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Effective Field Theory of Anisotropic Inflation and Beyond

We develop an effective-field-theory (EFT) framework for inflation with various symmetry breaking pattern. As a prototype, we formulate anisotropic inflation from the perspective of EFT and construct an effective action of the Nambu-Goldstone bosons for the broken time translation and rotation symmetries. We also calculate the statistical anisotropy in the scalar two-point correlation function for concise examples of the effective action.


I. INTRODUCTION
Cosmic inflation, which accounts for the origin of the large-scale structure of the universe, is strongly supported by observations of cosmic microwave background (CMB) [1][2][3][4][5]. As is often referred to as the cosmic no-hair conjecture [6], any classical memory is washed out by the exponential de Sitter expansion of the universe, hence primordial quantum fluctuations are responsible for the large-scale structure. Remarkably, the nature of the primordial fluctuations is understood by symmetries in inflation which can be summarized as follows: • Spatial translation and rotation symmetry The cosmic no-hair conjecture states that the universe will be homogenized and isotropized in a few Hubble expansion times once the inflationary epoch starts. The universe then enjoys the spatial translation and rotation symmetries. In particular, the universe is led to de Sitter space if the vacuum energy is dominant.
• de Sitter dilatation symmetry The metric for de Sitter space reads where the Hubble parameter H is constant. It is easy to find the dilatation symmetry: where ǫ being a constant parameter. Also, de Sitter space accommodates the special conformal symmetry accompanied by an appropriate transformation of the time coordinate.
• Shift symmetry of the inflaton field In addition to the above spacetime symmetries, there exists a symmetry in the field space for a single-field inflation with a canonical kinetic term. In order to have slow-roll inflation, we need a sufficiently flat potential, hence the inflaton field φ(x) is required to enjoy a shift symmetry, φ(x) → φ(x) + const.
The above symmetries can be promoted to statistical symmetries which determine the nature of primordial fluctuations. In general, we need n-point correlation functions to characterize the statistical nature of primordial fluctuations. For simplicity, we assume a single-field inflation with a standard kinetic term, which implies the shift symmetry in the field space. Then, the shift symmetry forbids nonlinearity, and hence the statistics of fluctuations should be Gaussian. *1 Thus, two-point functions carry all the statistical information.
Let us next implement the de Sitter space symmetry. For an illustration, we consider the curvature perturbation ζ. Because of the spatial translation symmetry, we can work in the Fourier space. Then, we have the power spectrum: where k and k ′ are the wavenumber vectors and k = |k|. The delta function stems from "momentum" conservation. Note that the dipole-type anisotropy is prohibited because where we have assumed that the modes are outside of the horizon so that ζ(k) and ζ(−k) commute with each other. The rotation symmetry further constrains the power spectrum as Namely, the direction dependence of the power spectrum is forbidden by the rotation symmetry. Finally, the de Sitter dilatation symmetry implies a scale-invariant power spectrum, because the curvature perturbation is conserved on large scales. Thus, the statistically homogeneous, isotropic, and scale-invariant Gaussian primordial curvature fluctuations are a consequence of the de Sitter symmetry. These predictions are robust and universal in conventional single-field inflationary scenarios. In fact, the above predictions have been almost confirmed by CMB observations [5]. However, when we look at the fine structure of the CMB data, we need to elaborate on the theoretical details. In fact, the cosmic expansion is not exactly de Sitter but quasi-de Sitter. Namely, there exists violation of the de Sitter dilatation symmetry due to the time-dependence of the inflaton background φ (x) =φ(t) = 0, which leads to a slight tilt of the power spectrum. Here, the bar denotes an expectation value. As the deviation from the de Sitter expansion can be characterized by the slow-roll parameter, the tilt should be of the order of the slow-roll parameter. The deviation from the de Sitter expansion also allows the model to have nonlinearity of the order of the slowroll parameter, namely, non-Gaussianity [7]. Moreover, if we consider multi-field inflation or a non-standard kinetic term, we can have nonlinearity, which leads to a sizable non-Gaussianity [8,9]. Remarkably, these fine structures of the primordial fluctuations have been observationally tested. The spectral tilt has been well confirmed [5] and non-Gaussianity has been strongly constrained [10].
Along this line of thought, it is natural to expect violation of the spatial rotation symmetry, which would lead to the statistical anisotropy [11,12]. Intriguingly, WMAP provided some hints of the statistical anisotropy [13][14][15][16]. Hence, there appeared serious studies of quadrupole anisotropy [17][18][19]. Indeed, significant quadratic anisotropy in the power spectrum was reported in [20,21]. Motivated by these studies, many theoretical mechanisms for realizing the statistical anisotropy have been proposed [22][23][24][25]. These attempts were challenges to the cosmic no-hair conjecture. A stable anisotropic inflation model was found in the context of supergravity [26], which gives rise to a clear counterexample to the cosmic no-hair conjecture [27]. There can be various extension of anisotropic inflation [28,29]. In these models, the statistical anisotropy appears in the form [30][31][32] P (k) = P (k) 1 + g * sin 2 θ , where P (k) is the isotropic part of the power spectrum P (k) and θ is the angle between the preferred direction n and the wavenumber vector k of fluctuations. Here, g * characterizes the magnitude of the statistical anisotropy. In fact, the way to make a preferred direction is not unique. For example, anisotropic inflation with two-form filed is possible [32][33][34]. The preferred direction n specified by a two-form field can be translated to the linear inhomogeneity of the scalar field through the dual transformation. This is nothing but the anisotropy in solid inflation [35,36]. A useful scheme for studying such models in a unified manner is the effective field theory (EFT) approach [37]. Actually, a direct application of EFT method to anisotropic inflation has been already performed [38,39]. As the theoretical investigation progressed, it turned out that the observed quadrapole anisotropy can be attributed to the anisotropy of the beam [40]. Now, we only have an upper bound g * = 0.002 ± 0.016 (68% CL) [41]. Of course, this does not mean the statistical anisotropy does not exist. Rather, the anisotropic inflation opens up a new direction of research. Therefore, it is worthy of study in detail. So far, we have not discussed the violation of spatial translation symmetry. However, since the Planck observations give a hint of the dipole anisotropy [42,43], we have a good motivation to consider this possibility. We can also consider the violation of Lorentz boost symmetry [44]. Apparently, we need a broader framework to incorporate these possibilities. The EFT approach [45] based on the spacetime symmetry viewpoint should be useful for this purpose. Indeed, the effective theory can accommodate more exotic models such as anisotropic k-inflation where the kinetic term is non-standard [46]. The aim of this paper is to develop an EFT formalism of anisotropic inflation from the perspective of general spacetime symmetry breaking. The formalism allows us to go beyond anisotropic inflation.
In this article, we illustrate our idea by formulating the EFT of anisotropic inflation. Following [45], we introduce general local Lorentz frames to describe spontaneous breaking of spacetime symmetries and construct EFTs in the unitary gauge. Then, we introduce three Nambu-Goldstone (NG) bosons (which will be called η 0 , η 1 , and η 2 ) for the broken spatial rotation symmetry on top of the one (called π) for the broken time translation symmetry. It should be noted that one of the NG modes corresponding to the broken rotation symmetry (i.e., η 0 ) is nondynamical for the conventional model of anisotropic inflation having the standard Maxwell kinetic term. Our EFT framework covers not only such models with nondynamical η 0 but also those with dynamical η 0 .
The rest of this article is organized as follows. In §II, we clarify the symmetry breaking patterns and specify the associated NG modes for a concrete model of anisotropic inflation. We also explain the unitary gauge conditions, under which all the NG modes are eaten by the vierbein and the gauge field. In §III, we construct the effective action in the unitary gauge and discuss the tadpole cancellation conditions. In §IV, we compute the scalar two-point correlation function based on the so-obtained EFT framework. Finally, we draw our conclusions in §V.

II. SYMMETRY STRUCTURES IN A CONCRETE MODEL
As was shown in [45], each broken local symmetry has one associated dynamical mode, which gives a correct identification of NG bosons with broken symmetries. In this section, we reformulate an anisotropic inflationary model based on such a local spacetime symmetry point of view and give a flavor of an EFT framework for anisotropic inflation and beyond.
Let us consider the following conventional action for a neutral scalar φ and an Abelian gauge field A µ : where M Pl is the reduced Planck mass and F µν := ∂ µ A ν − ∂ ν A µ is the field strength of the gauge field. If we introduce the vierbein e m µ and rewrite the gauge field as we can write down a local-Lorentz-invariant action.
Here and in what follows, we use Greek letters for the curvedspacetime indices (µ, ν, · · · = 0, 1, 2, 3) and Latin letters for the (local-)Minkowski indices (m, n, · · · =0,1,2,3). Let us now assume the spatially homogeneous anisotropic background: Here, without loss of generality we have given the anisotropy along the z-direction. Then, the background quantities are The independent components of the Einstein equation are given by The equation of motion for the vector field is while the one for the scalar field is automatically satisfied for the solution of (12)- (15). We now discuss perturbations around the background, emphasizing on its symmetry breaking pattern and the associated NG modes. Since the background (11) breaks the time translation symmetry and the rotation symmetry associated with the z-direction, we have two types of NG modes. To identify those degrees of freedom, it is convenient to decompose φ and A µ as Here, one linear combination of π and χ is the NG mode for the broken time translation. Since the U (1) symmetry is not spontaneously broken and hence the would-be associated NG mode is not eaten by a gauge field, the orthogonal linear combination of π and χ is not the NG mode but a gauge degree of freedom. Three NG modes, η m ( m = 0, 1, 2), for the broken rotation symmetry are encoded in Λ m n ∈ SO(3, 1) as A caveat should be added here. One could attempt to describe fluctuations of the gauge field A µ , e.g., as an "NG boson" for spatial diffeomorphism, but it turns out not so useful as we now explain. Suppose that the third component of the vector field A µ has a time-dependent background as Analogously to the NG boson for the spatial diffeomorphism in the third direction, one would introduce a scalar field χ as through a derivative. First of all, we cannot describe the four components of the vector fluctuations in terms of a single scalar χ. Also, the constant mode of χ does not generate any fluctuation of δA µ , essentially because the translation symmetry along the third direction is not broken. This is in sharp contrast to the standard NG bosons and indicates a wrong identification of NG bosons. On the other hand, if we introduce NG bosons for local Lorentz symmetries as above, the NG bosons provide appropriate degrees of freedom and also they are identified with vector fluctuations without involving derivatives. See, e.g., [45] for a more detailed discussion. We next introduce the action in the unitary gauge, where all the NG modes are eaten by the gauge field and the vierbein. We fix the gauge degrees of freedom for the time diffeomorphism and three of the six local Lorentz symmetries by π = 0 and η m = 0, respectively. Regarding χ, one can use the U (1) gauge degree of freedom to set χ = 0. In this gauge, the action reduces to the form: Note that the residual gauge symmetries in our gauge choice are spatial diffeomorphism symmetries and three local Lorentz symmetries along the0,1,2 directions, which are manifest in the above action. It is useful to introduce the quantity which is covariant under the unbroken symmetries and vanishes on the background. Here, the timelike unit vector n µ is defined by With this D µν and δg 00 = g 00 + 1, the total action can be recast as with c, Λ, and λ being Also, O(δ n ) denotes nth and higher-order terms in fluctuations. In (24), we have suppressed terms cubic or higher order in fluctuations. Thus, we have reformulated the original model of anisotropic inflation [26]. In the next section, we construct general EFTs of anisotropic inflation.

III. EFFECTIVE ACTION IN THE UNITARY GAUGE
As we illustrated in the previous section, the field contents and the residual gauge symmetries in the unitary gauge for our problem are e m µ , matters + spatial diffs , (2+1)-dim local Lorentz , where the matters are non-NG fields, i.e. the degrees of freedom which cannot be absorbed by gauge transformations. We then construct the effective action for the NG fields without matters. For simplicity, we impose the invariance under the Z 2 transformation e3 µ → −e3 µ as well. Under these assumptions, it is shown in Appendix A that the general unitary-gauge action up to the linear order in perturbations takes the form where Λ, c, and λ are scalar functions of time. Also, we took the Einstein frame to fix the coefficient of the Ricci scalar. Note that terms with more derivatives (e.g. those including the extrinsic curvature K µν ) can be converted to higher-order terms up to total derivatives (see [37] for details in the case of single-field inflation). For a generic choice of EFT parameters (Λ, c, λ), the action (29) contains tadpole terms. Requiring that the background satisfies the equation of motion, we remove them by the following tadpole cancellation conditions: which enables us to write down the EFT parameters in terms of background geometry. For the detail of the derivation, see Appendix A. Let us then incorporate higher-order terms in the perturbations. Since we are interested in the primordial twopoint functions which will be discussed in the next section, we focus on the second-order action, even though it is straightforward to generalize the analysis to higher-order terms, e.g., relevant to non-Gaussianities. We also focus on the leading order in derivatives where each NG mode carries at most one derivative. Under these assumptions, the general action in the unitary gauge up to the quadratic order in perturbations can be written as where the lowest-order EFT parameters (Λ, c, λ) follow from the tadpole cancellation conditions (30)- (32). Note in passing that we have omitted higher derivative terms with δK µν or δR µνλσ in (33). When taking into account such terms, one must be careful as the extrinsic curvature and the Riemann tensor now contain anisotropic contributions. Their background values,K µ ν andR µν λσ , are written as follows *2 : where h µν := g µν + n µ n ν is the induced metric on constant time slices. Their fluctuations in the unitary gauge are then defined as δK µ ν = K µ ν −K µ ν and δR µν λσ = δR µν λσ −R µν λσ in a covariant manner to respect the residual gauge symmetries in the unitary gauge.
In this section, we have succeeded in formulating EFTs with broken time translation and rotation symmetries. It should be stressed that the procedure performed in this section can be generalized to other inflationary models with various symmetry breaking patterns.

IV. STATISTICAL ANISOTROPY
In this section, we study the statistical anisotropy based on the effective action obtained in the previous section. To this end, we reintroduce the NG bosons π and η m via the Stückelberg trick, and then construct the quadratic action for the NG bosons under the decoupling limit. We first discuss the conventional model considered in §II, where η 0 is nondynamical. Then, we investigate the case with dynamical η 0 , taking a simple example.

A. Nondynamical η0
Let us focus on the simple model discussed in §II, for which we have the following effective action in the unitary gauge: Here, the parameters are written as Having introduced the action in the unitary gauge, now we recover the NG bosons (π and η m for the broken time diffeomorphism and the local rotation symmetry, respectively) via the Stückelberg trick. Note that χ is not the NG *2 Note that the background value of the Weyl tensor is calculated as , which vanishes in the isotropic case. Its fluctuation, δC µν λσ = δC µν λσ −C µν λσ , will also be useful in the EFT context because it describes propagating graviton modes which cannot be eliminated by field redefinition or equivalently by using the Einstein equation. mode and we do not need to recover the U (1) symmetry, which is not spontaneously broken. This can be achieved by the following replacements: It should be noted that η 0 is nondynamical in the present case with m 2 1 = m 2 2 . We study the case with dynamical η 0 in §IV B. In the decoupling limit where the metric fluctuations are neglected, the quadratic action for the NG bosons takes the following form: where the indices A and B run over 1, 2 and a prime denotes a derivative with respect to the conformal time τ . One easily observes that there is no kinetic term of η 0 and that it is really nondynamical. Note also that here we have used the slow-roll approximation and taken the small-anisotropy limit. The coefficients are given by Inserting our particular choices of the parameters given in (37)-(43) yields Here, we have moved to the Fourier space and taken the wavenumber vector to be k = (k 1 , 0, k 3 ), which does not lose generality thanks to the symmetry in the x-y plane. Note also that we have omitted terms that are higher order in ǫ H and I, where ǫ H := −Ḣ/H 2 is the Hubble slow-roll parameter and I := 3σ/(ǫ H H) characterizes the anisotropy. We require ǫ H > 0 and I > 0 so that the kinetic terms of π and η A have a correct sign. We summarize some useful formulae in the slow-roll and small-anisotropy approximations in Appendix B. The mass terms for π and η A have been neglected because they are respectively proportional to ǫ 2 H and I 2 ǫ 2 H . Since η 0 is nondynamical, we can eliminate it by use of its Euler-Lagrange equation. Before doing so, we perform a gauge transformation δA µ → δA µ + ∂ µ α such that δA 3 = 0 in the new gauge. This can be achieved by choosing the Fourier component of α as Accordingly, η 0 and η 1 are transformed respectively as (58) Note that η 2 is unchanged due to our choice of the wavenumber vector. Then, S mix changes as while the other parts of the action remain unchanged *3 . Now, the Euler-Lagrange equation for η 0 allows us to express η 0 itself as a function of π, η 1 , and their derivatives. In the Fourier space, we have Substituting this back into the action, we are left with the following expression: where we have defined the canonically normalized fields aŝ and θ k denotes the angle measured from the z-axis to k, i.e.
Thus, we see that the noninteracting part of the Lagrangian is that of free fields. We then quantize Q 0 :=π, Q 1 :=η 1 , and Q 2 :=η 2 as with the standard commutation relation: Here, the mode function u k is given by *3 Precisely speaking, the transformation (58) results in a nontrivial contribution on Sπ, but it is of order Iǫ H and thus can be neglected.
where we have used the de Sitter approximation aH = (−τ ) −1 . The interaction Hamiltonian is given by By use of the standard in-in formalism computation to compute the expectation value of an operator O(τ ), theπ-π correlation function can be calculated as Here, ∆ aniso (k) is the anisotropic part given by with v k (τ ) := (−τ ) −1 u ′ k (τ ) + 2(−τ ) −2 u k (τ ) and ℑ denoting the imaginary part of a complex number. Assuming that the mode is superhorizon (i.e. |kτ | ≪ 1), the main contribution to the integral in (72) comes from the region where |kτ 2 | ≪ 1. In this limit, |u k (τ )| 2 ≈ 1/(2k 3 τ 2 ) and the integrand in (72) can be approximated as Thus, we have with N k := − ln(−kτ ) being the e-folding number from the horizon exit to the end of inflation. Since I > 0, (74) predicts an oblate anisotropy. This is consistent with the result of [30][31][32]. For a large N k , we need a more careful analysis [47]. Moreover, we can extend the analysis to the bispectrum [48].

B. Dynamical η0
Now we proceed to the case where η 0 is dynamical. In order to illustrate the key ideas while avoiding technical complications, we consider the following action: where c, Λ, λ, m 1 , m 3 , and m 9 are assumed to be constant. In this case, the quadratic action for the NG bosons takes the following form:

V. CONCLUSIONS
By extending the idea of the EFT approach in isotropic inflation introduced in [37], we have developed an EFT of anisotropic inflation. The point is that we have introduced three NG bosons η m for the broken spatial rotation symmetry in addition to π for the broken time translation symmetry. After studying the simplest model (8), we have constructed the effective action in the unitary gauge up to quadratic order in the perturbations. The building blocks of the effective action in the unitary gauge are the metric g µν , the timelike unit vector n µ , the3-component of the vierbein e3 µ , and their derivatives. The lowest few terms in derivatives are presented in (33). Based on the so-obtained effective action, we have computed the statistical anisotropy in the scalar two-point correlation function. For the conventional model (36) having nondynamical η 0 , the anisotropy is given by (74), which reproduces the well-known result of [32]. On the other hand, for the model (75) with dynamical η 0 , the anisotropy is given by (89), which is in sharp contrast to the conventional case in the sense that the enhancement factor N 2 k is absent. In the present paper, we computed only the scalar two-point correlation function. It would be also intriguing to calculate the correlation function for the tensor modes and the cross-correlation between the scalar and tensor modes. The cross-correlation gives rise to the T B correlation in the CMB [51]. Moreover, for the simplest model (8), there exist consistency relations between these correlation functions [28]. If one could find such consistency relations for a broader class of models described by our EFT framework, they allow us to give a model-independent test of anisotropic inflation. We leave these issues for future study.
It should be stressed that our formalism allows us to go beyond anisotropic inflation. Indeed, we can discuss inflation with spontaneous breaking of the translation invariance, boost invariance, and a combination of these symmetry breaking patterns.
With the background metric (10) and the background vierbein (11), the the nonzero components of the Christoffel symbols are has the following nonzero components: The Ricci tensor, has the nonvanishing components: R 11 = a 2 e 2σ 3H 2 +Ḣ + 3Hσ +σ , (A6) from which we obtain the Ricci scalar as Note that the contribution of σ to the Ricci scalar is always positive. Finally, the nonzero components of the Einstein tensor, G µν = R µν − 1 2 g µν R, are given by

Useful formulae
To solve the background equations of motion, it is convenient to introduce quantities which are covariant under the unbroken symmetries and vanish on the background. Here, we introduce such quantities for covariant derivatives of the3-component of vierbein. Using the relation, we can compute its background value as where n µ = −δ 0 µ / −g 00 is the timelike unit vector. Then, if we introduce D µν as (22) it is covariant under the unbroken symmetries and vanish on the background. For later use, let us note several relations below.

Tadpole cancellation
Finally, let us solve the background equations of motion. For this purpose, we here write down the linear order action. With the field contents and symmetries being given by (28), we can decompose the effective action schematically as S = S P + S L + S P L . (A19) Here, S P breaks the time-diffeomorphism symmetry only, so that it contains the same operators as single-field inflation. Expanding up to the first-order fluctuations gives Pl 2 (δg 00 G 00 + δg 11 G 11 + δg 22 G 22 + δg 33 G 33 ) + ( c − Λ) − cδg 00 + O(δ 2 ) .
On the other hand, S L breaks the local Lorentz invariance, in the current case the rotation symmetry. In particular, S L represents terms which exist even if diffeomorphism symmetries are unbroken. Their lowest few terms in the fluctuations and their derivatives are It follows from the relation (A15) that the β 2 coupling starts from the second order in fluctuations. The β 1 coupling can be reformulated as where we used (A16) at the second equality. Using the relation e ν3 ∇ ν e3 µ = e ν3 D νµ + (H − 2σ)n µ , we also rewrite the β 3 coupling as In total, the rotation breaking sector can be written as We finally consider the mixing sector: S P L = d 4 x √ −g −γ 1 (t) e 03 2 + γ 2 (t)e 03 ∇ µ e3 µ + γ 3 (t)e 3µ n ν ∇ µ e ν3 + γ 4 (t)δg 00 e 3µ n ν ∇ µ e ν3 .
To summarize so far, we have obtained