^{3}(SO(4)) as isometry group is the duration and the CP

_{2}projection at homologically trivial geodesic sphere S

^{2}: the condition that the contribution from S

^{2}to g

_{rr}component transforms hyperbolic 3-metric to that of E

^{3}or S

^{3}metric fixes these cosmologies almost completely. Sub-critical cosmologies have one-dimensional CP

_{2}projection.

Do Robertson-Walker cosmologies have minimal surface representatives? Recall that minimal surface equations read as

D_{α}(g^{αβ} ∂_{β}h^{k}g^{1/2})= ∂_{α}[g^{αβ} ∂_{β}h^{k} g^{1/2}] + {_{α}^{k}_{m}}

g^{αβ} ∂_{β}h^{m} g^{1/2}=0 ,

{_{α}^{k}_{m}} ={_{l} ^{k}_{m}} ∂_{α}h^{l} .

Sub-critical minimal surface cosmologies would correspond to X^{4}⊂ M^{4}× S^{1}. The natural coordinates are Robertson-Walker coordinates, which co-incide with light-cone coordinates (a=[(m^{0})^{2}-r^{2}_{M}]^{1/2}, r= r_{M}/a,θ, φ) for light-cone M^{4}_{+}. They are related to spherical Minkowski coordinates (m^{0},r_{M},θ,φ) by (m^{0}=a(1+r^{2})^{1/2}, r_{M}= ar). β =r_{M}/m^{0}=r/(1+r^{2})^{1/2}) corresponds to the velocity along the line from origin (0,0) to (m^{0},r_{M}).

r corresponds to the Lorentz factor r= γ β=β/(1-β^{2})^{1/2}) .

The metric of M^{4}_{+} is given by the diagonal form [g_{aa}=1, g_{rr}=a^{2}/(1+r^{2}), g_{θθ}= a^{2}r^{2}, g_{φφ}= a^{2}r^{2}sin^{2}(θ)]. One can use the coordinates of M^{4}_{+} also for X^{4}.

The ansatz for the minimal surface reads is Φ= f(a). For f(a)=constant one obtains just the flat M^{4}_{+}. In non-trivial case one has g_{aa}= 1-R^{2} (df/da)^{2}. The g^{aa} component of the metric becomes now g^{aa}=1/(1-R^{2}(df/da)^{2}). Metric determinant is scaled by g_{aa}^{1/2} =1 → (1-R^{2}(df/da)^{2}^{1/2}. Otherwise the field equations are same as for M^{4}_{+}. Little calculation shows that they are not satisfied unless one as g_{aa}=1.

Also the minimal surface imbeddings of critical and over-critical cosmologies are impossible. The reason is that

the criticality alone fixes these cosmologies almost uniquely and this is too much for allowing minimal surface property.

Thus one can have only the trivial cosmology M^{4}_{+} carrying dark energy density as a minimal surface solution! This obviously raises several questions.

- Could Λ=0 case for which action reduces to Kähler action provide vacuum extremals provide single-sheeted model for Robertson-Walker cosmologies for the GRT limit of TGD for which many-sheeted space-time surface is replaced with a slightly curved region of M
^{4}? Could Λ=0 correspond to a genuine phase present in TGD as formal generalization of the view of mathematicians about reals as p=∞ p-adic number suggest. p-Adic length scale would be strictly infinite implying that Λ∝ 1/p vanishes.

- Second possibility is that TGD is quantum critical in strong sense. Not only 3-space but the entire space-time surface

is flat and thus M^{4}_{+}. Only the local gravitational fields created by topologically condensed space-time surfaces would make it curved but would not cause smooth expansion. The expansion would take as quantum phase transitions reducing the value of Λ ∝ 1/p as p-adic prime p increases. p-Adic length scale hypothesis suggests that the preferred primes are near but below powers of 2 p≈ 2^{k}for some integers k. This led for years ago to a model for Expanding Earth.

- This picture would explain why individual astrophysical objects have not been observed to expand smoothly (except possibly in these phase transitions) but participate cosmic expansion only in the sense that the distance to other objects increase. The smaller space-time sheets glued to a given space-time sheet preserving their size would emanate from the tip of M
^{4}_{+}for given sheet.

- RW cosmology should emerge in the idealization that the jerk-wise expansion by quantum phase transitions

and reducing the value of Λ (by scalings of 2 by p-adic length scale hypothesis) can be approximated by a smooth cosmological expansion.

- The cosmic recession velocity is defined from the redshift by Doppler formula.

z= (1+β)/(1-β)-1 ≈ β = v/c .

In TGD framework this should correspond to the velocity defined in terms of the coordinate r of the object.

Hubble law tells that the recession velocity is proportional to the proper distance D from the source. One has

v= HD , H= (da/dt)/a= 1/(g

_{aa}a)^{1/2}.

This brings in the dependence on the Robertson-Walker metric.

For M

^{4}_{+}one has a=t and one would have g_{aa}=1 and H=1/a. The experimental fact is however that the value of H is larger for non-empty RW cosmologies having g_{aa}<1. How to overcome this problem?

- To understand this one must first understand the interpretation of gravitational redshift. In TGD framework the gravitational redshift is property of observer rather than source. The point is that the tangent space of the 3-surface assignable to the observer is related by a Lorent boost to that associated with the source. This implies

that the four-momentum of radiation from the source is boosted by this same boost. Redshift would mean that the Lorentz boost reduces the momentum from the real one. Therefore redshift would be consistent with momentum conservation implied by Poincare symmetry.

g

_{aa}for which a corresponds to the value of cosmic time for the observer should characterize the boost of observer relative to the source. The natural guess is that the boost is characterized by the value of g_{tt}in sufficiently large rest system assignable to observer with t is taken to be M^{4}coordinate m^{0}. The value of g_{tt}fluctuates do to the presence of local gravitational fields. At the GRT limit g_{aa}would correspond to the average value of g_{tt}.

- There is evidence that H is not same in short and long scales. This could be understood if the radiation arrives along different space-time sheets in these two situations.

- If this picture is correct GRT description of cosmology is effective description taking into account the effect of local gravitation to the redshift, which without it would be just the M
^{4}_{+}redshift.

^{4}

_{+}representing particular sub-cosmology expanding in jerkwise manner.

- Many-sheeted space-time implies a hierarchy of cosmologies in different p-adic length scales and with cosmological constant Λ ∝ 1/p so that vacuum energy density is smaller in long scale cosmologies and behaves on the average as 1/a
^{2}where a characterizes the scale of the cosmology. In zero energy ontology given scale corresponds to causal diamond (CD) with size characterized by a defining the size scale for the distance between the tips of CD.

- For the comoving volume with constant value of coordinate radius r the radius of the volume increases as a. The vacuum energy would increase as a
^{3}for comoving volume. This is in sharp conflict with the fact that the mass decreases as 1/a for radiation dominated cosmology, is constant for matter dominated cosmology, and is proportional to a for string dominated cosmology.

The physical resolution of the problem is rather obvious. Space-time sheets representing topologically condensed matter have finite size. They do not expand except possibly in jerkwise manner but in this process Λ is reduced - in average manner like 1/a

^{2}.

If the sheets are smaller than the cosmological space-time sheet in the scale considered and do not lose energy by radiation they represent matter dominated cosmology emanating from the vertex of M

^{4}_{+}. The mass of the co-moving volume remains constant.

If they are radiation dominated and in thermal equilibrium they lose energy by radiation and the energy of volume behaves like 1/a.

Cosmic strings and magnetic flux tubes have size larger than that the space-time sheet representing the cosmology. The string as linear structure has energy proportional to a for fixed value of Λ as in string dominated cosmology. The reduction of Λ decreasing on the average like 1/a

^{2}implies that the contribution of given string is reduced like 1/a on the average as in radiation dominated cosmology.

- GRT limit would code for these behaviours of mass density and pressure identified as scalars in GRT cosmology in terms of Einstein's equations. The time dependence of g
_{aa}would code for the density of the topologically condensed matter and its pressure and for dark energy at given level of hierarchy. The vanishing of covariant divergence for energy momentum tensor would be a remnant of Poincare invariance and give Einstein's equations with cosmological term.

- Why GRT limit would involve only the RW cosmologies allowing imbedding as vacuum extremals of Kähler action? Can one demand continuity in the sense that TGD cosmology at p→ ∞ limit corresponds to GRT cosmology with cosmological solutions identifiable as vacuum extremals? If this is assumed the earlier results are obtained. In particular, one obtains the critical cosmology with 2-D CP
_{2}projection assumed to provide a GRT model for quantum phase transitions changing the value of Λ.

See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? of "Towards M-matrix" or article with the same title.

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

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