Stable small spatial hairs in a power-law k-inflation model

In this paper, we extend our investigation of the validity of the cosmic no-hair conjecture within non-canonical anisotropic inflation. As a result, we are able to figure out an exact Bianchi type I solution to a power-law {\it k}-inflation model in the presence of unusual coupling between scalar and electromagnetic fields as $-f^2(\phi)F_{\mu\nu}F^{\mu\nu}/4$. Furthermore, stability analysis based on the dynamical system method indicates that the obtained solution does admit stable and attractive hairs during an inflationary phase and therefore violates the cosmic no-hair conjecture.


I. INTRODUCTION
Cosmological principle, which states that our universe is just simply homogeneous and isotropic on large scales as described the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime, has played a central role in cosmology although it is not straightforward to observationally confirm this principle [1,2]. In fact, many theoretical predictions of the so-called cosmic inflation theory [3], which is basically based on the cosmological principle, have been well confirmed by the cosmic microwave background radiations (CMB) observations such as the Wilkinson Microwave Anisotropy Probe satellite (WMAP) [4] as well as the Planck one [5]. However, some anomalies of the CMB temperature such as the hemispherical asymmetry and the cold spot, which have been detected by the WMAP and then by the Planck, cannot be explained within the context of the cosmological principle [6]. In other words, understanding the nature of these exotic features might address a slight modification of the cosmological principle.
One possible modification we can think of is using the Bianchi metrics, which are homogeneous but anisotropic spacetimes, instead of the FLRW one in order to describe the early universe [7,8]. If the early universe was slightly anisotropic, an important question would be naturally addressed: what would the current state of our universe be ? In other words, would it be slightly anisotropic or completely isotropic ? It is worth noting that some observational evidences have claimed that the current universe might not be isotropic but anisotropic [9]. On the theoretical side, the so-called cosmic no-hair conjecture proposed by Hawking and his colleagues few decades ago [10] might provide a hint to this important question. Basically, this conjecture states that the late time universe should simply be homogeneous and isotropic, regardless of initial states, which might be inhomogeneous or/and anisotropic (a.k.a. spatial hairs). Hence, proving this conjecture is really an important task for physicists and cosmologists. Unfortunately, this is not a straightforward thing. In fact, a complete proof of this conjecture has been a great challenge since the first partial proof using the energy conditions for the Bianchi spacetimes by Wald [11][12][13][14].
Along with the partial proofs, some counterexamples to this conjecture have been claimed to exist in Refs. [15][16][17][18][19][20]. However, many of them have been shown to be unstable against field perturbations [18], except a recent supergravity motivated model proposed by Kanno, Soda, and Watanabe (KSW) [19,20]. In particular, this model has been shown to admit a Bianchi type I metric, which is homogeneous but anisotropic, as its stable and attractive solution during an inflationary phase, due to the existence of unusual coupling between scalar and electromagnetic (vector) fields −f 2 (φ)F µν F µν /4. This result indicates that the cosmic no-hair conjecture is really broken down in the KSW model. Consequently, many papers have appeared to discuss extensively this interesting model [21]. For example, a smoking gun of anisotropic inflation in the CMB such as the T B and EB correlations, which vanish in isotropic inflation, has been investigated systematically in Refs. [22][23][24]. In addition, primordial gravitational waves in anisotropic inflation have also been studied in Ref. [25]. Other cosmological aspects of the KSW mode can be found in recent interesting reviews [26].
It is worth noting that some non-canonical extensions of the KSW model, in which a canonical scalar field has been replaced by non-canonical ones such as the Dirac-Born-Infeld (DBI) [27], generalized ghost condensate [28], supersymmetric Dirac-Born-Infeld (SDBI) [29], and Galileon fields [30], have been proposed recently. As a result, the cosmic no-hair conjecture has been shown to be violated in all these non-canonical models. This might confirm the leading role of the unusual coupling −f 2 (φ)F µν F µν /4 in breaking down the validity of the cosmic no-hair conjecture. Hence, studying this conjecture in other non-canonical scalar field models is essentially important. Therefore, we would like to examine another scenario, in which a k-inflation model [32] is allowed to couple to the KSW model, in this paper to see whether the cosmic no-hair conjecture is violated or not. Note that the CMB imprints of anisotropic inflation of non-canonical scalar field, such as the T B and EB correlations, have been investigated in a recent paper [31]. Once these prediction were confirmed by more sensitive primordial gravitational wave detectors, we would be able to figure out the most viable non-canonical anisotropic model. The present model therefore would be a candidate for this classification.
As a result, the paper will be organized as follows: (i) A brief introduction of this study has been given in Sec. I. (ii) A basic setup of the proposed model will be shown in Sec. II. (iii) Then, exact anisotropic solutions will be presented in Sec. III. (iv) Stability analysis based on the dynamical system method of the obtained solution will be investigated in Sec. IV. (v) Finally, concluding remarks will be written in Sec. V.

II. SETUP
As a result, a general scenario of non-canonical KSW model is given by [28,31], with F µν ≡ ∂ µ A ν − ∂ ν A µ being the field strength of the vector field A µ used to describe the electromagnetic field. Note that the reduced Planck mass, M p , has been set as one for convenience. In addition, P (φ, X) is an arbitrary function of scalar field φ and its kinetic X ≡ −∂ µ φ∂ µ φ/2, which was firstly introduced in the so-called k-inflation [32]. It is clear that the KSW model of canonical scalar field is just the simplest case with P (φ, X) = X + V (φ). Moreover, if P (φ, X) takes the following form, then we will have the DBI extension of the KSW model with γ ≡ 1/ 1 +f (φ)∂ µ φ∂ µ φ being the Lorentz factor characterizing the motion of the D3-brane [27]. On the other hand, a supersymmetric DBI extension of the KSW model has been proposed in Ref. [29] with P (φ, X) being of the following form 3) It is clear that if we take a limit γ → 1, or equivalentlyf → 0, then both (S)DBI models will reduce to the canonical KSW model. In addition, another non-canonical extension of the KSW model has been proposed in Ref. [28] with P (φ, X) assumed to be the generalized ghost condensate form as where c > 0 and n ≥ 1 are all constants. It is noted again that all these models have been shown to admit counterexamples to the cosmic no-hair conjecture. And the CMB imprints of anisotropic inflation for arbitrary P (φ, X) have been studied in Ref. [31].
In this paper, we would like to seek analytical anisotropic solutions and investigate their stability during an inflationary phase for a specific k-inflation model [32] with P (φ, X) = K(φ)X + L(φ)X 2 coupled to the KSW model as follows here K(φ) and L(φ) are all functions of φ. It is noted that the potential V (φ) has not been introduced in the above action in a sense that the term L(φ)X 2 could play a similar role as V (φ) normally does in the slow-roll inflation scenario [32]. As a result, the corresponding Einstein field equation can be shown to be along with the corresponding equation of motion of the scalar field φ given by In addition, the corresponding field equation of the vector field turns out to be In order to seek anisotropic solutions to this model, we prefer using the following Bianchi type I metric along with the compatible vector field A µ chosen as A µ = (0, A x (t) , 0, 0) [19,20]. In addition, the scalar field is assumed to be homogeneous, i.e., φ = φ(t). Note that the scale factor σ(t) is regarded as a deviation from the spatial isotropy governed by the other scale factor α(t). This means that σ(t) should be much smaller than α(t) during an inflationary phase. Non-vanishing σ(t) will therefore correspond to the existing of spatial hairs [19,20].
As a result, Eq. (2.9) can be integrated directly to give a solution, [20]. Thanks to this solution, the Einstein field equation can be written explicitly as follows (see the Appendix A for a detailed derivation) (2.14) In addition, the corresponding field equation of the scalar field reads It straightforward to see that if K = −1 and L = exp[λφ] we will have the corresponding dilatonic ghost condensate model [28].

III. POWER-LAW SOLUTIONS
In this section, we would like to figure out analytical solutions to the derived field equations shown above by using the following ansatz such as [20] along with the exponential functions of scalar field given by where ζ, η, ξ, φ 0 , k 0 , l 0 , f 0 , λ, κ, and ρ are all additional parameters. Given this choice, the scale factors of the Bianchi type I metric will be the power-law function of time as follows Hence, an inflationary solution will require that ζ − 2η 1 and ζ + η 1. It is noted that η ζ according to the constraint σ(t) α(t) as mentioned earlier. Hence, ζ 1 will be required for inflationary solutions. It turns out thatα 2 = ζ 2 t −2 ,σ 2 = η 2 t −2 ,α = −ζt −2 , andσ = −ηt −2 . Hence, in order to have a set of algebraic equations from the above field equations, all terms in these field equations must have t −2 . As a result, this requirement will lead to the following constraints for the field parameters such as κ = 0, (3.6) λξ = 2, (3.7) ρξ + 2ζ + 2η = 1. (3.8) It turns out that the last two constraint equations imply that Hence, the requirements α σ, or equivalently ζ η, and ζ 1 for inflationary solutions, lead to two constraints: (i) λ < 0 provided that ρ > 0 and (ii) ρ |λ| such as ρ/|λ| 1. As a result, a set of algebraic equations can be defined from the field equations (2.12), (2.13), (2.14), and (2.15) to be 13) respectively. Here u and v are additional variables defined as follows As a result, v can be defined in terms of ζ and η according to Eq. (3.12) as Furthermore, u can be defined from Eqs. (3.11) to be ( 3.17) with the help Eq. (3.16). Given these useful results, solving either Eq. (3.10) or Eq. (3.13) gives us non-trivial solutions of ζ, Since ζ should be approximated to be −ρ/λ 1 according to the constraint equation (3.9), the suitable solution of ζ should be ζ − rather than ζ + , i.e., Hence, the following η turns out to be As a result, the corresponding anisotropy parameter Σ ≡σ/α is given by It is noted that k 0 has been regarded up to now as a free parameter, in contrast to Ref. [28], in which k 0 was initially fixed to be −1. As a result, the positivity λ 2 + 4λρ − 8ρ 2 − 8k 0 > 0 puts a constraint on k 0 as In addition, the positivity of v implies, according to Eq. (3.16), that η > 0, provided that ζ 1. As a result, this positivity of η leads to the following inequality On the other hand, we would like to have the following equality such that ζ −ρ/λ 1. Consequently, the corresponding value of k 0 should be 25) which does safisty the inequalities (3.22) as well as (3.25). Note that u is always positive during the inflationary phase. Consequently, the corresponding approximated value of ζ and η can be defined to be Note again that λ has been assumed to be negative definite, in contrast to ρ. Hence, the anisotropy Σ/H now takes an approximated value during the inflationary phase as Indeed, Σ should be smaller than one in order to be consistent with the current observations [19,20]. Let us provide here a simple comparison between the present model with the KSW model of canonical scalar field [20]. In particular, the field parameters will be chosen as λ = ±0.1 (the sign + for the KSW model and − for the present model) and ρ = 50 (for both models). Accordingly, it turns out that (Σ/H) KSW 0.0004 while Σ/H 0.0005. Hence, the anisotropy in the present model can be said to be similar to that of the KSW model.

IV. STABILITY ANALYSIS
So far, we have found the power-law Bianchi type I solution to the k-inflation model [32] in the presence of the unusual coupling between the scalar and electromagnetic fields, i.e., −f 2 (φ)F µν F µν /4. In this section, we would like to investigate the stability of this anisotropic solution during the inflationary phase in order to see whether the cosmic no-hair conjecture is violated or not. Following the previous studies [20,[27][28][29][30], the dynamical system method will be used to do this task. Note that there is another stability analysis approach based on power-law perturbations, i.e., δα = A α t n , δσ = A α t n , and δφ = A φ t n , which would lead to the same conclusion about the stability of the anisotropic solutions [27,29,30]. However, this method does not yield an information of attractive property of the anisotropic solutions.
As a result, introducing dynamical variables such as [20,[27][28][29][30] x =σ α ; y =φ along with two auxiliary variables [30] w κ = exp It is apparent that As a result, the field equations (2.13), (2.14), and (2.15) can be converted into the autonomous equations of the dynamical variables x, y, and z as follows (4.10) along with that of two auxiliary variables, Here dα =αdt can be understood as a new time coordinate [20,[27][28][29][30]. In addition, the following constraint equation coming from the Friedmann equation (2.12), 13) has been used to derive the above autonomous equations. Now, we would like to see whether this dynamical system admits anisotropic fixed points with x = 0. Mathematically, these fixed points are solutions of the following equations, dx/dα = dy/dα = dz/dα = dw λ /dα = dw κ /dα = 0. It is noted that the equation dw κ /dα = 0 implies the result that κ = 0, or equivalently w κ = 1. In addition, the equation dw λ /dα = 0 implies that (4.14) while the equation dz/dα = 0 leads to another relation Here, an isotropic fixed point solution with x = z = w λ = 0 is not our current interest. Hence, a relation between x and y can be figured out from these two equations as On the other hand, we can obtain from two equations dx/dα = 0 and dz/dα = 0 a relation as Thanks to these useful relations, both the equations dy/dα = 0 or dz/dα = 0, lead to a non-trivial equation for anisotropic fixed points x = 0 as As a result, this equation admits two solutions of x, 4.19) Comparing these solutions with the power-law solutions obtained in the previous section implies that only the solution, is a suitable solution. Indeed, it is straightforward to see that x + = Σ/H, which has been clearly defined in the previous section for the anisotropic power-law solution. And the corresponding value of the other dynamical variables y, z, and w λ can be defined according to Eqs. (4.16), (4.17), and (4.13), respectively. These results indicate that the anisotropic fixed point x = x + is indeed equivalent to the anisotropic power-law solution derived in the previous section. Consequently, the anisotropic fixed point and the anisotropic power-law solution share the same stability property. Hence, we will investigate the stability of the anisotropic fixed point during the inflationary phase from now on.
As discussed above, we have the following constraints as k 0 −3ρ 2 and |λ| ρ such that ζ ρ/|λ| 1 for the negative λ and positive ρ during the inflationary phase. Consequently, we are able to show the corresponding approximated value of the anisotropic fixed point as To see how small or large this fixed point is, we choose for example λ = −0.1 and ρ = 50. As a result, the corresponding anisotropic fixed point will be (x, y, z, w λ ) (0.0005, −0.04, 0.067, 2165). Now, we would like to perturb the autonomous equations (4.8), (4.9), (4.10), and (4.11) around the anisotropic fixed point. As a result, a set of the following perturbation equations turns out to be Here we have only considered the leading terms in these perturbation equations for simplicity. Following Ref. [20], we will take the exponential perturbations of dynamical variables given by  (4.32) here the sign of ω will determine the stability of the anisotropic fixed point. In particular, if a set of perturbation equations admits any positive ω, then exp[ωα] will blow up as α → +∞, causing an instability of the anisotropic fixed point. In contrast, the anisotropic fixed point will be stable if all solutions of ω turn out to be negative because exp[ωα] will tend to zero as α → +∞. As a result, we are able to obtain the following set perturbation equations, which can be written as a matrix equation as follows which can be defined to be a polynomial equation of ω as follows a 4 ω 4 + a 3 ω 3 + a 2 ω 2 + a 1 ω + a 0 = 0, (4.35) where a 4 = 1 > 0, a 3 10 > 0, Here, we have only kept the leading terms in the definition of a i (i = 0 − 3) for simplicity. It appears that the coefficients a i (i = 0 − 4) of Eq. (4.35) are all positive definite. Consequently, Eq. (4.35) admits only negative roots ω < 0, meaning that the anisotropic fixed point is indeed stable against the perturbations. More interestingly, the numerical calculations shown in the Fig. 1 indicate that the anisotropic fixed point is indeed an attractor solution to the dynamical system. These results, both analytical and numerical, strongly imply the violation of the cosmic no-hair conjecture in the present model.

V. CONCLUSIONS
We have shown that the non-canonical extension of the KSW model, in which the canonical scalar field has been replaced by the non-canonical field of the so-called k-inflation [32], does admit the Bianchi type I spacetime as its stable and attractor solution during the inflationary phase. This study together with the previous ones done in Refs. [20,[27][28][29][30] indicate that the cosmic no-hair conjecture proposed by Hawking and his colleagues [10] is extensively broken down in the KSW and its non-canonical extensions due to the existence of the unusual coupling between the scalar and electromagnetic field −f 2 (φ)F µν F µν /4. It is noted again that the CMB imprints of non-canonical anisotropic inflation have been investigated in Ref. [31]. Once these imprints were confirmed by more sensitive primordial gravitational wave observations, then the present paper would provide one more non-canonical anisotropic inflation scenario, which might be useful to figure out the most viable anisotropic inflation model.

ACKNOWLEDGMENTS
The author would like to thank Prof. W. F. Kao as well as Dr. Ing-Chen Lin very much for their fruitful collaborations on the previous works of anisotropic inflation.