Anisotropic inflation in Gauss-Bonnet gravity

We study anisotropic inflation with Gauss-Bonnet correction in presence of a massless vector field. In this scenario, exact anisotropic power-law inflation is realized when the inflaton potential, gauge coupling function and the Gauss-Bonnet coupling are exponential functions. We show that anisotropy becomes proportional to two slow-roll parameters of the theory and hence gets enhanced in presence of quadratic curvature corrections. The stability analysis reveals that anisotropic power-law solutions remain stable over a substantially large parameter region.


Introduction
The inflationary paradigm [1] in modern cosmology is immensely successful in resolving a number of limitations of hot big bang model. These drawbacks involve horizon problem, flatness problem, magnetic monopole problem, interpretations concerning sources of temperature fluctuations in CMB and large scale structure formation in universe [2]. As a result, the inflationary phase, characterized by a period of accelerated expansion is considered as an integral part of Standard model of cosmology. In addition, inflation also predicts a nearly scale invariant primordial power spectrum [3], almost adiabatic and Gaussian perturbations [4,5,6] on CMB as well as statistical isotropy thus confirming the validity of cosmological principle at large scales. Recent cosmological observations of WMAP and Planck [7,8] corroborate statistical isotropy, homogeneity and flatness of the universe however not all observational data comply with predictions, for example, WMAP data [9] indicate scale dependence of power-spectrum and deviations from Gaussian nature of primordial perturbations. These deviations being quite small, are thought of as corrections to zeroth order predictions of inflation. Nevertheless, zeroth order predictions match well with observational data. However, due to availability of high accuracy CMB data and other observational signatures, it is legitimate to look beyond zeroth order results and concentrate on precision cosmology which involves studying fine structures of primordial fluctuations such as spectral tilt, parity violation, low power of quadrupole [10]. Presently, it is known that universe does not exhibit exact de-Sitter nature ( [11] and references therein) and violations of both temporal and spatial de-Sitter symmetries occur. In fact, violation of spatial de-Sitter symmetry leads to statistical anisotropy which has been confirmed observationally by Planck collaboration [6,12]. Motivated by these observations, several theoretical propositions came up to address and realize anisotropy generated during the inflationary phase [13,14,15,16,17,18]. In [19] stable anisotropic inflation solutions are obtained as a consequence of the back reaction of a vector field in the context of supergravity.
The energy scale at which inflation is believed to occur suggests that quantum corrections of gravity must be taken into account. This in turn demands formulation of a more fundamental theory by incorporating the gravity quantization in order to describe Planck scale physics. To date, superstring theory provides the most consistent set-up for quantum gravity involving extra dimensions [20]. However, on the four dimensional world and in the low-energy limit, the reminiscence of the higher dimensional theories like superstring theory appear as corrections terms of higher orders in the curvature to the lowest order gravitational action. The simplest such correction in four dimensions is the Gauss-Bonnet term in the context of heterotic superstring effective theory [21,22]. Furthermore, the Gauss-Bonnet term is the first order correction term of the general extension of the Einstein's gravity commonly known as Lovelock theory [23] which gives rise to ghost-free theories. The Gauss-Bonnet term is topologically invariant in four dimensions and does not modify gravitational equations of motion unless it is non-minimally coupled to a scalar field. Therefore, when the Gauss-Bonnet term is non-minimally coupled to an additional scalar field, it influences dynamical equations in four dimensions. So, studying anisotropic effects during the inflationary phase in presence of Gauss-Bonnet term is therefore well-founded. In [19], anisotropic inflationary solutions are obtained with the help of a massless vector field whose kinetic term is coupled to the inflaton field. We shall follow this approach here. We mention that isotropic inflation with Gauss-Bonnet corrections has been extensively studied in past years [24,25,26,27,28]. Therefore, in the light of precision cosmology, in the present work, we investigate anisotropy effects during inflation in presence of Gauss-Bonnet gravity by coupling the inflaton field non-minimally to the Gauss-Bonnet term. The cosmological anisotropy is generated with the help of vector field which is also coupled to the inflaton. The back-reaction of the vector field contributes non-trivially to the inflation dynamics and exact power-law anisotropic solutions have been obtained when functional forms of the inflaton potential, gauge coupling function and the scalar-Gauss-Bonnet coupling have exponential dependences on the inflaton field. Although small in magnitude, these effects do not get diluted during slow-roll regime. The anisotropic solutions so obtained have been found to be stable. This outline of this paper is as follows. In section 2, we have showed how and under what conditions the anisotropic inflation can be realized in presence of Gauss-Bonnet term by taking the effect of a vector field and its back-reaction. We have derived a general relation where anisotropy is found to be proportional to slow-roll parameters of the theory. After constructing a general set-up for the anisotropic inflation, in section 3 we obtain exact anisotropic power-law solutions in the Gauss-Bonnet gravity scenario. We have estimated the anisotropy measure and compared with the results of non-Gauss-Bonnet case [30]. In section 4, we have performed the stability analysis of obtained anisotropic power-law solutions after determining fixed phase points. Finally, we accumulate all results in the last section.

Anisotropic inflationary solutions with Gauss-Bonnet gravity
In order to generate anisotropic effects during inflation with the help of a vector field in presence of Gauss-Bonnet correction term, we consider a massless vector field whose kinetic term is coupled to the inflaton field φ through the gauge coupling function f (φ) 2 . In this sense, f (φ) 2 may also be called the gauge kinetic function. Since we want to investigate the contribution of the Gauss-Bonnet term, it is therefore non-minimally coupled to the inflaton field through the coupling function ξ(φ). With these considerations, the four-dimensional action is given by where the Gauss-Bonnet term R 2 GB is Here, κ 2 is the four-dimensional gravitational constant, V (φ) is the inflaton potential and F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor. By varying the action (2.1) with respect to g µν , the equation of motion is given by where G µν is the Einstein's tensor and the Gauuss-Bonnet part in the equation of motion is given by The equation of motion of the inflaton field is given by where denotes derivative with respect to φ. It is to be noted that if the Gauss-Bonnet coupling ξ(φ) is constant, then ∇ α ∇ β ξ = 0 which implies that Gauss-Bonnet term adds up nothing to the equation of motion. We mention here that isotropic inflation corresponds to the situation when the gauge field A µ identically vanishes to zero. Without loss of generality, the x−axis is taken in the direction of the vector and for simplicity the direction of the vector field is considered to be time independent. Thus, in general the spatial isotropy is broken but due to our choice of the direction of the vector field the rotational symmetry exists in the y − z plane. With the gauge choice such that the temporal component of electromagnetic potential satisfies A 0 = 0, we can express A µ = (0, v(t), 0, 0) from which the electromagnetic field tensor F µν is constructed. We also assume that the inflaton field is a time-varying quantity i,e. φ = φ(t). Then, in order to find anisotropic scaling solutions in presence of Gauss-Bonnet term, we consider the following Bianchi-I metric [19] where t is the cosmic time. The isotropic scale factor is given by e α and σ specifies deviation from isotropy. The equation of motion of the vector field is given by which can be solved asv where c A is the constant of integration and 'dot' represents derivative with respect to time. Substituting (2.6), (2.8) in (2.3) and (2.5) together with (2.4) and F µν , we obtain gravitational field equations and equation of motion of the inflaton field as followṡ +3ξ (α +σ) α 3 −α 2σ +α(−2σ 2 +α) −σ(α + 2σ) , Let us now define the Hubble's expansion rate as H ≡α, then (2.9) can be expressed as Here, contributions of the vector field and the Gauss-Bonnet term add as effective potential in addition to the inflaton potential V (φ) and influence the dynamics of the inflation. The energy density of the vector field can be defined as 14) The conventional slow-roll inflation continues whenσ is suppressed over H i,e.σ H << 1 and two slow-roll conditions 1 2φ 2 << V (φ),φ << 3Hφ and additionally V (φ) >> ρ v are satisfied.
In presence of scalar coupled Gauss-Bonnet term, we have two additional slow-roll conditions i,e.ξH << 1 andξ <<ξH which hold during inflation. Under these conditions, (2.13) yields the Friedmann equation From these two relations one obtains, Thus for sustaining slow-roll inflation V (φ) >> ρ v . However the energy density of the vector field ρ v may still remain constant and may contribute to anisotropic effects. Under the approximation σ << α, ρ v can remain almost constant if the functional form of f (φ) from (2.14) can be expressed in the following form More generally, we may parametrize f (φ) such that where, c is a constant parameter. Then, in the slow-roll phase, the functional form of f (φ) using (2.17) becomes This shows that the general form of the gauge coupling function is related to both the inflaton potential and the Gauss-Bonnet coupling function and thus ρ v also depends on quadratic curvature corrections through Gauss-Bonnet term. We note that the functional form of f (φ) exactly reduces to the non-Gauss-Bonnet result when ξ(φ) = ξ (φ) = 0 [19]. So long as ρ v is negligibly small, conventional isotropic Gauss-Bonnet inflation is realized. However, as ρ v ∝ e 4(c−1)α , the vector field grows during inflation when c > 1 supported by Gauss-Bonnet corrections. Although ρ v can be neglected in (2.15) compared to inflaton potential, it can contribute non-trivially in the scalar field equation of motion to genetate anisotropy in the c > 1 regime. Using (2.20) we arrive at a condition which suggests that a vector hair can exist during the slow-roll phase of inflation in the Gauss-Bonnet gravity scenario provided any given set of f (φ), V (φ) and ξ(φ) satisfies the condition (2.21). Assuming that anisotropy attains a constant value such thatσ ≈ 0 in the anisotropic phase, the inflaton dynamics can be studied from the the equation of motion of the inflaton field (2.12) which is given bÿ where we have defined Σ ≡σ. Since in general anisotropic effects are small, we neglect higher orders of Σ H and the above equation using (2.20) reduces tö With the expansion of the universe, ρ v also increases with α and can no longer be neglected. From the scalar field equation, we observe that as universe expands, anisotropy is appreciable . If now during anisotropic phase, ρ v increases further i,e. 2c v R >> 1, the inflaton field never rolls down as can be seen from (2.23) and hence ρ v decreases so that inflation resumes again [19]. Therefore ρ v << V (φ) always remains valid such that inflation occurs while the vector field keeps contributing to anisotropy.
As ρ v grows in the slow-roll inflation regime characterized by approximationsφ << 3Hφ anḋ H << H 2 , the inflaton dynamics can be captured from the scalar field equation (2.25) We now integrate the above equation and neglect the variations of V ,V and σ with respect to α, so that where A is the constant of integration. Using (2.20), its derivative and (2.26) in the scalar field equation, we get, At this point, the energy density of the vector field ρ v → 0 thus suggesting vanishingly small contribution to the slow-roll inflation. On the other hand, as α → ∞, the quantity (1 + Ae −4(c−1)α−4σ ) −1 → 1. Therefore we can write, This relation as compared to (2.28) is now modified during the inflationary phase which is now accompanied with small but non-zero effects of anisotropy. We observe that in the modified slow-roll phase (2.29) is 1/c times reduced compared to (2.28). Now the energy density of the vector field can be expressed as, This relation suggests that the Gauss-Bonnet term contributes non-trivially to the energy density of the vector field where c > 1 must be taken. Let us now consider the anisotropy equation given by (2.10). Since during the anisotropic phase, the anisotropy is assumed to attain a constant value, we assumeσ <<ασ andΣ ≈ 0. Then under slow-roll conditionṡ ξH << 1 andξ <<ξH, the anisotropy equation may be written as, Dividing the above equation by 3H 2 , we have, . (2.32) NowḢ << H 2 and since Σ is suppressed over H, we assume that where the extra slow-roll parameter δ H = κ 2ξ H appears only in presence of Gauss-Bonnet correction. Now substituting (2.30) in the above equation, we obtain Let us now substitute (2.9) in the scale-factor equation given by (2.11). Under the approxi- 2 ,Σ ≈ 0,ξ <<ξH and neglecting higher powers of Σ H greater than 1, the scale factor equation becomes, Since −α α 2 = H , therefore the above equation can be written as, Using (2.29) and (2.14), the above expression using (2.15) becomes, Then the measure of anisotropy is given by, As H and δ H must be very small to sustain inflation, therefore up to first order, we have the following relation, Therefore when Gauss-Bonnet corrections are taken into consideration where the Gauss-Bonnet coupling is a function of inflaton field, we find that the anisotropy is proportional to slow-roll parameters of the theory namely H and δ H . We note that in absence of Gauss-Bonnet correction (2.39) reduces to the result obtained in [19].

Power law inflation with Gauss-Bonnet gravity : Exact solutions
After presenting a general set-up for realizing anisotropic inflation with Gauss-Bonnet correction term, we now aim to construct exact anisotropic power-law solutions. These solutions are obtained by taking combined effects of the inflaton potential, the gauge kinetic function and the Gauss-Bonnet coupling function. We assume that inflaton potential is exponentially dependent on the inflaton field such that, Similarly, we also assume that gauge coupling and Gauss-Bonnet coupling are exponential functions of the inflaton field where V 0 , f 0 , ξ 0 , λ, ρ and ω are constant parameters of the theory.

Isotropic inflation
Let us first discuss the isotropic power-law solution in presence of Gauss-Bonnet term for which we assume the following metric where e α(t) is the isotropic scale factor. The isotropic inflation corresponds to no anisotropy and hence inconsequential contributions of the vector field. Substituting the above metric in (2.3)-(2.5), the gravitational field equations and the equation of motion of the inflaton field are as followsα To find power-law isotropic solutions, we consider following ansatz where φ 0 is a constant quantity. The inflaton potential and Gauss-Bonnet coupling functions are V (φ) = V 0 e λκφ , ξ(φ) = ξ 0 e ωκφ . where we have defined u = κ 2 V 0 e κλφ 0 and n = κ 2 ξ 0 e κωφ 0 . We denote n as the Gauss-Bonnet parameter such that n → 0 limit leads to the non-Gauss-Bonnet case. The balance of amplitudes in (3.5) gives, Then substituting u from (3.10) in (3.11) together with (3.9), we obtain an equation for ζ as, This is a cubic polynomial equation of ζ which on solving gives three roots out of which two are imaginary and one is real. The real root of ζ is given by, (3.13) In the leading order of n plus higher order terms in n, the solution of ζ can be expressed as, Then the metric solution for isotropic inflation becomes where ζ is given by (3.14). However, for sufficiently fast power-law inflation we demand ζ >> 1. Then in the leading order of n, the following condition must be true. This condition suggests that n << 1 and can be treated as a correction term if λ << 1. As an estimation, if λ = 0.1 is taken then (3.16) implies n << 10 −4 on the other hand λ >> 1 implies n >> 1, the situation which is not desirable. So it is sufficient to consider terms in ζ up to leading order in n. We further note from (3.14) that in the limit n → 0, ζ = 2 λ 2 is exactly reproduced as obtained in [30] in absence of Gauss-Bonnet corrections.
(3.36) Now (3.31) and (3.32) suggest that for power-law anisotropic inflation to occur, ρ >> 1 while λ << 1 and n << 1. We emphasize here that the positivity of n is a requirement in our A is defined as a squared quantity, (3.35) shows that even a small negative value of Gauss-Bonnet parameter n leads to v < 0 over a large range of λ and ρ.
Since v cannot be negative for any value of λ and ρ, therefore we will always consider n > 0.
The quantity Σ H exactly reduces to the same relation obtained in [30] when n = 0 is substituted in (3.36). Therefore, our analysis gives a scope to compare anisotropy i,e. Σ/H between n = 0 case and the situation when the Gauss-Bonnet term is turned on. The following plot for anisotropy vs n clearly shows that for n > 0, Gauss-Bonnet correction enhances the anisotropy during inflation. From figure 1, we find that as ρ is increased, anisotropy also increases which is showed for three different values of ρ and for different values of n. The slow-roll parameters are given by, Since n > 0 and ζ > 0, δ H being proportional to n is positive in our case, so (2.39) indicates that the anisotropy increases when Gauss-Bonnet correction is included. Then an analogous relation of (2.39) in case of power-law anisotropic solutions can be expressed as which implies, Now using (3.31) and (3.32) we obtain, .
(3.40) For ρ >> 1, λ << 1 and n << 1, c is always positive and greater than 1. With c > 1, the vector field grows during slow roll inflation which is a requirement for anisotropic effects to be non-zero in presence of Gauss-Bonnet corrections. We note here above relation for c reduces to same expression obtained in [30] for non-Gauss-Bonnet case. In the following figure, the anisotropy Σ/H is plotted against α for given values of ξ 0 , V 0 , f 0 λ and c. The quantity Σ/H decreases with α. The nearly constant region ( of the order of 10 −14 ) in the plot corresponds to the isotropic phase. The universe then passes through an anisotropic inflationary phase when anisotropy becomes maximum and finally anisotropy decays down. The plot is obtained numerically using (2.

Stability analysis of anisotropic power-law solutions
We will now determine fixed phase points, then stability of inflationary solutions will be examined around these fixed points. Here, the e-folding number α is taken as the time coordinate so that dα =αdt. The equations of motion can be expressed in terms of dimensionless quantities which are defined as follows Using (2.8), (2.9) is expressed as, Since we assume a positive inflaton potential i,e. V (φ) > 0 therefore we have and we define N 2 = κ 2ξ . For Gauss-Bonnet power-law solutions, using (3.18), (3.20), we get N 1 = 2nζ and N 2 = 2n. Therefore, in absence of Gauss-Bonnet correction, N 1 and N 2 vanish and (2.10)-(2.12) reduce to those for non-Gauss-Bonnet case [30].

Determination of phase points
The fixed phase point corresponds to that point which does not evolve with time. Therefore in the given set-up fixed points in the phase space are determined by solving

Isotropic phase points
Since isotropy implies η = 0 therefore we have X = 0. Substitution of X = 0 and N 1 = N 2 ζ in (4.4) leads to Z = 0. Now with X = 0 = Z, N 1 = N 2 ζ = 2nζ and using conditions (4.7), (4.5) yields It is a cubic equation which gives three roots of Y . Since the inflaton potential V (φ) > 0, therefore the root of Y which satisfies positivity of V (φ) is taken and other two roots are discarded. The particular root of Y satisfying the condition (4.3) is given by where p 1 = −λ 2 − 6ζn − 6n − 18 and p 2 = 2λ(−λ 2 + 54 + 81ζ 2 n + 72ζn − 9n). The complete solution of Y is obtained by substituting ζ i,e. (3.13) in (4.9). Now, Y can be expressed in the leading order of n plus higher order terms in n as The phase point Y exactly reduces to Y = −λ when n = 0 which is the non Gauss-Bonnet counterpart of Y obtained in [30]. We mention here that other two roots of Y by solving (4.8) reproduces Y = ± √ 6 in n → 0 limit but we will not consider them here as these solutions do not satisfy (4.3). So the isotropic fixed point is given by (0, Y, 0). It may be seen that this isotropic fixed phase point corresponds to isotropic power-law inflation given by (3.9) and (3.13). In Figure. 2 we have plotted the variation of Y with λ for four different values of n to compare the deviations in the phase point Y due to Gauss-Bonnet corrections. The deviations are small but particularly appreciable for lower λ.
Since anisotropic power-law inflation requires n << 1, λ << 1 and ρ >> 1, the roots of X can be computed for given values of n, λ, ρ satisfying these conditions. Then using roots of X determined from (4.16) in (4.14) and (4.15) Y and Z 2 are evaluated. Here, we have taken ρ = 40 and λ = 0.1 for our numerical calculations. Figure 3(a) is the plot of (4.16) vs X in which the point of intersection on the X-axis for a particular value of n gives the root of X for that value of n which can be seen from Figure 3a. When n = 0, for ρ = 40 and λ = 0.1, (4.16) X = 4.1 × 10 −4 . Slowly the Gauss-Bonnet correction is turned on and n is increased from zero. It is found that deviation of X occurs from n = 0 to n = 10 −5 where X increases by an order of 10 as compared to its value at n = 0. We mention here that the parameter n due to Gauss-Bonnet correction is restricted up to the value of 10 −5 (where X = 17.3 × 10 −4 ) beyond which X shifts completely from n = 0 result so that Gauss-Bonnet gravity cannot be treated as correction. The phase points X as plotted in Figure 3(b) are obtained as roots of (4.16). Using these values of X, Y and Z can be determined using (4.14) and (4.15) respectively. These values of anisotropic fixed phase points Y and Z 2 for different values of n with ρ = 40 and λ = 0.1 are shown in Figure 4 and Figure 5 respectively. These plots present the corresponding deviations of Y and Z 2 from n = 0 results. We mention here that relations of X, Y and Z 2 reduce to non-Gauss-Bonnet results obtained in [30].

Stability analysis
We now investigate the linear stability of isotropic and anisotropic fixed phase points in order to investigate the impact of Gauss-Bonnet term on the stability of the obtained solutions. The linearized equations necessary for the stability analysis are determined from (4.4)-(4.6) and are given by δX − n (X + 1) 2 + 2X(X + 1) δX, (4.19)

Anisotropic case
In order to examine stability of anisotropic inflationary solutions, we again concentrate on linearized equations. Substituting ζ and η from (3.31) and (3.32) in (4.17)- (4.19) and in the leading order in n, we obtain n 6ρ 6X(4X + 3)ZδX + (X + 1) 8X 2 + X − 1 δZ λ + 48X(4X + 3)ZδX + 8(X + 1) 8X 2 + X − 1 δZ λ(λ + 2ρ) (4.24) Now stability of anisotropic solutions are studied from (4.22),(4.23) and (4.24). Since Gauss-Bonnet gravity describes higher order corrections so leading order terms in n give most dominant contribution compared to sub-leading orders which we have also observed earlier from our study. As ρ >> 1 and λ << 1 are required conditions for inflation, we have determined phase points for ρ = 40, λ = 0.1 in the range n = 0 to n = 10 −5 using (4.16), (4.14) and (4.15) respectively. The evolution of the phase points with n are then plotted in Fig. 3(b), Fig. 4 and Fig. 5 respectively. Using these phase points, we shall investigate the stability of anisotropic power-law solution. For our analysis, we again take ρ = 40 and λ = 0.1. We will consider different values of n in the increasing order of magnitude and solve Starting from a small value i,e. n = 10 −7 , when n is increased up to n = 10 −5 , we observe that anisotropic power-law solutions have stable fixed points (as Ω has negative eigenvalues). However, our study suggests anisotropic power-law solutions will possess stable fixed points with further increase in n provided ρ >> 1, λ << 1, n << 1 and (λ 2 + 2λρ) 1 + 2n λ 4 (2 + λ 2 ) > 4 hold. Since Gauss-Bonnet term is treated as higher order corrections of curvature in our analysis, it is not desirable to increase n to a large value which may drift away stable anisotropic fixed points from corresponding non-Gauss-Bonnet (n = 0) fixed point solutions. For example, in Section 4.1.2, it was found that with ρ = 40, λ = 0.1, the anisotropic point X increases by an order of 10 for n = 10 −5 compared to X evaluated at n = 0. 1 This consideration led us to restrict our study till n = 10 −5 .

Concluding remarks
In the present work, we have studied anisotropic inflation in the backdrop of quadratic curvature corrections. Anisotropic inflationary solutions are obtained in presence of Gauss-Bonnet correction term and a massless U (1) gauge field whose kinetic part is coupled to the inflaton field. The vector field plays a significant role in generating anisotropic effects during slow-roll inflation. To bring about non-zero contribution of the Gauss-Bonnet term, it is non-minimally coupled to the inflaton field. In this scenario, we have showed that exact anisotropic power-law inflationary solutions can be constructed when the inflaton potential, gauge kinetic function and the Gauss-Bonnet coupling are exponential functions of the inflaton field. In presence of Gauss-Bonnet corrections, we have derived a general relation which shows that anisotropy is proportional to slow-roll parameters of the theory namely H and δ H as a consequence, anisotropy gets enhanced as compared to the non-Gauss-Bonnet case. We have performed stability analysis and showed that obtained solutions are stable. Thus our study shows that if Gauss-Bonnet corrections are taken into account, vector hair may persist giving rise to enhanced anisotropic effects during slow-roll regime. This suggests that observational signatures of anisotropic Gauss-Bonnet inflation is worth studying.