^{k}. The flux tubes would thicken until the string tension as energy density would reach stable minimum. After this a phase transition reducing the cosmological constant would allow further thickening of the flux tubes. Cosmological expansion would take place as this kind of phase transitions .

This would solve the basic problem of cosmology, which is understanding why cosmological constant manages to be so small at early times. Time evolution would be replaced with length scale evolution and cosmological constant would be indeed huge in very short scales but its recent value would be extremely small.

I have however not really understood how this evolution could be realized! Twistor lift seems to allow only a very slow (logarithmic) p-adic length scale evolution of Λ . Is there any cure to this problem?

- The magnetic energy decreases with the area S of flux tube as 1/S∝ 1/p≈ 1/2
^{k}, where p^{1/2}defines the transversal length scale of the flux tube. Volume energy (magnetic energy associated with the twistor sphere) is positive and increases like S. The sum of these has minimum for certain radius of flux tube determined by the value of Λ. Flux tubes with quantized flux would have thickness determined by the length scale defined by the density of dark energy: L∼ ρ_{vac}^{-1/4}, ρ_{dark}= Λ/8π G. ρ_{vac}∼ 10^{-47}GeV^{4}(see this) would give L∼ 1 mm, which would could be interpreted as a biological length scale (maybe even neuronal length scale).

- But can Λ be very small? In the simplest picture based on dimensionally reduced 6-D Kähler action this term is not small in comparison with the Kähler action! If the twistor spheres of M
^{4}and CP_{2}give the same contribution to the induced Kähler form at twistor sphere of X^{4}, this term has maximal possible value!

The original discussions in treated the volume term and Kähler term in the dimensionally reduced action as independent terms and Λ was chosen freely. This is however not the case since the coefficients of both terms are proportional to (1/α

_{K}^{2})S(S^{2}), where S(S^{2}) is the area of the twistor sphere of 6-D induced twistor bundle having space-time surface as base space. This are is same for the twistor spaces of M^{4}and CP_{2}if CP_{2}size defines the only fundamental length scale. I did not even recognize this mistake.

- The induction of the twistor structure by dimensional reduction involves the identification of the twistor spheres S
^{2}of the geometric twistor spaces T(M^{4})= M^{4}× S^{2}(M^{4}) and of T_{CP2}having S^{2}(CP_{2}) as fiber space. What this means that one can take the coordinates of say S^{2}(M^{4}) as coordinates and imbedding map maps S^{2}(M^{4}) to S^{2}(CP_{2}). The twistor spheres S^{2}(M^{4}) and S^{2}(CP_{2}) have in the minimal scenario same radius R(CP_{2}) (radius of the geodesic sphere of CP_{2}. The identification map is unique apart from SO(3) rotation R of either twistor sphere possibly combined with reflection P. Could one consider the possibility that R is not trivial and that the induced Kähler forms could almost cancel each other?

- The induced Kähler form is sum of the Kähler forms induced from S
^{2}(M^{4}) and S^{2}(CP_{2}) and since Kähler forms are same apart from a rotation in the common S^{2}coordinates, one has J_{ind}= J+RP(J), where R denotes a rotation and P denotes reflection. Without reflection one cannot get arbitrary small induced Kähler form as sum of the two contributions. For mere reflection one has J_{ind}=0.

**Remark:**It seems that one can do with reflection if the Kähler forms of the twistor spheres are of opposite sign in standard spherical coordinates. This would mean that they have have opposite orientation.

One can choose the rotation to act on (y,z)-plane as (y,z)→ (cy+ sz, -sz+ cy), where s and c denote the cosines of the rotation angle. A small value of cosmological constant is obtained for small value of s. Reflection P can be chosen to correspond to z→ -z. Using coordinates (u= cos(Θ),Φ) and their primed counterparts and by writing the reflection followed by rotation explicitly in coordinates (x,y,z) one finds v= -c u-s (1-u

^{2})^{1/2}sin(Φ), Ψ=arctan[(su/(1-u^{2})^{1/2})cos(Φ)+ ctan(Φ)]. In the lowest order in s one has v= - u-s (1-u^{2})^{1/2})sin(Φ), Ψ=Φ+ s cos(Φ)(u/(1-u^{2})^{1/2})).

- Kähler form J
^{tot}is sum of unrotated part J=du∧dΦ and J'=dv∧dΨ. J' equals to the Jacobian determinant ∂(v,Ψ)//∂(u,Φ). A suitable spectrum for s could reproduce the proposal Λ ∝ 2^{-k}for Λ. The S^{2}part of 6-D Kähler action equals to (J^{tot}_{θφ})^{2}/det(g))^{1/2}and is in the lowest order proportional to s^{2}. For small values of s the integral of Kähler action for S^{2}over S^{2}is proportional to s^{2}.

One can write the S

^{2}part of the dimensionally reduced action as S(S^{2})= s^{2}F^{2}(s). Very near to the poles the integrand has 1/[sin(Θ) +O(s)] singularity and this gives rise to a logarithmic dependence of F on s and one can write: F= F(s,log(s)). In the lowest order one has s≈ 2^{-k/2}, and in improved approximation one obtains a recursion formula s_{n}(S^{2},k)= 2^{-k/2}/F(s_{n-1},log(s_{n-1}) giving renormalization group evolution with k replaced by anomalous dimension k_{n,a}= k+2 log[F(s_{n-1},log(s_{n-1})] differing logarithmically from k.

- The sum J+RP(J) defining the induced Kähler form in S
^{2}(X^{4}) is covariantly constant since both terms are covariantly constant by the rotational covariance of J.

- The imbeddings of S
^{2}(X^{4}) as twistor sphere of space-time surface to both spheres are holomorphic since rotations are represented as holomorphic transformations. Also reflection as z→ 1/z is holomorphic. This in turn implies that the second fundamental form in complex coordinates is a tensor having only components of type (1,1) and (-1,-1) whereas metric and energy momentum tensor have only components of type (1,-1) and (-1,1). Therefore all contractions appearing in field equations vanish identically and S^{2}(X^{4}) is minimal surface and Kähler current in S^{2}(X^{4}) vanishes since it involves components of the trace of second fundamental form. Field equations are indeed satisfied.

- The solution of field equations becomes a family of space-time surfaces parameterized by the values of the cosmological constant Λ as function of S
^{2}coordinates satisfying Λ/8π G = ρ_{vac}=J∧(*J)(S^{2}). In long length scales the variation range of Λ would become arbitrary small.

- If the minimal surface equations solve separately field equations for the volume term and Kähler action everywhere apart from a discrete set of singular points, the cosmological constant affects the space-time dynamics only at these points. The physical interpretation of these points is as seats of fundamental fermions at partonic 2-surface at the ends of light-like 3-surfaces defining their orbits (induced metric changes signature at these 3-surfaces). Fermion orbits would be boundaries of fermionic string world sheets.

One would have family of solutions of field equations but particular value of Λ would make itself visible only at the level of elementary fermions by affecting the values of coupling constants. p-Adic coupling constant evolution would be induced by the p-adic coupling constant evolution for the relative rotations R combined with reflection for the two twistor spheres. Therefore twistor lift would not be mere manner to reproduce cosmological term but determine the dynamics at the level of coupling constant evolution.

- What is nice that also Λ=0 option is possible. This would correspond to the variant of TGD involving only Kähler action regarded as TGD before the emergence of twistor lift. Therefore the nice results about cosmology obtained at this limit would not be lost.

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

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