Charged Vector Inflation

We present a model of inflation in which the inflaton field is charged under a triplet of $U(1)$ gauge fields. The model enjoys an internal $O(3)$ symmetry supporting the isotropic FRW solution. With an appropriate coupling between the gauge fields and the inflaton field, the system reaches an attractor regime in which the gauge fields furnish a small constant fraction of the total energy density. We decompose the scalar perturbations into the adiabatic and entropy modes and calculate the contributions of the gauge fields into the curvature perturbations power spectrum. We also calculate the entropy power spectrum and the adiabatic-entropy cross correlation. In addition to the metric tensor perturbations, there are tensor perturbations associated with the gauge field perturbations which are coupled to metric tensor perturbations. We show that the correction in primordial gravitational tensor power spectrum induced from the matter tensor perturbation is a sensitive function of the gauge coupling.


Introduction
Models of inflation based on a single scalar field with a flat potential are well consistent with cosmological observations [1,2]. Among the basic predictions of models of inflation are that the primordial perturbations are nearly scale invariant, nearly adiabatic and nearly Gaussian, in very good agreements with observations. Having said this, there is no unique realization of inflation dynamics in the context of high energy physics or beyond Standard Model (SM) of particle physics. For example, what is the nature of the inflaton field(s)? What mechanism keeps the inflationary potential flat enough to sustain a long enough period of inflation to solve the flatness and the horizon problems?
It is generally believed that there may exist many fields during inflation which can play some roles. If the fields are very heavy compared to the Hubble scale during inflation, then they are not expected to play important roles. However, if the fields are light or semi-heavy they can have non-trivial effects on cosmological observables such as the power spectrum and bispectrum, see for example [3,4,5]. In addition, there is no reason that only scalar fields play important roles during inflation. Specifically, the gauge fields and vector fields are essential ingredients of SM and any theory of high energy physics. Therefore, it is quite natural to look for the imprints of the vector fields during inflation. One issue with the vector fields in background is that they have preferred directions so in general models of inflation with background vector fields are anisotropic. The second issue with the vector fields is that because of the conformal invariance, they are quickly diluted in an expanding background, so their effects become rapidly insignificant during inflation.
Anisotropic inflation is a model of inflation based on a U (1) gauge field dynamics. To remedy the second issue mentioned above, the gauge kinetic coupling in these models is a function of the inflaton field so the conformal invariance is broken. By choosing an appropriate form of the gauge kinetic coupling, the electric field energy density becomes nearly constant so the gauge field survives the expansion till end of inflation [6]. In addition, the gauge field perturbations become nearly scale invariant and can take parts in generating cosmological perturbations. In particular, quadrupolar statistical anisotropies are generated in these models which can be tested in CMB maps. For various works on anisotropic inflation and their cosmological imprints see [7]. The anisotropic inflation model [6] has been extended to the case where the scalar field is charged under the U (1) gauge field in [8,9,10] while its isotropic realization containing a triplet of U (1) gauge fields has been studied in [11,12] . In this work we consider the isotropic extension of [6] in which the inflaton is charged under a triple of U (1) gauge fields. We show that the model has some interesting features such as it contains entropy mode in addition to the adiabatic mode and the gravitational tensor modes are sourced by the tensor modes coming from the gauge fields.
The rest of the paper is organized as follows. In Section 2 we present our model of isotropic charged vector inflation and then study its background dynamics in Section 3. In Section 4 we study the cosmological perturbations in this setup while the power spectra of the adiabatic and entropy perturbations and their cross correlations are studied in Section 5. The tensor perturbations of the metric and the matter fields are studied in Section 6 followed by the summaries and discussions in Section 7. The analysis of quadratic action are relegated into the Appendix.

The Model
In this Section we introduce our model in which we extend the model of anisotropic inflation to the setup which can support isotropic FRW solution. A realization of this was studied in [11,12] in which the model contains a triplet of U (1) gauge fields with an additional global internal O(3) symmetry. The internal O(3) symmetry allows one to obtain isotropic FRW solution [13]. In this work, we extend the setup of [11,12] containing a triplet of U (1) gauge fields which are charged under complex scalar fields. In a sense our setup is the isotropic realization of the model of anisotropic charged inflation studied in [8,9,10].
We consider a model consisting of a triple of U (1) gauge fields which may be thought as three independent copies of the U (1) scalar electrodynamics. The desired gauge symmetry is U (1) × U (1) × U (1) and the scalar sector is defined by a triplet Φ in which φ (a) , a = 1, 2, 3 are complex scalar fields which are charged under the gauge field A µ through the covariant derivative The gauge coupling constant e assigns the same charges to each scalar field. Similar to the original model of anisotropic inflation [6], the action of the model is given by where M P is the reduced Planck mass, R is the Ricci scalar, |Φ| = √ Φ † Φ, V is the potential, f is the conformal factor, and is the curvature (field strength tensor) defined in the spirit of the covariant derivative (2.2).
To simplify the setup, we have assumed that V and f are only functions of the magnitude |Φ|.
We have three independent U (1) gauge fields and therefore we should demand that the three generators τ a of the algebra u(1)×u(1)×u(1) being independent. In the matrix notation, we choose the following representation The above matrices are clearly independent, and further satisfy µ τ a . Due to the abelian structure (2.7) of the algebra u(1)×u(1)×u(1), the gauge coupling e did not appear in the above curvature tensor which confirms that we deal with three independent copies of U (1) gauge fields. The where Λ is a general matrix in the field space. More specifically, the matrix Λ can be expressed in terms of the basis as Λ = λ (a) τ a which, after substituting from Eq. (2.5), takes the form Λ = diag(λ 1 , λ 2 , λ 3 ). The gauge transformations (2.9) then implies where we have used the fact that Tr(τ a τ b ) = δ ab as can easily be deduced from Eq. (2.5). We see that the Maxwell kinetic term enjoys an internal O(3) symmetry, i.e. it is invariant under are the components of the O(3) rotation matrices. Therefore, the Maxwell term can support an isotropic FRW background solution. On the other hand, the the kinetic term of the scalar sector in unitary gauge where all φ a are real is given by where in the second line we have substituted from Eq. (2.1) and the summation rule on the repeated index a is understood.
rotation so in general an isotropic FRW background may not be supported by this model. In order to obtain an isotropic solution we consider a subset of the model in which φ 1 = φ 2 = φ 3 ≡ φ/ √ 3 where the kinetic term (2.11) takes the isotropic form [14] ( Putting all these together the action (2.3) takes the following isotropic form As expected, the action (2.13) has the same form as in models of anisotropic inflation [6] but the gauge fields here enjoy an additional internal O(3) symmetry. Thanks to this new symmetry the model is no longer anisotropic and it admits FRW background solution. As in [6] the conformal coupling f (φ) will be chosen such that to prevent the dilution of the gauge field energy density in the inflationary background. Before closing this Section, let us compare our model with the other inflationary models that are constructed by means of U (1) gauge fields. The isotropic extension of the anisotropic inflation [6] is suggested in [11,12] by means of a triplet of U (1) gauge fields and the charged extension of it is considered in [8]. Here we have constructed the charged isotropic extension of anisotropic inflation [6]. In other words, our model is the charged generalization of [11,12] and isotropic extension of [8]. In addition, it is shown in [15], see also [16], that with a large multiplet of U (1) gauge fields and with appropriate form of the conformal factor f (φ), the FRW solution is the attractor limit of arbitrary initial conditions with background anisotropies. Correspondingly, one may ask how in our model with scalar fields charged under three U (1) gauge fields isotropic configurations can be supported? The model considered in [11,12] has local U (1) × U (1) × U (1) symmetry while it enjoys global O(3) symmetry. When one is interested in a charged extension of the model [11,12] there are two different ways to think about it: i) it is a model with O(3) symmetry. Therefore the charged model is nothing but the model with a gauge symmetry homomorphic to O(3). For instance, the model considered in [17] can be thought as the charged extension of [11,12]. ii) It is a model with U (1) × U (1) × U (1) symmetry. In this picture, one considers three independent U (1) gauge fields and then charge the inflaton field under these new symmetry. This latter picture is what we have adopted in this work.

Background Equations
Since the action (2.13) is O(3) invariant, the model admits the flat FRW cosmological background with the ansatz [13] A (a) The model behaves like three mutually orthogonal U (1) gauge fields and ansatz (3.2) assigns the same magnitudes A(t) to each gauge field [16]. Note that the ansatz (3.2) is not the only solution. Indeed, one can imagine a situation in which the initial amplitudes of the gauge fields are not equal to each other, A (a) In this case, the spacetime metric will be in the form of Bianchi type I Universe. However, as shown in [18], one expects the isotropic FRW background to be the attractor solution of the system so the spacetime rapidly approaches the FRW background and the gauge field amplitudes become equal.
Varying the action (2.13) with respect to the gauge fields, we obtain the associated Maxwell equation where a dot indicates derivative with respect to the cosmic time t.
The variation of the action (2.13) with respect to the scalar field gives the Klein-Gordon equationφ where ,φ denotes the derivative with respect to the scalar field. Note the important effects of the gauge field back-reactions on the scalar field as captured by the source term in the right hand side of the above equation. Finally, the corresponding Einstein equations are The right hand side of Eq. (3.5) is the total energy density while the expression in the parentheses on the right hand side of Eq. (3.6) is the total pressure. In the absence of e, from the above relations we see that the pure gauge fields contributions behaves like radiation thanks to the conformal symmetry. Let us consider the effects of the gauge coupling e. We see from the second term in the right hand side of Eq. (3.4) that the interaction e 2 φ 2 A 2 a −2 induces a time-dependent mass for the inflaton. However, the exponential time-dependence of this induced mass makes its main effect to occur towards the end of inflation where the exponential growth of the gauge field has its main influence. Thus, to have a long enough period of inflation, the back-reaction e 2 φ 2 A µ A µ is negligible during much of the period of inflation and it only controls the mechanism of end of inflation [8,9]. In this approximation one can easily solve the Maxwell equation (3.3) to obtaiṅ where q 0 is an integration constant. Now, as in anisotropic inflation model [6], it is convenient to define the ratio of the energy density of the gauge fields to that of the inflaton as In order to obtain a long period of inflation with a dS-like background, we expect that the contribution of the gauge field to the total energy density to be small. This is because, as just mentioned above, the gauge fields' contributions are like radiation and cannot support inflation by themselves. In other words, as in conventional models of slow-roll inflation, we expect that inflation to be driven predominantly by the scalar field. As a result, we require R 1 in order to obtain a long period of inflation. The dynamics of the background is very similar to the setup of anisotropic inflation. During the early stage of inflation, the gauge fields do not drag enough energy from the inflaton field so the parameter R is much smaller than the slow-roll parameters. In this limit, we can safely neglect the contributions of the gauge fields in total energy density and pressure and solve the system as in single field slow-roll models with Therefore, in the slow-roll limit, and for a given potential V (φ), the above equations provide the solution Now, as inflation proceeds, the gauge fields drag more and more energy from the inflaton field via the conformal coupling f (φ). As shown in [6] the system reaches an attractor limit in which the fraction of the gauge field energy density to total energy density reaches a constant value. During the attractor stage R becomes at the order of slow-roll parameter and it stays nearly constant till end of inflation.
In order for R reach a constant value, from Eq.
with a constant parameter c.
As the roles of the gauge fields become important, they back-react on the inflaton dynamics as given by the source term in Eq. (3.4). Taking into account the back-reactions of the gauge fields on the inflationary trajectory fix the relation between R and the slow-roll parameter .
The scalar field equation in the slow-roll limit is given by (3.12) Using Eqs. (3.9) and (3.12) we obtain the following equation for φ in terms of the number of e-folds ln a = N (setting M P = 1 for simplicity) Now, it is suitable to rearrange Eq. (3.13) to the following form (3.14) Defining G(N ) ≡ e 4N e 4c (V /V ,φ )dφ , the above equation takes the following form One can solve this differential equation in slow-roll limit to obtain where C is a constant of integration. We see that for sufficiently small values of q 2 0 C, the last term in the above bracket falls off during inflation and Eq. (3.16) implies Consequently, ρ A becomes nearly constant during the second phase of inflation, and after straightforward calculations, we obtain Substituting Eq. (3.17) into the modified slow-roll equation (3.12), we obtain This shows that during the second phase of inflation the effective mass squared of the inflaton field m 2 is reduced by the factor 1/c compared to the first stage of inflation [6]. Moreover, from Eqs. (3.5), (3.6) and (3.17), one can also obtain the slow-roll parameter as follows (3.20) Therefore, we find in which we have defined the parameter I ≡ (c − 1)/c. Interestingly the relation between R and given in Eq. (3.21) is the same as in anisotropic inflation.  In the left panel of Fig. 1, the phase space plot of (φ,φ) for the potential V = 1 2 m 2 φ 2 for a fixed value of e and for three different values of c are plotted. In the right panel of Fig.1 the behaviour of (φ,φ) as a function of the number of e-folds N is plotted. As we see from the plots, initially the inflaton field evolves independent of the effects of the gauge field so all three curves coincide during the first phase of inflation. However, as the gauge fields drag enough energy from the background, they kick in and after a short transient period, the system reaches the attractor phase. The attractor phase starts sooner for the larger value of c. This is understandable, since the larger is the value of c, the more energy is pumped into the gauge field from the inflaton field. We also see that the attractor phase and the total number of e-folds are longer for the larger values of c. This can be seen from our equations too. Starting from N = − Ḣ φ dφ, and using Eq.
Therefore, the total number of e-fold increases by increasing the value of c.
In Fig. 2 the phase space plot of (φ,φ) (left panel) and their dependence on N (right panel) are plotted for the same potential as in Fig. 1, but this time c is held fixed while e is varied. As can be seen from the plots, e does not play important roles during much of the period of inflation. However, its effect become important during the final stage of inflation, modifying the total number of e-folds slightly. More specifically, the coupling e induces an effective mass m 2 ∼ e 2 A 2 e −2N for the inflaton field. When this induced mass becomes comparable to H, then the slow-roll conditions are violated and inflation ends abruptly. During the attractor phase A ∝ e (4c−1)N so the induced mass scales like e 2 e (4c−3)N . Consequently, the total number of e-folds depends only logarithmically on e. In other words, holding other parameters such as c fixed while varying e, as in Fig. 2, the total number of e-folds changes as Although e does not play important roles during the inflation background, but it has important effects on curvature perturbations power spectra and other cosmological observables.

Cosmological Perturbations
In this section, we present the perturbations of our model based on action (2.13). From now on, we work with the conformal time τ defined as dτ = dt/a(t).
The metric perturbations around the background geometry (3.1) are given by where α, β, ψ and E are scalar modes, B i and F i are vector modes while h ij are the tensor perturbations which satisfy the following transverse and traceless conditions The gauge fields enjoy internal O(3) symmetry and the perturbations should be defined in the spirit of O(3) symmetry as [13] δA (a) where (Y, δQ, M, U ) are scalar modes, (Y a , M a , U a ) are vector modes, and (t ia ) label the tensor modes associated with the gauge field perturbations which are subject to the transverse and traceless conditions In addition to the above perturbations, we also have the inflaton perturbations δφ.
The gauge freedom associated with the four-dimensional diffeomorphism invariance fixes two scalar modes and two vector modes of metric perturbations. For the scalar modes, we work in the spatially flat gauge in which while for the vector perturbations we fix the gauge by setting F i = 0. Apart from the diffeomorphism invariance, the gauge fields enjoy the U (1) gauge invariance given by Eq. (2.10). But, we have already fixed the U (1) gauge in choosing the scalar fields to be real, i.e. going to the unitary gauge, yielding to the action (2.13).
In summary, after fixing the gauges associated with the diffeomorphism invariance and local U (1) invariance, we have seven scalar degrees of freedom (α, β, δφ, δQ, Y, U, M ), eight vector degrees of freedom (B i , U a , Y a , M a ) and four tensor perturbations (h ij , t ij ). In total we have 19 physical degrees of freedom.
Since the model with the action (2.13) enjoys O(3) symmetry, the scalar, vector and tensor perturbations decouples at the linear order of perturbations. Moreover, since our setup is isotropic, the vector perturbations decay as usual in an expanding Universe and we will not consider them from now on.

Scalar Perturbations
Working in spatially flat gauge (4.5) and fixing local gauge symmetry (2.10), we deal with seven scalar modes (α, β, Y, δQ, U, δφ, M ). Direct calculations shows that α, β appear with no time derivatives in the quadratic action and therefore they can be substituted from their algebraic equations of motion. Moreover, the contribution coming from these non-dynamical modes are slow-roll suppressed [9,10] and we therefore neglect them.
The quadratic action for the remaining modes (Y, δQ, U, δφ, M ) is presented in Appendix A. As discussed there, the contributions of the perturbations Y and M are suppressed during much of the period of inflation and therefore can be neglected. Therefore, the quadratic action for the remaining light scalar perturbations in Fourier space is given by in which a prime indicates the derivative with respect to the conformal time, τ e is the time of end of inflation and we have defined the canonically normalized fields We have ignored pure slow-roll corrections i.e. terms containing the slow-roll parameters and its derivative without the factor I since they are the same as those coming from the gravitational back-reactions and can be absorbed into the power spectrum in the absence of gauge fields. In addition, as we shall show later on, I 1 so we have kept the leading terms of I in the action (5.1) which turns out to be proportional to √ I. Form the action (5.1), we see that the field U is decoupled from the other fields. In addition, it did not exist at the background level. Therefore, the field U is a pure isocurvature mode. This is unlike the mode δQ which is the perturbations associated with the diagonal component of A contributes to the background energy and interact with each other. In this view, we are dealing with a multiple field model of inflation which is studied vastly in the literature. In particular, similar to the logic of [19], we expect that a combination of the fields (δφ, δQ) to play the roles of the adiabatic mode while a different combination to play the role of the entropy perturbations.

Adiabatic and entropy decompositions
In order to find the adiabatic and entropy modes, we first find the comoving curvature perturbations R from the standard definition where ψ measures the spatial curvature and δu is the velocity potential which is defined as δT t i = (ρ + p)∂ i δu. Calculating the energy-momentum tensor at the linear order of perturbations, and noting that we work in spatially flat gauge Eq. (4.5), the comoving curvature perturbation takes the following form We need to substitute the non-dynamical perturbation Y in the above relation from Eq. (A.2). As discussed in Appendix A, the contribution of Y in curvature perturbation is subleading during the inflationary stage. Therefore, to leading order, the curvature perturbation takes the following simple form The above formula is interesting showing that the contribution of each field into the total curvature perturbation is weighted by the fraction of the corresponding field into the total energy density [20,21]. Since I 1, the dominant contribution into curvature perturbations is given by the inflaton field perturbations δφ. But we expect to have subleading contributions from the diagonal component of A (a) i which is given by the fraction √ I in the above formula. Following the logic of [19], the scalar modes δφ c and δQ c can be decomposed into the adiabatic and entropy components as follows δσ c = cos θδφ c + sin θδQ c , (5.6) where we have defined The canonical variables δσ c and δs σ are related to the standard adiabatic and entropy perturbations defined in [19] via δσ c = a δσ δs c = a δs . (5.9) Using the decomposition Eq. (5.6) into Eq. (5.5), the comoving curvature perturbations is given by In the limit I → 0, we have cos θ = 1 and Eq. (5.6) gives δσ = δφ in which we find the well-known result R = − Ḣ φ δφ for the curvature perturbations. Correspondingly, we define the associated normalized entropy perturbation via Our final aim is to find the power spectrum for the observable quantities R and S. For this purpose, we rewrite the quadratic action (5.1) in terms of the adiabatic and entropy modes, yielding We see that the adiabatic and entropy modes are coupled to each other with the couplings proportional to √ I.
We calculate the power spectra of P R and P S and their cross-correlation P RS in next subsections. However, before that, let us consider the perturbation U which is a pure isocurvature mode and does not couple to other modes. Decomposing U into the creation and the annihilation operators with the Minkowski (Bunch-Davies) initial condition, we have Correspondingly, the dimensionless power spectrum for U = U c /a, defined as usual via , on super-horizon scales is given by The above result shows that the scalar mode U behaves like an spectator field with the amplitude H/2π.

Curvature perturbations power spectrum
In this subsection we calculate the curvature perturbation power spectrum P R . From Eq. (5.10) the power spectrum of curvature perturbation at the end of inflation τ e is given by 14) The leading contribution to curvature perturbation power spectrum comes from the adiabatic mode δσ. However, the adiabatic and the entropy modes are coupled to each other with the interactions given by the last two terms in the action (5.12). Therefore, we also have to calculate the corrections from the entropy mode in P R . Since we assume I 1, this analysis can be done perturbatively using the standard in-in formalism [22].
The two-point function for the adiabatic mode is then given by To calculate the corrections in curvature perturbations power spectrum we need to obtain the interaction Hamiltonians. In addition to the two interactions which directly couple the fields δσ and δs (the last line in action (5.12) containing √ I ) we also have new interactions in the action from the second and third lines of Eq. (5.12) containing I. Note that we treat I as the parameter of the perturbations so any term containing this parameter should be treated as interaction compared to the free theory. In total, we have seven interaction Hamiltonians for the scalar perturbations, H s ij , in which the indices i, j are for H s i (τ 1 ) and H s j (τ 2 ) respectively. The free wave function for M ik = δσ c (k), δs c (k) with the Bunch-Davies initial condition, is given by To simplify the notation, let us pull out the factor (2π) 3 δ (3) (k − k ) and denote the corresponding correlations by ∆ . Then, the leading order corrections in ∆ δσ 2 are obtained to be Im (5.26) where N e = − ln(−kτ e ) is the number of e-folds at the end of inflation and ∆ (1) δσ c The parameter β measures the effects of the gauge coupling e 2 . With M P /H ∼ 10 5 , we have β 1 for e 10 −3 . For large value of e the function F (β) grows like β 2 . Interestingly, the correction from the gauge field dynamics in curvature perturbations in Eq. (5.27) has the same form as in [10] studied in the context of charged anisotropic inflation model. However, in the model of [10] with a single copy of U (1) gauge field, the gauge field corrections in power spectrum induce statistical anisotropy ∆P R /P R (0) = g * cos 2 (k ·n) with the quadrupolar amplitude g * = −24IF (β)N 2 e in whichn is the preferred direction (direction of anisotropy) in the sky. Note that when e = β = 0, then F (β) = 1 and one recovers the well known results [23,24,9,25] e . In order to be consistent with the observational constraints |g * | 10 −2 [26,27], one then requires I 10 −7 . However, in our setup with internal O(3) symmetry, we have three orthogonal gauge fields with equal amplitude so there is no statistical anisotropy. As a result, we have less stringent constraint on the value of I.
Having calculated the corrections in curvature perturbation power spectrum, we can also calculate the corrections in the spectral index ∆n s , given by (5.29) in which the subscript * represents the time of horizon crossing for the mode of interest k.
In order to have a nearly scale invariant power spectrum we require ∆n s to be at the order of the slow-roll parameters. As a result, we conclude that I /10N e . This justifies our assumption in taking I 1. However, the above result also indicates that I is parametrically at the order I ∼ 10 −2 ∼ 10 −4 assuming that is at the order of few percent. This is less restrictive compared to constraint imposed on the magnitude of I in models of anisotropic inflation discussed above.
The smallness of I may raise concerns about the existence of the background attractor regime [28,29]. One may require some fine-tunings on the combination q 2 0 C in order to neglect the last term in the brackets in (3.16). To be specific, for the chaotic inflation with V = 1 2 m 2 φ 2 , the condition 1 requires This indicates the level of fine-tuning required in order for the gauge field dynamics to actually reach the attractor phase.

P S and P RS
In this subsection we calculate the power spectrum of entropy mode P S and its crosscorrelation with the curvature perturbation P RS . For the cross-correlation, we find (5.31) We see that, unlike in previous integrals, the cross-correlation is proportional to √ I. The reason is that we did not have to calculate a nested integral. Correspondingly, the crosscorrelation of the entropy and the curvature perturbation is given by To calculate P S we can perform similar in-in integrals as in the case of curvature perturbations in previous subsection. However, there is a less cumbersome way to obtain P S as we describe below. Let us first look at the interaction Hamiltonians The last correction to the power spectrum of the entropy mode comes from the interaction Hamiltonians H s 2 and H s 3 . Performing an integration by part, it is easy to see that the appropriate identification will be which gives (5.40) In the same manner we can easily see ∆ (2) δs c 2 13 = ∆ (2) δs c 2 23 = 0. All of these results can also be confirmed from the direct in-in calculations. Summing up all the above corrections, we find where β and F (β) are defined in Eq. (5.28).

Tensor Perturbations
There are two different types of tensor perturbations in our model. One is the usual tensor perturbations of the metric h ij . The other one is t ij coming from the matter sector of the O(3) gauge fields in Eq. (4.3). We therefore have four tensor modes in our model.
Using the transverse and traceless conditions, the quadratic action in Fourier space is obtained to be where we have defined the canonically normalized fields as follows It is convenient to write the tensor modes in terms of their polarizations. In order to do this, we note that the traceless and transverse conditions implyh ii = k ihij =t ii = k itij = 0. Consequently, we can express them in terms of the polarization tensor ash ij = +,×h λ e λ ij andt ij = +,×t λ e λ ij where we have e λ ii = k i e λ ij = 0 and e λ ij e λ ij = 2δ λλ . The interaction terms in (6.1) are proportional to √ I . In the previous section, we have seen that I 10 −2 and therefore √ I /10 which is small. On the other hand, the interactions in (6.1) have the same form as the interactions in (5.12). Therefore, from our results for the scalar modes, the leading corrections in tensor correlations are at the order I N 2 e . The wave functions for the free tensor modes N ik = {h λ (k),t λ (k)} are given by 3) The interaction Hamiltonians associated with the quadratic action (6.1) in the interaction picture are given by Similar to the analysis of entropy power spectrum in subsection 5.3, we do not need to explicitly perform the cumbersome in-in calculations since we can simply model the above interaction Hamiltonians to those we had in the case of scalar perturbations given in Eq. (5.17) via the following identifications Using the above identifications and the results obtained from Eq. (5.20) to Eq. (5.26), we can easily obtain the nonzero corrections to the power spectrum of the tensor modes as follows Summing up all the above corrections we find Note the important effect that the charge coupling interaction induces 1/ enhancement to the tensor power spectrum which is the specific feature of this model. This is similar to the results obtained in model of charged anisotropic inflation [10] where the statistical anisotropy induced in tensor power spectrum is more pronounced compared to statistical anisotropy induced in the scalar power spectrum.
To calculate the power spectrum of the gauge field tensor mode, we note that it appears exactly the same as entropy mode. Therefore, upon making the appropriate identifications of the interaction Hamiltonians, we find the following results with ∆ (2) (t λ ) 2 13 = ∆ (2) (t λ ) 2 23 = 0.
Summing the above corrections, we see that they cancel one another and, similar to the case of P S , there is no I N 2 e correction to the two-point function oft λ and we have to keep the I N e corrections.
What remain is the cross-correlation betweenh λ andt λ . Keeping the above identifications in mind, looking at Eq. (5.31), we see that the first term in the last line comes from the interaction Hamiltonians H s 1 and H s 2 in Eq. (5.17) while the second term comes from H s 3 in Eq. (5.17). Therefore, from the identifications (6.5), we easily find which can also be justified from the direct in-in calculation.
Having obtained the two point function ofh λ andt λ and their cross-correlation we can obtain the power spectra. The power spectrum of the gravitational tensor modes as usual are defined via , which after substituting from Eq. (6.6), we obtain the following expression for the power spectrum of the gravitational tensor modes in which is the standard tensor power spectrum for gravitons. The function F (β) is defined as in Eq. (5.28) with the new dimensionless parameterβ given in terms of β aŝ Interestingly, the corrections induced from the gauge fields dynamics in gravitational tensor power spectrum in Eq. (6.9) has the same form as statistical anisotropy induced in tensor power spectrum in model of charged anisotropic inflation [10]. As discussed before, with e 10 −3 we have β 1 and therefore one can easily haveβ 100. In order for our perturbative approach to be valid, we require that 16I N 2 e F (β) 1. Using the form of the function F (β) and the definition ofβ, this is translated into in which the approximations I 10 −4 , ∼ 10 −2 and H/M P ∼ 10 −5 have been used to obtain the final result. In conclusion, for e > 10 −3 or so, the corrections induced from the gauge field into the gravitational tensor power spectrum becomes large and our perturbative approximations break down. This conclusion is in line with the result obtained in [10].
Similarly, for P t and P ht , we find h N e 1 −β . (6.14) We see interesting similarities between P t and P S in Eq. (5.41) and between P ht and P SR in Eq. (5.32).
Having calculated the curvature perturbation and the gravitational tensor power spectra in Eqs. (5.27) and (6.9), the ratio of the tensor to scalar power spectra, denoted by the parameter r, is given by For large enoughβ, the last term above dominates over the second term and we will have a positive contribution for r, modifying the standard result r = 16 in single field slow-roll models of inflation. For example, if we take e such thatβ ∼ 10, then the last term above is at the order of unity in chaotic model. A large value of r is disfavoured in light of the recent constraint r 0.07 [30].

Summary and Conclusions
In this work we considered a model of inflation containing a triplet of U (1) gauge fields charged under complex scalar fields with gauge coupling e. The gauge fields enjoy an internal O(3) symmetry associated with the rotation in field space. In a sense this model is a hybrid of models of anisotropic inflation and models based on non-Abelian gauge fields [31,32,33,34,35,36,37]. Similar to anisotropic inflation models, with appropriate coupling of the gauge fields to the inflaton field, the system reaches an attractor phase in which the energy density of the gauge fields reaches a constant fraction of the total energy density and the gauge field perturbations become scale invariant. We have decomposed the scalar perturbations into the adiabatic and entropy modes. The corrections from the gauge fields into the curvature perturbations are given by Eq. (5.27) where the effects of gauge coupling is captured by the function F (β). As expected, it has the same structure as in models of anisotropic inflation, i.e. being proportional to IN 2 e . However, because of the background isotropy, no quadrupolar statistical anisotropy is generated. We have also calculated the corrections in spectral index. Requiring a nearly scale invariant curvature perturbation power spectrum requires I /10N e ∼ 10 −4 . This should be compared to models of anisotropic inflation in which the amplitude of quadrupolar anisotropy g * is given by g * = 24IN 2 e and demanding |g * | 10 −2 from CMB observations requires I 10 −7 . We have calculated the tensor power spectra of the model. In addition to tensor perturbations coming from the metric sector, we also have new tensor perturbations from the gauge fields sector. The interactions between the matter and metric tensor perturbations induce corrections into the primordial gravitational wave spectra given by Eq. (6.9). We have shown that the effects of gauge coupling e are more pronounced in tensor power spectrum, controlled by the function F (β). For example, in simple model of chaotic inflation with H/M P ∼ 10 −5 , we require e 10 −3 in order for the corrections in tensor power spectrum to be perturbatively under control. This is originated from the interaction e 2 g µν A (a) µ A (a) ν φ 2 as in Higgs mechanism. In large field model with φ > M P , large interactions between the tensor perturbations and gauge field perturbations are generated inducing large corrections in tensor power spectrum. We also calculated the power spectrum of the matter tensor perturbation and the cross correlation between the matter and metric tensor perturbations, given respectively by Eqs. (6.13) and (6.14).
There are a number of directions in which the current study can be extended. One natural question is the non-Gaussianity of the model. In particular, in models of anisotropic inflation large anisotropic non-Gaussianities are generated. Correspondingly, we expect that observable local type non-Gaussianity to be generated in our model. In addition, there will be cross correlation between tensor-scalar-scalar correlations which may have observable implications such as for the fossil effects [38,39,40,41,42,43,44]. Another open question in our model is the reheating mechanism which is not specified. One simple mechanism, as in standard mechanism of reheating, is that at the end of inflation the gauge fields simply transfer all their energies to conventional radiation i.e. photons and other degrees of freedom in Standard Model. Another option is that the gauge fields do not decay. In this case its energy density has the form of radiation which will be quickly diluted in subsequent expansion of the Universe. Another open question in our setup is the roles of the entropy perturbations. This question is also linked to the previous question about the mechanism of reheating. Observationally, there are stringent constraints on entropy perturbations. Therefore, the model should not generate too much entropy perturbations. To study this question, we have to specify how the reheating mechanism works in this model and whether or not the gauge fields decay to photon, baryons etc. Finally, in this work we did not elaborate on the observational implications of the model. It is an interesting question to study the predictions of the model for the CMB temperature perturbations and polarizations. The contributions of the entropy modes and the corrections in primordial tensor power spectrum can have interesting observational implications in the light of the Planck CMB data.
substituting the result into the action (A.1) we find f A 2 f δφ 2 + f 2 1 + 1 6k 2 f 2 e 2 a 2 φ 2 δQ 2 − f 2 k 2 1 + 1 2f 2 k 2 a 2 e 2 φ 2 δQ 2 + 1 6 e 2 k 2 a 2 φ 2 M 2 − k 2 M 2 + f 2 k 2 U 2 − f 2 k 4 1 + 1 3f 2 k 2 a 2 e 2 φ 2 U 2 +4f A f 1 + 1 6k 2 f 2 e 2 a 2 φ 2 δQ δφ − 2a 2 Ae 2 φδQδφ We now consider the field redefinitionM = k 2 M − δQ in terms of which the above action takes the following form f A 2 f δφ 2 + f 2 δQ 2 − f 2 k 2 1 + 1 3f 2 k 2 a 2 e 2 φ 2 δQ 2 + 1 6k 2 e 2 a 2 φ 2 M 2 − k 2M 2 + f 2 k 2 U 2 − f 2 k 4 1 + 1 3f 2 k 2 a 2 e 2 φ 2 U 2 +4f A f δQ δφ − 4 3 a 2 Ae 2 φδQδφ + 2e 2 3 a 2 φAδφM − 2e 2 3k 2 f a 2 φ 2 f A δφM . (A.5) The advantages of working withM is that not only the quadratic action takes a more simple form but also that this mode is heavy during most of the inflationary era and we can therefore neglect it. To see this, we compare the two scalar modes δQ andM in the above action as ∼ k 2 f 2 e 2 a 2 φ 2 1 , (A. 6) which clearly shows that the contribution from the modeM is negligible during much of the period of inflation. Now, neglecting the subleading slow-roll corrections containing and its derivative and working to linear order in I we obtain the action (5.1). In principle we could calculate the quadratic action non-perturbatively in terms of the parameter I (i.e. to all orders in powers of I). However, as demonstrated in subsection 5.2, requiring a nearly scale invariant corrections from the gauge field into curvature perturbation power spectrum requires I 1, justifying our approximation in keeping only terms linear in I in quadratic action (5.1).
In obtaining the action (5.1), we have used the following formula