Chromo-Natural Model in Anisotropic Background

In this work we study the chromo-natural inflation model in the anisotropic setup. Initiating inflation from Bianchi type-I cosmology, we analyze the system thoroughly during the slow-roll inflation, from both analytical and numerical points of view. We show that the isotropic FRW inflation is an attractor of the system. In other words, anisotropies are damped within few $e$--folds and the chromo-natural model respects the cosmic no-hair conjecture. Furthermore, we demonstrate that in the slow-roll limit, the anisotropies in both chromo-natural and gauge-flation models share the same dynamics.


I. INTRODUCTION
The early epoch of almost exponential expansion, known as inflation, was proposed to resolve some of standard Big Bang cosmology puzzles such as the horizon and the flatness problems [1]. Besides these remarkable successes, inflation is also able to describe the cosmological data and CMB temperature anisotropies [2]. Moreover, the inflationary paradigm has the powerful theoretical and model building feature of almost relaxing the dependence of late time physics on the pre-inflation initial conditions [3]- [5]. One particular set of such conditions is anisotropic but homogeneous metric; the Bianchi cosmology models. In this work, we are interested in this family of pre-inflationary initial conditions and in particular Bianchi type I.
For inflationary models consist of (only) scalar fields, the anisotropic part of the metric is not sourced by the (scalar) matter field. Hence, anisotropies are (exponentially) damped by the inflationary expansion of the background. However, this picture may change in the presence of vector (gauge) fields, where the gauge field can source the anisotropic part of the metric. More precisely, inflationary extended cosmic no-hair theorem [6] states that for general inflationary systems of all Bianchi types, anisotropies can grow during inflation, in contrast to the cosmic no-hair conjecture (and also Wald's cosmic no-hair [7]). Nevertheless, inflation puts an upper bound on the enlargement of the anisotropies. Assuming a (slowroll) quasi-de Sitter expansion, the enlargement of anisotropies is at most of order of the slow-roll parameter [6].
Introducing a general class of non-Abelian gauge field inflationary models minimally coupled to Einstein gravity, gauge-flation opens a new venue for building inflationary models closer to particle physics models. In [8,9], it was shown that non-Abelian gauge field theory can provide a setup for constructing isotropic and homogeneous inflationary background. For an extensive review on this topic see [10]. Another interesting inflationary scenario involving non-Abelian gauge fields is the chromo-natural model [11]. In this model, inflation is driven by an axion field, while the non-Abelian gauge field plays the important role in flattening the axion potential, without requiring (unnatural) super-Planckian axion decay constant [11]- [17]. In other words, the axion-gauge field interaction makes the model technically natural. It is worth to mention that both gauge-flation and chromo-natural models have been disfavored by recent CMB observational data [2,18,19].
Due to the vector nature of the gauge fields, inflationary models involving gauge fields may or may not respect the cosmic no-hair conjecture [6,20]. Since the gauge field sources anisotropies in (homogeneous but anisotropic) Bianchi background, it is not clear that the isotropic setup is a stable solution in these models. Hence, it is necessary and important to study the stability of the isotropic background against the initial anisotropies. This matter has been studied in [21] for the gauge-flation model. There, it was shown that although the gauge field is turned on in the background, the isotropic configuration is an attractor solution of the model and the initial anisotropies are damping within few e-folds.
Considering Bianchi type I, we tackle the same issue in the chromo-natural model and we study the classical stability of this model against the initial anisotropies during the slow-roll inflation.
The rest of this paper is organized as follows. In Sec. II, we study the chromo-natural model in Bianchi type I cosmology. In Sec. III, we analyze the system during the slow-roll inflation, both from analytical and numerical points of view. Finally, we conclude in Sec. IV.

II. SETUP
Consisting of an axion field χ and a non-Abelian SU (2) gauge field A a µ , the chromonatural model [11] is an inflationary model with the following action whereF µν a = µνλσ F a λσ and the gauge field strength tensor F a µν , is given by Here a, b, c = 1, 2, 3 are used for the indices of algebra while µ, ν = 0, 1, 2, 3 represent the space-time indices. The above action has 4 parameters, dimensionless parameters λ and g and dimensionful parameters µ and f , where f is the axion decay constant.
The homogeneous and isotropic FRW cosmology of the chromo-natural model has been widely studied in [11]- [23]. In this model, the axion field is the inflaton, while the non-Abelian gauge field is a secondary field which is coupled to the axion through its Chern-Simons interaction. In the absence of the non-Abelian gauge field, this model reduces to the natural inflation [24,25]. In this model, the slow-roll inflation is obtained for super-Planckian f parameter, which is not a natural scale within particle physics models. However, in the chromo-natural model, the non-Abelian gauge field slows down the inflaton's evolution and leads to slow-roll inflation even with the natural values of f (f M Pl ).
In the FRW background, the gauge field has the following homogeneous and isotropic where e a i | FRW are the spatial triads of the FRW metric, a(t) is the scale factor and ψ is a scalar field. Using the axion field equation and imposing the slow-roll condition, ψ is determined in terms of χ as following where H is the Hubble parameter. Note that in the homogeneous and isotropic FRW background, H, χ and ψ are evolving slowly during the slow-roll inflation. For a thorough review on this issue see [10].
Since the chromo-natural model includes a non-Abelian gauge field in the background, a question which may arise naturally is the classical stability of the isotropic inflationary trajectory against initial anisotropies. Since the gauge field sources anisotropies in (homogeneous but anisotropic) Bianchi background, it is not clear if the isotropic setup is a stable solution. Here, considering the chromo-natural model in the homogeneous but anisotropic Bianchi type I background, we study the behavior of anisotropies during inflation.
The space-time metric in the homogeneous but anisotropic (axially symmetric) Bianchi type I metric is given by where e α(t) is the isotropic scale factor and the anisotropy in the metric is represented by (the time derivative of) σ(t). Choosing the temporal gauge, the consistent truncation for the gauge field reads as where the spatial triads have the following explicit forms e a 1 = e α−2σ δ a 1 , e a 2 = e α+σ δ a 2 and e a 3 = e α+σ δ a 3 .
Note that, due to the axial symmetry of the metric in y − z plane, we have ψ 2 = ψ 3 . As a result, the explicit form of our ansatz is given by For mathematical convenience, which will be clear soon, we introduce the following field where ψ represents the isotropic field and β parametrizes anisotropy in the gauge field.
Thus the deviation of β 2 from one parametrizes the amount of anisotropy in the gauge field.
Determining the energy-momentum tensor and plugging the metric (II.5) and the ansatz (II.6) into it, we have a diagonal homogeneous tensor in which ρ is the energy density, while P andP are the isotropic and anisotropic pressures, respectively. By decomposing the energy density as ρ = ρ χ + ρ YM , where ρ χ and ρ YM are contributions of axion field and Yang-Mills term respectively 1 , we have Isotropic and anisotropic pressures are given by Interestingly, the anisotropic pressureP , is only originated from the Yang-Mills term and the other terms do not contribute. Note that, for the isotropic case in which β 2 = 1 (wherė σ = 0 andβ = 0), the anisotropic pressure vanishes.
From the Einstein equations, the inflation condition (e α) > 0 leads to which implies that the inflation happens only if the axion potential V (χ) = µ 4 1 + cos( χ f ) , is the dominant term in the energy density.
At this point, inserting the metric (II.5) and the ansatz (II.6) into the action (II.1), one obtains the explicit form of the action Interestingly, the fieldsσ and β only appear in the Yang-Mills term of the matter action.

III. SLOW-ROLL REGIME
Slow-roll inflation is quantified in terms of slow-roll parameters, and η where H =α is the Hubble parameter and slow-roll conditions require , |η| 1. Using the Einstein equations, the first slow-roll parameter is which demanding the slow-roll inflation, leads to following conditions Hereafter, we simplify and analyze the dynamical field equations assuming quasi-de Sitter inflation, in a sense that ψ is given by Eq. (II.15), whileψ/(αψ),χ/α andσ/α are very small during inflation (Eq. (III.3)). It is clear that all of the trajectories approach the isotropic fixed points; i.e. β 2 = 1. In Eq. (III.5), one can distinguish the following three regions for different values of β.
• Starting from a very small β 2 , we have the following solution for β which shows that |β| is growing very rapidly and escaping quickly from the vicinity of zero. Note that this solution is only valid in first few e-folds where β 2 is far from one.
Once |β| gets close to one, it starts damped oscillations around the attractor solution.
As we see here, β monotonically increases for all possible values of B 1 and B 2 , but this is not necessarily the case forσ. In particular, for two different initial conditions in which (i) B 1 = 0 and (ii) B 2 = 0, we havė where β 0 is the initial value of β.
• Initiating from a very large |β|, we have β C 1 e −2αt + C 2 e −αt , (III.11) which has an exponentially damping behavior and when |β| gets close to one, it starts damped oscillations around the attractor solution. Note that Eq. (III.11) is identical to Eq. (III.9) that governs the evolution of 1/β 2 in the limit |β| 1. For all possible values of C 1 and C 2 in (III.11), β monotonically decreases, however this is not necessarily the case forσ. In fact, for two initial conditions in which (i) C 1 = 0 and (ii) where β 0 is the initial value of β. In a system which undergoes quasi-de Sitter inflation (i.e. very small ), we see that regardless of initial β values, all solutions converge to β 2 = 1 (the isotropic solutions), within the first few e-folds. In other words, the isotropic inflation is the attractor solution in the chromo-natural model. Furthermore, we demonstrate that initiating from |β| 1 (|β| 1), β decreases (increases) exponentially fast and gets close to one in few e-folds.
Interestingly, this is not necessarily the case forσ. From Eqs. (III.10) and (III.12), we realize that having a gauge (vector) field in the system, |σ/α| can grow for one or two e-folds during the slow-roll inflation and saturates its theoretical upper bound |σ/α| ∼ [6] 3 . However, β gets close to one quickly, and eventually |σ/α| starts damping exponentially fast after the first few e-folds.

Numerical analysis:
At this point, we present the numerical analysis of the system. We have studied the system for different values of β, and different χ/f values numerically. Note that, demanding slow-roll inflation, one can choose the initial value of χ as χ 0 /f ∈ (0, π), where χ 0 is the initial value of the axion field [14]. In the limit that χ 0 /f π, the chromo-natural model reduces to the gauge-flation model [13]. Our numerical study also confirms the analytical

IV. CONCLUSIONS
Considering the chromo-natural model in the Bianchi type I background, we have studied the classical stability of isotropic slow-roll inflation with respect to initial anisotropies.
For the inflationary models which consist of (only) scalar fields, the anisotropies are (exponentially) damped due to the inflationary expansion of the universe and they respect the cosmic no-hair conjecture [7]. However, this is not necessary the case in the presence of vector (gauge) fields. More precisely, inflationary extended cosmic no-hair theorem [6] states that for general inflationary systems of all Bianchi types, in contrast to the cosmic no-hair conjecture (and also Wald's cosmic no-hair theorem [7]), anisotropies can in principle grow during inflation. However, assuming a (slow-roll) quasi-de Sitter expansion, inflation puts an upper bound on the enlargement of the anisotropies and enforces them to be of the order of the slow-roll parameter .
We parametrized the anisotropy in the metric with σ(t), while β(t) is a parameter that represents anisotropy in the gauge field and the isotropic setup is described byσ = 0 and β = ±1. We showed that although the gauge field is turned on in the background, the isotropic solution is an attractor solution in the chromo-natural model and the anisotropies are damping within first few e-folds. Here, for simplicity in the analytical study, we considered an axial symmetric version of Bianchi type I. However, our numerical investigations confirmed that this attractor behavior is a generic property of the system, i.e. the attractor domain includes all the initial anisotropies. In other words, the chromo-natural model respects the cosmic no-hair conjecture. In particular, we demonstrated that starting from |β| 1 (|β| 1), β decreases (increases) exponentially fast and gets close to the isotropic configuration in few e-folds. However, this is not necessarily the case forσ. Since we have a gauge (vector) field in the system, |σ/α| may increase for one or two e-folds during the slow-roll inflation and saturates its theoretical upper bound |σ/α| ∼ [6]. However, as β approaches one, |σ/α| damps exponentially fast after the first few e-folds.
We showed that the anisotropies in both chromo-natural and gauge-flation models share the same dynamics during the slow-roll inflation. In [10], it has been shown that these two models share the same vector (and also tensor) mode perturbations around the isotropic and homogeneous FRW background. The reason for the above correspondence between these two models is that, both vector perturbations and the background anisotropies are originated from the Yang-Mills term in the action which is common in chromo-natural and gauge-flation models. Moreover, during the slow-roll inflation, ψ and χ undergo slow-roll transition and the Hubble parameterα, is almost constant. In fact, in both chromo-natural and gauge-flation models, the quasi-de Sitter expansion is globally stable against the initial anisotropies and the anisotropies are damped within first few e-folds, in agreement with the cosmic no-hair conjecture.