# Inflation with an antisymmetric tensor field

## Abstract

We investigate the possibility of inflation with models of antisymmetric tensor field having minimal and nonminimal couplings to gravity. Although the minimal model does not support inflation, the nonminimal models, through the introduction of a nonminimal coupling to gravity, can give rise to stable de-Sitter solutions with a bound on the coupling parameters. The values of field and coupling parameters are sub-planckian. Slow roll analysis is performed and slow-roll parameters are defined which can give the required number of e-folds for sufficient inflation. Stability analysis has been performed for perturbations to antisymmetric field while keeping the metric unperturbed, and it is found that only the sub-horizon modes are free of ghost instability for de-Sitter space.

## 1 Introduction

Inflation as a theory, has been successfull in describing the structure and evolution of our universe [1, 2]. As ordinary matter or radiation can not source inflation, several models have been built to describe inflation where a hypothetical field may it be scalar, vector or tensor drives the inflation [3]. Many theories have considered the scalar field called “inflaton” as the source for inflation and are able to describe the cosmology of universe [4, 5, 6, 7, 8, 9]. Most of the scalar field models having simple form of potential are ruled out as they are not compatible with the Planck’s observational data for the cosmic microwave background [3, 10, 11]. Another class of models considers a vector field as an alternative to the inflaton [12, 13, 14, 15, 16]. But almost all of these models suffer from instabilities like ghost instability [17] and gradient instability [18] which leads to an unstable vacuum.

As the quantum corrections in cosmology and their possible phenomenological implications are becoming relevant [19, 20, 21], models with connections to high energy theories like the string theories provide an interesting alternative to traditional inflation model building. A particular theory of interest is that of a rank-2 antisymmetric tensor field, which appears in all superstring models [22, 23]. Antisymmetric tensors have been studied before in several aspects, including phase transitions, strong-weak coupling duality [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] and even some astrophysical aspects [36]. More recently, quantum aspects of antisymmetric fields in different settings have been studied [37, 38, 39, 40, 41]. However, efforts for cosmological studies with antisymmetric tensors were rare until the past decade. A few pertinent works with regard to inflation scenarios with *N*-form fields in anisotropic spacetime was carried out in Refs. [42, 43] and near a Schwarzschild metric in Ref. [44]. More recently, two-form fields have been studied in the context of anisotropic inflation [45] and gravitational waves [46].

*R*coupling and possibility of ghosts was found, the present analysis is different in the following ways: (1) the spacetime is isotropic and homogeneous; (2) background structure of \(B_{\mu \nu }\) is specified; and (3) choice of parameter space takes into account the conditions for slow-rolling inflation.

This work is organized as follows. In Sect. 2, we introduce background structures of the metric and the antisymmetric tensor, and establish the general setup of our analysis through a simple model of a massive antisymmetric tensor field minimally coupled to gravity. It is shown that minimal model cannot give rise to inflation. Three cases of nonminimally coupled extensions of this model are considered in Sect. 3. The conditions for inflation and the de-Sitter space solutions have been derived. In Sect. 4, we check the stability of possible de-Sitter space. In Sect. 5, the slow roll parameters for the nonminimal models are constructed and the number of e-folds are calculated. Sect. 6 presents stability analysis for perturbations to antisymmetric tensor field, while keeping the metric unperturbed.

## 2 Minimal model and the setup

### 2.1 Setup

- 1.
a de-Sitter space solution exists,

- 2.
the de-Sitter space should be stable, i.e. perturbations to solutions must decay with time, and

- 3.
more than 70 efolds of slow-roll inflation.

*a*(

*t*) is the scale factor for expansion of universe. The Riemann Christoffel tensor, Ricci tensor and Ricci scalar in terms of metric components in Eq. (2) are given by,

### 2.2 Minimal model

*V*(

*B*) is the potential term. Rest of the symbols have their usual meanings, with

*g*being the metric determinant,

*R*the Ricci scalar and \(\kappa \) the inverse square of Planck mass \(M_{pl}\). For the present problem, we will only consider quadratic potential of the form \(m^{2}B_{\mu \nu }B^{\mu \nu }/4\), though some of the expressions (especially for slow roll analysis) will be written in terms of

*V*(

*B*) for generality.

Here onwards, we omit the arguments of functions and functionals (*a*(*t*), *B*(*t*), *V*(*B*), etc.) for notational convenience and their functional dependence is assumed until stated otherwise.

*a*(

*t*) is negative and hence the minimal model does not support the possibility of inflation. Equation (16) provides an insight into what modifications could be made to the action (9) to allow inflation. A straightforward solution for positive acceleration would be to incorporate additional parameter in the rhs of Eq. (16) such that \(\ddot{a}\) has nontrivial solutions. In subsequent sections, we consider an extension of this model consisting of nonminimal coupling of \(B_{\mu \nu }\) with gravity that resolves this issue.

## 3 Nonminimal models

### 3.1 The models

*R*and \(R_{\mu \nu }\) respectively. The parameters \(\xi \) and \(\zeta \) have dimensions of \(M_{pl}^{-2}\).

#### 3.1.1 Case: \(\mathcal {L}_{NM}=\frac{1}{2\kappa }\xi B^{\mu \nu } B_{\mu \nu } R\)

#### 3.1.2 Case: \(\mathcal {L}_{NM} = \frac{1}{2 \kappa } \zeta B^{\lambda \nu } B^{\mu }_{\ \nu } R_{\lambda \mu }\)

### 3.2 de-Sitter solutions

*R*,

The de-Sitter space solutions of \(\phi \) and *H*, along with the condition on parameters \(\xi \) and \(\zeta \) corresponding to *R* and \(R_{\mu \nu }\) coupling terms respectively

\(\mathcal {L}_{NM}\) | \(\phi _{0}^{2}\) | \(H_{0}^{2}\) | Condition |
---|---|---|---|

\(\dfrac{1}{2\kappa }\xi B^{\mu \nu } B_{\mu \nu } R\) | \(\dfrac{1}{6\xi }\) | \(\dfrac{\kappa m^{2}}{4(6\xi - \kappa )}\) | \(\xi > \dfrac{\kappa }{6}\) |

\(\dfrac{1}{2 \kappa } \zeta B^{\lambda \nu } B^{\mu }_{\ \nu } R_{\lambda \mu }\) | \(\dfrac{1}{3\zeta }\) | \(\dfrac{\kappa m^2}{2(3 \zeta - 2 \kappa )}\) | \(\zeta > \dfrac{2 \kappa }{3}\) |

It is worth noting that value of \(\phi \) is sub-planckian in both cases. An interesting observation in the context of theories (9) and (17) is that adding a nonminimal coupling gives rise to de-Sitter solutions, which are otherwise absent in minimal model. This is a unique feature of antisymmetric field models in contrast to the nonminimal models of scalar field inflation (see [49] and references therein).

## 4 Stability analysis of the de-Sitter background

*H*(

*t*) about de-Sitter solutions \(H_{0}\) and \(\phi _{0}\). The condition for stability is that the perturbations \(\delta \phi (t)\) and \(\delta H (t)\) must decay over time. The corresponding perturbations are given by,

*A*is a \((2\times 2)\) square matrix, given by,

*A*can be calculated from its trace and determinant, which are

### 4.1 Case: \(\mathcal {L}_{NM} = \frac{1}{2 \kappa } \zeta B^{\lambda \nu } B^{\mu }_{\ \nu } R_{\lambda \mu }\)

*A*of Eq. (32),

## 5 Slow roll parameters

Relation between the slow-roll parameters \(\epsilon \) and \(\delta \) for each case of nonminimal coupling

\(\mathcal {L}_{NM}\) | \(\epsilon \) |
---|---|

\(\dfrac{1}{2\kappa }\xi B^{\mu \nu } B_{\mu \nu } R\) | \(\epsilon \approx \dfrac{\delta }{(6\xi - 2\kappa )^{-1}\phi ^{-2} + 1}\sim \delta \) |

\(\dfrac{1}{2 \kappa } \zeta B^{\lambda \nu } B^{\mu }_{\ \nu } R_{\lambda \mu }\) | \(\epsilon \approx \dfrac{\delta }{1- (2 \kappa \phi ^2)^{-1} } \sim \delta \) |

*H*is nearly constant which says that \(\dot{\delta }\) is approximately zero or \(\delta \) is nearly constant during inflation. Now, the number of e-folds can be calculated to be,

## 6 Stability of perturbations to \(B_{\mu \nu }\)

*R*coupling was performed in Ref. [42] in anisotropic spacetime. In the present case, we consider both couplings, i.e. \(\mathcal {L}_{NM}=\frac{1}{2\kappa }\xi B^{\mu \nu } B_{\mu \nu } R + \frac{1}{2 \kappa }\zeta B^{\lambda \nu } B^{\mu }_{\ \nu } R_{\lambda \mu }\), and the spacetime background is homogeneous and isotropic. The choice of background structure of \(B_{\mu \nu }\) remains the same as in Eq. (8). The perturbed field is given by \(B_{\mu \nu } + \delta B_{\mu \nu }\), where

*z*-axis along the direction of three-momentum \(\vec {k}\), so that

*f*, \(\tilde{f}\equiv \tilde{f}(t,\vec {k})\) and \(\tilde{f}^{\dagger }\equiv \tilde{f}(t,-\vec {k})\). Now, varying the action with respect to \(E_{x}^{\dagger }\), \(E_y^{\dagger }\), \(E_z^{\dagger }\), their equations of motion are found to be,

## 7 Conclusion

We study the possibility of inflation with minimal and nonminimal models of rank-2 antisymmetric tensor fields. We find that the minimal model does not support inflation. Interesting features appear when a model with non-minimal coupling to gravity is considered, as a way to introduce a new parameter in the form of couplings \(\xi \) and \(\zeta \). It is possible to have solutions for de-Sitter space in nonminimal model that can support inflation. A simple bound on the couplings \(\xi \) and \(\zeta \) has been obtained from the de-Sitter solutions, and can support stable de-Sitter space under certain conditions. A detailed fixed point analysis will be carried out in future to ascertain the issue of stability of de-Sitter solutions. To study inflation, the slow roll analysis has been performed, and corresponding slow roll parameters \(\epsilon \) and \(\delta \) have been obtained. Validity of slow roll conditions has been checked. A notable feature of the present analysis is that the values of \(\xi \), \(\zeta \) and \(\phi \) are sub-planckian in these models.

The ghost instability analysis has been performed for perturbations to \(B_{\mu \nu }\) (keeping the metric unperturbed). We find that while the longitudinal modes are ghost free, the transverse modes may admit ghosts. For a special case of exact de-Sitter space and \(\zeta = 0\), only the sub-horizon modes are ghost free. It is noteworthy that the conditions encountered here are common in vector field models as well [17, 50].

The structure of Eqs. (53) and (54) hints towards the kind of modifications one would have to include in action (17) to build a successful model of inflation with antisymmetric tensor field. An interesting possibility arises by adding a *U*(1) symmetry breaking kinetic term to Eq. (17): there are kinetic coulings between \(E_{i}\) and \(M_{i}\) modes, and any claim about instabilities cannot be made until one solves the coupled dynamical equations. This will be the subject of our subsequent study, and we speculate that possibly, instability problems could be resolved. In a future work, the full perturbation theory for such models may be developed, which will allow for phenomenologically relevant calculations. Possible extensions of this study include considering more combinations of coupling terms involving Ricci tensors and scalars, particularly \(R^{2}\) coupling to tackle possible instabilities. Further studies may also involve the study of spontaneous Lorentz violation with antisymmetric fields in cosmological context and could provide significant insights for investigating signatures of new physics.

## Notes

### Acknowledgements

The manipulations in Sect. 6 were done using \(\hbox {Maple}^{\mathrm{TM}}\) [51] and cross-checked by hand. This work is partially supported by DST (Govt. of India), Science and Engineering Research Board (Grant no. SERB/PHY/2017041). The authors thank Tomi Koivisto for pointing out the results of Ref. [42].

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