Our hypothesis about primordial particles [1] - [5] introduces the concept of quantized-velocity space, in which there exist fundamental particles, to which, in their ground state, we can assign masses equal to the Planck mass, and which exist before the creation of our space and time (For the sake of a more concise expression, in further work, we will simply call the mass attributed to these primary particles “masses of primary particles”). The transition of one of them into the domain of velocities smaller than or equal to the speed of light is interpreted as quantum tunneling that causes the creation of our space and time through the Big Bang.

On the other hand, quantum gravity (especially Loop Quantum Gravity and Loop Quantum Cosmology) [6] [7] offers a framework for quantizing space and time and removing the Big Bang singularity, replacing it with a quantum bounce (the Big Bounce).

In this paper, we connect these two approaches into one coherent theoretical framework and consider the experimental consequences of such a model.

We define the Lagrangian for a particle moving in its own space and time with an effective quantum of velocity, assuming: 1D motion of a particle of mass $m_p$ with a parameterized time parameter $\tau$:

$ℒ=\frac{1}{2}{m}_p{\left(\frac{\text{d}x}{\text{d}\tau }\right)}^2-V\left(x\right).$(5)

To describe tunneling, we introduce the potential barrier that protects our universe from particles from an external space as:

$V\left(x\right)=\{\begin{array}{ll}0,\hfill & x<0\left(\text{external space}\right)\hfill \\ V_0,\hfill & 0<x<a\left(\text{potential barrier}\right)\hfill \\ 0,\hfill & x>a\left(\text{our universe}\right).\hfill \end{array}$(6)

This barrier can represent “boundary conditions” between dimensionally different spaces.

In a way analogous to classical mechanics, tunneling can be described by solving the Schrödinger equation through the barrier:

$\psi \left(x\right)~{\text{e}}^{-\kappa x},\kappa =\sqrt{\frac{2m_p\left(V_0-E\right)}{{\hslash }^2}},$(7)

and the probability of tunneling (the transmission coefficient) is:

$T\approx {\text{e}}^{-2\kappa a},$

where the above-mentioned values are:

$E$: Energy of the particle in its own space and time.

$V_0$: The barrier between dimensions (e.g., the gravity wall or velocity threshold).

$a$: The width of the barrier in the corresponding coordinates (e.g., in the velocity space, if it is modelled).

We will use the space of quantized velocities (1) and (8) according to our model, and a simple quasi-classical (WKB) approximation, setting some realistic parameter values.

The quantum amplitude for tunneling (in the WKB approximation) through the velocity barrier is:

$\mathcal{A}~{\text{e}}^{-2{\displaystyle {\int }_{v_f}^{v_i}\kappa \left(v\right)\text{d}v}},$(14)

where is:

$\kappa \left(v\right)=\frac{1}{\hslash }\sqrt{2m_p\left[V\left(v\right)-E\left(v\right)\right]}.$(15)

We shall assume that: - The velocity ranges from $v_i>c$ to $v_f=c$ (the boundary of our universe), - $E\left(v\right)=\frac{1}{2}{m}_p{v}^2$, - $V\left(v\right)=V_0=\text{constant}$ in the barrier (for $v\le v_i$).

Tunneling occurs in the region $v\in \left[c,v_i\right]$, where $E\left(v\right)<V_0$.

We will assume the following values as numerical settings:

- The Planck mass, i.e. the mass of the primordial particle: $m_p=m_{\text{Planck}}=2.18\times {10}^{-8}\text{kg}$.

- Planck’s constant: $\hslash =1.055\times {10}^{-34}\text{J}\cdot \text{s}$.

- The critical velocity (the boundary velocity for our universe): $c=3\times {10}^8\text{m}/\text{s}$.

- The initial velocity of the primordial particle: $v_i=5\times {10}^8\text{m}/\text{s}$.

- The quantum velocity: ${\epsilon }_v={10}^7\text{m}/\text{s}$.

- The height of the barrier: Let:

$V_0=\frac{1}{2}{m}_p{v}_b^2,v_b=5.5\times {10}^8\text{m}/\text{s}$

Thus, the barrier is slightly higher than the energy of the particle, and this creates the conditions for tunneling. Tunneling between $v_f=c=3\times {10}^8$ and $v_i=5\times {10}^8$, in steps of ${\epsilon }_v={10}^7$:

That is, a total of $N=20$ steps.

At each step, we approximate:

${\kappa }_n=\frac{1}{\hslash }\sqrt{2m_p\left(V_0-\frac{1}{2}{m}_p{v}_n^2\right)}$

Then:

$S\approx 2\underset{n=1}{\overset{N}{{\displaystyle \sum }}}{\kappa }_n\cdot {\epsilon }_v\Rightarrow \mathcal{A}~{\text{e}}^{-S}$

Result (calculated):

Using these values:

- $m_p=2.18\times {10}^{-8}\text{kg}$ - ${\epsilon }_v={10}^7\text{m}/\text{s}$ - $\hslash =1.055\times {10}^{-34}\text{J}\cdot \text{s}$ - velocity: $v_n=c+n\cdot {\epsilon }_v$, $n=1,\cdots ,20$

We calculate:

$S\approx \underset{n=1}{\overset{20}{{\displaystyle \sum }}}\frac{2\times {10}^7}{1.055\times {10}^{-34}}\cdot \sqrt{2\cdot 2.18\times {10}^{-8}\cdot \left(V_0-\frac{1}{2}{m}_p{v}_n^2\right)}$

We include $V_0=\frac{1}{2}{m}_p{\left(5.5\times {10}^8\right)}^2$.

We get:

$S\approx 1.4\times {10}^5\Rightarrow \mathcal{A}~{\text{e}}^{-1.4\times {10}^5}\approx 0.$

Interpretation:

- The probability of tunneling is extremely small, but not zero, which is very important, because from this we can draw the following two conclusions: - If there is an infinite number or quasi-continuum of primary particles in the external quantized-velocity space, at least one can tunnel, and that event would trigger the Big Bang. - This low probability supports the idea of a spontaneous and rare origin of the universe as an extremely rare but inevitable fluctuation in the quantized-velocity space.

Physical meaning:

- If ${\mathcal{A}}_E$ is not zero, there is a non-zero probability that the particle passes from the world with $v>c$ to our world with $v\le c$. - This enables spontaneous quantum “breaking” of the velocity barrier and entering the space where classical relativity rules, i.e. our universe.

This is also confirmed by the total probability with given parameter values (Planck mass, barrier height, quantum velocity, etc.), which is:

$\boxed{P_{\text{total}}\approx 0}$

The goal of this model is to create a combination: primordial particle hypothesis: primordial particles in quantized-velocity space  tunneling  creation of the universe; and on the other hand quantum gravity: quantization of space and time (e.g., Loop Quantum Cosmology, LQC)  dynamic metrics  Big Bounce or quantum Big Bang.

Assumption: The velocity-quantized space of primordial particles

$E_n=\frac{1}{2}{m}_p{v}_n^2$

- Certain velocities $v_n>c$ are allowed in that space.

Tunneling is the passage through an effective barrier:

$V\left(v\right)=\frac{1}{2}{m}_p{v}_b^2$

where $v_b>c$, and the barrier acts as a quantum threshold for entry into our universe.

The tunneling leads into space and time where: - Velocity limit $v\le c$, - Beginning of spatial and temporal geometry.

When the particle “passes” the barrier: - It is no longer just a particle, but becomes a source of curvature of space and time, - Quantum gravity is engaged: space and time are quantum generated around it.

In Loop Quantum Cosmology (LQC), quantum cosmological dynamics is described by a modified Friedman equation (4):

$H^2=\frac{8\pi G}{3}\rho \left(1-\frac{\rho }{{\rho }_c}\right),$

where: $\rho$ is the energy density of the primordial particle, ${\rho }_c~{\rho }_{\text{Planck}}$ is the critical density from quantum gravity.

This equation predicts the Big Bounce, not the singularity.

We hypothesized that primordial particles exist in their own flat space and time while moving at velocities much faster than that of light. This “outer space” is a quantized-velocity space. In this space, “collisions” of these particles are possible, and with some of them there exists the possibility that one primordial particle slows to a velocity slightly higher than that of light, i.e. for quantum velocity ${\epsilon }_v$ higher than $c$. At the same time, its energy increases rapidly and it tunnels through the quantum barrier (exponentially suppressed probability).

Upon entering the domain where $v\le c$, it becomes a source of quantum curvature. A quantum-generated space and time is initiated via the LQC formalism, so that our universe emerges with Planck conditions as a boundary.

Our hypothesis provides a natural mechanism for initial entropy and time.

It is possible for many other universes to arise in a similar way (each “Big Bounce” is a consequence of the tunneling of some new primordial particle).

We expect that experimental confirmation of our hypothesis is possible through a quantum spectral signature in the cosmic microwave background (CMB).

We combine: