# Three Duality Symmetries between Photons and Cosmic String Loops, and Macro and Micro Black Holes

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## Abstract

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## 1. Introduction

## 2. Duality Symmetry between Cosmic String Loop Thermodynamics and Photon Thermodynamics

**Figure 1.**Plot of the dimensionless entropy density $y=(S/V)({{l}^{\prime}}_{P}^{3}/{k}_{B})$ in terms of the dimensionless temperature $\theta =T/{T}_{P}$ for photons (straight line) and cosmic string loops (hyperbolic line). For the sake of qualitative illustration, in this figure we have taken ${\beta}_{1}$ and ${\beta}_{2}$ as unity in Equation (11) and in Equation (12).

- 1)
- cosmic string loops lead to vanishing gravitational effects because $(U/V)+3p=0$, thus implying a cosmic expansion with constant rate, whereas photons slow down the cosmic expansion because $(U/V)+3p=2(U/V)>0$, according to Equation (8);
- 2)
- in an adiabatic reversible expansion, i.e., expansion with $S(V,T)=constant$, and assuming $V\sim {R}^{3}$ with R the cosmic scale factor, we have ${T}^{3}V\sim {\left(TR\right)}^{3}\sim constant$ for photons, and ${T}^{3}/V\sim {(T/R)}^{3}\sim constant$ for cosmic string loops, according to Equation (10); this also leads to a generalized Wien’s law [26];
- 3)
- in view of the former result and from the behaviour of Equation (9) in $U(T,V)$ we have $U/V\sim {R}^{-4}$ for photons and $U/V\sim {R}^{-2}$ for cosmic string loops.

**Figure 2.**Dimensionless mass m ($M/{M}_{P}$) as function of dimensionless temperature θ ($T/{T}_{P}$) given by Equation (14) for $b=0$ (Hawking expression) in the upper curve, for $b=\pi /2$ (solid line), and the second term of Equation (16) with $b=\pi /2$ (tilted right line). The vertical line corresponds to a given value of T. The intersecting points of it with the continuous line define the respective thermally dual black holes. Figure from Ref. [11].

## 3. Thermal Duality Symmetry between Macro and Micro Black Holes

## 4. Duality Symmetry of Thermal Equilibrium Photons/Macro Black Holes, Cosmic Loops/Micro Black Holes

#### 4.1. Macro Black Holes and Photon Gas

#### 4.2. Micro Black Holes and Loop Gas

## 5. Concluding Remarks

**Figure 3.**Sketch of the three thermal duality symmetries considered in this paper: (1) duality between photon gas and cosmic string gas; (2) duality between thermal equilibrium (photons/macro black holes) and thermal equilibrium (cosmic string loops/micro black holes); (3) duality between macro black holes and micro black holes.

- 1)
- Since the evaporation of micro black holes would be much slower than in the classical theory, this could allow for the survival of a number of primordial black holes and could explain why the bright remnants of their explosions are not observed. Thus, if supersymmetric particles of masses around 150 GeV (providing one of the candidates to the WIMPS black matter) are not found in the LHC (Large Hadron Collider) experiments currently at CERN (European Organization for Nuclear Research), micro black holes could provide an alternative candidate. Of course, this is only possible if micro black holes evaporate much more slowly than in the Hawking theory, as it is the case in the present theory. Otherwise, micro black holes of 150 GeV would last only a few Planck times.
- 2)
- Our analysis suggests that the temperature of the universe cannot be identified with the temperature of photons, since the temperature of the different cosmic components (photons, cosmic string loops) will change in different forms along expansion. Indeed, here we have seen that ${T}_{ph}\sim {R}^{-1}$ whereas ${T}_{cs}\sim R$.
- 3)
- Having a duality symmetry between the thermodynamics of photons and of cosmic string loops points a connection between a radiation-dominated universe in the early stages, with R increasing with time as ${t}^{1/2}$, and a loop-dominated universe at the long-time behaviour, with R increasing with time as t (i.e., without slowing down of the expansion). It does so by just adding a duality symmetry as a hypothetical fundamental ingredient in this model universe.
- 4)
- The duality symmetry would be of interest not only to relate early states with late states of the Universe, but, at every moment of its evolution, it would establish a symmetry between photons and cosmic string loops, in spite of the different behaviour of their temperatures along the cosmic expansion.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Jou, D.; Sciacca, M.; Mongiovì, M.S.
Three Duality Symmetries between Photons and Cosmic String Loops, and Macro and Micro Black Holes. *Symmetry* **2015**, *7*, 2134-2149.
https://doi.org/10.3390/sym7042134

**AMA Style**

Jou D, Sciacca M, Mongiovì MS.
Three Duality Symmetries between Photons and Cosmic String Loops, and Macro and Micro Black Holes. *Symmetry*. 2015; 7(4):2134-2149.
https://doi.org/10.3390/sym7042134

**Chicago/Turabian Style**

Jou, David, Michele Sciacca, and Maria Stella Mongiovì.
2015. "Three Duality Symmetries between Photons and Cosmic String Loops, and Macro and Micro Black Holes" *Symmetry* 7, no. 4: 2134-2149.
https://doi.org/10.3390/sym7042134