Does TGD allow description of accelerated expansion in terms of cosmological constant?The introduction of cosmological constant seems to be the only manner to explain accelerated expansion and related effects in the framework of General Relativity. As summarized in the previous posting, TGD allows different explanation of these effects. I will not however go to this here but represent comments about the notion of vacuum energy and the possibility to describe accelerated expansion in terms of cosmological constant in TGD framework. The term vacuum energy density is bad use of language since De Sitter space is a solution of field equations with cosmological constant at the limit of vanishing energy momentum tensor carries vacuum curvature rather than vacuum energy. Thus theories with nonvanishing cosmological constant represent a family of gravitational theories for which vacuum solution is not flat so that Einstein's basic identification matter = curvature is given up. No wonder, Einstein regarded the introduction of cosmological constant as the biggest blunder of his life. De Sitter space is representable as a hyperboloid a^{2}u^{2}= R^{2}, where one has a^{2}=t^{2}r^{2} and r^{2}=x^{2}+y^{2}+z^{2}. The symmetries of de Sitter space are maximal but Poincare group is replaced with Lorentz group of 5D Minkowski space and translations are not symmetries. The value of cosmological constant is Λ= 3/R^{2}. The presence of nonvanishing dimensional constant is from the point of view of conformal invariance a feature raising strong suspicions about the correctness of the underlying physics. 1. Imbedding of De Sitter space as a vacuum extremal De Sitter Space is possible as a vacuum extremal in TGD framework. There exists infinite number of imbeddings as a vacuum extremal into M^{4}×CP_{2}. Take any infinitely long curve X in CP_{2} not intersecting itself (one might argue that infinitely long curve is somewhat pathological) and introduce a coordinate u for it such that its induced metric is ds^{2}=du^{2}. De Sitter space allows the standard imbedding to M^{4}×X as a vacuum extremal. The imbedding can be written as u= ±[a^{2}+R^{2}]^{1/2} so that one has r^{2}< t^{2}+R^{2}. The curve in question must fill at least 2D submanifold of CP_{2} densely. An example is torus densely filled by the curve φ = αψ where α is irrational number. Note that even a slightest local deformation of this object induces an infinite number of selfintersections. Spacetime sheet fills densely 5D set in this case. One can ask whether this kind of objects might be analogs of D>4 branes in TGD framework. As a matter fact, CP_{2} projections of 1D vacuum extremals could give rise to both the analogs of branes and strings connecting them if spacetime surface contains both regular and "brany" pieces. It might be that the 2D Lagrangian manifolds representing CP_{2} projection of the most general vacuum extremal, can fill densely D> 3dimensional submanifold of CP_{2}. One can imagine construction of very complex Lagrange manifolds by gluing together pieces of 2D Lagrangian submanifolds by arbitrary 1D curves. One could also rotate 2Lagrangian manifold along a 2torus  just like one rotates point along 2torus in the above example  to get a dense filling of 4D volume of CP_{2}. The M^{4} projection of the imbedding corresponds to the region a^{2}>R^{2} containing future and past lightcones. If u varies only in range [0,u_{0}] only hyperboloids with a^{2} in the range [R^{2},R^{2}+u_{0}^{2}] are present in the foliation. In zero energy ontology the spacelike boundaries of this piece of De Sitter space, which must have u_{0}^{2}>R^{2}, would be carriers of positive and negative energy states. The boundary corresponding to u_{0}=0 is spacelike and analogous to the orbit of partonic 2surface. For u_{0}^{2}<R^{2} there are no spacelike boundaries and the interpretation as zero energy state is not possible. Note that the restriction u_{0}^{2}>R^{2} plus the choice of the branch of the imbedding corresponding to future or past directed lightcone is natural in TGD framework. 2. Could negative cosmological constant make sense in TGD framework? The questionable feature of slightly deformed De Sitter metric as a model for the accelerated expansion is that the value of R would be same order of magnitude as the recent age of the Universe. Why should just this value of cosmic time be so special? In TGD framework one could of course consider spacetime sheets having De Sitter cosmology characterized by a varying value of R. Also the replacement of one spatial coordinate with CP_{2} coordinate implies very strong breaking of translational invariance. Hence the explanation relying on quantization of gravitational Planck constant looks more attractive to me. It is however always useful to make an exercise in challenging the cherished beliefs.
For details see the chapter Quantum Astrophysics.
