Effects of Gravitational Chern-Simons during Axion-SU(2) Inflation

In this paper, we examine the viability of inflation models with a spectator axion field coupled to both gravitational and SU(2) gauge fields via Chern-Simons couplings. Requiring phenomenological success of the axion-SU(2) sector constrains the coupling strength of the gravitational Chern-Simons term. We find that the impact of this term on the production and propagation of gravitational waves can be as large as fifty percent enhancement for the helicity that is not sourced by the gauge field, if the cut-off scale is as low as $\Lambda$ = 20H. The effect becomes smaller for a larger value of $\Lambda$, while the impact on the helicity sourced by the gauge field is negligible regardless of $\Lambda$.

The upcoming LiteBIRD [25,26] and CMB Stage-4 experiments [27] will provide further constraints on the axion-gauge fields [28,29]. However, before the predictions of these models are taken seriously, it is important to check if these models are viable both phenomenologically and theoretically.
From the phenomenological point of view, it is important to see if couplings of SU(2) gauge fields to other fields lead to particle production and their backreaction on the axiongauge field backgrounds do not spoil the model setup. In [30], a charged scalar field is coupled to the SU(2) gauge field, and production and backreaction of pairs of charged particles is studied in de Sitter spacetime. In [31], the backreaction of the extra spin-2 field in this setup is analytically studied for all the inflationary models involving the SU(2) gauge field. In [32,33] a pair of massive Dirac fermions are coupled to the SU(2) gauge field. The coupling to a massless fermion is studied in [34]. In all these cases there exists a parameter space in which the backreaction of the particles on the SU(2) background is negligible. The nonlinear impact on the scalar perturbations of the chromo-natural inflation and the spectator sector during inflation has been studied in [35,36]. Anisotropic initial conditions are discussed in [37].
From the theoretical point of view, it is important to consider all different classes of parity-violating terms that arise from the same physics. The existence of both FF and RR is the result of one phenomenon. As an example, if coupling to a massive degree of freedom, such as axially coupled heavy fermions, is considered, then radiative fermion loops generate not only FF , but also RR [38]. String theory predicts the existence of axions that couple to both terms simultaneously [39][40][41]. Hence, these two parity-violating terms arise at the same time and should be effectively considered on the same level in the theory. Moreover, given that RR can introduce a ghost instability, it is important to check the stability of these models up to their cut-off scales.
In this paper we consider the axion-SU(2) gauge field spectator sector [42] together with the gravitational Chern-Simons term coupled to the axion field. In this setup we have the parity-violating terms on both sides of the equation of motion for the tensor metric perturbations; the left-hand side due to the gravitational Chern-Simons term and the righthand side due to the axion-SU(2) sector. There are many studies focused on the cosmological signatures of the gravitational Chern-Simons term [38,[43][44][45][46][47][48][49][50][51][52][53][54]. In this paper, we extend these studies to the axion-SU(2) models. In [55][56][57][58], the authors have considered both FF and RR originating from string theory. The non-abelian gauge field in their consideration does not share the same vacuum expectation value as in our case, hence it does not source gravitational waves linearly. This paper is organized as follows. In section 2 we present the model and analyse the background evolution. In section 3 we present the second-order Lagrangian for tensor perturbations and compare the gravitational waves with and without the gravitational Chern-Simons term. We discuss the stability and the cut-off scale of the model in section 4. We conclude in section 5.

Action
We consider the following action where S EH is the Einstein-Hilbert action and S ϕ is the inflaton sector action given by The spectator sector action S SP EC contains axion and SU(2) gauge fields, where χ is an axion field with potential U (χ) and a decay constant f : is the field strength tensor of the SU(2) gauge fields, with g A being the self-coupling constant and abc the three dimensional anti-symmetric symbol. The last term in S SP EC is the Chern-Simons interaction, where λ 1 /f parametrizes its coupling strength andF aµν ≡ ε µναβ F a αβ /2 is the dual of F a µν . ε µναβ is defined as ε µναβ ≡ µναβ / √ −g, where µναβ is the totally anti-symmetric symbol with 0123 = 1. The S SP EC is invariant under the local SU(2) transformation. FF is a total derivative and for χ = const, it reduces to a surface term. Hence, we can write FF as The last term in (2.1) is the gravitational Chern-Simons term coupled to the axion, with coupling strength of λ 2 /f : is the Riemann tensor and the dual of the Riemann tensor is (2.10) We can also write RR as The gravitational Chern-Simons term is also a total derivative. For χ = const, it reduces to a surface term.

Background Evolution
In this section, we describe the evolution of the background. As RR vanishes in a Friedmann-Lemaître-Robertson-Walker (FLRW) background, there is no contribution to the background. The vacuum expectation value of the gauge field is given by [1,2] A a 0 = 0, A a i = δ a i a(t)Q(t) . (2.13) The 00-component of the Einstein equations is [42] 14) The equations of motion for the axion and gauge fields are given by [3,42] are all much smaller than unity. Also we define the following dimensionless parameters The fourth term in the left hand side of (2.16) becomes 2m 2 Q H 2 Q; thus m Q can be regarded as the mass of Q (divided by H).
In the slow-roll approximation, the following relation holds between m Q and ξ 1 [42] To prevent instabilities of the scalar perturbations (lower bound) and strong backreaction on the gauge background (upper bound), we consider √ 2 < m Q 4 [11,14,31,42]. This implies where we have used H/M pl 2 10 −9 . Thus, a sizeable ξ 2 , e.g., ξ 2 10 −2 , requires large λ 2 /λ 1 , e.g., λ 2 /λ 1 = 10 7 . A large hierarchy between λ 2 and λ 1 i.e., λ 2 /λ 1 1, is in principle allowed, since all the degrees of freedom are coupled to the gravitational Chern-Simons term, but only the charged ones are coupled to the SU(2) Chern-Simons term. Specifically, λ 2 ∝ (f /Λ)N where N is the number of integrated out degrees of freedom and Λ is the cut-off scale of the effective field theory, e.g., the mass of fermions in the loops. Note that this holds assuming that we are integrating out the massive fermions as an example at the same energy scale to get FF and RR simultaneously.

Tensor Perturbations
Now, we consider tensor perturbations during inflation at linear level. The tensor perturbations are amplified due to the tachyonic instability, whereas the scalar and vector perturbations are not amplified for m Q > √ 2 [4,9,11,42] in the spectator axion-SU(2) sector. We have four tensor degrees of freedom: two metric tensor degrees of freedom that represent the gravitational waves and two additional tensor degrees of freedom associated with the SU(2) gauge field. We consider a perturbed FLRW metric as follows where τ −1/aH is the conformal time andh ij is a transverse and traceless tensor, i.e. ∂ ih ij =h i i = 0. We define the Fourier transformed right and left-handed circular polarization states ash where e A ij is the polarization state tensor for the right (A = R) and left-handed (A = L) circular polarization states and satisfies the relation where ab c is the three dimensional anti-symmetric symbol. For simplicity, we assume that the gravitational waves are propagating along the z spatial direction We write the tensor perturbations of the gauge field as δA a i = at a i , wheret a i is chosen to be transverse and traceless, i.e. ∂ it a i =t ai i = 0. We write the gauge tensor perturbations as . For our convenience we work with the canonically normalised tensor perturbations We define the left and the right helicities as [42] h Now we write the second order action for the tensor perturbations.
where s = −1, 1 for the left-and right-handed helicities respectively. A prime denotes the derivative with respect to the conformal time τ , and H ≡ a /a.
Using the parameters defined in (2.19) in the action, we find The equations of motion for the tensor modes up to leading order in slow-roll parameters are Next we calculate the four tensor modes numerically. 1 Only the right-handed helicity mode of the tensor perturbations of the SU(2) gauge field t R is amplified unlike the gravitational waves. In the left panel of Figure 1 we show the amplification of |t R | = t † R t R (green line) around the horizon crossing (|kτ | ∼ 1), assuming m Q , ξ 1 , ξ 2 , H, B and E are constant. In all the plots in this section we use the following parameters

Without FF
To understand the effect of RR, we first consider the case where ξ 1 = 0 and m Q = 0 in (3.10). The last term on the left hand side of (3.10) acts as a friction term for h L which prevents it from decaying, whereas it acts as an anti-friction term for h R , which makes h R decay faster. We plot the metric tensor mode functions for different values of ξ 2 in Figure 2. The difference between the right-and left-handed helicity modes are negligible for a small value of ξ 2 , as shown in the top-left panel of Figure 2. As ξ 2 becomes larger, the difference between the two helicity modes becomes more visible (middle-and bottom-left panels).
In the right panels of Figure 2, we show the ratios of the right-and left-handed helicity mode functions with respect to those for ξ 1 = 0, m Q = 0 and no gravitational Chern-Simons term, labelled as h ξ 2 =ξ 1 =0 . Contrary to the case where gauge fields are present, the lefthanded helicity of the metric tensor perturbations is amplified. This difference depends on the relative sign between the coefficients of the parity-violating terms FF and RR, i.e., λ 1 in (2.4) and λ 2 in (2.8). The effect of RR on the enhancement/suppression is nearly symmetric as shown in the right panels of Figure 2. This enhancement/suppression occurs already deep inside the horizon. On the other hand, amplification of the right-handed helicity of the gauge field occurs near horizon crossing (see the green dotted line in the left panel of Figure 1). This difference becomes important in section 3.2.

With FF
We turn on the FF term with ξ 1 = 3.3 (m Q = 3). To capture the effect of RR in axion-SU(2) gauge field models, we plot the ratio of metric tensor mode functions for different values of ξ 2 with respect to those without the gravitational Chern-Simons term, i.e. ξ 2 = 0 in Figure 1 and 3. In the right panel of Figure 1 for ξ 2 = 4.5 × 10 −6 , the contribution of the gravitational Chern-Simons term is small given such a small value of ξ 2 . In Figure 3 we have plotted the same as Figure 1 for larger values of ξ 2 . After considering different configurations, we conclude that the contribution from the gravitational Chern-Simons term on the left-handed helicity modes is about fifty percent amplification for ξ 2 = 4.5 × 10 −2 as shown in Figure 3 while the right-handed helicity modes are largely unaffected. This value of ξ 2 requires a large hierarchy between λ 2 and λ 1 , as noted at the end of section 2.2.
For completeness, the right-and left-handed helicity mode functions for four different cases: with FF and RR, without RR, without FF , and without both terms, for different values of ξ 2 are shown in Figure 4.
The right-handed helicity modes are unaffected by the gravitational Chern-Simons coupling because they are sourced by the gauge field after horizon crossing, while the gravitational Chern-Simons coupling affects mode functions already deep inside the horizon.

Stability Analysis
For k > H/ξ 2 , the sign of the kinetic term of h R in the equation (3.8) becomes negative and, consequently, ghost instabilities may, in principle, be introduced into the model [49,50]. Existence of ghosts does not necessarily translate to catastrophe in a model but translates to the breaking of the effective theory. Let us rewrite the first term in (3.8), (1 − kξ 2 /H) (this is the only problematic term we have to analyse), in physical coordinates. It is given by where k phy ≡ k/a is the physical wave number. To show that the gravitational Chern-Simons term in this model is ghost-free, i.e., stable, we have to show that the effective field theory cut-off, Λ, on the physical wave number, k phy , is below H/ξ 2 . Note that we have two new free parameters in this model, the gravitational Chern-Simons coefficient λ 2 in (2.8) and the cut-off Λ. As there are no a priori constraints on λ 2 , our strategy is to work our way backwards. Specifically, relying on independently motivated constraints on ξ 1 , we ask what constraint is imposed on λ 2 in order to guarantee that the theory is healthy. Once this question is answered, we will ask how stringent or natural the resulting constraint is.
Let us first take a look at (2.19) and write the relation for λ 2 To remain in the ghost-free regime we need the cut-off Λ on k phy to be the following: and a more radical case where it is around 20H. 2 • Conservative case: The inequality in (4.2) boils down to ξ 2 < H/M pl [44,53], given the assumption that Λ does not exceed M pl . Using this in (4.1), we have: • More radical case: The inequality in (4.2) boils down to ξ 2 < 1/20, given the assumption that Λ is around 20H. Using this in (4.1), we have: On the right hand side of both inequalities above we have ξ 1 ∼ O(1), which guarantees a slow variation of the gauge field, λ 1 ∼ O(10), and there is an upper bound on the tensorto-scalar ratio r ≡ (P h /P ζ ) < 0.06 from not observing tensor modes in the CMB [59] where P h and P ζ are the power spectra of tensor and curvature perturbations, respectively. In our model both the vacuum fluctuations of the metric and the sourced gravitational waves contribute to P h . Using the upper bound on r, the measurement of the dimensionless power spectrum of scalar fluctuations, ∆ ζ ≡ k 3 P ζ /2π 2 ≈ 2.2 × 10 −9 , and the expression for the dimensionless power spectrum of tensor fluctuations only from the metric vacuum fluctuations ∆ hvac ≡ k 3 P hvac /2π 2 = 2H 2 /(π 2 M 2 pl ), we get a bound on the last term (M pl /H) 2 1.5×10 9 . Therefore, in both (4.3) and (4.4), the right side is expected to be a very large number. As there is no stringent constraint on the free parameter λ 2 in our model, the model is not disfavoured by fine-tuning arguments.

Discussion
We have studied the effect of the gravitational Chern-Simons term coupled to the axion field on production and propagation of gravitational waves during inflation with the spectator axion-SU(2) sector [42]. Both parity-violating terms RR and FF exist simultaneously. We find that the effect of the RR term on chiral gravitational waves can be as large as fifty percent amplification for the left-handed helicity mode functions compared to the case without the RR term for ξ 2 = 4.5 × 10 −2 . The effect is smaller for smaller values of ξ 2 . The right-handed helicity mode functions are unaffected regardless of the values of ξ 2 . Moreover, using the existing bounds on m Q and ξ 1 from the spectator axion-SU(2) gauge field sector, and requiring that the cut-off scale of the theory, Λ, is in the conservative case Λ = M pl and in a more radical case Λ = 20H, we put constraints on the new free parameter λ 2 in our model to remain in the ghost-free regime. Consequently, values of ξ 2 are related to the cut-off scale of the theory, Λ. ξ 2 = 4.5 × 10 −2 is allowed when Λ = 20H and ξ 2 = 4.5 × 10 −6 is allowed when Λ = M pl .
We conclude that the inflation models with the spectator axion-SU(2) sector remain phenomenologically viable in the presence of the gravitational Chern-Simons term.

A Quantisation of the tensor modes and initial conditions
Since the tensor-like perturbations in the gauge fields and the tensor metric perturbations are linearly coupled, we expand both in terms of the same pair of creation and annihilation operators [60] h A (τ, k) = n=h,t a A n,k h A,n + a A † n,−k h * A,n , t A (τ, k) = n=h,t a A n,k t A,n + a A † n,−k t * A,n , where we have the standard commutators a A n,k , a B † m,q = δ n,m δ A,B δ (3) (k − q) .

(A.3)
The solution h A,n=h can be interpreted as the vacuum gravitational wave, whereas h A,n=t as the sourced one (by the vacuum gauge field, t A,n=t ). While choosing a cut-off for the gravitational Chern-Simons term we should note where the tachyonic instability in the gauge sector exists for a given m Q . Considering m Q = 3 in the equation of motion for the gauge fluctuation, the tachyonic instability takes place around x ∼ 10. A reasonable cut-off must be chosen far enough to capture the effects of the instability completely. The cut-off Λ = 20H is acceptable considering this criteria.