** Basic differences between inflationary and TGD inspired cosmology**

Inflationary and TGD inspired cosmologies make remarkably similar predictions. There are also important differences following from quantum criticality of TGD Universe.

- The flatness of 3-space during very early cosmology is the basic prediction of inflationary cosmology. TGD predicts a period of quantum critical cosmology with flat 3-space. There are two options. The critical period can follow cosmic string dominated period during which space-time sheets do not exist yet and end up with a transition to sub-critical cosmology. Alternatively, cosmic string dominated period can be followed by overcritical period followed by a critical phase transition period leading to sub-critical cosmology. Only subcritical cosmologies have global imbeddings and asymptotic cosmology is necesssary subcritical. The Robertson-Walker type metric during both critical and over-critical periods is fixed apart from the parameter fixing its duration and the metric component g
_{aa}has exactly the same form for the two cases (a corresponds to lightcone proper time and the scale factor of RW cosmology denoted often by R). Hence TGD inspired early cosmology is extremely predictive. - In inflationary cosmology the temperature fluctuations reflect primordial density fluctuations amplified in scale during the exponential expansion. In TGD framework exponential expansion is replaced by an extremely sluggish logarithmic expansion during both overcritical phase (if present) and critical phase, and temperature fluctuations correspond to quantum critical long range fluctuations associated with the phase transition to subcriticality. The hierarchy of quantum coherent dark matters characterized by arbitrarily large values of hbar would be an essential piece of picture.

** 1. Fluctuations of microwave background as source of information about deviations from global homogenuity**

The fluctuations of the microwave background temperature are due to the un-isotropies of the mass density: enhanced mass density induces larger red shift visible as a local lowering of the temperature. Hence the fluctuations of the microwave temperatures spectrum provide statistical information about the deviations of the geometry of the 3-space from global isotropy and homogenuity. The symmetries of the fluctuation spectrum can also provide information about the global topology of 3-space and for over-critical topologies the presence of symmetries is easily testable.

The first year Wilkinson microwave anisotropy probe observations allowed to deduce the angular correlation function. For angular separations smaller the 60 degrees the correlation function agrees well with that predicted by the inflationary scenarios and deriving essentially from the assumption of a flat 3-space (due to quantum criticality in TGD framework). For larger angular separations the correlations however vanish, which means the existence of a preferred length scale. The correlation function can be expressed as a sum of spherical harmonics. The J=1 harmonic is not detectable due to the strong local perturbation masking it completely. The strength of J=2 partial wave is only 1/7 of the predicted one whereas J=3 strength is about 72 per cent of the predicted. The coefficients of higher harmonics agree well with the predictions based on infinite flat 3-space.

Later some interpretational difficulties have emerged: there is evidence that the shape of spectrum might reflect local conditions. There are differences between northern and southern galactic hemispheres and largest fluctuations are in the plane of the solar system. In TGD framework these anomalies could be interpreted as evidence for the presence of galactic and solar system space-time sheets.

** 2. Dodecahedral cosmology?**

The WMAP result means a discrepancy with the inflationary scenario and explanations based on finite closed cosmologies necessarily having Ω>1 but very near to Ω=1 have been proposed. J. P. Luminet (arXiv:astro-ph/0310253) has proposed that Poincare dodecahedral space, which is globally homogenous space obtained by identifying the points of S^{3} related by the action of dodecahedral group, or more concretely, by taking a dodecahedron in S^{3} (12 faces, 20 vertices, and 30 edges) and identifying opposite faces after 36 degree rotation, could explain the weakness of lowest partial waves. It was found to fit quadrupole and octupole strengths for 1.012<Ω <1.014 without an introduction of any other parameters than Ω .

However, according to Tegmark (arXiv:astro-ph/0310723), the quadrupole and octupole moments have a common preferred spatial axis along which the spectral power is suppressed so that dodecahedron model seems to be excluded. The analysis of Cornish et al (astro-ph/0310233) led to the same result. According to the recent Physics Web article of Luminet, the situation is however not yet completely settled, and there is even some experimental evidence for the predicted icosahedral symmetry of the thermal fluctuations.

The possibility to imbed also a very restricted family of over-critical cosmologies raises the question whether it might be possible to develop a TGD based version of the dodecahedral cosmology. The dodecahedral property could have two interpretations in TGD framework.

- A space-time sheet with boundaries could correspond to a fundamental dodecahedron of S
^{3}. If temperature fluctuations are assumed to be invariant under the so called icosahedral group, which is subgroup of SO(3) leaving the vertices of dodecahedron invariant as a point set, the predictions of the dodecahedral model result. - An alternative interpretation is that the temperature fluctuations for S
^{3}decomposing to 120 copies of fundamental dodecahedron are invariant under the icosahedral group. S^{3}would correspond to two space-time sheets containing 60 dodecahedrons on top of each other and glued along boundaries.

For neither option topological lensing phenomenon is present since icosahedral symmetry is not due to the identification of points of 3-space in widely different directions but due to symmetry which is not be strict. An objection against both options is that there is no obvious justification for the G invariance of the thermal fluctuations. The only justification that one can imagine is in terms of quantum coherent dark matter.

The finding of WMAP that the ratio Ω of the mass density of the Universe to critical mass density is Ω= 1+g_{aa}=1+ε , ε=0.02&plusm; 02. This is consistent with critical cosmology. If only slightly overcritical cosmology is realized, there must be a very good reason for this.

The WMAP constraint implies that the value of a which corresponds to the value of cosmic time a_{s} which characterizes the thermal fluctuations must be such that g_{aa}=ε holds true. The inspection of the explicit form of g_{aa} deduced in the subsection "Critical and over-critical cosmologies" of TGD and Cosmology requires that a_{s} is extremely near to the value a_{0} of cosmic time for which g_{aa}=0 holds true: the deviation of a from a_{0} should be of order (R/a_{0})×R (R is CP_{2} size about 10^{-30} meters), and most of the thermal radiation should have been generated at this moment.

Since gravitational mass density approaches infinity at a→a_{0} one can imagine that the spectrum of thermal fluctuations reflects the situation at the transition to sub-criticality occurring for Ω =1+ε. Thermal fluctuations would be identifiable as long ranged quantum critical fluctuations accompanying this transition and realized as a hierarchy of space-time sheets inducing the formation of structures. The scaling invariance of the fluctuation spectrum generalizes in TGD framework to conformal invariance. This means that the correlation function for fluctuations can have anomalous scaling dimension for which there is some evidence (astro-ph/9611208).

The transition k=1→0→-1 would involve the change in the shape of the S^{2}⊂ CP_{2} angle coordinate Φ as a function f(r) of radial coordinate of RW cosmology. The shape is fixed by the value of k=1,0,-1 . In particular, Φ would become constant in the transition to subcriticality. k=1→0 phase transition would be accompanied by the increase of the maximal size of space-time sheets to infinite in accordance with the emergence of infinite quantum coherence length at criticality. Whether this could be regarded as the TGD counterpart for the exponential expansion during inflationary period is an interesting question. In the transition to subcriticality also the shape of Θ as function of a necessarily changes since sin(Θ(a>a_{0}))>1 would be required otherwise.

**3. Hyperbolic cosmology with finite volume?**

Also hyperbolic cosmologies allow infinite number of non-simply connected variants with 3-space having finite volume. For these cosmologies the points of a=constant hyperboloid are identified under some discrete subgroup G of SO(3,1) . Also now fundamental domain determines the resulting space and it has a finite volume.

It has been found that a hyperbolic cosmology with finite-sized 3-space based on so called Picard hyperbolic space (astro-ph/0403597), which in the representation of hyperbolic space H^{3} as upper half space z>0 with line element ds^{2}= (dx^{2}+dy^{2}+dz^{2})/z^{2} can be modelled as the space obtained by the identifications (x,y,z)=(x+ma, y+nb,z). This space can be regarded as an infinitely long trumpet in z -direction having however a finite volume. The cross section is obviously 2-torus. This metric corresponds to a foliation of H^{3} represented as hyperboloid of M^{4} by surfaces m^{3}=f(ρ) , ρ^{2}= (m^{1})^{2}+(m^{2})^{2} with f determined from the requirement that the induced metric is flat so that x,y correspond to Minkowski coordinates (m^{1},m^{2}) and z a parameter labelling the flat 2-planes corresponds to m^{3} varying from -∞ to +∞.

This model allows to explain the small intensities of the lowest partial waves as being due to constraints posed by G invariance but requires Ω=.95 . This is not quite consistent with Ω=1.02&plusm; .02 .

Also now two interpretations are possible in TGD framework. Thermal photons could originate from a space-time sheet identifiable as the fundamental domain invariant under G . Alternatively, a=constant hyperboloid could have a lattice-like structure having fundamental domain as a lattice cell with thermal fluctuations invariant under G . The shape of the fundamental domain interpreted as a surface of M^{4} is rather weird and one could argue that already this excludes this model.

Quantum criticality and the presence of quantum coherent dark matter in arbitrarily long length scales could explain the invariance of fluctuations. If Ω reflects the situation after the transition to subcriticality, one has Ω=g_{aa}-1=.95 . This gives g_{aa}=1.95 which is in conflict with g_{aa}<1 holding true for the imbeddings of all hyperbolic cosmologies. Thus Ω must correspond to the critical period and one should explain the deviation from Ω=1 . A detailed model for the temperature fluctuations possibly fixed by conformal invariance alone would be needed in order to conclude whether many-sheeted space-time might allow this option.

** 4. Is the loss of correlations due to the finite size of the space-time sheet? **

One can imagine a much more concrete explanation for the vanishing of the correlations at angles larger than 60 degrees in terms of the many-sheeted space-time. Large angular separations mean large spatial distances. Too large spatial distance, together with the fact that the size of the space-time sheet containing the two astrophysical objects was smaller than now, means that they cannot belong to the same space-time sheet if the red shift is large enough, and cannot thus correlate. The size of the space-time sheet defines the preferred scale. The preferred direction would be most naturally defined by cosmic string(s) in the length scale of the space-time sheet. For instance, closed cosmic string would define an expanding 3-space with torus topology and thus having symmetries. This option would explain also the anomalies as effects due to galactic and solar space-time sheets.