Wednesday, May 23, 2018

The universe in the golden ratio: A model based on the number φ

This is the observable universe

“The good, of course, is always beautiful, and the beautiful never lacks proportion.”

1.1 Grasping the scales of the universe

This work is the result of a preoccupation of mine, for some period of time, with Drake’s equation, and the possibility of intelligent life elsewhere in the universe. I personally believe that the existence of intelligent life in the universe is not just highly probable but certain. The reason for this is that we exist. If we exist then other beings will also exist. If we have the privilege (or bad luck) of being alone in the universe, then this seems to be unnatural, since the universe which created us will not be able to create other beings, thus life elsewhere, thus ourselves in the first place.

This simple thought, which is based on the coincidence between ourselves and the universe, and which also has its deepest roots in the anthropic principle, led me to seek for ratios and analogies between nature and the way own mind works. After analyzing the parameters and the notions related to habitable zones around solar systems, I then wondered if there could be an analogous galactic habitable zone for solar systems in our Milky Way. This analysis was made with respect to the number φ. After I saw that my results matched with current estimations, I then made a step forward to assume a habitable zone of galaxies within the universe. This finally led me to some estimate about the distance of our galaxy from the center of the universe, and about the approximate size of the universe.

But does the universe has a center? According to the theory of the Big Bang, this is impossible, or even pointless. However if we assume that the universe was created by accretion, in the same way galaxies or solar systems form, then not only will the universe have a center, but also the universe will be rotating. It is the analogy which should lead us to such a conclusion. There is further evidence suggesting such a fact. Stars, for example, which have the same age with the universe. The mysterious Hubble constant, which in fact has units of angular frequency (i.e. rotation). Large structures, for example the Hercules-Corona Borealis Great Wall at the fringes of the universe, suggesting that structure begins from the outside towards the inside. Other large structures, such as the Huge- Large Quasar Group, pointing towards a hyper-active nucleus at the center of the universe.

But above all is the geometry which should be considered; that structure cannot exist without geometry, also that our own thought cannot exist without geometry and structure. It is always the same principles which hold true both for nature and for us who think about nature. Thus even if the way I used the number φ proves to be wrong, this number has to be there, everywhere in nature and the universe.

1.2 The number φ

Line segments in the golden ratio

The number φ can be defined using the following identity,

Some of the basic (also interesting) properties of φ are the following,

The number φ (which is also called the golden ratio) can be found in all aspects of nature and human activity. An example from art is Leonardo da Vinci’s ‘Vitruvian man,’


Another example from astronomy, is the Earth-Moon system,


The number φ is also intimately related to the Fibonacci sequence,

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…

In that sequence each next number is the sum of the two previous numbers, so that if a is a number, and n is its position in the sequence, then,

Remarkably enough, the ratio of two successive terms in the Fibonacci sequence approaches the number φ, an aspect which was discovered by Kepler, among others,

For example,

The last ratio gives the number φ with an accuracy of 3 decimal digits (φ=1.6180…).

2.1 Habitable zones

The habitable zones (green) for stars that are like the Sun (middle), hotter than the Sun (top) Kepler mission/Ames Research Center/NASA.

This is a definition of the habitable zone, in relation to the previous picture:

Habitable zone, the orbital region around a star in which an Earth-like planet can possess liquid water on its surface and possibly support life. Liquid water is essential to all life on Earth, and so the definition of a habitable zone is based on the hypothesis that extraterrestrial life would share this requirement. This is a very conservative (but observationally useful) definition, as a planet’s surface temperature depends not only on its proximity to its star but also on such factors as its atmospheric greenhouse gases, its reflectivity, and its atmospheric or oceanic circulation. Moreover, internal energy sources such as radioactive decay and tidal heating can warm a planet’s surface to the melting point of water. These energy sources can also maintain subsurface reservoirs of liquid water, so a planet could contain life without being within its star’s habitable zone. Earth, for instance, has a thriving subsurface biosphere, albeit one that is composed almost exclusively of simple organisms that can survive in oxygen-poor environments. Jupiter’s moon Europa has a liquid water ocean tens of kilometers below its surface that may well be habitable for some organisms.

2.1.1 Circumstellar Habitable Zone

An example of a system based on stellar luminosity for predicting the location of the habitable zone around various types of stars. Planet sizes, star sizes, orbit lengths, and habitable zone sizes are not to scale.

This is an explanation of the CHZ (Circumstellar Habitable Zone),

In astronomy and astrobiology, the circumstellar habitable zone (CHZ), or simply the habitable zone, is the range of orbits around a star within which a planetary surface can support liquid water given sufficient atmospheric pressure. The bounds of the CHZ are based on Earth’s position in the Solar System and the amount of radiant energy it receives from the Sun. Due to the importance of liquid water to Earth’s biosphere, the nature of the CHZ and the objects within is thought to be instrumental in determining the scope and distribution of Earth-like extraterrestrial life and intelligence.

The habitable zone is also called the Goldilocks zone, a metaphor of the children’s fairy tale of Goldilocks and the Three Bears, in which a little girl chooses from sets of three items, ignoring the ones that are too extreme (large or small, hot or cold, etc.), and settling on the one in the middle, which is ‘just right.’

The range of published estimates for the extent of the Sun’s CHZ. The conservative CHZ is indicated by a dark-green band crossing the inner edge of the aphelion of Venus, whereas an extended CHZ, extending to the orbit of the dwarf planet Ceres, is indicated by a light-green band.

Whether a body is in the circumstellar habitable zone of its host star is dependent on the radius of the planet’s orbit (for natural satellites, the host planet’s orbit), the mass of the body itself, and the radiative flux of the host star. Given the large spread in the masses of planets within a circumstellar habitable zone, coupled with the discovery of super-Earth planets which can sustain thicker atmospheres and stronger magnetic fields than Earth, circumstellar habitable zones are now split into two separate regions- a ‘conservative habitable zone’ in which lower-mass planets like Earth or Venus can remain habitable, complemented by a larger ‘extended habitable zone’ in which super-Earth planets, with stronger greenhouse effects, can have the right temperature for liquid water to exist at the surface.

The inner edge of the HZ is the distance where runaway greenhouse effect vaporize the whole water reservoir and, as a second effect, induce the photo-dissociation of water vapor and the loss of hydrogen to space. The outer edge of the HZ is the distance from the star where adding more carbon dioxide to the atmosphere fails to keep the surface of the planet above the freezing point.

Estimates for the habitable zone within the Solar System range from 0.5 to 3.0 astronomical units, though arriving at these estimates has been challenging for a variety of reasons. According to extended habitable zone theory, planetary mass objects with atmospheres capable of inducing sufficient radiative forcing could possess liquid water farther out from the Sun.

This is a table which gives the distances of the planets from the Sun in our solar system:


Distance from Sun
0.387 A.U.
0.723 A.U.
1.000 A.U.
1.524 A.U.
5.203 A.U.
9.539 A.U.
19.18 A.U.
30.06 A.U.

Figure: Hypothetical structure of the inner solar system. The two green circles represent the limits of the CHZ (Circumstellar Habitable Zone), with the Earth in the middle, at point , and the Sun at the center, at point O.

While it would be ambiguous trying to define the edge of our solar system, we may still suppose that the Earth is not found ‘in the middle between the extremes,’ but in the golden ratio. Thus we will use the following notations, according to the previous sketch:

Ο: Position of the Sun.
: Position of the Earth.
Ε: Edge of the (inner) solar system.
This limit is hypothetical.
α=OA: Outer radius of the CHZ.
β=OB: Inner radius of the CHZ.
d=α-β: size (width) of the CHZ.
The Earth will be in the middle of the CHZ.
RE=OO΄: Distance of the Earth from the Sun.
This distance is 1 AU by definition.
RSS=OE: Hypothetical radius of our inner solar system.
γ=O΄E: Distance between the Earth and the hypothetical edge of our inner solar system.
As we shall see, this edge can be identified with the asteroid belt.

The radii α and β will be in the golden ratio:

The distances RE and γ will also be in the golden ratio:

This is a table with the results:



The inner and outer radii of the CHZ (β=0.7639AU, and α=1.236AU respectively) give the right proportions of a possible CHZ. Venus (0.723 AU) and Mars (1.524 AU) are marginally located outside the CHZ.

The problem with the solar system is that it has no definite edge. However RSS=2.618AU (the hypothetical radius of the inner solar system) can be identified with the asteroid belt:

Because the asteroid belt is between the Mars and Jupiter orbits, it is around 2.2 to 3.2 Astronomical Units (AU) from the Sun.

(Thus 2.7AU on average.)

We should also mention that the value of d=0.4729AU (the width of the CHZ) is fixed, since it is the difference of two numbers (α and β) in the golden ratio. This width is d=0.1806Rss, about 18% of the radius of the inner solar system (up to the asteroid belt).

Therefore the results based on number φ are in good accordance with current estimates.

2.1.2 Galactic Habitable Zone


The previous picture offers an idea of our position in the Milky Way. The following article gives an estimation about our distance from the center of the Milky Way:

Although the light year is a commonly used unit, astronomers prefer a different unit called the parsec (pc). A parsec, equal to 3.26 light years, is defined as the distance at which 1 Astronomical Unit subtends an angle of 1 second of arc (1/3600 of a degree). When we use the parsec for really large distances, we often put a prefix in front of it- like kiloparsecs (kpc), which are equal to 1000 parsecs- or Megaparsecs (Mpc), equal to a million parsecs.

The Milky Way is about 1,000,000,000,000,000,000 km (about 100,000 light years or about 30 kpc) across. The Sun does not lie near the center of our Galaxy. It lies about 8 kpc from the center on what is known as the Sagittarius arm of the Milky Way.

The following article explains the meaning of the GHZ (Galactic Habitable Zone):

The galactic habitable zone (GHZ) is an annular region of our Galaxy in which it has been hypothesized that conditions are best suited to the development and survival of life as we know it. The GHZ was first proposed in 1991, and has subsequently been endorsed by a number of researchers. Outside the galactic habitable zone, the theory goes, various factors make the existence of complex (multicellular) life difficult if not impossible. The current GHZ is said to extend from 7 to 9 kiloparsecs (23,000 to 29,000 light-years) from the galactic center, is widening with time, and is composed of stars that formed between 4 and 8 billion years ago. The GHZ is analogous to the much more well-established concept of the habitable zone of a star.

According to the GHZ hypothesis, the width of the GHZ is controlled by two factors. The inner (closest to the center of the galaxy) limit is set by threats to complex life: nearby transient sources of ionizing radiation, including supernovae and gamma-ray bursts, and comet impacts. Such threats tend to increase close to the galactic center. The outer limit is imposed by galactic chemical evolution, specifically the abundance of heavier elements, such as carbon.

Figure: Hypothetical structure of the Milky Way, with its center lying at point O. The two white dotted circles represent the limits of the GHZ (Galactic Habitable Zone). The Sun is supposed to be located in the middle of the GHZ, at point O΄.
Inset picture: Artist’s impression of the Milky Way Galaxy, as seen from above the galactic ‘North pole.’ Credit: NASA

Here we will consider again two pairs of distances presumably in the golden ratio. This is an explanation of the values, according to the following picture,

O: COG (Center Of our Galaxy).
: Position of the Sun.
E: EOG (Edge Of our Galaxy).
β=OB: Inner radius of the GHZ.
α=OA: Outer radius of the GHZ.
Presumably α and β will be in the golden ratio.
d=α-β: Size (width) of the GHZ.
Presumably the Sun will be located in the middle of the GHZ.
RS=OO΄: Distance of the Sun from the COG.
By definition RS=8Kpc=26,000ly.
γ=O΄E: Distance of the Sun from the EOG.
Presumably γ and RS will also be in the golden ratio.
RG=OE: Radius of the galaxy.
The current estimation is roughly RG=50,000ly.

Thus, we have,

This is a table with the results,

The Galaxy
(Distances in light years)
RS (Distance of the Sun from the COG)
α (Outer radius of the GHZ)
β (Inner radius of the GHZ)
d (Width of the GHZ)
γ (Distance of the Sun from the EOG)
RG (Radius of the galaxy)


These results (b=19,863ly, a=32,138ly,) are in good accordance with current estimates of the GHZ (7-9kpc, about 23,000-29,000ly).

If this is true then the radius RG of the galaxy also has to be bigger, since it will be in the golden ratio:

That the radius of the galaxy (RG= 68,068ly) is in fact bigger than current estimations (RG=50,000ly), can be supported by recent evidence:

The disk of the Milky Way Galaxy disk may actually be rippled

Two ring-like structures of stars wrapping around the Milky Way’s outer disk now appear to belong to the disk itself. The results, outlined in a new study, show that the disk is about 60 percent larger than previously thought. Not only do the results extend the size of the Milky Way, they also reveal a rippling pattern, which raises intriguing questions about what sent wavelike fluctuations rippling through the disk. The researchers said the likely culprit was a dwarf galaxy. It might have plunged through the Milky Way’s center long ago, sparking the rippling patterns astronomers have now detected for the first time.

Roughly 15 years ago, Heidi Newberg, an astronomer at the Rensselaer Polytechnic Institute in New York, and her colleagues found a group of stars beyond the disk’s outermost edge. The so-called Monoceros Ring is about 60,000 light-years from the galactic center (just beyond where the disk was thought to end at 50,000 light-years).

Over the years, astronomers were divided into two camps regarding the origins of the ring. Some argued that it was simply a tidal stream: The debris of a dwarf galaxy that fell into the Milky Way and was stretched in the process. Others argued that the ring is a part of the disk. The issue, however, is that the ring is slightly above the plane of the disk. So astronomers in the latter camp attributed that to the fact that the disk flares up toward the edge.

Enter Yan Xu, an astronomer at the National Astronomical Observatories of China, Newberg and colleagues took a second look at the problem using data from the Sloan Digital Sky Survey. With improved data compared to previous studies, they found four total structures in and just outside what is currently considered the Milky Way’s outer disk. The third structure was the highly debated Monoceros ring, and the fourth structure was the Triangulum Andromeda Stream, located 70,000 light-years from the galactic center.

All four structures alternated with respect to the disk. They went from above it, to below it, to above it, to below it. Newberg, who was in the tidal stream camp, was surprised that the ring and three other structures were actually a part of an oscillating disk.

“We didn’t know how a disk could go up and down,” said Newberg. Luckily, computer simulations by various teams showed that a dwarf galaxy falling into the Milky Way might create a similar pattern. “When it goes through, it can disturb the disk just like a pebble disturbs water in a puddle,” said Newberg. “And that wave can propagate through the disk from that event.”

This new picture makes sense, said Newberg. It even matches observations of the gases in the disk, which have long been observed as rippled. But the implications extend far beyond a corrugated disk.

“If it’s true that the Monoceros Ring and the Triangulum Andromeda structure are part of this oscillatory pattern, then the stellar disk goes out way further than the textbook tells us it ought to be,” said Newberg. Instead of extending nearly 100,000 light-years from one side to the other, it would be more like 160,000 light-years wide.

This brings the Milky Way’s size up to that of Andromeda. The Milky Way’s small radius in comparison to Andromeda’s larger radius has always puzzled astronomers, because the two galaxies have roughly the same mass.

Such a galactic radius RG≈70,000ly (up to the Triangulum Andromeda Stream) perfectly matches the radius in the golden ratio, RG=68,068ly.

Therefore, amazingly enough, the results we took with respect to the golden ratio fit very well current observational data.

Finally we may also note that, as we did in the case of the CHZ (Circumstellar Habitable Zone), we have that,

Doing the same calculation with respect to the relative areas, we have that,

Thus the GHA (Galactic Habitable Area) is somewhat less than the GHZ (Galactic Habitable Zone).

2.1.3 Universal Habitable Zone

Up till now we have seen two different manifestations of the golden ratio. The first one had to do with the position of the Earth in the solar system, and the second one with the position of the Sun in the galaxy. Considering this, we may now go a step further and assume a similar zone in the universe, thus a UHZ (Universal Habitable Zone).

According to Wikipedia, the current measurement of the age of the universe is 13.799±0.021 billion years.

If our distance from the Big Bang, thus the radius of the HOU (Horizon of the Observable Universe), is 13.8Gly, and the universe appeared 13.8Gly ago, then the age of the universe and our distance from the COU (Center Of the Universe), supposedly the Big Bang, will numerically coincide.

However, the numerical coincidence between the age of the universe and our distance from the COU (Center Of the Universe) is not necessarily true. In fact, most likely it is not. Thus, here, we will assume a different distance of the COG (Center Of our Galaxy) from the COU (Center Of the Universe), in such a way that it will be in the golden ratio, while the COU will not be a virtual point of reference with respect to the ‘Big Bang,’ but a real place in the universe. About the possibility of the existence of such a COU, we will talk later on. For the moment, this is a list of the relative sizes according to the following picture, and in accordance with what we have mentioned earlier,

Figure: Hypothetical structure of the universe, with point O representing the center, and point representing the position of the Milky Way. The two dotted white circles represent the limits of the UHZ (Universal Habitable Zone).
Note: The inset picture is a true image of the Milky Way, but here it is used as a model of the whole universe. Link to the inset picture:

O: Position of the COU (Center Of the Universe).
: Position of the COG (Center Of our Galaxy).
E: Position of the EOU (Edge Of the Universe).
H: Hypothetical point where the HOU (Horizon of the Observable Universe) crosses the outer circle of the UHZ (Universal Habitable Zone).
α =OA: Outer radius of the UHZ.
β =OB: Inner radius of the UHZ.
Presumably, α and β will be in the golden ratio.
d=α-β: Size (width) of the UHZ.
We will suppose that the COG is located in the middle of the UHZ.
RH=O΄H: Radius of the HOU.
By definition, RH=13.8Gly.
RG=OO΄: Distance between the COU and the COG.
We will suppose that, approximately, RH =O΄H and RG=OO΄ form the two sides of a right triangle, with αOH being the hypotenuse (see previous picture).
γ=O΄E: Distance between the COG and the EOU.
Presumably, γ and RG will also be in the golden ratio.
RU=OE: Radius of the universe.
OD: Distance between the HOU and the COU.

Thus we have,

This is a table with the results,

The Universe
(Distances in billion light years)
RH: Radius of the HOU
RU: Radius of the Universe
RG: Our distance from the COU
α: Outer radius of UHZ
β: Inner radius of UHZ
d: Width of the UHZ
γ: Our distance from the EOU
OD: Distance of the HOU from the COU

1st Note:

If we supposed that our distance from the COU (Center Of the Universe), the ‘Big Bang,’ was numerically identical to the age of the universe, so that RG=RH=13.8Gy, then it would be,

This would give a very small radius to the universe. In fact, the proper radius of the universe is estimated to be about 47Gly:

According to calculations, the comoving distance (current proper distance) to particles from which the CMBR was emitted, which represent the radius of the visible universe, is about 14.0 billion parsecs (about 45.7 billion light years), while the comoving distance to the edge of the observable universe is about 14.3 billion parsecs (about 46.6 billion light years).

This estimated value of 46.6Gly is in good accordance with the result we got, RU=49.7Gly.

2nd Note:

Figure: The distances OO΄RG (our distance from the COU), OH= α (outer radius of UHZ), and O΄H=RH (distance to the HOU), can be considered to form a right triangle.

The approximation we used (previous sketch),

is valid as long as the radius RH=O΄H of the observable horizon crosses the outer circle of the UHZ at the point H approximately vertically, so that, if θ is the angle between the radius of the event horizon RH=O΄H and the radius RG=O΄O which connects the COG (point Ο΄) with the COU (point O), and α=OH is the hypotenuse of the triangle O΄ΟΗ, then, taking the law of cosines,

Figure: Here the line O΄H=RH (distance to the HOU) crosses the center of the UHZ approximately at point H, which in turn is vertically projected on the inner circle of the UHZ at point B.

Another approximation (previous sketch), would be if we assumed that the radius RH=O΄H of the observable horizon crosses the center of the UHZ at a point H, in such a way that if OH=OO΄RG is the radius which connects the COG (point Ο΄) with the COU (point Ο), OB=β is the inner radius of the UHZ, and h=BH is the height of the triangle O΄ΟΗ at point B, then we have that

This would give us a distance of the COG (Center Of our Galaxy) from the COU (Center Of the universe) about 1Bly bigger.


The calculation with respect to the number φ gave us a result for the radius of the universe (RU=49.7Gly) approximately equal to currently estimated radius (46.6Gly). If this is true then it will also be true that our distance from the COU will be RG=19Bly, since

This result, that our distance from the COU (Center Of the universe) is 19Bly, leaves us 5.2Bly away from the COU. This means that the light from the ‘zero-point’ at the center of the universe will take another 5.2 Bly to reach us. This time will coincide with the death of our own Sun. The COU however is not a point but it covers some region, in the same sense that the core (the bulge) of our own galaxy has dimensions. Even if the light has not reached us yet from the very center, the light from the outer region of the central core of the universe may have reached us. If the core extends 5.2Bly from the center, then the light from that region will be reaching us right now. If the bulge extends even further, then we will have already taken a glimpse of what is going on there. The HLQG (we will say more about this later on) may be a good candidate.

2.2 Further notes and clues

These are some of the clues which helped me realize that our current view about the universe is rather limited and egocentric.

2.2.1 The age of our galaxy

Night sky from a hypothetical planet within the Milky Way 10 billion years ago

This is an article of Wikipedia related to the age of our galaxy:

Globular clusters are among the oldest objects in the Milky Way, which thus set a lower limit on the age of the Milky Way. The ages of individual stars in the Milky Way can be estimated by measuring the abundance of long-lived radioactive elements such as thorium-232 and uranium-238, then comparing the results to estimates of their original abundance, a technique called nucleocosmochronology. These yield values of about 12.5 ± 3 billion years for CS 31082-001 and 13.8 ± 4 billion years for BD +17° 3248. Once a white dwarf is formed, it begins to undergo radiative cooling and the surface temperature steadily drops. By measuring the temperatures of the coolest of these white dwarfs and comparing them to their expected initial temperature, an age estimate can be made. With this technique, the age of the globular cluster M4 was estimated as 12.7 ± 0.7 billion years. Age estimates of the oldest of these clusters gives a best fit estimate of 12.6 billion years.

Several individual stars have been found in the Milky Way’s halo with measured ages very close to the 13.80-billion-year age of the Universe…

The age of stars in the galactic thin disk has also been estimated using nucleocosmochronology. Measurements of thin disk stars yield an estimate that the thin disk formed 8.8 ± 1.7 billion years ago. These measurements suggest there was a hiatus of almost 5 billion years between the formation of the galactic halo and the thin disk.


By comparing the age of stars in the halo of our galaxy (which are as old as the universe itself) with the age of stars in the thin disk (which are about 5 billion years younger) we realize that our galaxy (at least some parts of it) appeared simultaneously with the universe. Such an observation makes the theory of the Big Bang very doubtful, as it must have taken the first stars some time to appear. This observation also made me think that the universe may have formed in the same way galaxies do, by accretion.

2.2.2 Formation of galaxies

Credit: NASA, ESA, R. Ellis (Caltech), and the UDF 2012 Team

This is an article which explains how galaxies may form:

The largest structures in the universe bind billions or even trillions of stars in their massive gravitational yokes. Cosmic dust and vast clouds of gas fill galaxies, too, along with the planets and other matter that may orbit stars.

Galaxies got their start nearly 14 billion years ago, with one unimaginably hot, dense and tiny pinpoint. According to the big bang theory, this singularity was the universe in its entirety. Then it exploded, cooling and expanding in the process. Imagine a balled-up piece of paper unfolding into a giant map, and you have a very crude model of what happened.

Following the big bang, the primordial universe consisted of only radiation and subatomic particles. How did it evolve into more than 100 billion galaxies? Scientists have two kinds of theories, both of which hinge on the gravitational effects of collapsing gas in the early galaxy.

First, there are the bottom-up theories, in which the gas collapsed and compressed into clumps the size of a million suns (that’s starting small for something the size of the universe). These clumps then merged to build galaxies. Top-down theories, on the other hand, start big. This school of thought argues that the resulting clumps were each the size of multiple galaxies, which in turn broke down into individual galaxies. These latter theories would explain why galaxies occur in clusters.

Either way- bottom-up or top-down- the resulting clumps then collapsed into protogalaxies consisting of dark matter and hydrogen gas. The hydrogen then fell toward the center of the protogalaxy while the dark matter remained as an outer halo surrounding it.

Astronomers recognize two main galaxy types: elliptical and spiral. These differences in shape, according to one theory, are due to star formation. Stars develop inside a protogalaxy when clouds of gas mix and collide. If the stars in a protogalaxy form all at once, then the mature galaxy essentially retains the spherical shape of the protogalaxy and becomes an elliptical galaxy.

Spiral galaxies occur when the stars inside the protogalaxy arise at different intervals. The gas between developing stars continues to collapse and the resulting gravitational differences manhandle the protogalaxy’s stars, dust and gas. This motion forces everything into a rotating disc, and additional differences in gravity result in the spiral arms.

Additional changes can occur when galaxies drift too close to one another or collide. Astronomers believe that the merger of two galaxies always results in an elliptical galaxy. As such, the Milky Way has probably never merged with another galaxy, while the massive elliptical galaxies found at the center of galaxy clusters are likely the result of multiple cosmic mash-ups.

This is an article of Wikipedia related to galaxy formation:

Many of the properties of galaxies (including the galaxy color- magnitude diagram) indicate that there are fundamentally two types of galaxies.
These groups divide into blue star-forming galaxies that are more like spiral types, and red non-star forming galaxies that are more like elliptical galaxies.
Spiral galaxies are quite thin, dense, and rotate relatively fast, while the stars in elliptical galaxies have randomly-oriented orbits.
The majority of mass in galaxies is made up of dark matter, a substance which is not directly observable, and might not interact through any means except gravity…

Elliptical galaxies (such as IC 1101) are among some of the largest known thus far. Their stars are on orbits that are randomly oriented within the galaxy (i.e. they are not rotating like disk galaxies). They have central supermassive black holes, and the masses of these black holes correlate with the galaxy’s mass. Also, they are more likely found in crowded regions of the universe (such as galaxy clusters)…

One observation that must be explained by a successful theory of galaxy evolution is the existence of two different populations of galaxies on the galaxy color-magnitude diagram. Most galaxies tend to fall into two separate locations on this diagram: a ‘red sequence’ and a ‘blue cloud.’ Red sequence galaxies are generally non-star-forming elliptical galaxies with little gas and dust, while blue cloud galaxies tend to be dusty star-forming spiral galaxies. Galaxies tend to evolve from spiral to elliptical structure via mergers. However, the current rate of galaxy mergers does not explain how all galaxies move from the ‘blue cloud’ to the ‘red sequence.’ It also does not explain how star formation ceases in galaxies.

This is a list of the most distant astronomical objects,

Distance (Gly)
Ursa Major
Irregular galaxy
1×10^9 M

Ursa Major
Dwarf galaxy
1.0×10^9 M
GRB 090423
Gamma-ray burst

Lyman-break galaxy

Dwarf galaxy
1.7×10^9 M

Ursa Major
Irregular galaxy

Piscis Austrinus

ULAS J1120+0641

A1703 zD6

Lyman-break galaxy

Piscis Austrinus



The relative abundance of dwarf and irregular galaxies as the most distant astronomical objects, suggests that at the edges of the universe (at least in some directions) ordinary matter was never sufficient to build up regular galaxies (spiral galaxies for example). Furthermore spiral galaxies, like the Milky Way, do not form by mergers of dwarf and irregular galaxies. Therefore dwarf and irregular galaxies are there, at the edges of the universe, to stay.

Another observation is that elliptical galaxies consist of mostly old stars, in comparison to the younger stars of spiral galaxies. Although such an aspect may be attributed to the low star- forming activity of elliptical galaxies, on the other hand it may suggest that elliptical galaxies, at least in some cases, can be older than spiral galaxies. If this is so then mergers cannot explain their origin. Instead the oldest elliptical galaxies may have originated in denser regions of the universe since the beginning.

Such a stratification, with dwarf and irregular galaxies at the edges of the universe, and large elliptical galaxies (as well as quasars) towards the center of the universe, with most of the spiral galaxies located inbetween, an aspect which incidentally casts doubt on the cosmological principle (problem of homogeneity), will be discussed later on.

2.2.3 Stars older than the universe

This is a list of the oldest stars:

Age (in billion years)
23 stars identified as ‘from the cosmic dawn in the bulge of the Milky Way.’

HD 140283 (Methuselah star)
14.46 ± 0.8
BD +17° 3248
13.8 ± 0.4
SMSS J031300.36-670839.3
HE 1523-0901
BPS CS22892-0052 (Sneden’s Star)


This is an article about the oldest star:

HD 140283

HD 140283, informally nicknamed Methuselah star, is a metal-poor subgiant star about 190 light years away from the Earth in the constellation Libra. Its apparent magnitude is 7.223. The star has been known to astronomers for over a century as a high-velocity star. It is one of the closest Population II stars to us.

Because HD 140283 is neither on the main sequence nor a red giant, its position in the Hertzsprung-Russell diagram can be interpreted with theoretical models of stellar evolution to infer the stellar age. For field stars (as opposed to stars in clusters) it is rare to know a star’s luminosity, surface temperature and composition precisely enough to get a well-constrained value for the stellar age; because of their relative scarcity, this is even rarer for a Population II star like HD 140283. One recent study used the Fine Guidance Sensors of NASA’s Hubble Space Telescope to measure a precise parallax (and therefore distance and luminosity) for the star, and employ this information to estimate an age for the star of 14.46 ± 0.8 billion years.

Very low but non-zero metallicities of stars like HD 140283 indicate the star was born in the second generation of stellar creation; their heavy-element content is believed to have come from truly zero-metal stars (Population III stars), none of which have yet been identified. The first stars are thought to have been born a few hundred million years after the Big Bang, and they died in supernova explosions after only a few million years. A second generation of stars, the generation in which HD 140283 is theorized to have been born, could not have coalesced until gas, heated from the supernova explosions of the earlier stars, cooled down. The age of HD 140283 indicates that the time it took for the gases to cool was likely only a few tens of millions of years.


In the previous list of the oldest stars, the oldest are 23 stars in the bulge of the Milky Way. Equally old stars are expected to be found in the halo of the Milky Way, because both the bulge and the halo formed first. In all likelihood Methuselah star originated in one of these two regions, before it migrated at a distance of just 190ly from us. This is because the thin disk, in which we are located, formed later on (5 billion years later than the bulge and the halo of the Milky Way). But since the bulge and the halo of our galaxy are as old as the universe itself (since they contain stars as old as the universe) this may also suggest that the bulge and the halo of the universe formed first, 5 billion years earlier than the thin disk of the universe.

2.2.4 Large Quasar Groups

This is in short the definition of a quasar:

A region at the center of a galaxy that produces an extremely large amount of radiation.

In more words,

A massive and extremely remote celestial object, emitting exceptionally large amounts of energy, which typically has a starlike image in a telescope. It has been suggested that quasars contain massive black holes and may represent a stage in the evolution of some galaxies.

This is the definition of a LQG (Large Quasar Group): A large quasar group (LQG) is a collection of quasars (a form of supermassive black hole active galactic nuclei) that form what are thought to constitute the largest astronomical structures in the known universe. LQGs are thought to be precursors to the sheets, walls and filaments of galaxies found in the relatively nearby universe.

The Huge Large Quasar Group, is the biggest of all Large Quasar Groups: The Huge Large Quasar Group, (Huge-LQG, also called U1.27) is a possible structure or pseudo-structure of 73 quasars, referred to as a large quasar group, within the vicinity of the constellation Leo. It measures about 4 billion light-years across. At its discovery, it was identified as the largest and the most massive known structure in the observable universe, though it has been superseded by the Hercules-Corona Borealis Great Wall at 10 billion light-years.

The Huge-LQG was estimated to be about 1.24 Gpc (4Gly) in length, by 640 Mpc (2Gly) and 370 Mpc (1.2Gly) on the other dimensions, and contains 73 quasars, respectively. It has the approximate binding mass of 6.1×10^18 M☉. The Huge-LQG was initially named U1.27 due to its average redshift of 1.27, placing its distance at about 9 billion light-years from Earth.

According to the same article of Wikipedia, The Huge-LQG is 615Mpc (2Bly) away from the Clowes-Campusano LQG:

The Clowes-Campusano LQG (CCLQG; also called LQG 3 and U1.28) is a large quasar group, consisting of 34 quasars, and measures about 2 billion light-years across, and about 1 billion light years wide. It is one of the largest known superstructures in the observable universe, located in the constellation of Leo, 1.8 billion light-years away from the larger Huge-LQG. It was named U1.28 because of its average redshift of 1.28, lying at a distance of 9.5 billion light years away.

Because of its proximity to the Huge-LQG, it has been suggested that the two structures are really a single structure in itself, and only connected by hidden intergalactic filament; however, no such evidence has been found.

Another LQG (smaller than the HLQG but bigger than CCLQG) is U1.11: U1.11 is a large quasar group located in the constellations of Leo and Virgo. It is one of the largest LQG’s known, with the estimated maximum diameter of 780 Mpc (2.2 billion light-years) and contains 38 quasars. Until the discovery of the Huge-LQG in November 2012, it was the largest known structure in the universe, beating Clowes-Campusano LQG’s 20-year record as largest known structure at the time of its discovery.

The structure is is at redshift z = 1.11, hence its name, corresponding to a distance of approximately 8.8 billion light years away. It is adjacent to the CCLQG, approximately 2° away, and is relatively close to U1.54, another LQG.

Sky distribution of the 73 quasars of the Huge-LQG (circles) is shown, together with that of the 34 quasars of the CCLQG (crosses). The members of each LQG are connected at the linkage scale of 100 Mpc. The area shown is approximately 29.5º×24.0º.


The previous picture shows that the HLQG and the CCLQG are adjacent to each other, with a separation of about 2Bly. The third LQG we mentioned, namely U1.11, apparently (according to the previous picture) is not located between the other two, but it is adjacent to the CCLQG. If we suppose that these three LQG constitute one unique megastructure, then adding the sizes of those three LQG, including the filament between the HLQG and the CCLQG, we take,

This size is comparable to the size of the biggest of all cosmological megastructures, the Great GRB Wall, which will be mentioned soon afterwards.

All three structures are located in the constellation of Leo (with U1.11 extending into the constellation of Virgo).

The mass of the HLQG is estimated at 6.1×1018 solar masses.

One solar mass (M☉) is equal to approximately 1.99×1030 kilograms.

The mass of the observable universe is:
4.4506×1052 kg as estimated by NASA, or
6×1052 kg, as estimated by the National Solar Observatory.

Thus, calling MHLQG the mass of the HLQG, and MU the mass of the observable universe, we have,

The HLQG is composed of 73 quasars, while the CCHLG is composed of 34 quasars, and the U1.11 quasar is composed of 38 quasars. Supposing that these three quasars constitute one megastructure (and that each quasar has on average the same mass), then the total mass of that megastructure will be double as much.

Thus we have the HLQG, in the direction of the constellation Leo, with a mass of about 1/10,000 the mass of the observable universe.

This is what made me think that if there is a center of the universe, then this must be it.

Incidentally a similar ratio can be taken by comparing the mass of the Milky Way with the mass of Sagittarius A* (the central black hole):

Mass of the Milky Way:
5.8×1011M☉ (solar masses)

Mass of Sagittarius A*:

This ratio is approximately of the same order. Thus all aspects of the HLQG fit the description of a truly supermassive black hole located at the center of the universe.

2.2.5 Great GRB Walls

This is an article of Wikipedia about the Hercules- Corona Borealis Great Wall, the largest known structure in the observable universe:

Hercules- Corona Borealis Great Wall or the Great GRB Wall is a massive galactic superstructure in a region of the sky seen in the data set mapping of gamma-ray bursts (GRBs) that has been found to have an unusually higher concentration of similarly distanced GRBs than the expected average distribution. It is the largest known formation in the universe, exceeding the size of the prior Huge-LQG by about two times.

This overdensity lies at the Second, Third and Fourth Galactic Quadrants (NQ2, NQ3 and NQ4) of the sky. Thus, it lies in the Northern Hemisphere, centered on the border of the constellations Draco and Hercules. The entire clustering consists of around 19 GRBs with the redshift ranges between 1.6 and 2.1…

With a mean size in excess of 2 billion to 3 billion parsecs (6 to 10 billion light-years), the Hercules- Corona Borealis Great Wall contains many billions of galaxies, depending on how they are counted.

The current most plausible explanation for the existence of the clustering is a supercluster within the region that shows a high rate of star formation. Since GRBs are linked with massive stars, such stars form only on regions with more matter. Although large superclusters are known in the universe, a supercluster would have to be exceptionally immense to explain the clustering, perhaps 30 to 50 times larger and 200 times the volume of expected typical superclusters. It would be 10 billion light-years away and about 10 to 18 billion light-years across, and would perhaps be even more improbable to form in the universal large-scale structure than the GRB clustering would be on the gamma-ray sky.

This is another article of Wikipedia about the possible origin of GRBs: Because of the immense distances of most gamma-ray burst sources from Earth, identification of the progenitors, the systems that produce these explosions, is challenging. The association of some long GRBs with supernovae and the fact that their host galaxies are rapidly star-forming offer very strong evidence that long gamma-ray bursts are associated with (collapsing) massive stars…

The massive-star model probably does not explain all types of gamma-ray burst. There is strong evidence that some short-duration gamma-ray bursts occur in systems with no star formation and no massive stars, such as elliptical galaxies and galaxy halos. The favored theory for the origin of most short gamma-ray bursts is the merger of a binary system consisting of two neutron stars… Numerous other models have also been proposed to explain short gamma-ray bursts, including the merger of a neutron star and a black hole, the accretion-induced collapse of a neutron star, or the evaporation of primordial black holes.


Perhaps the Hercules- Corona Borealis Great Wall is a region in the sky with a high rate of star formation. But it is also possible that this structure is a clustering of old dying stars and evaporating black holes. Such a megastructure could be an analogue of the globular clusters which are located in the halo of our galaxy and revolve around the Milky Way in random orbits. Due to their location in the halo of the galaxy, such regions have never been star-forming regions. In analogy megastructures like the Hercules- Corona Borealis Great Wall can be a collection of such ‘globular’ superclusters of galaxies this time, perhaps dwarf and irregular galaxies loosely related to each other, located in the halo of the universe, remnants of an early stage in the history of the universe.

2.2.6 The cosmological principle

This is a list of the largest cosmological structures:

Size (in billions of light years)
Distance from Earth (in billion light years)
Hercules–Corona Borealis Great Wall
Discovered through gamma-ray burst mapping, and is the first structure to exceed 10 billion light-years.
Giant GRB Ring
Discovered through gamma-ray burst mapping. Largest-known regular formation in the observable Universe.
Decoupling of 73 quasars. Largest-known large quasar group and the first structure found to exceed 3 billion light-years.
U1.11 LQG
Involves 38 quasars. Adjacent to the Clowes-Campusano LQG.
Clowes- Campusano LQG
Grouping of 34 quasars. Discovered by Roger Clowes and Luis Campusano.
Sloan Great Wall
Discovered through the 2dF Galaxy Redshift Survey and the Sloan Digital Sky Survey.
(Theoretical limit)

Structures larger than this size are incompatible with the cosmological principle according to all estimates.

In the same articles about the HLQG (Huge Large Quasar Group), and the U1.11 Large Quasar Group, Wikipedia also mentions that,

According to the cosmological principle, the random distribution of matter and energy within the different parts of the universe must be approximately homogeneous and isotropic, and that random overdensities of these objects must be small if projected on a large enough scale. Some scientists project that the maximum structural sizes are somewhere around 260 h/Mpc, while others give values of 70-130 h/Mpc. More recent calculations suggest values within 370 Mpc (1.2Gly). However, some structures, such as Hercules- Corona Borealis Great Wall, exceed the scale by a factor of 8. Given also the proximity of the Huge-LQG, to U1.11, CCLQG and U1.54 (other Large Quasar Groups), it will be a big contradiction to the modern cosmological model.

Apparently, at distances greater than about 9-10Gly the universe is no longer homogeneous. This is at least what the data suggests (see previous table). This is also what the model we have put forward suggests. Here I will give some examples:

With respect to our own galaxy, the Milky Way, this is some related data about its dimensions,

Astronomers have found a thick disc in the Andromeda Galaxy for the first time

The thin disk is a structural component of spiral and S0-type galaxies, composing of stars, gas and dust. The Milky Way’s thin disk is thought to have a scale height of around 300-400 parsecs (980-1,300ly) in the vertical axis perpendicular to the disk, and a scale length of around 2.5-4.5 kiloparsecs (8.2-14.7kly) in the horizontal axis, in the direction of the radius. The thin disk contributes about 85% of the stars in the Galactic plane and 95% of the total disk stars.

The Galactic thin disk of the Milky Way is estimated to have been formed 8.8±1.7 billion years ago. It is considered to be considerably younger than the thick disk…

The thick disk is one of the structural components of about 2/3 of all disk galaxies, including the Milky Way. It is supposed to dominate the stellar number density between 1 and 5 kiloparsecs (3.3 and 16.3 kly) above the Galactic plane and, in the solar neighborhood, is composed almost exclusively of older stars. Compared to the thin disk, thick disk stars typically have significantly lower levels of metals- that is, the abundance of elements other than hydrogen and helium…

The Galactic Center is the rotational center of the Milky Way. The estimates for its location range from 7.6 to 8.7 kiloparsecs (about 25,000 to 28,000 light years) from Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest. There is strong evidence consistent with the existence of a supermassive black hole at the Galactic Center of the Milky Way…

In the inner few kpc (around 10,000 light-years radius) is a dense concentration of mostly old stars in a roughly spheroidal shape called the bulge.

This is a table with the data about our galaxy (approximate values):

Aspect of the galaxy
Respective size
Ratio (with respect to the radius of the galaxy)
Size (height) of thin disk on each side
Size (height) of thick disk on each side
Radius of central bulge
Size (width) of GHZ


The concentration of old stars in the bulge and the thick disk of our galaxy (in contrast to the younger stars of the thin disk) suggests that both the bulge and the thick disk were formed earlier than the thin disk. In fact the age of the thin disk is estimated at 9-10 billion years ago, while the galaxy as a whole (i.e. the bulge and the thick disk, together with the halo) is almost as old as the universe.

Thus by analogy we may assume a similar structure for the universe as a whole. This is a related table:

Aspect of the universe
Ratio (with respect to the radius of the universe)
Respective size
Radius of the universe
Height of thin disk on each side
Height of thick disk on each side
Radius of central bulge
Size (width) of UHZ

If the central bulge of the universe has a radius of approximately RCB=10Gly, and within this radius are located the LQGs (Large Quasar Groups), then since our distance from the COU (center of the universe), presumably, is RG= 19Gly≈ 20Gly, then RG-RCB≈ 10Gly will be the distance of the first LQG of the central bulge from us.

On the other hand, the Hercules- Corona Borealis Great Wall (Great GRB Wall) may be something analogous to the globular clusters of our galaxy, located at the edges of the thick disk of the universe, 10Gly from us.

This same size of ≈10Gly- which is about equal to the width of the UHZ (Universal Habitable Zone)- may be considered the size of homogeneity. Beyond this limit are located the LQGs on one side (towards the center), and the GRB Walls on the other side (towards the top). This size does not refer to the size of the structure itself, but to the limit after which homogeneity breaks down, in the same sense that homogeneity breaks down in a galaxy, towards the sparser edges, or towards the denser center.

Therefore the true meaning of the cosmological principle is found in such analogies, or coincidences, from the smallest structure to the largest, within the symmetry of the universe as a whole. Still it is remarkable that we are able not only to think about, but also to understand those coincidences. Why? We will say more about this in the section about the anthropic principle.

2.2.7 Inflation theory

According to the Big Bang theory, the universe began as a point of infinite density, and then rapidly expanded outwards. This is a related article:

But what is the Inflation Theory?

The Inflation Theory proposes a period of extremely rapid (exponential) expansion of the universe during its first few moments. It was developed around 1980 to explain several puzzles with the standard Big Bang theory, in which the universe expands relatively gradually throughout its history.

While the Big Bang theory successfully explains the "blackbody spectrum" of the cosmic microwave background radiation and the origin of the light elements, it has three significant problems:

The Flatness Problem:
WMAP has determined the geometry of the universe to be nearly flat. However, under Big Bang cosmology, curvature grows with time. A universe as flat as we see it today would require an extreme fine-tuning of conditions in the past, which would be an unbelievable coincidence.

The Horizon Problem:
Distant regions of space in opposite directions of the sky are so far apart that, assuming standard Big Bang expansion, they could never have been in causal contact with each other. This is because the light travel time between them exceeds the age of the universe. Yet the uniformity of the cosmic microwave background temperature tells us that these regions must have been in contact with each other in the past.

The Monopole Problem:
Big Bang cosmology predicts that a very large number of heavy, stable ‘magnetic monopoles’ should have been produced in the early universe. However, magnetic monopoles have never been observed, so if they exist at all, they are much more rare than the Big Bang theory predicts.

How does inflation solve this problems?

The Inflation Theory offers solutions to these problems and several other open questions in cosmology. It proposes a period of extremely rapid (exponential) expansion of the universe prior to the more gradual Big Bang expansion, during which time the energy density of the universe was dominated by a cosmological constant-type of vacuum energy that later decayed to produce the matter and radiation that fill the universe today.

The Flatness Problem:
Imagine living on the surface of a soccer ball (a 2-dimensional world). It might be obvious to you that this surface was curved and that you were living in a closed universe. However, if that ball expanded to the size of the Earth, it would appear flat to you, even though it is still a sphere on larger scales. Now imagine increasing the size of that ball to astronomical scales. To you, it would appear to be flat as far as you could see, even though it might have been very curved to start with. Inflation stretches any initial curvature of the 3-dimensional universe to near flatness.

The Horizon Problem:
Since Inflation supposes a burst of exponential expansion in the early universe, it follows that distant regions were actually much closer together prior to Inflation than they would have been with only standard Big Bang expansion. Thus, such regions could have been in causal contact prior to Inflation and could have attained a uniform temperature.

The Monopole Problem:
Inflation allows for magnetic monopoles to exist as long as they were produced prior to the period of inflation. During inflation, the density of monopoles drops exponentially, so their abundance drops to undetectable levels.

Inflation was both rapid, and strong. It increased the linear size of the universe by more than 60 ‘e-folds,’ or a factor of ~1026 in only a small fraction of a second! Inflation is now considered an extension of the Big Bang theory since it explains the above puzzles so well, while retaining the basic paradigm of a homogeneous expanding universe. Moreover, Inflation Theory links important ideas in modern physics, such as symmetry breaking and phase transitions, to cosmology.

As a bonus, Inflation also explains the origin of structure in the universe. Prior to inflation, the portion of the universe we can observe today was microscopic, and quantum fluctuation in the density of matter on these microscopic scales expanded to astronomical scales during Inflation. Over the next several hundred million years, the higher density regions condensed into stars, galaxies, and clusters of galaxies.


The Flatness Problem:
The universe is so big that it seems to be flat (no matter what the theory which describes the universe may be). Also according to an accretion model the universe will be flat due to rotation.

The Horizon Problem (the problem of homogeneity):
We have already seen that, according to observations, the universe seizes to be homogeneous at scales greater than ≈10Gly. If inflation theory is based on homogeneity, then the same observations contradict this theory.

The Monopole Problem:
Without being an expert on the subject, I imagine that the existence of magnetic monopoles (if any) may be linked to dark matter, regardless of whether the universe is expanding or contracting (although as the universe contracts dark matter may decrease, and vice-versa).

The problem of dark matter I believe is mostly significant, and there won’t be an answer until it is found out what dark matter is. But I guess that during the accretion process the dominant form of matter is dark matter, while as the universe accretes and spins around dark matter becomes hotter and hotter, till it gives its place to heat and visible (ordinary) matter. According to such a model (an accreting model), the universe never needed to be ‘inflated,’ because it covered the whole area since the beginning.

2.2.8 The accreting universe

Figure: A hypothetical structural division of the universe into three main different zones: 1) The grey zone is the halo of the universe, and includes the Great GRB Walls of aged and dissolved dwarf and irregular galaxies. 2) The green area is the Universal Habitable Zone, and includes most of the spiral galaxies like our own galaxy. 3) The yellow area is the center of the universe, and includes the Huge-Large Quasar Groups (and perhaps great elliptical galaxies). The white dot represents the center of our galaxy.

So why an accreting model of the universe?

Structure in the universe seems to be in accordance with both bottom- up and top- down theories. On one hand, the universe began as a conglomeration of particles of dark matter, which gradually condensed in order to produce the visible universe as we know it. On the other hand, structure in the universe appeared simultaneously everywhere, because the universe began accreting as a whole (although it seems that the thin disk of the universe appeared about 5 billion years after the outer layers and the central bulge).

Such an assumption (a model of accretion) is based on a vast range of evidence. Independently of my calculation about the distance of our own galaxy from the center of the universe, we have the following clues:

Stars which are as old as the universe. Our own galaxy (the halo and the central bulge) is as old as the universe. Thus there would be no sufficient time for stars and galaxies to form according to the Big Bang model (no matter how fast the universe may have inflated, according to Inflation Theory).

Large cosmological megastructures fit perfectly into a model of the universe consisting of an inhomogeneous structure:

On one hand, the Hercules- Corona Borealis Great GRB Wall, hanging at the edges of the universe, like globular clusters do in the case of a galaxy.

On the other hand, the Huge LQG has all the credentials to constitute the central bulge of the universe, towards the constellation of Leo- we also know the direction to search for. Thus the universe may have a center in the same sense that galaxies do.

Such a model offers us a unique view of the universe as a complete structure, so that we have the opportunity to better estimate the scales, and understand the universe as a whole.

Such an opportunity is also an advantage offered by the cosmological principle, a set of analogies which can be transferred from the personal, everyday scale, to the cosmological scale. Thus the cosmological principle is an expansion of the anthropic principle, the notion of which will be the subject of the epilogue of this document, in the next section.

3.1 Hubble’s constant

This is an article related to Hubble’s constant:

The Big Bang model was a natural outcome of Einstein’s General Relativity as applied to a homogeneous universe. However, in 1917, the idea that the universe was expanding was thought to be absurd. So Einstein invented the cosmological constant as a term in his General Relativity theory that allowed for a static universe. In 1929, Edwin Hubble announced that his observations of galaxies outside our own Milky Way showed that they were systematically moving away from us with a speed that was proportional to their distance from us. The more distant the galaxy, the faster it was receding from us. The universe was expanding after all, just as General Relativity originally predicted! Hubble observed that the light from a given galaxy was shifted further toward the red end of the light spectrum the further that galaxy was from our galaxy.

Expanding raisin bread loaf moves the raisins apart at different speeds with greater distance between raisins. The specific form of Hubble’s expansion law is important: the speed of recession is proportional to distance. Hubble expressed this idea in an equation- distance/time per megaparsec. A megaparsec is a really big distance (3.26 million light-years). The expanding raisin bread model at left illustrates why this proportion law is important. If every portion of the bread expands by the same amount in a given interval of time, then the raisins would recede from each other with exactly a Hubble type expansion law. In a given time interval, a nearby raisin would move relatively little, but a distant raisin would move relatively farther - and the same behavior would be seen from any raisin in the loaf. In other words, the Hubble law is just what one would expect for a homogeneous expanding universe, as predicted by the Big Bang theory. Moreover no raisin, or galaxy, occupies a special place in this universe- unless you get too close to the edge of the loaf where the analogy breaks down.

The current WMAP results show the Hubble Constant to be 71.0 ± 2.5 (km/sec)/Mpc. If the WMAP data is combined with other cosmological data, the best estimate is 70.4 ± 1.4 (km/sec)/Mpc.

This is an average value of Hubble’s constant:

Fit of redshift velocities to Hubble’s law. Various estimates for the Hubble constant exist. The HST Key H0 Group fitted type Ia supernovae for redshifts between 0.01 and 0.1 to find that H0≈71kms−1Mpc−1, while Sandage et al. find H0=62. kms−1Mpc−1.

Let’s keep this average value (previous graph):

It would be convenient here (also elucidating) to change the units of Hubble’s constant, in the following way,

It is straightforward to realize that Hubble’s constant has units of angular frequency (one over time). Thus it is a ‘constant’ which measures rotation.

Now, if we calculate the tangent speed of an object which is located at a distance equal to the radius RH of our event horizon (the horizon of the observable universe), from the formula which relates the tangent speed v of an object, its angular frequency ω, and the distance r of the object from the center of rotation,

we take

Thus, at a distance equal to the radius of our cosmological horizon an object will have a speed equal to the speed of light. This result shows that no observation was necessary to infer what is self-evident: Supposing that at the horizon of the observable universe an object will be moving at the speed of light, and knowing what this distance is (RH=13.8Gly), the angular frequency of this object (Hubble’s constant) can be calculated in advance.

Beyond this, a recent observation that the tangential speed of stars in the Milky Way seems to be constant, implies that the angular frequency with which the stars rotate decreases with distance. If what is true for stars in galaxies, is also true for galaxies in the universe, then the angular frequency with which galaxies are rotating (i.e. Hubble’s constant) should decrease with distance. Such an assumption would give us back the ‘static’ universe which Einstein so strongly envisioned- although in our case the universe, fundamentally, will not be expanding but rotating.

3.2 The golden spiral


Some have assumed that the GHZ (Galactic Habitable Zone) expands with time. Thus, what if the same were true for the UHZ (Universal Habitable Zone)? Basically, there is no reason to suggest such a kind of expansion. This is because the center of the galaxy is not expanding outwards, neither the halo of the galaxy is contracting inwards. Thus the GHZ is a stable zone, fixed in distance and time. The same will be true for the UHZ. However the notion of a ‘variable’ φ may turn out to be useful. Of course what changes with time will not be the value of φ itself, but the parameters related to this number. An example is offered through the notion of the golden spiral.

Here are some related articles,

Firstly, the notion of the Archimedean (linear, or arithmetic) spiral:

The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

with real numbers a and b. Changing the parameter a will turn the spiral, while b controls the distance between successive turnings.

The generalization of the Archimedean (linear or arithmetic) spiral, is the logarithmic (exponential or geometric) spiral:

A logarithmic spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, ‘the marvelous spiral.’

In polar coordinates (r,θ) the logarithmic curve can be written as

with e being the base of natural logarithms, and a and b being arbitrary positive real constants.

In parametric form, the curve is

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

The golden spiral is a special case of the logarithmic spiral, if the base is the number φ instead of e:

A golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. A golden spiral with initial radius a=1 has the following polar equation:

The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:

Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies or phyllotaxis of leaves. A recent analysis of spirals observed in mouse corneal epithelial cells indicated that some can be characterized by the golden spiral, and some by other spirals. It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both φ and the Fibonacci series. In truth, spiral galaxies and nautilus shells (and many mollusk shells) exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral. This pattern allows the organism to grow without changing shape.

First of all we may notice that the arithmetic spiral

is a linear approximation of the logarithmic spiral,

Τhe fact that the distances between the turns of a logarithmic spiral increase in geometric progression, can be seen as follows. Setting for simplicity,

Thus the distance r from the center, r0=a=1, increases by a factor of e at each cycle or turn:

where An is the sequence, and Sn is the sum of the successive terms.

In order to change the base from e to φ, we use the formula which transforms logarithms from one base to another,

so that

Setting again for simplicity,

where the common ratio here is φ instead of e.

Here we will make the following change. We will use a negative exponential in the previous formula:

By doing this, and using the same simplification,

we take

This series converges to the number φ.

If we add another term for θ=0, the series will be

In this case the golden spiral turns from the outside towards the inside (following an accreting mode). Also the formula with the negative exponential is more convenient to determine the boundary conditions. First we change the angle θ to time t, setting θ=ωt.

Choosing boundary conditions, we have

Thus the spiral turns inwards, from its full extent (equal to RU=49.7Gly, the radius of the universe) at time t=0, till it reaches us at time t=13.8Gy, at a distance RG=19Gly from the COU (center of the universe).

The same result could have been taken with the positive exponential by reversing the order: t=0, a=19Gly, and t=13.8Gy, r(t)= 49.7Gly. But a has the meaning of the initial radius of the universe, which is best explained by the negative exponential.

Now since,

we take,

Comparing this result about Ω, with Hubble constant H,

we realize that we may have found an analytical method to take Hubble constant, since Ω is identical to H.

This is a related graph:

Figure: Representation of the universe as a golden spiral

The previous graph gives the function (red dotted line)

in the parametric form

in the following way,

The two black circles have the parametric equations

representing the limits of the UHZ, with α=14.5Gly, and β=23.5Gly.

The ratio Ω/b was set equal to 1 for simplicity.

The previous graph represents an overall picture of the universe (the two axes are the coordinates x and y), while the time t is a parameter running on the spiral (red dotted line). In that sense the area enclosed by the two black circles (the UHZ) doesn’t change with time. This is expected if the universe has the same structure as a galaxy does. We should expect that the UHZ is located not too close to the hot center, not too far away in the sparse halo of the universe. Thus what can really change is not the number φ, but our understanding of the universe, in relation to the same number.

3.3 The anthropic principle


If our distance from the COU is 19Gly, and the universe has an age of about 14Gy, then the light from the ‘Big Bang,’ which is the center of the universe, will reach us in about 5Gy. This is exactly the time when our own Sun will be dying as an aged star.

This is another example of a coincidence based on the cosmological principle, or especially on the anthropic principle. According to Wikipedia, the anthropic principle can be defined as follows:

The anthropic principle is a philosophical consideration that observations of the Universe must be compatible with the conscious and sapient life that observes it.

The meaning of this principle is not that we know everything, but that we have the capacity of knowing, because we are made of the same laws and ratios which also made the universe possible. The number φ is an example of such a ratio.

The anthropic principle also unites nature and logic. Thus we may say,

The cosmological principle:
“What is a star to the galaxy is a galaxy to the universe.”

The anthropic principle:
“What is a star to the galaxy, or a galaxy to the universe, is a sparkle of thought to the human mind.”

Thus the cosmological principle is an extension of the anthropic principle from the anthropic to the cosmological level.

In such a sense,
“The center of the universe is not the center of ourselves.”

Towards the next universe

Extending the anthropic comparison, we may also include the following data:
The distance of the Andromeda galaxy (the closest galaxy to us) is about 2.5 million light years.

The Canis Major Dwarf Galaxy is classified as an irregular galaxy and is now thought to be the closest neighboring galaxy to the Earth’s location in the Milky Way, being located about 25,000 light-years away from the Solar System.

This is a table with the data:

Astronomical object
Distance of astronomical object from us
Ratio: Distance from us/Radius of the galaxy (50kly)
Andromeda galaxy
Canis Major Dwarf galaxy

And by analogy:

Astronomical object
Ratio: Distance from us/Radius of the universe (50Gly)
Distance of astronomical object from us
Closest universe
Closest dwarf universe

If the closest dwarf universe is at a distance of 25 billion light years from us, will it ever be possible to observe it? If that universe was created simultaneously with ours, will we have to wait about another 11 billion years till its light reaches us?

If nothing can travel faster than light, then we will never be able to observe such distant objects. But perhaps when gravitational astronomy advances (telescopes receiving gravitational waves instead of light) things will change. Gravitational lensing for example may be proved to be a technique by which an object can be observed even if its light had no sufficient time to reach us.

This is a rough example. If an astronomical object is located 25 billion years away from us (e.g. the closest dwarf universe), and the object was created 25 billion years ago, its light will be reaching us right now. But if the object was created simultaneously with us (our universe), we may suppose that the gravitational wave which connects that object to us will also be reaching us right now. Thus we may use light to estimate the distance, and the gravitational wave to explain the connection.

A more elaborate demonstration of such an aspect is related to the problem of the brachistochrone. This is a rough estimation. The time of the brachistochrone (which is also the tautochrone) is given by the following formula,

If g is the intensity of gravity in a region of space (thus also the intensity of the gravitational wave at the same region), and R is the distance between two points in the same region (presumably the wavelength of the gravitational wave), then the time T it takes the gravitational wave (a particle running on that wave) to connect the two points is approximately given by the previous formula.

The intensity of the gravitational wave in turn is given by the following formula

where G=6.674×10-11m3kgs-2 is Newton’s gravitational constant.

If the mass M(r) is a function of the distance from the center of the gravitational field, then

where ρ is the linear density.

If MU is the mass of the observable universe, and RH is the radius of the observable universe, then the density (if it is a constant) can be defined as

so that the acceleration of gravity can also be written as

In order to use this formula in units of light, we change the units of Newton’s gravitational constant G into units of light,

Thus the product will be

Here we may note the following coincidence with respect to Hubble’s constant. If we take the value of g at the horizon of the observable universe, it will be

Assuming that the maximum speed v an object can reach is the speed of light c, and that an object reaches the speed of light if it is located on the horizon of the observable universe, at a distance equal to the radius RH of the observable universe, then

The associated acceleration will be

The last value is exactly the same with the value of g(RH). This is why we have already said that Hubble’s constant can be estimated from first principles, independently of any observational data.

Now, to return where we have left, let’s assume that a gravitational wave has a wavelength λ equal to one light year, if we use the unit of light to measure the gravitational wave. If RH=13,8Gly is the radius of the observable universe, then the mass Mλ corresponding to a gravitational wave with a wavelength λ equal to one light year will be

where MU≈1053kg is the total mass of the observable universe.

Thus the intensity of gravity per wavelength λ of the gravitational wave will be

This result is remarkable on its own, because even if we have arbitrarily assumed that the wavelength λ of a gravitational wave is 1ly, the fact that the product is equal to 1 is not self-evident.

Here is another anthropic coincidence which further supports the result. The acceleration of gravity on Earth g=9.807m/s2 is equal to 1ly/y2:

The difference of about 3% between an acceleration of gravity exactly equal to 1ly/y2 and the previous value can be attributed either to some statistical error with respect to the previous values, or, perhaps, to the presence of the Earth as a gravitating body itself.

Nevertheless, the aspect that that the acceleration of gravity on the Earth’s surface is, more or less, equal to 1ly/y2, can be interpreted in two ways. One way is to assume that we have the privilege, due to some preferred location in the universe, so that the acceleration of gravity will be about equal to 1ly/y2 with respect to us. The other way is to assume that this anthropic coincidence is also a cosmological coincidence, based on a universal law. Thus we may assume that everywhere in the universe such an acceleration is constant and equal to g=1ly/y2. This value will also be the same per wavelength of a gravitational wave, if its wavelength is equal to 1ly.

Furthermore, it is possible to generalize the result for any size of wavelength. Instead of assuming that the density ρ of the universe is constant, we may suppose that it is also a function of distance,


The quantity σ will be the density ρ=Mλ, which we earlier mentioned, divided by distance, and it will be a constant. If Newton’s gravitational coefficient G is always a constant, then the acceleration g0= will also be a constant.

Thus we can say that the acceleration of gravity on Earth is equal to g0=1ly/y2 because such an acceleration is the same everywhere in the universe, independently of the distance between any two points in space and time. Thus what will stay constant is not the speed of the gravitational wave, but its acceleration.

In such a sense we may simulate the whole universe with a gravitational wave of wavelength equal to the linear size of the universe. Even more, we may equivalently imagine the whole distance between the center of our universe and the center of the nearest universe as a huge gravitational wave whose acceleration of gravity will be constant and equal to g=1ly/y2. If this distance is RNU=2.5Tly, then the time it will take for a gravitational wave from the nearest universe to reach us will approximately be

This result is simple (that the time is proportional to the square root of the distance), and gives us the chance to be able to detect another universe. Whether such an acceleration refers to gravity or some other, yet unknown, force, is another question. Still the fundamental problem is not only how we detect such huge wavelengths and analyze them, but also how we conceptualize the vast scales and acknowledge the underlying ratios.

What is Astronomy?

Quotes related to Phi

Golden ratio

Explore Golden Ratio, The Human Body, and more!

Phi and the Solar System

Habitable zone

Circumstellar habitable zone

The Planets in Our Solar System


What is the position of the sun relative to the center of the Galaxy?]

The Milky Way

Galactic habitable zone


Size of the Milky Way Upgraded, Solving Galaxy Puzzle

Age of the universe

Milky Way

Observable universe

Milky Way: Age and cosmological history

The Farthest Visible Reaches of Space

How do galaxies form?

Galaxy formation and evolution

List of the most distant astronomical objects

Oldest star

HD 140283

Definition of quasar

Quasar (definition)

Large quasar group


Clowes-Campusano LQG


Huge LQG- the Huge Large Quasar Group

Solar mass

Orders of magnitude (mass)

Is there a black hole at the center of the Galaxy?

Hercules-Corona Borealis Great Wall

Gamma-ray burst

Scientists hope disc of older stars found in Andromeda will reveal the building blocks of our own galaxy
[ galaxy.html]

Thin disk

Thick disk

Galactic Center

Milky Way: Galactic Center

What is the Inflation Theory?

Tests of Big Bang: Expansion

Hubble's law

Golden ratio

Archimedean spiral

Logarithmic spiral

Golden spiral

Observable universe logarithmic illustration

Anthropic principle

Andromeda Galaxy: Distance to Earth

Canis Major Overdensity

Orders of magnitude (mass): 1042 kg and greater

‘The universe in the golden ratio: A model based on the number φ,’
© 2017 Chris C. Tselentis
Last revised: May 2018


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