Cosmic evolution of the radius of the fiber of the twistor space of space-time surface

I have continued the little calculations inspired by the surprising finding that twistorial lift of Kähler action based dynamics immediately leads to the identification of cosmological length scales as fundamental classical length scales appearing in 6-D Kähler action, whose dimensional reduction gives Kähler action plus small cosmological term with correct sign to explain together with magnetic flux tube tension accelerating cosmic expansion. Whether Planck length emerges classically from from quantum theory remains still an open question.

For a fleeting moment I thought that for the twistor space of Minkowski space the 2-D fiber could be hyperbolic sphere H² (t²-x²-y² =-R_H²) rather than sphere S² as it is for CP₂ with Euclidian signature of metric. I however soon realized that the infinite area of H² implies that 6-D Kähler action is infinite and that there are many other difficulties.

The correct manner to define Minkowskian variant of twistor space is by starting from the generalization of complex and Kähler structures for M⁴= M²+ E² of local tangent space to longitudinal (defined by light-like vector) and to transversal directions (polarizations orthogonal to the light-like vector. The decomposition can depend on point but the distributions of two planes must integrated to 2-D surfaces. In E² one has complex structure and in M² its hyper-complex variant. In M² has decomposition of replacing complex numbers by hyper-complex numbers so that complex coordinate x+iy is replaced with w=t+ie, i²=-1 and e²=-1.

It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure Hamilton-Jacobi structure! The twistor fiber is defined by projections of 4-D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of light-like vector and its dual in M². This part of tensor could be called magnetic. Th magnetic part of the tensor defines a direction and one has natural metric making the space of directions sphere S² with metric having signature (-1,-1). This requires that twistor space has metric signature (-1,-1,1,-1,-1,-1) (I also considered seriously the signature (1,1,1,-1,-1,-1) so that there are three time-like coordinates) .

The radii of the spheres associated with M⁴ and CP₂ define two fundamental scales and the scaling of 6-D Käler action brings in third fundamental length scale. On possibility is that the radii of the two spheres are actually identical and essentially equal to CP₂ radius. Second option is that the radius of S²(M⁴) equals to Planck length, which would be therefore a fundamental length scale.

The radius R_D of the 2-D fiber of twistor space assignable to space-time surfaces is dynamical. In Euclidian space-time regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S²(X⁴)→ S²(M⁴) and S²(X⁴) → S²(CP₂). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something non-trivial about cosmic evolution of R_D as function of cosmic time. This seems to be the case.

Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.

1. The first term is essentially the ordinary Kähler action multiplied by the area of S²(X⁴), which is compensated by the length scale, which can be taken to be the area 4π R²(M⁴) of S²(M⁴). This makes sense for winding numbers (w₁,w₂)=(n,0) meaning that S²(CP₂) is effectively absent but S²(M⁴) is present.
   
   
2. Second term is the analog of Kähler action assignable assignable to the projection of S²(M⁴) Kähler form. The corresponding Kähler coupling strength α_K (M⁴) is huge - so huge that one has
   

   α_K (M⁴)4π R²(M⁴)== L² ,
   

   where 1/L² is of the order of cosmological constant and thus of the order of the size of the recent Universe. α_K(M⁴) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as α_K(M⁴) ∝ p≈ 2^k, p prime, k prime.
   
   

3. The Kähler form assignable to M⁴ is not assumed to contribute to the action since it does not contribute to spinor connection of M⁴. One can of course ask whether it could be present. For canonically imbedded M⁴ self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If α_K(M⁴) is given by above expression, then this contribution is extremely small.
   
   

Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M⁴) and T(CP₂) but with different values of α_K if one has (w₁,w₂)=(n,0). Also other w₂≠ 0 is possible but corresponds to gigantic cosmological constant.

Given the parameter L² as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive. One can actually get estimate for the evolution of R_D as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology.

1. Assume critical mass density so that one has
   

   ρ_cr= 3H²/8π G .
   

   
   

2. Assume that the contribution of cosmological constant term to the mass mass density dominates. This gives ρ≈ ρ_vac=Λ/8π G. From ρ_cr=ρ_vac one obtains
   

   Λ= 3H² .
   
   

3. From Friedman equations one has H²= ((da/dt)/a)², where a corresponds to light-cone proper time and t to cosmic time defined as proper time along geodesic lines of space-time surface approximated as Friedmanncosmology. One has
   

   Λ= 3/g_aaa²
   

   in Robertson-Walker cosmology with ds²= g_aada²-a² dσ₃².
   
   

4. Combining this equations with the TGD based equation
   

   Λ= 8π²G/L²R_D²
   

   one obtains
   

   8π²G/L²R_D²= 3/g_aaa².
   
   

5. Assume that quantum criticality applies so that L has spectrum given by p-adic length scale hypothesis so that one discrete p-adic length scale evolution for the values of L. There are two options to consider depending on whether p-adic length scales are assigned with light-cone proper time a or with cosmic time t
   

   T= a (Option I) ,

   T=t (Option II).

   Both options give the same general formula for the p-adic evolution of L(k) but with different interpretation of T(k).
   

   L(k)/Lnow= T(k)/Tnow ,

   T(k)= L(k) = 2(k-151)/2× L(151) ,

   L(151)≈ 10 nm .

   Here T(k) is assumed to correspond to primary p-adic length scale. An alternative - less plausible - option is that T(k) corresponds to secondary p-adic length scale L₂(k)=2^k/2L(k) so that T(k) would correspond to the size scale of causal diamond. In any case one has L ∝ L(k). One has a discretized version of smooth evolution
   

   L(a) = L_now × (T/T_now) .
   
   

Consider now the predictions.

1. Feeding into the formula following from two expressions for Λ one obtains an expression for R_D(a)
   

   R_D/l_P= (8/3)^1/2π× (a/L(a)× g_aa^1/2
   

   This equation tells that R_D is indeed dynamical, and becomes very small at very early times since g_aa becomes very small. As a matter of fact, in very early cosmic string dominated cosmology g_aa would be extremely small constant (see this). In late cosmology g_aa→ 1 holds true and one obtains at this limit
   

   R_D(now)= (8/3)^1/2π× (a_now/L_now) × l_P ≈ 4.4 ×(a_now/L_now) × l_P .
   
   

2. For T= t option R_D/l_P remains constant during both matter dominated cosmology, radiation dominated cosmology, and string dominated cosmology since one has a∝ tⁿ with n= 1/2 during radiation dominated era, n= 2/3 during matter dominated era, and n=1 during string dominated era (see this). This gives
   

   R_D/l_P=(8/3)^1/2π× at (g_aa^1/2(t(end)/L(end)) = (8/3)^1/2π×(1/n)(t(end)/L(end)) .
   

   Here "end"> refers the end of the string or radiation dominated period or to the recent time in the case of matter dominated era. The value of n would have evolved as R_D/l_P∝ (1/n (t_end/L_end), n∈ [1,3/2,2}. During radiation dominated cosmology R_D ∝ a^1/2 holds true. The value of R_D would be very nearly equal to R(M⁴) and R(M⁴) would be of the same order of magnitude as Planck length. In matter dominated cosmology would would have R_D ≈ 2.2 (t(now)/L(now)) × l_P .
   
   

3. For R_D(now)=l_P one would have
   

   L_now/a_now =(8/3)^1/2π≈ 4.4 .
   

   In matter dominated cosmology g_aa=1 gives t_now=(2/3)× a_now so that predictions differ only by this factor for options I and II. The winding number for the map S²(X⁴)→ S²(CP₂) must clearly vanish since otherwise the radius would be of order R.
   
   

4. For R_D(now)= R one would obtain
   

   a_now/L_now =(8/3)^1/2π× (R/l_P)≈ 2.1× 10⁴ .
   

   One has L_now=10⁶ ly: this is roughly the average distance scale between galaxies. The size of Milky Way is in the range 1-1.8 × 10⁵ ly and of an order of magnitude smaller.
   
   

5. An interesting possibility is that R_D(a) evolves from R_D ∼ R(M⁴) ∼ l_P to R_D ∼ R. This could happen if the winding number pair (w₁,w₂)=(1,0) transforms to (w₁,w₂)=(0,1) during transition to from radiation (string) dominance to matter (radiation) dominance. R_D/l_P radiation dominated cosmology would be related by a factor
   

   R_D(rad)/R_D(mat)= (3/4)(t(rad,end)/L(rad,end))×(L(now)/t(now))
   

   to that in matter dominated cosmology. Similar factor would relate the values of R_D/l_P in string dominated and radiation dominated cosmologies. The condition R_D(rad)/R_D(mat) =l_P/R expressing the transformation of winding numbers would give
   

   L(now)/L(rad,end) =(4/3) (l_P/R) (t(now)/t(rad,end)) .
   

   One has t(now)/t(rad,end)≈ .5× 10⁶ and l_P/R =2.5× 10^-4 giving L(now)/L(rad,end)≈ 125, which happens to be near fine structure constant.
   
   

6. For the twistorial lifts of space-time surfaces for which cosmological constant has a reasonable value , the winding numbers are equal to (w₁,w₂)=(n,0) so that R_D=n^1/2 R(S²(M⁴)) holds true in good approximation. This conforms with the observed constancy of R_D during various cosmological eras, and would suggest that the ratio t(end)/L(end) characterizing these periods is same for all periods. This determines the evolution for the values of α_K(M⁴).
   
   

R(M⁴)∼ l_P seems rather plausible option so that Planck length would be fundamental classical length scale emerging naturally in twistor approach. Cosmological constant would be coupling constant like parameter with a spectrum of critical values given by p-adic length scales.

For background see the article From Principles to Diagrams or the chapter From Principles to Diagrams of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.