Emergence of branched electron flow from weak scattering across a random potential. (C) Tobias Kramer
Our universe displays various mass concentrations of matter and is not a homogeneous density soup of particles as assumed in the simplest cosmological models where isotropy is assumed (cosmological principle). Interestingly recent supernova data (see Colin et al Evidence of anisotropy of cosmic acceleration A&A 631 L13, also on the arxiv) shows deviations of the angular distribution of matter as seen from Earth. One possibility (also not taken into account in the standard model) is the topology of the cosmos, discussed in this blog by Peter Kramer before. But back to the “small structures” (< 100 Mega parsec): How do mass concentrations arise and how get small fluctuations amplified?
One universal mechanism at work across many domains of physics is structure formation and concentration into branches by random weak scattering. The key-point is: despite randomness this type of scattering does not smear out the density as expected by diffusion processes. This important mechanism has been independently discovered in various domains of physics, but has been rarely discussed and further explored within a consistent framework. Interestingly the fundamental branching behavior can be seen in both quantum mechanics and classical physics. Together with Rick Heller and Ragnar Fleischmann we have written a short overview article (available on the arxiv: “Branched flow”) which provides some background information and serves as a starting point for further exploration. We also provide a short python script to generate intricate patterns out of random deformations (script available on github).
Sunlight creating a web of caustics at the bottom of a swimming poll. Picture (C) Tobias Kramer.
One example discussed in this blog is the formation of “dust concentrations” in the near nucleus coma of comets from a homogeneously emitting cometary surface. The structure formation of the universe (the cosmic web) is discussed by Y. Zeldovich and reviewed by P.J.E. Peebles in his monograph The Large-Scale Structure of the Universe (Princeton University Press, 1980) in the chapter “caustics and pancakes”. A more accessible incarnation of the caustics are the ripples of sunlight at a pool bottom.
Johannes Kepler published his book Harmony of the world 400 years ago contains what is now commonly known as “Kepler’s third law”:
Sed res est certissima exactissimaque, quòd proportio quae est inter binorum quorumcunque Planetarum tempora periodica, sit praecisè sesquialtera proportionis mediarum distantiarum, …
Johannes Kepler Harmonices Mundi 1619, see p. 302 of the Kepler edition by Max Caspar.
Expressed in equations: the ratio of orbital periods T1 : T2 is proportional to the semimajor axes a11.5 : a21.5 to the power of 1.5 for any two planets 1 and 2. What is more difficult to predict (and still unknown today!) is why the planets in our solar system are at their respective orbit. With the vast statistics of thousands of extra solar planets obtained from the Kepler mission and other surveys we might soon find out if there is an empirical relation hidden in the formation of planetary systems.
Kepler did make several attempts to find a law behind the distances of the 5 known planets Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Most known is his book Mysterium Cosmographicum published already 1597.
In the first rendering, I show the elliptical orbits of the five planets Mercury, Venus, Earth, Mars, Jupiter, Saturn, in black. The spherical shells have radii corresponding to the perihelion and aphelion distances of the planets. The red tetrahedron has as insphere a ball of radius the aphelion distance of mars and a circumsphere with radius equal to Jupiter’s perihelion distance. The cube has as insphere a ball with radius aphelion distance of Jupiter and circumscribed a sphere with the perihelion distance of Saturn.
Kepler’s 1597 study of the proportions of the planetary orbits in relation to the platonic bodies. (C) for this visualization: Tobias Kramer 2019
The second figure shows the inner planets in more detail. The black ellipses are the actual orbits of Mercury, Venus, Earth, and Mars. The innermost yellow octahedron encloses the orbit of Mercury (inscribed in the square shaped plane), then the construction proceeds by connecting the outer sphere of the yellow octahedron to the Venus shell (perihelion distance), then put around the aphelion shell of Venus the green icosahedron. The sphere circumscribed around the icosahedron has the radius of the Earth’s perihelion distance. The blue dodecahedron harbors an insphere with radius of the Earth’s aphelion distance and its circumsphere yields the perihelion sphere of Mars. The larger eccentricities of the Mars and Mercury orbits yield thicker shells for these planets. Note that the construction takes the eccentricities to be known and does not rely on circular trajectories.
Kepler was well aware of the eccentricity of the planets and did not assume circular orbits. He also noted that the theory agrees only within few percent with observations. In Harmonices Mundi he investigates if alternatively the planetary distances are encoded in musical proportions.
The following post is contributed by Peter Kramer.
Shown are two faces of a hyberbolic dodecahedron. The red line from the family of shortest lines (geodesics) connects both faces. Adapted from CRM Proceedings and Lecture Notes (2004), vol 34, p. 113, by Peter Kramer.
The new Planck data on the cosmic microwave background (CMB) has come in. For cosmic topology, the data sets contain interesting information related to the size and shape of the universe. The curvature of the three-dimensional space leads to a classification into hyperbolic, flat, or spherical cases. Sometimes in popular literature, the three cases are said to imply an inifinite (hyperbolic, flat) or finite (spherical) size of the universe. This statement is not correct. Topology supports a much wider zoo of possible universes. For instance, there are finite hyperbolic spaces, as depicted in the figure (taken from Group actions on compact hyperbolic manifolds and closed geodesics, arxiv version). The figure also shows the resulting geodesics, which is the path of light through such a hyperbolic finite sized universe. The start and end-points must be identified and lead to smooth connection.
Recent observational data seem to suggest a spherical space. Still, it does not resolve the issue of the size of the universe.
Instead of a fully filled three-sphere, already smaller parts of the sphere can be closed topologically and thus lead to a smaller sized universe. A systematic exploration of such smaller but still spherical universes is given in my recent article Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis.
In physics, it is important to give specific predictions for observations of the topology, for instance by predicting the ratio of the different angular modes of the cosmic microwave background. It is shown that this is indeed the case and for instance in a cubic (still spherical!) universe, the ratio of 4th and 6th multipole order squared are tied together in the proportion 7 : 4, see Table 5. On p. 35 of ( the Planck collaboration article) the authors call for models yielding such predictions as possible explanations for the observed anisotropy and the ratio of high and low multipole moments.
The following article is contributed by Peter Kramer.
Einstein’s fundamental theory of Newton’s gravitation relates the interaction of masses to the curvature of space. Modern cosmology from the big bang to black holes results from Einstein’s field equations for this relation. These differential equations by themselves do not yet settle the large-scale structure and connection of the cosmos. Theoretical physicists in recent years tried to infer information on the large-scale cosmology from Cosmic microwave background radiation (CMBR), observed by satellite observations. In the frame of large-scale cosmology, the usual objects of astronomy from solar systems to galaxy clusters are smoothed out, and conditions imprinted in the early stage of the universe dominate.
Fig 2: The planar Möbius crystal cm
In mathematical language one speaks of cosmic topology. Topology is often considered to be esoteric. Here we present topology from the familiar experience with the twisted Möbius strip. This strip on one hand can be seen as a rectangular crystallographic lattice cell whose copies tile the plane, see Fig. 2. The Möbius strip is represented as a rectangular cell, located between the two vertical arrows, of a planar crystal. A horizontal dashed line through the center indicates a glide-reflection line. A glide reflection is a translation along the dashed line by the horizontal length of the cell, followed by a reflection in this line. The crystallographic symbol for this planar crystal is cm. In three-dimensional space the planar Möbius crystal (top panel of Fig. 1) is twisted (middle panel of Fig. 1). The twist is a translation along the dashed line, combined with a rotation by 180 degrees around that line. A final bending (bottom panel of Fig. 1) of the dashed line and a smooth gluing of the arrowed edges yields the familiar Möbius strip.
Fig 3: Cubic twist N3.
Given this Möbius template in two dimension, we pass to manifolds of dimension three. We present in Fig. 3 a new cubic manifold named N3. Three cubes are twisted from an initial one. A twist here is a translation along one of the three perpendicular directions, combined with a right-hand rotation by 90 degrees around this direction. To follow the rotations, note the color on the faces. The three neighbor cubes can be glued to the initial one. If the cubes are replaced by their spherical counterparts on the three-sphere, the three new cubes can pairwise be glued with one another, with face gluings indicated by heavy lines. The complete tiling of the three-sphere comprises 8 cubes and is called the 8-cell. The gluings shown here generate the so-called fundamental group of a single spherical cube on the three-sphere with symbol N3. This spherical cube is a candidate for the cosmic topology inferred from the cosmic microwave background radiation. A second cubic twist with a different gluing and fundamental group is shown in Fig. 4. Here, the three twists combine translations along the three directions with different rotations.
The key idea in cosmic topology is to pass from a topological manifold to its eigen- or normal modes. For the Möbius strip, these eigenmodes are seen best in the planar crystal representation of Fig. 2. The eigenmodes can be taken as sine or cosine waves of wave length which repeat their values from edge to edge of the cell. It is clear that the horizontal wavelength of these modes has as upper bound the length L of the rectangle. The full Euclidean plane allows for infinite wavelength, and so the eigenmodes of the Möbius strip obey a selection rule that characterizes the topology. Moreover the eigenmodes of the Möbius strip must respect its twisted connection.
Fig 4: Cubic twist N2.
Similarly, the eigenmodes of the spherical cubes in Fig. 3 must repeat themselves when going from cube to neighbor cube. It is intuitively clear that the cubic eigenmodes must have a wavelength smaller than the edge length of the cubes. The wave length of the eigenmodes of the full three-sphere are bounded by the equator length of the three-sphere. Seen on a single cube, the different twists and gluings of the manifolds N2 and N3 shown in Figs. 3 and 4 form different boundary value problems for the cubic eigenmodes.
Besides of these spherical cubic manifolds, there are several other competing polyhedral topologies with multiple connection or homotopy. Among them are the famous Platonic polyhedra. Each of them gives rise to a Platonic tesselation of the three-sphere. Everitt has analyzed all their possible gluings in his article Three-manifolds from platonic solids in Topology and its applications, vol 138 (2004), pp. 253-263. In my contribution Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules, Class. Quant. Grav., vol. 27 (2010), 095013 (freely available on the arxiv) I display them and present a full analysis of their corresponding eigenmodes and selection rules.
Since terrestrial observations measure the incoming radiation in terms of its spherical multipoles as functions of their incident direction, the eigenmodes must be transformed to a multipole expansion as done in my work. New and finer data on the CMB radiation are expected from the Planck spacecraft launched in 2009. These data, in conjunction with the theoretical models, will promote our understanding of cosmic space and possible twists in its topology.
which repeat their values from edge to edge of the cell. It is clear that the horizontal wavelength 
Trespassing allowed: from the classical to the quantum world (and back again) 