Gravitational Wave from Axion-SU(2) Gauge Fields: Effective Field Theory for Kinetically Driven Inflation

Phenomenological success of cosmic inflation [1, 2] requires a flat potential for a slowly-rolling scalar field $\phi$ [3, 4]. Since the seminal work by Freese, Frieman and Olinto [5], shift symmetry, symmetry under a constant shift of $\phi\to\phi+c$, has often been used to construct the necessary flat potential. In this setup, a pseudo Nambu-Goldstone boson, an axion field, is identified as the inflaton field, and the flat potential emerges as a consequence of softly broken shift symmetry (e.g., by the instanton effect).

Another approach is to drive inflation by kinetic terms, ${\cal L}=K(\phi)X+L(\phi)X^{2}+\dots$, with $X\equiv-(\partial\phi)^{2}/2$ [6]. While $X$ is shift symmetric, the coefficients $K$ and $L$ may not be. Nevertheless, we can demand softly broken shift symmetry by requiring $K$ and $L$ to depend on $\phi$ only weakly.

In this paper, we construct an effective Lagrangian for a pseudo scalar field with shift symmetry. The basic idea follows from Weinberg’s effective field theory for inflation [7]; namely, the number of spacetime derivatives is less than or equal to four. We then retain terms that are shift symmetric. A novel feature of our Lagrangian is that we also add the shift symmetric Chern-Simons coupling to (non-Abelian) gauge fields, $\phi{F\tilde{F}}$ [8, 9]. We find that our construction predicts a lower bound for the tensor-to-scalar ratio of $r\gtrsim 5\times 10^{-3}$ with partially chiral and non-Gaussian gravitational waves. In our setup, shift symmetry breaking effects can explain the tilt of the scalar curvature power spectrum, $n_{\rm s}<1$, discovered by the cosmic microwave background (CMB) experiments [10, 11, 12].

The rest of this paper is organized as follows. In section 2, we explain our construction of the effective Lagrangian for a pseudo scalar field with shift symmetry. In section 3, we derive the background equations of motion for the scalar and gauge fields and find approximate solutions with softly broken shift symmetry. In section 4, we analyze the scalar and tensor perturbations and calculate observables such as the scalar spectral tilt and non-Gaussianity as well as the tensor-to-scalar ratio, chiraity, and non-Gaussianity of the primordial gravitational wave. In section 4.3, we constrain the model parameter space using sizes of backreaction of spin-2 particles produced by the gauge field. We conclude in section 5.

We start with the kinetic Lagrangian with no more than four spacetime derivatives [6, 7]:

$\displaystyle{\cal L}_{0}=\sqrt{-g}\left[(M_{\rm p}^{2}/2)R+a_{1}X+a_{2}X^{2}\right]\ ,$

(2.1)

where $g\equiv\det{g_{\mu\nu}}$, $X\equiv-\partial_{\mu}\phi\partial^{\mu}\phi/2$, $R$ is the Ricci scalar, $M_{\rm p}=(8\pi G)^{-1/2}$ is the reduced Planck mass, and $a_{1}$ and $a_{2}$ are coefficients characterized by the mass scale $M$ of theory. To achieve inflation, we need $a_{1}<0$ and $a_{2}>0$. Then the cosmic expansion rate $H\equiv\dot{a}/a$ ($a$ is the cosmic scale factor and an over-dot is a time derivative) is given by $M_{\rm p}^{2}H^{2}\sim a_{2}X^{2}\sim M^{4}$ and $\epsilon\equiv-\dot{H}/H^{2}=0$; a phase of the exact de Sitter inflation is realized without a potential [6, 13, 14, 15]. The configuration $a_{1}<0$ and $a_{2}>0$ has been used to achieve “ghost inflation” [15] by forming “ghost condensate” [14]. If $a_{1}<0$ and $a_{2}<0$, the system is unstable since the Hamiltonian is not positive-definite.

Any additional derivatives acting on $\partial_{\mu}\phi$ or on the metric yield factors of order $H\sim M^{2}/M_{\rm p}\ll M$, which guarantees that (2.1) is the leading terms in the low-energy effective field theory for inflation, and any correction terms are suppressed by factors of $H/M$. The leading correction to (2.1) consists of a sum of all generally covariant and shift symmetric terms with four spacetime derivatives. It can be put in the form

$\displaystyle\Delta{\cal L}_{1}=\sqrt{-g}\left[a_{3}X\Box\phi+a_{4}G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+a_{5}\phi R_{\rm GB}\right]+a_{6}\phi\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu}{}^{\kappa\lambda}R_{\rho\sigma\kappa\lambda}\ ,$

(2.2)

where $\Box\phi\equiv g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi$, $G^{\mu\nu}\equiv R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R$ is the Einstein tensor, $R_{\rm GB}\equiv R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ is the Gauss-Bonnet scalar, and $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor density with $\epsilon^{0123}\equiv+1$. The coefficients $a_{i}$ ($i=3,4,5,6$) are characterized by the mass scale $M$ (or higher scale like $M_{\rm p}$ depending on underlying ultra-violet theory).

The correction terms given in (2.2) are the most general ones with four spacetime derivatives [7]. The other terms such as $(\Box\phi)^{2}$, $R^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$, $R\Box\phi$, and so on, are eliminated by the field equations. We can estimate sizes of the correction terms as $a_{3}X\Box\phi\sim HM^{3}$, $a_{4}G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\sim H^{2}M^{2}$, $a_{5}\phi R_{\rm GB}\sim H^{3}M$, and $a_{6}\phi\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu}{}^{\kappa\lambda}R_{\rho\sigma\kappa\lambda}\sim H^{3}M$, with $X\sim M^{4}$ and $\phi\sim\dot{\phi}\Delta t\sim M^{2}/H$. Since terms with six spacetime derivatives like $(\Box\phi)^{3}$ would be on the order of $H^{3}M$, we ignore the last two terms in (2.2); moreover, they can be rewritten as bilinear in the Weyl tensor that vanishes at the background level in a conformally flat spacetime [7].

The first three terms in (2.2) introduce no auxiliary field (known as the Ostrogradski ghost mode) and theory is stable on a cosmological background [16, 17, 18], while the last term in (2.2) causes instability if ${\cal L}_{0}+\Delta{\cal L}_{1}$ is taken as the full Lagrangian [19]. The last term is parity violating in a nontrivial background of $\phi$; thus, it results in chiral gravitational waves with different amplitudes of right- and left-handed helicities [20]. Chirality can be as large as several tens of percent when the cut-off scale of theory, $M$, is as low as $M=20H$ [21]; the effect becomes smaller for a larger cut-off (e.g., Planck scale [22, 23]). Specifically, we assume $M\sim\sqrt{HM_{\rm p}}$ in this paper. The (perturbative) strong coupling scales in the gravity-scalar sector were studied in [24] where they found $M=\epsilon^{1/4}\sqrt{HM_{\rm p}}$ if (2.1) dominates among others.

Finally, we add another shift symmetric term to the Lagrangian, a Chern-Simons coupling term between $\phi$ and gauge fields. In this paper we focus on SU(2) gauge fields, as they (or a SU(2) subgroup of non-Abelian gauge fields) acquire an isotropic and homogeneous background solution during inflation when conformal invariance is broken by a four derivative operator $(F\tilde{F})^{2}$ [25, 26] or by a Chern-Simons interaction $\phi F\tilde{F}$ [9], where $F$ is the gauge field strength tensor. The former can be obtained as an effective Lagrangian of the latter by integrating out the massive $\phi$ on energy scales below its mass scale [27, 28, 29]. We assume that a global symmetry breaking scale $f$ lies in a range of $H<f<M_{\rm p}$. The gauge sector is given by

$\displaystyle\Delta{\cal L}_{2}=-\frac{1}{4}\sqrt{-g}F_{\mu\nu}^{a}F^{a\mu\nu}-\frac{\lambda}{8f}\phi\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^{a}F_{\rho\sigma}^{a}\ ,$

(2.3)

where $F^{a}_{\mu\nu}\equiv\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+g_{A}\epsilon^{abc}A^{b}_{\mu}A^{c}_{\nu}$ is the field strength tensor, $g_{A}$ is the gauge coupling constant, superscripts $a$, $b$, $c$ are the SU(2) group indices, and summation is assumed for repeated indices. $\lambda$ is a dimensionless coefficient associated to microphysics of the axion [30].

As perturbations in the SU(2) field around the homogeneous and isotropic vacuum expectation value contain tensor modes [25, 26], it can source gravitational waves at linear order. The resulting signal is chiral [31, 32, 29, 33] and non-Gaussian due to self-coupling of SU(2) gauge fields [34, 35, 36, 37].¹¹1A similar phenomenology is obtained with a Chern-Simons coupling with a U(1) gauge field [8]. The sourced gravitational wave is chiral [38] and non-Gaussian because it is sourced non-linearly by the quadratic term in the stress-energy tensor [39, 40, 41].

In summary, our effective action is given by

	$\displaystyle S=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm p}^{2}}{2}R+a_{1}X+a_{2}X^{2}+a_{3}X\Box\phi+a_{4}G^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\right.$
	$\displaystyle\left.-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}-\frac{\lambda}{8f\sqrt{-g}}\phi\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^{a}F_{\rho\sigma}^{a}\right]\ ,$		(2.4)

where we have set $a_{5}=a_{6}=0$ based on the estimates of their magnitudes given above. If we impose symmetry under parity, the term with $a_{3}$ is absent. In the present model, parity symmetry will be spontaneously broken by the Chern-Simons interaction once the gauge field acquires a nontrivial background value. Thus, we consider the case $a_{3}\neq 0$ as well.

Here we provide comparison with the previous work on the scalar field Lagrangian non-minimally coupled to gravity. A canonical scalar field corresponds to $a_{1}=1$. The terms with $a_{1}$ and $a_{4}$ are included in “UV-protected inflation” [42, 43] with $a_{1}=1$ and $a_{4}=1/(2M^{2})$ in their notation. The terms with $a_{1}$, $a_{2}$, and $a_{3}$ are included in “G-inflation” [44] if we set their free functions to $K=a_{1}X+a_{2}X^{2}$ and $G=-a_{3}X$, which is equivalent to “kinetic gravity braiding” [45] if $K=a_{1}X+a_{2}X^{2}$ and $G=a_{3}X$ (or $a_{1}=1$ and $a_{3}=-M(\phi)$ in the notation of [46]). The terms with $a_{1}$ and $a_{3}$ are included in “galileon inflation” [47] if we set their coefficients to $c_{2}=a_{1}$ and $c_{3}\Lambda^{-3}=-a_{3}/2$. In “generalized G-inflation” [18], our scalar field Lagrangian is realized when $K=a_{1}X+a_{2}X^{2}$, $G_{3}=-a_{3}X$, $G_{4}=M_{\rm p}^{2}/2$, and $G_{5}=-a_{4}\phi$, which is equivalent to $G_{4}=M_{\rm p}^{2}/2+a_{4}X$ up to total derivative. The terms with $a_{4}$ and $a_{5}$ are included in “the Fab Four” [48] if we set their free functions to $V_{\rm george}=M_{\rm p}^{2}/2$, $V_{\rm john}=a_{4}$, $V_{\rm ringo}=a_{5}\phi=0$, and $V_{\rm paul}=0$. Our aim in this paper is not to work with the most general Lagrangian for a scalar field with shift symmetry, but to work with the Lagrangian that is valid in the low-energy effective field theory.

Shift symmetry results in the exact de Sitter expansion, which fails to explain a small but non-zero tilt of the scalar curvature power spectrum [10, 11, 12]. In our setup, shift symmetry may be broken in three ways. The first possibility is to introduce a potential, e.g., $V(\phi)=V_{,\phi}\phi$, where $V_{,\phi}$ is nearly constant. The specific potential form is not important for realizing inflation in our model, since inflation is assumed to be driven by the kinetic terms given in (2.1). The operators $\phi$ and $\phi^{2}$ are protected by nonrenormalization theorem; thus, shift symmetry is softly broken [47]. Any periodic potential arisen from the instanton mechanism breaks shift symmetry without receiving large quantum corrections. The second possibility is to introduce weak $\phi$ dependence on the coefficients, $a_{i}$. Since (2.1) is the leading part, we shall assume $a_{1}=a_{1}(\phi)$, $a_{2}=a_{2}(\phi)$, $a_{3}=$ const., and $a_{4}=$ const. for simplicity; thus, ${\cal L}_{0}\to\sqrt{-g}\left[(M_{\rm p}^{2}/2)R+a_{1}(\phi)X+a_{2}(\phi)X^{2}-V(\phi)\right]$. The third possibility is backreaction of particle production by the gauge field on the equation of motion for $\phi$, which turns out to be too small to break shift symmetry effectively (section 4.3). In any case, the symmetry breaking terms should be understood as small.

We take the flat, homogeneous and isotropic background such that $ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)d{\bf x}^{2}$, $\phi=\phi(t)$, $A_{0}^{a}=0$, and $A_{i}^{a}=\delta_{i}^{a}a(t)Q(t)$ [25, 26]. This homogeneous and isotropic configuration of the gauge field is an attractor solution during inflation [49, 50]. After finding the Hamiltonian constraint (i.e., the Friedmann equation), we set the background lapse function to $N(t)=1$.

The field equations for $\phi$ and $Q$ are given by

	$\displaystyle\dot{J}$	$\displaystyle+3HJ-K_{,\phi}=-3\frac{g_{A}\lambda}{f}Q^{2}(\dot{Q}+HQ)\ ,$		(3.1)
	$\displaystyle J$	$\displaystyle\equiv\dot{\phi}(a_{1}+a_{2}\dot{\phi}^{2}-3a_{3}H\dot{\phi}+6a_{4}H^{2})\ ,\quad K\equiv a_{1}X+a_{2}X^{2}-V\ ,$
	$\displaystyle\ddot{Q}$	$\displaystyle+3H\dot{Q}+(\dot{H}+2H^{2})Q+2g_{A}^{2}Q^{3}=\frac{g_{A}\lambda}{f}\dot{\phi}Q^{2}\ ,$		(3.2)

where $K_{,\phi}\equiv\partial K/\partial\phi\$. The last term in the left hand side of (3.2) gives an effective mass term for the gauge field background. We write this as $2g_{A}^{2}Q^{3}=2m_{Q}^{2}H^{2}Q$ with $m_{Q}\equiv g_{A}Q/H$.

The flat Friedmann equations are given by

	$\displaystyle 3M_{\rm p}^{2}H^{2}$	$\displaystyle=\rho_{\phi}+\rho_{A}\ ,$		(3.3)
	$\displaystyle-3M_{\rm p}^{2}H^{2}-2M_{\rm p}^{2}\dot{H}$	$\displaystyle=p_{\phi}+p_{A}\ ,$		(3.4)
	$\displaystyle\rho_{\phi}$	$\displaystyle=\dot{\phi}J-K+6a_{4}H^{2}X\ ,$
	$\displaystyle p_{\phi}$	$\displaystyle=K+2a_{3}X\ddot{\phi}-6a_{4}H^{2}X-4a_{4}\dot{H}X-4a_{4}H\dot{X}\ ,$
	$\displaystyle\rho_{A}$	$\displaystyle=\frac{3}{2}(\dot{Q}+HQ)^{2}+\frac{3}{2}g_{A}^{2}Q^{4}\ ,\quad p_{A}=\frac{1}{3}\rho_{A}\ .$

Combining equations (3.3) and (3.4), we get

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}J}{2H^{2}M_{\rm p}^{2}}+\frac{a_{3}X\ddot{\phi}}{H^{2}M_{\rm p}^{2}}+2\frac{a_{4}X\epsilon}{M_{\rm p}^{2}}-2\frac{a_{4}\dot{X}}{HM_{\rm p}^{2}}+\frac{2\rho_{A}}{3H^{2}M_{\rm p}^{2}}\ .$

(3.5)

Noting a relation $\epsilon=3(1+w)/2$ with $w\equiv p/\rho$ being the equation of state ($p$ and $\rho$ are total pressure and energy density, respectively), we would expect $w\simeq 1/3$ and the universe becomes radiation dominated if the last term in (3.5) becomes dominant. However, we will show in the following that this is not the case for non-vanishing $\lambda$, $g_{A}$, and $Q$, and the universe is inflationary with $w\simeq-1$ in a quasi-stationary state regardless of the fraction $\rho_{A}/\rho$.

We solve the $\phi$ field equation (3.1) iteratively using shift symmetry. Ignoring the $\phi$ dependence (i.e., $K_{,\phi}$), we get the zeroth iterative solution:

$\displaystyle J^{(0)}=-\frac{g_{A}\lambda}{f}Q^{3}+\frac{C}{a^{3}}\ ,$

(3.6)

where the second term is a decaying solution and $C$ is an integration constant. Here, $J$ is a conjugate momentum of $\phi$ in the absence of the gauge field background $Q$. In the presence of a nontrivial $Q$, the conserved charge associated to shift symmetry is $C=a^{3}\left(J+\frac{g_{A}\lambda}{f}Q^{3}\right)$. If $Q$ is constant, the solution implies $\dot{\phi}$ and $H$ are constant and the exact de Sitter expansion ($w=-1$) is realized.

Plugging (3.6) into (3.1), we get the first iterative solution:

$\displaystyle J^{(1)}=-\frac{g_{A}\lambda}{f}Q^{3}+\frac{K_{,\phi}}{3H}+\frac{C}{a^{3}}\ ,$

(3.7)

where we have assumed that $H$ and $K_{,\phi}$ are nearly constant. The first iterative solution is enough to obtain a quasi de Sitter expansion for a slightly tilted spectrum of curvature perturbations.

A nontrivial, stationary value of $Q$ can be obtained from the $Q$ field equation (3.2). To see the appearance of a nontrivial $Q$, we define an effective potential for $Q$:

$\displaystyle U_{\rm eff}(Q)\equiv\frac{1}{2}(\dot{H}+2H^{2})Q^{2}+\frac{1}{2}g_{A}^{2}Q^{4}-\frac{g_{A}\lambda}{3f}\dot{\phi}Q^{3}\ ,$

(3.8)

which acquires a nontrivial minimum for $\dot{\phi}\neq 0$:

$\displaystyle Q_{*}=\frac{\lambda\dot{\phi}}{4g_{A}f}+\frac{\lambda}{4g_{A}f}\sqrt{\dot{\phi}^{2}-16\frac{f^{2}H^{2}(1-\epsilon/2)}{\lambda^{2}}}\ ,$

(3.9)

where $|\lambda\dot{\phi}/f|>4H$ must be satisfied and $\lambda\dot{\phi}/(4g_{A}f)<Q_{*}<\lambda\dot{\phi}/(2g_{A}f)$ for positive values. Equivalently, we can solve the stationary condition $U_{\rm eff}^{\prime}(Q)=0$ for $\dot{\phi}$:

$\displaystyle\dot{\phi}_{*}=\frac{2fg_{A}Q}{\lambda}+\frac{2fH^{2}(1-\epsilon/2)}{\lambda g_{A}Q}\ .$

(3.10)

Plugging (3.7) and (3.10) into (3.5), we find that the first and last terms in (3.5) nearly cancel, finding a value of $\epsilon$ at stationary trajectory as

$\displaystyle\left(1-\frac{2a_{4}X}{M_{\rm p}^{2}}-\frac{Q^{2}}{2M_{\rm p}^{2}}\right)\epsilon_{*}=\frac{\dot{\phi}K_{,\phi}}{6H^{3}M_{\rm p}^{2}}+\frac{a_{3}X\ddot{\phi}}{H^{2}M_{\rm p}^{2}}-\frac{2a_{4}\dot{X}}{HM_{\rm p}^{2}}+\frac{\dot{Q}^{2}}{H^{2}M_{\rm p}^{2}}+\frac{2Q\dot{Q}}{HM_{\rm p}^{2}}\ .$

(3.11)

As stated before, quasi de Sitter expansion ($w\simeq-1$) is realized for the attractor solution $\dot{\phi}=\dot{\phi}_{*}$. In the limit $K_{,\phi}\to 0$, the solutions indicate $\dot{\phi}\to$ const., $Q\to$ const., and then $\epsilon\to 0$ (i.e., $w\to-1$). Since we are interested in the quasi-static state, we assume that the field values change slowly as

$\displaystyle\left|\frac{\ddot{\phi}}{H\dot{\phi}}\right|\ll 1\ ,\quad\left|\frac{\dot{Q}}{HQ}\right|\ll 1\ .$

(3.12)

So far, we have not taken into account backreaction of particle production by the gauge field on the background equations of motion, which modifies the solutions (3.7) and (3.11). We shall take backreaction into account in section 4.3.

The clock of the system is set by the uniform energy density of $\phi$ when $\rho_{\phi}\gg\rho_{A}$. In the unitary gauge $\delta\phi=0$ at all orders, we write the spatial metric as $g_{ij}=a^{2}e^{2\zeta}[e^{h}]_{ij}=a^{2}e^{2\zeta}(\delta_{ij}+h_{ij}+h_{ik}h_{kj}/2+\dots)$, where $h_{ij}(t,{\bf x})$ is a symmetric, traceless and divergence-free tensor.

When the effective mass of the gauge field, $m_{Q}=g_{A}Q/H$ (3.2), is small and gauge scalar perturbations do not decouple from the system, instability appears in the scalar perturbation for $m_{Q}<\sqrt{2}$ [33, 31]. Thus, we shall assume $m_{Q}>\sqrt{2}$ so that gauge scalar perturbations decouple.

However, tensor perturbations do not decouple and one of the $\pm 2$ helicity states undergoes an exponential amplification due to instability. This tensor mode instability is essential to obtain chiral [31, 32, 29, 33] and non-Gaussian [34, 35, 36, 37] gravitational waves.