Sudhir Aggarwal, Ph.D., Independent Researcher

(Published on April 18, 2023)

Abstract

The Big Bang theory posits the universe's origination from an inexplicable singularity. Moreover, it requires speculative inflation and does not explain the nature of dark matter and dark energy. We propose the universe's origin from an initial cluster of dark matter particles to overcome these limitations. The initial cluster is an explicit state of dark matter having a minimum size at a very high temperature. We postulate that dark matter is constituted by Planck-mass particles conforming to the universe's absolute scale of the Planck units. The conversion of dark matter particles to dark energy initiates the universe's expansion while maintaining a constant dark energy density. The theory rationalizes the value of the Planck density with the observed value of the vacuum density while it conserves the universe's mass-energy. A solution of Friedmann's equation shows the universe's age of 13.8 billion years, consistent with the observed distribution of dark matter, matter, energy, and dark energy.

1 INTRODUCTION 1

2 OBSERVATIONS & METHODOLY 1

2.1 The Absolute

Unit Scale of the Universe 1

2.2 Dark matter 1

2.3 Dark Matter to Dark Energy Conversion 1

2.4 A Model of the Universe 1

3 RESULTS AND DISCUSSION 1

3.1 Early Universe 1

3.2 Matter and Antimatter Creation 1

3.3 Formation of Planets

and Stars 1

3.4 Formation of Galaxies 1

4 CONCLUSION 1

References 1

Initial State of The Universe

INTRODUCTION

The most accepted Big Bang theory assumes the universe's origination from a singularity. It can explain several phenomena, such as the galaxies' origin and the large-scale structures' formation. However, the Big Bang theory has several limitations. For example, it posits the initial singularity where space and time become ill-defined. Also, the composition of dark matter and the source of dark energy are not explicit [1]. Furthermore, it assumes a speculative inflationary regime [2]. Here, we propose a model for the observable universe that overcomes these limitations of the Big Bang theory.

We will begin with an introduction to our proposal in Section 2. After that, to justify the basis of the proposed concept, we will reformulate quantization and the uncertainty principle from the absolute unit Planck scale of the universe. It is described in Section 2.1. Then, we postulate the quantization of length, the mass of dark matter particles, and time to the absolute unit scale. It allows us to justify that the dark matter consists of the Planck-mass particles, as presented in Section 2.2. Finally, we propose the conversion of dark matter to dark energy by introducing a dimensionless universal constant in Section 2.3. Thus, it enables the continuum of all types of mass-energy while conserving mass-energy. Consequently, we will present the proposed model for the universe in section 2.4.

We will present the discussion and results for the universe's origin from an initial cluster based on the model in Section 3. We postulate that the initial cluster's scattered clumps and dipoles of dark matter are the starting points for forming planets and galaxies. The model predicts the universe's expansion with the present-day matter and energy distribution consistent with the observations. To conclude, we will summarize the main features of the proposed initial state.

OBSERVATIONS & METHODOLY

We know that matter, energy, dark matter, and dark energy constitute the universe [3]. In our proposal, we assume the universe's initial state consists of dark matter only. Further, the dark matter is made of Planck mass particles with the whole mass squeezed in the volume having a diameter of Planck length. Each particle is spinning with a unit momentum and a surface temperature of about 10³² degrees Kelvin. These particles are held together with gravitational interaction only. Thus, the whole universe occupies a tiny volume. There is no space except dark matter. A perturbation causes a dark matter particle to state transition to create space and time, marking the universe's beginning. With space availability, the dark matter particles get scattered all over. The temperature of the space falls with expansion. The state transition of the dark matter particles continues, and space temperature keeps dropping. Under favorable temperatures, matter and antimatter particles are continuously created and annihilated due to quantum fluctuations of the vacuum. Thus, the baryonic matter is created from the vacuum. The collapse of matter particles gives rise to galaxies and other universe structures we see today. The continuous state transition of the dark matter particle keeps generating a vacuum and pushing the galaxies further. Thus, the expansion of the universe continues.

According to the proposal, the numerous black holes of various sizes are constituted by scattered clumps of dark matter particles. Most small black holes exist as binary systems with a companion. These black holes feed on the companion, thus converting the baryonic matter into dark matter. Therefore, dark matter particles are in the beginning state as well as the end state.

We will expand on this concept further, starting from the basis of
dark matter particles.

The Absolute Unit Scale of the Universe

The composition and fundamental interaction of the universe's constituents provide indicators for the universe's origin. Therefore, we will review the constituents' components and the interactions essential for understanding the universe's origin [4]. We can express the relationship of matter, energy, and interactions by equations involving a few constants. Also, these constants hold their value everywhere in the universe. These universal constants are the gravitational constant G, the velocity of light c, and the reduced Planck constant ℏ. So far, we know of only these three universal constants. However, we propose one more dimensionless constant that we will introduce later.

We can write the numerical values for the three constants in SI units as below.

G = 6.674\ \times \ 10\ ^{- 11}\ \frac{m\ ^{3}}{kg\ \cdot \ s\ ^{2}\ }

c\ = \ 2.998\ \times \ 10^{8}\ \frac{m}{s}

\hslash\ = \ 1.055\ \times \ 10^{- 34}\ \frac{kg.\ \ m^{2}}{s}

The values of these constants turn out to be too big or too small as we have adopted standards for the basic unit of length, mass, and time on some convenient-to-use references. Alternatively, we can assign unit values to the fundamental constants. We can derive the basic unit of length, mass, and time from the assigned unit values. The system based on these derived units is the Planck scale. These units are the Planck length, $l_{P}$ , the Planck time, $t_{P}$ , and the Planck mass, $m_{P}$ .

We can find the values for these units by solving the following equations where a value of unity is assigned to each constant.

G = 1.\ \ \frac{l_{P}^{3}}{m_{P}\ \cdot \ t_{P}^{2}\ }

\hslash = 1.\ \ \frac{m_{P}\ \cdot l_{P}^{2}}{\ t_{P}}

We can solve these equations for $l_{P}$ , $t_{P}$ , and $m_{P}$ in terms of G,

c, and h. Solving these equations gives the following value for the Planck units of length, mass, and time.

$l_{P} = \ \sqrt{\frac{G.\hslash}{c^{3}}}$ = 1.616 × 10⁻³⁵ meter

$t_{P} = \ \sqrt{\frac{G.\hslash}{c^{5}}}$ = 5.391 × 10⁻⁴⁴ second

$m_{P} = \ \sqrt{\frac{c.\hslash}{G}}$ =

2.176 × 10⁻⁸ kilogram

Using Boltzmann's constant, we can get the Planck temperature $T_{P}$ corresponding to the

energy of mass $m_{P}$ as follows.

Solving it, we get the value for the Planck temperature as follows.

$T_{P}\$ = 1.416 × 10³² °K

We will adopt the scale based on the Planck unit values as the absolute scale of the universe. The Planckian scale leads to quantization and uncertainty, as explained below.

Quantization

Every measurable quantity of the universe can be quantified in the Planck units. We can assign the Planck units to the minimum length, time, and mass. Further, we can measure all larger quantities as the integer multiple of these units. The restriction on the larger values to only integral multiples is quantization. Thus, we postulate the quantization of every physical entity in the universe to the Planck units with the following implications.

The Planck time is the shortest time we can resolve in the universe. The universe clock ticks, with each tick equal to the Planck time. The state of the universe is defined only at the ticking moment of the clock. Thus, time is quantized in the absolute units of Planck time, $t_{P}$ .

The Planck length is the smallest possible length and is the absolute measurement unit. Each spatial dimension is measured in the units of the Planck length. It leads to the quantization of spatial dimensions with unit length, $l_{P}$ .

The minimum values of time and length as Planck units are minimal and are beyond the precision of most accurate physical measurement systems. Thus, these can be assumed to be the minimum units. However, the Planck mass is quite significant. All the elementary particles have masses much smaller than the Planck mass for the baryonic matter. Since the Planck mass is a unit, it can be a pointer for the primary constituent of the dark matter. Therefore, we postulate the Planck mass is the minimum mass for a dark matter particle. It also suggests that the dark matter particle is the primary of all other types of matter particles. Thus, we propose a particle of Planck mass is the basic unit of dark matter. The clumping of these Planck-size particles forms dark matter and black holes. It implies that the mass of the black holes is quantized.

Uncertainty

Quantizing the Planck units' time, length, and mass leads to the uncertainty principle. Since time is quantized, there is uncertainty associated with each event happening within the Planck time interval because the state of the event is unknown between two successive ticks. We can write a time increment, $\mathrm{\Delta}t_{P}\ ,\$ between two consecutive ticks equals one unit of the Planck time, as given below.

\mathrm{\Delta}t_{P} = \ \sqrt{\frac{G.\hslash}{c^{5}}}

( 1 )

Similarly, we can write the incremental energy ∆E from the quantization of mass as below.

m_{P}c^{2} = \ \sqrt{\frac{c^{5}.\hslash}{G}}

( 2 )

Taking the product of incremental time and energy gives us the following relationship.

\mathrm{\Delta}t_{P}.\ \mathrm{\Delta}E = \ \hslash

( 3 )

The above equation is derived with the condition that the minimum change in ∆E or $\mathrm{\Delta}t_{P}\$ is one unit. Further, assuming that we can resolve a unit value of less than half of a unit to zero and a greater than half to one unit, we can write the above equation as follows.

\mathrm{\Delta}t_{P}.\ \mathrm{\Delta}E \geq \ \frac{\hslash}{2}

( 4 )

This equation is the well-known uncertainty principle. An alternate version of this equation is derived below.

We can write the quantization of spatial dimension $\mathrm{\Delta}x$ as below.

\mathrm{\Delta}x = c.\mathrm{\Delta}t_{P} = \ \sqrt{\frac{G.\hslash}{c^{3}}}

( 5 )

In the same way, the quantization of the momentum is given below.

\mathrm{\Delta}p = m\mathrm{\Delta}v = m\frac{\mathrm{\Delta}x}{\mathrm{\Delta}t} = \ \sqrt{\frac{c^{3}.\hslash}{G}}

( 6 )

Thus, the product of the unit increment in spatial dimension (5) and the momentum (6) leads to the uncertainty principle.

\mathrm{\Delta}x.\ \mathrm{\Delta}p = \ \hslash

( 7 )

It can also be written below following the argument used for equation (4)

\mathrm{\Delta}x.\ \mathrm{\Delta}p \geq \ \frac{\hslash}{2}

( 8 )

In deriving the two alternative equations of the uncertainty principle, we have used one unit of the Planck mass, length, or time for the minimum change in any of these quantities. Thus, it validates our postulates about quantizing mass, length, and time to Planck units. Furthermore, the uncertainty principle is found to hold in all the experiments. Therefore, we can say that the validity of the uncertainty principle confirms the quantization of Planck units for the universe.

As mentioned earlier, it also affirms to postulate of Planck mass particle as a candidate for dark matter. So now, we will explore the properties of the Planck mass particles.

Dark matter

Dark matter is not visible as it does not interact with electromagnetic energy. However, it exerts a considerable gravitational interaction implying that it is made of something massive. Several types of weakly interacting massive particles (WIMPs), massive compact halo objects (MACHOs), and primordial black holes (PBHs) have been considered possible candidates for dark matter [5, 6, 7, 8]. Earlier, Planck mass black holes were also considered [9]. More recently, it has been shown again that Planckball is a good candidate for dark matter [10].

Here, we also postulate that Planck mass particles constitute dark matter. Therefore, we can enlist the essential characteristics of dark matter particles (DMPs).

Each particle has a mass equal to Planck mass concentrated in a sphere of a radius of Planck length.
The particles are electrically neutral but interact gravitationally.
The particles can spin. When spinning, each particle has angular momentum equal to $\hslash$ . Since the momentum is conserved and quantized, a particle is either spinning or resting.
An individual particle may behave like a Planck mass black hole (PMBH). It is known that Planck mass black hole can decay in Planck time by Hawking radiations [11]. However, a significant number of these can constitute big chunks of dark matter that can exist for long periods.
For the Planck-mass black hole $m_{P}$ the Schwarzschild radius 'R_s' is given below [12].

R_{s} = \frac{2Gm_{P}}{c^{2}} = 2.\sqrt{\frac{G.\hslash}{c^{3}}} = 2l_{P}

( 9 )

where G is the gravitational constant, c is the velocity of light,

$\hslash$ is the reduced Planck constant, and $l_{P}$ is the Planck length.

Two Planck-mass particles at rest in close vicinity do not merge into one. It can be explained with a diagram, as shown in Figure 1. When two such particles approach each other, their event horizons will touch at a certain point. At that point, one particle will pull the other with the same force as it is being pulled by the other. Therefore, these will be in stable equilibrium.

Figure 1. The gravitational field of a dipole formed by dark matter particles (DMPs)

An arrangement of two particles separated by their event horizons acts as a gravitational dipole [13]. We expect a gravitational dipole with positive and negative mass at the opposite ends [14]. However, the negative mass does not exist. Therefore, we define dark matter dipole as consisting of two centers of masses separated by a distance. When the two particles touch each other, the center of mass is separated by a certain distance. The left and right-side centers of mass act as a center of gravity for the matter on the left and right sides, respectively. Any point in the middle at equidistance from the two centers of mass will have gravitational interaction equally from both sides. As shown in Figure 1, the midline is a neutral for gravitational interaction.
A spinning particle will have its event horizon shrink to a value equal to its radius l_P from the value of 2l_P when it is not spinning.
When two spinning particles approach each other, we can determine the characteristics of the combination as the two-body problem.
Dark matter is made of combinations of two or more particles. These combinations of particles are held together by gravitational interaction only. Two or more particles adjoining each other constitute dark matter. When the particles are spinning with their spin axis aligned in the same direction, they can be close to almost touching each other. As the mass and angular momentum is quantized, two possible arrangements of the spinning particles are shown in Figure 2. The particles will align in horizontal or vertical lines without losing momentum. However, if the particles are at rest, these can co-exist, separated by their event horizons.

Figure 2. Possible arrangements of the dark matter particles.

Further, we postulate that these particles or clumps of aligned particles are spread all over the universe. A clump can have any number of particles. As explained earlier, we can envisage a dipole of two dark matter clumps like the dipole of two particles. A dipole of a clump of particles is possible if the two clumps have a precisely equal mass which is possible as mass is quantized for dark matter. Such dipoles or clumps with a large number of particles are essential for understanding the universe's origin [15, 10].

Dark Matter to Dark Energy Conversion

Matter, energy, and dark matter exist in the universe's space, known as vacuum or dark energy. Dark energy is the most significant constituent of the universe. Dark energy has the property that it exerts negative pressure. Thus, it causes the expansion of the universe. The energy density of the vacuum $\rho_{DE}\$ is about 5.8×10⁻²⁷ kilograms per cubic meter [16]. The vacuum energy density is assumed to remain constant even though the universe expands with time. Thus, it points to the time-dependent creation of vacuum energy with the expansion of space [17]. We will present a possible mechanism for creating dark energy as follows.

From equation (3), it can be shown that a unit of Planck time is identical to a quantum of angular momentum per unit of Planck mass-energy.

For a Planck mass particle, the total energy is given by the rest mass energy and the angular momentum.

E\ = \sqrt{(m_{P}^{2}c^{4} + p^{2}c^{2})}

For a spinning Planck mass particle, the angular momentum is one unit. Therefore, the total energy of a Planck mass particle can be considered a unit of energy for creating one unit of spacetime. The rest mass energy creates a unit of space, while the angular momentum creates a unit of time. Thus, we postulate that a dark matter particle can be converted to dark energy. We will explain it further as follows.

Let us consider a case of two spinning particles in the vicinity, as shown in Figure 3. We know that each particle can act as a tiny black hole. If the particles are perturbed, these can stop spinning, and their event horizon will expand to '2l_p'. As a result, the event horizon will overlap with the other particle in the close vicinity. In this condition, each particle will extract mass from the other because the event horizon will touch the surface of the other particle. Thus, the two particles will rip apart each other. Since we have imposed the condition of quantization of mass, the particles will rip each other in a process similar to spaghettification around a black hole. We postulate that the ripping of the particles will create vacuum energy giving rise to space. We call this transition of particles from the solid state to the vacuum state.

Figure 3. Spinning dark matter particles surrounding a particle at rest with its event horizon overlapping

In general, several particles can undergo a state transition to create a vacuum under suitable conditions. Alternatively, we can say that the DMPs exist in two states, solid and vacuum. Transitioning from solid to vacuum implies dark matter to dark energy conversion. Thus, particles of Planck mass undergo a state transition to convert dark matter to dark energy. This dark energy conversion results in creating a large volume of space.

We can estimate the value of the space created. As an approximation, assuming a DMP is a sphere having a radius of the minimum length $l_{P}$ . We can calculate dark matter or Planck density by dividing its mass by the volume of the sphere, as shown below.

\rho_{c} = \ \ \frac{m_{P}\ }{\ \frac{4}{3}{\text{p}\text{\ }l}_{P}^{3}\ }

( 10 )

Substituting the values of m_p and l_p, we get the dark matter density as below.

\rho_{c} = 1.2\ \times \ 10\ ^{96}\ \frac{kg\ }{\ m\ ^{3}\ }

We postulate the following equation to convert dark matter with a density $\rho_{c}\$ into dark energy of density $\rho_{DE}$ .

( 11 )

In equation (11), we introduce a new universal constant for converting dark matter density to vacuum energy density. We denote this constant as $S_{a}$ . It is a dimensionless constant, and its value can be found as a fitting parameter and is given below.

Substituting the dimensionless constant value in equation (11), we find that the $\rho_{DE}$ and $\rho_{c}\$ are related as below.

\rho_{DE} = 4.82\ \times \ 10^{- 123}\ \rho_{c}

( 12 )

Thus, the dark matter density, $\rho_{c}\ of\ value\ \ \ 1.2\ \times \ 10\ ^{96}\ \frac{kg\ }{\ m\ ^{3}\ }\$ is converted to the vacuum density or space

energy density, $5.8\ \times \ 10^{- 27}\frac{kg\ }{\ m\ ^{3}\ }$ . These values are in close agreement with those commonly encountered in astronomy literature. Physically, it means that the transition of a particle from a solid state to a vacuum creates a vast amount of space. For example, a spherical dark matter particle with a radius of Planck's length will create a vacuum sphere with approximately a million meters radius.

Now, we will discuss one more feature of the universe that helps in understanding the origin of the universe.

Conservation and Continuum

We have presented the dark matter to dark energy conversion. We know that the creation of matter and antimatter happens from the vacuum energy at suitable temperatures due to quantum fluctuations of the vacuum [18]. For example, quantum fluctuation can create a positron and an electron. Also, the wave energy of two gamma rays can give rise to an electron and a positron.

Dark matter converts to dark energy, which creates matter and antimatter at high temperatures. We know that stars are made by the accretion of matter in space. In the cores of the stars, the transformation of matter to energy occurs. Also, matter is converted into the dark matter when stars collapse at the end of their lives. Thus, we find a complete cycle for the dark matter converting from one state to another, making the continuum of all the universe's constituents, as shown in Figure 4. The total mass-energy is conserved in transforming one component into another due to a continuum of the universe's constituents. Dark matter to dark energy conversion leads to expansion.

Figure 4. An illustration representing the constituents' continuum.

We have explained the basis for the dark matter particles as the primary component of the universe. Now, we can propose a model for the origin of the universe.

A Model of the Universe

We propose a model of the universe based on the Planck mass dark matter particles. For the universe model, our objective is to find a relationship between the physical parameters that describe our universe's origin and its expansion and growth with time.

The Initial State of the Universe

We propose that all the matter and energy of the universe is in a cluster of dark matter particles, which is the starting state of the universe. Also, each particle is of the Planck mass with its mass concentrated in the sphere of Planck length radius. We assume that all the particles are in close vicinity of each other. Thus, the whole universe occupies a small volume as a cluster with a density approaching the Planck density. Furthermore, the surface temperature of each particle is exceptionally high. Thus, the entire cluster is extremely hot. Thus, the whole universe is well-ordered and can be assumed to have the minimum entropy.

Let D be the total mass-energy of the universe. In the beginning, this mass-energy is in the form of dark matter. Let us assume an initial number of dark matter particles as I₀. Thus, the universe's total initial dark matter as energy is given below.

( 13 )

Expansion of the Universe

We will measure the time in units of Planck time $t_{P}$ . The state of the

universe is defined only at the end of each period $t_{P}$ as shown in Figure 5. A small perturbation of the cluster of the particles at a time t equal to zero starts the universe. Consequently, the spinning particles stop spinning one by one. The event horizons of the non-spinning particles overlap the adjacent particles. Therefore, some of the particles in the cluster undergo a state transition from the solid to the vacuum state. Thus, it begins the expansion of a new universe.

Figure 5. An illustration of the initial cluster & time at the origin of the present universe.

The initial state transition of some of these particles creates enormous space for the remaining particles in the cluster. Due to the vacuum's negative pressure, all other cluster particles get scattered to occupy the newly created space. The state transition of initial particles initiates a chain reaction to create more space rapidly, and the remaining particles are dispersed further over the space. Thus, the particles pervade all over in all possible sizes of clumps.

A significant space creation continues with each tick of the universe clock. The number of particles available for the state transition decreases with each clock period of the universe. Dark matter decreases with time, while dark energy increases. Dark matter conversion to dark energy continues to expand the universe.

We can assume the universe to be homogeneous and isotropic for the model. Formulating the universe's expansion by the scale factor allows us to study its growth with time. For an isotropic expansion about a sphere's center with a comoving radius 'r' normalized to present-day value 'r₀', we can define

( 14 )

where a(t) is the time-varying scale factor. It is assumed to be unity at the present time.

Since time is quantized with fundamental Planck time t_P, any time 't' can be written as integer 'n' multiple ticks of Planck time.

( 15 )

As the radius of the sphere expands with time, we can also write any time 't' in terms of the present time 't₀',

( 16 )

As we described earlier, all the quantities of the universe are quantized with the absolute scale of Planck units. Strictly speaking, we need discrete mathematics to model the universe. However, we will have the mathematical formulation with continuous variables for simplicity and to maintain continuity with the existing literature. From the model formulated with the continuous variables, we can quantize the values of the targeted parameters afterward.

Friedmann's equation represents the rate of growth of the scale factor in terms of normalized densities of the various constituents of the universe. Therefore, we can use the Friedmann equation to describe the universe's expansion [19]. Furthermore, this equation can be modified for application to the proposed model. In the proposed model, the universe's initial state is dark matter. All other universe components, matter, antimatter, energy, and dark energy, are created from the initial dark matter.

We can write the Friedmann equation by modifying the effective contribution of various components with a normalized density as below.

H^{2}(a) = \left( \ \frac{\dot{a}}{a} \right)^{2}\

H^{2}(a) = H_{0}^{2}\lbrack(\Omega_{c} + \Omega_{b})a^{- 3} + \Omega_{rad}a^{- 4} + \Omega_{k}a^{- 2} + \Omega_{DE}a^{- 3(1 + w)}\rbrack

( 17 )

All the densities are normalized in the equation, and the various symbols are as follows.

Ω_cis effective dark matter density at present. In our case, it is a time-dependent function.

Ω_bis effective baryonic matter density

today,

Ω_rad is the radiation density today,

Ω_kis the spatial curvature density today,

Ω_DEis the dark energy density today and is time-independent,

H₀ is the Hubble rate parameter today.

Dark energy governs the universe scale factor 'a' in the proposed model. As explained earlier, a vast dark energy space is created even when one particle of the dark matter is converted into a vacuum. Since Ω_DE remains constant with the expansion, the dark matter creates the additional dark energy required to keep its density constant.

We can numerically solve Friedmann's equation with the constraints set by equations (11) & (13). However, it can be simplified for an analytical solution with approximations for a more likely scenario that we will consider here.

The Model Solution

At the beginning of the universe, we have $\Omega_{rad},\ {\ \Omega}_{b},{and\ \Omega}_{DE}$ , all equal to zero since all the universe's

energy is in the dark matter within a tiny volume. Assuming the flat universe, the effect of curvature $\Omega_{k}$ can be considered equal to zero. Also, the equation of state parameter 'w' for the dark energy can be assumed equal to -1. With these conditions, equation (17) can be simplified. Further, we can simplify the equation by considering that baryonic matter and radiation are created from dark energy. Thus, taking the baryonic matter and radiation energy as part of the dark energy will have little impact on the overall solution. We can write the Friedmann equation with these conditions as below.

H^{2}(a)\ = H_{0}^{2}\lbrack(\Omega_{c})a^{- 3} + \Omega_{DE}\rbrack

( 18 )

In our case of the dark matter to the dark energy conversion, we can write the dark matter density as

\Omega_{DE} = \lbrack 1 - \Omega_{c}\rbrack

( 19 )

Substituting this in equation (18), we have

H^{2}(a) = \left( \ \frac{\dot{a}}{a} \right)^{2}\ = H_{0}^{2}\lbrack\Omega_{c}a^{- 3} + (1 - \Omega_{c})\rbrack

( 20 )

\left( \ \frac{\dot{a}}{a} \right)\ = H_{0}\sqrt{\lbrack\Omega_{c}a^{- 3} + (1 - \Omega_{c})\rbrack}

( 21 )

\frac{da}{dt} = a^{- 1/2}\ H_{0}\ \sqrt{\lbrack\Omega_{c} + (1 - \Omega_{c})a^{3}\rbrack}

This equation can be solved by integrating with respect to time from 0 to t with the condition, a $\widetilde{=}$ 0 at t=0, as below.

\int_{0}^{t}{H_{0\ }dt} = \int_{0}^{a}\frac{\sqrt{a}.da}{\sqrt{\lbrack\Omega_{c} + (1 - \Omega_{c})a^{3}\rbrack}}

( 22 )

The solution gives

t = \frac{2}{3\ H_{0\ }\sqrt{1 - \Omega_{c}}}

{\sinh^{- 1}(\sqrt{\frac{{1 - \Omega}_{c}}{\Omega_{c}}}}{\ a^{3/2}})

( 23 )

Alternatively, we can express the scale factor as a function of time t as below.

a = \lbrack{(\ \frac{\Omega_{c}}{1 - \Omega_{c}})}^{\frac{1}{2}}

{{{sinh(}{\frac{3}{2}\ H_{0\ }}\sqrt{\left( 1 - \Omega_{c} \right)}}t)\rbrack}^{\frac{2}{3}}

( 24 )

Equation (24) was derived assuming the constant dark matter density as usually fixed to the present-day value. However, the dark matter density decreases with time in the proposed model. Assuming the initial density $\Omega_{c0}$ , we can write the rate of decrease of the dark matter density as a function of time f(t) given below.

( 25 )

In the equation, the function f(t) can be modeled to fit the present-day value of the dark matter density with the initial value of $\Omega_{c0}$ equal to unity.

It is known that the temperature 'T' of the universe decreases linearly with expansion. Thus, we can assume it to be inversely proportional to the scale factor 'a'.

( 26 )

We can write equation (26) after substituting 'a' and a constant of proportionality 'θ 'as below.

T = \ \frac{\theta}{\lbrack{(\ \frac{\Omega_{c}}{1 - \Omega_{c}})}^{\frac{1}{2}}\ {{{sinh(}{\frac{3}{2}\ H_{0\ }}\sqrt{\left( 1 - \Omega_{c} \right)}}t)\rbrack}^{\frac{2}{3}}}

( 27 )

From equations (24) and (27), we can get the estimates of the scale factor and temperature during the various stages of the universe.

RESULTS AND DISCUSSION

Based on the model, we will briefly present the initial stages of the universe's origin. After that, we will explain the formation of matter, planets, and galaxies.

From the total mass-energy of the present universe, an estimate of the initial number of dark matter particles is of the order of $10^{62}.$ The estimated mass, M, of the universe is as below.

With a dark matter density of $1.2\ \times \ 10\ ^{96}\ \frac{kg\ }{\ m\ ^{3}\ }$ , we can estimate the universe's initial volume, V, as below.

V = \ 3.3\ \times \ 10\ ^{- 42}\ m\ ^{3}

It is a tiny volume at the time of the universe's origin. In the proposed model, at the beginning of the universe, we can assume that the temperature of the ensemble of the dark matter particles is close to the PMBHs surface temperature, as all the particles are nearby. The surface temperature of the PMBHs can be estimated from the equation below [11].

T = \frac{\hslash c^{3}}{\text{8}\text{p}{k\ G\ m}_{P}} = 5.6\ \times \ 10^{30}\ {^\circ}K

( 28 )

As mentioned earlier, we measure the time in the units of Planck time from the start of the present universe. From equation (24), we can estimate the scale factor at first Planck time $5.4 \times \ 10\ ^{- 44}$ second of the universe. It comes to be about the same order as estimated from the volume above.

The above discussion shows that the initial mass, volume, and temperature are defined at the first Planck time. Thus, the universe's initial state is more specific in the proposed model as compared to the initial singularity of the Big Bang theory.

With the state transition of dark matter particles to space or vacuum, the dark matter density is converted from the Planck density to the present-day vacuum density value as given by equation (12).

Thus, the proposed model implicitly rationalizes the value of the Planck density with the value of the vacuum density, a conflict frequently mentioned in the existing literature. The constancy of the vacuum density throughout the universe's history is inherent in the proposed model.

We can plot the scale factor given by equation (24) by substituting the known values of the parameters in the equation. However, to get the estimates, we need values of the present-day Hubble constant and the dark energy density. We have taken the value of dark energy density as 68.5% from the Planck 2019 data [20]. We have taken the value for the Hubble constant as 70.88 Km/s/Mpc since recent data from the year 2021 to 2022 on the Hubble constant indicates its value ranging from 69.8 [21] to 73.4 Km/s/Mpc [22]. The sum of the present-day baryonic matter and radiation density is 5% approximately. According to the proposed model, matter and radiation are also created from the vacuum created by the initial dark matter. Therefore, in the proposed model equation (24), we have used the dark energy density as a sum of all three, i.e., dark energy, matter, and radiation. These imply that the total converted dark energy is 73.5%, and the remaining dark matter is 26.5%.

A plot of the scale factor with the values of Hubble constant as 70.88 Km/s/Mpc and dark energy density as 0.735 is shown in Figure 6. From the graph, we can estimate the age of the universe. It can be observed from the plot that the value of the scale factor is one at 13.8 billion years. It closely agrees with the value of the universe's age obtained from other methods [20].

Figure 6. A plot of the scale factor with time for expansion of the universe

We have also included a scale factor with dark matter density decreasing with time. It can be observed from the graph that the slope of the scale factor near the origin is very sensitive to the model of the time-varying decrease in dark matter. Therefore, equation (27) is exposed to dark matter decrease modeling. However, the proposed model can match the initial temperature of dark matter particles with the estimated temperature of the early universe.

Early Universe

Now, we will expand on the initial few moments of the universe.

The initial time interval between t=10⁻⁴⁴ second to t=10⁻¹³ second (initial 10³¹ t_p)

In our proposed theory, it is assumed that there is no space at the beginning. The whole universe is in a tiny volume of dark matter consisting of spinning dark matter particles. At the start time, the dense collection of particles is in thermal equilibrium. A perturbation causes a spinning particle to lose momentum. The universe clock starts as soon as one of the spinning particles stops spinning, thus creating the first unit of Planck time.

Also, we know from equation (9) that even a single Planck mass dark matter particle will create a volume of vacuum with a radius of about one million meters. From equation (24), we find that the radius of the universe is about one million meters after a lapse of 10³¹ tp. Furthermore, we know that dark matter and time are quantized. Therefore, we can assume that the scale factor remains constant during this time interval. The first change in the value of the scale factor occurs after a lapse of 10³¹ t_p. In this manner, we can quantize the values of the continuous scale factor with time to an integral multiple of t_p.

During the initial 10³¹t_p time-lapse, the spinning particles' momentum is used for creating the time. At the end of this interval, a few particles undergo a state transition to make space. With the first state transition, the space becomes available for the remaining particles to occupy. As the vacuum energy exerts negative pressure, the particles spread out over all the space in various clumps. The spreading of the particles is controlled by the gravitational interaction that comes into play with separating the particles. At the end of this period, the temperature decreases, but it is still above 10²⁰ °K. The strong nuclear interaction, the weak nuclear interaction, and electromagnetic interactions are all unified.

With each period of t_p, there is a creation of space by a significant amount. The space expands rapidly; thus, smooth eternal inflation is implicit in this model [23].

The time interval between t=10⁻¹³ second to t=10⁻⁵ second (after the lapse of 10³¹ t_p)

The temperature continues to fall with continuous space created from converting dark matter to dark energy. At the end of this time interval, the temperature drops to 10¹⁵ °K, as estimated from equation (27). This temperature is favorable for creating elementary matter particles and antiparticles. All the space gets occupied by a soup of hot ionized quark-gluon plasma. At the end of this period, the weak nuclear interaction gets separated from the strong and electromagnetic interactions.

The time interval between t=10⁻⁵ seconds to t=10⁴ seconds (after the lapse of 10³⁹ t_p)

The space temperature drops further to 10⁹ °K at the end of this interval, as estimated from equation (27). The quark-gluon plasma starts coalescing into fermions. Thus, we have a soup of matter and antimatter particles. These particles undergo creation and annihilation continuously.

Matter and Antimatter Creation

The expansion of the universe results in the fall of temperature in space. As the universe expands, its temperature falls to about 3000 °K after the lapse of about 300,000 years, as estimated from equation (27). This temperature is suitable for the existence of matter particles such as protons, neutrons, and electrons.

At this temperature, free electrons start falling into the nucleus. Thus, these form neutral atoms for the first time. This process of recombination continues for many years. We know that the photons are emitted during the recombination process. The photons can move freely as the recombination also creates voids. As a result, the universe becomes transparent. These radiations are observed as cosmic microwave background radiation (CMBR) presently.

Based on the model, we can estimate the amount of dark matter converted into dark energy at the end of 300,0000 years. The estimate for the converted dark energy is 11.5% of the initial dark matter at this time. According to the model, baryonic matter and radiations are created from dark energy. For a few years after this time, conditions remain suitable for creating baryonic matter and antimatter. Therefore, only a part of 11.5% of dark energy gets converted further into matter and radiation. It explains why we have only a small percentage (5%) of baryonic matter and radiation in the universe.

The created matter and antimatter are used in forming the galaxies, stars, and planets. Now, we will explain the formation of the stars and galaxies.

Formation of Planets and Stars

As mentioned earlier, dark matter clumps of all sizes are spread all over space with the ongoing conversion of dark matter. The number of particles in these clumps varies a lot. Arbitrarily, we can classify these clumps into small, medium, and large sizes according to the number of DMPs. We specify the first category with particles ranging from 10²¹ to 10³⁰ as a 'small clump.' In the second category, we have a number of particles ranging from 10³¹ to 10⁴⁰, called 'medium clumps.' The third category, 'large clump,' consists of particles ranging from 10⁴¹ to 10⁵⁰. Any two clumps of equal mass that happen to be in close vicinity can form a gravitational dipole.

As soon as matter and antimatter particles get created, these are attracted to the nearest clump of DMPs. Thus, the matter accumulates in small clumps or small-size dipoles. As the matter accretes, it causes the rotation of the small clump or the dipole. A small dipole can revolve either about the y-axis or the x-axis, as shown in Figure 7. In the case of rotation on the y-axis of the dipole, the y-axis is a line of zero gravitational force on the matter. Thus, matter can move or slide freely along the y-axis.

It is also possible for a small dipole to revolve around the x-axis. In this case, the y-axis will sweep a circular disk. The gravitation interaction is neutral on the disk for the matter. Therefore, matter can swirl freely as a disc or ring around the dipole. In such cases, we have planets with rings, such as Saturn. Thus, we assume that Saturn-like planets have a dipole at their center, which revolve on the x-axis of the dipole.

Figure 7. A conceptual diagram illustrating the starting point for the formation of the planets (not to scale)

Similarly, medium size clumps or dipoles also accumulate matter. These medium-sized clumps or dipoles also attract small clumps of particles. Thus, these medium-sized dipoles become stars. The accretion of the matter at the two ends results in the spinning of the dipoles. These spinning stars attract planets in the vicinity. Therefore, we have a star's revolving disk and the planets swirling around the nearest medium size clump or dipole. In this manner, stars form with the planets revolving around them.

The surface temperature of the star-forming clumps is very high as it consists of Planck-sized dark matter particles. Therefore, the matter accreted by the stars falls on the surface at a very high temperature. Hydrogen is the first element that forms in space which is accreted by these hot clumps. These are ideal conditions for a fusion reaction we observe during a star's active lifetime.

Formation of Galaxies

Galaxies are formed around large clumps of dark matter particles. We can illustrate the formation of a galaxy with a diagram, as shown in Figure 8. As large-clump dipoles have a strong gravitational field, the newly created stars get attracted to the ends of the nearest large dipoles. Thus, a large dipole becomes an active galactic nucleus of a galaxy.

Figure 8. A conceptual diagram illustrating the formation of a galaxy and the active galactic nucleus with stars around it (not to scale)

Several stars get pulled by the dipole's ends in the galactic nucleus. As a result, the galactic nucleus starts swirling around due to the accretion of more and more stars at both ends of the dipole. Thus, the entire galaxy revolves around the y-axis. Furthermore, the y-axis is a point of neutral interaction for the matter. Therefore, we find that the matter particles can move freely along the y-axis. Consequently, we observe a jet of particles extending to huge distances at the active galactic nucleus.

All the galaxies with billions of stars form in this manner. However, some stars far from a galaxy's active nucleus continue floating, giving the intracluster light (ICL) [24].

CONCLUSION

In the proposed model, the universe's initial state is more specific as compared to the singularity of the Big Bang theory. The universe has a minimum size, with all the particles of dark matter arranged in a cluster at a very high temperature. Also, the model conforms to the absolute scale of the universe based on the Planck units. The model proposes the Planck mass particles as the constituents of dark matter. Also, the continuum of all the universe's components establishes dark matter particles as the primary constituent of the universe. It provides the basis for the origin of dark energy from dark matter by correlating their densities with a dimensionless universal constant. Thus, it justifies the constant density of dark energy with the universe's expansion. It rationalizes the value of the Planck density and the observed value of the vacuum density. It maintains the conservation of mass energy during all the universe states.

The universe's initial expansion is inherent in the model and does not require speculative inflation. It also explains the amount of the baryonic matter-energy to be a small fraction of the total mass-energy content of the universe. Finally, it predicts the planets' possible shapes based on the gravitational dipole of dark matter, including the planets with rings like Saturn. Thus, it provides pointers to the forms of galaxies and the shape of the active galactic nucleus.

We have presented the universe's origin by converting dark matter to

dark energy. The proposed model overcomes some of the Big Bang theory limitations and agrees with the observed mass-energy distribution for the universe's age.

Acknowledgment

The author is grateful to Dr. S.C. Rustagi, Dr. R. Singh, Dr. Y. Rohinkumar, L. Aggarwal, and P. Tyagi for their insightful comments and suggestions.

References

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