Minimal surface cosmology

Before the discovery of the twistor lift TGD inspired cosmology has been based on the assumption that vacuum extremals provide a good estimate for the solutions of Einstein's equations at GRT limit of TGD . One can find imbeddings of Robertson-Walker type metrics as vacuum extremals and the general finding is that the cosmological with super-critical and critical mass density have finite duration after which the mass density becomes infinite: this period of course ends before this. The interpretation would be in terms of the emergence of new space-time sheet at which matter represented by smaller space-time sheets suffers topological condensation. The only parameter characterizing critical cosmologies is their duration. Critical (over-critical) cosmologies having SO3× E³ (SO(4)) as isometry group is the duration and the CP₂ projection at homologically trivial geodesic sphere S²: the condition that the contribution from S² to g_rr component transforms hyperbolic 3-metric to that of E³ or S³ metric fixes these cosmologies almost completely. Sub-critical cosmologies have one-dimensional CP₂ projection.

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

D_α(g^αβ ∂_βh^kg^1/2)= ∂_α[g^αβ ∂_βh^k g^1/2] + {_α^k_m}
g^αβ ∂_βh^m g^1/2=0 ,

{_α^k_m} ={_l ^k_m} ∂_αh^l .

Sub-critical minimal surface cosmologies would correspond to X⁴⊂ M⁴× S¹. The natural coordinates are Robertson-Walker coordinates, which co-incide with light-cone coordinates (a=[(m⁰)²-r²_M]^1/2, r= r_M/a,θ, φ) for light-cone M⁴₊. They are related to spherical Minkowski coordinates (m⁰,r_M,θ,φ) by (m⁰=a(1+r²)^1/2, r_M= ar). β =r_M/m⁰=r/(1+r²)^1/2) corresponds to the velocity along the line from origin (0,0) to (m⁰,r_M).
r corresponds to the Lorentz factor r= γ β=β/(1-β²)^1/2) .

The metric of M⁴₊ is given by the diagonal form [g_aa=1, g_rr=a²/(1+r²), g_θθ= a²r², g_φφ= a²r²sin²(θ)]. One can use the coordinates of M⁴₊ also for X⁴.

The ansatz for the minimal surface reads is Φ= f(a). For f(a)=constant one obtains just the flat M⁴₊. In non-trivial case one has g_aa= 1-R² (df/da)². The g^aa component of the metric becomes now g^aa=1/(1-R²(df/da)²). Metric determinant is scaled by g_aa^1/2 =1 → (1-R²(df/da)^21/2. Otherwise the field equations are same as for M⁴₊. 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 M⁴₊ carrying dark energy density as a minimal surface solution! This obviously raises several questions.

1. 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 M⁴? 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.
   
   
2. 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 M⁴₊. 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.
   
   
3. 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 M⁴₊ for given sheet.
   
   
4. 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.
   
   

One should understand why Robertson-Walker cosmology is such a good approximation to this picture. Consider first cosmic redshift.

1. 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_aaa)^1/2 .
   

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

   For M⁴₊ 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?
   
   

2. 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 M⁴ coordinate m⁰. 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.
   
   

3. 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.
   
   
4. 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 M⁴₊ redshift.
   
   

Einstein's equations for RW cosmology should approximately code for the cosmic time dependence of mass density at given slightly deformed piece of M⁴₊ representing particular sub-cosmology expanding in jerkwise manner.

1. 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/a² 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.
   
   
2. 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 a³ 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/a².
   

   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 M⁴₊. 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/a² implies that the contribution of given string is reduced like 1/a on the average as in radiation dominated cosmology.
   
   

3. 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.
   
   
4. 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 CP₂ projection assumed to provide a GRT model for quantum phase transitions changing the value of Λ.
   
   

If this picture is correct, TGD inspired cosmology at the level of many-sheeted space-time would be extremely simple. The new element would be many-sheetedness which would lead to more complex description provided by GRT limit. This limit would however lose the information about many-sheetedness and lead to anomalies such as two Hubble constants.

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.

Articles and other material related to TGD.