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In an original quantum cosmology model, the scale factor evolution describing 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.

This research note is motivated to address problems of time and quantum gravity [

The reader is referred to Hu et al. [

The following section sets out the formulation of an original quantum cosmology model in which the scale factor evolution [

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 [

As a preliminary condition, let us assume information is processed in a quantum-classical network [

G ↦ { D U (2.1)

The dark rendering, D , (2.1) is placed above the perceived rendering, U , that is our emergent physical universe, to emphasise that D is dominant. Hoffman and co-workers [

Let us now define and consider the sets T, (2.2) and H, (2.3):

T = { Δ t , 2 Δ t , 3 Δ t , ⋯ , n Δ t − 2 Δ t , n Δ t − Δ t , n Δ t } (2.2)

H = { Δ t , 2 Δ t , 3 Δ t , ⋯ , t H − 2 Δ t , t H − Δ t , t H } (2.3)

t H = 1 / H 0 (2.4)

t = | T | / | H | , t ∈ ℚ , n ∈ ℤ (2.5)

where Δ t is a minimum computational timestep in the evolution of the Graphiverse; Hubble time, t H , is the reciprocal of the present value of the Hubble parameter, H 0 (2.4); and | T | and | H | are respectively the integer cardinalities (number of elements) of the sets T and H. Their ratio, | T | / | H | , defined as system time, t, (2.5) is an element of the set of rational numbers, ℚ , and is dimensionless and countable. The number n is an element of the set of integers, ℤ .

The four-colour theorem [

The four temporal registers, or times, are:

1) system time, t ;

2) classical complexity time, t Δ C C ;

3) quantum-classical correlation (or discord) time, t Δ Q C ; and

4) quantum coherence time, t Δ Q Q

An original formulation of cosmological scale factor a ( t ) is conjectured:

a ( t ) = t Δ C C ( t Δ Q C t Δ Q Q ) (2.6)

where spontaneous self-synchronisation [

t = t Δ C C = t Δ Q C = t Δ Q Q (2.7)

From (2.6) and (2.7) the main equation of this paper (2.8) is boxed:

a ( t ) = t t t (2.8)

where the first derivative a ˙ ( t ) :

a ˙ ( t ) = t t t ( t t ( log ( t ) + 1 ) log ( t ) + t t − 1 ) (2.9)

and second derivative a ¨ ( t ) :

a ¨ ( t ) = t t t ( t t − 1 + t t log ( t ) ( log ( t ) + 1 ) ) 2 + t t t ( t t − 1 log ( t ) + t t − 1 ( log ( t ) + 1 ) + t t − 1 ( t − 1 t + log ( t ) ) + t t log ( t ) ( log ( t ) + 1 ) 2 ) (2.10)

which are plotted in Figures 1-7.

Equation (2.6) is a right-associative third tetration, or iterative exponentiation, of registered counts of timesteps. The value of a ( t ) is calculated right to left, or top to bottom. In the first exponentiation, top right bracket, t Δ Q C is the base and t Δ Q Q is its exponent. In the second exponentiation, t Δ C C is the base and ( t Δ Q C t Δ Q Q ) 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 (Fermat’s principle of least time) with a loss in symmetry. (Note a similarity to emergent optical refraction). Nonetheless, the loss in symmetry during evolution of the Graphiverse in system time creates a rich and existential perceived rendering, U , for its multiple agents. Furthermore, the corresponding right-associative iterative exponentiation of time greatly compounds the perceived expansion of cosmological scale, (2.6) and (2.8). The formulation (2.8) models the expansion of our perceived universe, with early acceleration, mid-time deceleration, and late acceleration (Figures 1-7).

Integration of the complex wave packet (

lim t → − ∞ ( t + ∫ t 0 t t t d t ) ≈ 0.62284 + 0.54701 i (2.11)

lim t → − ∞ ( t + ∫ T 0 | t t t | 2 d t ) ≈ 1632.76 (2.12)

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 t = 0 is given by:

t t t ≈ t + t 2 log 2 ( t ) + 1 2 t 3 log 3 ( t ) ( log ( t ) + 1 ) + 1 6 t 4 log 4 ( t ) ( log 2 ( t ) + 3 log ( t ) + 1 ) + 1 24 t 5 log 5 ( t ) ( log 3 ( t ) + 6 log 2 ( t ) + 7 log ( t ) + 1 ) + 1 120 t 6 log 6 ( t ) ( log 4 ( t ) + 10 log 3 ( t ) + 25 log 2 ( t ) + 15 log ( t ) + 1 ) + O ( t 7 ) (2.13)

Model Time t | a ( t ) | a ˙ ( t ) | a ¨ ( t ) | Redshift z = 1 a ( t ) − 1 | Interpretation |
---|---|---|---|---|---|

t ≲ − 4 | R e [ a ( t ) ] ≈ 1 I m [ a ( t ) ] ≈ 0 | Far negative time. | |||

− 4 ≲ t < 0 | C o m p l e x [ a ( t ) ] wave packet | C o m p l e x [ a ˙ ( t ) ] wave packet | C o m p l e x [ a ¨ ( t ) ] wave packet | Oscillation of scale factor in a localized interval of near negative time. | |

t = 0 | 0 | 1 | ∞ | ∞ | Big Bang, a ¨ ( t ) explodes. |

0 < t ≲ 0.064 | Early accelerating expansion. Scale factor monotonically increasing with the arrow of time. | ||||

t ≈ 0.064 | ≈+0.10 | Max ≈ +1.71 | 0 | ≈9.03 | End of the Dark Ages, reionization and large-scale structuration of the universe. |

0.064 ≲ t ≲ 0.162 | Decelerating expansion. Matter dominated. | ||||

t ≈ 0.162 | ≈+0.26 | ≈+1.47 | Min ≈ −3.25 | ≈2.87 | Deceleration extremum. Matter dominated. |

0.162 ≲ t ≲ 0.668 | Decelerating expansion. Matter dominated. | ||||

t ≈ 0.668 | ≈+0.73 | Min ≈ +0.71 | 0 | ≈0.36 | End of deceleration. |

0.668 ≲ t < 1 | Late accelerating expansion. “Dark Energy” dominated. | ||||

t = 1 | 1 | 1 | 2 | 0 | Time now. Late accelerating expansion. “Dark Energy” dominated. |

1 < t ≲ 4 | Near positive time. Late accelerating expansion. “Dark Energy” dominated. | ||||

t = 4.0708 | ≈8 × 10^{184} | a ( t ) ≈ Number of Planck volumes in observable universe | |||

t ≳ 4 | Far positive time. Not physical. |

The Fourier transform of scale factor evolution a ( t ) (2.8) with frequency variable ω is given by:

F t [ t t t ] ( ω ) = 1 2 π ∫ − ∞ ∞ t t t e i ω t d t (2.14)

Numerical analyses of a ( t ) , a ˙ ( t ) and a ¨ ( t ) and a study of the plots in Figures 1-7 leads to

The scale factor evolution a ( t ) during model times less than negative 4 from the Big Bang has complex solutions. The real part, R e [ a ( t ) ] ≈ 1 , and the imaginary part, I m [ a ( t ) ] ≈ 0 . Far negative time is the most quiescent stage of the model cosmology, in terms of scale factor dynamics.

The scale factor evolution a ( t ) during model times less than zero and greater than negative 4 from the Big Bang has complex solutions. The real part, R e [ a ( t ) ] , and the imaginary part, I m [ a ( t ) ] , form a complex wave packet (

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.

The Big Bang occurs at model time t = 0 . It is interpreted to be characterised by the explosion of a ¨ ( t ) (

In contrast to the complex wave packet scale factor evolution in negative time, the Big Bang event at t = 0 heralds monotonically increasing scale factor evolution through positive time. The scale factor evolution in (2.8) is strongly asymmetric about t = 0 . The scale factor evolution becomes spontaneously real valued for t > 0 and according to Barbour et al. a gravitational arrow of time [

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. [

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 t = 0 and ends at t ≈ 0.064 after the Big Bang, at redshift z ≈ 9.03 ) (

Cosmological decoherence (from quantum coherence to quantum-classical correlation) [

The scale factor evolution a ( t ) during model times between at t ≈ 0.064 after the Big Bang, redshift z ≈ 9.03 and t ≈ 0.668 after the Big Bang, redshift z ≈ 0.36 ) is marked by decelerating expansion. The extremum of deceleration occurs at model time t ≈ 0.162 after the Big Bang, redshift z ≈ 2.87 (

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 tetration of time, a ( t ) = t t t . The end of this stage occurred during the Hadean Eon on our planet Earth, when the chemical conditions for abiogenesis emerged.

We are presently ( t = 1 , redshift

One might reasonably consider that late accelerating expansion in far positive time

This quantum cosmology model proposes that scale factor evolution is defined by the third tetration of time,

The tetration of time is considered as compounding information symmetry-breaking in the Graphiverse. The right-associative process runs through the Graphiverse and comprises 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 accumulates then quantum coherence symmetries break (decoherence), quantum-classical correlation symmetries break (decorrelation) and classical symmetries break (complexity evolution), in an iterative exponentiation. In the perceived rendering, since the Big Bang, we observe emergent space undergoing early accelerating expansion, mid-time decelerating expansion, and late accelerating expansion. The model also provides solutions for cosmological scale evolution before the Big Bang. It is conjectured that a complex wave packet in the scale factor evolution in near negative time represents a hyper-massive quantum proto-universe in superposition before the Big Bang.

The gravitational dynamics of the cosmos are emergent in this model. The model formulation of

This original research is self-funded and I thank my reviewers and editors for their valuable support.

The author declares no conflicts of interest regarding the publication of this paper.

McCoss, A. (2020) Quantum Cosmological Tetration of Time. Journal of High Energy Physics, Gravitation and Cosmology, 6, 225-237. https://doi.org/10.4236/jhepgc.2020.62016