# Big Bounce Genesis and Possible Experimental Tests: A Brief Review

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## Abstract

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## 1. Motivation and Overview

- In terms of the primordial power spectrum, a complete duality between the inflation scenario and BUS has been well established [49,50,51,54,55]. It enables both inflation and BUS to generate a scale-invariant primordial power spectrum with the same probability from the unified parameter space. In short, if a primordial power spectrum can be generated in an expanding phase of cosmological evolution, it can also be generated in a contracting phase with the same scale dependence and time dependence [51]. Literally, they are degenerate in the leading-order signatures of CMB spectra;
- Currently, all models in these two scenarios are utilizing some undetected classical/quantum fields to drive the inflation or big bounce at the very early stage of cosmic evolution. Hence, their predictions of the CMB spectrum and the scalar-tensor ratio, both built on the linear perturbation theory of these unconfirmed fields, are still questionable.

- Thermal equilibrium production: DM particles with large cross-sections are produced very efficiently, so that their abundance increases rapidly and achieves the thermal equilibrium value even in the pre-bounce contracting or post-bounce expanding phases. Then, the abundance of DM tracks the thermal equilibrium value before the freeze-out takes place,$$Y\left(t\right)={Y}_{eq},\phantom{\rule{1.em}{0ex}}t<{t}_{f},$$
- Out-of-chemical equilibrium production [56]: In a given cosmological background, if the cross-section of DM is small enough, the production of DM should be inefficient. Therefore, its abundance cannot achieve the thermal equilibrium value during the production process,$$Y\left(t\right)\ll {Y}_{eq},\phantom{\rule{1.em}{0ex}}t<{t}_{f}.$$

- Strong freeze-out: If plenty of DM particles have been produced before, the backward reaction of Equation (3) dominates, as the forward reaction of Equation (3) is suppressed exponentially. The backward reaction decreases the abundance of DM very efficiently until the number density of DM is too low to keep thermal contact in the expanding phase. Therefore, after such a strong freeze-out, the relic abundance of DM is significantly lower than that before freeze-out and is inverse to the DM cross-section,$${Y}_{f}\propto \frac{1}{\langle \sigma v\rangle},$$
- Weak freeze-out: If the abundance of DM is very low, the backward reaction in Equation (3) is always negligible. When the forward reaction in Equation (3) is suppressed exponentially, both the production and annihilation of DM end. Therefore, the relic abundance of DM is equal to the abundance of DM at the end of the production phases and is generically proportional to the DM cross-section,$${Y}_{f}=Y\left({t}_{f}\right)\propto \langle \sigma v\rangle .$$

## 2. Dark Matter Production and Evolution in the Bounce Universe Scenario

- Initial abundance of DM takes ${n}_{\chi}^{i}=0$, i.e., the number density of DM is set to be zero at the onset of the pre-bounce contraction phase in which $T\ll {m}_{\chi}$.
- The matching condition on the bounce point is ${n}_{\chi}^{+}\left({T}_{b}\right)={n}_{\chi}^{-}\left({T}_{b}\right)$, i.e., the number density of DM, ${n}_{\chi}$, at the end of the pre-bounce contraction (denoted by −) is equal to the initial abundance of the post-bounce expansion (denoted by +), given that the entropy of the universe is conserved around the bounce point [63].

#### 2.1. Type I: Scalar Dark Matter in a High Temperature Bounce

- Thermal equilibrium production: For the upper line in Equation (14), with the large value of $\langle \sigma v\rangle {m}_{\chi}$, DM is produced in abundance, which has reached the thermal equilibrium before the end of the production phases.
- Out-of-chemical equilibrium production: Additionally, for the lower line in Equation (14) with the small value of $\langle \sigma v\rangle {m}_{\chi}$, the production is mostly one-way, and thermal equilibrium cannot be established, so that its abundance is much lower than the value of the thermal equilibrium state even at the end of he production phases.

- Strong freeze-out: If ${Y}_{+}{|}_{x=1}\gg {\left(4{\pi}^{4}f\langle \sigma v\rangle {m}_{\chi}\right)}^{-1}$, the initial abundance of DM at the onset of the freeze-out process is large enough for pair-annihilation of DM particles during the thermal decoupling, so that the relic abundance of DM becomes irrelevant of the initial abundance. Particularly, it is inversely proportional to the cross-section,$${Y}_{f}=\frac{1.71\times {10}^{-29}e{V}^{-1}}{\langle \sigma v\rangle {m}_{\chi}}.$$
- Weak freeze-out: If, on the other hand, ${Y}_{+}{|}_{x=1}\ll {\left(4{\pi}^{4}f\langle \sigma v\rangle {m}_{\chi}\right)}^{-1}$ and the density of DM is too low to pair-annihilate during the thermal decoupling. The relic abundance of DM after freeze-out in this limit is just the initial abundance at the onset of the freeze-out process,$${Y}_{f}={Y}_{+}{|}_{x=1}.$$

#### 2.2. Type III and IV: Bosonic and Fermionic Dark Matter in a Low Temperature Bounce

#### 2.3. Type II: Fermionic Dark Matter in a High Temperature Bounce

## 3. Thermal Fluctuations of Dark Matter in the Bounce Universe Scenario

- Step I (inside horizon): Computing the energy density of the sub-horizon modes of thermal fluctuations of DM, $\delta {\rho}_{k}{|}_{k\ge \left|aH\right|}$, by utilizing the traditional thermodynamics,The energy density of sub-horizon thermal fluctuation takes [73,74]:$$\delta {\rho}_{L}^{2}=\frac{{\langle \delta {E}_{\chi}\rangle}^{2}}{{\left(aL\right)}^{6}}=\frac{{\mu}^{2}{e}^{\beta \mu}}{{\pi}^{2}{\beta}^{3}}{\left(aL\right)}^{-3}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{1.em}{0ex}}aL\le {\left|H\right|}^{-1}\phantom{\rule{3.33333pt}{0ex}},$$$$\delta {\rho}_{k}^{2}=\frac{6{\pi}^{2}\delta {\rho}_{L}^{2}}{{k}^{3}}=\frac{6{\mu}^{2}{e}^{\beta \mu}}{{\left(a\beta \right)}^{3}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{1.em}{0ex}}k\ge \left|aH\right|$$
- Step II (beyond horizon): Getting the solution of the energy density of the super-horizon modes of thermal perturbations, $\delta {\rho}_{k}{|}_{k\le \left|aH\right|}$, by deriving and solving their equation of motion in the long wavelength limit and leaving the initial amplitude of these long wavelength perturbations undetermined.Being different from the sub-horizon thermal fluctuations originating from the thermal uncertainties and correlations in the grand ensemble, the super-horizon thermal perturbations describe how the energy density varies with the spatial variance of underlying physical quantities, such as local temperature and chemical potential. Therefore, the starting point for investigating the super-horizon mode of DM thermal fluctuation can be taken as:$$\delta {\widehat{\rho}}_{\chi}(\mathrm{x},t)=\delta {n}_{\chi}(\mathrm{x},t){\u03f5}_{\chi}\left(t\right)+{n}_{\chi}\left(t\right)\delta {\u03f5}_{\chi}(\mathrm{x},t),$$$$\tilde{\beta}=\beta +\delta \beta (\mathrm{x},t)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{2.em}{0ex}}\delta \mu (\mathrm{x},t)=0\phantom{\rule{3.33333pt}{0ex}},$$$$\delta {\widehat{\rho}}_{\chi}(\mathrm{x},t)=-{n}_{\chi}\mu \left(\mu -3{\beta}^{-1}\right)\delta \beta (\mathrm{x},t)\phantom{\rule{3.33333pt}{0ex}},$$It is clear that if $\delta \beta (\mathrm{x},t)$ is determined, one can figure out $\delta {\widehat{\rho}}_{\chi}(\mathrm{x},t)$ with Equation (35) immediately. By expanding the Boltzmann equation, Equation (9), up to the first order, and simplifying it with the relation $\partial \left(\delta \beta \right)/\partial t=Hy\phantom{\rule{3.33333pt}{0ex}}\partial \left(\delta \beta \right)/\partial x$ in a radiation-dominated background, we can obtain:$$\frac{\partial \left(\delta \beta \right)}{\partial x}+\frac{\mathsf{\Theta}}{xH}\delta \beta =0.$$$$\begin{array}{c}\hfill \mathsf{\Theta}\equiv \left\{{e}^{-g\left(x\right)}\tilde{\langle \sigma v\rangle}\frac{{m}_{\chi}^{3}}{{\pi}^{2}{x}^{3}}\left[1+{e}^{2g\left(x\right)}+6{(g\left(x\right)-3)}^{-1}\right]-[1-\frac{dg\left(x\right)}{dx}x{(g\left(x\right)-3)}^{-1}]H\right\}\end{array}$$Here, we focus on the Type I model of BBG, i.e., bosonic DM in a high temperature bounce, for illustration. From Equation (12), we have:$${n}_{\chi}=\frac{1-{e}^{-\mathsf{\Lambda}(1\mp x)}}{1+{e}^{-\mathsf{\Lambda}(1\mp x)}}{n}_{\chi}^{eq}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{1.em}{0ex}}\mathsf{\Lambda}\equiv 2{\pi}^{2}f\langle \sigma v\rangle {m}_{\chi}\phantom{\rule{3.33333pt}{0ex}},$$$$\delta \widehat{\rho}\left(t\right)={\left(\frac{\beta \left({t}_{i}^{\mp}\right)}{\beta \left(t\right)}\right)}^{4}\delta \widehat{\rho}\left({t}_{i}^{\mp}\right)\phantom{\rule{1.em}{0ex}}\u27f9\phantom{\rule{1.em}{0ex}}\delta {\widehat{\rho}}_{k}\left(t\right)={\left(\frac{\beta \left({t}_{i}^{\mp}\right)}{\beta \left(t\right)}\right)}^{4}\delta {\widehat{\rho}}_{k}\left({t}_{i}^{\mp}\right).$$
- Step III (matching on horizon crossing): During the contraction of the universe, the effective horizon ${\left|aH\right|}^{-1}$ shrinks, so that the previously sub-horizon modes will become super-horizon after horizon crossing. Then, the sub-horizon mode and the super-horizon mode can be matched on the moment of horizon crossing, $k=\left|aH\right|$, to determine the initial amplitude of the super-horizon thermal perturbations. Afterwards, the evolution of super-horizon thermal perturbation is fully determined during the contacting phase.The sub-horizon modes with k cross the effective horizon at different times. Additionally, the horizon crossing condition is,$$k={\left|aH\right|}_{t={t}_{i}^{-}}\phantom{\rule{3.33333pt}{0ex}},$$After horizon crossing, each sub-horizon mode becomes a super-horizon one. Therefore, the initial value of each super-horizon mode is determined by the value of sub-horizon mode at horizon crossing,$$\delta {\widehat{\rho}}_{k}^{2}\left({t}_{i}^{-}\right)={\left.\delta {\rho}_{k}^{2}\right|}_{k=\left|aH\right|}={\left.\frac{6{\mu}^{2}{e}^{\beta \mu}}{{\left(a\beta \right)}^{3}}\right|}_{t={t}_{i}^{-}}\phantom{\rule{3.33333pt}{0ex}},$$Substituting these two matching conditions, Equations (40) and (41), on the horizon crossing into Equation (39), the evolution of the super-horizon mode of thermal perturbation in the contracting phase is obtained,$$\begin{array}{c}\hfill \delta {\widehat{\rho}}_{k}\left(t\right)=\left\{\begin{array}{c}{\displaystyle {\left(\frac{1}{\beta \left(t\right)}\right)}^{4}\frac{\sqrt{6}}{{\mathcal{C}}_{0}^{\frac{3}{2}}}\frac{\mathsf{\Lambda}}{2}ln\left(\frac{2}{\mathsf{\Lambda}}\right),\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Lambda}\ll 1}\hfill \\ \\ {\displaystyle {\left(\frac{1}{\beta \left(t\right)}\right)}^{4}\frac{\sqrt{6}}{{\mathcal{C}}_{0}^{\frac{3}{2}}}2{e}^{-\mathsf{\Lambda}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathsf{\Lambda}\gg 1}\hfill \end{array}\right..\end{array}$$
- Step IV (matching on bounce point): Eventually, the universe is bouncing from the contracting phase to the expanding phase. By assuming the entropy of the cosmological background is conserved before and after the bounce point, the matching conditions at the bounce point are obtained. By utilizing these matching conditions, the evolution of super-horizon thermal perturbation can be also fully determined during the expanding phase.By assuming the entropy of the bounce is conserved [63], we have an additional pair of matching conditions on the bounce,$$\beta \left({t}_{f}^{-}\right)=\beta \left({t}_{i}^{+}\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{2.em}{0ex}}\delta {\widehat{\rho}}_{k}\left({t}_{f}^{-}\right)=\delta {\widehat{\rho}}_{k}\left({t}_{i}^{+}\right)\phantom{\rule{3.33333pt}{0ex}},$$$$\begin{array}{c}\hfill \delta {\widehat{\rho}}_{k}\left(t\right)=\left\{\begin{array}{c}{\displaystyle {\left(\frac{1}{\beta \left(t\right)}\right)}^{4}\frac{\sqrt{6}}{{\mathcal{C}}_{0}^{\frac{3}{2}}}\frac{\mathsf{\Lambda}}{2}ln\left(\frac{2}{\mathsf{\Lambda}}\right),\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Lambda}\ll 1}\hfill \\ \\ {\displaystyle {\left(\frac{1}{\beta \left(t\right)}\right)}^{4}\frac{\sqrt{6}}{{\mathcal{C}}_{0}^{\frac{3}{2}}}2{e}^{-\mathsf{\Lambda}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathsf{\Lambda}\gg 1}\hfill \end{array}\right..\end{array}$$

## 4. Direct Detections of Dark Matter to Test the Big Bounce Genesis

- The elementary DM-nucleon cross-section computed in quantum field theory;
- The knowledge of the relevant nuclear matrix elements obtained with as reliable as possible many body nuclear wave functions;
- The knowledge of the density of DM in our vicinity and its velocity distribution;

## 5. Summary

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The exit and entrance of physical perturbations at the Hubble horizon in the inflation (left) and in the bounce (right) scenarios.

**Figure 2.**The cross-section $\langle \sigma \upsilon \rangle $ as a function of the scalar dark matter (DM) mass. In the standard cosmology, it is a constant (solid line), but it varies considerably in the bounce universe scenario (dash-dotted line) [56].

**Figure 3.**The breakdown of the Big Bounce period into a pre-bounce contraction (Phase I), a post-bounce expansion (Phase II) and the freeze-out of the DM particles (Phase III).

**Figure 4.**A schematic plot of the time evolution of DM in a generic bounce universe scenario. Two pathways of producing DM yet satisfying current observations’ thermal production (which is indistinguishable from standard cosmology) and non-thermal production (characteristic of the bounce universe) are illustrated. The horizontal axis indicates both the time, t, as well as the temperature, T, of the cosmological background [56].

**Figure 5.**Contour plots of the predicted relic abundance in the (${x}_{b}$-${\tilde{\sigma}}_{0}$) plane in the high temperature bounce case. Here, we take ${m}_{\chi}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}$ [62].

**Figure 6.**Low temperature bounce: ${x}_{b}$ as a function of a to reproduce the present relic abundance ${\mathsf{\Omega}}_{\chi}{h}^{2}\simeq 0.1$ for various ${m}_{\chi}$ [62].

**Figure 7.**The quark-scalar DM scattering mediated by a scalar particle [61].

**Figure 8.**The nucleon cross-section as a function of the DM mass in the case of a scalar DM particle: as predicted by our model (thick solid line), consistent with the low DM mass CRESSTII experiment [94] and with the quartic coupling λ adjusted to fit the limit of the Xe100 experiment, for a DM mass of 50 GeV, i.e., ${\sigma}_{p}={10}^{-8}$ pb (dashed line).

**Figure 9.**The quark-scalar dark matter scattering mediated by a scalar particle [61].

Thermal Equilibrium Production | Out-of-Chemical Equilibrium Production | |
---|---|---|

Strong freeze-out | Route I | — |

Weak freeze-out | Route III | Route II |

High Temperature Bounce | Low Temperature Bounce | |
---|---|---|

${\mathit{T}}_{\mathit{b}}\mathbf{\gg}{\mathit{m}}_{\mathit{\chi}}$ | ${\mathit{T}}_{\mathit{b}}\mathbf{\ll}{\mathit{m}}_{\mathit{\chi}}$ | |

Bosonic Dark Matter | Type I | Type III |

Fermonic Dark Matter | Type II | Type IV |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Cheung, Y.-K.E.; Li, C.; Vergados, J.D.
Big Bounce Genesis and Possible Experimental Tests: A Brief Review. *Symmetry* **2016**, *8*, 136.
https://doi.org/10.3390/sym8110136

**AMA Style**

Cheung Y-KE, Li C, Vergados JD.
Big Bounce Genesis and Possible Experimental Tests: A Brief Review. *Symmetry*. 2016; 8(11):136.
https://doi.org/10.3390/sym8110136

**Chicago/Turabian Style**

Cheung, Yeuk-Kwan Edna, Changhong Li, and Joannis D. Vergados.
2016. "Big Bounce Genesis and Possible Experimental Tests: A Brief Review" *Symmetry* 8, no. 11: 136.
https://doi.org/10.3390/sym8110136