Curved and finite or flat and infinite?

ΛCMD model provides the standard model for cosmology. The model assumes flat 3-space cold dark matter realized in terms of dark matter halos at galactic level. Cosmological constant Λ parameterizes dark energy density. The gives a rather satisfactory description of the CMB data in terms of 6 parameter but there are however some anomalies. There are deviations from the predictions from the actual CMB data. The problems related to the notions of dark matter and energy itself represent deeper problems.

Now a new problem - even crisis - has emerged (I have the feeling that it has been always in crisis). Alessandro Melchiorri of Sapienza University of Rome Eleonora di Valentino of the University of Manchester and Joseph Silk, principally of the University of Oxford, published in Nature Astronomy a paper Planck evidence for a closed Universe and a possible crisis for cosmology. Here is the abstract of the article.

The recent Planck Legacy 2018 release has confirmed the presence of an enhanced lensing amplitude in cosmic microwave background power spectra compared with that predicted in the standard Λ cold dark matter model, where Λ is the cosmological constant.

A closed Universe can provide a physical explanation for this effect, with the Planck cosmic microwave background spectra now preferring a positive curvature at more than the 99 per cent confidence level.

Here, we further investigate the evidence for a closed Universe from Planck, showing that positive curvature naturally explains the anomalous lensing amplitude, and demonstrating that it also removes a well-known tension in the Planck dataset concerning the values of cosmological parameters derived at different angular scales. We show that since the Planck power spectra prefer a closed Universe, discordances higher than generally estimated arise for most of the local cosmological observables, including baryon acoustic oscillations. The assumption of a flat Universe could therefore mask a cosmological crisis where disparate observed properties of the Universe appear to be mutually inconsistent. Future measurements are needed to clarify whether the observed discordances are due to undetected systematics, or to new physics or simply are a statistical fluctuation.

A lensing amplitude 5 per cent larger than expected is observed. Concretely this means following.

  1. Gravitational lensing is essentially scattering of the incoming light in the gravitational field of matter between the detector an sources so that for a single object between the source the light seems to becoming from a circular object rather than point like object.
  2. The light of CMB entering detector has experienced a large number of lensings and the images of various features in the data become blurred. The larger the lensing, the more blurred the object image. Lensing amplitude is proportional to the curvature of 3-space and if the curvature vanishes as in flat cosmology, lensing amplitude should be very small and due to fluctuations. There should be for other reasons.
  3. ΛCDM predicts flat 3-space so that the observed lensing is anomaly if real effect. The obvious explanation would be that the density of matter is about 5 per cent higher than the model predicts. The density would become overcritical and the infinite flat 3-space would close to sphere. This would be a dramatic change in the topology of 3-space.
As the abstract tells, positive curvature would also remove the well-known tension in the Planck dataset concerning the values of cosmological parameters derived at different angular scales.

There are however objections. For instance, inflation theory favours infinite 3-space. There is also second manner to deduce 3-curvature. Lensing reconstruction measures correlations from sets of four points in the sky to deduce 3-curvature. The results are in accordance with flatness.

Various empirical inputs force flat 3-space so that one cannot just add to ΛCDM the curvature as 7:th parameter. For instance, it would very difficult to understand how this modification could be consistent with inflation theory involving flat expanding Universe. Therefore one can say that cosmology is in crisis.

What says TGD?

Something new is needed. Hyperbolic or flat Universe seems to be a natural assumption in TGD framework assuming that at fundamental level space-times are surfaces in M4×CP2.

  1. General Relativity emerges as a long length scale approximation in which space-time surfaces are assumed to be Einsteinian in long scale, that is having 4-D M4 projection. There are also non-Einsteinian space-time surfaces. CP2 type extremals with 1-D light geodesic as M4 projection, and string like objects with 2-D M4 projection.
  2. Cosmic strings are basic entities of the primordial cosmology in TGD Universe and have 2-D M4 projection as string world sheets: there is no Einsteinian space-time and it emerges during the TGD analog of the inflationary period. Cosmic strings are unstable against thickening of the M4 projection of cosmic strings so that they tend to thicken to flux tubes. Cosmic strings and flux tubes are present also during the Einsteinian era. One can speak of topological condensation of cosmic strings to space-time surface. In GRT based cosmology these objects have no counterpart. Cosmic strings appear in two varieties depending on whether the quantizes magnetic flux associated with their closed cross section is monopole flux or not (vanishes). What is important that monopole fluxes require no currents, which solves several problems of cosmology and astrophysics.
  3. The model for the formation of various astrophysical objects relies on the thickening of cosmic string portions to monopole flux tubes. One must therefore include them. In accordance with the notion of many-sheeted space-time, cosmology must be replaced with a fractal hierarchy of cosmologies with length scale dependent parameters such as cosmological constant.

    Remark: String world sheets appear also as singularities of space-time surfaces as minimal surfaces appearing as extremals of also 4-D Kähler action obtainedas a dimensional reduction of the twistor lift of 4-D K\"ahler action. These strings are like edges of 3-space.

  4. Cosmic strings carry dark energy and dark matter. Dark energy is sum of magnetic and volume contributions. The latter is proportional to cosmological constant Λ, which depends on the size scale of the space-time sheets and comes in some negative powers of 2 corresponding to preferred p-adic length scales. ΛCDM would correspond to length scale for space-time sheets of order recent size of cosmos.
In this framework the most recent crisis of cosmology would reflect the failure of the Einsteinian space-time concept basically due to the existence of cosmic strings and other objects with lower-D projection.

The observed too large lensing amplitude is intuitively easy to understand in terms of cosmic strings. The incoming CMB photon would express additional scattering from long cosmic strings along its route. Second manner to deduce the curvature is by measuring correlations from sets of 4 points in the sky. The correlations should not change if TGD is correct.

The conceptual and technical challenge is to understand how two treat cosmic strings in the QFT-GRT limit of TGD based fractal cosmology.

  1. For space-time surfaces with 4-D M4 projection the GRT space-time is obtained as 4-D piece of M4 with gauge potentials and the deviation of metric from flat M4 metric as a sum of those for space-time sheets. One actually obtains a length scale hierarchy of space-times assignable to causal diamonds (CDs) and standard cosmology should correspond to the limit of CD size, which is about the size of the recent cosmology. CMB data indeed suggest length scale dependence of CMB parameters.
  2. At every step in hierarchy 3-space is finite corresponding to the radius of CD at give value of Minkowski time. The actual 3-surfaces are always closed and can be though of as pairs of 3-surfaces with boundary at boundary of CD glued together along their boundaries. Covering of flat 3-space would be in question. Could this somehow relate to why closed Universe seems to work in some respects? The hierarchy brings in new elements such as hierarchy of cosmological constants with a concrete interpretation discussed elsewhere.
  3. But what should one do with the cosmic strings and flux tubes? The addition of them as singular string world sheets in space-time carrying concentrated a string world sheet is one possibility which comes into mind. GRT based cosmology relies on pointlike objects and infinitely long cosmic strings are not pointlike objects in any scale.

    A more plausible approach would be in terms of tensor nets with cosmic strings and flux tubes defining the connections between the nodes of the net. The identification of dark matter as heff=n× h0 phases of ordinary matter would make possible quantum coherence in arbitrarily long length scales. Fractal tensor net would be genuinely quantal object in all scales, and provide very different view about the role of quantum dynamics in cosmology and astrophysics.

See the chapter More TGD inspired cosmology or the article Cosmology in crisis.