# Entropy Balance in the Expanding Universe: A Novel Perspective

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## Abstract

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## 1. Introduction

## 2. The Role of Entropies

#### 2.1. Information

#### 2.2. Thermodynamic Entropy

## 3. Cosmic Expansion Comes into Play

^{−35}s after the Big Bang [31]. Vacuum quantum fluctuations (dictated by the Heisenberg energy-time uncertainty principle) could have been able to cause, through an inflaton-based mechanism, the occurrence of the Big Bang, characterized by very high density and temperature state [32,33,34,35]. At the very beginning, 1

^{−43}s after the Big Bang, our Universe was equipped with an energy of 10

^{19}GeV and a temperature of 10

^{32}K; its horizon was 10

^{−25}cm large and the density 10

^{96}kg/m

^{3}. By then, the temperature halved every double expansion. At 10

^{−36}s, the energy lowered to 10

^{16}GeV, and at 10

^{−32}s the temperature decreased to 10

^{28}K. The cosmic inflationary expansion at 10

^{−35}s stands for the standard explanation for experimentally detected cosmic features such as isotropicity, homogeneity, symmetry and zero curvature. It is noteworthy that the Universe underwent a rapid expansion so that, from the above-mentioned horizon diameter of 10

^{−25}at 10

^{−43}, it reached the size of about onemeter diameter at 10

^{−32}s. Another gentler inflationary period started approximately 4.5 billion years ago [36]. Currently, 13.79 billion years after its birth, our Universe is still accelerating, slowly proceeding towards thermal death [37]. In our cosmic era, from our standpoint of local observers, the visible cosmological horizon is 10

^{29}cm, the cosmic density is 10

^{−29}gr/cm

^{3}, the matter corresponds to one atom/m

^{2}and the space is expanding at a speed of about 67–74 km/s per megaParsec, with slightly different values according to different techniques [38].

## 4. Linking Cosmic Expansion, Information and Thermodynamic Entropy

_{sys}is the cosmic thermodynamic entropy detectable by the observer, A is the area of the local observer’s horizon, E is the energy including matter (the total mass–energy of the Universe consists of about 10

^{69}Joule), ħ is the reduced Planck constant, c is the speed of light, k is the Boltzmann constant, ζ is a factor such that 0 < ζ < 1.

_{sys}, we are allowed to quantify the thermodynamic information, by partitioning the factor into a relative information component (ζ

_{I}= 1 − ζ

_{S}) and a relative entropy component (ζ

_{S}= 1 − ζ

_{I}) [41]:

## 5. Entangled Spacetime and Comoving Horizons: An Unexpected Link

^{2}, just by embedding them in a higher dimensional S

^{3}space [48,49]. These authors view quantum entanglement as the simultaneous activation of signals in a 3D space mapped into a S

^{3}hypersphere. The particles are entangled at the S

^{2}level and un-entangled at the S

^{3}hypersphere level, therefore a composite system is achieved, in which each local constituent is equipped with a pure state. It is noteworthy that the two issues of a comoving horizon and entanglement on a hypersphere are assessable through the framework described by the Borsuk–Ulam theorem, which states that “every continuous map $f:{S}^{n}\to {R}^{n}$ must identify a pair of antipodal points”—diametrically opposite points on an n-sphere [50,51].This means that at least some of the entropy values detected at opposite sides of the spherical comoving horizon display matching description, i.e., they are entangled.

## 6. Conclusions

^{90}causally-disconnected quantum “seeds” [31] led to the experimentally detected homogeneity and isotropy. Still, inflation would have amplified minute quantum fluctuations (pre-inflation) into slight density ripples of over- and under-density (post-inflation).Here the concept of hyperuniformity comes into play, i.e., the anomalous suppression of density fluctuations on large length scales occurring in amorphous cellular structures of ordered and disordered materials [55].The evolution of a given set of initial points takes place when, through Lloyd iterations, each point is replaced by the center mass of its Voronoi cell. This corresponds to a gradient descent algorithm which allows a progressive, general convergence to a random minimum in the potential energy surface. Klatt et al. [55] report that systems equipped with different initial configurations (such as, e.g., either hyper-fluctuating, or anisotropic, or relatively homogeneous point sets), converge towards the same high degree of uniformity after a relatively small number of Lloyd iterations (about 10

^{5}).This means that, in the systems’ final states, independent of the initial conditions, the cell volumes become uniform and the dimension less total energy converges towards values comparable to the deep local energy minima of the optimal lattice. Therefore, we are allowed to describe the cosmic evolution suddenly after the Big Bang in terms of Lloyd iterations, where the initial quantum seeds stand for initial point sets, progressively converted to point sets with a centroidal Voronoi diagram. In other words, the tiny perturbations in the primeval Universe which seed the later formation of cosmic macro-structures might stand for the starting points of the subsequent processes described in terms of Voronoi cells. This would permit observers to achieve, starting from countless different possible conformations of the primeval Universe, the detected isotropic and homogeneous Cosmic Background Radiation. Indeed, after just 10

^{5}iterations, every possible initial system must converge towards an hyperuniform state, where observers perceive energy as very low and uniformity degree as very high.

## Author Contributions

## Funding

## Conflicts of Interest

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Tozzi, A.; Peters, J.F.
Entropy Balance in the Expanding Universe: A Novel Perspective. *Entropy* **2019**, *21*, 406.
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Tozzi A, Peters JF.
Entropy Balance in the Expanding Universe: A Novel Perspective. *Entropy*. 2019; 21(4):406.
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**Chicago/Turabian Style**

Tozzi, Arturo, and James F. Peters.
2019. "Entropy Balance in the Expanding Universe: A Novel Perspective" *Entropy* 21, no. 4: 406.
https://doi.org/10.3390/e21040406