We need the Plank-length $l_P$, Plank-time $t_P$, Planck-mass $m_P$ and Planck-mass-density ${\varrho }_P$: The following relations between them are utilized: The last one is obtained according to Besides the dimensional quantities $a=$ (essentially a radius of space curvature), $t=$ time, and $V=$ volume of space, we use corresponding dimensionless quantities x, $\tau$, and v resp., defined by Comparing the last two relations yields

Our model of space-time is based on the assumptions that the space is uniformly curved, thus closed, and consists of DE of uniform mass density $\varrho \left(t\right)$, i.e. DE is space energy. In the framework of GR, a space-time of this kind must satisfy the classical (=non-quantum mechanical) FL equation Regarding $\varrho \left(t\right)$ we assume as in Refs. [4]–[5] that the activity of the cosmological constant $\mathsf{\Lambda }$ remains in force during the space expansion following the PS, but is impeded and thus weakened by the information transfer to the newly added space elements. As already mentioned, we take this into account by adding a friction term $-f\dot{a}\left(t\right)$ (linear for reasons of simplicity) in the cosmological momentum equation $\ddot{a}\left(t\right)=(8\pi G/3){\varrho }_{\mathsf{\Lambda }}a$, which results by time derivation from Equation (6) with $\varrho \left(t\right)\equiv {\varrho }_{\mathsf{\Lambda }}$, thus obtaining where $f>0$ is a constant, and ${\varrho }_{\mathsf{\Lambda }}$ is the (constant) mass density of $\mathsf{\Lambda }$. Multiplying this equation with $\dot{a}\left(t\right)$ and integrating it with respect to t from 0 to t yields In order to remain within the framework of recognized physics, we require that in spite of the added friction term Equation (6) is still valid, whence $Z_0=0$ or This means that Equation (7) implies the validity of the FL-equation (6), if the latter is satisfied for $t=0$, or, in other words, the solution of Equation (6) can be obtained from Equation (7), if Equation (6) is only used as an initial condition at $t=0$ for the latter. Inserting Equation (4) and using the relations (3), Equations (6) and (7) become where Equation (9) yields the initial condition at $\tau =0$ for Equation (10). From Equations (8) follows For the system under consideration, the quantization according to the WDW method is performed at the FL equation. As already mentioned in paragraph 2. of the Introduction, we don’t do this with regard to the scale factor a as usual, but rather with regard to the volume V. Hence we still specify how the FL equation reads in terms of the volume of a uniformly curved and closed three-dimensional space. For this purpose, we only have to insert the relation $a=V^{1/3}/{\left(2{\pi }^2\right)}^{1/3}$ in Equation (6) to obtain The corresponding equation in dimensionless quantities is obtained by inserting $x=v^{1/3}$ (according to Equation (5)) into Equation (9) and is

For computing the space expansion $a\left(t\right)$ and in the WDW equation for the PS, the friction term plays no role in the FL equation, because it disappears in both cases. In the first case this holds, because only the initial condition (11) is needed, and in the second, because in the PS $\varrho ={\varrho }_{\mathsf{\Lambda }}$ is not reduced. However, friction losses play an important role in the energy balance, which can be derived from the FL equation. Let us take a closer look.

Equation (7) can also be viewed as the motion equation for a solid body of mass 1, accelerated by a force $\sim a$ and decelerated by a frictional force $\sim \dot{a}\left(t\right)$. Under suitable initial conditions it can be integrated to the same energy law as that in Equation (8) with $Z_0=0$, i.e. The terms on the left represent the total energy of the body, and the term on the right the accumulated friction losses, so overall there is no conservation of energy. The latter can be achieved by including the surroundings (e.g. air) of the body, where the friction losses appear as heat energy.

In our cosmological case, there is no environment to absorb energy losses, and moreover, ${\dot{a}}^2\left(t\right)$ is no kinetic energy, so the considered equation must be interpreted quite differently. For this purpose we reformulate it into The first term on the left is the primordial DE density, and the second represents losses of it caused by the friction force and accumulated during the time t. The negative quantity $g\left(t\right)c^2$ is the energy density of the gravitational field caused by the rest mass of the DE and is also reduced by the friction losses, according to amount just as much as ${\varrho }_{\mathsf{\Lambda }}{c}^2$. This means that the same equation as above now implies conservation of energy as required due to the closedness of the system space as a whole. These relationships enable a simple graphical representation, shown in Figure 1.

From the last considerations it follows, that v can only assume discrete values which on average differ by $\Delta v_n=v_n-v_{n-1}$. (In principle, with lower probability other differences from the interval $0<\Delta v<2\Delta v_n$ would be possible, but in order not to unnecessarily complicate our calculations, we use the most probable values $\Delta v_n$ in the further course.) As explained in paragraph 5. of the Introduction, the volume of the PS must be extremely large for being able to store the information about all laws of nature. In Equation (18) the second term in parentheses can be neglected against the first for $v\gg {\delta }^{-3/2}=0.041$ according to Equation (36a), which is the case for almost all v, given the large size of the primordial volume we find out later (Section 5). Under these circumstances, Equation (18) simplifies to Of the two independent solutions, a suitable one is where for the amplitude we could arbitrarily chose the value 1, because every multiple of a solution is also a solution. The probability density ${\mathsf{\Psi }}^2\left(v\right)$ has the period $\omega \Delta v=\pi$, from which with $V=2{\pi }^2{l}_P^{3}v$ it follows, that for sufficiently large v or V resp. the most probable discrete volumes that the PS can assume are with $v^{*}$ (or $V^{*}$) is a small correction term in v (or V), which results from the fact that $\Delta v$ is slightly larger for small volumes than for large ones due to the stronger influence of space curvature. For the numerical values the later result (36a) for $\delta$ was used. The (very large) volume on which we base the PS in our model is denoted by where $1/2\ll N_Q$, $v^{*}\ll N_Q{v}_q$ and $V^{*}\ll N_Q{V}_q$ where used for the approximations obtained in the last step. The exact value of $N_Q=maxn\gg 1$ has yet to be determined. The extent of the corresponding quantum state, the PS, is from which follows that from $\psi \left(v_Q\right)=sin\left(\omega v_Q\right)=sin\left(\omega N_Q{v}_q\right)=sin\left(N_Q\pi \right)=0$ and from ${\psi }^{\prime }\left(v_Q\right)=\omega$$cos\left(\omega v_Q\right)=\omega cos\left(N_Q\pi \right)$ according to Equations (21) and (23), at the upper boundary we get the boundary conditions whereby, due to the extreme size of $N_Q$ we could assume without noticeable error that $N_Q$ is an odd integer.

From the discreteness of the possible PS volumes can be concluded that $V_Q$ (or $v_Q$ in dimensionless variables) is composed of elementary volumes whose average and also most probable size is $V_q$ (or $v_q$). Because both the FL and the WDW equation employed for deriving Equations (22) only describe space (via $V=2{\pi }^2{a}^3$) as a whole, a quantum structure of it can only become apparent through discrete values of the total volumes V. This interpretation is also supported by the fact that the differences between neighboring discrete volumes of sufficiently large size are all the same. As an example, consider a stack of bricks of height h. A stack of one brick has height h, of two height $2h$ etc. If we only know the total heights $nh$, we can regard the composition of bricks as a plausible hypothesis, see Figure 2. Admittedly, our interpretation can only be assessed as an indirect proof.4 However, any other interpretation of the discreteness of possible PS volumes (the “quantum staircase” shown in Figure 2) makes no sense, and we will see that our interpretation leads to a coherent overall concept.

In the following, we will refer to $V_q$ (or $v_q$) as the uniform volume of space quanta of which $V_Q$ (or $v_Q$) is composed, “uniform” for the sake of simplicity as already noted above. This simplification does not matter for most applications in this study, because – except for a rough determination of the primordial mass density ${\varrho }_{\mathsf{\Lambda }}$ in the next section – we always have to deal with very large numbers of space quanta, for which only their average volume matters. (This is due to the fact, that differences in size and probability, which we are actually dealing with, cancel each other out in the sum.) However, at the end of Section 3.2.1 we consider – if only qualitatively – the possibility of distinguishable space quanta that could turn out useful for the information storage so important for our model.5 Numerical values for $v_Q$ and the corresponding extent of the PS are derived in Section 5.1.4.

From Equations (23) results with the definition of $\delta$ in Equation (10c) i.e. the size $V_q$ of the space quanta is a function of the mass density ${\varrho }_{\mathsf{\Lambda }}$ of the DE in the PS and is not fixed by the WDW quantization. However, by comparing solutions of Equation (18) for different values ${\varrho }_{\mathsf{\Lambda }}$ we will come to a choice that is acceptable although not stringent. It appears sensible to limit our search to the range where quantum and GR phenomena are equally important, from which it follows that ${\varrho }_{\mathsf{\Lambda }}$ must be located in the vicinity of ${\varrho }_P$. Accordingly, we must allow for the space quanta in the PS to be tiny black holes. In the next section, we want to roughly find out from which point on this is the case. For this, we content ourselves with a simple calculation, ignoring both quantum and GR effects. We then numerically determine the solution of Equation (18) for different values of $\delta$ or ${\varrho }_{\mathsf{\Lambda }}$ resp. and decide which one seems most suitable for our purposes. The preceding rough calculation will help us to get a particularly illustrative picture of the space quanta.

Although Equation (9) must no longer be taken into account, we can still use it to calculate $\varrho \left(\tau \right)$ or $\rho \left(\tau \right)$ resp., i.e. If we were to use Equation (39) for calculating $\dot{x}\left(\tau \right)$, according to Equations (9b) and (40) the initial value of $\varrho \left(\tau \right)$ in the CS would be $\varrho \left(0\right)=3({\gamma }^2+1/x_Q^2){\varrho }_P/\left(8\pi \right)\approx 1.63·{10}^{-42}{\varrho }_P$ which contradicts Equations (12a) and (36b). In order to remove this discrepancy, for calculating $\dot{x}\left(\tau \right)$ we must instead employ Equation (38) to get With Equation (39) and the (very precise) approximation $\delta /\gamma +\gamma =\delta /\gamma$ follows from this so we finally get With this result the discrepancy disappears, because for $\tau =0$, neglecting $\gamma$ and $1/x_Q$ versus $\sqrt{\delta }$ (based on Equations (36a), (54) and (80b)), we get $\rho \left(0\right)=\delta$ and from Equations (9b) and (10c) $\varrho \left(0\right)=3\delta {\varrho }_P/\left(8\pi \right)={\varrho }_{\mathsf{\Lambda }}$ as required.

For sufficiently large values of $\delta \tau /\gamma$, the second bracket term in Equation (43) becomes negligible versus $\gamma$, and we obtain To find out more precisely when this is the case, we determine when $\sqrt{\delta }{e}^{-\delta \tau /\gamma }\le {10}^{-3}\gamma$ or $e^{\delta \tau /\gamma }$$\ge {10}^3\sqrt{\delta }/\gamma$ holds. This is the case for

with

where Equations (1), (4), (36) and (54) were used for the numerical values. After just $1.77·{10}^{-60}$ Planck times or $\approx {10}^{-103}$ seconds resp., i.e. practically instantaneously $\varrho$ decreases from the huge value $\varrho \left(0\right)={\varrho }_P$ to the much smaller value (coming about due to $\gamma \ll 1/x_Q$). This density decrease without equal represents a practically instantaneous and exorbitantly huge crash. Detailed reasons for its occurrence are discussed further down. It is particularly noteworthy that due to $x\left(0\right)=x_Q$ the collapse of the density leads almost exactly towards the density $\varrho \left(0\right)=3{\varrho }_P/\left(8\pi x_Q^2\right)$ that results from the FL equation (9) for the equilibrium $\dot{x}\left(\tau \right)=0$ at $\tau =0$. The solution belonging to $\theta =-\sqrt{\delta }$ (lower sign) even leads exactly to this point, because Equation (41) results in and $\varrho \left({\tau }_e\right)=3{\varrho }_P/\left(8\pi x_Q^2\right)$ according to Equations (9b), (41) and (43). In relation to our model, however, this is not an equilibrium, because from Equation (10) follows $\ddot{x}\left(\tau \right)=\delta x\left(\tau \right)>0$ for $\dot{x}\left(\tau \right)=0$. Because the solutions for both signs behave essentially the same, we further on restrict ourselves to that with the upper sign, i.e. $\theta =\sqrt{\delta }$.

We can now use Equations (9b) and (44) to calculate $\gamma$ by inserting for $\varrho$ today’s value ${\varrho }_0=\varrho \left(t_0\right)$ and resolving for $\gamma$, where $x_0=x\left({\tau }_0\right)$ is the present value of $x\left(\tau \right)$. Since in $\mathcal{U}$, DE contributes 68.3 percent of the total mass density and the latter equals the critical density ${\varrho }_{c0}=3H_0^2/\left(8\pi G\right)=3/\left(8\pi G{t}_{H0}^2\right)$ with $H_0=$ present Hubble parameter and $t_{H_0}=$ present Hubble time, we have Here, $t_{H0}=14.12·{10}^{9}y=4.45·{10}^{17}\mathrm{s}$ was plugged in, a value that is slightly higher than the age of $\mathcal{U}$, but slightly below the last values obtained from measurements for $t_{H0}$. (This value was chosen because it yields a particularly handy result for $\gamma$.) The value of $x_0$ can be freely specified, but only above $1/\sqrt{8\pi {\varrho }_0/\left(3{\varrho }_P\right)}$ due to Equation (49). For illustration we relate it to the radius $R_0=c{t}_0=13.8·{10}^9\mathrm{ly}=1.306·{10}^{26}\mathrm{m}$ (slightly below present the Hubble radius $c{t}_{H_0}=14.12\mathrm{ly}$) by setting $a_0=R_0\zeta$ or Inserting all this into Equation (49), we get with ${(8\pi {\varrho }_0/\left(3{\varrho }_P\right))}^{1/2}={10}^{-61}$ from which the condition results for obtaining real values of $\gamma$. For the space curvature to be below the maximum value compatible with measurements, according to Ref. [19] the condition $|{\mathsf{\Omega }}_k|\le 0.005$ must be satisfied. In Appendix C it is shown, that in terms of our parameters this amounts to $\zeta \ge 14.5$. In $\mathcal{U}$, the DE is usually attributed to a cosmological constant. Since the DE of our model is the substrate of space it should share this attribute, so we assume that at least during $\mathcal{U}$’s present lifetime the density $\varrho \left(\tau \right)$ of the DE is largely constant. For this reason we choose $\zeta$ still well above $14.5$. The value $\zeta =100$ seems sufficient to us and shall serve as a reference value. With this, due to $1.54/{\zeta }^2=1.54·{10}^{-4}$, we get from Equation (52) with very little error With the specification of $\gamma$ now also $\varrho \left(\tau \right)$ can be calculated. According to Equations (9b), (39) and (44) the further evolution of the density after its crash (i.e. for $\tau \ge {\tau }_c$) is given by Characteristic values, calculated with this, Equations (36), (51), (54), (80) and $\zeta =100$ are given in Table 2.

We are additionally interested in how the expansion rate $\dot{a}\left(t\right)$ changes over time t. For this we first calculate $x\left(\tau \right)$ and $\dot{x}\left(\tau \right)$ for some times $\tau$. From Equations (39), (42), (54) and (80) we get and where the relation $\sqrt{\delta }{e}^{-\delta {\tau }_c/\gamma }={10}^{-3}\gamma \ll \gamma$, used for defining ${\tau }_c$, was employed to calculate $\dot{x}\left({\tau }_c\right)$. For $\tau \ge {\tau }_c$ Equation (39) can be used for calculating $\dot{x}\left(\tau \right)$, and together with the above relationships, Equation (62a), and $\dot{x}\left(\tau \right)=1$ for $\tau =-{\gamma }^{-1}ln\left(\gamma x_Q\right)=9.34·{10}^{62}$, we get the values given in Table 3.

The values $\dot{a}\left(t\right)$ corresponding to $\dot{x}\left(\tau \right)$ are obtained from so simply by multiplying $\dot{x}\left(\tau \right)$ by the speed of light.

According to Equations (57), Table 2 and Table 3, we have (the = signs holding with high precision but not exactly), and due to $x\left({\tau }_c\right)=x\left(0\right)$ this state can be considered as start of the space expansion.

The causes of the initial density crash can best be understood on the basis of Equation (40). The values of the quantities on the right before and after the crash are given in Equations (56) and (57) resp. $\dot{x}\left(0\right)$ must even be larger than the already rather large extent $x_Q$ of the PS (consisting of very many space quanta according to Equation (89)) in order that $\varrho \left(0\right)$ can assume the high primordial value ${\varrho }_{\mathsf{\Lambda }}$. (If it consisted of only one or a few space quanta, $\dot{x}\left(0\right)$ would be almost zero, and then the term $1/x^2\left(0\right)$ in Equation (40) would provide the required high density.) From Equations (56) and (57) follows that essentially $x\left({\tau }_c\right)/x\left(0\right)=1$ whereas $\dot{x}\left({\tau }_c\right)/\dot{x}\left(0\right)=3.46·{10}^{-62}$, i.e. $x\left(\tau \right)$ changes virtually not at all during the crash, while $\dot{x}\left(\tau \right)$ changes by almost 62 orders of magnitude. In short, the huge extent of the PS in combination with the large initial value of $\varrho$ requires a large initial value of $\dot{x}\left(\tau \right)$ (see Equation (9a)), but both are pushed down nearly instantaneously by many orders of magnitude, $\varrho \left(\tau \right)$ by 42 (see Table 2) and $\dot{x}\left(\tau \right)$ by almost 62 (see Table 3).

The reason for the latter arises from Equations (10b) and (A6b). Solving the latter with respect to $\sigma$ and taking advantage of the smallness of gamma we obtain The high value of this friction coefficient combined with the high velocity $\dot{x}\left(0\right)$ leads to the extreme friction force against which the force can easily be neglected. Equation (10) thus results in The extremely fast change of $\dot{x}\left(\tau \right)$ at $\tau =0$ leads there to an extreme kink-like curvature of $x\left(\tau \right)$.

The huge velocity $\dot{x}\left(0\right)$ at the initiation $x=x_Q$ of the CS means that the space is in an extreme disequilibrium. Obviously, this affects the density $\varrho \left(t\right)$ much worse than an instability, where first a velocity $\dot{x}\left(t\right)$ or $\dot{\varrho }\left(t\right)$ must gradually be built up. Now, from a classical point of view, the initial CS is the same as the PS, and one might assume that the latter is also in a disequilibrium and therefore decays. In the PS, however, time does not even exist, and hence the question of equilibrium or disequilibrium is irrelevant. Even so, we will see later that the density crash, initiated by the emergence of time, affects ${\mathrm{ES}}_1$, the first of the quantum states ${\mathrm{ES}}_i$ to be derived later, in such a way that a huge increase in state probability comes about.

To determine the age of space-time, we first obtain from Equation (39) With this and Equations (1), (51) and (80) we get for the present age of space-time For the reference value $\zeta =100$ follows from this where is the age of $\mathcal{U}$.

Now we pursue the question of how it is about the required constancy of $\varrho$ during the lifetime of $\mathcal{U}$. For this, using we first go in Equation (39) from $\tau$ to a larger time scale in which the age of $\mathcal{U}$ becomes $\Delta T=1$. In this way we obtain from Equations (9b) and (44) with Equation (39) where for the numerical evaluation Equations (54), (62)–(64) and (80b) were used. If future measurements should show that the $0.6$ per mil decrease in $\varrho$ found herewith is too much (or too little), our reference value of $\zeta$ could accordingly be adapted.

Despite the simplification achieved by the transformation (68), the initial value problem posed by Equations (69) and (70) still does not have a consistent analytical solution, and because of the huge factor $\sqrt{3\beta }\sim {10}^{61}$, a numerical solution is not manageable either. Handling the problem, even analytically, succeeds, however, if $\mathsf{\Psi }\left(u\right)$ is decomposed into two branches: one, the main or residual branch, with values of $u-1$ large enough for neglecting the term $\sim u^{-\beta }$ against 1 in Equation (69), and another, the initial branch, with values of $u-1$ so small that $\mathsf{\Psi }$ is practically indistinguishable from its initial value $\mathsf{\Psi }\left(1\right)=0$.