# Bell Violation in Primordial Cosmology

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## Abstract

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## 1. Introduction

## 2. Bell’s Inequality and its Violation in Quantum Mechanics

## 3. Cosmological Bell Violating Setup

- Choose a toy model of universe that will make the job easy.
- Here we are not claiming that this is the unique model of universe using which one can design the setup.
- Since we do not have any direct observational evidence, we also cannot claim that this toy model is our known universe. However, it may be that in the future this will be tested.
- In this toy model of the universe, we test the validity of Bell’s inequality with primordial fluctuations.
- In this computation, massive particles with spin “s” ($s=$ 0, 2, and >2 allowed) and additionally “isospin” quantum number plays important role.
- We provide an example for Stringy axion which has “isospin”.
- Time dependent mass profile with dependence on “isospin” makes the job easy. Axions have such profile.
- Such time dependence in mass profile of the massive particles (ex. axion) produces classical perturbations on the inflaton. As a result, hot spots produced in CMB by curvature fluctuations and all such massive particles are visible today.

## 4. Role of Massive New Particles

## 5. Important Note and Results in Cosmological Perturbation Theory

## 6. Analogy with Axion Fluctuations from String Theory

## 7. Role of Isospin Breaking Interaction and Detection

**even**and

**odd**contributions in the interaction picture, here we chose I. ${\lambda}_{\pm}\left(\mathsf{\eta}\right)/H=\sqrt{{\gamma}_{\pm}{\left(\frac{\mathsf{\eta}}{{\mathsf{\eta}}_{0}}-1\right)}^{2}+{\delta}_{\pm}}$, II. ${\lambda}_{\pm}\left(\mathsf{\eta}\right)/H=\frac{{m}_{0\pm}}{\sqrt{2}H}\sqrt{\left[1-tanh\left(\mathsf{\rho}\frac{ln(-H\mathsf{\eta})}{H}\right)\right]}$, III. ${\lambda}_{\pm}\left(\mathsf{\eta}\right)/H=\frac{{m}_{0\pm}}{H}\mathrm{sech}\left(\mathsf{\rho}\frac{ln(-H\mathsf{\eta})}{H}\right)$. To simplify the calculations, we introduce new parameters as: ${\gamma}_{\pm}={\gamma}_{\mathbf{even}}\pm {\gamma}_{\mathbf{odd}}$, ${\delta}_{\pm}={\delta}_{\mathbf{even}}\pm {\delta}_{\mathbf{odd}}$, ${m}_{0\pm}=\sqrt{{m}_{\mathbf{even}}^{2}\pm {m}_{\mathbf{even}}^{2}}$. See Figure 6, where we have shown the conformal time dependent behaviour of three toy models of heavy particle mass profile for two eigenstates. To make the eigenbasis stable, eigenvalues of the mass matrix are always positive definite. At late time scales ${\mathcal{M}}_{\mathbf{even}}^{2}\left(\mathsf{\varphi}\right)\sim {\mathcal{M}}_{\mathbf{odd}}^{2}\left(\mathsf{\varphi}\right)$. Hence, the eigenvalue of the mass-matrix increases. Another important requirement is that eigenvalues of the mass matrix should be on the order of ${M}_{p}$. The eigenvalue of the mass matrix is $\lambda \left(\mathsf{\varphi}\right)\sim {\mathcal{M}}_{\mathbf{even}}\left(\mathsf{\varphi}\right)$ when we know that all the isospin breaking interactions are absent from the effective Lagrangian. This is also one criterion of significant importance that can be observed with $SU\left(2\right)$ isospin singlet state during the time of heavy mass particle creation. Here the angular parameter $\mathsf{\theta}$ and its functional dependence on the background plays an extremely important role to set up the Cosmological Bell violating setup. To make the case easy, one can assume that $\mathsf{\theta}$ is a constant. Here, if the particle mass eigenvalue is ${\lambda}_{\pm}\left(\mathsf{\varphi}\right)$, then the antiparticle mass eigenvalues are given by ${\lambda}_{\pm}\left(\mathsf{\varphi}\right)$. The sign of the eigenvalue of the antiparticle mass eigenstate may flip if the angular parameter $\mathsf{\theta}$ is not a constant quantity but background dependent. After the period of inflation, we can set the axion potential to be zero to avoid the problem of domain wall formation. We have not yet observed any signatures of these heavy fields or the axion; therefore, we can consider such heavy fields or the axions corresponding to a component of dark matter, and we can treat its corresponding density fluctuations as isocurvature fluctuations.

## 8. Role of Spin for New Particles

## 9. Conclusions

- We provide a toy model for Bell’s inequality violation in cosmology.
- The model consists of inflaton and additional massive field with time-dependent behaviour.
- For each model, massive particle creation in “isospin” singlet state plays a crucial role.
- The prescribed methodology is consistent with axion fluctuations appearing in the context of String Theory.
- The signature of the Bell violation is visualized from non-zero one-point function of curvature fluctuation. Additionally, we provide the result of a two-point function which has direct observational consequence.
- Finally, we also provide the mass bound on the new particle in terms of the arbitrary spin $\mathcal{S}$.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of Bell’s inequality example for a spin system. Here pictures are taken from [3].

**Figure 2.**(

**a**) Successful vs. (

**b**) Unsuccessful setup for Bell’s inequality violation in Quantum Mechanics and Cosmology [4].

**Figure 3.**Toy model mass profiles for massive particles. (

**a**) Mass profile for A with $\gamma =1$ and $\delta =1$; (

**b**) Mass profile for B with $\frac{{m}_{0}^{2}}{2{H}^{2}}=1$; (

**c**) Mass profile for C with $\frac{{m}_{0}^{2}}{{H}^{2}}=1$. Here pictures are taken from [3].

**Figure 4.**Particle creation for mass profile A for two cases: $m\approx H$ and $m>>H$. (

**a**) For $m\approx H$; (

**b**) For $m\approx H$; (

**c**) For $m>>H$; (

**d**) For $m>>H$. Here pictures are taken from [3].

**Figure 5.**Behaviour of the axion effective potential and time-dependent decay constant. (

**a**) Various parts of potential; (

**b**) Total potential; (

**c**) Axion decay constant profile. Here pictures are taken from [3].

**Figure 6.**Conformal time scale dependent behaviour of heavy particle mass profile for two eigenstates. (

**a**) Here we set ${\gamma}_{\mathbf{even}}=1={\delta}_{\mathbf{even}}$, ${\gamma}_{\mathbf{odd}}=0.5={\delta}_{\mathbf{odd}}$; (

**b**) Here we set ${m}_{\mathbf{even}}=3$, ${m}_{\mathbf{odd}}=1.5$ and $\mathsf{\rho}/H=1$; (

**c**) Here we set ${m}_{\mathbf{even}}=2$, ${m}_{\mathbf{odd}}=1.5$ and $\mathsf{\rho}/H=1$. Here pictures are taken from [3].

Characteristics | New Particle | Axion |
---|---|---|

Action | ${S}_{new}=\frac{1}{2}\int d\mathsf{\eta}{d}^{3}x\frac{2\mathsf{\u03f5}{M}_{p}^{2}}{{\tilde{c}}_{S}^{2}{H}^{2}}\left[\frac{{\left({\partial}_{\mathsf{\eta}}\zeta \right)}^{2}-{c}_{S}^{2}{\left({\partial}_{i}\zeta \right)}^{2}}{{\mathsf{\eta}}^{2}}\right]$ | ${S}_{axion}=\int d\mathsf{\eta}\phantom{\rule{3.33333pt}{0ex}}{d}^{3}x\left[\frac{{f}_{a}^{2}\left(\mathsf{\eta}\right)}{2{H}^{2}}\frac{\left[{\left({\partial}_{\mathsf{\eta}}a\right)}^{2}-{\left({\partial}_{i}a\right)}^{2}\right]}{{\mathsf{\eta}}^{2}}-\frac{U\left(a\right)}{{H}^{4}{\mathsf{\eta}}^{4}}\right].$ |

$-\int \frac{d\mathsf{\eta}}{{\tilde{c}}_{S}H}m\left(\mathsf{\eta}\right){\partial}_{\mathsf{\eta}}\zeta (\mathsf{\eta},\mathbf{x}=0)$ | $-\int \frac{d\mathsf{\eta}}{{f}_{a}H}{m}_{axion}{\partial}_{\mathsf{\eta}}\overline{a}(\mathsf{\eta},\mathbf{x}=0)$ | |

Mass parameter | $\frac{{m}^{2}}{{H}^{2}}=\left\{\begin{array}{cc}{\displaystyle \gamma {\left(\frac{\mathsf{\eta}}{{\mathsf{\eta}}_{0}}-1\right)}^{2}+\delta \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \phantom{\rule{3.33333pt}{0ex}}\mathbf{Case}\phantom{\rule{3.33333pt}{0ex}}\mathbf{I}\hfill \\ {\displaystyle \frac{{m}_{0}^{2}}{2{H}^{2}}\left[1-tanh\left(\mathsf{\rho}\frac{ln(-H\mathsf{\eta})}{H}\right)\right]\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \phantom{\rule{3.33333pt}{0ex}}\mathbf{Case}\phantom{\rule{3.33333pt}{0ex}}\mathbf{II}\hfill \\ {\displaystyle \frac{{m}_{0}^{2}}{{H}^{2}}{\mathrm{sech}}^{2}\left(\mathsf{\rho}\frac{ln(-H\mathsf{\eta})}{H}\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \phantom{\rule{3.33333pt}{0ex}}\mathbf{Case}\phantom{\rule{3.33333pt}{0ex}}\mathbf{III}.\hfill \end{array}\right.$ | $\frac{{m}_{axion}}{{f}_{a}}=\left\{\begin{array}{cc}{\displaystyle \sqrt{-\frac{{\Lambda}_{C}^{4}}{{f}_{a}^{2}}cos\left({sin}^{-1}\left(\frac{{\mathsf{\mu}}^{3}{f}_{a}}{{\Lambda}_{C}^{4}}\right)\right)}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathbf{total}\phantom{\rule{3.33333pt}{0ex}}U\left(a\right)\hfill \\ \\ {\displaystyle \sqrt{\frac{{\Lambda}_{C}^{4}}{{f}_{a}^{2}}{(-1)}^{m+1}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathbf{osc}\mathbf{.}\phantom{\rule{3.33333pt}{0ex}}U\left(a\right).\hfill \end{array}\right.$ |

Rescaled mode | ${h}_{\mathbf{k}}=-\frac{\sqrt{2\mathsf{\u03f5}}}{H\mathsf{\eta}{\tilde{c}}_{S}}{M}_{p}{\zeta}_{\mathbf{k}}$ | ${\vartheta}_{\mathbf{k}}=\frac{{f}_{a}^{2}}{{H}^{2}{\mathsf{\eta}}^{2}{M}_{p}^{2}}{\overline{a}}_{\mathbf{k}}$ |

Scalar mode | ${h}_{\mathbf{k}}^{\u2033}+\left({c}_{S}^{2}{k}^{2}+\frac{\left(\frac{{m}^{2}}{{H}^{2}}-\delta \right)}{{\mathsf{\eta}}^{2}}\right){h}_{\mathbf{k}}=0$ | ${\partial}_{\mathsf{\eta}}^{2}{\vartheta}_{\mathbf{k}}+\left({k}^{2}-\frac{{\partial}_{\mathsf{\eta}}^{2}\left(\frac{{f}_{a}^{2}}{{H}^{2}{\mathsf{\eta}}^{2}}\right)}{\left(\frac{{f}_{a}^{2}}{{H}^{2}{\mathsf{\eta}}^{2}}\right)}+\frac{{m}_{axion}^{2}}{{f}_{a}^{2}{H}^{2}{\mathsf{\eta}}^{2}}\right){\vartheta}_{\mathbf{k}}=0$ |

Equation | where $\delta =\frac{{z}^{\u2033}}{z}=\left\{\begin{array}{cc}{\displaystyle 2\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathbf{dS}\hfill \\ {\displaystyle \left({\nu}^{2}-\frac{1}{4}\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathbf{qdS}\mathbf{.}\hfill \end{array}\right.$ | where $\frac{{\partial}_{\mathsf{\eta}}^{2}\left(\frac{{f}_{a}^{2}}{{H}^{2}{\mathsf{\eta}}^{2}}\right)}{\left(\frac{{f}_{a}^{2}}{{H}^{2}{\mathsf{\eta}}^{2}}\right)}\approx \left\{\begin{array}{cc}{\displaystyle \frac{6}{{\mathsf{\eta}}^{2}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\eta}\sim {\mathsf{\eta}}_{c}\hfill \\ {\displaystyle \frac{6}{{\mathsf{\eta}}^{2}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathbf{early}\phantom{\rule{3.33333pt}{0ex}}\mathbf{and}\phantom{\rule{3.33333pt}{0ex}}\mathbf{late}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\eta}\hfill \\ {\displaystyle \frac{6+{\Delta}_{c}}{{\mathsf{\eta}}^{2}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}}\hfill & \mathbf{for}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\eta}<{\mathsf{\eta}}_{c}.\hfill \end{array}\right.$ |

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**MDPI and ACS Style**

Choudhury, S.; Panda, S.; Singh, R.
Bell Violation in Primordial Cosmology. *Universe* **2017**, *3*, 13.
https://doi.org/10.3390/universe3010013

**AMA Style**

Choudhury S, Panda S, Singh R.
Bell Violation in Primordial Cosmology. *Universe*. 2017; 3(1):13.
https://doi.org/10.3390/universe3010013

**Chicago/Turabian Style**

Choudhury, Sayantan, Sudhakar Panda, and Rajeev Singh.
2017. "Bell Violation in Primordial Cosmology" *Universe* 3, no. 1: 13.
https://doi.org/10.3390/universe3010013