_{1}

^{*}

The nature and properties of dark matter and dark energy in the universe are among the outstanding open issues of modern cosmology. Despite extensive theoretical and empirical efforts, the question “what is dark matter made of?” has not been answered satisfactorily. Candidates proposed to identify particle dark matter span over ninety orders of magnitude in mass, from ultra-light bosons, to massive black holes. Dark energy is a greater enigma. It is believed to be some kind of negative vacuum energy, responsible for driving galaxies apart in accelerated motion. In this article we take a relativistic approach in theorizing about dark matter and dark energy. Our approach is based on our recently proposed Information Relativity theory. Rather than theorizing about the identities of particle dark matter candidates, we investigate the relativistic effects on large scale celestial structures at their recession from an observer on Earth. We analyze a simplified model of the universe, in which large scale celestial bodies, like galaxies and galaxy clusters, are non-charged compact bodies that recede rectilinearly along the line-of-sight of an observer on Earth. We neglect contributions to dark matter caused by the rotation of celestial structures (e.g., the rotation of galaxies) and of their constituents (e.g., rotations of stars inside galaxies). We define the mass of dark matter as the complimentary portion of the derived relativistic mass, such that at any given recession velocity the sum of the two is equal to the Newtonian mass. The emerging picture from our analysis could be summarized as follows: 1) At any given redshift, the dark matter of a receding body exists in duality to its observable matter. 2) The dynamical interaction between the dark and the observed matter is determined by the body’s recession velocity (or redshift). 3) The observable matter mass density decreases with its recession velocity, with matter transforming to dark matter. 4) For redshifts z < 0.5, the universe is dominated by matter, while for redshifts z > 0.5 the universe is dominated by dark matter. 5) Consistent with observational data, at redshift z = 0.5, the densities of matter and dark matter in the universe are predicted to be equal. 6) At redshift equaling the Golden Ratio (z ≈ 1.618), baryonic matter undergoes a quantum phase transition. The universe at higher redshifts is comprised of a dominant dark matter alongside with quantum matter. 7) Contrary to the current conjecture that dark energy is a negative vacuum energy that might interact with dark matter, comparisons of our theoretical results with observational results of ΛCDM cosmologies, and with observations of the relative densities of matter and dark energy at redshift z ≈ 0.55, allow us to conclude that
*dark energy is the energy carried by dark matter*. 8) Application of the model to the case of rotating bodies, which will be discussed in detail in a subsequent paper, raises the intriguing possibility that the gravitational force between two bodies of mass is mediated by the entanglement of their dark matter components.

The nature of Dark Matter is one of the most pressing open problems of modern cosmology [

Another popular class of dark matter candidates are baryonic matter tied up in brown dwarfs or in chunks of massive compact halo objects, or MACHOs [

Dark energy remains a complete mystery. The common prejudice is that it is some unknown substance, with an enormous anti-gravitational force, which drives the galaxies of our universe apart in accelerated motion [^{120} times the measured ratio of pressure to energy density. An alternative explanation argues that dark energy is an unknown dynamical fluid, i.e., one with a state equation that is dynamic in time, Λ = Λ(t). This type of explanation is represented by theories and models that differ in their assumptions regarding the nature of the state equation dynamics [

Given the lack of knowledge about the nature of dark matter and dark energy, most experts contend that understanding the content of the universe and its cosmic acceleration requires nothing less than discovering a new physics. As an example, the Dark Energy Task Force (DETF) summarized its 2006 comprehensive report on dark energy by stating that there is consensus among most physicists that “nothing short of a revolution in our understanding of fundamental physics will be required to achieve a full understanding of the cosmic acceleration” [

Here we take a completely different approach than all contemporary physics to explain dark matter, dark energy, and the interaction between matter and dark matter in the universe. Our approach is relativistic, based on our recently proposed Information Relativity theory (previously termed Complete Relativity theory) [

The following sections are organized as follows: In section 2 we summarize the main tenets of Information Relativity theory, and depict its main transformations. In section 3 we apply the theory to a simplified model of the Universe, in which large-scale celestial bodies, like galaxies and galaxy clusters, are compact bodies receding rectilinearly along the line-of-sight of an observer on Earth. In our simplistic model we neglect contributions to dark matter caused by the rotation of celestial structures (e.g., the rotation of galaxies), and of their constituents (e.g., rotation of stars inside galaxies). For such a grossly simplified universe, we define dark matter, and derive exact terms for the dynamics between matter and dark matter densities, and of their respective energy distributions in the universe, as functions of the recession velocity. In section 4 we express the derived terms as functions of the redshift z, and utilize the theoretical results to explain the GZK cosmic rays cutoff at z ≈ 1.6 [

A complete formulation of information relativity theory (IR) and its applications to various field in physics, including small particles physics, quantum mechanics, and cosmology, are detailed elsewhere [

Before describing the model, and its application to the cosmology of the universe, it is in order here to caution the reader that he or she will probably find difficulty in the absence of the terminology used in the literature. Primarily, we do not use the concept of spacetime. Space and time are treated by us independently, just like Newton did. We refrain from using variables that are not completely understood physically like vacuum energy, or arbitrary parameters like the cosmological constant. All the terms used to construct the theory are observable physical variables.

In principle, information relativity theory is nothing more than “relativizing” Newtonian physics, which we accomplished by accounting for the time travel of information from one reference frame to another. As will become clear from the derivation of the theory’s transformations, the scale of the system is of no importance. In several previous articles we showed that not only does the theory reproduce quantum theoretical results, including entanglement [

We note that unlike Special Relativity theory, in which the relativity of time is achieved by axiomatizing the constancy of light velocity, relativizing time, and other physical entities in Information Relativity theory is a force majeure of the fact that information does not pass between two points in space instantaneously, but rather suffers delay, which depends on the spatial distance between the two points and the velocity of the information carrier.

The rationale behind the theory is extremely simple and straightforward. It can be illustrated as follows: Consider the case where information from a “moving” body, is transmitted to a “stationary” observer by light signals. Assume that the start and end of an occurrence on the body’s reference frame are indicated by two signals sent from the body’s “moving” reference frame to the “stationary” observer. Because light’s velocity is finite, the two signals will arrive to the observer’s reference frame with delays, determined by the distances between the body and the observer at the time when each signal was transmitted. Suppose that the “moving” body is distancing from the observer. In this case, the termination signal will travel a longer distance than the start signal. Thus, the observer will measure a longer occurrence duration than the occurrence duration at the body’s reference frame (time dilation). For approaching bodies, the termination signal will travel a shorter distance than the start signal. Thus, the observer will measure a shorter duration than the duration at the body’s reference frame (time contraction). Notice that for the above scenario, no synchronization of the clocks at the two reference frames is required.

For the simple case of rectilinear motion with constant velocity v, expressing the above mentioned example in the language of mathematics detailed elsewhere [

Δ t Δ t 0 = 1 1 − β (1)

where Δ t is the occurrence’s time duration as measured by the observer, Δ t 0 is the occurrence’s time duration at the body’s rest-frame, and β is the relative

velocity, β = v c . Derivations of the transformations of length, mass, and energy,

are detailed in Appendix A. The resulting transformations are depicted in the middle column of

Variable | Transformation in terms of velocity | Transformation in terms of redshift |
---|---|---|

Time interval Δ t Δ t 0 | 1 1 − β | z + 1 (1') |

Length l l 0 | 1 + β 1 − β (2) | 2z + 1 (2') |

Matter density ρ M ρ 0 | 1 − β 1 + β (3) | 1 2 z + 1 (3') |

Matter energy density e M e 0 | 1 − β 1 + β β 2 (4) | z 2 ( z + 1 ) 2 ( 2 z + 1 ) (4') |

Dark matter density ρ D M ρ 0 | 2 β 1 + β (5) | 2 z 2 z + 1 (5') |

Dark matter energy density e D M e 0 | 2 β 3 1 + β (6) | 2 z 3 ( z + 1 ) 2 ( 2 z + 1 ) (6') |

The dark (unobservable) matter density term is defined in the present framework as the difference between the Newtonian and relativistic matter density terms, or:

ρ D M ρ 0 = 1 − ρ M ρ 0 = 1 − 1 − β 1 + β = 2 β 1 + β (5)

Similarly, the energy density carried by the dark matter is defined as the difference between the Newtonian and relativistic matter energy densities:

e D M e 0 = β 2 − 1 − β 1 + β β 2 = 2 β 3 1 + β (6)

The picture emerging from our analysis thus far suggests that at any given recession velocity, matter and dark matter co-exist in a dynamic duality, with matter decreasing in density and dark matter increasing in density as functions of the recession velocity β. Thus, in our model matter and dark matter are interchangeable, accelerating the velocity of a receding body of mass is predicted to increase its dark matter at the expense of matter, vice versa, decelerating a receding body of mass is predicted to increase the portion of its matter at the expense of dark matter.

It is worth noting that as β → 0, the density of dark matter and its associated energy ( ρ D M , e D M ) approach zero, and Newtonian mechanics under constant rectilinear velocity is recovered. Also, since no restrictions were put on the size of the moving mass, the derived results apply equally well to mesoscopic and microscopic rectilinearly receding bodies of mass. Generalization of the results for motion in a gravitational field is detailed in [

To construct a simple cosmology of the universe we assume a grossly simplified model in which galaxies and galaxy structure are infinitesimal, uncharged, and non-rotating masses, receding from us. We take this as an approximation of the universe at high enough redshifts. For any celestial body receding from us with velocity β,

As could be seen, the densities of matter and dark matter are mirror images of

one another. They coincide when the recession velocity is exactly β = v c = 1 3 , with matter dominating at recession velocities β < 1 3 , and dark matter dominating at recession velocities β > 1 3 . As examples, for a recession velocity β =

0.01 the model predicts a composition of about 98% matter and 2% dark matter, whereas as for a high relativistic velocity of β = 0.99 the predicted composition is 0.5% matter and 99.5% dark matter. Metaphorically, we can think of matter and dark matter as two strictly competitive Siamese twins, with one increasing its density at the expense of the other. The ratio of dark matter to matter densities depends only on the recession velocity as depicted in the following equation (see also

ρ D M ρ M = 2 β 1 − β (7)

_{cr}, and then decreases to zero at β = 1. The emerging type of non-monotonicity is quite surprising. No less surprising is the value of β, at which the matter energy density achieves its unique maximum. Differentiating Equation (4) with respect to β, and equating the result to zero yields:

β 2 + β − 1 = 0 (8)

which solves for:

β c r = 5 − 1 2 = φ ≈ 0.618 (9)

where φ is the famous Golden Ratio [

( e M ) max = e 0 φ 2 1 − φ 1 + φ (10)

From Equation (8) we can write: φ 2 + φ − 1 = 0 , which implies 1 − φ = φ 2 and 1 + φ = 1 φ . Substitution in Equation (10) gives:

( e M ) max = φ 5 e 0 ≈ 0.09016994 e 0 (11)

the relative matter energy density at this point, ( e M ) max e 0 , is equal (to the eighth

decimal digit) to Hardy’s maximum probability of obtaining an event that contradicts local realism [

Since our objective is to apply the theory to the cosmology of the universe, the theory transformations in terms of redshift are added in the right column in

z = β 1 − β (12)

And the inverse relationship is:

β = z z + 1 (13)

Substituting Equation (13) in the transformations depicted in the middle column in

As shown in

z < 0.5 (corresponding to recession velocities β < 1 3 ), and dark matter is predicted to dominate at redshifts z > 0.5.

Our model proposes a completely different, and far more plausible explanation about the nature of dark energy. It is not a mysterious negative vacuum energy.

It is the energy carried by the dark matter. Luckily, the name given to dark matter fits well with our explanation. As a post hoc test of our explanation of dark energy, we calculated the proportions of the energies of matter and dark matter at a given redshift range z and compared it with observed results based on ΛCDM cosmologies.

From Equations (4’) and (6’) in

e M e M + e D M = z 2 ( z + 1 ) 2 ( 2 z + 1 ) z 2 ( z + 1 ) 2 ( 2 z + 1 ) + 2 z 3 ( z + 1 ) 2 ( 2 z + 1 ) = 1 1 + 2 z (14)

And,

e D M e M + e D M = 2 z 3 ( z + 1 ) 2 ( 2 z + 1 ) z 2 ( z + 1 ) 2 ( 2 z + 1 ) + 2 z 3 ( z + 1 ) 2 ( 2 z + 1 ) = 2 z 1 + 2 z (15)

The ratio of the energy densities of the dark to matter is:

e D M e M = 2 z (16)

The energy of matter and dark matter in any redshift range ( z 1 , z 2 ) are obtained by integrating expressions 14 and 15 with respect to z, yielding:

e M ( z 1 − z 2 ) e 0 = 1 2 ln ( 2 z 2 + 1 2 z 1 + 1 ) − z 2 − z 1 ( z 2 + 1 ) ( z 1 + 1 ) (17)

And,

e D M ( z 1 − z 2 ) e 0 = ( z 2 − z 1 ) + 2 z 2 − z 1 ( z 2 + 1 ) ( z 1 + 1 ) − 2 ln ( z 2 + 1 z 1 + 1 ) − 1 2 ln ( 2 z 2 + 1 2 z 1 + 1 ) (18)

Calculations based on the above expressions are in good agreement with observations. As an example, it was concluded by Wittman et al. (2000) [

e M e t o t = e M e M + e D M = 0.0300775 0.0300775 + 0.0486354 ≈ 0.382 ( ≈ 38.2 % ) (19)

And,

e D M e t o t = e D M e M + e D M = 0.0486354 0.0300775 + 0.0486354 ≈ 0.618 ( ≈ 61.8 % ) (20)

which are in good agreement with the observations based ΛCDM model with

( Ω m = 1 3 , Ω Λ = 2 3 ). For the entire range of semi-classical matter (0 ≤ z < 1.618) we obtain: e M e 0 ≈ 0.1038 , and e D M e 0 ≈ 0.3420 , yielding:

e M e M + e D M = 0.138 0.138 + 0.3420 ≈ 0.233 (or 23%) (21)

And,

e D M e M + e D M = 0.3420 0.138 + 0.3420 ≈ 0.767 (or 76.7%) (22)

Which are in excellent agreement with the ΛCDM cosmology with Ω matter = 0.23 , Ω Λ = 0.77 [

Another supporting evidence for our conjecture that dark energy is the energy carried by dark matter, is the recent observations of what is termed the “coincidence problem”, namely, the question why “now” at a redshift z ≈ 0.55 the densities of matter and dark energy are equal [

equal at z = 0.5 (corresponding to a recession velocity of β = 1 3 ). Considering

the simplified assumptions made in the discussed model of the universe, the theoretical prediction is in good agreement with the ‘coincidence problem” observational data.

The nature and properties of dark matter and dark energy in the universe are among the outstanding open issues of modern cosmology. The tremendous effort to answer the question “what is dark matter made of?” has so far been unsatisfactory. Most effort was, and is still being, invested in searches for particle dark matter candidates, partly within the Standard Model of elementary particles, but mostly among its supersymmetric extensions.

The different approach taken here is based on our recently proposed Information Relativity theory [

1) For a body in recession, matter and dark matter co-exist in a dynamic duality, with matter decreasing in density, and dark matter increasing in density as functions of the recession velocity β (and Redshift z) (see

2) Contrary to the current conjecture that dark energy is something different than dark matter, a mysterious negative vacuum energy of unknown source, our model suggests that dark energy is the energy carried by dark matter. With respect to the discovery that the universe expands with acceleration [

unknown source, but rather the aftermath of the decrease (as function of 1 r 2 ) in

the gravitation pulls between galaxies as they become more distant from one another. Metaphorically speaking, we contend that the universe is accelerating, not because someone is pressing on the gasoline pedal, but because he or she is gradually releasing the pressure from the brake pedal.

3) At a recession velocity β equaling the Golden Ratio (≈ 0.618), or equivalently z ≈ 1.618, matter undergoes a critical quantum phase transition (see

4) Matter dominates the universe at z < 0.5, and dark matter dominates at z > 0.5. In good agreement with observational data, the two components are predicted to be equal at z = 0.5. This could provide a physical explanation to the “coincidence problem”, namely the question why “now”, at a redshift z ≈ 0.55, the densities of matter and dark energy are equal [

5) The emergence of the Golden Ratio symmetry in our analysis as a key player in the structure of the Universe, echoes nicely with many discovered Golden Ratio symmetries in nature, science, technology, and the arts, including in the structure of plants [

We note that the emerging duality between matter and dark matter holds much in common with the realistic matter-wave duality in quantum mechanics, first conjectured by Louis de Broglie [

It is worthwhile to stress that the cosmology proposed here is based on a very simplified model, in which galaxies, or even galaxy-clusters, are represented as point masses receding rectilinearly along the line of sight of an observer on Earth. Real galaxies and galaxy structures are in rotation relative to an Earth’s observer, and the stars and planets in each galaxy are also in continuous rotation. The observed rotations of all celestial structures are major sources of dark and quantum matter. The process of their production due to the structures’ rotational motion could be analyzed in a similar way to the current analysis. We have already embarked on the analysis of the production of dark and quantum matter due to rotational motion, and of the spatial entanglement between the dark matter dispersed spherically at their halos. We conjecture that the physical entanglement of dark matter in space is medium by which gravitational forces between bodies are enacted. It is well accepted that dark matter is the “gravitational glue” that keeps galaxies and galaxy structures bound together. Our preliminary analysis confirms this.

I wish to thank Dr. Raid Saabni and three anonymous reviewers for their helpful remarks.

The authors declare no conflicts of interest regarding the publication of this paper.

Suleiman, R. (2018) A Model of Dark Matter and Dark Energy Based on Relativizing Newton’s Physics. World Journal of Condensed Matter Physics, 8, 130-155. https://doi.org/10.4236/wjcmp.2018.83009

In sections 1a-4a we derive the theories transformations of time, length, matter density, and energy density, as functions of the recession velocity β.

Consider two reference frames, F and F ′ , moving with constant velocity v with respect to each other. A “stationary” observer in frame F defines events with coordinates t, x, y, z. Another observer in F ′ defines events using the coordinates t', x', y', z'. For simplicity, assume that the coordinate axes in each frame are parallel (x is parallel to x', y to y', and z to z'), and that the two systems are synchronized, such that at t = t ′ = 0 , ( x , y , z ) = ( x ′ , y ′ , z ′ ) = ( 0 , 0 , 0 ) . Now consider the case in which a physical occurrence starts at the point of origin in F ′ at t ′ 1 = 0 , and lasts for a period of Δ t ′ = t ′ 2 − t ′ 1 = t ′ − 0 = t ′ as measured by an observer at the occurrence rest frame in F ′ . We use the term occurrence to denote a physical time-dependent process. By this we seek to differentiate it from the term “event”, which in physics came to mean an occurrence that is sharply localized at a single point in space and instant of time. The observer at F has no way of knowing when the occurrence at F ′ ended, unless information is sent to him/her from the observer at F ′ indicating the termination of the occurrence. Such information could be sent by any type of information carrier as long as its velocity, V_{c}, exceeds the relative velocity v at which F ′ is departing from F, i.e., V_{c} should satisfy V_{c} > v. After t seconds for an observer in F, the reference frame F ′ will be at distance x = vt. Thus, the information about the termination of the physical occurrence will arrive to the observer at F with a delay of:

t d = x V c = v t V c . (1a)

The termination time registered by the observer at F will be

t = t ′ + v t V c , (2a)

which could be written as:

t = 1 1 − v V c t ′ = 1 1 − β t ′ , (3a)

where β = v V c .

Quite interestingly, Equation (3a), is quite similar to the Doppler’s Formula derived for the frequency modulation of waves emitted from traveling bodies. Importantly, in both cases the direction of motion matters. In the Doppler Effect a wave emitted from a distancing body will be red-shifted (longer wavelength), whereas a wave emitted from an approaching body with be blues-shifted (shorter wavelength). The degree of red, or blue shift is an increasing function of the body’s velocity. The same applies to the time duration of an occurrence taking place at a stationary point on a moving frame. If the frame is distancing from F, the time measured at F will be dilated relative to the time measured at frame F ′ , whereas if the frame is approaching F, the time measured at F will contract.

It is important to note further that the above derived transformation applies to all carriers of information, including the commonly employed acoustic and optical communication media. For the case in which information is carried by light, with velocity c, or by electromagnetic waves with equal velocity, Equation (3a) becomes:

Δ t Δ t ′ = 1 1 − v c = 1 1 − β (4a)

where β = v c .

Evidently, the derived result does not agree with the time dilation prescribed by Special Relativity under the assumption of constancy of the velocity of light in vacuum. We prefer here to avoid the above mentioned inconsistency, by strictly limiting our theorizing to cases in which the information carrier wave propagates in velocities c < c_{0}, where c_{0} is the velocity of light in vacuum. Putting such a limit does depreciate the value of our result, since it will still apply to all physical situations of interest.

To derive the distance transformation, assume that F ′ has on board a rod placed along its x’ axis between the points x ′ = 0 and x ′ = x ′ 2 , and that the observer in F ′ uses his clock to measure the length of the rod (in its rest frame) and communicates his measurement to the observer in F. Assume that the information carrier from frame F ′ to frame F travels with constant velocity V c (as measured in the source rest frame). To perform the measurement of the rod’s length, at t ′ 1 = t 1 = 0 , a signal is sent from the rare end of the rod, i.e., from x ′ = x ′ 2 to the observer at the point of origin x ′ = 0 . The length in F ′ (the rest frame) is calculated as:

l 0 = x ′ 2 = V c t ′ 2 . (5a)

Denote by Δ t c is the time duration in the signal’s rest frame for its arrival to the observer in F ′ . Using Equation (3a), t ′ 2 as a function of Δ t c can be expressed as

t ′ 2 = 1 1 − − v V c Δ t c = 1 1 + v V c Δ t c , (6a)

which could be rewritten as:

Δ t c = ( 1 + v V c ) t ′ 2 . (7a)

Because F ′ is departing F with velocity v, the signal will reach an observer in F at time t_{2} equaling

t 2 = Δ t c + v t 2 V c = Δ t c + v V c t 2 . (8a)

Substituting the value of Δ t c from Equation (7a) in Equation (8a) yields:

t 2 = ( 1 + v V c ) t ′ 2 + v V c t 2 , (9a)

which could be rewritten as:

t 2 = ( 1 + v V c ) ( 1 − v V c ) t ′ 2 . (10a)

Substituting the value of t ′ 2 from Equation (5a), we get:

t 2 = ( 1 + v V c ) ( 1 − v V c ) l 0 V c . (11a)

Thus, the observer in F will conclude that the length of the rod is equal to

l = V c t 2 = ( 1 + v V c ) ( 1 − v V c ) l 0 (12a)

or,

l l 0 = 1 + β 1 − β , (13a)

where β = v V c .

Regardless of the value of V c , the above derived relativistic distance equation predicts distance contraction only when the two reference frames approach each other (i.e., for −1 < β ≤ 0). On the other hand, in contradiction of the famous Lorentz contraction, for distancing frames (i.e., 0 < β < 1) Equation (13a) predicts length extension.

Let us assume that that the rod has a total rest-mass m 0 distributed uniformly along the x axis. According to Equation 13a an approaching rod will contract causing the mass density along the x axis to increase. On the other hand, a distancing rod will extend, causing its mass density along the x axis to dilute. The

body’s mass density in its rest-frame is ρ 0 = m 0 A l 0 , where A is the area of the body’s cross section, perpendicular to the direction of movement. In F the density is given by: ρ M = m 0 A l , where l is the object’s length in F. Using the length transformation (Equation (13a)) we can write:

ρ M = m 0 A l = m 0 A l 0 ( 1 + β 1 − β ) = ρ 0 1 − β 1 + β (14a)

Or,

ρ M ρ 0 = 1 l / l 0 = 1 − β 1 + β (15a)

For the case in which information is carried by waves traveling at the velocity

of light we set β = v c . As could be seen from Equation (15a), the relativistic

mass density is inversely proportional to the relativistic length. It is predicted to increase for approaching bodies and a decrease for distancing bodies. For applications to cosmology, the unobservable (Dark) matter is defined as the difference between the Newtonian, and relativistic mass densities, or:

ρ D M ρ 0 = 1 − 1 − β 1 + β = 2 β 1 + β (16a)

at velocity of recession equaling exactly β = v c = 1 3 . The ratio of dark matter to

matter densities, is given by:

ρ D M ρ M = 2 β 1 − β (17a)

The matter energy density of a moving body with velocity v, and rest density ρ_{0} is given by:

e M = 1 2 ρ v 2 = 1 2 ρ 0 c 2 1 − β 1 + β β 2 = e 0 1 − β 1 + β β 2 (18a)

where e 0 = 1 2 ρ 0 c 2 . For β → 0 (or v ≪ c ) Equation (18a) reduces to e = 1 2 ρ 0 v 2 , which is the classical Newtonian expression.

β 2 + β − 1 = 0 (19a)

which solves for:

β c r = 5 − 1 2 = φ ≈ 0.618 (20a)

where φ is the famous Golden Ratio [1a] [2a]. Substituting β c r in Equation (18a) yields:

( e M ) max = e 0 φ 2 1 − φ 1 + φ (21a)

From Equation (19a) we can write: φ 2 + φ − 1 = 0 , which implies 1 − φ = φ 2 and 1 + φ = 1 φ . Substitution in Equation (21a) gives:

( e M ) max = φ 5 e 0 ≈ 0.09016994 e 0 (22a)

The results in Equations (20a) and (22a) are quite amazing, given the key role played by the Golden Ratio in nature, in nature, technology and the arts, including in the structure of plants, physics, the structure of the human brain, music, aesthetics, the social sciences, and more (see relevant references in the main text).

The energy carried by dark matter is given by:

e D M = 1 2 ρ D M v c 2 = 1 2 ρ D M c 2 β 2 = ( 1 2 ρ 0 c 2 ) 2 β 1 + β β 2 = ( 1 2 ρ 0 c 2 ) 2 β 3 1 + β (23a)

Or:

e D M e 0 = 2 β 3 1 + β (24a)

where e 0 = 1 2 ρ 0 c 2 .

The predicted dark matter energy density increases rapidly with velocity (see

at β = 1 3 . The model predicts that at β c r = φ = 0.618 a receding body undertakes a phase transition, seizing to behave classically. The value of the matter energy density at this point reaches a peak equal to φ 5 e 0 , which amounts to ≈ 0.09016994e_{0}. The dark matter energy density at this critical point becomes equal to 2 φ 3 1 + φ = 2 φ 4 ≈ 0.29179607 . These are aesthetically beautiful results, which could be put to direct experimental test.

To derive the expression of redshift in its dependence on the recession velocity, consider an observer on earth who receives redshifted waves emitted from a receding celestial object (e.g., A star, galaxy center, etc.). Assume that the recession velocity of the celestial object at the time the light wave was emitted was equal to v. Using Doppler’s formula, we can write:

z = λ o b − λ e m λ e m = f e m − f o b f o b (1b)

where λ e m ( f e m ) is the wavelength (frequency) of the wave emitted by the object and λ o b ( f o b ) is the wavelength (frequency) measured by the observer.

We also have f e m = 1 Δ t e m and f o b = 1 Δ t o b , where Δ t e m and Δ t o b are the time intervals corresponding to f e m and f o b , respectively. Substitution in Equation (1b) gives:

z = 1 Δ t e m − 1 Δ t o b 1 Δ t o b = Δ t o b Δ t e m − 1 (2b)

From Equation (1) in the main text we have: Δ t o b Δ t e m = 1 1 − β , where β = v c . Substitution in Equation (2b) yields:

z = 1 1 − β − 1 = β 1 − β (3b)

And the recession velocity in terms of redshift is:

β = z z + 1 (4b)

For blue-shift the same equation holds except that we must replace β by − β.