# Antisymmetric Tensor Fields in Modified Gravity: A Summary

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## Abstract

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## 1. Introduction

- In [113] an antisymmetric tensor field ${B}_{\mu \nu}$ identified to be the Kalb–Ramond field was shown to act as the source of spacetime torsion.

- Why is the present universe practically free from the observable footprints of the higher rank antisymmetric tensor fields despite getting the signatures of scalar, fermion, vector and spin-2 massless graviton, while they all originate from the same underlying Lorentz group?
- Do the higher rank antisymmetric tensor fields have a considerable impact during early stage of the universe, in particular during inflation and on the inflationary parameters?

- Sitting in present day universe, how do we confirm the existence of the Kalb–Ramond field which has considerably low energy density (with respect to the other components) in our present universe but has a significant impact during the early universe?

## 2. Antisymmetric Tensor Fields in 4D Higher Curvature Gravity

#### 2.1. Suppression of Antisymmetric Tensor Fields: A Non-Dynamical Way

#### 2.2. A Different Non-Dynamical Method for the Suppression of Antisymmetric Tensor Fields by the “Scalaron Tunneling”

#### 2.3. Cosmological Scenario

## 3. Kalb–Ramond Field in Randall–Sundrum Braneworld Scenrio

## 4. Cosmology with Kalb–Ramond in Higher Curvature Warped Spacetime

- Sitting in present day universe, how do we confirm the existence of the Kalb–Ramond field which has considerably low energy density (with respect to the other components) in our present universe but has a significant impact during the early universe?

## 5. Cosmological Quantum Entanglement with Kalb–Ramond Field

- The effects of $\xi $ on the entanglement entropy can be understood from the left panels of Figure 7 and Figure 8, where $\alpha $ has the fixed value zero. The left panels clearly demonstrate that in the conformal coupling case i.e., for $\xi =1/6$, the entanglement entropy vanishes for $m=0$, while in the case of weak coupling (i.e., $\xi =0$) the entanglement entropy acquires a non-zero value even at $m=0$. This is a consequence of the fact that for $\xi =1/6$, the action of a massless scalar field becomes conformally invariant in four dimensional spacetime, unlike to $\xi =0$ for which the corresponding conformal invariance is broken. Moreover it is evident that the maximum value of the entanglement entropy is larger for $\xi =1/6$ in comparison to that of the weak coupling case.
- The effects of $\alpha $ on the entanglement entropy can be understood from the left and right panels of Figure 7 where $\xi $ has a fixed value $1/6$, or from the left and right panels of Figure 8 having $\xi =0$ (recall the left and right panels of the figures are plotted for $\alpha =0$ and $\alpha \ne 0$ respectively). Figure 7 demonstrates that in absence of the interaction between the Kalb–Ramond and scalar field (i.e., for $\alpha =0$), the entropy vanishes at $m=0$, while a non-zero KR field coupling parameter (i.e., $\alpha \ne 0$) leads to a non-zero entropy even for $m=0$. Again this is related to the conformal symmetry of the scalar field, in particular for $\xi =1/6$, $m=0$ and $\alpha =0$, the scalar field becomes conformally invariant in 4 dimensional spacetime, however the condition $\alpha \ne 0$ breaks the conformal invariance of a massless scalar field even for $\xi =1/6$ and thus the corresponding entanglement entropy becomes non-zero. Moreover Figure 7 also depicts that without the KR field, the entanglement entropy is bounded by $S\lesssim 0.17$ (in the unit of ${k}_{B}=1$, see the left panel), while due to the presence of KR field the upper bound of entropy goes beyond $0.17\phantom{\rule{0.166667em}{0ex}}{k}_{B}$ and reaches up to $S\lesssim 2\phantom{\rule{0.166667em}{0ex}}{k}_{B}$ (see the right panel). Similarly Figure 8 reveals that for $\xi =0$, the maximum value of the von-Neumann entropy is given by $S\lesssim 0.032\phantom{\rule{0.166667em}{0ex}}{k}_{B}$ in absence of the KR field, however the Kalb–Ramond field indeed affects the situation, in particular the maximum entanglement entropy reaches upto $S\lesssim 1.5\phantom{\rule{0.166667em}{0ex}}{k}_{B}$ due to $\alpha \ne 0$. Therefore the maximum value of the entanglement entropy becomes larger due to the coupling between Kalb–Ramond and scalar field, in comparison to the case when the coupling parameter $\alpha $ is zero.

## 6. Conclusions

- Why the present universe is practically free from any noticeable footmarks of higher rank antisymmetric tensor fields, despite having the signatures of scalar, vector, fermion as well as symmetric rank 2 tensor field in the form of gravity?
- What are the possible roles of the Kalb–Ramond field during early universe?
- If the Kalb–Ramond field has considerable impact during early universe, then an immediate question will be—sitting in present day universe, how do we confirm the existence of the Kalb–Ramond field which has considerably low energy density (with respect to the other components) in our present universe but has a significant impact during early universe?

## 7. Brief Discussions on Future Perspectives

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**${n}_{s}$ vs. r for $10\le |{\sigma}_{0}|\le 14$ and $0.003\le \frac{{\kappa}^{2}{h}_{0}}{{m}^{2}}\le 0.004$.

**Figure 3.**Left figure: $\xi $ vs. $\tau $, right figure: q vs. $\tau $. In both the figures, we take $\frac{{\kappa}^{2}{h}_{0}}{{m}^{2}}=0.0035$, ${\sigma}_{0}=-10$ and $m={10}^{-5}$ (in reduced Planckian unit).

**Figure 4.**${\rho}_{KR}$ vs. $\tau $ for $\frac{{\kappa}^{2}{h}_{0}}{{m}^{2}}=0.0035$, ${\sigma}_{0}=-10$ and $m={10}^{-5}$ (in reduced Planckian unit).

**Figure 5.**$1\sigma $ (yellow) and $2\sigma $ (light blue) contours for Planck 2018 results [147], on ${n}_{s}-r$ plane. Additionally, we present the predictions of the present bounce scenario with $({\sigma}_{0},{\kappa}^{2}{h}_{0}\sqrt{\alpha})=(-5,0.003)$ (blue point), $({\sigma}_{0},{\kappa}^{2}{h}_{0}\sqrt{\alpha})=(-5,0.03)$ (black point) and $({\sigma}_{0},{\kappa}^{2}{h}_{0}\sqrt{\alpha})=(-5,0.1)$ (red point).

**Figure 6.**${\chi}_{{\phi}_{0}}^{(0)}(t,\phi )$ vs. t for $\gamma =0.15$, $\kappa {v}_{v}=\frac{\sqrt{{\Omega}_{0}}}{{M}^{2}}\simeq {10}^{-7}$, $\frac{{m}_{\Phi}}{k}=0.2$ and ${\Psi}_{0}=36$ (in 4D reduced Planckian unit).

**Figure 7.**Left part: S (along y axis) vs. m (along x axis) for $\xi =1/6$; $k=0.01$ in absence of KR field. Right part: 3D plot of S with respect to mass ($0\le m\le 1$ along x axis) and KR field energy density ($0\le \alpha {h}_{0}\le 0.7$ along y axis) for $\xi =1/6$; $k=0.01$. The quantities m, k and $\alpha {h}_{0}$ (having mass dimensions [+1], [+1] and [+2] respectively) are taken in the reduced Planckian unit.

**Figure 8.**Left part: S (along y axis) vs. m (along x axis) for $\xi =0$; $k=0.01$ in absence of Kalb–Ramond KR field. Right part: 3D plot of S with respect to mass ($0\le m\le 1$ along x axis, in Planckian unit) and KR field energy density ($0\le \alpha {h}_{0}\le 0.7$ along y axis for $\xi =0$; $k=0.01$. The quantities m, k and $\alpha {h}_{0}$ (having mass dimensions [+1], [+1] and [+2] respectively) are taken in the reduced Planckian unit.

**Table 1.**The couplings of rank 2 (onwards) antisymmetric tensor field(s) to matter fields for a considerable range of $\alpha \beta $.

$(1-\mathit{\alpha}\mathit{\beta})$ | ${\mathit{\lambda}}_{\mathit{KR}-\mathit{fermion}}$ | ${\mathit{\lambda}}_{\mathit{KR}-\mathit{U}(1)}$ | ${\mathsf{\Omega}}_{\mathit{X}-\mathit{fermion}}$ | ${\mathsf{\Omega}}_{\mathit{X}-\mathit{U}(1)}$ |
---|---|---|---|---|

${10}^{-2}$ | ${10}^{-1}/{M}_{Pl}$ | ${10}^{-1}/{M}_{Pl}$ | ${10}^{-1}/{M}_{Pl}$ | ${10}^{-2}/{M}_{Pl}$ |

${10}^{-4}$ | ${10}^{-2}/{M}_{Pl}$ | ${10}^{-2}/{M}_{Pl}$ | ${10}^{-2}/{M}_{Pl}$ | ${10}^{-4}/{M}_{Pl}$ |

${10}^{-8}$ | ${10}^{-4}/{M}_{Pl}$ | ${10}^{-4}/{M}_{Pl}$ | ${10}^{-4}/{M}_{Pl}$ | ${10}^{-8}/{M}_{Pl}$ |

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Paul, T.
Antisymmetric Tensor Fields in Modified Gravity: A Summary. *Symmetry* **2020**, *12*, 1573.
https://doi.org/10.3390/sym12091573

**AMA Style**

Paul T.
Antisymmetric Tensor Fields in Modified Gravity: A Summary. *Symmetry*. 2020; 12(9):1573.
https://doi.org/10.3390/sym12091573

**Chicago/Turabian Style**

Paul, Tanmoy.
2020. "Antisymmetric Tensor Fields in Modified Gravity: A Summary" *Symmetry* 12, no. 9: 1573.
https://doi.org/10.3390/sym12091573