Resonance instability of primordial gravitational waves during inflation in Chern-Simons gravity

We investigate a kind of inflationary models where the Chern-Simons term is coupled to a periodic function of the inflaton. We find that tensor perturbations with different polarizations are amplified in different ways by the Chern-Simons coupling. Depending on the model parameters, the resonance amplification results in a parity-violating peak or a board plateau in the energy spectrum of gravitational waves, and the sharp cutoff in the infrared region constitutes a characteristic distinguishable from stochastic gravitational wave backgrounds produced by matter fields in Einstein gravity.


I. INTRODUCTION
Primordial gravitational waves (GWs) from quantum fluctuations of the tensor modes of the spacetime metric were stretched outside the horizon during inflation, and were then frozen on super-Hubble scales. In Einstein gravity, the amplitude of the power spectrum of primordial tensor perturbations produced in the single-field slow-roll inflationary models is proportional to the energy density of the Universe [1,2], and it can be measurable by CMB polarization experiments since primordial GWs can lead to the B-mode polarization of the cosmic microwave background (CMB) anisotropies. The Planck 2018 result combined with the BICEP2/Keck Array BK14 data gives an upper limit on the tensor spectrum which is quantified by the tensor-to-scalar ratio r 0.002 < 0.064 [3] at the CMB scales. Inflationary models with quartic and cubic potentials which predict strong primordial GWs are strongly disfavored by the Planck data [4]. However, at the scales much smaller than the CMB scales, the constraints on tensor perturbations are relatively loose on a board range of frequencies.
Different from the GWs sourced by matter fields, primordial tensor perturbations could be enhanced by Lorentz-violating massive gravity [5], non-attractor phase in generalized Ginflation [6], quantum gravitational inflation [7] and the thermal history of the early Universe [8,9].
The parameter resonance amplification of scalar perturbations during inflation has been widely discussed [10][11][12]. In this paper, we investigate the similar amplification of tensor perturbations with parity-violation caused by the Chern-Simons coupling while they are deep inside the horizon during inflation. The Chern-Simons coupling commonly arises in the string axiverse as the Chern-Pontryagin density [13,14], which can affect tensor perturbations during their propagation [15][16][17][18][19][20][21][22]. Due to the weakness of gravitational interaction, the string corrections are weakly constrained. The energy spectrum of the amplified GWs has a parameter dependent characteristic sharp peak or a board plateau with cutoffs in both the infrared and the ultraviolet regions. In Einstein gravity, a power-law slope appears in the infrared region of the GW energy spectrum sourced by the transverse-traceless component of the matter fields [23,24]. Detecting the sharp infrared cutoff of the GW energy density provides an important clue to the Chern-Simons coupling, as well as extra dimensions predicted in string theory. This resonance peak might be observed by ground-based and space-based GW detectors, such as aLIGO [25], LISA [26], Taiji [27], and pulsar timing arrays, such as SKA [28]. It is worthy to mention that, in some parameter space, the string-inspired resonance peak is similar to the noise in the measured sensitivity curve of LIGO [29,30], and has the possibility to explain the unknown measured noise which also has a sharp-peak profile. In turn, if the noise is reduced in the aLIGO further observation runs, the absence of such peaks will provide stringent constraints on the Chern-Simons coupling during inflation.
The paper is organized as follows. In Sec. II, we briefly introduce Chern-Simons gravity and the equation of motion (EOM) of tensor perturbations. In Sec. III, we study the dynamics of resonant instability of tensor perturbations caused by the Chern-Simons coupling. In Sec. IV, we present the numerical results of the amplified tensor perturbations with parity-violation in Starobinsky model as an example. Section V is devoted to conclusions.

GRAVITY
In this section, we give a brief introduction to the Chern-Simons gravity, whose action has the following form where κ −1 ≡ M p = 2.4 × 10 18 GeV is the reduced Planck mass, R is the Ricci scalar, L CS is the Lagrangian which contains a Chern-Simons term coupled to a scalar field, and L φ is the Lagrangian for the scalar field, which is non-minimally coupled to gravity. As a simple example, we consider the action of the scalar field as Here V (φ) denotes the potential of the scalar field. The Chern-Simons Lagrangian can be written in the form with ε ρσαβ being the Levi-Civitá tensor defined in terms of the antisymmetric symbol ρσαβ In the flat Friedmann-Robertson-Walker Universe, the background metric is given by where a(t) denotes the scale factor of the Universe and t represents the cosmic time. In this paper, we assume that the Chern-Simons term in the action (1) has a negligible effect on the background evolution. We further assume that the Universe is dominated by the scalar field φ which plays the role of the inflaton field during the slow-roll inflation. In this case, the Friedmann equation, which governs the background evolution, takes exactly the same form as that in General Relativity (GR), i.e., where H denotes the Hubble parameter. The evolution of the inflaton field φ is also the same as that in GR,φ Now, let us turn to study the propagation of tensor perturbations on a homogeneous and isotropic background. With tensor perturbations, the spatial metric is written as where h ij represents the first-order transverse and traceless metric perturbations. In order to derive the equation of motion for tensor perturbations, we substitute h ij into the action (1) and expand it to the second order. After tedious calculations, we find where Then taking the variation of the action with respect to h ij , we obtain the equation of motion In Chern-Simons gravity, the propagation equations for two circular polarization modes of gravitational waves are decoupled. To study the evolution of h ij , we expand it over spatial Fourier harmonics, where R and L represent the right-handed and left-handed polarization, respectively. Here e A ij denote the circular polarization tensors and satisfy the relation with ρ R = 1 and ρ L = −1. Then one can write Eq. (11) in the Fourier space as where and quantity ν A describes the modification of the friction term of gravitational waves, which induces the amplitude birefringence effect of gravitational waves.

III. RESONANCE INSTABILITY OF GRAVITATIONAL WAVES
For the convenience of the following discussion, we define a new variable X A which satisfies the following relation and then Eq. (14) can be rewritten as where Assuming |(k/a)θ| 1, one can obtain In this paper, we consider that ϑ(φ) has the following functional form, where M and Λ that have the dimension of mass characterize the magnitude and the oscillation period of ϑ(φ), respectively. Here Θ is the Heaviside theta function, φ s and φ e denote the field values at the starting and ending point that the Chern-Simons term works. In the subsequent discussions, any quantity with subscript s means that its value is obtained when φ = φ s , likewise for subscript e. Next, we focus on the period when the inflaton goes through from φ s to φ e . In some parameter space, the dominant term ofF A /F A in Eq. (20) is the d 3 ϑ/dφ 3 one, and then the equation of motion given in Eq. (17) can be reduced tö Considering the case of |φ e − φ s | M p , the evolution of the inflaton and the scale factor can be simply described as φ = φ s +φ s (t − t s ) and a = a s e Hs(t−ts) respectively during the period from t s to t e . After introducing a new time variable 2z = φ s /Λ +φ s (t − t s )/Λ + π(1 + ρ A )/2, the above equation can be cast in the form of the Mathieu equation where with k s = |φ s /(2Λ)|. In the case of |(k/a)θ| 1, as q denotes the magnitude of |(k/a)θ| and thus q 1, the resonance bands are located in some narrow ranges near A k n 2 (n = 1, 2, ...) and each resonance band has a width of order k ∼ q n . Since the first band (n = 1) is the widest and most enhanced band, in the following analysis we focus on this instability band, 1 − q < A k < 1 + q. Moreover, in the present paper we consider k s H, which means that the modes are deep inside the Hubble horizon when they stay in the resonance band.
For the first instability band with q 1, the Floquet exponent µ k which describes the rate of exponential growth is given by Thus the resonance occurs for where . For a given k mode, after it goes through the resonance band, the corresponding X A will be amplified by where t I and t F , which satisfy t s ≤ t I < t F ≤ t e , represent the time when the k mode enters and exits the resonance band, respectively. Defining A k (t) = k/(k s a(t)), we have . (28) Considering k + a e > k − a s , we can classify the resonant k mode into following three groups: Then the corresponding A k (t I ) and A k (t F ) can be calculated as for k + a s < k ≤ k − a s ,  for k − a s < k < k + a e , and for k + a e ≤ k < k − a e . It is interesting to note that the magnification R k is independent of k for the second group. The resulting tensor power spectrum, which is evaluated at the horizon crossing [k = aH], can be written as At the CMB scales, the amplitude of the tensor power spectrum is limited to less than the order of 10 −10 . For the smaller scales, if tensor perturbations can be enhanced by a certain order of magnitude through the resonance, the predicted GWs may be probed by the future GW experiments. In the next section, we will study a concrete example by numerical method.

IV. RESULTS
In this section, we present the numerical results by considering the Starobinsky model of inflation as an example, which is now regarded as one of the most popular inflationary models and whose potential has the following form We set the e-folding number from the time when the pivot scale k * = 0.05Mpc −1 exits the horizon to the end of the inflation as N * = 60, and thus, µ is fixed at 1.14 × 10 −5 M p by the amplitude of the curvature perturbations P R 2.1 × 10 −9 [4]. Table I shows the three parameter sets we choose for predicting a testable GW background in the future. Taking case 1 as an example, we plot the tensor spectra P h from the analytically approximate result given in Eq. (33) and the exact numerical result by solving Eq. (14) in Fig. 1. It is easy to see that the numerical result matches the analytical one very well for modes in the k < k − a s and k > k + a e regimes. For modes within k − a s < k < k + a e , the amplitude of the exact tensor spectrum is larger than that of the approximate one, and it is interesting to observe that the exact tensor spectrum exhibits an irregular structure, instead of the smooth plateau structure predicted by the analytic method. The ripples on the plateau are caused by the phases of X A at the time when X A leaves the resonance band.
For the GWs produced during inflation, the current energy spectrum can be approximately written as [31,32] Ω GW,0 (k) 1.08 × 10 −6 P h (k) (35) for the frequency f > 10 −10 Hz, and one can relate the current frequency f and the comoving gravitational-wave projects summarized in [35]. The shaded regions represent the present existing constraints on GWs [36,37].
wave number k through Figure 2 shows the current energy spectrum of GWs obtained by numerically solving Eq. (14) for the three cases in Table I. In case 1, the peak of the GW energy spectrum locates in the sensitive region of SKA [28]. The peak of the GW energy spectrum for case 2 is above the sensitivity curve of LISA [26]. For case 3, the predicted GW spectrum could be simultaneously detected by the deci-hertz interferometer GW observatory (DECIGO) [33] and the big bang observer (BBO) [34]. Figure 3 shows the ratio of power spectrum of the right-handed GWs P R h = k 3 |h R (k)| 2 /(2π 2 ) to that of the left-handed GWs P L h = k 3 |h L (k)| 2 /(2π 2 ) for case 1. From this figure, one can find that the power spectra of right-handed and left-handed GWs have markedly different amplitudes for the resonated modes, and the ratio rapidly oscillates with k. Similar results can also be found in the other two cases. This phenomenon originates from the phase difference of π in the EOM of two polarization modes, according to Eq. (14).
This provides an opportunity to directly detect the parity violation of the gravitational interaction in the future GW experiments.

V. CONCLUSIONS
We have investigated the amplification of the parity-violating tensor perturbations during inflation with the Chern-Simons coupling. The Chern-Simons term is coupled to a periodic function ϑ(φ) of the inflaton, so that the EOM of tensor perturbations can be transformed into the Mathieu equation. Tensor perturbations with modes in the narrow resonance band are exponentially amplified, which results in a narrow peak or a board plateau in Ω GW,0 , depending on the duration length of ϑ(φ). The cutoff of Ω GW,0 in the infrared region is a characteristic distinguishable from the stochastic GW background sourced by matter fields in Einstein gravity. Since the frequency of GWs depends on the comoving length scale at reentry, the amplified GWs can be detected by detectors sensitive to different frequencies.
Moreover, the detection of parity-violating GWs provides another evidence of the Chern-Simons gravity.
In the first Advanced LIGO observational run (aLIGO O1), the sum of all known noise sources cannot explain the measured sensitivity curve noise, especially below 100Hz [29].
This discrepancy has been reduced significantly in the aLIGO O2 [30], but some noise peaks remain unknown, for example the peak around 300Hz observed by Livingston, Hanford and Virgo detectors. Interestingly, this noise peak could be explained by the resonance peak induced by the Chern-Simons term. This possibility should be examined by the next Advanced LIGO observational run with more detailed noise analysis.