Monday, January 24, 2011

Large-Scale Structures - Superclusters and Supervoids

This section, in fact this entire text, is nothing more than a documentary covering the evolution of the universe. It is designed to uncover the truth about what is transpiring around us. In presenting empirical evidence objectively and in an informative manner, without inserting highly speculative material (or dark energy) to make observations fit the theory, the goal is that a better understanding of nature (in the widest possible sense of the term) can be obtained. The ultimate goal is the understanding of the essence of the physical universe and its evolution in time.

It may turn out the truth is embodied in big bang cosmology. Or it may not. If the evidence were indisputable I would certainly not, as a pragmatic empiricist (and Bright), be wasting time writing about an alternative model, let along one that is static. The fact is, there are no indisputable facts inherent in big bang cosmology. Redshift z is not necessarily a relativistic Doppler effect, nor is the CMB necessarily a relic of a hot dense phase of the early universe. There may not have been an epoch of primordial creation at the outset, or even an outset at all.

The fact is too, there exists at least one viable alternative consistent with general relativity (where gravitation is treated as a curved spacetime phenomenon) that is dynamic and evolving, and is consistent with observations (to the extent that those observations have been interpreted within the outline of the model).

Let me sum up what has been discussed so far in the context of a general relativistic static universe model that possess constant Gaussian curvature of the global spacetime continuum. A few concepts not extensively covered so far will be explored below regarding the problems and solutions of diverging gravitational potential in a homogeneous stationary universe inherent in gravitational theory since the time of Newton.

Spacetime Curvature

Curvature is the central theme of theoretical and observational cosmology. Spacetime curvature in the presence of matter, introduced by Einstein in the general theory of relativity, is the physical concept that replaced the profound mystery of Newtonian action at a distance. To a large extent today the concept of spacetime curvature is a profound mystery.

The concept of curvature found its way into general relativity following the lead of Saccheri, Lobachewsky, Gauss, Bolyai, Clifford. Subsequently, Riemann, Christoffel, Ricci and Levi-Citiva developed non-Euclidean geometry into a robust analytical discipline.

The premise upon which all observational cosmology rests was built by Gauss in his 1827 "Disquisitiones Generales Circa Superficies Curvas." Curvature (a number) can be calculated for any arbitrary manifold from data obtained by measurement of the metric properties internal to that manifold. The departure from flatness is called curvature. See Gauss' Theorema Egregium.

The surface of a sphere is a 2-space example with constant positive curvature. Curvature remains the same along the entire surface in all directions (every point is the same). It is often the difficulty of visualizing this intrinsic 2-space curvature in 3-space that causes intuition to fail. I've added a schematic diagram below to help in this visualization (a cross section of a static four-dimensional universe: Figure ESU).

Ultimately, the curvature of 3-space must be expressed by equations that contain quantities observed telescopically: Fluxes (apparent magnitudes), redshift z (the spectral shift relative to laboratory wavelengths of a particular spectral line), angular diameter and surface brightness. All of these observables are affected by Gaussian curvature of 3-space. Note that in a spacetime manifold of the type under study here—a static universe of positive Gaussian curvature k = 1—the volume at a given distance interval appears smaller than that of a Euclidean or hyperbolic 3-space (k = 0, k = -1). In addition, time intervals appear to run slower with increasing distance from the observer (from any arbitrary origin of the manifold centered on an observer). Again, spacetime curvature is determined by the mass-energy density of the manifold, in accord with general relativity.

The point of the current discussion (see above), is in part to show that photons propagate along 'true' geodesic paths in a four-dimensional spacetime continuum (three space and one time), consistent with great circle arcs (in reduced dimension, i.e., on a spherical surface of constant positive Gaussian curvature). Electromagnetic radiation is redshifted due to the geodesic path. Thus, in a static universe of constant positive Gaussian curvature cosmological redshift z is observed to increase with distance from the observer in the look-back time. Redshift is a measure of geodesic distance between the emitting source and the observer in a homogeneous and isotropic curved spacetime. Redshift is due to a geodesic distortion along the path of the photon wave between two epochs. Cosmological redshift along with the associate time dilation factor is a direct measurement of constant intrinsic Gaussian curvature of the spacetime continuum.

The universe remains non-expanding and non-collapsing (i.e., static) due to the physical properties of a spacetime continuum of constant positive curvature. The fact that massive objects do not follow the same geodesic trajectory as photons—they are indeed restricted to local gravitational field geodesic motion—implies that all objects in the universe are not impelled to accelerate, gravitate or free-fall towards one another.

Massive objects are bound to follow local geodesics determined by the gravitational fields of neighboring objects (such as the orbit of the earth around the sun, the sun around the Galaxy, the Galaxy relative to the Local Group, the Local Group relative to the Local Supercluster). The question is whether galaxy superclusters are stable gravitationally bounded systems. Unfortunately, the dynamics of galaxy clusters cannot be directly observed since the rotational times are typically hundreds of millions (or billions) of years. Galaxy clusters and superclusters appear as motionless systems, but their morphology, orientation relative to neighbors, and physical connections in a web-like or sponge-like network imply otherwise.

Equilibrium Solutions

Fortunately, methods for computing the N-body problem have grown exponentially in computational efficiency. These methods have been used to simulate the dynamics of systems with as many as 10 billion particles. (Source). Not only do clusters persist as clusters when they fall into a larger host, but groups of clusters retain their identity for long periods within larger host superclusters. Line-of-sight velocity dispersions for individual clusters show large variance which depends on viewing angle. Kinematics is strongly viewing angle dependent. The point is that clusters of galaxies, even after merging with a larger host cluster (or supercluster), can retain their identity for several Gyr. Equilibrium solutions are possible for scales compatible with superclusters. Exceedingly large tangent velocities are not required in order that superclusters remain in stable equilibrium relative to one another. The larger the scale of the clusters, the greater the distance that separates clusters, and as the mass density profile becomes less concentrated, the slower the clusters need to move relative to one another in order for the maintenance of stability. Whatever the initial orbital shapes of superclusters relative to one another, isotropization of their relative orbits can (and here argued, does) occur. Observed should be a lack of significant evolution in the orbital anisotropy. It is essential to tighten the current constraints on the orbital evolution of superclusters (both nearby and high-z) in the future, and to re-assess such dynamics as a function of both time and supercluster mass. Only from this type of analysis will it be possible to obtain a thorough understanding of the dynamics and hierarchical assembly history (evolutionary trends) of galaxy superclusters as bounded gravitating systems. Within a supercluster, the individual galaxy clusters orbit the supercluster's center of mass, roughly as the sun orbits the Galaxy. Of course, such an orbital velocity would be difficult to observe, since both the intrinsic redshift and blueshift due to orbital motion would be dwarfed by the cosmological redshift (i.e., a blushift induced by rotation towards the observer would simply be seen as a redshift, and the intrinsic redshift would make objects looks further away when coupled with cosmological redshift z).

Locally, where cosmological redshift is negligible, investigations into cluster dynamics is more robust. Observations inferring total mass in galaxies and clusters, based on the distribution of orbital velocities of components within these systems, suggest that much of the mass is non-luminous. Dynamical mass estimates are usually 5-10 times larger than mass estimates derived by summing the luminous components (and hot gas, in the case of clusters). The existence of nonbaryonic cold dark matter is hypothesized to explain such observations (Postman 2006, Distribution of Galaxies, Clusters, and Superclusters). Of course the possibility that much of this dark matter is solely baryonic exists (e.g., in the form of neutron stars, white dwarfs, brown dwarfs, massive planets, dilute plasma, neutral gas). But something still appears to be missing. Certainly, in a static universe, times scales are sufficiently large to explain the presence of vast quantities of non-luminous objects. An additional possibility (which should be compounded with the greater baryonic density than current estimates) is that the dynamics of these systems is determined to a large extent by the larger systems still, within which are embedded the objects in question. The larger the scale, the greater the effect. For example, gravitational effect on the solar system due to the Galaxy are small. Effects on the dynamics of the Galaxy due to the Local group are small but relatively larger. The dynamical gravitational effects on the Local Group due to the Local Supercluster would be large in comparison. This would mean that the density perturbations associated with superclusters affects the dynamics of its components. Very large and extremely massive superclusters would have a greater affect on the dynamics of their subsystems than would smaller less massive superclusters. Orbital velocities or rotational curves should appear faster than gravitational effects of luminous matter would permit for components of very large massive systems. Whether this mechanism is operational remains to be tested. But is seems that the CDM contribution may not be necessary if this hypothesis is valid. The question of how best to use observations regarding excessively large orbital velocities and rotational curves to obtain optimal mass estimates from the relevant equations is one I hope to address in future work.

On large scales, beyond which the lumpiness seen in the large-scale structure of the universe appears isotropized and homogenized as per the cosmological principle, the distances that separates systems (great walls, filaments, supervoids and associated superclusters) are so large and time-scales so vast that orbital velocity is insignificant (or irrelevant). Orbital velocity tends to zero as distance and time-scale increases, up to the scale where the universe appears homogeneous and isotropic. That would be so due to the finite speed of the gravitational interaction (something not present in Newtonian gravitation). And on scales compatible with the observable section of the universe, no orbital velocity is required for the maintenance of stability.

The terms void and supervoid here corresponds to the lowdensity regions on the physical scales of clusters and supercusters respectively. Supervoid structures are roughly 100-150 h^1 Mpc (h is the Hubble constant in units of 100 km s^−1 Mpc^−1); a scale below which the universe is inhomogeneous, and above which the matter distribution tends toward homogeneity.

The Local Group

Only with respect to the Local Group and the Local Void is it possible to distinguish between the redshift of galaxies due to the Gaussian curvature of the universe and the local relative motions caused by the way matter is clustered together, with its consequential gravitational effects. It has been found that galaxies flow in streams, with coherent flows caused by large distant attractors and eddies caused by modest nearby attractors. The currently favored standard model of the universe (Lambda-CDM) with dark matter and dark energy does not allow for voids that are as large as inferred for the Local Void. (Source).

The distribution, in supergalactic coordinates, of galaxies in the region of the Milky Way.

(Source: Institute for Astronomy University of Hawaii and Our Peculiar Motion Away from the Local Void, ApJ 676, 184)

Each dot represents a galaxy (on average containing 100 billion stars). The colors indicate the relative motions: green and blue indicate motions toward the Milky Way. Shades of yellow, orange and red indicate motions away from us. The closest galaxies have small relative motions (see right panel). The Galaxy, along with all the Local Group, are moving together toward the lower right corner of each diagram. As a result, all galaxies in the lower right appear to be moving toward us, while those in the upper left appear to be moving away. The motion of the Galaxy is represented by the orange arrow. There are two suspected causes for this motion. The Virgo Cluster on the right of the figures causes an attraction indicated by the blue vector (in the exploded panel on the right). The red vector is the remainder which represents a motion of ~260 km s−1 (600,000 miles-per-hour) away from the Local Void. "given the velocities expected from gravitational instability theory in the standard cosmological paradigm, the distance to the center of the Local Void must be at least 23 Mpc from our position. The Local Void is extremely large."

Such a problem does not arise in this general relativistic static model, since cosmological redshift is not interpreted as resulting from expansion, i.e., a large component of the observed redshift is not due to radial motion away from the local supervoid.

The WMAP Cold Spot, a cold region in the microwave sky, is highly improbable under the Lambda-cold dark matter model (ΛCDM). This supervoid could be the cause the cold spot, but it would have to be improbably large: a billion light-years across, according to some estimations.

The Large-scale Structures

The differentiation between models must be made by studying the characteristics that would arise in the case where superclusters are gravitationally bound, in comparison with properties in the case where they are not gravitationally bound. Recall, it is currently believed, by virtue of expansion, that superclusters are not gravitationally bound systems (so there should be little observed interaction, if any at all, between these groups). The observed nonlinearity in the redshift and rise times of distant Type Ia supernovae interpreted as an acceleration of expansion is usually attributed to a mysterious negative pressure (dark energy) that causes the gravitational potential between large-scale structures to decay, lowering the amount of interaction between superclusters. The supervoids that separate these structures grow with time. Regularity in the structure of large-scale density perturbations would be disrupted with time (with expansion), resulting in irregularity in the distribution of superclusters.

Conversely, in a static universe where these large-scale structures are gravitationally bound there should be observed a great deal of interaction and regularity in the distribution of superclusters. After all, these objects would be connected gravitationally, just as clusters themselves (in either model). There should be a straightforward relation between clusters and the supercluster substructure, and there should be a relation between superclusters themselves, e.g., there should be observed a tendency for superclusters to be alined with their closest neighbor as well as with other superclusters that reside in the local vicinity (the friends of friends approach), as well as large-scale filamentary structures connecting superclusters. There should be a relation between the dynamic internal state of superclusters and the large-scale environment. The morphology of superclusters must be consistent with systems that exhibit rotation on smaller scales (e.g., flattening). In sum, there should be observed a coherent orientation effect and a strong link between the dynamical state of superclusters and their large-scale environment. The size of the supervoids that separate superclusters remains relatively constant over time. In other words, there should be observed a regularity on scales of superclusters (which are considered gravitating systems in a static universe), i.e., the supercluster-supervoid network should be highly structured, just as there exists non-random highly structured regularity in the distribution of gravitating systems on smaller scales, where systems are known to be gravitationally bound. In fact, in a static universe all systems are bound gravitationally. Albeit, gravitational influences are strongest for those objects that neighbor one another, but non-negligible for friends of friends. Structural coherence should be observed across a vast region of the visible universe.

Einasto et al (1997, The supercluster-void network II. An oscillating cluster correlation function) compared correlation functions derived for popular CDM-models of structure formation (as well as simple geometrical models with randomly and regularly located superclusters) with the empirically observed cluster correlation function. The quantitative tests showed that overdense regions in the universe punctuated by rich galaxy clusters are distributed more regularly than expected, and that, consequently, "our present understanding of structure formation needs revision." Note: the discrepancy between galaxy formation models and observation would be even greater in a universe dominated by dark energy. "The fact that the amplitude of oscillations near the last maximum is still rather large suggests that the coherence of positions of high-density regions extends over very large separations (at least 10% of the diameter of the observable Universe)." Einasto et al pose a good question: "Can the observed correlation function of clusters of galaxies be reproduced by conventional models of structure evolution? If not, what changes in models are needed to reproduce the observed function?"

Early models of structure formation based on a universe dominated by CDM had predicted a trend, according to Marc Postman, towards higher correlation lengths as the mass of galaxies and clusters increased. But "these same models did not come close to predicting the actual observations in which the richest clusters are 10-20 times more strongly clustered than are galaxies." (Postman 2006)

In contrast to the standard expanding picture, the dynamical times of superclusters in a static universe are very large. Some, if not all, superclusters should be relaxed, thereby bearing the imprints of systems in dynamic equilibrium. Too, since expansion is not ripping these systems apart, the large-scale structures should bear the signature of the physical density fluctuations that were dominant during the formation process.

Certainly, the merging or colliding of components of superclusters (even of superclusters themselves) as well as the scattering or dispersion of superclusters are ongoing processes (this can occur in superclusters just as it does in smaller clusters). According to the standard model galaxy clusters (not superclusters) are the largest systems known to have reached dynamic equilibrium. In a non-expanding universe gravitationally bound structures exist on larger scales. Indeed, in a static universe superclusters, supervoids, great walls, sheets and filaments (some of which, in excess of 10^16 solar masses, are known to span 200 million light-years, or several hundred Mpc: see here for example) have had amply sufficient time to reach dynamic equilibrium. In addition, these structures need not be governed by the gravitational potential of ubiquitous nonbaryonic cold dark matter.

"Although early versions of structure formation theory based upon a universe dominated by cold dark matter (CDM) did predict a trend towards higher correlation lengths as the galaxy and cluster mass increased, these same models did not come close to predicting the actual observations in which the richest clusters are 10–20 times more strongly clustered than are galaxies." Source

All the material objects in the universe are subject to their own mutual gravitational field interactions. The fields of massive objects affect the local motion of other neighboring objects, proportionally to the gradients of the local gravitational fields. Inhomogeneities in the mass distribution induce motions (peculiar velocities) of galaxies, clusters and superclusters that are unrelated to the expansion of space, and unrelated to the globally non-Euclidean geometry in the case of a static universe where the cause of redshift z is attributed to the geodesic distortion in the photon path in a manifold of constant Gaussian curvature.

Preliminary evidence suggests that there do exist bulk flows of 600-800 (±150-300) km s−1 that extend to scales exceeding 100 Mpc. Though these ongoing surveys have not yielded conclusive directions for these flows. To date, there appears no inconsistency with the conclusion that superclusters are gravitationally bound systems in relative motion with respect to one another.

"If very largescale bulk flows turn out to be a reality, then present models for structure formation in the universe will require substantial revision (if not outright rejection)." Source

The impetus that all neighboring galaxy superclusters should merge gravitationally over time is vacated. Global collapse to a Big Crunch-like event is not an option, even in the absence of an expanding regime.

Let's take a break and look at this breathtaking 4-D simulation of the large-scale structure evolution. (Note, this model universe is not rotating. The geometrical setup of the camera path, the observers viewpoint, is in motion relative to the structures.)

This simulation uses a uniform time-dependent background radiation field, revealing the superclusters, voids, sheets, walls and filamentary structures formed so far. This video is shown in the Virtual Reality facility of the new Turin Planetarium. (See K Dolag, M Reinecke, C Gheller and S Imboden, Max-Planck-Institute for Astrophysics, Splotch: visualizing cosmological simulations). The compressed demo version of this simulation can in principle be download here, along with a host of other simulations.

The importance of all this is fundamental, as it shows that in a static four-dimensional universe, where positive curvature is due to the mass-energy density of the universe, objects do not all coalesce geodesically towards one massive big crunch. Global curvature does not cause gravitational instability, since objects are not bound by the same geodesic path as the photon. In another way, the global topology of the manifold does not affect the motion of gravitating systems within it. Yet the global topology is determined by the mass-energy content.

The example was given that the earth has settled into a stable orbit. So too the other planets of the solar system. Galaxies, though at time exhibit chaotic behavior, are just about everywhere present in the universe, i.e., they are relatively stable gravitating systems. Galaxy custers are relatively stable bound self-gravitating systems. These objects do not all gravitate towards one another, since their motion (angular momentum) is in general sufficient for the maintenance of dynamic equilibrium (see references below). So far so good.


Superclusters, on the other hand, are thought to be dynamically unstable systems, not bounded gravitationally. The reason given is usually because the universe is thought to be expanding, i.e., superclusters are moving away from one another, radially, in all directions. It is thought too that the velocity at which they are moving away from one another is sufficient to prevent them form collapsing back onto each other gravitationally. I will argue that superclusters are most definitely gravitationally bound systems.

In a static universe it has been suggested that the orbital velocity of superclusters is insufficient to prevent wholesale collapse of such structures. But this conjecture is not justified by empirical evidence. In addition, the larger the scale considered, the greater is the distance that separates objects, and the smaller is the force of gravity between them. This is like the classic analogy of an figure skater spinning on her axis. While her arms extend outwards the velocity of rotation decreases. In the case of superclusters, distances that separate galactic components (and often superclusters themselves) are so vast that orbital velocities sufficient for dynamic equilibrium are practically negligible. Large orbital velocities would not be required for stability to be maintained, even for the most massive superclustes (though again, interactions or merging will occur when in close proximity and when there is insufficient relative velocity). Indeed, the distances that separate galaxies in a cluster are much larger (in general) than distance that separate stars in a galaxy. In other words, distances that separate superclusters (on average) are relatively much larger than the distances that separated clusters. The larger the system under consideration the slower the orbital velocity required for equilibrium to be maintained (or the longer the collapse time). This is why we would not see (even if we could) peculiar velocities of superclusters in excess of peculiar velocities observed for clusters. See below, this hierarchal structuring of the cosmos has been used to undermine the diverging Newtonian gravitational potential in a homogeneous universe. But it is not the only method for dealing with the problem. Other methods have been shown more conclusive.

See below too for a further explanation of stability on scales compatible with the observable universe (or the universe in its entirety).

So the conclusion that over large distances there are no stable orbits because there are no tangent velocities sufficient to attain orbit is not justified by empirical evidence.

Empirical evidence confirms that the orientation of galaxies inside superclusters are generally alined in the directions of the major axes of the galaxies. Galaxies are significantly correlated. It has been known for many decades (Brown, 1964, 1968) that the minor axis of superclusters are oriented along the direction of the flattening of the supercluster as a whole. The same phenomenon is observed in the Local Supercluster, associated with the rotation established by de Vaucouleurs (1959). The broad conclusion that superclusters rotate is permitted. The rotation on these large scales cannot be explained by local effects, or tidal effects. Rotation on such scales is exceedingly difficult to incorporate into the standard model of galaxy formation, which suggests clusters are formed relatively rapidly via density fluctuations as the universe expands (Source). In a static universe this problem does not arise since density fluctuations have no time limit, i.e., pregalactic, precluster and protosupercluster density inhomogeneities can grow freely and interact in a variety of ways over extended (virtually limitless) time periods.

Due to motion, the distances involved, and due to non-instantaneous interaction (or finite velocity) of gravity, superclusters during the formation process and therafter remain gravitationally bound systems free of the need of to attain large peculiar velocities (their angular momentum is sufficiently large and operational), and free of the propensity to grossly collapse. On scales compatible with the observed superclusters, supervoids, sheets, filaments and walls, the universe is in dynamic equilibrium. At least, that possibility remains a viable option, in principle. Though again, to establish the rotational velocity of distance superclusters is quite a difficult task, since we would have to disentangle the intrinsic redshift (due to the rotation towards us) and the blueshift (rotation away from our viewpoint) from cosmological redshift z (which at great distances dwarfs the effect of peculiar motion). The observed redshift contains contributions from both gravitationally induced velocity and a cosmological component (whatever its cause). In another way, orbital velocities distort the spatiotemporal positions determined from cosmological redshift measurements, resulting in a highly elongated appearance of clusters and superclusters oriented toward the center of 2-D and 3-D maps of the sky. This is an artifact, dubbed "the fingers of God," of large orbital velocities (close to 1000 km s^-1) around the center of clusters. If distances could be plotted independent of redshift the fingers of god would vanish. (Source: Distribution of Galaxies, Clusters, and Superclusters, IoP).

Implications from the relevant data are that the larger the correlation length (i.e., the greater the mean separation between objects) and the larger the mass of the system, the stronger the clustering. (See source directly above). The question is whether this trend continues to scales of superclusters. My hunch is that it does. The standard model for structure formation dominated by CDM had predicted a this correlation, but the predictions didn't come close to actual observations, which show that the richest clusters are between 10 and 20 times more strongly clustered than galaxies.

On scales exceeding 150-200 Mpc the universe tends toward homogeneity, so the idea that velocities are required to maintain equilibrium configurations is no longer relevant. There is no need to stipulate that larger systems still (ad infinitum) and greater velocities are required for stability to be attained. Or simply, there appear to be no structures larger than 150 Mpc that would need to rotate around something. In principle, this fact (if indeed it is a fact) would set an upper limit on the peculiar velocity of objects in the field.

There is a recent tendency for larger structures to be recognized. So it's quite possible that the top of clustering hierarchy has not yet been established, i.e., the largest inhomogeneities in the universe may be, to date, unidentified as individual structures. The Great Wall, for example, and its full extent is still being assessed. Whenever a volume of space is sampled, "there always seems to be structure with a dimension comparable to that of the volume surveyed. This has led to considerations of fractal structures (identical forms repeated on ever-increasing scales) occurring in the universe. If this is correct, although one would gain a geometrical interpretation to the nature of the structures, it would make a physical explanation extremely difficult." (Source: Fairall). The discovery of larger structures still would bolster the prediction that smaller structures (superclusters in this case) are gravitationally bound systems.

On scales where inhomogeneities in the mass distribution are observed, motions of clusters and superclusters are induced by mutual gravitational forces (or the local spacetime curvature) in proportion to the local gradients in the composite fields. If matter were distributed homogeneously locally no motion would be induced, since the gravitational forces would cancel on average. However, matter is not quite homogeneously distributed on scales the size of a supercluster.

Observations of the CMB and the large-scale structures indicate the universe is relatively homogeneous on scales in excess of 150 Mpc and up. The fluctuations become weaker the larger the scale. Weak fluctuations imply that the cluster distribution is very close to homogeneous. The larger the scale, the weaker the fluctuation, as homogeneity is approached.

By symmetry considerations (discussed below) we will se that the universe in its entirety is a stable system in dynamic equilibrium.

Mounting Evidence for Gravitationally Bound Superclusters 

Direct evidence that superclusters rotate was obtained by Vauclouleurs (1958), when it was observed a high degree of flattening of the Local Supercluster. The link between flattening and orbital motion had been found by Oort with respect to the Milky Way many years prior to the suggestion by Vaucouleurs that the same principle holds for clusters and superclusters.

Indirect evidence was found by Reinhard and Roberts (1972) when it was observed that the orientation of the minor axis of spiral galaxies correlates with the direction of the poles corresponding to the flattening of the Local Supercluster. Said differently, the angular momentum vectors tend to coincide with the rotational axis of the Supercluster itself (again; Source).

A significant flattening of a large number of other superclusters supports the hypothesis of supercluster rotation, in addition to observed correlated orientations of galaxies along the direction of extension of superclusters, the nonrandom correlation between the main axis of radiosourses with the central supergiant cD-galaxies, and the rotation (and flattening) of clusters within superclusters: for example, the direction of rotation (indicated by the orbital planes of spiral galaxies within clusters) coincides with the apparent rotation of superclusters.

The shape of the Draco supercluster, a very isolated, extremely rich supercluster, resembles a pancake with axis ratios 1:4:5 (Einasto et al 1997 and references therein); precisely what would be expected of a rotating system.

Note, when using the word 'rotation' relative to superclusters (or even clusters), one should not imagine orbits like that of 3-body system (say, the earth-moon orbiting the sun), to be smooth, quasi-circular and of relatively constant velocity. These rotations or motions would be not be isolated.

After all, some are conglomerates roughly 10^15-10^17 solar masses (representing thousands of galaxies). Superclusters are typically separated by 100 Mpc, which means there are about 10 million superclusters in the visible section of the universe. From their size and mass inferences it has been calculated an average free-fall time of 40 billion years. According to the standard model the universe is only 13.7 Gyr, which implies that superclusters are dynamically young. A typical galaxy cluster will not have passed through the system for the first time (A. Jerkstrand, Superclusters).

Superclusters form part of a web-like network of extensive walls, supervoids, clouds and filaments, all interacting gravitationally with their neighbors in a vast and extensive series of local minima and maxima in the combined gravitational fields of an exceedingly complex N-body system (or systems) extending perhaps 100 Mpc or more. Even the regions of galaxy underdensity (supervoids) are nonspherical in nature. To give an idea of the scales we're talking about, consider that the Local Group, a loose aggregate of several dozen galaxies (including the Milky Way), resides within a volume of radius about 1 Mpc. The Local Group itself lies asymmetrically on the outskirts of a flattened structure with a radius of about 15 Mpc, known as the Local Supercluster, centered on a rich cluster of galaxies; the Virgo cluster.

To complicate the matter significantly, all of these systems are embedded in a manifold that is either expanding nonlinearly or of constant positive Gaussian curvature globally. Note the topologically complex structure pictured in the diagram below. No simple characterization describes the distribution of galaxies or superclusters:

Space density and velocity fields derived from the PSCz survey. The amplitude of the velocity vectors is on an arbitrary scale.Source

It can be seen from these simulations that the density and velocity fields of N-body systems are very complex. There are no clear-cut orbits (as that of the earth around the sun). The point is that superclusters can be in motion relative to one another, close to dynamic equilibrium, without possessing a well defined orbital path or velocity relative to neighboring superclusters. Yet these large-scale structures may still be gravitationally bound systems.

The distribution of superclusters, according to Einasto et al (1997) is far from homogeneous. "Most of very rich superclusters are located along rods of a quasi-rectangular cubic lattice with almost constant step, and form elongated structures - chains. These chains are almost parallel to axes of the supergalactic coordinate. [...] Several data sets suggest that giant structures seen in the Southern and Northern sky may be connected, and superclusters form sheets or planes in supergalactic coordinates. One example of such connection is the Supergalactic Plane, which contains the Local Supercluster, the Coma Supercluster, the Pisces-Cetus and the Shapley superclusters [...] This aggregate separates two giant voids - the Northern and Southern Local supervoids." (Einasto et al 1997, The Supercluster-Void Network I, A&A Suppl. Ser. 123, 119-133).

Evidence that superclusters are connected clearly does rule out the possibility that these systems are gravitationally bound. Quite the contrary. These findings are consistent with the large-scale dynamics of a non-expanding universe, whereby all neighboring objects are gravitationally bound (and so too friends of friends). Tully et al (1992) had already described the the supercluster-void network as a 3-dimensional chessboard, due to the presence of superclusters nearly orthogonal to the Supergalactic plane. Einasto et al (1997) find that these structures (delineated by rich superclusters) are not only orthogonal but distributed quite regularly. Too, isolated clusters and poor superclusters are distributed very closely to rich superclusters, belonging to outlying portions of superclusters (i.e., they do not form a random population in voids). Superclusters form intertwined systems that appear regularly spaced. "Thus the mean separation of high-density regions across  voids is almost identical for all observed samples." (Einasto et al 1997) The characteristic separation scale of rich superclusters resides around 120 h^-1 Mpc, corresponding to the distance between superclusters across the supervoids. Evidence that the supercluster-void network is regularly structured is indicated by the small scatter of this characteristic distance. Einasto et al also conclude that there exists no larger preffered scale in the universe, i.e., superclusters and supervoids should be the upper end of the hierarchy of galaxy distribution. Rich superclusters are often found in pairs (e.g., Fornax-Eridanus and the Caelum superclusters, the supercluster in the Aquarius complex and others), a surprising feature, for any expanding universe model. The pairing of gravitationally bound objects is observed across a very wide range of scales, from planets and satellites to binary stars on up to superclusters.

This manifestation of hierarchical (ordered) structure from galactic scales up to the scale of superclusters would indeed be a very surprising feature in a universe blowing apart at the seams, where one would expect a discontinuity for systems not gravitationally bound.

In fact, the ordered structure is observed in simple two- and three-body systems. Arguably, the classic geometric configurations discovered by Lagrange operational on scales compatible with planets and satellites can be observed on scales of the largest known structures. See for example Coldcreation 2008 The Physical Mechanism of Gravity - The Spatiotemporal Ground-State.

Dynamic Equilibrium of the universe and Subsystems, 2008: In Théorie des fonctions analytiques (1796), more than one hundred years before Albert Einstein and Hermann Minkowski, Joseph-Louis Lagrange referred to dynamics as a “four-dimensional geometry.” Without doubt, such illustrations aimed at their audiences, with physical intent, were certainly not geometric in any ordinary sense; but unquestionably they were geometric in their concern with planimetric space of gravitating systems, and in the fundamentals of their relationship with matter. He believed now another basic factor was to be understood and exploited: space. He saw the understanding of space as a legacy left by the invention of Newton. In Lagrange’s system imbued with the notion of dynamic continuity, is stressed the importance of unbroken rhythms and completed movements by circular field lines; now with the spatial factor introduced.

It is difficult to overestimate the importance of the Lagrange discovery. Not only does it confine the number of possible structures that can exist, it demonstrates a regularity and pattern among those systems that do exist. The Lagrange system reveals how interacting gravitational fields of massive bodies generate periodicity, just as the Pauli principle generates the periodicity among atomic elements in the case of electrons orbiting atomic nuclei—as in the Mendeleev periodic table of elements—and the systematic pattern among quark clusters that represent the subsistence of a profound layer of reality on the smallest scales. (The forces that cluster quarks together are not yet fully understood but some of the patterns and features have already been identified). Our ability to recognize that Nature forms regular patterns at all scales (from quarks to superclusters) and limits the number of available structures, rather than giving way to disorganized chaos, is essential if we are to make any progress at all in cosmology.

Limits of the Cosmological Principle

On the scales of superclusters the cosmological principle does not hold. Deviations from pure Hubble expansion are expected, just as deviations from constant Gaussian curvature would be expected in the static model. Nevertheless, for convenience, it can be argued that any observer can consider the universe as homogeneous and isotropic on sufficiently large scales. There is no preferred spatiotemporal location or epoch.

With increasing scale galaxy-clustering and cluster-clustering become weaker. At some scale the correlation function (characterizing the distribution of galaxies) tends to zero. On scales of superclusters, it has been found a series of almost regularly spaced maxima and minima, corresponding to superclusters and supervoids (around 115 h^-1 Mpc). The correlation function of galaxy clusters, on large scales, depends on the geometry of the distribution of superclusters. The correlation function has an oscillatory behavior when superclusters are distributed quasi-regularly. (Einasto et al, 1997, The supercluster-void network III. The correlation function as a geometrical statistic).

Redshift Surveys and the Large-scale Structure

The web-like (or sponge-like) structure of the cluster-void network was discovered when the first 3D maps of the universe were made in the late 1970's based on galaxy redshift surveys. In 1976 Stephen A. Gregory and Laird A. Thompson decided to create a 3D map of a large volume of space in a slice across the sky that stretched 21 degrees from the Coma cluster to the cluster Abell 1367. They hypothesized that if the two galaxy clusters (Coma and Abell 1367) are members of the same supercluster, the 21 degree span between them should be filled with galaxies that bridge the gap between the two clusters. Gregory and Thompson measured the redshifts at Kitt Peak National Observatory, made a 3D map, and were astonished to find that not only was their hypothesis confirmed regarding the "bridge of galaxies connecting Coma and Abell 1367, but more importantly, the galaxy distribution within the entire 21 degree slice of the sky was very filamentary with large empty regions of space throughout the survey volume." (see Gregory S. and Thompson L.). Astrophysical Journal published their work in June 1978. In this manuscript, Gregory and Thompson introduced the word "voids" to describe (for the first time) the large empty regions seen in the 3D redshift map. Based on previous studies, Gerard de Vaucouleurs properly suggested that the spatial distribution of the local galaxy distribution (out to a redshift of about 2000 km/s) was quite irregular. But the true nature of the supercluster-void network wouldn't become clear until large-scale surveys were projected onto a 3D map.

Gregory and Thompson wrote in their 1978 manuscript: "It is an important challenge for any cosmological model to explain the origin of these vast, apparently empty regions of space. There are two possibilities: (1) the regions are truly empty, or (2) the mass in these regions is in some form other than bright galaxies. In the first case, severe constraints will be placed on theories of galaxy formation because it requires a careful (and perhaps impossible) choice of both omega (present mass density/closure density) and the spectrum of initial irregularities in order to grow such large density irregularities. If the second case is correct, then matter might be present in the form of faint galaxies, and an explanation would have to be sought for the peculiar nature of the luminosity function."

Redshift surveys are important in that the measurement of peculiar velocities (and bulk motion) allows the reconstruction of the underlying density field, which can be compared with that derived from the distribution of the visible galaxies. Measurements to date suggest that either (a) there exists a large component of invisible cold dark matter and that superclusters are not gravitationally bound systems (in the expanding case), or that (b) clusters and superclusters are gravitationally bound and that these large-scale structures are moving relative to one another (in the static case).

In the latter scenario, where superclusters are gravitationally bound systems, there should be observed interrelated connections between superclusters (e.g., in the form of luminous bridges of interconnected clusters).

In the past few decades large galaxy redshift surveys have revealed that there are connections between superclusters. In 2005, Proust et al (Structure and dynamics of the Shapley Supercluster, A&A 447, 133–144 (2006), DOI: 10.1051/0004-6361:20052838, ESO 2006) present results of our wide-field redshift survey of galaxies in a 285 square degree region of the Shapley Supercluster (SSC), based on a set of 10,529 velocity measurements (including 1,201 new ones) on 8,632 galaxies obtained from various telescopes and from the literature. The data reveals that the main plane of the SSC extends further than previous estimates, filling the whole extent of our survey region of 12 degrees by 30 degrees on the sky. There is also a connecting structure associated with the slightly nearer Abell 3571 cluster complex. These galaxies seem to link two previously identified sheets of galaxies and establish a connection with a third one. The remarkably rich Shapley Supercluster is one of the most massive agglomerations of galaxies in the local universe. Background structure appear to be linked to the SSC by filaments (see Figure 4 of Proust et al 2006). Too, superclusters SCL146 and SCL266 seem to be associated with radial extensions of the SSC. The Shapley Supercluster is clearly linked to other huge superstructures, as shown in the Figs. 9 and 10 of Jones et al. (2004, MNRAS, 355, 747). Apart from the radial connection to the Hydra-Centaurus complex, a tangent bridge of galaxies extends in the direction of the Sextans supercluster. The inferred mass of the Shapley supercluster is large enough to effect gravitationally the observed motion of the Local Group. Its morphology is generally flat, and extends further than previously estimated, linking the Hydra-Centaurus foreground region (D. Proust, et al, 2006).

The fact that superclusters tend to be grouped together, linked by filaments and walls forming a weaved fabric throughout space, suggests that superclusters are indeed gravitationally bound systems moving relative to one another in the static universe.

Bacall (1990, ASPC, 21, 281B) finds a weak correlation for r=100-150 Mpc, suggesting superclusters may be themselves grouped together with a characteristic separation of this scale.

Superclusters are not completely isolated in vacuo (Einasto et al 1997, The Supercluster-Void Network I, A&A Suppl. Ser. 123, 119-133), separated by voids or supervoids, as could be expected in an expanding universe where superclusters are not gravitationally bound systems. Neighboring superclusters are connected by galaxy and cluster filaments forming a single network (Einasto et al 1997), just as would be expected in a static universe where superclusters are bound gravitationally.

Recently, it was suggested that the relative orientation of neighboring clusters within superclusters reflects an underlying formation mechanism. Numerical simulations have demonstrated that clusters formation is intrinsically connected with supercluster systems that characterize the large-scale structure of the universe. There has been found a correlation between the relative orientation (parallel or filamentary alignment along the major elongation axes) of neighboring clusters and the angular momenta of clusters. Major elongation axes and the lines pointing towards neighboring clusters shows a strong deviation from random orientation (Faltenbacher et al, 2002, Correlations in the orientations of galaxy clusters). Angular momentum of ellipsoidal systems tend to align with minor the axis. Closely neighboring cluster pairs tend to have similar or higher angular momenta compared to clusters further removed (or to the global average).

The observed distribution and dynamics of matter in space may hold the key to understanding the essence of the physical universe and its evolution in time. The large-scale structures and spatial distribution can place important constraints on formation and evolution as function of time. That is, the spatiotemporal distribution of galaxies, groups of galaxies, clusters and superclusters within the bubble-like networks of sheets, filaments, walls, voids and supervoids (along with spectrographic and morphological properties) place important constraints on how these systems formed, and can be used to test the viability of cosmological models, whether static or unstable. Much progress has been made but much has yet to be explored.

The evidence to date relative to the kinematics of the large-scale structures seems to suggest that at least a few (if not all) superclusters are gravitationally bound systems. Observations both direct and indirect do not contradict this possibility. Again, in the static case these large-scale structures (some in excess of 100 Mpc) have had ample time to reach dynamic equilibrium (without the need of cold dark matter).

Interestingly, the CMB measurements imply density fluctuations of 0.001%. According to theory, a characteristic time for the gravitational settling at the center of a clump where the density fluctuation is 1.7% would be 1 billion years. A fluctuation of 0.3% would require 13 Gyr. And for fluctuations consistent with the CMB (0.001%) the settling time would be 1000 times greater than the supposed age of the universe. Only by adding cold dark nonbaryonic matter to enhance the fluctuation can the gross inconsistency be appeased. In 2004 (Large Structure Formation, Comparison of Models) several measurements of galaxies and clusters deep in the look-back time (in the "early" universe) indicate that the structures are larger than predicted by the standard dark-energy cosmology (lambda-CDM). The problem revolves around the inability of a dark-energy dominated universe to create such large structures within such a short time; 1/5 of the present age. (Source).

Further observations (with the next generation of telescopes) may refute the dark-energy model if large-scale structures are observed as far back in time as our telescopes can see (i.e., more research will likely refute the dark energy model. Obviously the prediction here is that superclusters are present at all redshift distances; all the way to the horizon. :)

Time-scales and the large structures

So far, observations appear to show large-scale structuring at High-redshift. For example, high-z quasars and radiosources show clustering above correlation levels of galaxies, which means large-scale structures exist at early times, in sharp contrast with standard model predictions. Strong indications for the existence of both clusters and superclusters at high-z have been known for several decades. Quasar clustering had already been quantified in 1988 (Iovino and Shaver). This was shown to be associated with superclustering by Bacall and Choksi (ASPC, 21, 281B). It was found (West, 1991, ApJ, 379, 19W) that radio galaxies, in addition to clustering, show spin axis alignment pointing in the direction of nearby radio galaxies and quasars. Pencil-beam surveys at intermediate redshift (z < 2) indicate that superclustering is not a recent development. At redshift z = 2.38 a 100 Mpc long string of galaxies was observed, in sharp contrast, too, with time-scales considered in hierarchical computer simulations. Observations of large-scale structure at very high-z is not compatible with the time-scales typically required by ΛCDM model. The universe at early times is simply not old enough to account for observations (A. Jerkstrand). This insufficient time-scale would be a problem for both the hierarchical and monolithic models of formation.

Large-scale structures are potentially very old. Tully (1886) suggests that the large dimensions of these objects (or groups of objects), some possibly in excess of 300 Mpc, must have taken a hundred billion years to form. Tully finds that the entire Local Supercluster conglomeration has an elongated dimension of at least 386 Mpc (for H0 = 70 km s−1 Mpc−1. A similarly large dimension is also found for the sheet-like configuration attached to an agglomeration of superclusters in the region of Aquarius (Tully, 1986 Astrophys. J. 303, 25).

These elongated cellular walls formed by supercluster complexes are perhaps indeed ancient, as suggested too by Lerner (1991). His conclusion is that these cellular walls are at least one hundred billion years old. (Lerner, E.J., The Big Bang Never Happened). (See too Hoyle, F., Burbidge, G., Narlikar, J.V., 1997, Mon. Not. R. Astron. Soc. 286, 173).

All high redshift surveys indicate that large-scale structures are present at great distances in the look-back time. These structures either formed exceedingly rapidly after the big bang (even though expansion dilutes density fluctuations, thereby the ability to cluster), or they've been around for much longer than currently suspected. No doubt, I'm banking on the latter. Forthcoming high-z data in the next decade should rule out one model or the other.

The existence of rotation (or bulk motion) on scales compatible with superclusters cannot be ignored by any model of structure formation. Indeed, data regarding the dynamical age of the Local Supercluster (and others) present serious difficulties for the standard model, with its hierarchical model of galaxy formation from primeval density fluctuations (a 'bottom-up' scheme). In fact, the existence of the observed structures, from their size alone, seems to rule out the standard model, i.e., their hasn't been enough time since "creation" for such large-scale structures to have formed (gravitationally bound systems or not) from minute density clumps in the Radiation Era which would have acted as seeds for the growth of galaxies early in the Matter Era.

Clearly, the large-scale structures including the supervoids (not just locally but at great distances in the look-back time) are far more advanced than predicted by the concordance model (ΛCDM).

Cosmic Structure Formation: Hierarchical and monolithic collapse vs the top-down model in a static universe

One of the beauties of the static model of constant Gaussian curvature, contrarily to the standard hierarchical model or the picture of monolithic collapse, is that the formation of the large-scale structures and galaxies themselves occurs over very long time periods and can do from the top-down. In other words, the density fluctuations from which individual galaxies and clusters are formed cover vast regions of space (perhaps 100 Mpc or more). These density fluctuations are on scales of superclusters. The over-dense regions would splinter into smaller substructures consistent with cluster scales, and subsequently individual galaxies would form with well defined spatial orientations relative to neighboring galaxies, the clusters themselves, and  the parent superclusters. The location, orientation, morphology, and dynamics of galaxies (from the early formation process onwards) are all related to the topological shape and size of the supercluster denstity fluctuation, as well as the relative proximity of neighboring superclusters, which can affect all of the above properties. The fact that (i) planes of galaxies tend to be lined up perpendicularly with the primary plane of the parent supercluster, and that (ii) projections of rotational axes on the primary plane of superclusters are oriented toward the primary structure, and that both the above can be interpreted as perpendicularity of the galaxies plane to the radius vector, lend support to the top-down scenario; where supercluster density fluctuations are formed prior to those of galaxies (P. Flin, 1996, The Alignment of Galaxies in Superclusters, Astronomical & Astrophysical Transactions, 10:2, 153 - 159). Certainly there is little evidence, if any, that would suggest otherwise.

The point is that density fluctuations occur on a variety of scales. The smaller fluctuations that lead to the formation or stars and planets and satellites, star clusters, galaxies and galaxy clusters, are superimposed on the largest fluctuations, i.e., these smaller fluctuations form part of the larger ones from the outset. Superclusters are relatively old objects from which galaxies and clusters are formed.

Resulting from shocks generated by supersonic motion and adiabatic compression during virialization and shell collapse a gas (in the form predominantly of hydrogen) permeating the superclusters gravitational potential well is heated and ionized and emits through bremsstrahlung in the X-ray band, consistent with observations. Baryonic and leptonic matter represents 100% of the mass content of the universe; much more in the form of baryonic dark matter than currently suggested.

[Note: It's worth pointing out that it is precisely this combined total mass-energy density of the universe to which is attributed the globally homogeneous gravitational field of constant positive Gaussian curvature].

Coherent orientation effects from the scale of superclusters to that of galaxies support the contention that superclusters are gravitationally bound systems within which clusters, smaller groups (such as the Local Group) and individual galaxies form via the top-down scenario.

Certainly there are characteristic features that will overlap between the two scenarios (the top down and the bottom up). That is because primordial density fluctuations come is all sizes up to a scale beyond which the universe tends toward homogeneity and isotropy. Within large overdense regions (compatible with the scales of 150-200 Mpc or more) of the underlying matter field at very early times there are, too, overdense regions on scales all the way down to individual atoms.  Gravitational influences due to the large-scale fluctuations of the matter field impel individual atom to move closer together, thus overdensity increases across a wide range of scales as a function of time. This accretion process responsible for the formation material structures is simultaneously responsible for the formation of voids. As some areas of the primordial substratum condense under gravitational influences, other areas become underdense. The less dense these regions become, the less accretion will occur. Eventually, most of the material in these regions will have migrated gravitationally towards denser areas. According to this scenario of gravitational instability for the evolution of the observed cosmic structures the universe (perhaps hundreds or thousands of billion years ago) was practically smooth, except for spatially extensive density variations (which were small in amplitude) with respect to the overall background. The current size and mass of superclusters observed today is directly related to the scale of the fluctuations. The scales and variations in amplitude of the observed thermal CMB blackbody spectrum would mimic the large-scale distribution of these density fluctuations. The CMB would be (not a relic of a hot dense phase) a product of baryonic matter and its spatial distribution in time. The thermalization of the blackbody spectrum in a static universe will be discussed in a separate dedicated post, as well as the formation of hydrogen and the other elements.

It is also predicted by some hierarchical models (Plionis, 1994) the alignment of galaxy clusters with their closest companions (since these are gravitationally bound) as well as with other clusters that reside within the same supersluster. Alignment is suspected to occur as a result of a property of Gaussian random fields which interact between density fluctuations of various scales (West et al 1991, Faltenbacher et al 2002, and others). Flatter systems tend to show lower velocity dispersions (the range of velocities about the mean velocity). However the dynamical evolution of clusters and the intergalactic medium, along with post-merging relaxation time corresponding to clusters remain open issues, i.e., the existence of substructure does not imply that clusters are dynamically young (Plionis 2001, and references given therein).

There are primary difference between the two views on the origin of cosmic structures: The Coldcreation top-down view is that superclusters are not young cosmic objects that formed after the merging of smaller systems (cannibalism), whereby larger systems form from smaller ones, and the baryonic matter does not follow some other dominant specie of matter (CDM) during the collapse phase. In other words, density perturbations on stellar and galactic scale do not collapse to form stars and galaxies first, then subsequently merge to form galaxy systems followed by the large-scale filamentary superstructures. Nor do galaxies form by monolithic collapse in a relatively recent given epoch in cosmic history (the "epoch of galaxy formation" just after the "dark age"). The process is long, drawn-out and ongoing.

Star formation is thought to be triggered primarily by violent merging events, where by the gravitational fields of dense cluster regions vary rapidly. Active galaxies usually reside in high-density environments, so merging may not be the sole contribution to star formation activity. An interesting feature is that increased cluster sphericity appears to be associated with rich massive systems where the velocity dispersion is large. The increase in cluster velocity dispersion is larger for systems that are further removed from one another (Ragone et al 2004). In analogy to galaxy formation from larger density fluctuations associated with protoclusters, individual stars are formed in regions of larger density fluctuations associated with protogalaxies. So too are objects such as planets formed from larger fluctuations consistent, in this case, with the scale of the protoplanetary disc itself, and fragmentations thereof.

It should be noted that a perfectly smooth homogeneous background was never an option. The very early universe, whether expanding or not, whether infinite or not, whether formation of the structures occurred from the top-down or the bottom-up, could not have been perfectly homogeneous and isotropic. This would have been so from the large-scale all the way up to the smallest. It could not be said that that the universe was inhomogeneous on one scale and not the other. So what we really have is both a top-down and a bottom-up formation process which occurs relatively simultaneously. The important thing to recall is that gravitational interactions are not instantaneous. It takes a relatively long time for a large-scale fluctuations to produce significant contraction on scales compatible with components that would form stars and galaxies in a static universe. But it would take a lot longer to produce the observed large-scale structures (from the bottom-up or top-down) in an expanding universe.

Notice the evolution pictured in the simulation below:

Figure FCLSF

The formation of clusters and large-scale filaments in the Cold Dark Matter model with dark energy. The frames show the evolution of structures in a 43 million parsecs (or 140 million light years) box from redshift of 30 to the present epoch, corresponding to redshift in accord with the Hubble law (upper left z=30 to lower right z=0). Source.

And here is the first frame modified by Coldcreation:

I modified the first frame because it showed a homogeneous distribution of hot-spots. Notice in the original frame (top left) the color of the background is identical to the color of condensed areas of the other frames. In a static universe these hot-spots, or condensed areas are not cold dark matter, but galaxies, clusters and superclusters. What we see now in the first frame, with its modification, is the absence of hot-spots; there are no stars or galaxies yet. But what we see too is that the background is not perfectly smooth. There are inhomogeneities (over- and underdense regions) on all scales. The large-scale inhomogeneities are already present. The same density fluctuations present in the first modified frame are also present in the last (something that would not be observed in an expanding hierarchical model where galaxies merge to form clusters and subsequently superclusters). These large-scale fluctuations will determine where and how galaxies are located and oriented relative to others. The background in the modified frame consists of a very cool hydrogen gas (the origin of which will be discussed in a subsequent post). With time, the overdense regions become denser, and the underdense regions less dense. As the process of accretion continues with time the structures move closer toward dynamic equilibrium. Though just as a perfectly smooth homogeneous background is unattainable in the past, so too is a perfect dynamic equilibrium unattainable in the future. [Note, this simulation was performed with dark matter in mind, but one can just as well picture these structures as being composed of ordinary baryonic matter.]

The above simulation (Source) confirms the scenario presented here. The original large-scale density fluctuations within which galaxies are embedded at later times are present from the outset. The change in size of these fluctuations over time is negligible (there is no expansion in this case), but the density within both the voids and condensing regions does change slowly with time, relative to the mean density. Galaxies are formed within the density fluctuations present at very large physical scales, in contrast to the hierarchical model.

Top down model of galaxy formation in a static universe

The top-down scenario proposed here is also in contrast to the monolithic collapse saga (a kind of wham-bam thank you ma'am approach) since the latter postulates that all galaxies were formed by gravitational collapse of a primordial gas cloud in a single event shortly (in relative terms) after the big bang. This is far more inappropriate than casual serendipitous tidal encountersram pressure stripping or galaxy harassment.

Large galaxies located at the cluster core need not form by merging or cannibalistic processes (pathological colliding galaxies). The location, size and densities of large active galaxies (along with the rate at which they will evolve) depends upon favorable conditions present within the large-scale Gaussian density fluctuations. These are not linearly uniform spherically symmetric fluctuations that cause the free-fall of galaxies towards the central massive dominant galaxy of a given cluster. Nor do clusters ultimately collapse for similar reasons. Just as the rotational characteristics of spiral and elliptical galaxies differ, so too the rotational characteristics of clusters and superclusters differ. While some clusters and supercluster exhibit organized rotational structure, others will be dominated by random motion. The stability of superclusters relative to themselves and adjacent structures is not put into question, since their motions (orbital or random), implied by morphologically flattened structure, would be sufficient for the maintenance of stability. The typical dispersion velocity is consistently fitted by a Gaussian distribution for systems in dynamic equilibrium. For poor clusters (20-30 members) the velocity dispersion is on the order of 500 kms^-1, while for rich systems, such as the Virgo cluster (about 2500 galaxies), the typical velocity distribution is about 1000 kms^-1 (Popesso, 2006, The RASS-SDSS Galaxy Cluster Survey. Correlating X-ray and optical properties of Galaxy Clusters).

So far, all observational attempts to distinguish between the two competing big bang models for the formation of galaxies and the large-scale structures have failed. The question of how galaxies form and evolve with time thus remains one of the most important unanswered questions of contemporary astrophysics (Popesso 2006).

Because superclusters are often large (up to 10% of the horizon scale) they are extremely difficult to study statistically. The answers to the questions of how superclusters form, how their morphology evolves, multiplicity (richness), sizes and structures evolve as a function of time, are not yet known (Wray and Bahcall, et al, 2006). The lack of statistical data regarding these large-scale objects makes it difficult to determine whether they are gravitationally bound systems or not. Increasing the linking length in numerical simulations often causes neighboring superclusters to be connected (joined into one), resulting in many large, complex structures with central cores joining multiple lower density filaments. Long linking length find higher percentages of of large rich superclusters. Conversely, shorter linking lengths show small superclusters with low multiplicity. At early times, in accord with the hierarchical model (wherein clusters migrate toward one another gravitationally over time) there is a reduction in supercluster density. With increasing redshift there is a significant drop in the simulated number density of superclusters. Abundance of rich superclusters drops off very quickly. (Wray and Bahcall 2006). The problem, at the telescope, is that the resolution of high-z clusters drops off with distance, and the identification of these faint distant clusters as members of superclusters becomes more and more difficult with increasing z, until they become unidentifiable (very faint galaxies cannot be observed). Massive clusters are more easily identified in high-z surveys, yet they will appear less rich with increasing z. Generally thin filamentary superstructures will tend to be undetectable too with higher z. This drop-off with distance can easily mimic significant evolution in the look-back time. One should not be misled in this respect when statistical uncertainties are so large.

In short, it seems rather natural, given sufficient time scales, that large-scale fluctuations lead to the formation of small-scale structures such as galaxies. Density fluctuations of characteristic scale between 100 - 200 h^-1 Mpc are related to superclusters and supervoids. Short density fluctuations of wavelength consistent with several Mpc give rise to the formation of small galaxy systems and individual galaxies, interspersed by small voids. Intermediate scale density fluctuations give rise to galaxy clusters and voids. Perturbations larger than superclcusters and supervoids (if indeed they exist, and they likely do) are of much lower amplitude, and therefor only modulate densities and masses of smaller gravitating systems (Frisch et al 1995, A&A 296, 611). Thus, as scale increases, the amplitude of the perturbation decreases. Taking this to the extreme, we could conjecture: as scale tends to infinity, the fluctuation amplitude(s) tends to zero.

Properties derived from large-scale high resolution cosmological simulations (alternatively with static coordinates in place of comoving coordinates) and more sophisticated high resolution observations to be carried out in the coming years should provide direct information on the physical size, richness, morphology and evolution of superclusters, along with testable predictions that can distinguish between static and expanding models; between models that describe these systems as dynamically and gravitationally bound conglomerations and models which do not.

Just as one would expect observational evidence to support the conclusion that superclusters are gravitationally bound systems in a static universe, one would expect supervoids and voids to form hierarchical systems resembling the hierarchy formed by superclsuters, galaxy clusters and galaxies. We would also expect, in a static universe, an upper limit of this void hierarchy to be compatible with scales of the largest material structures (high-density regions, or supervoids).

Voids and Supervoids

What is the role played by voids and supervoids in the cosmic dance? How can the structure and dynamics of supervoids be used to distinguish between competing cosmologies?

Supervoids are large underdense voids (with diameters of about 100 h^-1 Mpc) interspersed with superclusters, defined as low-density regions of the universe that do not contain rich galaxy clusters, though they can contain poor clusters and small galaxy systems. They are completely devoid of certain types of objects, e.g., galaxies of a specified morphological type or luminosity limit, or rich or poor galaxy clusters. These are the largest known voids in the universe, and they are connected in a regular chess board-like structural network (Linder et al 1997). In other words, supervoids are not necessarily completely empty, nor are they isolated. The size and properties of supervoids, along with smaller lowdensity regions (voids) and void-walls are related, i.e., voids form hierarchical systems that depend on the large-scale environment, void diameter and the luminosity of galaxies contained within. The presence of Multi-branching systems in the galaxy distribution appear to be the principle mechanism that creates the hierarchy in the void and supervoid distribution (see Linder et al 1995, The Structure of Supervoids, Astron. Astrophys. 301, 329-347).

Recall that de Vaucouleurs (1970) had advocated the presence of a hierarchy of galactic systems. That voids, too, form hierarchical structures has been confirmed since. Supervoids may be divided into a network of faint galaxy systems and smaller voids (3 - 10 time smaller than supervoides, which may exceed 100 Mpc). Comparing structural differences (if any) of galaxies found in high- and lowdensity regions may provide insight on galaxy formation and the dependence on environment. The walls interspersed between superclusters consist of numerous smaller poorer galaxy systems that appear structured, forming a thin web of filaments. These are not large structureless clouds of galaxies, or a smooth distribution of isolated galaxies. The size, morphology and luminosity of galaxies (from poor systems to rich superclusters) are related to the environmental properties of the system within which they reside (Linder et al 1995).

Note that hierarchical structuring is not confirmation of the hierarchical model (the bottom-up scenario) of galaxy formation. It simply means that populations of clusters, voids, superclusters and supervoids are not randomly distributed. For both high- and lowdensity regions there is observed the presence of a fine structured network of system consisting alternatively of both poor and rich clusters. The void-filament structure is observed on a wide variety of physical scales, up to at least 130 Mpc. Voids located in lowdensity environments of superclusters are larger than voids located in highdensity environments. Any realistic galaxy formation scenario must explain these observed properties (Linder et al 1995).

One of the current views is that while overdense regions of superclusters continue to collapse, the underdense regions interspersed amongst them, supervoids, continue to grow, becoming increasingly underdense and larger with time. These superclusters and supervoids are, respectively, undergoing collapse and expansion. One consequence of this interpretation is that superclusters cannot be gravitationally bound systems; a property that distinguishes them for their constituent cluster counterparts. The physical scale of voids that separate these constituent clusters remains constant in time (since cluster are bound gravitationally), or changes insignificantly.

The empirical evidence is consistent with the scenario that large-scale (primordial) density fluctuations (maxima and minima on scales of superclusters and supervoids) splinter in course of time to form clusters and subvoids at later epochs, i.e., there was less substructure at early epochs than observed tonight. The existence of (i) hierarchical structure on scales compatible with superclusters and supervoids, (ii) the morphology and physical distribution of objects within the supercluster-supervoid network, and (iii) the existence of faint structures inside supervoids, supports the conclusion that superclusters are gravitationally bound systems in a non-expanding universe, i.e., observations are consistent with the dynamics and evolution of structures expected in a static universe.

The average physical scale of large underdense supervoid regions is about 100 h^1 Mpc; a scale above which the universe approaches homogeneity (Granett et al 2008).

The distribution of these large-scale structures affects the cosmic microwave background (CMB) in both expanding models and static models, by heating or cooling photons as they travel through over-and underdense regions; a phenomenon described as a late-time integrated Sachs-Wolfe effect (ISW). The effect on the CMB is small variations (anisotropies) in temperature between regions of corresponding to 100 Mpc. The consensus in the relevant literature is that the presence of dark energy (DE) in a geometrically flat ΛCDM universe induces the ISW effect on the CMB via the decay of gravitational potentials as the DE dominated universe expands and accelerates.

It's of particular interest to note that the presence of hot spots and cold spots in the CMB correspond to regions of over- and underdensity (superclusters and supervoids). This is exactly what would be predicted in a static universe where the CMB is produced by hydrogen burning stars over time-scales that dwarf the suspected age of the universe. (The physical mechanism operating that thermalizes the radiation released through hydrogen burning will be discussed in a forthcoming post). The reason is straightforward: in vast underdense regions there are fewer stars emitting radiation, thus the thermal spectrum of these areas appears cooler, and visa versa for the dense regions.

This is exactly what would be predicted, too, in a static universe where the top-down scenario for galaxy formation is operational. The large-scale structures form from large-scale density fluctuations which determine the morphology, orientation and location of galaxies and clusters.

Detecting the ISW effect in the framework of a static universe of constant positive Gaussian curvature has nothing to do with dark energy. By definition, hot and cold spots are present throughout the cosmos in overdense and underdense regions respectively, though observing such anisotropies at high-redshift would remain a daunting task. Low- to moderate-redshift superclusters and supervoids would dominate the spectrum, explaining most of the thermal fluctuations observed on the CMB. Foreground objects skew background fluctuations.

The larger the light-crossing time the greater the deviation from smoothness. On scales of clusters and voids such an effect would be present but smaller. Photons passing though regions of high and low gravitational potentials (through deep gravitational wells and areas of negligible local curvature) become slightly hotter or colder. The larger the system and the deeper the well, the greater the observed correlation between galaxy catalogs and the CMB anisotropies. Local effects should be extracted in order to reveal the imprints of individual superclusters and supervoids at significant cosmological distances. Without local signal extraction (or averaging) the CMB fluctuations will appear larger than superclusters and supervoids.

Figure 1. A map of the microwave sky over the SDSS area (diagram from Dark Energy with Supervoids and Superclusters Granett et al 2008).

Einasto et al 1994 suggested supercluster and voids [supervoids] form a rather regular network characterized by peak distance separations 110 - 140 h^-1 Mpc. (See too Einasto et al 1997, The Supercluster-Void Network I, A&A Suppl. Ser. 123, 119-133).

For distances that are small in comparison to the characteristic scale of a given density fluctuation, with respect to a globally homogeneous manifold of constant Gaussian curvature, gravitational phenomenon are virtually indistinguishable from those in Minkowski spacetime. The properties of the spacetime manifold, globally, are homogeneous; meaning that (i) there are no privileged points in space or time, (ii) there are no privileged directions, and (iii) there are no privileged inertial frames. There is a generalized equivalence of all arbitrary frames of reference relative to the background spacetime. The background spacetime is a Riemannian (or semi-Riemannian) manifold of quadratic differential form whose symmetry group is the group of arbitrary transformations (it is thus not the absolute space of Newton), it is an Einstein manifold, where gravity is represented by its curvature and its metric tensor is governed by the Einstein field equations.

Material structures present in the manifold are thus not dynamically affected by the geometric properties intrinsic to the manifold of constant Gaussian curvature. I.e., the dynamical character of the metric are irrelevant. All objects, and groups of objects (galaxies, clusters, superclusters), since they are all considered at the origin of the manifold where the magnitude of curvature asymptotically approaches zero (and attain it), are freely-floating relative to the homogeneous background curvature. The globally homogeneous gravity field is everywhere locally Minkowskian. All objects can be, and must be, considered at the origin of a polar coordinate system. In passing, all clocks tick at a slower rate, relative to clocks at the origin, wherever the observer is located, and do so at an increasingly slower rate with increasing distance. (Cosmological redshift z, an affect on clocks and light signals, is a visual embodiment of this global spacetime curvature; which results from the total mass-energy density of the universe).

However, locally, spacetime has a nontrivial topology, i.e., it is not asymptotically flat. The metric is governed by the locally anisotropic inhomogeneous matter distribution through the gravitational field equations. Therefor the curvature of the metric is altered in the vicinity of massive objects. The metrical spacetime structure acts on the matter distribution, and the matter distribution acts back on the metric structure. Though gravity technically has an infinite range, the gravitational field of, say, the Sun, extends slightly beyond the solar system (around one parsec). For the Galaxy this 'limit' is relatively closer, extending to the outer reaches of the Local Supercluster (approximately 100 Mpc). Beyond this range the gravitational field is dominated by other superclusters (see Sokolowski 2008, Stability of a Metric f(R) Gravity Theory Implies the Newtonian Limit).

Re-Redshift z:

The origin of the observed redshift z in the spectra of distant objects is attributed to the fact that photons propagate along geodesic paths (great circle arc equivalents in reduced dimension) in a non-expanding, non-contracting, four-dimensional spacetime continuum. These paths are the shortest distance between two points (from any source, to any observer). These paths appear straight along the line of sight, but there is a constant distortion that increases with distance and look-back time, just as the curvature of the earth increases with distance (e.g., the distortion along a great arc from LA to Paris is larger than the distortion between LA and NY, along a great circle arc). The distortion, or curvature we are talking about begins to manifest itself as redshift z at the outer limits of the traditional boundary of the Local Group, beyond which redshift increases with distance proportionally with the constant positive curvature of the manifold.

Figure ESU

Figure ESU represents an equatorial slice of the visible universe (on an oblique angle) with the observer centered at the origin of the past light cone. A slice through any point of origin, and at any angle will be identical. All distances are in look-back time from the observers view-point. Each concentric circle around O represents about 2 Gly. This universe has a spherical topology, where light is redshifted due to the propagation of photons along (great circle arcs in reduced dimension) geodesic paths (shown here schematically as straight lines converging towards the origin). Stability is maintained locally due to motion. Massive objects move along local geodesic paths determined by spacetime curvature in the local vicinity of the objects under consideration, just as the earth remains stable against gravitational collapse into the sun. (Local curvature is not shown in this diagram: see Figure PRM-LSM for an example of local curvature, inhomogeneities or deviations in a negatively curved spacetime). Stability is maintained globally because all points in the Riemannian manifold of constant positive curvature are equivalent.

Massive objects remain unaffected by the non-Euclidean geometry of the background field since the origin (anywhere in the universe) is effectually flat. There is no motion or acceleration imparted on massive objects resulting for the topology of the spacetime continuum, again, since all points and directions are the same in the globally homogeneous field of constant positive Gaussian curvature. The radius of curvature, the scale factor or the size of the universe does not change with time (i.e., there is no expansion or collapse) since the mass-energy density remains constant, i.e., the conserved quantity ensures a constant radius of curvature over time (excluding for now evolution in the look-back time). This equilibrium is not of the unstable kind, e.g., a pencil balancing on it's point, or a roller coaster at the summit of the track poised to steal the world speed record. To elaborate on the latter analogy, this equilibrium is stable since the roller coaster track is as if curved around the surface of the world (on what amounts to a great circle arc in reduced dimensions): the coaster is essentially on a flat surface relative to the geodesic arc. Hills and valleys that induce acceleration are only present locally (inhomogeneities in the local spacetime surrounding massive objects). Note, we still need to discuss and solve the problem of diverging integrals of the Newtonian gravitational potential.

Notice too in Figure ESU light propagates in what appears to be straight lines in the Euclidean sense. Curvature no longer appears as it does on the surface of a sphere, where light follows the curve of the surface along great circle arcs. Now, in a four-dimensional relativistic spacetime, the photon paths are essentially straight geodesic lines (excluding local gravitational effects, such as lensing and deflection in the vicinity of massive bodies). The curvature, or distortion, occurs along the path itself. Though the path is straight from the viewpoint of the observer, there is a distortion, plainly visible in the schematic diagram above. And the spatiotemporal distortion becomes increasingly apparent the further the distance considered (increasingly with look-back time). This is exemplified by the cross-sections of the spherical shells centered on the observer, which appear to become closer together with distance. The volume of this positively curved universe appears smaller than those of its Euclidean or hyperbolic counterparts.

Recall that in reduced dimension, i.e., on the surface of a sphere, light propagates along the very same geodesic lines towards the origin (centered on the any observer, always located at what would look like a North or South pole: compare with Figure PRMCC). The distortion in the path is the cause of cosmological redshift z in a static Einstein universe. There is a loss of energy associated with increasing distance of propagation from the observer in the non-Euclidean manifold: the result is redshift z and time dilation. This is exactly what would be observed from the rest-frame of any observer located anywhere in the Einstein static four-dimensional manifold of constant positive intrinsic Gaussian curvature.

The fact that redshift occurs in such a universe appears intuitively far less problematic than the maintenance of equilibrium globally in such a universe. Locally there seems to be no problem, since objects such as the solar system, the Galaxy, the Local Group and so on, are observed to be relatively stable systems due to a process of 'environmental selection' (objects of orbital velocity superior to, or less than required in order for stability to be maintained either disperse of gravitationally collapse). Fortunately for us many gravitationally bound systems manage to survive.

So the problem seems to be with regard to global stability (intuitively): How can the universe itself, if curved positively, negatively, or curved not at all, remain stable in the face of gravity, an attractive force a la Newton, or a curved spacetime phenomenon a la Einstein that causes objects to merge along geodesic paths? Well, we've already seen how gravitationally bound systems remain locally stable. Notice, here we assume galaxy superclusters are gravitationally bound systems as well. So what about the universe itself? Isn't a non-expanding universe a gravitationally bound system subject to the same laws and principles as local systems, meaning that the universe too should eventually gravitationally collapse into one great massive fireball?

Yes to the former, and no to the latter. The universe is an agglomeration of bounded gravitating systems and as such can be treated as a gravitationally bound system unto itself. Yes too, the universe itself is subject to the same laws and principles as local systems (local physics is global physics) and it is precisely for this reason that the universe itself remains free of global instability. And no, the universe doesn't collapses due to the globally curved spacetime.

Globally, equilibrium is maintained as mentioned previously not just because its parts are in motion, thus avoiding gravitational collapse up to scales consistent with groups of superclusters, great walls, and supervoids, but because on larger scales still, compatible with the visible universe (and beyond) the curvature of the universe itself plays a fundamental role in the maintenance of equilibrium. Locally, the universe is inhomogeneous. Objects move geodesically relative to gravitational fields of neighboring massive objects. Globally, the universe is homogeneous, and the larger the scale considered, the more homogeneous it becomes. The gravitational field that permeates the entire spacetime continuum is homogeneous and isotropic, while locally the fields of individual objects and clusters of objects remains inhomogeneous. This difference is key to understanding the essence of the physical universe and its evolution in time.

A homogeneous field of this type is the same at all points. All points are equivalent in a gravitational field of constant curvature. There is no "inward" direction towards which all material objects will collapse. There is no global geodesic path upon which all objects will gravitate, as for photons. Curvature of the manifold vanishes locally (i.e., on scale compatible with galaxy clusters the topology is virtually flat) just as curvature vanishes locally on the surface of the earth.

Figure SUCGC

Static general relativistic Einstein universe of constant positive Gaussian curvature. This universe is non-expanding. Redshift z is a curved spacetime effect. This is a reduced dimension schematic diagram. Galaxies, clusters and superclusters reside on the surface of the reduced dimension sphere. The actual structure of the spacetime continuum is positively curved in four dimensions: 3 spatial and 1 temporal.

While locally, a test particle relative to the inhomogeneous gravitational field of a nearby object will freely-fall, or accelerate towards the gravitating object, globally this does not happen. There is no free-fall or acceleration is any particular direction relative to a homogeneous gravitational field of constant curvature. And since massive objects do not follow the same geodesic path as photons (the equivalent of great circle arcs on a reduced dimension spherical manifold of constant positive Gaussian curvature) all objects do not impart on a journey towards one another leading to a big crunch.

In another way, each supercluster is embedded inside what amounts to a four-dimensional Minkowski spacetime relative to the globally homogeneous field of constant positive curvature. Essentially, curvature vanishes locally and does so everywhere (again, just as curvature vanishes on the surface of the earth locally, at all locations). Interestingly, this would mean that the geometric structure of universe (e.g., a homogeneous gravitational field of constant positive Gaussian curvature) imparts no acceleration in any direction whatsoever on any object in the manifold. Massive objects are freely-floating relative to the background topology. Notice here that Ernst Fischer's "tension" (the physical embodiment of lambda) is not present in the discourse. We'll come back to this shortly.

It is of interest to note too that space tells matter how to move, and matter tells space how to curve (Misner et al, 1973, p. 5) in this model. Or, more precisely, locally inhomogeneous gravitational fields impel matter to move, while the globally homogeneous field of constant curvature does not. And matter tells space how to curve both locally and globally.

The universe remains stable against gross expansion or wholesale deflation.

In sum, cosmological redshift z in a static universe is caused by the geodesic trajectory of photons in a curved spacetime. Generally speaking, the large-scale geometric structure of the spacetime continuum distorts the wavelength of every photon in direct proportion to the curvature of space in the elapsed time interval. Redshift is interpreted as a function of both distance and time. According to any observer located at the origin O (the rest-frame of an observer) in a static curved spacetime continuum, the distance that separates the source and O will not equal the proper distance (or actual distance) between the emitting source and O. The wavelength of each photon is lengthened (as if 'stretched') between emission and reception. In another way, the energy of every photon is reduced, degraded, in direct proportion with the spatial distance traveled and the amount of time it takes for the photon to arrive at the observer.

In this static model, redshift occurs not because galaxies a moving away from the observer, nor because of the expansion of space itself, but because of the curvature of spacetime in accord with Einstein's geometric interpretation of gravitation. The curvature of this Einstein manifold is directly related to the gravitating mass-energy density contained in the universe.

Olbers' paradox is solved in such a non-Euclidean universe because of the relation between energy-loss and distance. The greater the distance, the greater the energy-loss associated with the lengthening of the photon wave packet. The energy carried per photon upon arrival is always less than the energy upon emission. What seemed before a profound cosmological statement, and method of determining whether the universe was static or expanding, is actually a trivial observation. The fact that the night-sky is dark was used as a viable argument (by Hawking and others) for ruling out all static universe models. But in reality, the only static model that failed to resolve the paradox would be geometrically flat (k = 0).

Converging Integrals in a Homogeneous Universe

Structurally similar to Olber's paradox (the darkness of the night sky) is a problem that had affected the cosmology of both Newton and Einstein.

Recall briefly the problem: The gravitational force exerted on a test body in a Newtonian universe (a linear theory of gravitation) is the resultant of the forces exerted by the totality of the masses present in the universe, which are assumed to be homogeneously distributed. The force is computed by an integration over all masses. The integration does not converge. Any value may be obtained depending on how the limit of integration over all of space is approached. (Norton 1999, The Cosmological Woes of Newtonian Gravitation Theory).

In other words, if an infinite 3-dimensional Euclidean universe is filled with a uniform and isotropic distribution of mass the net gravitational force on a test mass located at any point can take on any nominated magnitude and direction. (See also Norton 2002, A Paradox in Newtonian Gravitation Theory II). In laymen's term a homogeneous Newtonian universe would be gravitationally unstable. I.e., Based on diverging potential and the Poisson equation, no Newtonian cosmology with a homogeneous distribution of matter is possible without some non-local extension of Newtonian theory.

Although this problem was known by Newton (as shown in the exchange of letters with Bentley) it was not clearly stated until the late 19th Century by Seeliger. Einstein tackled the problem within the context of his own theory of general relativity but the  inconsistency had not yet disappeared. Milne and McCrea (1934) discovered that relativistic cosmologies gave results similar to that of Newton.

One way of dealing with the problem, historically, had been to simply ignore it. This had arguably been Newton's tactic. It had also been this author's tactic, to some extent, just as Olbers' paradox could be ignored in the context of a static non-Euclidean universe. But the problem continued to raise it's ugly head and so had to be addressed a new.

According to Norton (2002) there are three types of responses by theorists to the inconsistency of Newtonian cosmology: (1) They are unaware of the inconsistency and derive their results without impediment. (2) They are aware of the problem but ignore the possibility of deriving results that contradict those that seem appropriate. (3) They find the inconsistency unacceptable and attempt to modify assumptions of the theory in order to restore consistency.

The latter had been the most common approach. "At one time or another, virtually every supposition of Newtonian cosmology has been a candidate for modification in the efforts to eliminate the inconsistency. These candidates include Newton's law of gravitation, the uniformity of the matter distribution, the geometry of space and the kinematics of Newton's space and time itself." (Norton 2002)

An expanding universe solves the problem (albeit not of instability), just as it solves Olbers' paradox, but it is not the only solution.

A spherical mass distribution of arbitrarily large yet finite size (in an otherwise empty infinite universe) solves the problem by avoiding the complications of the infinite mass case.

As part of an ambitious analysis of Newtonian gravitation in non-Euclidean spaces, Josef Lense (1917) postulated a closed spherical-eliptical spatial geometry with a uniform (homogeneous) mass distribution, corresponding to a finite total mass. Convergent integrals resulted from Lense's work, which gives constant values for the potential and tidal forces, and a vanishing gravitational force for the first dependence. The second law gives a constant value for the potential and vanishing gravitational and tidal forces. (see Norton 1999, section 10, pp. 314-315)

Similarly, Ernst Fischer solves the problem by postulating a finite homogeneous mass distribution in a universe of constant positive curvature.

In the case of homogeneous, isotropic cosmologies where the mass distribution is non-expanding, symmetry considerations would require the vanishing of the net gravitational force on a test particle, regardless of its location. Seeliger (1895, 1896) noted that adding an attenuation factor to the inverse-square law of gravitation would be sufficient to solve the problem. The larger the distances, the faster the force of gravity would fall-off relative to the standard inverse-square law. Such an attenuation factor would be negligible effects on scales compatible with the solar system, making empirical verification very difficult (Norton 2002), but not impossible.

The removal of flat (absolute) space (Newtonian or otherwise) along with the instantaneous propagation of gravity (absolute time) and its replacement with a geometrically curved structural spacetime manifold and the influence of gravitation proceeding at finite speed consistent with general relativity provides a natural and compelling path to the removal of the problem of diverging integrals inherent in Newtonian cosmology. The assumptions of homogeneity, isotropy, and static mass distribution in an infinite (or finite) universe can be retained, and in fact allowable, provided the manifold is of constant Gaussian curvature. This is how stability is maintained on scale above and beyond superclusters. Or rather, this is why the instability associated with a static Newtonian universe does not carry over to a homogeneous and isotropic general relativistic cosmology.

There is another (related) transient point to make, to be discussed in a detailed subsequent post, regarding evolution, homogeneity, and infinity. Philosophically, the universe may be considered homogeneous at any given cosmic time. But cosmic time does not exist in the physical universe. A universe that evolves from initial density fluctuations to the observed structures (in either expanding or static models) is necessarily inhomogeneous over time. In the static case, such would be equivalent to an "island universe" where the matter density (in the form of stars, galaxies, clusters and so on) diminishes in the look-back time, leading to an epoch dubbed the "dark age." In an expanding model this epoch occurs relatively shortly after the big bang. In a static (Coldcreation) model, this epoch transpires hundreds of billions of years ago. One of the consequences of such evolution in the past is that the gravitational potential diminishes with distance (in the past), since the mass-density diminishes in the past. So too does the magnitude (or radius) of Gaussian curvature. As time tends to minus infinity, the gravitating mass-energy density, along with the magnitude of curvature, tends toward zero (without ever attaining such a value). The Newtonian problem diverging potentials is resolved in this model by virtue of cosmic evolution. Here we tread treacherously beyond the domain of empirical verifiability, since the conjecture resides well beyond the limits of the observable universe. However, there are means by which such a comprehensive model can be tested; thus the precipitous boundary (superfluous as it often seems) that separates science and metaphysics can be avoided.

Indeed, this is a kind of "finite infinity' proposal for the general relativistic case of a Newtonian universe. GFR Ellis et al write in a similar context: "that is, to isolate the considered local system by a sphere that is far enough away to be regarded as infinity for all practical purposes but, because it is at a finite distance, can be investigated easily and uses as a surface where boundary condition can be imposed (and the residual influence of the outer regions on the effectively 'isolated' interior can thus be determined)" The gravitational potential fades away outside some bounded region, at some epoch, allowing a boundary condition at the expense of denying the philosophical assumption of spatial homogeneity (Ellis et al 2008, Newtonian Evolution of the Weyl Tensor).

(How this deviation from the inverse-square law affects the problem of rotational curves and excessive orbital velocities of galaxies within clusters remains to be explored, as a possible resolution of the missing-mass problem).

An interesting feature to emerge from this scenario is that the need to introduce the cosmological constant into the Einstein field equations (as either a repulsive force of the vacuum a la de Sitter, or tension associated with the mass-energy density a la Fischer) is explicitly excluded.

Symmetry considerations are still valid (per the cosmological principle), yet due to evolution as a function of time there exists an asymmetry between past and future (i.e., we have an 'arrow of time').

Homogeneous (background) gravitational field in general relativity, the problem of diverging integrals, and the inverse-square law of gravity

Consider the situation of test particles placed in a four-dimensional manifold of constant positive Gaussian curvature in reduced dimension. The surface of a sphere represents our manifold. We begin by placing one test particle, A, at rest on the surface of the sphere. Since all points are the same on our manifold particle A is not impelled to move in any direction on the background field of constant curvature. It remains at rest. We then introduce another particle, B, in motion relative to test particle A, at some other point on the manifold. Depending on various factors, particle B will either collide with A, disperse away form A (pending a potential close encounter at a later date), or find a configuration whereby the two particles can remain bound in a quasi-equilibrium configuration. In the latter case we'll say these particle are gravitationally bound. Now we place another test particle, C, arbitrarily in motion into the manifold. This particle too will be confronted with the three possibilities: collision, dispersion, or gravitational binding. We could continue this process until billions, (or even trillions) of test particles are present on the manifold. Some particles will form groups, or clusters and superclusters, others will not (pending potential close encounters at later dates).

Notice, during this entire process, the background field of constant positive Gaussian curvature had no affect on the outcome of the experiment. The dynamics of all particles is determined by local gravitational effects, the relative directions and relative velocities of particles placed in the field, in relation to others. At no time does the spherical surface influence (gravitationally) the entire population of particles to coalesce at one point.

The other important point to make is that the gravitational force between each particle on our spherical surface is not the same as the gravitational force between particles located on a flat, Euclidean, Minkowskian, or Newtonian manifold. Just as the propagation of light is affected by the Gaussian curvature, the gravitational force between particles diminishes with distance at a greater rate than predicted by the inverse-square law (ISL). In another way, the gravitational potential exerted on a test particle located in a manifold of constant positive Gaussian curvature with a homogeneous distribution of material particles converges with increasing volume. Integrals, rather than diverging to some arbitrarily large value (or even to infinity) as would possibly occur on a flat manifold, converge due to the geodetic propagation of force on the background manifold of constant positive intrinsic Gaussian curvature. There is a diminution of gravitational force with increasing distance, greater than inversely proportional to the square of the distance.

Just as a curved spacetime manifold solves Olbers' paradox by modifying the inverse-square law for the propagation of light, so too does a curved spacetime manifold solve the problem of diverging integrals by modifying the inverse-square law of gravitation. It is no miracle that the quantity by which the inverse-square law for the propagation of light is affected is exactly identical the quantity by which the inverse-square law of gravitation is affected. The deviation from the inverse-square law (for light and gravity) is directly proportional to the Gaussian curvature.

In another way, there is a divergence from the inverse-square law of gravitation (1/r^2) directly proportional to the magnitude of curvature of the manifold, and thus with increasing distance from the origin. The problem of diverging integrals inherent in a geometrically flat Newtonian manifold differs thus from the situation in an Einsteinian manifold of constant positive Gaussian curvature in such a way that the entire problem vanishes naturally.

At sufficiently small distances, where spacetime curvature of the manifold is negligible, the ISL holds, to a good approximation (i.e., there will be found no deviation from Newtonian physics locally, e.g., within the range of the Local Group, beyond which cosmological redshift manifests itself). The deviation from 1/r^2 manifests itself increasingly with distance. The greater the distance from the origin, the greater the deviation from 1/r^2.

Einstein's views on Newtonian cosmology was expressed in a 1917 paper (Cosmological Considerations on the General Theory of Relativity). "...the density of matter becomes zero at infinity...the mean density p must decrease towards zero more rapidly than 1/r^2 as the distance r from the center increases. In this sense, therefor, the universe according to Newton if finite, although it may possess an infinitely great total mass."

Of course, viz the cosmological principle, such a universe described by Einstein is homogeneous and isotropic at any given cosmic time. The 'island universe' surrounded by empty space is not at all incompatible, as it would first appear, with homogeneity and isotropy. This is simply a (static) universe that evolves in time from a relatively smooth background ('empty space') to the observed structures. The universe described by Einstein in 1917 could be observed, in principle, only in the look-back time and only if one could 'see' that far.

Of course too, if it were possible to see the empty space at infinity now (in cosmic time), it would appear as does the local universe, with a similar galaxy distribution. Such a universe has no center. The density of stars, galaxies, clusters and superclusters always appears to be at its maximum from the point of view of an observer. All observers appear as if located at the center of an island universe. And as we proceed outward from the origin (the observer) the density of galaxies and clusters diminishes in the look-back time, and continues to diminish, until eventually, at distances far beyond the range of telescopes (and at times far beyond the suspected age of the universe), the mass-density galaxies and clusters (or rather proto-galaxies, proto-clusters and proto-superclusters) tends towards zero. The stellar universe is thus a finite island in an "infinite ocean of space," static, homogeneous, isotropic and evolving with time. I might add that such a static universe possess a curved spatiotemporal manifold, i.e., this is a 4-dimensional static universe with constant positive Gaussian curvature (by virtue of the nonzero mass-energy density) and of infinite spatiotemporal extent. It is consistent with both Newtonian theory and Einsteinian theory in that there exists a globally homogeneous gravitational field.

So far, Einstein's analysis of Newtonian cosmology is impeccable. What we have here is a solution to the problem of diverging integrals in a static Einsteinian universe compatible with homogeneity and isotropy considerations. The gravitational force converges. Gravitational force diminishes with increasing distance from the origin (and asymptotically vanishes as time tends to minus infinity). Einstein then fell into error. Ironically, he criticized such a universe as unsatisfactory, speculating that (i) an island universe would lose radiation to infinite space and that (ii) the energy of motion distributed statistically among stars of the 'island' would impel stars (once they acquire enough velocity) to escape the island's gravitational pull. Such a universe, in contradiction with the presumed static nature of matter on large-scales, would 'evaporate' Einstein suggested, using Boltzmann's analysis of the statistical physics of gas molecules in a gravitational field. (See John D. Norton, 1999, The Cosmological Woes of Newtonian Gravitation Theory, Goenner et al. (Eds.), The Expanding Worlds of General Relativity, Einstein Studies, volume 7, pp. 271-323, The Center for Einstein Studies).

The problem Einstein laid out (either the mass distribution was concentrated in an island or it was homogeneously distributed) was a false dilemma. The "island" would only be apparent (in principle) due to the limited speed of light in vacuo. An observer situated at the "edge" of the island, relative to an observer situated at the origin, would see the universe as if, herself, centered on the island. In other words there is no center. All points are equivalent. The "island" exist only as a function of time, and thus of distance. With cosmic time there is no island at all (i.e., when all clocks in the universe are synchronized the universe is homogeneous and isotropic).

There would be no loss of radiation to infinite space. By symmetry considerations there would be no stars escaping the island's gravitational pull, no evaporation. Such a universe was indeed consistent with the presumed static nature of matter on large-scales. It's difficult to comprehend how the inventor of general relativity (amongst numerous other contributions to science) could have made such a mistake. Perhaps Einstein leveled this criticism against Newtonian cosmology in the context of a geometrically flat Euclidean manifold with absolute space and absolute time, in which case he would have been correct after all: the island universe (i.e., Newtonian cosmology) is untenable in the absence of general relativistic principles.

Newtonian theory (along with special relativity, Minkowski spacetime) would be a good approximation only in the local neighborhood of an observer, i.e., when distances are small and time scales short.

Einstein would attempt to resolve the problem of Newtonian gravitation by modifying Poisson's equation, thereby admitting an acceptable cosmological solution with a homogeneous distribution of matter. The result was a spatially closed relativistic model of spacetime with a static mass distribution (known as the Einstein universe). In order for such a model to satisfy the field equations of general relativity Einstein felt it necessary to introduce a supplementary term, "perfectly analogous to the extension of Poisson's equation". Consistent with symmetry considerations the average gravitational force (the force per unit volume at a point) was zero everywhere. Though perfect the cosmological constant was not. Einstein's initial satisfaction with these modifications were twofold: the integral would converge rather than diverge, and Einstein's dream of a Machian universe would be realized. What Einstein had (apparently) not seriously contemplated was how such a static universe with a uniform mass distribution would evolve as a function of time. Einstein had shown that a slight modification of the inverse-square law of gravitation permitted a cosmology consistent with staticity and homogeneity (dissimilar in form but with the same result as Seeliger 1895, 1896, 1909, and Neumann 1896). The fact too that Newtonian gravitation could be adjusted to readmit cosmology by other means was immediately apparent (Norton 1999, pp. 297-303).

The most obvious solution was to modify cosmological assumptions and leave Newtonian gravitation unchanged, rather than visa versa. Such was the solution of Wilsing (1895). Charlier (1908) developed a scheme that led to a vanishing mass density when averaged over space that required no preferred center, free of gravitational divergence. The idea was to consider the hierarchical grouping of matter into clusters on different scales. The spacing between each scale of clusters would increase as the scale of the cluster considered increased, indefinitely. The escape from the gravitational problem, and Olbers' paradox in passing, occurred as the mean mass density would vanish over infinite space, i.e., there is a dilution of mass density with increasing volume (Norton 1999, pp. 304-309).

Certainly, the big bang model thwarts the problem of instability associated with a homogeneous distribution of matter. But at what price? The universe is simply unstable; expanding, and loaded profusely with an unidentified form of dark matter along with an unphysical form of energy or pressure.

Fortunately, the big bang model is not the only solution; there have been many viable solutions over the relevant years. One of these solutions is consistent with a wide variety of observations, ranging from phenomena such as the formation of large-scale structures, cosmological redshift z, the CMBR, and the abundance of light elements and their isotopes.

In sum, a variety of solutions had emerged for both the homogeneity of Newtonian cosmology and the gravitational potential problem, including the kinematic solution (the expanding model of Milne and McCrea, 1934, that mimics the dynamics of a relativistic Friedmann universe).

"Perhaps the most astonishing part of our story" according to Norton, "is that both of the greatest figures of cosmology and gravitation, Newton and Einstein, stumbled on the same problem. When presented with the problem, Newton seemed so sure that his cosmology would be well behaved that he saw no need to think the problem through. Einstein also was overly hasty, seeing in the problem a dilemma for Newtonian cosmology that others showed to be a false dilemma." (Norton 1999, pp. 315, 316)

In any case, what we are left with is a cosmology (or several cosmologies) the consequences of which are testable in principle. The one that stands out here is a static general relativistic universe of constant positive Gaussian curvature. This universe is homogeneous and isotropic at any given time on very large scales and possibly infinite in spatiotemporal extent (in both past and future directions of time), i.e., there is no beginning or end. The universe does not collapse or expand as a function of time. Stability is maintained on the largest scales since all points are the same; there is no preferred direction in the 4-dimensional spacetime manifold of constant curvature. Material objects are not impelled to move due to a gradient of the global manifold, since the gradient is the same everywhere locally (equal to zero at every point), just as the curvature of the surface of the earth (in reduced dimension) is zero at all points. Curvature manifests itself with increasing distance and time as measured from the rest frame of any observer (the origin). Cosmological redshift z is the result of this curvature, since photons are obliged to travel a geodesic path in a curved spacetime manifold.

On scales relatively smaller than the observable universe (somewhere around or below 300-200 Mpc), homogeneity and isotropy break-down to form a hierarchical network of superclusters-supervoids, cluster-voids, galaxy groups and smaller voids, galaxies, star clusters, stars, planets and so on down to the realm of quantum mechanics, where clumps of matter and voids are present still. It would have been nice to delve a little deeper into the quantum world here, for there is much to be said and written in relation to all that has been discussed so far, but that will be for a subsequent post. Perhaps just as a foretaste for what will come let it just be said that phenomena such as Bose-Einstein condensation (BEC), Brownian motion, superfluidity, supercondictivity, zero-point energy ZPE, Casimir force, are all of interest and to some extent on-topic here. Though there still exists a rather large discrepancy between GR and QM, there are laws of nature common to both the macro- and micro-scopic domain that permit both reversible and irreversible phenomena, but irreversibility is the rule rather than the exception. Symmetry-breaking brings the universe from a static geometrical configuration to one whereby space and time are shaped by the events and objects in the evolving system. Order and coherence of the system are the extraordinary features to emerge from the spontaneous transition from the simple vacuum to the complex structures observed today (a process that has transpired over many hundreds of billions of years). Despite the random fluctuations of thermal motion performed by a large number of particles, the coherent self-organization leads to the emergence of complex behavior and far-from-equilibrium states. It is this self-accelerating process that has opened the gateway to systems capable of self-reproduction: the most fundamental property of life. Yes, life, in animal and plant form, humans, are all part a cosmology. The history of the universe is the history of life, its origins, its evolution.

At the heart of nature lies the deep concepts of symmetry and geometry, the competition between order and disorder, of energy and entropy, self-organization, erratic and coherent behavior at large length scales, frictionless, resistanceless flow of ground-energy, phase transitions, density fluctuations at  many scales, and the critical points at which new phenomena occur.


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