Quantum Cosmological Tetration of Time

In an original quantum cosmology model, the scale factor evolution de-scribing Hubble expansion is solely determined by the third tetration of time. The model exhibits early accelerating expansion, mid-time decelerating expansion, and late accelerating expansion. The substrate of reality, coined the “Graphiverse”, is a quantum-classical information processing network, represented by a learning deep generative graph. It comprises two complementary sub-graphs which are the substrates of a perceived rendering and a dark rendering of the emergent physical universe. Four temporal registers that count change (i.e. dimensions of time) are defined: system time, classical complexity time, quantum-classical correlation (or discord) time, and quantum coherence time. The cosmological scale factor evolves through a right-associative iterative exponentiation of these times. In the first (right) exponentiation, quantum-classical correlation (or discord) time is the base and its exponent is quantum coherence time. In the second (left) exponentiation, classical complexity time is the base and its exponent is the first (right) exponentiation. The four temporal registers that count change self-synchronize and equalize the four dimensions of time. The model provides a nexus for a new discussion about time and quantum gravity.


Introduction
This research note is motivated to address problems of time and quantum gravity [1]. Quantum theory and general relativity are each an accurate formalization of physics, however their unification remains obscure. Time is absolute in quantum theory, whilst dynamical in general relativity. The passage of time is a familiar experience, yet leading scientists and philosophers continue to debate its meaning and significance [2] [3].
The reader is referred to Hu et al. [4] for a review and contemporary bibliography of quantum coherence and quantum correlations. Seminal insights on this subject are presented in the works of Zurek [5] [6]. For an introduction to the theory of complex systems see Thurner et al. [7] and for further background refer to Zurek's compilation on complexity, entropy and the physics of information [8].
The following section sets out the formulation of an original quantum cosmology model in which the scale factor evolution [9], ( ) a t , is solely determined by the third tetration of time, t t t . That section also provides a physical, quantum cosmological, interpretation of the formulation and discusses how the universe is experienced via agents' perceptual interfaces. Interpretations of the chronology of the model evolution are given in the subsequent section and succinct conclusions wrap up this note.

Formulation
The standard cosmological scale factor is the ratio of the proper distance, say between two galaxies, at a time counted from the Big Bang, divided by the distance at the reference time, now. The scale factor, spatial curvature and the energy density of the universe are related by the Friedmann equations, derived from Einstein's field equations of gravitation, where gravity is a geometric property of space and time [9]. An original model cosmological scale factor evolution is defined here which provides a nexus to relate these successful equations of gravitation to the quantum foundations of time.
As a preliminary condition, let us assume information is processed in a quantum-classical network [10], represented by a learning deep generative graph [11], that is the substrate of reality. This substrate, here coined the "Graphiverse",  , comprises two complementary sub-graphs which are the substrates of: 1) a specific perceived rendering,  , that is our emergent physical universe, explored by its multiple agents; and 2) a dark rendering,  , which is hidden from the specific sensory, technological and neural interactions of the multiple agents in  : The dark rendering,  , (2.1) is placed above the perceived rendering,  , that is our emergent physical universe, to emphasise that  is dominant. Hoffman and co-workers [12] make a reasoned claim that Darwin's evolution by natural selection tunes the perceptual interfaces of agents to maximise their fitness in  , over attaining enlightenment about all truths in  . The perceived rendering,  , that appears as our virtual physical universe, only provides a tantalising glimmer of that deep enlightenment, whilst keeping us pragmatically focused on passing on our genes, knowledge and values. Future scientific progress, particularly in high energy physics and cosmology, is expected to unveil the dark rendering,  , and hence reveal more of the Graphiverse,  , that where t ∆ is a minimum computational timestep in the evolution of the Graphiverse The four-colour theorem [13] states the vertices of every planar graph can be identified with at most four labels (generalisation of colours) where no adjacent vertices have the same label. Let us consider the four labels as four 1-dimensional temporal registers that count change, as times on rational number lines, during the information symmetry-breaking evolution of the Graphiverse.
The four temporal registers, or times, are: 1) system time, t ; 2) classical complexity time, ΔCC t ; 3) quantum-classical correlation (or discord) time, ΔQC t ; and 4) quantum coherence time, ΔQQ t An original formulation of cosmological scale factor ( ) a t is conjectured: where spontaneous self-synchronisation [14] causes all registers to equalize with system time: From (2.6) and (2.7) the main equation of this paper (2.8) is boxed: and second derivative ( ) a t  : which are plotted in Figures 1-7 is its exponent.
Consider this tetration as compounding information symmetry-breaking in the Graphiverse. The right-associative process runs through the entire quantum-classical computational network, represented by a learning deep generative graph, comprising counted changes in quantum coherence, compounding counted changes in quantum-classical correlations (or discord), that all together compound counted changes in classical complexity. As system time, t, accumulates then quantum coherence symmetries break (decoherence), quantum-classical correlation symmetries break (decorrelation) and classical symmetries break (complexity evolution).
In the Graphiverse, self-synchronisation of temporal registers (2.7) with concomitant complexity growth (2.6) relates to a gain in computational efficiency The integrand in (2.11) is the scale factor evolution given by the third tetration of time (2.8), and the integrand in (2.12) is the square of the magnitude of the third tetration of time.
A generalised Puiseux series expansion of (2.8) at 0 t = is given by:   Table 1 for numerical analyses.     Table 1 for numerical analyses.  Table 1 for numerical analyses.   Table 1 for numerical analyses.   Table 1 for numerical analyses.
Oscillation of scale factor in a localized interval of near negative time.
Early accelerating expansion. Scale factor monotonically increasing with the arrow of time. (2.14)

Model Cosmology Chronology
Numerical analyses of ( ) a t , ( )  Table 1 which sets out the model cosmology chronology. The model cosmology chronology is characterised by 7 stages (below, sub-sections 3.1 to 3.7). Model time is anchored by definition:

Far Negative Time
The scale factor evolution ( ) a t during model times less than negative 4 from the Big Bang has complex solutions. The real part, Re a t ≈     , and the imaginary part, Far negative time is the most quiescent stage of the model cosmology, in terms of scale factor dynamics.

Near Negative Time
The scale factor evolution ( )  Figure 7). This stage of the development of the universe is conjectured to involve the superposition of multiple scale factor oscillations, such as given in a generalised Puiseux series expansion (2.13), or by the Fourier transform (2.14), or some other wave packet decomposition in time, that combine to establish a single wave packet in a localised interval of near negative time.

Big Bang
The Big Bang occurs at model time 0 t = . It is interpreted to be characterised by the explosion of ( ) It is conjectured here that the complex wave packet scale factor evolution in near negative time represents a hyper-massive quantum proto-universe in superposition before the Big Bang. The constants (2.11) (2.12) at the limit in far negative time, of the integrals of the complex wave packet, may relate to the conjectured mass-energy of the hyper-massive quantum proto-universe.
Whether any such hyper-massive quantum proto-universe before the Big Bang could lead to entanglement of temporal orders between time-like events is a profound question for modern physics and the reader is referred to Zych et al. [16] for a seminal discussion on Bell's theorem for temporal order. Researchers are encouraged to build on their important work, perhaps towards new physics. As Zych and her co-workers emphasise, pre-defined local variables cannot be used to describe temporal order and the classical concept of a causal structure is unsound in any framework that respects the basic principles of general relativity and quantum mechanics. Extending the principles of these great theories into the realm of negative time before the Big Bang may help us formulate any such new physics.

Early Accelerating Expansion
In the model cosmology, in positive time immediately following the Big Bang, the evolution of ( ) a t is characterised by early accelerating expansion. The homogeneous and isotropic nature of the present universe (cosmological principle) is interpreted to be a consequence of this primordial exponential expansion, which starts at 0 t = and ends at 0.064 t ≈ after the Big Bang, at redshift 9.03 z ≈ ) ( Figure 3).
Cosmological decoherence (from quantum coherence to quantum-classical correlation) [4] [5] [6] is the dominant mechanism during this stage. The end of this stage marks the end of the Dark Ages, reionization and large-scale structuration of the early universe.

Decelerating Expansion
The scale factor evolution ( )  Figure 6). This model stage corresponds to a matter dominated interval in the evolution of the universe when emergent gravitational attraction appears to constrain the rate of expansion. The foundational constraint is however the third