Anisotropic power-law k-inflation
aa r X i v : . [ h e p - t h ] N ov KOBE-TH-13-09
Anisotropic power-law k-inflation
Junko Ohashi, Jiro Soda, and Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science,1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan Department of Physics, Kobe University, Kobe 657-8501, Japan (Dated: November 1, 2018)It is known that power-law k-inflation can be realized for the Lagrangian P = Xg ( Y ), where X = − ( ∂φ ) / φ and g is an arbitrary function in termsof Y = Xe λφ/M pl ( λ is a constant and M pl is the reduced Planck mass). In the presence of avector field coupled to the inflaton with an exponential coupling f ( φ ) ∝ e µφ/M pl , we show that themodels with the Lagrangian P = Xg ( Y ) generally give rise to anisotropic inflationary solutions withΣ /H = constant, where Σ is an anisotropic shear and H is an isotropic expansion rate. Providedthese anisotropic solutions exist in the regime where the ratio Σ /H is much smaller than 1, they arestable attractors irrespective of the forms of g ( Y ). We apply our results to concrete models of k-inflation such as the generalized dilatonic ghost condensate/the DBI model and we numerically showthat the solutions with different initial conditions converge to the anisotropic power-law inflationaryattractors. Even in the de Sitter limit ( λ →
0) such solutions can exist, but in this case the nullenergy condition is generally violated. The latter property is consistent with the Wald’s cosmicconjecture stating that the anisotropic hair does not survive on the de Sitter background in thepresence of matter respecting the dominant/strong energy conditions.
I. INTRODUCTION
The inflationary paradigm, which was originally proposed in [1], is now widely accepted as a viable phenomenologydescribing the cosmic acceleration in the very early Universe. The simplest inflationary scenario based on a singlescalar field predicts the generation of nearly scale-invariant and adiabatic density perturbations [2]. This prediction isin agreement with the temperature fluctuations of the Cosmic Microwave Background (CMB) observed by the WMAP[3] and Planck [4] satellites.The WMAP data showed that there is an anomaly associated with the broken rotational invariance of the CMBperturbations [5]. This implies that the statistical isotropy of the power spectrum of curvature perturbations is broken,which is difficult to be addressed in the context of the simplest single-field inflationary scenario. Although we cannotexclude the possibility that some systematic effects cause this anisotropy [6], it is worth exploring the primordialorigin of such a broken rotational invariance.If the inflaton field φ couples to a vector kinetic term F µν F µν , an anisotropic hair can survive during inflation fora suitable choice of the coupling f ( φ ) [7]. In such cases, the presence of the vector field gives rise to the anisotropicpower spectrum consistent with the broken rotational invariance of the CMB perturbations [8, 9] (see also Refs. [10]-[23] for related works). In addition, the models predict the detectable level of non-Gaussianities for the local shapeaveraged over all directions with respect to a squeezed wave number [24, 25]. In the two-form field models wherethe inflaton couples to the kinetic term H µνλ H µνλ the anisotropic hair can also survive [26], but their observationalsignatures imprinted in CMB are different from those in the vector model [27].For a canonical inflaton field with the potential V ( φ ), the energy density of a vector field can remain nearlyconstant for the coupling f ( φ ) = exp[ R V / ( M V ,φ ) dφ ] [7], where V ,φ = dV /dφ . For the exponential potential V ( φ ) = ce − λφ/M pl the coupling is of the exponential form f ( φ ) = e − φ/ ( λM pl ) , as it often appears in string theory andsupergravity [28]. In this case there exists an anisotropic power-law inflationary attractor along which the ratio Σ /H is constant [29], where Σ is an anisotropic shear and H is an isotropic expansion rate. For general slow-roll models inwhich the cosmic acceleration comes to end, the solution with an anisotropic hair corresponds to a temporal attractorduring inflation [7].There exists another inflationary scenario based on the scalar-field kinetic energy X = − ( ∂φ ) / P ( φ, X )– dubbed k-inflation [30]. The representative models of k-inflation are the (dilatonic) ghost condensate [31, 32]and the Dirac-Born-Infeld (DBI) model [33]. In such cases the evolution of the inflaton can be faster than that of thestandard slow-roll inflation, so the coupling f ( φ ) with the vector field can vary more significantly. It remains to seewhether the anisotropic hair survives in k-inflation. This is important to show the generality of anisotropic inflation.In Refs. [32, 34] it was found that in the presence of a scalar field and a barotropic perfect fluid the condition for theexistence of scaling solutions restricts the Lagrangian of the form P ( φ, X ) = Xg ( Y ), where g is an arbitrary function interms of Y = Xe λφ/M pl and λ is a constant. On the flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) backgroundthere exists a scalar-field dominated attractor responsible for inflation under the condition λ < ∂P/∂X [35, 36]. Infact, the Lagrangian P ( φ, X ) = Xg ( Y ) covers a wide class of power-law inflationary scenarios such as the canonicalscalar field with the exponential potential ( g ( Y ) = 1 − cM /Y ), the dilatonic ghost condensate ( g ( Y ) = − cY /M ),and the DBI model ( g ( Y ) = − ( m /Y ) p − Y /m − M /Y ). There is also another power-law inflationary scenariostudied in Ref. [37].In the presence of a vector kinetic term F µν F µν with the coupling f ( φ ) = f e − µφ/M pl , the canonical scalar field withthe exponential potential V ( φ ) = ce − λφ/M pl gives rise to stable anisotropic inflationary solutions under the condition λ + 2 µλ − > P ( φ, X ) = Xg ( Y ). Remarkably, if anisotropicinflationary fixed points exist, they are stable irrespective of the forms of g ( Y ) in the regime where the anisotropy issmall (Σ /H ≪ P ( φ, X ) on theanisotropic cosmological background. In Sec. III we obtain anisotropic fixed points for the Lagrangian P = Xg ( Y )and discuss the stability of them against the homogenous perturbations. In Sec. IV we apply our general results toconcrete models of power-law inflation and numerically confirm the existence of stable anisotropic solutions. Sec. Vis devoted to conclusions. II. BACKGROUND EQUATIONS OF MOTION
Let us consider the theories described by the action S = Z d x √− g M " M R + P ( φ, X ) − f ( φ ) F µν F µν , (1)where g M is the determinant of the metric g µν , R is the scalar curvature, and P ( φ, X ) is a function with respect tothe inflaton φ and its derivative X = − (1 / g µν ∂ µ φ∂ ν φ . The field φ couples to a vector kinetic term F µν F µν , wherethe vector field A µ is related to F µν as F µν = ∂ µ A ν − ∂ ν A µ .Choosing the gauge A = 0, we can take the x -axis for the direction of the vector field, i.e., A µ = (0 , v ( t ) , , v ( t ) is a function of the cosmic time t . Since there is the rotational symmetry in the ( y, z ) plane, we take theline element of the form ds = −N ( t ) dt + e α ( t ) h e − σ ( t ) dx + e σ ( t ) ( dy + dz ) i , (2)where N ( t ) is the Lapse function, e α ≡ a and σ are the isotropic scale factor and the spatial shear, respectively. Forthis metric the action (1) reads S = Z d x e α N (cid:20) M ( ˙ σ − ˙ α ) + N P ( φ, X ( N )) + 12 f ( φ ) e − α +4 σ ˙ v (cid:21) , (3)where a dot represents a derivative with respect to t , and X ( N ) = ˙ φ N − /
2. The field equation of motion for thefield v following from the action (3) is integrated to give˙ v = p A f ( φ ) − e − α − σ , (4)where p A is an integration constant. Varying the action (3) with respect to N , α , σ , φ , and setting N = 1, it followsthat H = ˙ σ + 13 M (cid:20) XP ,X − P + p A f ( φ ) − e − α − σ (cid:21) , (5)¨ α = − σ − M (cid:20) XP ,X + p A f ( φ ) − e − α − σ (cid:21) , (6)¨ σ = − α ˙ σ + p A M f ( φ ) − e − α − σ , (7)( P ,X + 2 XP ,XX ) ¨ φ + 3 P ,X ˙ α ˙ φ + P ,Xφ ˙ φ − P ,φ − p A f ( φ ) − f ,φ ( φ ) e − α − σ = 0 , (8)where H ≡ ˙ α = ˙ a/a is the Hubble expansion rate, and P ,X ≡ ∂P/∂X etc. We define the energy densities of theinflaton and the vector field, respectively, as ρ φ ≡ XP ,X − P , ρ A ≡ p A f ( φ ) − e − α − σ . (9)In order to sustain inflation, we require the condition ρ φ ≫ ρ A . Since the shear term Σ ≡ ˙ σ should be suppressedrelative to H , Eq. (5) reads H ≃ ρ φ M . (10)On using the slow-roll parameter ǫ ≡ − ˙ H/H , Eq. (6) can be written as ǫ = 3 ˙ σ H + XP ,X M H + 2 ρ A M H . (11)Each term on the r.h.s. of this equation needs to be much smaller than unity. In particular, if the contributions of theshear and the vector-field energy density are negligible, Eq. (11) reduces to the standard relation ǫ ≃ XP ,X / ( M H )of k-inflation [30].From Eq. (7) the shear term obeys ˙Σ = − H Σ + 2 ρ A M . (12)If Σ converges to a constant value, it follows that Σ H ≃ ρ A ρ φ , (13)where we used Eq. (10). If the evolution of ρ A is proportional to ρ φ , the ratio Σ /H remains constant. This actuallyhappens for anisotropic inflationary attractors discussed in the next section. III. POWER-LAW K-INFLATION AND THE STABILITY OF ANISOTROPIC FIXED POINTS
On the flat isotropic FLRW background the power-law k-inflation can be realized by the following general Lagrangian[35, 36] P ( φ, X ) = X g ( Y ) , Y ≡ Xe λφ/M pl , (14)where g is an arbitrary function of Y , and λ is a constant. Originally, the Lagrangian (14) was derived for the existenceof scaling solutions in the presence of a barotropic perfect fluid [32, 34]. Under the condition λ < P ,X there existsa power-law inflationary solution for any functions of g ( Y ) [35].For the choice g ( Y ) = 1 − cM /Y , where c is a constant, the Lagrangian (14) reduces to P = X − cM e − λφ/M pl [40], in which case the dynamics of anisotropic inflation was studied in Ref. [29]. The dilatonic ghost condensatemodel P = − X + ce λφ/M pl X /M [32] corresponds to the choice g ( Y ) = − cY /M . If we choose the function g ( Y ) = − ( m /Y ) p − Y /m − M /Y , we recover the DBI Lagrangian P = − h ( φ ) − p − h ( φ ) X + h ( φ ) − − V ( φ )with h ( φ ) − = m e − λφ/M pl and V ( φ ) = ( M + m ) e − λφ/M pl .In the following we study inflationary solutions for the Lagrangian (14) on the anisotropic background given by themetric (2). A. Anisotropic fixed points
For the Lagrangian (14) the field equation of motion (8) reads¨ φ + 3 HA ( Y ) P ,X ( Y ) ˙ φ + λXM pl { − [ g ( Y ) + 2 g ( Y )] A ( Y ) } − f ,φ f ρ A A ( Y ) = 0 , (15)where g n ( Y ) = Y n dg n ( Y ) dY n , P ,X ( Y ) = g ( Y ) + g ( Y ) , A ( Y ) = [ g ( Y ) + 5 g ( Y ) + 2 g ( Y )] − . (16)The quantity A = ( P ,X + 2 XP ,XX ) − is related to the sound speed c s , as c s = P ,X A [32, 41].In order to study the dynamics of anisotropic power-law k-inflation, it is convenient to introduce the followingdimensionless variables x = ˙ φ √ HM pl , x = M pl e − λφ M pl √ H , x = ˙ σH , x = √ ρ A √ HM pl . (17)The variable Y is related to x and x via Y /M = x /x . (18)From Eq. (5) there is the constraint equation x = 1 − x − x ( P ,X + g ) , (19)whereas Eq. (6) gives ˙ HH = − − x − x ( P ,X − g ) . (20)On using Eqs. (7), (15), (19), and (20), we obtain the following autonomous equations x ′ ( N ) = 12 x h x − √ λx + 2 x ( P ,X − g ) i − √ A h √ x P ,X − x ( P ,X + g )( λ + 2 µ ) + 2(1 − x ) µ i , (21) x ′ ( N ) = 12 x h x − √ λx + 2 x ( P ,X − g ) i , (22) x ′ ( N ) = ( x − x −
2) + x [ P ,X ( x − − g ( x + 1)] , (23)where N = ln a , x ′ i ( N ) = dx i ( N ) /dN ( i = 1 , , µ = − M pl f ,φ /f . In the following we focus on the case ofconstant µ , i.e., the coupling f ( φ ) = f e − µφ/M pl , (24)where f is a constant.The fixed points responsible for the cosmic acceleration correspond to non-zero values of x and x . Setting ther.h.s. of Eqs. (21)-(23) to be 0, we obtain the following two fixed points • (i) Isotropic fixed point P ,X ( Y ) = λ √ x , g ( Y ) = 6 − √ λx x , x = 0 , x = 0 . (25) • (ii) Anisotropic fixed point P ,X ( Y ) = ( λ + 2 µ )[2 √ − ( λ + 6 µ ) x ]8 x , g ( Y ) = [2 √ − ( λ + 2 µ ) x ]( √ − λx )8 x ,x = √
64 ( λ + 2 µ ) x − , x = 18 [3( λ + 2 µ ) x − √ √ − λx ) . (26)Provided g ( Y ) is given, the quantities Y and x are known by solving the first two equations of (25) or (26). In theAppendix we discuss more explicit expressions of isotropic and anisotropic solutions corresponding to the fixed points(25) and (26), respectively. For both the isotropic and anisotropic fixed points, the slow-roll parameter is simply givenby ǫ = − ˙ HH = √ λx , (27)where we used Eq. (20). If λx >
0, the power-law inflation a ∝ t / ( √ λx ) is realized. Violation of the condition λx > H >
0. Then, the condition forthe cosmic acceleration with a decreasing Hubble parameter is given by0 < λx < √ . (28)The presence of the anisotropic fixed point (ii) implies that x >
0. This translates to3( λ + 2 µ ) x > √ , (29)where we used (28). Under the condition (29) we also have x > φ , the ghost is absent for P ,X >
0. For the anisotropic fixed point (ii),the condition P ,X > λ + 6 µ ) x < √ , (30)where we employed the fact that, from Eq. (29), the signs of x and λ + 2 µ are the same.From Eqs. (5) and (6) the total energy density and pressure are given by ρ t = 2 XP ,X − P + p A f ( φ ) − e − α − σ / P t = P + p A f ( φ ) − e − α − σ /
6, respectively. Then we have ρ t + P t = 2 H M (3 x P ,X + 2 x ) . (31)If P ,X >
0, then the null energy condition (NEC) ρ t + P t > P ,X <
0. Substituting Eq. (26) into Eq. (31), it follows that ρ t + P t = 2 H M h √ µ + 2 λ ) x − λ + 2 µ ) x / − i . (32)Then, the NEC translates to2 √
69 4 λ + 6 µ − p λ (7 λ + 12 µ )( λ + 2 µ ) < x < √
69 4 λ + 6 µ + p λ (7 λ + 12 µ )( λ + 2 µ ) , (33)whose existence requires that λ (7 λ + 12 µ ) >
0. Let us consider the case where λ > µ >
0. As long as the upperbound of Eq. (33) is larger than the value 2 √ / [3( λ + 2 µ )], there are some values of x consistent with both (29) and(33). This is interpreted as the condition λ + p λ (7 λ + 12 µ ) >
0, which is in fact satisfied for λ > λ → x > √ / (3 µ ), while the region (33) shrinks to the point x = √ / (3 µ ). When λ = 0, Eq. (32) reads ρ t + P t = − H M ( √ − µx ) , (34)which is negative for x > √ / (3 µ ). Notice that, from Eq. (27), the limit λ → H . Hence the NEC is generally violated on the de Sitter solution. The violation of the NEC meansthat the dominant energy condition (DEC; ρ t ≥ | P t | ) as well as the strong energy condition (SEC; ρ t + P t ≥ ρ t + 3 P t ≥
0) are not satisfied [42]. This property is consistent with the Wald’s cosmic no-hair conjecture [43] statingthat, in the presence of an energy-momentum tensor satisfying both DEC and SEC, the anisotropic hair does notsurvive on the de Sitter background.In summary, for λ > µ >
0, the anisotropic fixed points satisfying both P ,X > √ λ + 2 µ ) < x < √ λ + 6 µ , (35)whose upper bound (which comes from P ,X >
0) gives a tighter constraint than that in Eq. (33) (which comes fromthe NEC). As long as λ >
0, there are some allowed values of x which exist in the region (35). Under the condition(35) the anisotropic parameter x = Σ /H is in the range0 < x <
21 + 6 µ/λ , (36)whose upper limit is determined by the ratio µ/λ . For compatibility of the two conditions (28) and (29) we requirethat µ/λ > /
2. Hence the anisotropic parameter is generally constrained to be x < / B. Stability of the anisotropic fixed point
We study the stability of the anisotropic inflationary solution by considering small perturbations δx , δx , and δx about the anisotropic critical point (ii) given by ( x ( c )1 , x ( c )2 , x ( c )3 ), i.e., x i = x ( c ) i + δx i ( i = 1 , , . (37)We expand the function g ( Y ) around Y c = ( x ( c )1 /x ( c )2 ) M , i.e., g ( Y ) = g c + g ′ ( Y c )( Y − Y c ) + g ′′ ( Y c )2 ( Y − Y c ) + · · · , (38)where g c ≡ g ( Y c ) and g ′ ( Y ) = dg ( Y ) /dY . Taking the terms up to the second order of Y − Y c , we have δP ,X =(2 g ′ c + Y c g ′′ c ) δY and δg = ( g ′ c + Y c g ′′ c ) δY . Note that g ′′ c and δY can be expressed as g ′′ c = ( A − − g c − Y g ′ c ) / (2 Y c )and δY /M = 2[ x ( c )1 δx / ( x ( c )2 ) − ( x ( c )1 ) δx / ( x ( c )2 ) ]. In the following we omit the subscripts “ c ” and “( c )” for thebackground quantities.Perturbing Eqs. (21)-(23) around the critical point (ii), we can write the resulting perturbation equations in theform ddN δx δx δx = M δx δx δx , (39)where M is the 3 × x , Y , A , λ , and µ . Using the relations (26), the three eigenvaluesof the matrix M , which determine the stability of the anisotropic point (ii), are γ = √ λx − , γ = √ λx −
32 + 18 √D , γ = √ λx − − √D , (40)where D = 16 h − √ λ + 3 µ ) x i + 3 A ( λ + 2 µ ) h λ + 2 µ ) x − √ i h(cid:0) λ + 28 µλ + 36 µ (cid:1) x − √ λ + 14 µ ) i . (41)As long as the condition (28) of the cosmic acceleration is satisfied, we have that γ <
0. The term √ λx / − / γ and γ is also negative under the same condition. If D is negative, then the anisotropic fixed point is a stablespiral. For positive D the eigenvalue γ is negative. When x = 2 √ / [3( λ + 2 µ )], the eigenvalue γ vanishes for thesame signs of λ and µ . In order to see this more precisely, we substitute x = 2 √ / [3( λ + 2 µ )] + δ into the eigenvalue γ , where δ is a small parameter. It then follows that γ = − √
616 ( λ + 2 µ ) [4 + A ( λ + 2 µ )( λ + 4 µ )] δ + O ( δ ) . (42)Provided that A >
0, we have γ < λ > µ > δ > λ < µ < δ <
0. Then theanisotropic fixed point is stable for 3( λ + 2 µ ) x > √
6, which is exactly equivalent to the condition (29). Plugging x = 2 √ / [3( λ + 2 µ )] + δ into P ,X of Eq. (26), we obtain P ,X = 14 λ ( λ + 2 µ ) − √
632 ( λ + 2 µ ) δ + O ( δ ) , (43)which is positive at x = 2 √ / [3( λ + 2 µ )] for the same signs of λ and µ . If P ,X > A > c s = P ,X A is positive, so that the Laplacian instability of small-scale perturbations can be avoided. For x away from 2 √ / [3( λ + 2 µ )], the quantity P ,X can be negative. In order to avoid this, we require the condition (30).We also note that A can change its sign at some value of x . Since this depends on the forms of the function g ( Y ),we shall study this property in several different models in Sec. IV.We recall that x and x exactly vanish at 3( λ + 2 µ ) x = 2 √
6. In order to keep the small level of anisotropies( x ≪ x ≪ x is only slightly larger than the critical value 2 √ / [3( λ + 2 µ )] for positive λ and µ . In this regime the stability of the anisotropic fixed point is ensured for A > µ λ ε <0.5 ε <0.1 FIG. 1: The parameter space in the ( λ, µ ) plane for the model P = X − cM e − λφ/M pl . The two solid curves, which determinethe minimum values of µ for large and small λ , correspond to the bounds (46) and (47), respectively. The two dotted curvescorrespond to ǫ = 0 . ǫ = 0 .
5. In order to realize ǫ ≪
1, we require that µ/λ ≫ IV. CONCRETE MODELS OF POWER-LAW INFLATION
In this section we study the existence of anisotropic fixed points as well as their stabilities in concrete models ofpower-law inflation. For simplicity we shall focus on the case of the positive values of λ and µ . A. Canonical field with an exponential potential
Let us first consider the model P = X − cM e − λφ/M pl ( c = constant) , (44)i.e., the function g ( Y ) = 1 − cM /Y . Solving the first two equations of (26) for this function, we obtain the followinganisotropic fixed point x = 2 √ λ + 2 µ ) λ + 8 µλ + 12 µ + 8 , cx = 6(2 + 2 µ + µλ )(8 + 12 µ + 4 µλ − λ )( λ + 8 µλ + 12 µ + 8) ,x = 2( λ + 2 µλ − λ + 8 µλ + 12 µ + 8 , x = 3( λ + 2 µλ − µ + 4 µλ − λ )( λ + 8 µλ + 12 µ + 8) , (45)which agree with those derived in Ref. [29]. The upper bound of Eq. (28) translates to8 + 12 µ − µλ − λ > , (46)which is satisfied for µ ≫ λ . The condition (29) for the existence of the anisotropic fixed point is interpreted as λ + 2 µλ − > . (47)Since P ,X = 1 > x is smaller than the upper bound of Eq. (35). In this model the quantity A is 1, so that thestability of the anisotropic inflationary solution is ensured under the condition (47) in the regime where x is not faraway from the value 2 √ / [3( λ + 2 µ )]. Even for x ≫ √ / [3( λ + 2 µ )] the determinant D appearing in γ of Eq. (40)becomes negative and hence the fixed point is a stable spiral. This means that the anisotropic inflationary solution isan attractor under the condition (47) [29].In Fig. 1 we show the viable parameter space in the ( λ, µ ) plane satisfying the two bounds (46) and (47). Thestable anisotropic inflation can be realized for the parameters in the shaded region. We also plot the two curvescorresponding to ǫ = 0 . ǫ = 0 .
5. For λ and µ satisfying the conditions µ ≫ λ and µ ≫
1, we approximately have x ≃ √ / (3 µ ) from Eq. (45) and hence ǫ ≃ λ/µ from Eq. (27). The slow-roll parameter ǫ of the order of 10 − can berealized for µ/λ = O (10 ). If µ/λ = 10 , for example, the condition λ + 2 µλ − > µ = 10 λ > λ →
0, the condition λ + 2 µλ − > ǫ → f ( φ ) in Eq. (24) to give rise to anisotropic solutions. B. Generalized ghost condensate
The second model is the generalized ghost condensate given by the Lagrangian P = − X + cM n pl e nλφ/M pl X n +1 ( c, n = constant with n ≥ , (48)in which case g ( Y ) = − c ( Y /M ) n . The diatonic ghost condensate model [32] corresponds to the case n = 1.From the first two equations of (26) we find that x = −√ λ + 2 µ + (5 λ + 6 µ ) n ] ± p λ + 2 µ + (5 λ + 6 µ ) n ] + 48( n + 1)[2(4 − λ − λµ − µ ) n − λ ( λ + 2 µ )]2[2(4 − λ − λµ − µ ) n − λ ( λ + 2 µ )] . (49)Since the plus sign of Eq. (49) can give positive values of x , we use this solution in the following discussion. Thenthe anisotropic parameter x reads x = 3 λ + 10 µ + ( λ + 6 µ ) n − p (9 λ − λµ − µ + 64) n + 2(3 λ − λµ − µ + 32) n + ( λ − µ ) λ + 2 µ + (5 λ + 6 µ ) n + p (9 λ − λµ − µ + 64) n + 2(3 λ − λµ − µ + 32) n + ( λ − µ ) . (50)The condition (35) translates to µ < µ < µ , where µ ≡ p (2 n + 1) λ + 24 n ( n + 1) − ( n + 2) λ n + 1) , µ ≡ λ + r λ + 96 nn + 1 ! . (51)From the condition (28) the variable µ is bounded to be µ < µ , where µ ≡ (2 n + 1) λ + p n − (11 n + 26 n − λ n . (52)For the determinant of Eq. (49) to be positive, we require that µ < µ , where µ ≡ p n (5 n + 1)( n + 1) λ + 4 n ( n + 1)(15 n + 18 n − − (5 n + 10 n + 1) λ n + 18 n − . (53)In Fig. 2 we plot the parameter space in the ( λ, µ ) plane satisfying the conditions (51)-(53) for n = 1 and n = 4.In the limit that λ →
0, the region described by (51) shrinks to the point µ = p n/ [3( n + 1)]. As we see in Fig. 2,the region (51) tends to be wider for larger λ . The condition (52) gives upper bounds of λ and µ . The intersectionpoint of the curves µ = µ and µ = µ is given by ( λ, µ ) = ( p n/ ( n + 2) , p n/ [2( n + 2)]), whereas the curves µ = µ and µ = µ intersect at the point ( λ, µ ) = ( p n/ [5( n + 1)] , p n/ [6( n + 1)]). For n ≥ λ and µ arein the range 0 < λ < r nn + 2 , r n n + 2) < µ < s n n + 1) . (54)We note that the condition (53) does not provide an additional bound. From Eq. (54) the parameter µ is of the orderof 0 . µ = p / n → ∞ ). µ λµ = µ µ = µ µ = µ µ = µ µ λµ = µ µ = µ µ = µ µ = µ FIG. 2: The parameter space in the ( λ, µ ) plane for the generalized ghost condensate model with n = 1 (left) and n = 4 (right).The four curves correspond to the borders given in Eqs. (51), (52), and (53). In the shaded region, all the conditions (51)-(53)are satisfied. In Fig. 3 we plot the phase space trajectories in the two-dimensional plane ( x , x ) for n = 1, λ = 0 .
35, and µ = 0 . µ is close to the lower bound µ = µ , the anisotropic parameter x = Σ /H is much smallerthan 1. For increasing µ the anisotropy gets larger. In the numerical simulation of Fig. 3 the slow-roll parameter is ǫ = 0 .
521 along the anisotropic attractor. In order to realize ǫ of the order of 10 − , we require that λ = O (10 − ).For µ close to its upper bound, it can happen that the stability of the anisotropic fixed point is subject to change.In fact, the parameter A = 1 / [ c ( n + 1)(2 n + 1)( Y /M ) n −
1] diverges at c ( Y /M ) n = 1 / [( n + 1)(2 n + 1)]. This leadsto the sign change of the determinant (41) from negative to positive by passing the singular point at µ = µ . If n = 1then we have µ = √ λ + 8 / − λ/
6, so the anisotropic fixed point is stable for µ < p λ + 8 / − λ/ . (55)This does not give an additional bound to those given in Eqs. (51)-(53). When n > n = 1. It is worth mentioning that, for n = 1, the condition(55) is equivalent to x < x is reached for ( λ, µ ) = ( p n/ [5( n + 1)] , p n/ [6( n + 1)]). Substituting these values intoEq. (50) we have x = 1 /
3, which corresponds to the upper bound of (36) with µ/λ = 5 /
6. Hence the anisotropicparameter is constrained to be Σ /H < / , (56)which holds independent of n . This bound comes from the combination of the conditions P ,X > λx < √ / ρ t + P t > P ,X >
0, the upper bound (56) gets larger. However, such a largeanisotropy is not accepted observationally.In summary, for n ≥
1, there exist the allowed parameter spaces satisfying all the conditions (51)-(53). In order torealize the sufficient amount of inflation ( ǫ ≪
1) with the suppressed anisotropy ( x ≪ λ ≪ µ is close to the lower bound µ . C. DBI model
The DBI model is characterized by the Lagrangian [33] P = − h ( φ ) − p − h ( φ ) X + h ( φ ) − − V ( φ ) , (57)0 x x FIG. 3: The phase space in the two-dimensional plane ( x , x ) for the dilatonic ghost condensate model with the Lagrangian P = − X + e λφ/M pl X /M . The model parameters are chosen to be λ = 0 .
35 and µ = 0 . x = 1 . x and x . The solutions finally converge to the anisotropic fixed point( x , x ) = (5 . × − , . × − ) with x = 1 . x = 1 . ǫ = 0 . where h ( φ ) and V ( φ ) are functions of φ . For the choice g ( Y ) = − ( m /Y ) p − Y /m − M /Y , where m and M are constants having a dimension of mass, we obtain the Lagrangian (57) with h ( φ ) − = m e − λφ/M pl and V ( φ ) =( M + m ) e − λφ/M pl . The ultra-relativistic regime corresponds to the case where the quantity Y /m is close to 1 / c M ≡ M /m is much largerthan 1 [36].Since P ,X = [1 − h ( φ ) X ] − / > A = (1 − Y /m ) / , so that there is no divergence associated with the determinant (41) in theregime Y /m < /
2. From the first two equations of (26) we find that the anisotropic fixed point satisfies the fourthorder equation of x , but it is not analytically solvable for general values of λ , µ , and c M . However, substituting thelower bound of Eq. (35) into the fourth order equation of x , we obtain the following constraint µ > √ λ + 12 c M λ + 36 − λ λ . (58)In the ultra-relativistic regime the quantity Y /m is close to 1 /
2, so that P ,X = (1 − Y /m ) − / is much larger than1. Using the bound (29), the anisotropic fixed point of Eq. (26) satisfies the relation P ,X < λ ( λ + 2 µ ) /
4, i.e., r − Ym > λ ( λ + 2 µ ) . (59)In order to realize the situation where Y /m is close to 1 /
2, we require that λ ( λ + 2 µ ) ≫
1. As x is away from thevalue 2 √ / [3( λ + 2 µ )], there is a tendency that the anisotropic fixed point deviates from the ultra-relativistic regimebecause of the decrease of P ,X . In the following we focus on the situation where x is close to 2 √ / [3( λ + 2 µ )], inwhich case the anisotropic fixed point is stable with a small anisotropy.For x ≃ √ / [3( λ + 2 µ )], the slow-roll parameter is given by ǫ ≃ λ/ ( λ + 2 µ ) from Eq. (27). In order to realize ǫ ≪
1, we need the condition µ ≫ λ . Then, the condition λ ( λ + 2 µ ) ≫ µλ ≫
1. From Eq. (58) the condition µλ ≫ c M λ ≫
10, in which case Eq. (58) reduces to µ > √ c M /
3. When c M = O (100), for example, we have µ & O (10) and λ & O (1).For compatibility of the two conditions (28) and (29), we require that µ > λ/
2. If λ > λ m ≡ q c M + 2 p c M + 3,the condition µ > λ/ λ < λ m , as λ gets larger around the lower1 x x Y/m FIG. 4: The three-dimensional phase space (
Y /m , x , x ) for the DBI model with the Lagrangian P = − m e − λφ/M pl q − Xe λφ/M pl /m − M e − λφ/M pl . The model parameters are chosen to be c M = M /m = 500, λ = 1,and µ = 26 with the initial condition Y /m = 10 − and several different initial values of x and x . The trajectories withdifferent initial conditions converge to the anisotropic fixed point ( Y /m , x , x ) = (4 . × − , . × − , . × − )with x = 3 . × − and ǫ = 3 . × − . bound of µ given in Eq. (58), the slow-roll parameter also increases and it reaches the value ǫ = 1 at λ = λ m . Then,the realization of anisotropic inflation demands the condition λ < r c M + 2 q c M + 3 . (60)When c M = 500, for example, the condition (60) translates to λ < .
7. As long as λ is much smaller than the upperbound of Eq. (60), anisotropic inflation with ǫ ≪ µ close to the lowerbound of Eq. (58).In Fig. 4 we show the trajectories of solutions in the three-dimensional phase space ( Y /m , x , x ) for c M = 500, λ = 1, and µ = 26. In this case the solutions with several different initial conditions converge to the anisotropic fixedpoint with constant values of x , x satisfying x ≪ x ≪
1. The attractor is in the ultra-relativistic regime(
Y /m close to 1 /
2) with ǫ of the order of 0.01. It is also possible to realize stable anisotropic inflation for λ = O (10)and µ = O (10), but in such cases the slow-roll parameter ǫ is not much smaller than 1.In summary, the stable anisotropic DBI inflation can be realized in the ultra-relativistic regime under the conditions(58) and (60) for µ close to the lower bound (58). V. CONCLUSIONS
We have studied the dynamics of anisotropic power-law k-inflation in the presence of a vector kinetic term F µν F µν coupled to the inflaton field φ . Such a power-law k-inflation can be accommodated for the general Lagrangian P = Xg ( Y ), where Y = Xe λφ/M pl . The cosmological dynamics in the anisotropic cosmological background is knownby solving the autonomous equations (21)-(23).Without specifying the functional forms of g ( Y ), we have shown that anisotropic inflationary solutions exist for theexponential coupling (24). The anisotropic fixed point satisfying Eq. (26) is present for 3( λ + 2 µ ) x > √
6, where2 x = ˙ φ/ ( √ HM pl ). The condition for the cosmic acceleration translates to λx < √ /
3. Provided the conditions3( λ + 2 µ ) x > √ A = ( P ,X + 2 XP ,XX ) − > x is close to 2 √ / [3( λ + 2 µ )]. This property holds irrespective of the forms of g ( Y ) and hence theanisotropic hair survives whenever the anisotropic power-law inflationary solutions are present.The quantity A is related to the sound speed c s as c s = AP ,X , so that the Laplacian instability can be avoidedfor A > P ,X >
0. For the models in which P ,X can be negative, it happens that the NEC ρ t + P t > λ →
0) we found that the NEC is alwaysviolated for anisotropic solutions. This is consistent with the Wald’s cosmic no hair conjecture. As long as λ is not0, there are some parameter spaces in which the NEC is satisfied.In Sec. IV we applied our general results to concrete models of k-inflation such as the generalized ghost condensateand the DBI model. In the generalized ghost condensate we showed that there are allowed parameter spaces in the( λ, µ ) plane where stable anisotropic inflationary solutions with P ,X > A > λ = O (0 .
1) and µ = O (0 .
1) occurs, but if the slow-roll parameter ǫ is of the order of 10 − , it followsthat λ = O (10 − ). In the DBI model there exists stable anisotropic inflationary solutions in the ultra-relativisticregime ( Y /m ≃ /
2) for µ close to the lower bound of Eq. (58) and λ satisfying the bound (60) (see Fig. 4). Themodel parameters are typically of the order of λ = O (1) and µ = O (10) to realize ǫ = O (10 − ).While we focused on the vector field coupled to the inflaton in this paper, we expect that the similar propertyshould also hold for the two-form field models studied in Ref. [26] in the context of potential-driven slow-roll inflation.It is also known that in k-inflation the non-Gaussianities of scalar metric perturbations can be large for the equilateralshape due to the non-linear field self-interactions inside the Hubble radius [44]. It will be of interest to study howthe non-linear estimator f NL of the single-field k-inflation is modified by the interactions between inflaton and thevector/two-form fields. We leave these issues for future work. Acknowledgments
This work is supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science andCulture of Japan (Nos. 23 · Appendix
In this Appendix we provide more explicit analysis for the properties of isotropic and anisotropic solutions given inEqs. (25) and (26).The power-law inflationary solution corresponds to ˙ α = H = ζ/t , where ζ is a constant larger than 1. Sincethe quantities x = ˙ σ/H and x = ˙ φ/ ( √ HM pl ) are constant along the fixed points, we have that ˙ σ = η/t and˙ φ/M pl = ξ/t , respectively, where η = ζx and ξ = √ x ζ . Then, the evolution of α , σ , and φ is characterized by α = ζ log tt , σ = η log tt , φM pl = ξ log tt , (61)where t is a constant. For Y = Xe λφ/M pl to be constant, we need to require λξ = 2 . (62)For the solutions (61) satisfying the relation (62), the dimensionless variables defined in Eq. (17) read x = 2 √ λζ , x = M pl t √ ζ , x = ηζ , x = W ζ , (63)3where W = p A t / ( M f ). Substituting the solutions (61) into Eqs. (5)-(8), we obtain µξ − ζ − η = − , (64) ζ = η + 2 P ,X − g ξ + W , (65) ζ = 3 η + P ,X ξ + W , (66) η = 3 ζη − W , (67) ξ − AP ,X ζξ − λ − AP X ) ξ + λ P ,X − g ) Aξ − µAW = 0 . (68)Notice that Eq. (64) follows from the demand to have the time dependence t − for the last term of Eq. (5). Pluggingthe relation (62) into Eq. (68), it follows that W = 2 λµ [ P ,X (2 − ζ ) − g ] . (69)First, let us seek isotropic solutions. In this case, Eq. (64) is absent and η = W = 0. From Eqs. (66) and (65) weobtain the following relations ζ = 2 P ,X λ , P ,X − λ P ,X + λ g = 0 , (70)respectively. Note that these are consistent with Eq. (69). On using the correspondence (63), we find that the tworelations (70) are equivalent to the first two of Eq. (25).Now, we move on to anisotropic power-law solutions. From Eq. (64) we have ζ + η = 1 / µ/λ . CombiningEqs. (66) and (67), it follows that ζ + η = 3 η ( ζ + η ) + P ,X ξ /
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