Avoiding instabilities in antisymmetric tensor driven inflation

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 by considering the most general kinetic term for antisymmetric tensor. 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 also perform a check for gradient instability when this new kinetic term is dominant and find that the perturbations are free of gradient instability.


I. 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], 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. [8].
Among the theories not involving scalar fields, in particular those with vector fields [9][10][11][12][13], constructing successful models is often marred by ghost and gradient instabilities [14,15] that lead to unstable vacua. 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 [16,17]. There are no observational signatures of antisymmetric fields in the present universe [18], but it is interesting to study them in the early universe when their presence may become signigicant [19].
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 [20,21]. Even with an unperturbed FriedmannLematreRobertsonWalker (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 [22], and one can in principle consider a model with modifications to the kinetic part of action.
In this letter, we show that by working with a general kinetic term, it is possible to avoid ghost and gradient instabilities in antisymmetric tensor driven inflation in an unperturbed FLRW metric. The most general kinetic term for an even-parity antisymmetric tensor B µν upto quadratic order in field components and derivatives, is which is equivalent to, upto some constant coefficients c i . The first term in Eq. (2) is the standard gauge invariant kinetic term, while the second term is new and is taken into account in the present analysis.

II. BACKGROUND COSMOLOGY
We begin by briefly reviewing the results of Ref. [21], where the authors first considered the possibility of inflation driven by a rank-2 antisymmetric tensor. A typical action for antisymmetric inflation model has the form, where V (B) is the potential, which in our case is quadratic, m 2 B µν B µν /4, and L N M is a nonminimal coupling term. the metric signature (−+++). H λµν (B) = ∇ λ B µν +∇ µ B νλ +∇ ν B λµ (∇ µ is the covariant derivative) constitutes the kinetic term and admits gauge invariance under the transformation A peculiar characteristic of antisymmetric tensor models is that while minimally coupled models generically fail to support inflation, those with nonminimal coupling can give rise to stable de-Sitter solutions and support slow-rolling inflation. The choice of nonminimal coupling term does not affect the extent of support and is only restricted by theoretical constraints like stability near a Schwarzschild metric [23]. Specifically, upto quadratic order in B µν and second order metric derivative, allowed choices for L N M are B µν B µν R and B λν B µ ν R λµ . However, for any choice of L N M in action (3) the perturbations to B µν in FLRW background admit ghosts [20,21] induced by the presence of nondynamical modes of perturbation. This rather generic problem has hindered the progress towards building inflation models with antisymmetric tensor, and remains to be addressed before any serious effort for analysing the full perturbation theory, including metric perturbations.
Although all possible couplings upto quadratic order have been exhausted, and it might be tempting to explore higher order couplings of B µν and R for a resolution to ghosts, modifications to the kinetic term of action (3) have not been explored. Therefore, we start with constructing the most general kinetic term upto quadratic order in B µν , which yields a new gauge-symmetry braking kinetic term in addition to the gauge invariant kinetic term already present in action (3) [22], The action that we work with is then given by, Of course, a whole new class of terms arise if one also takes into account the parity-odd dual But we restrict ourselves to only parity-even terms for the sake of simplicity and because our goal is to show that it is indeed possible to avoid instabilities in models with antisymmetric tensor.
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 τ term.
The background metric g µν is FLRW, with its components given by, Our choice of the background structure of B µν is motivated by the spacetime symmetries as well as calculational convenience, and is given by along with a rescaling B(t) = a(t) 2 φ(t), where a(t) is the scale factor.
The contribution of τ term (second in Eq. (6)) to the background cosmology is through the modifications in Einstein equation viz. the corresponding energy-momentum tensor, T τ µν , given by Remarkably, upon substituting the background value of the metric and B µν in Eq.(10), one finds that This implies, there is no additional contribution to the background cosmology of action (3) and all results for theory (3) follow from Ref. [21], leading to the following conclusions: (i) de-Sitter solutions exist, and (ii) Slow roll inflation is supported.
As a side note, we point out that the vanishing T τ µν is specific to the choice of background Eq. (9). It is certainly of academic interest to check for other choices of background, and we leave it as a future project.

III. PERTURBATIONS
The interesting part however is when B µν is perturbed. Surely, the perturbed modes have nontrivial contributions from the τ 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 [20,21].
Adding a perturbation δB µν to B µν , the perturbed action has the form, where, terms have been segregated according to the order of perturbations. We are interested in the part of action that is quadratic in perturbations, S 2 , since it leads to evolution equations of perturbed modes. Another trick that we use for our convenience is to Fourier transform the spatial part of all modes δB µν , so that all spatial derivatives in the action get replaced algebraic factors of k. In our calculations, we also utilize the freedom to choose the coordinate axis (z−axis) along momentum vector k so that all spatial derivatives along x− and y−axes vanish. As a notation, we choose 'tilde' to denote Fourier transforms. The resulting quadratic part of action,S 2 , in general has a form,S whereL 2 is the corresponding Lagrangian density expressed in terms of Fourier transformed modes δB µν .
There are a total of six modes of perturbation to the fieldB µν , which we represent as, Here and throughout this letter, the momenta ( k) and time (t) dependence of all perturbed modes are understood but not explicitly displayed, to save space. Going forward, for convenience we will work with all perturbed modes simultaneously using an array ∆ given by, The T matrix is found to be, Note that there are no off-diagonal terms present and there are no non-dynamical modes iñ S 2 , which leads us to conclude that there are no ghosts provided that the coupling τ satisfies a simple no-ghost condition: Clearly, when τ = 0, modes E i (i = 1, 2, 3), become nondynamical and would lead to ghosts [21]. The matrix X consists of coefficients of terms involving coupling between modes upto first order in time derivative, The complex nature of X (and hence the presence of X † ) arises from the factors coming from spatial derivatives of perturbed modes, and the hermiticity of action ensures that equations of motion are real.
The rest of terms inS 2 , including other couplings ξ and ζ, are encompassed in O. The full expression for O is quite complicated and irrelevant for questions about ghost instability.
However, to check for gradient instability one needs to find the equation of motion forS 2 and thus matrices X and O do contribute. Since we are particularly interested in the effect of τ term and since it has direct contribution to the kinetic term of action, we consider a dominant τ scenario. This essentially translates to re-writing the integrand in action (17) as, where all components of T τ , X τ and O τ are O(τ ) ∼ O(m 2 ) and rest of the terms are O(ε) ≪ τ . In our calculation, we neglect contributions from O(ε) terms for convenience.
This does not pose any technical problems for the ghost instability issue, and also simplifies the problem of checking for gradient instability.
T τ and X τ can be read off of Eqs. (18) and (20) respectively, where the only nonzero components would be those containing τ . O τ too takes a simplified form, and is given by, where time dependence of scale factor a is understood. The equation of motion for action (17), after using (21), is obtained by varying with respect to ∆ † : where we have neglected O(ǫ) terms. A reasonable assumption in the deep subhorizon is to take ∆ ∝ exp[−i t c s k/a(t ′ )dt ′ ] e as the solution to eigenvector equation (23), where c s is the sound speed and is treated as constant (c s ≪ k), and e is a constant vector [24].
Substituting this ansatz into Eq. (23), the condition for existence of nontrivial solutions reads, where, H is the hubble parameter. A theory suffers from gradient instability when the speed of sound, c s (defined in the relativistic fluid approximation, see Ref. [25]), becomes imaginary. Hence, to avoid gradient instability one must demand that c 2 s > 0. Solving Eq. (24) yields a third degree polynomial equation in c 2 s . In contrast, if one were to solve for the full action (17), the equivalent of Eq. (24) would be a sixth degree polynomial equation in c 2 s . We leave a detailed analysis for future investigations including metric perturbations. Our current interest is limited to the high momentum limit, because instabilities in low energy limit can be characterized as Jeans-like instability and may be under control [26]. Hence, in Eq. (24), we will further take k → ∞ and consider only highest power of τ terms. The polynomial equation (24) is given by, where, For all real and positive roots for c 2 s , the following set of conditions must be satisfied [24] γ > 0, δ > 0, ǫ > 0, It is straightforward to see that the first three conditions are satisfied by Eqs. (26), since beside the reality and positivity of k 2 , H, τ , a(t) and m,ä > 0 during inflation. Since γ = δ in this particular case, the third condition is satisfied when γ > 3. This puts a lower bound on τ , and is not significant for super-horizon modes. Likewise, the final condition is also satisfied by mere power counting of k without imposing further constraints on τ .

IV. CONCLUSION
To conclude, we highlight a few important points. First, we showed that by including a new kinetic (τ ) term in the action (3) it is possible to avoid ghost instabilities in perturbations. For our choice of background B µν , the τ term does not affect background cosmology.
Moreover, for a dominant τ scenario the theory is free from gradient instability in the high momentum limit.
The results of this analysis present a strong case for more detailed investigations of ghost and gradient instabilities for perturbations including the metric part, and should motivate further directions in inflation model building. An important aspect of academic interest is to study the effect of different choices of background structure of B µν . 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 µν and gravity may also be explored.