# Avoiding instabilities in antisymmetric tensor field driven inflation

## Abstract

Models of inflation with antisymmetric tensor studied in the past are plagued with ghost instability even in an unperturbed FRW background. We show that it is possible to avoid ghosts in an unperturbed FRW background by considering the most general kinetic term for antisymmetric tensor field. The kinetic part acquires a new gauge symmetry violating term whose effect on perturbed modes is to prevent the appearance of nondynamical modes, and thus avoid ghosts. For completeness, we perform a check for gradient instability and derive the conditions for perturbations to be free of gradient instability.

## 1 Introduction

Inflation as a paradigm to explain horizon and flatness problem of early universe was first introduced by Guth [1], and since has led to more than three decades of effort to build models of inflation that fit well with the observed CMB data (see Ref. [2] for a review). With the advent of high-precision observational data (like the recent Planck 2018 results [3]), majority of scalar field driven inflation models have been ruled out while the ones in agreement are tightly constrained. More recently, new set of theoretical conditions called the Swampland criteria arise from the requirements for any effective field theory to admit string theory UV completion [4, 5, 6, 7, 8], and further constrain scalar field potentials. There is thus a genuine interest to explore inflationary scenario with alternative driving fields. Some major programs include multiple fields, vector and/or gauge fields. For a comprehensive review, see Ref. [9].

Among the theories not involving scalar fields, in particular those with vector fields [10, 11, 12, 13, 14], constructing successful models is often marred by ghost and gradient instabilities [15, 16] that lead to unstable vacua. Inflation with non-Abelian gauge fields have been shown to be free from these instabilities [17, 18, 19, 20], but are in tension with Planck data and hence ruled out [21]. Our endeavour is to explore inflation models with rank-2 antisymmetric tensor fields. Also referred to as the Kalb–Ramond fields, they appear naturally in the low energy limit of superstring models [22, 23]. There are no observational signatures of antisymmetric fields in the present universe [24], but it is interesting to study them in the early universe when their presence may become significant [25].

Past attempts at studying inflation with antisymmetric tensor have not been successful because of the possibility of ghosts as a generic feature of the theory [26, 27]. Even with an unperturbed Friedmann Lematre Robertson Walker (FLRW) metric background, the perturbations to field components admit ghosts and this result remains unaffected for different choices of couplings and potential. The cause of this instability can be traced to the presence of nondynamical modes for some components of the field, which in turn is due to the structure of the gauge invariant kinetic term in these models. It turns out that the choice of kinetic term is indeed not general [28], and one can in principle consider a model with modifications to the kinetic part of action.

The organization of this letter is as follows. In Sect. 2, we study the effect of modifying the kinetic part on the background cosmology for a particular choice of background structure of \(B_{\mu \nu }\). We then study perturbations to \(B_{\mu \nu }\) and subsequently the ghost and gradient instability in an unperturbed FRW spacetime, in Sect. 3. We conclude with a few remarks on future directions in Sect. 4.

## 2 Background cosmology

*V*(

*B*) is the potential, which in our case is quadratic, \(m^{2}B_{\mu \nu }B^{\mu \nu }/4\), and \(\mathcal {L}_{NM}\) is a nonminimal coupling term. the metric signature (\(- + + +\)). \(H_{\lambda \mu \nu }(B) = \nabla _{\lambda } B_{\mu \nu } + \nabla _{\mu } B_{\nu \lambda } + \nabla _{\nu } B_{\lambda \mu }\) (\(\nabla _{\mu }\) is the covariant derivative) constitutes the kinetic term and admits gauge invariance under the transformation

*R*for a resolution to ghosts, modifications to the kinetic term of action (3) as yet remain unexplored. Therefore, we start with constructing the most general kinetic term upto quadratic order in \(B_{\mu \nu }\), which yields a new gauge-symmetry breaking kinetic term in addition to the gauge invariant kinetic term already present in action (3) [28],

Apart from ghost instability, inflationary solutions are prone to gradient instability, which occurs when the speed of sound becomes imaginary. Gradient (in)stability has not been checked explicitly for the model(s) (3) before. For completeness, the gradient instability check has been performed for action (6) in later part of this work, albeit in a relevant limit suited to check the effect of \(\tau \) term.

*a*(

*t*) is the scale factor.

*viz.*the corresponding energy-momentum tensor, \(T^{\tau }_{\mu \nu }\), given by

- (i)
de-Sitter solutions exist, and

- (ii)
Slow roll inflation is supported.

## 3 Perturbations

The interesting part however is when \(B_{\mu \nu }\) is perturbed. Surely, the perturbed modes have nontrivial contributions from the \(\tau \) term, as we shall see. A full perturbation analysis, where perturbations to both metric and field are considered, is ideally required to investigate the viability of an inflation theory. However, as a starting point and because of the complexity of full perturbation theory (involving a total of 10(metric) \(+ 6\)(field) \(= 16\) perturbed modes), it is useful to check the stability of just the field perturbations while keeping the metric unperturbed. In several past studies, instabilities have been found at this stage [26, 27].

*k*. In our calculations, we also utilize the freedom to choose the coordinate axis (\(z-\)axis) along momentum vector \(\vec {k}\) so that all spatial derivatives along \(x-\) and \(y-\)axes vanish. As a notation, throughout this paper, the coordinate (\(\vec {x}\)), time (

*t*) and momenta (\(\vec {k}\)) dependence of all perturbed modes and their Fourier transforms are understood but not explicitly displayed, to save space. The resulting quadratic part of action, \(\tilde{S}_{2}\), in general has a form,

*u*and

*v*are scalar fields. It can be shown that using Eq. (18) in Eq. (16), the scalar and vector parts of decomposition (18) get decoupled, and \(S_2\) can be written as,

### 3.1 Ghost instability

*T*which reads,

### 3.2 Gradient instability

## 4 Conclusion

We showed that by including a new kinetic (\(\tau \)) term in the action (3) it is possible to avoid ghost instabilities in perturbations. This can be attributed to the absence of nondynamical modes that otherwise lead to ghosts [26, 27]. For our choice of background \(B_{\mu \nu }\), the \(\tau \) term does not affect background cosmology. We performed gradient instability analysis for perturbations in \(B_{\mu \nu }\) and derived conditions on \(\tau \) to avoid gradient instability. In the high momentum limit (\(k\rightarrow \infty \)), the theory is trivially free from ghost and gradient instabilities for all positive \(\tau \).

The results of this analysis present a strong case for more detailed investigations of ghost, gradient and other instabilities for perturbations including the metric part, and should motivate further directions in inflation model building. The choice of kinetic term (2) also motivates further analysis of the physical degrees of freedom, that can be addressed through Hamiltonian analysis using \(3+1\) ADM decomposition. An important aspect of academic interest is to study the effect of different choices of background structure of \(B_{\mu \nu }\). Another interesting problem is to explore the cosmology and viability of parity-odd terms, which the authors plan to pursue in future. Additionally, studies involving the higher order terms of \(B_{\mu \nu }\) and gravity may also be explored.

## Notes

### Acknowledgements

Some manipulations in this work were done using Maple\(^{\textsc {tm}}\)[36] and most of them have been cross-checked by hand. This work is partially supported by DST (Govt. of India) Grant No. SERB/PHY/2017041. S.A. wishes to thank ICTS (Bengaluru), where a part of this work was completed, for hospitality.

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