Anisotropic Constant-roll Inflation

We study constant-roll inflation in the presence of a gauge field coupled to an inflaton. By imposing the constant anisotropy condition, we find new exact anisotropic constant-roll inflationary solutions which include anisotropic power-law inflation as a special case. We also numerically show that the new anisotropic solutions are attractors in the phase space.


I. Introduction
It is widely believed that the accelerated expansion of the universe in the early universe explains cosmological observations such as the large scale structure of the universe and the temperature anisotropy of the cosmic microwave background radiation. Such rapid expansion is called inflation [1]- [4] which is driven by an inflaton field φ. Conventionally, we impose slow roll conditions, namely −Ḣ H 2 ,φ Hφ 1, where H is the Hubble parameter and an overdot denotes a derivative with respect to the cosmic time. This is required to realize a sufficiently long (∼ 60 e-foldings) quasi de Sitter phase and to make the predictions consistent with the observational data.
Recently, constant-roll inflation characterized by a constant-roll condition,φ Hφ = const., instead of the slow roll conditions has been discussed. Under the constant-roll condition, one can solve equations exactly [5]. Thus, one can systematically find a certain class of constant-roll inflationary models which are compatible with the observational data [5]- [7].
Moreover, one can extend an idea of constant-roll inflation to teleparallel f (T ) gravity where the inflaton is coupled to the f (T ) [8]. The generalization of the constant-roll condition to f (R) gravity has been done [9,10]. Since f (R) gravity models can be related to scalar-tensor theory, scalar fields play the central role in all of these models.
In this paper, we study constant-roll inflationary models in the presence of a gauge field coupled to an inflaton. Now, we can expect anisotrpy of the expansion of the universe.
Indeed, anisotropically expansing inflationary models have been studied in the context of the supergravity and called anisotropic inflation [12]. There are various extensions of anisotropic inflationary models [13]- [40] and various predictions for the observations [41]- [55]. Here, we provide a new way to investigate anisotropic inflation. Remarkably, we find exact anisotropic constant-roll inflationary solutions which would be attractors in the phase space.
In section II, we present inflationary models we consider in the present paper. In section III, we study the constant-roll condition in anisotropic inflation. In section IV, we show specific examples to illustrate how to solve the system. In section V, we investigate the phase space dynamics to show that anisotropic solutions are attractors in the phase space.
The final section is devoted to summary.

II. Inflation with a gauge field
We consider an inflationary universe with a gauge field. The action is given by where g is the determinant of the metric g µν , M pl is the reduced Plank mass, R is the Ricci scalar and the inflaton field is represented by φ with a potential V (φ). The field strength of the gauge field F µν is defined by a gauge potential A µ as F µν = ∂ µ A ν − ∂ ν A µ and it couples to the inflaton field through a coupling function f (φ).
For the gauge field, we take an homogeneous ansatz A µ = (0, v A (t), 0, 0) where the specific direction is along with the x-direction. In the presence of the non-trivial gauge field, we can not take an isotropic metric. Instead, we take the anisotropic ansatz where a(t) describes the average expansion of the universe and b(t) represents the anisotropic expansion of the universe. Now, we can derive equations of motion. It is easy to solve the equation for the gauge field asv where p A is an integration constant. With Eq.(3), one can obtain the Hamiltonian constraint where H a ≡˙a a , H b ≡˙b b . The rest of Einstein equations arė The field equation for the inflaton is which is parameterized by a constant α. On top of the constant-roll condition, it is reasonable to seek inflationary solutions with the constant anisotropy condition where n is a constant. From Eqs. (5), (6) and (9), we obtain Note that this reduces to the conventional one in the isotropic case n = 0. Since one can regard H a as a function of φ, Eq.(10) can be rewritten as This can be solved with respect toφ aṡ Differentiating it with respect to t and using (8), we get Here, let us recall the case of an isotropic universe (n = 0). In this case, Eq. (13) is One can immediately solve Eq. (14) as This is the general solution of isotropic constant-roll (inflation) [5]. ansatz Substituting this ansatz (16) into Eq. (13), we obtain the following algebratic equations In the case of C 1 = 0 or C 2 = 0, they correspond to anisotropic power-law inflation [15,16]. Moreover, n = 0 means isotropic constant-roll inflation and α = −3 represents the de Sitter universe. Thus, interesting and new solutions correspond to the case n = 3+α 6 . This tells us that we can reach the de Sitter limit (α = −3) by taking an isotropic limit n → 0.
It is consistent with Wald's cosmic no-hair theorem [57], roughly speaking, which claims an initially homogeneous and anisotropic spacetime approaches the de Sitter spacetime in the presence of a positive cosmological constant. It implies that any fields can not survive and the spacetime becomes isotropic eventually. In contrast, anisotropic inflation can be realized in slow-roll inflation because it deviates from an exact de Sitter spacetime. The relation n = 3+α 6 means that anisotropy is related to the deviation from a de Sitter universe, this is a feature of anisotropic inflation [12].
Hereafter in this paper, we focus on the non-trivial case which determines the degree of the anisotropy of the spacetime. In this case, Eq.(17) yields and Eq. (16) becomes Since we have obtained H as a function of φ, the remaining equations are completely integrable.

IV. Specific examples
Let us illustrate how to solve the remaining equations. We consider a special cases of (20) as C 1 = C 2 = M/2, then From Eq.(12), one can expressφ by φ By solving Eq. (22), we can express φ as a function of t. Then the scale factor a(φ) is immediately found from the definition H =˙a a as a ∝ 1 + sin The potential V (φ) and the coupling function f (φ) are obtained from Eqs. (4) and (6) as and where we assumed −9 < α < −3 . Taking a look at Eq.(12), we see that there are apparently two solutions for a given Hubble parameter H a . However, since the two branches stemming from Eq.(12) are related by the field redefinition of the scalar field, there is only one solution.
Taking the parameter close to that for the slow roll inflation α −3, we can realize slowroll inflation with anisotropic expansion in principle. The anisotropic expansion becomes a prolate type because n is negative. However, for inflation to occur, we have to require V (φ) to be positive, which makes the gauge field a ghost, namely f (φ) becomes imaginary.
Interestingly, in this case, we can numerically confirm that this anisotropic constant-roll solution can not be an attractor.
Next, we consider another example C 1 = M/2, C 2 = −M/2, namely, The solution (26) gives rise tȯ and a ∝ ∓ 1 + cosh Thus, we can determine the following functional forms and where we assumed −3 < α. Again, inflation can occur taking the parameter close to that of the slow roll inflation α −3. In this case, anisotropic expansion has an oblate type because n is positive. It turns out that these are viable models. In Fig.1, we plotted a trajectory in the phase space of the inflaton. From Fig.2, we can see the anisotropy converges to a constant value n. In Fig.3, we see that the rateφ Haφ and the anisotropy n = H b Ha become constant after about e-folding number 10 in the phase space. This is certainly the anisotropic constant-roll inflationary solution. Indeed, the trajectory Thus, from Figs.1∼3, it turned out that the solution is an attractor. We also confirmed the behavior is independent of initial conditions. Note that the gauge field suffers from a strong coupling problem (f (φ) < 1) around φ = 0. Thus, inflation needs to finish before φ approaches the origin. On the other hand, in the case that we choose the lower sign in Eq. (29), there is no strong coupling problem. Instead, when α −3, the potential is negative around φ = 0 and then inflation needs to finish before φ approachs the origin even for this case. Otherwise, the solution is useful. Indeed, we confirmed that this anisotropic constant-roll inflationary solution is also an attractor.

VI. Summary
We studied constant-roll inflation in the presence of a gauge field coupled to an inflaton.
We imposed the constant-roll condition,φ Hφ = const., and the constant anisotropy condition.
Then, we looked for anisotropic inflationary solutions. Remarkably, we found exact solutions which describe anisotropic constant-roll inflation. Our result includes anisotropic power-law inflation [15], [16] as a special case. We also numerically demonstrated that our new solutions are attractors in the phase space.
Although we have considered anisotropic inflation with a gauge field, our method to find exact solutions can be easily extended to the anisotropic inflationary models with a twoform field [16]- [18]. In the case that both of a gauge field and a two-form field coexist, it is known that there appears an exact solution where both fields survive and isotropic universe is realized for a particular parameter set [16]. This is because a gauge field and a two-form field produce opposite anisotropy. It is also interesting to apply our method to anisotropic inflation with multi-gauge field (two-form field) models [19], [20]. We leave these issues for future work.