Production and backreaction of spin-2 particles of SU(2) gauge field during inflation

Primordial SU(2) gauge fields with an isotropic background lead to the production of spin-2 particles during inflation. We provide a unified formalism to compute this effect in all of the inflation models with isotropic SU(2) gauge fields such as Gauge-flation and Chromo-Natural inflation with and without spectator axion fields or the mass of the gauge field from the Higgs mechanism. First, we calculate the number and energy densities of the spin-2 particles. We then obtain exact analytical formulae for their backreaction on the background equations of motion of SU(2) and axion fields in (quasi) de Sitter expansion, which were calculated only numerically for one particular model in the literature. We show that the backreaction is directly related to the number density of the spin-2 field. Second, we relate the number density of the spin-2 particles to the power spectrum and the energy density of the gravitational waves sourced by them. Finally, we use the size of the backreaction to constrain the parameter space of the models. We find that the tensor-to-scalar ratio of the sourced gravitational waves can at most be on the order of that of the vacuum contribution to avoid a large backreaction on slow-roll dynamics of the gauge ssand axion fields in quasi-de Sitter expansion.


Introduction
Inflation [1][2][3][4] with SU(2) gauge fields [5][6][7][8] has a rich phenomenology that is not shared by canonical single-scalar-field inflation models (see [9] for a review). As was first discovered by one of the authors (A.M.), when the conformal symmetry of Yang-Mills theory is broken by an effective (FF ) 2 term in the Lagrangian, non-Abelian gauge fields acquire an isotropic and homogeneous background (vacuum expectation value; VEV) solution during inflation [5,6]. This VEV produces a copious amount of spin-2 particles which, in turn, linearly mix with tensor perturbations in the metric, i.e., gravitational waves. The same JHEP05(2019)174 phenomenology is obtained when the conformal symmetry is broken by a Chern-Simons interaction with an axion field ϕFF [7,8].
In addition to the original models, 1 there are several more inflationary models with the SU(2) VEV which share the above features [11][12][13][14][15][16]. Despite differences in details of the models, their tensor sector can be presented in a unified manner. As the sourced tensor power spectrum is proportional to the density parameter of the gauge field during inflation, these models violate the Lyth bound [17,18]. Depending on the details of the slow-roll dynamics of the gauge field VEV, e.g., a form of the axion potential, the tensor spectral index n T can be negative or positive, and thus violates the conventional consistency relation of single-field slow-roll inflation, n T = −r/8. Moreover, parity-violating interactions in linear perturbations make this spin-2 field chiral and hence generate observable circularly polarized gravitational waves as well as parity-odd correlations of cosmic microwave background (CMB) anisotropies, i.e., non-zero T B and EB [19]. Due to self-interactions of gauge fields, the gravitational waves can be highly non-Gaussian, yielding a large tensor bispectrum with approximately an equilateral shape [20][21][22]. Finally, chiral gravitational waves can generate baryon asymmetry via a gravitational anomaly [13,[23][24][25], and can serve as a natural leptogenesis mechanism during inflation to explain the observed baryon asymmetry in the Universe.
All of these signatures are robust consequences of having gauge fields during inflation and carry important information about the matter content of the early universe. The stochastic background of gravitational waves can be within reach of future CMB experiments [26][27][28], and that of n T > 0 can be within reach of future gravitational wave interferometers [13,19,29]. As none of these features exists in canonical single-scalar-field inflation models, we can use them to distinguish the particle physics models of inflation. For example, these models could be embedded in supergravity [30] and string theory [31].
In this paper, we take a closer look at the phenomenology of the spin-2 particles generated from SU(2) gauge fields. Particularly significant is the backreaction of spin-2 particles on the background equations of motion of the gauge and axion fields, as it could spoil significant properties of inflation with SU(2) gauge fields. We also gain better insights into the power spectrum and energy density of primordial gravitational waves by relating them to the number density of the spin-2 particles. This paper is organized as follows. In section 2, we briefly review the existing models of inflation with an SU(2) gauge field in a unified approach. We study the spin-2 particle production in section 3. In section 4, we compute the backreaction of this spin-2 field on the background field equations. We then relate the power spectrum and energy density of the sourced gravitational waves to the number density of the spin-2 particles in section 5. In section 6, we use the size of the backreaction to constrain the parameter space of the models. Finally, we conclude in 7. In appendix A, we discuss the symmetry structure of the background SU(2) gauge field. In appendix B, first, we discuss the action of the transversetraceless perturbed SU(2) field around its VEV in a unified approach. In B.1, we prove that JHEP05(2019)174 a perturbed SU(2) field around its VEV has a spin-2 field. The details of our analytical study as well as some necessary mathematical tools are presented in appendices C-E.

Review of theory
Consider an inflationary model with a Friedmann-Lemaître-Robertson-Walker (FRLW) background that can support slow-roll inflation with a slowly varying SU(2) gauge field VEV given by [5,6]Ā where {T a } are the generators of the su(2) algebra with a = 1, 2, 3 where I n δ ab is the identity matrix and abc is the totally antisymmetric matrix. In appendix A, we show that ansatz (2.2) is the general background solution for a gauge field with an isotropic and homogeneous energy-momentum tensor. We unify all the inflation models with an SU(2) field in the literature in the following Lagrangian

4)
where L A is the gauge field theory sector (with possibly an axion field ϕ), L 0 is a (possible) scalar theory and L H is a (possible) Higgs sector which makes the gauge field massive. The parameters α s and α H classify models as (see table 1) α s = 0 (axion)-SU(2) gauge field inflaton, 1 spectator (axion)-SU(2) gauge field , and α H = 0 massless models, 1 Higgsed models .
In these models, the conformal symmetry of Yang-Mills theory is broken by either adding a (FF ) 2 effective term to the gauge theory, e.g. Gauge-flation [5,6], or by coupling the gauge field sector to an axion field ϕ with slow-roll dynamics, e.g. chromo-natural [7,8].
The former models are given by [5,6] while the latter are given by 2 [7,8]

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Model α s α H L A Original references 0 0 L Gf [5,6] L Cn [7,8] Spectator 1 0 L Gf − L Cn [15] Higgsed 0 1 L Gf [12,16] L Cn [14] Spectator-Higgsed where F µν = T a F a µν is the field strength tensor whileF µν ≡ 1 2 µνλσ F λσ , ϕ is the axion and V (ϕ) is the axion potential. 3 Moreover, in the Higgsed version of the models, the gauge field becomes massive by a Higgs field and we have an extra term for the dynamics of the Goldstone boson which in the Stueckelberg form is [33,34] where U = exp(ig A π) and π is the Goldstone mode corresponding to the Higgs fluctuations around its VEV, π = π a T a . We relate these models to the literature in table 1. The (FF ) 2 term in (2.5) comes as an effective theory of (2.6) by integrating out the massive axion on energy scales below the mass of the axion, M µ 2 /f . In that case, the parameter κ is given as κ = 3λ 2 µ 4 [9,35,36]. Therefore, gauge-flation models are effectively equivalent to chromo-natural models in the limit that the axion is very massive, and they have the same tensor and vector perturbations [9]. (See also appendix B.) These models can be specified in terms of three dimensionless parameters, ξ A , ξ and ξ Z 0 , which are defined in the following. First, the almost constant gauge field configuration 3 In the original chromo-natural model, the potential is the standard cosine potential, V (ϕ) = µ 4 (1 + cos( ϕ f )), with f M Pl and λ 10 3 f M Pl . 4 Here, the full Higgs field theory is with a VEV given asZA = Z0(t)δ a A Ta in which A = 1, 2, 3 is the field's internal index, and Dµ = ∂µ −igAAµ is the covariant derivative. However, we are interested in the limit that the Higgs mass is much greater than the Hubble scale. Therefore, the only relevant sector is the Goldstone boson part given in (2.8). Notice that LH is gauge invariant and it can be written as LH = −g 2 A Z 2 0 tr(U DµU −1 /(−igA)) 2 .

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of the form (2.2) leads to a slowly-varying dimensionless parameter Validity of perturbation theory in the scalar sector of the SU(2) gauge field requires [8,10,17] ξ A > √ 2. (2.10) A scalar mode in these models would have a negative frequency at k a ∝ (2 − ξ 2 A ) which is unstable for ξ A < √ 2. In refs. [15,20,21], ξ A has been called m Q . The second dimensionless parameter is In the Higgsed version of the models (α H = 1), we also have Another important quantity in this setup is which is the contribution of the gauge field to the total slow-roll parameter and equals twice the ratio of the energy density of the gauge field backgroundρ YM to the total energy densityρ. In refs. [15,20,21] The background field equation of the gauge field is given by the µ = i component of the following equation 5D where a bar denotes a background quantity and D ν is the covariant derivative andᾱ A is a function of the background fields which, depending on the form of L A , is given as 6ᾱ (2.16) Note that the zeroth component of (2.15) is a constraint equation which is equivalent to zero for our background ansatz. 5 The explicit form of the background field equation ofĀµ is (2.14)

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Assuming slow-roll dynamics in the background, equation (2.15) relates ξ A , ξ Z 0 , anḋ In the L Cn models, the background field equation of the axion is

Tensor perturbations
The existence of a spin-2 degree of freedom in the gauge field is a unique feature of the SU(2) inflation models. This is the primary focus of our work. Once we have a slowroll background dynamics, the tensor perturbations in this family of models are entirely determined by the background quantities ξ, ξ A , ξ Z 0 , and by the perturbed gauge field sector of the model, L A + α H L H . The vector and tensor perturbations in L Gf and L Cn are the same. Let us first briefly review the spin-2 part of the perturbed SU(2) gauge field. More details are presented in appendix B. See [5,6] for the full decomposition of the field into the scalar, vector, and tensor perturbations. Once we perturb the metric and the SU(2) gauge field around their homogeneous and isotropic solutions (2.2), we have the following spin-2 fluctuations where δ T denotes the spin-2 subsector of the perturbed field. In appendix B.1, we prove thatγ ij = B ij /a is a (pseudo) spin-2 field. Nonetheless, throughout this paper, we shall call B ij a spin-2 field. In Fourier space, the vacuum (free) γ ij and B ij can be expanded as where h σ and B σ are the canonically normalized fields, e ij (±,k) are the polarization tensors associated with the ±2 helicity states, 7 which are normalized as e ij (σ, k)e * ij (σ , k) = 2δ σσ , andâ σ ( k) andb σ ( k) are the annihilation operators of the spin-2 modes of the metric and gauge field, respectively, satisfying The polarization tensor of the spin-2 field in the directionk = −r is given as wherer,θ andφ are the local orthogonal unit vectors in the directions of increasing r, θ, and φ. Note that eij(σ, k) = e * ij (σ, − k), and k × e(±1, k) = ∓ik e(σ, k).

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L Cn [7,8,11,13,15] ( Table 2. Definition of parameters in the equations of tensor perturbations in (2.24)-(2.25) in terms of ξ A , ξ and ξ Z0 for each type of models. The last row shows the parameters in the unified form for the generic action given in (2.4).

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(see (C.12)) (2.28) Using the above in the field equation of h σ (2.24), we find the sourced part of the gravitational waves.
Here, we summarize the main features of the spin-2 field with the field equation of (2.25) and the quadratic action of (B.6).
• B ± evolves as a massive field in de Sitter space with a parity breaking linear derivative interaction term, ∓δ c kHB ± , with δ c given by table 2. In terms ofᾱ A (2.16), we can write it in a unified form, δ c = 2(ξ A +α A H ).
• The first term in δ c (2ξ A ) comes from the interaction of B σ with the VEV of the gauge field through the covariant derivative D µ , and is due to the self-interactions of gauge field in Yang-Mills theory.
• The second contribution in δ c (2α A H ) is a time derivative ofᾱ A . As shown in (2.16), for L A = L Gf , this parameter is due to the VEV of FF while for L A = L Cn , it is due to the derivative interaction with the VEV of the axion field.
• The sound speeds of B ± field and GWs are unity in all of the models in this family. 9 • Due to the self-interactions of the gauge field, the B σ is massive with a mass term • In a similar Abelian field case (see (B.5)), the first derivative interaction as well as the mass term are missing. Thus, the non-Abelian nature of the gauge field makes i) a more efficient particle production, while making ii) the transverse field massive and therefore decaying after horizon crossing.
• As we will see in section 3, these derivative interactions are responsible for production of the spin-2 particle by the background fields.
• In SU(2) gauge field setups, the right hand side of the field equation of the gravitational waves in (2.24) is non-zero and is given by an anisotropic inertia proportional to ψ M Pl . Therefore, the efficiency of the mixing between the spin-2 field and the gravitational waves is specified by the VEV of the SU(2) gauge field.
• This anisotropic inertia is parametrized in terms of θ c , β c , and a small mass term for the graviton, m 2 H 2 , given in table 2.
• The term β c is the coefficient of a linear derivative interaction which is equal to ξ A regardless of the model. The other parameter can be written as θ c = ξ 2

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• All the interaction and parameters in the tensor perturbation sector are specified only by L A and L H , and therefore independent of whether the gauge field sector is a spectator or not.
In this work, we assume (quasi) de Sitter expansion and keep terms up to first order in slow-roll. The slow-roll time evolution of ξ A and ξ Z 0 , which is model dependent, contributes to the spectral tilt of the sourced gravitational waves. Depending on the details of the evolution of the gauge field VEV, the spectral tilt of the sourced gravitational waves can be positive or negative [37]. Since we are interested in the number density and the backreaction of the B σ particle as well as in the amplitude of the sourced gravitational waves which are model independent, we neglect this effect in this paper.
3 Spin-2 Schwinger-type particle production In this section, we study the spin-2 particle production due to their interactions with the VEV of the background fields. The background fields act as a classical source for the quantum fluctuations analogous to the well-known Schwinger effect [38]. However, unlike the standard Schwinger process in which the quantum field is sourced only by a background gauge field, here the spin-2 quantum field is sourced by both the backgrounds of axion and gauge fields. The derivation given in this section follows closely section 3.2 of [39].
To have a better qualitative understanding of the particle production process, let us write the field equation of Fourier modes using the (normalized) physical momentum where ω σ (τ ) is the (time-dependent) effective frequency of the modes In the limits that the effective frequency is slowly varying and the particle production is zero and the modes are in an adiabatic vacuum state. Then the solution can be well-approximated by the WKB form, are the positive and negative frequency modes respectively. The WKB approximation is the exact solution of  Figure 1. Effective frequency squared (ω 2 ) and deviation from adiabaticity (|Ω|). The left panel shows ω 2 (3.2) as a function ofτ while the right panel shows |Ω ± (τ )| (3.6) for massless systems (α H = 0) with ξ A = √ 2 (red line) and 10 (blue line). For comparison, the dotted black line shows the vacuum gravitational waves. Here, the solid red and blue lines show the plus polarization modes while the dashed blue and red lines show the minus polarization. The shaded areas in the right panel show the particle production regime. where Ω(τ ) is defined as which quantifies the deviation of our mode function from the exact adiabatic solution (see figure 1). We find that Ω − is always very small and thus B − (τ, k) remains adiabatic. However, Ω + becomes large around the roots of ω + and the system experiences a large deviation from adiabaticity. The above roots are presented in figure 2.

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When the adiabatic conditions hold, i.e., |Ω| 1, we have a well-defined adiabatic vacuum, and the field excitation about it describes particles. Deviations from adiabaticity in the asymptotic past and future are ifτ → 0 . (3.8) In the asymptotic past,τ → ∞, the adiabaticity conditions are satisfied and the WKB solution (3.4) is the Bunch-Davies vacuum (2.28). At later times the adiabaticity conditions are violated and particles are produced. In the asymptotic future, recalling that ξ A > √ 2 and using table 2, we have limτ →0 |Ω + (τ )| 10 −2 . Thus, the positive frequency modes (vacuum mode functions) in the asymptotic future, v σ (τ, k), are given by the WKB solution in which we used µ = i|µ| and limτ →0 ω(τ ) |µ|/τ . Using (C.12), the asymptotic future vacuum mode functions can be well approximated by the M -Whittaker functions as Using (C.14), the asymptotic past and future vacuum modes are related as Therefore, the spin-2 field can be either expanded in terms of the positive frequency modes in the asymptotic past (2.28) as in (2.22), or in terms of the positive frequency modes in the asymptotic future (3.10), as whereb σ ( k) andb † σ ( k) are the annihilation and creation operations of a particle with respect to the asymptotic future vacuum respectively. By definition, we have where |0 in and |0 out are the vacuum states in the asymptotic past and future of the (quasi) de Sitter spacetime, respectively. Using Bogoliubov transformation, we can writeb where α σ, k and β σ, k are Bogoliubov coefficients which satisfy the normalization condition .

(3.17)
Having the β σ, k coefficients, we are ready to determine the particle number density as well as the vacuum-vacuum transition amplitude. The efficiency of the particle production for σ = +2 is given by the exponent |κ| − |µ|, which, recalling (2.10) and using (2.27), can be approximated as This quantity is presented in figure 3 as a function of ξ A and ξ Z 0 and has the following asymptotic forms The number density of the created particles with a given comoving momentum, k, in the asymptotic future is which has a k-independent spectrum for each polarization state. As we see, there is a large pair production in the plus polarization while it is almost zero for the minus state n + ( k) e ξ A π and n − ( k) e −6ξ A π . (3.21)

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The total particle creation from the asymptotic past to the asymptotic future, therefore, is which is divergent since it expresses the number of pairs created for all times. The physically meaningful quantity, however, is the pair production rate, i.e., the number of pairs produced per unit time per unit physical volume To calculate the derivative we need to convert the wavenumber integral into a time integral. It has been shown in the right panel of figure 1 that the system has two sharp deviations from adiabaticity around the roots of ω + . Therefore, pairs of particles with +2 helicity state of a given comoving momentum, k, are produced mostly around (3.7) Note thatτ = k/(aH) −kτ . As a result, the total particle creation at τ 1 (k) and τ 2 (k) are given respectively by The corresponding production rates are We find that the particle production during τ 1 is much more efficient than during τ 2 , i.e.
Therefore, we can neglect the burst of particles created at τ 2 . After integrating eq. (3.23), we find that the physical number densities of pairs created up to time τ are also time independent i.e., gravitational and Schwinger-type particle production are exactly balanced by the gravitational redshifting. We can approximate the above as The particle production increases exponentially with (|κ|−|µ|) and has the following asymptotic forms which may cause a large backreaction on the background VEV fields. This is the next subject of our study, given in section 4. The number density of the created particles is presented in figure 4.
Another interesting quantity to compute is the vacuum-vacuum transition amplitude defined by where Υ vac is the vacuum decay rate. Using eq. (3.20) and (3.24), we obtain , (3.34) which is well approximated by implying a sizable vacuum decay rate. Finally, in the Minkowski limit with H → 0, this setup with an isotropic SU(2) gauge field does not experience Schwinger-type particle production [39].

Backreaction
In this section, we compute the induced current and backreaction of the spin-2 field on the inflationary background.

Induced current and backreaction
The continuous global SU(2) symmetry (A.5) leads to the conserved Noether current as where δ B denotes quadratic order action with respect to B ij field, a bar denotes background quantities, and D µ ≡ ∂ µ − ig A A µ is the covariant derivative. The current affects the background equations of motion as (see (2.15)) whereᾱ A is a function of background fields given in (2.16). The expectation value of the Thus, the background field equation of ψ in (2.14) is sourced by J A defined by In the L A = L Cn models with axion-gauge field coupling, B ij induces another backreaction term of the form ∇ µ P µ ϕ where with P i ϕ = 0. The divergence of the zero-competent then backreacts on the background axion field asφ Note that P µ ϕ is not a Noether current. In figure 5, we plotted J A and P ϕ with respect to ξ A . Both J A and P ϕ can be written in terms of the following momentum integral We find  where we have usedα A (2.17). We work out the integral of K[X] and its renormalization (using the adiabatic subtraction technique) in appendix D. Here we present its final regularized form We find that K reg [X] is proportional to e 2(|κ|−|µ|)π 1 (see figure 3). In the limit that |κ| − |µ| 1, we can further simply K reg [X] to in which we used (C.8) to expand ψ (0) (z). This completes our derivation of the analytical formulae for the backreaction terms. 10 Using (3.31), we can relate K reg [X] to the number density of the spin-2 field as 10 As we see in figure 3, the generated pair particle number is very large and thus we are in the classical regime. Therefore, we can estimate the induced gauge field current, J µ A , by semi-classical approximations. Specifically, we can approximate it as JA ∼ 2gAnpairsv. Assuming that the particles travel with the speed of light v ∼ 1, we can approximate the induced gauge field current as JA ∼ 2gAH 3 6π 2 |κ| + |κ| 2 − |µ| 2 3 e 2(|κ|−|µ|)π , (4.14) which is in agreement with the result of our exact solution in (4.9). where I BR [X] is of order unity and given as

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The backreaction is also directly related to n pairs . In particular, the backreaction to the field equation of the gauge field is where is of order unity as presented in figure 6. Similar to J A , the backreaction to the axion field equation is also directly given by the number density of the spin-2 field, i.e. P ϕ ∼ λH f n pairs . This relation is valid for all the models considered in this paper.
Equations (4.9)-(4.11) and (4.17) are the first main results of this paper. Equations (4.9)-(4.11) have been estimated only numerically for one model in this family previously in [40] which are in agreement with our formulae over the region where the comparison is possible. The relation between the backreaction and the number density of the spin-2 field in (4.17) is derived here for the first time.

Energy density of spin-2 fields
The extra spin-2 field has a sizable energy-momentum density. The expectation value of its energy density adds to the total energy density in the background,ρ, as Perturbing the energy momentum tensor and considering only terms quadratic in B ij , we have δ B T 00 = −ḡ 00 δ B ρ and δ B T ij =ḡ ij δ B P. JHEP05(2019)174 The contribution of B ij to the energy density, δ B ρ, is The contribution to the isotropic pressure is In the absence of interaction with the Higgs field, α H = 0, B ij field has the equation of state of radiation. On the other hand, for a massive gauge field with α H = 1, the B ij field gets closer to dust. We compute the normalized energy density in appendix D.2, and only show the result here. Using (D.41), we find the energy density fraction in the spin-2 field B σ as We show the energy density as a function of ξ A and ξ Z 0 in figure 7. Let us summarize the main features of the energy density: • The energy density of B − is always positive while the energy density of B + can be negative.
• The total energy density in B ij is negative for ξ Z 0 9 5 ξ A .
• The energy density fraction is of order • Validity of perturbation theory requires δ B ρ ρ 10 −5 , which constrains the parameter space of the models as a function of the energy scale of inflation.

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Although δ B ρ reg is negative in most of the parameter space, it is always a small part of the total energy density of the setup. The total energy-momentum tensor satisfies null and weak energy conditions, while the B + field violates both. The reason underlying the negative energy of the plus polarization is the existence of a short phase of instability for each k-mode of B + around horizon crossing. However, this phase ends as soon as the mode exists the horizon; thus, the existence of the cosmic horizon evades (dangerous) infinite energy extraction of negative energy systems, unlike in flat space.

Gravitational waves
Each polarization state of the spin-2 field B ij mixes with the corresponding polarization of the gravitational waves, γ ij (see (2.24)). In particular, in the presence of the gauge field, we have where γ vac σ (t, x) is the vacuum gravitational waves (i.e. by quantum fluctuations of the spacetime [41,42]) with helicity σ while γ s σ (t, x) is the part sourced by the spin-2 field. Here, we compute the power spectrum and the energy density of γ s σ in terms of ξ A and ξ Z 0 and relate them with the number density of B σ , n pairs . We work out the exact form of the sourced gravitational waves in (quasi) de Sitter in appendix E. The result in the super horizon limit is where the explicit form of G σ (ξ A , ξ Z 0 ) is given in (E.6) and shown in figure 8. It can be well approximated as .

(5.3)
Taking the (classical) limit, |µ| 1, and using (E.8), we have (5.5) Figure 8 shows |A + | 2 as a function of ξ A and ξ Z 0 . We find that |A + | 2 oscillates between zero and unity as a function of ξ A .  Figure 8. The prefactors |G + | 2 (left) and |A + | 2 (right) as functions of ξ A and ξ Z0 . The (black) dashed line shows unity. Finally, we can write the ratio of the power spectra of sourced and vacuum gravitational waves in term of the spin-2 fields number density as

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is the power spectrum of γ x + where x = (s, vac). Moreover, γ vac + ( k)γ vac + ( k ) = H 2 (2π) 3 k 3 where aγ + ( k ) is a canonically normalized field. In the right panel of figure 9, we show 10 −2 M Pl /ψ 2 P s T /P vac T . This is the second main result of this paper.
The next interesting quantity is the energy density of the gravitational waves. The energy-momentum tensor of the gravitational wave is In the left panel of figure 9, we show 10 −8 M 2 Pl /Hψ 2 δρ GW s /ρ. The energy density can be written as where I GW (ξ A , ξ Z 0 ) is roughly Comparing to the number density of the (gauge field's) spin-2 particles in (3.31), we find Hn pairs . (5.12) The above relation is correct up to order unity coefficients. The energy density in the sourced gravitational wave is proportional to ψ M Pl 2 , the number density of B + , Hn pairs , and the inverse ofτ 1 in (3.7).

Constraints on the parameter space
In this section, we use the size of the backreaction to constrain the parameter space of the models. Validity of perturbation theory requires that the backreaction terms be much smaller than the other terms in the field equations. To do this, we normalize J A and P φ by H 2 ψ and λHg A ψ 3 /f respectively to construct two dimensionless backreaction terms where the explicit forms of K[X] for X =α A /H (1 + ξ 2 A + α H 2 ξ 2 Z 0 )/ξ A and X = ξ A are given in (4.11) and a good approximation is given in (4.13). Another backreaction is the contribution of the spin-2 field, B σ particle, to the total energy density which is presented in figure 7 and is subleading comparing to the above quantities. As we see, both B A and B ϕ are proportional to the scale of inflation as (H/M Pl ) 2 and hence decrease by reducing the scale of inflation. These dimensionless backreaction terms are shown in figure 10. We require both B A and B ϕ to be lower than 0.01 so that they are smaller than the slow-roll suppressed terms in the background field equation. Figure 11 shows the available parameter space corresponding to each scale of inflation.
Among the three backreactions, B A , B ϕ , and δ B ρ ρ , the first one is the largest.  given ψ and a bound on B A , lowering H by a factor of 10 enlarges the acceptable domain of ξ A and ξ Z 0 by one unit (see figure 11). For instance, for a ψ M Pl ∼ 10 −2 and a GUT scale inflation with H ∼ 10 −6 M Pl , we find that 0 ≤ ξ Z 0 < 4 and √ 2 < ξ A 3.5 (lower bound comes from stability of the scalar sector) are allowed. However, for a lower scale of inflation with H ∼ 10 −7 M Pl , we find that 0 ≤ ξ Z 0 < 5 and √ 2 < ξ A 4.5 are allowed. In this region, the energy density fraction of the spin-2 field is δ B ρ ρ 10 −7 and decreases linearly with the decrease of H.

Parameter space of massless models
Up to now, our formulae have been presented in the unified form and are valid for any models assuming slow-roll dynamics of the VEV of the gauge field and quasi-de Sitter expansion. In this section, we use our backreaction formulae to further constrain the massless models, i.e., ξ Z 0 = 0 [11,15,40].
The total slow-roll parameter, , is given by where ϕ and A are respectively the contributions of the axion and the gauge fields to the slow-roll parameter, whereas χ is the contribution of the inflaton field in the spectator case (with α s = 1). In massless models with ξ Z 0 = 0 and in the regime ξ A 2.5, the extra scalar fields have little effect on the scalar power spectrum and scalar tilt, and thus we have the standard result given by P ζ 1 2 H 2πM Pl 2 and n s − 1 −2(3 − η). On the other hand, adding the sourced tensor power spectrum given in (5.6) to the standard vacuum part, we have the total tensor power spectrum given by (6.4) The tensor-to-scalar ratio, r ≡ P T /P ζ , is where r vac = 16 is the standard vacuum value, whereas r source is the contribution from the gauge field. Using P ζ = 2.2 × 10 −9 and A in (2.13), we write r and B A in terms of r vac , A / and ξ A as and The exact form of K[ ξ A ] is given in (4.11). It is approximately given by K[ In this section we constrain the parameter space of models in B -ξ A plane, following refs. [21,39].
. We can relate this to the backreaction term as which puts a lower bound on B from backreaction. We restrict the parameter space by imposing the following constraints: JHEP05(2019)174 • The BICEP2/Keck and Planck (BKP) upper bound on the tensor-to-scalar ratio, r < 0.07 (95% C.L) [43]; • The Planck upper bound on the tensor non-Gaussianity parameter, f tens NL [21,44]; • Small backreaction given by B A < 10 −2 or 10 −1 ; • Small Schwinger pair-creation of scalar fields [39]; • Consistency of the slow-roll parameter, A < ; We show the constraints in figure 12 and 13 for r vac = 10 −2 , 10 −3 , and 10 −4 . We find that large parameter space is excluded already. In particular, we find that r source cannot be much greater than r vac . To see this in more detail, in figure 14 we show the ratio r source /r vac = (r − r vac )/r vac as a function of ξ A for given values of A / and r vac . Imposing the bound on the size of the backreaction, B A < 10 −2 , as well as on A / < 0.9, we find that the maximum possible value of (r−r vac )/r vac can be at most 5 for r = 3×10 −3 , and smaller for smaller r vac because of r source /r vac ∝ r vac for a given A / ; see (6.6). The constraints weaken when we impose a weaker bound on the backreaction, B A < 10 −1 . For larger values of r vac = 3 × 10 −2 , the strongest constraint comes from the BKP upper bound on r < 0.07 rather than from backreaction, yielding (r − r vac )/r vac < 1.3. The allowed parameter space we find in this section is much more constrained than that found in the literature for the spectator axion-SU(2) model [15,40]. The reason is two folds. First, the upper limit of B 10 −2 adopted by the previous study is too conservative to satisfy the consistency of the slow-roll parameters given by B / = 16 B /r vac < 1 for a given value of r vac < 0.07. Second, the region of the strong backreaction was defined as J A /(g A λψ 2φ /f ) < 1, which is too conservative to satisfy the slow-roll dynamics of the gauge field. The field equation of ψ for the massless case is given in (2.14)

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Slow-roll demands V eff,ψ (ψ) 1, while each of the terms in the right hand side can be much larger, e.g., g A λφ f ψ 2 /V eff,ψ 1. On the other hand, J A should be at most on the order of the slow-roll suppressed terms, i.e., J A H 2 ψ ≡ B A 1, which is more restrictive. Before we leave this section, let us comment on the Higgsed models, e.g., Higgsed gauge-flation and Higgsed chromo-natural models [12,14,16]. Both scalar and tensor perturbations are amplified by the Higgs VEV in this set up, with the scalar ones being more strongly amplified; thus, the Higgs VEV, quantified by ξ Z 0 , reduces the tensor-toscalar ratio [14,16]. On the other hand, in section 4, we find that the size of the backreaction is much stronger in the Higgsed models with ξ Z 0 = 0. Hence, requiring slow-roll dynamics in the gauge field sector during (quasi)-de Sitter expansion, the Higgsed models cannot evade r/r vac < O(1) bound either.

Conclusion
A background of axion and SU(2) gauge fields produces a copious amount of spin-2 particles during inflation. In this paper, we have calculated the number [eq. (3.31)] and energy densities [eq. (4.22)] of the spin-2 particles as well as their backreaction on the equations of motion of the axion and gauge field backgrounds [eqs. (4.9) and (4.10)]. We provided analytical formulae which are valid for all the inflation models with SU(2) gauge fields studied in the literature (see eq.s (2.24) and (2.25) for the definition of the model parameters and table 2 for their correspondence to the literature). The former results are new. The latter results were presented only numerically for a single model in this family but in [40]. With that exception, this is the first time that the backreaction constrains this family of models on the equations of motion. The analytical formulae derived in this paper allow us to easily estimate the importance of the backreaction for any parameters and constrain the parameter space that is consistent with perturbation theory. Moreover, it enables us to relate the backreaction in the gauge field and axion background equations to the number density of the spin-2 field as J A ∼ g A n pairs and P ϕ ∼ λH f n pairs respectively. These spin-2 particles mix with gravitational waves. We related the number density of the spin-2 particles to the power spectrum [eq. (5.6)] and energy density [eq. (5.12)] of primordial gravitational waves from inflation as well as to the size of the backreaction [eq. (4.17)]. The relation to the energy density δρ GW s is intuitive: δρ GW s is given by the number density of spin-2 particles times particle's physical momentum at horizon crossing (i.e., k/a ∼ H), times the coupling strength squared, i.e., δρ GW s ∼ (ψ/M Pl ) 2 Hn pairs . Moreover, the ratio of the power spectra of sourced and vacuum gravitational waves is also proportional to the number density of the spin-2 field and the VEV of the SU(2) gauge field, i.e., P s T /P vac T ∼ (ψ/M Pl ) 2 (n pairs /H 3 ). That gives us a physical insight into how the strength of gravitational waves from SU(2) gauge fields is determined.
Finally, we constrained the parameter space of the massless models in this class of inflationary scenarios. We find that the backreaction and the consistency of the slow-roll condition exclude most of the parameter space. In particular, the tensor-to-scalar ratio of the gravitational waves sourced by the gauge field can at most be on the order of that of JHEP05(2019)174 the standard vacuum contribution. Going beyond the massless models, we argue that the Higgsed models cannot evade this conclusion either.
The analytical study presented in this paper is based on quasi-de Sitter expansion and slow-roll dynamics of the background gauge field. For more general situations, full numerical analysis is required. At second order in perturbation, the spin-2 field couples to the scalar sector and contributes to the scalar power spectrum and non-Gaussianity. This non-linear effect may be important [45]. We expect that the loop contribution to the scalar power spectrum is related to the number density of the spin-2 field, n 2 pairs /H 6 , that we computed in this paper.

A Symmetry of the VEV SU(2) field
In section 2, we present the metric and the VEV of the gauge field in a specific coordinate system in which the spatial metric is a 2 δ ij . Here we present the general solution of the VEV of the gauge field which generates a homogeneous and isotropic energy-momentum tensor in general spatial coordinates.
Fixing the time-function, t, under a global rotation, we have where Λ 0 ν = 0 and the tetrad fields (η αβ = e α µ e βµ ) transform as Since we choose to use the tetrad system with e a 0 (t) = 0 and e a i (t) = aδ a i , under the action of (A.1), we have e a 0 (t) = 0. The general form of the gauge field's VEV in the temporal gauge that can generate a homogeneous and isotropic energy-momentum tensor (hence respect the symmetries of the FLRW background) is given bȳ More precisely, using (A.2) and the above solution, the field strength tensor is which leads to a homogeneous and isotropic energy-momentum tensor for the gauge field sector. In addition to the above global rotational symmetry, there is a residual gauge symmetry, a continuous global SU(2) symmetry, as well U (α) = exp(iα a T a ), (A.5)

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which respects the temporal gauge and the form of the VEV in (A.3) It is straightforward to see that the VEV gauge field in (A.3) also satisfies Finally, action has a Z 2 symmetry, Parity, as ψ → −ψ (and for L A = L Cn ϕ → −ϕ), which can be spontaneously broken by the VEV while it is still the symmetry of the background energy-momentum tensor.

B Transverse-traceless field
In this part, we expand the gauge field's action around the VEV as 11 where A SV µ (x ν ) parameterizes the scalar and vector modes in A a µ (longitudinal modes), 12 while B µ (x ν ) is the transverse part of the gauge field where Q, Z, andz are scales, vi, wi andγij are transverse fields andγij is symmetric. However,γij is the only field that contributes to the transverse part of the A a i . 13 For an Abelian field with action (2.6), the quadratic action of Bµ is

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where δ B denotes terms quadratic order in B µ ,ᾱ A is defined in (2.16), while the explicit form of the covariant derivative,D µ , and the mass terms are 14 Note that (M µ B ν ) 2 cancels the term proportional toĀ µĀν in the (D µ B ν ) 2 . Therefore, the only actual masses are the one proportional toᾱ A and α H . One can determineα as a function ofĀ µ by the gauge field background equation in (2.15). Notice that a given value ofĀ µ and Z 0 gives the same value forᾱ A regardless of whether L A is L Gf or L Cn . That indicates that their tensor sectors are the same at the linear order.
We emphasize that unlike the U(1) field in which its transverse part is a spin-1 field, the transverse part of A a µ (in the temporal gauge) can be written as In the following, we prove thatγ ij is a (pseudo) spin-2 field.

B.1 (pseudo) spin-2 in perturbed SU(2) field?
At first, it may come as a surprise that there is a (pseudo) spin-2 degree of freedom in a spin-1 SU(2) gauge field as in (2.20). In this appendix, we show that once the gauge field is perturbed around its isotropic and homogeneous solution (2.2), there is a sector in the perturbed field that transforms as a spin-2 field under rotations and is an odd eigenstate of parity. 15 It is convenient to write the fields in the complex spherical coordinates (r, z,z) which are related to the standard spherical coordinates (r, θ, φ) as z = e iφ tan θ 2 andz = e −iφ tan θ 2 . (B.11) The FLRW background geometry in this coordinate system is Moreover, consider a choice of the spatial triads (e α i e αj = g ij ) as e α r = a(t)δ α r and e α z = a(t)δ α z , (B.14) 14 Note that comparing to the background covariant derivativeDµ = ∇µ − igAĀµ, the gauge field has a factor of 2 in (B.7). That comes from the fact that [XµYν ] a = 1 2 [Xµ, Yν ] a and therefore ([Dµ, Bν ]) a = (DµBν ) a . 15 Another way to see the pseudo-tensor nature of the spin-2 degree of freedom is the fact that it always couples with the tensor metric perturbation with a factor of ψ which is a pseudo-scalar.

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where α is the index of the SO(3) algebra and runs from 1 to 3. One can write B a i in (B.10) as B a i =γ ij (t, x)e αj δ a α .

(B.15)
Then in the (r, z,z) coordinates and for a wave propagating in the directionk = −r, we have B a r = 0 while B a z and B ā z are non-zero. Moreover, given the fact that tr(γ ij ) = 0, the only non-zero components ofγ ij areγ zz andγzz. Under the action of a rotation around the x 3 −direction, we have Thus,γ ij is a pseudo-tensor. This completes the proof thatγ ij is a (pseudo) spin-2 degree of freedom.

C Mathematical supplement
Here, we present some mathematical formulae and relations which we need throughout this work including some properties of Gamma and Whittaker functions as well as the asymptotic form of the Meijer G-functions in the large argument limit. The Gamma function has simple poles for non-positive integers

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The derivative of the Γ-function can be written as a polygamma function which has the following series representation which holds for d > 1 and any complex z not equal to a negative integer. Therefore, the harmonic series can be written as Furthermore, the asymptotic series which is valid in the sector |arg(z)| < π, leads to the following asymptotic expansion of the digamma function For complex values of z in which z = x + iy with finite real x and y → ∞, we have [46] Γ(x + iy) √ 2π|y| x− 1 2 e −πy/2 e −i(y+ π 2 ( 1 2 −x)) . (C.9) Finally, in the limit that z goes to zero, Gamma functions satisfy lim z→0 zΓ(−z − n) = − (−1) n n! and lim z→0 d dz (zΓ(−z − n)) = (−1) n n! ψ (0) (n + 1), (C.10) where n ∈ N. For later convenience, we recall that in the complex analysis, if f (z) has a pole of order k at z = z 0 then the residues are given as (C.11) The W and M Whittaker functions, which are the solutions of (2.26), have the asymptotic expansions for |z| → 0, (C. 13) implying that W /M functions correspond to positive frequency modes in the asymptotic past/future limits of de Sitter, respectively.

D Computation of the induced currents
In this appendix, we compute the momentum integral K[X] in (4.8) which is necessary for the induced currents and energy density. 16 First, we work out the total integral which is divergent. In appendix D.1, we regularize this current by using adiabatic subtraction. Finally, in appendix D.2, we compute the energy density of the spin-2 fluctuations.

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It is convenient to decompose K[X] in terms of polarizations as Using (2.28) and (4.8), we can write K σ [X] as whereτ is a rescaled physical momentum and Λ is the physical UV cutoff which in the end will be sent to infinityτ ≡ k aH and Λ = k UV aH .
Note that in our setup, both κ and µ are pure imaginary. In the following, for notational convenience, we drop the argument X of K σ [X] and σ subscript in κ σ and λ σ , unless otherwise stated. Upon using the integral representation of Whittaker functions in (C.15), we find The integrand has singularities at s = − 1 2 ± µ − n, κ + n, 3 − s, and 2 − s. Moreover, it is proportional to Λ 2−s−s which vanishes for Re(s ) > 3 − Re(s) in the limit Λ → ∞. Upon choosing the contour of s such that Re(s) > −1 and closing the s -contour in the right-half plane without passing through the poles, 17 we are left with the following six poles s 1 = κ, s 2 = κ + 1, s 3 = κ + 2, s 3 = κ + 3, s 5 = 2 − s, and s 6 = 3 − s.
Doing the s -integral, we obtain 17 Note that the integral of (D.2) over a finite path along the real axis vanishes at lim Im(s ) → ±∞.

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Adding (D.21) and (D.22), we finally find the desired K σ as It has divergent terms of the order 3, 2, 1 and log of Λ. Summing over the polarization states, we find Re (e 2|κ|π +e 2|µ|π ) sinh(2|µ|π) We find that K is real and has divergent terms of the order 2 and log of Λ. Before renormalizing K and removing Λ terms, let us take a closer look at the finite terms in (D.24). Recalling (2.10) and (3.18) (|κ| > 3.5), we realize that the dominant finite terms are proportional to e 2(|κ|−|µ|)π > e πξ A 1. Thus, we can approximate K[X] as where O(ξ 3−m A ξ m Z 0 ) with m = 0, 1, 2, 3 are the next leading terms and hence negligible. The quantity (|κ| − |µ|) is presented in figure 3. In the next section, we renormalize the above and find the physical quantity, K reg [X], as where K c.t.
[X] is the counter-term.

D.1 Regularized current
In this section, we use the adiabatic subtraction technique in curved QFT to remove the divergent terms in the current. The mode function B σ, k (τ ) has the following WKB form  where O(ξ 3−m A ξ m Z 0 ) with m = 0, 1, 2, 3 are the next leading terms and hence negligible. 19 Note that the instantaneous frequency squared is negative in the intervalτ ∈ (τ2,τ1) in (3.7). Therefore, technically, we have to consider an IR cut-off for the momentum integral, ΛIR τ1. In principle it can be a problem since ΛIR explicitly appears in the finite terms. However, this effect and the other finite terms in K c.t. are at most of the order O(ξ 3−m A ξ m Z 0 ) with m = 0, 1, 2, 3 and are subleading comparing with the dominant finite terms of K in (D.25). As a result, a more careful regularization process would not improve our results. as a function of ξ A and ξ Z0 . Hereτ IR is the physical momentum in which the deviation from adiabaticity is Ω + (τ IR ) = 0.1. The ratio ∆ for the minus polarization is smaller with the same order of magnitude.

E Sourced graviational waves
In this appendix, we work out the analytical form of the gravitational waves sourced by the gauge field for the general action (2.4). The derivation given here follows closely [11]. The inhomogeneous solution of (2.24) sourced by B σ is given as