Chirality oscillation of primordial gravitational waves during inflation

We show that if the gravitational Chern-Simons term couples to a massive scalar field ($m>H$), the primordial gravitational waves (GWs) will show itself the chirality oscillation, i.e., the amplitudes of the left- and right-handed GWs modes will convert into each other and oscillate in their propagations. This oscillation will eventually develop a permanent difference of the amplitudes of both modes, which leads to nearly opposite oscillating shapes in the power spectra of the left- and right-handed primordial GWs. We discuss its implication to the CMB B-mode polarization.


JHEP03(2017)024 2 Chirality oscillation
We begin with the quadratic action for the tensor perturbation where τ = dt/a and ′ = d/dτ , M p is the Planck scale and L HD is the higher-order derivative corrections with the parity violation. Considering only the leading order terms, we could write where ǫ ijk is the Levi-Cevita symbol, ξ 1 and ξ 2 are time-dependent parameters, and M is the cutoff scale. The gCS term corresponds to ξ 1 = −ξ 2 [13,23,30], while W W corresponds to ξ 1 = 0 [4,5,12], with W being the Weyl tensor.
In the Fourier space, we have where the polarization tensors p Here, ξ 1 < a is required, otherwise the ghost modes will appear at the cutoff scale k/a = M, see [31].
Initially, the perturbations are deep inside their horizon, i.e., c T k k. The power spectrum of primordial GWs is The chiral parameter ∆χ = (P (L) T ) reflects the intensity of parity violation of primordial GWs.

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When ξ 1 or ξ 2 = 0, γ (s) k with different polarizations will have different evolutions, the primordial GWs are chiral. In sting theory, ξ 1 and ξ 2 actually are moduli-dependent, see [3], and the massive moduli fields (m > H) are ubiquitous. The nontrivial evolution (especially the oscillation) of the moduli field indicates the nontrivial variations of ξ 1 and ξ 2 , which may induce oscillation of the chiral term c (s) T k or z (s) ′′ /z (s) in eq. (2.4). It is this physics that induces the chirality oscillation of the primordial GWs, similar to the case in refs. [10,11,21] where the nontrivial variation (such as oscillation) of the propagating speed of GWs induces oscillations in the GWs modes as well as in the GWs power spectrum.
Generally, the single field slow-roll inflation is thought as the effective model after all massive modes are integrated out. However, during inflation, the massive fields could be excited and oscillate around its minimum. This will inevitably induce the chirality oscillation of the primordial GWs, which could be encoded in the power spectrum of primordial GWs, as will be showed.

The model
We will illustrate the idea of the chirality oscillations with a workable model in detail, in which the background is set as the inflation with ǫ = −Ḣ/H 2 ≪ 1, and φ is a massive modulus field with where R ∧ R = ǫ αβγδ R αβµν R µν γδ is the gCS term. Thus we have and c (s) that the background is unaffected by φ.
In this model, if φ oscillates around the minimum of its potential during inflation, the chiral term z (s) ′′ /z (s) in eq. (2.4) could experience nontrivial oscillation, which will induce the chirality oscillation of primordial GWs, as explained in section 2. Here, we would like to point out that the oscillation of φ around the minimum of V (φ) is a sufficient condition rather than a necessary one for producing the chirality oscillation of GWs as long as f is φ-dependent, since what plays the key role is z (s) ′′ /z (s) , which depends also on the shape of f (φ). For instance, a trivially evolved φ may also generate the chirality oscillation of GWs if f (φ) is delicately designed. However, in the following, we will focus on the case where it is the oscillation of φ that generates the chirality oscillation of primordial GWs.

Numerical results
Below, we will numerically show the evolutions of the left-and right-handed GWs modes and the corresponding spectrum. We define α = ln(a/a 0 ), where there may be a difference JHEP03(2017)024 of a constant between α and the usually used e-folding number N . Then, eq. (2.4) can be rewritten as and the subscript ", α" is the derivative with respect to α, and the Hubble parameter is written as H(α) = H 0 e −ǫα with H 0 set by the amplitude of curvature perturbation. Generally, f (φ) ∼ φ, see [3] for a review. To avoid the ghost mode, we require that the parity-violation only occur in the region around φ = 0. Thus we set where A f and c are dimensionless, which gives f ,α ≈ gφ ,α for φ → 0, in which g = A f /c is the coupling coefficient of φ to the gCS term, and f ,α ≈ 0 for |φ| ≫ √ c. During inflation, the massive modulus field might be excited and then relax towards its local minimum, and oscillate rapidly around it. Thus for the simulation, we set where Λ has the mass dimension. When |φ| ≫ Λ/M p , is constant, which insures that the inflation background is unaffected by φ. Additionally, the initial condition of φ is set by φ(α ini ) = φ ini and φ ,α (α ini ) = 0. Note that, since in our model it is the oscillation of φ that induces the chirality oscillation of primordial GWs, any potential with a minimum could do the job. Additionally, the chirality oscillation would present for any f (φ) ∼ φ when φ is small. To demonstrate this point, we will also show the oscillation of ∆χ with f (φ) = A f φ/(φ + c). Although the shape of the chirality oscillation could be a little model-dependent (which should be attributed to the transition of f (φ) around φ = ± √ c or ±c), the phenomenon of chirality oscillation of primordial GWs is quite general. We plot the evolutions of φ, ξ 1 (i.e., f ′ ) and z (s) ′′ /z (s) in figure 1, and plot |u (s) k | and |γ (s) k | in figure 2 for the mode k = 10 −3 Mpc −1 , which is at the CMB scale and whose corresponding frequency is about 10 −17 Hz. The chirality oscillations of the primordial GWs modes start at around α = 8, since φ starts oscillating and z (s) ′′ /z (s) starts its nontrivial oscillation at that time. We see that the chirality oscillation of primordial GWs is encoded in its power spectrum, as was showed in figure 3(a), see also [32] for the GWsgauge field oscillation. The amplitudes of the left-and right-handed GWs mode oscillate almost (but not exactly) symmetrically, and will eventually arrive at different values. This difference with respect to the comoving wave number is reflected in the oscillation of ∆χ, see figures 3(c) and 3(d). It is also noticed that the broken symmetry of the evolutions of left-and right-handed GWs modes, i.e., |γ (L) | + |γ (R) | = 2|γ (0) |, actually also imprint an oscillating fingerprint in P T = P  The chiral GWs will induce non-vanishing TB/EB-mode correlation at CMB last scatting surface, C ] (e.g., [27,28]), if the modulus field happens to oscillate at the time of ∼ 60 efolds before the end of inflation. 1 For parameters used in the numerical calculation, the perturbation mode with k ∼ 10 −4 Mpc −1 exits horizon at about α = 8, where k ∼ aH is used for the estimation. Thus after a 0 and H 0 are fixed, there is some fine-tuning of the parameters φ ini and α ini , which set the initial condition of φ, so that φ starts oscillating at about α = 8. We plot the TB/EB-mode spectrum in figure 4. We see that the chirality oscillation of primordial GWs will bring obvious wiggles in the TB/EB-mode spectrum, which is a novel phenomenon has not been uncovered before.

Analytic estimation
With (3.2) and the numerical calculation, we approximately have (3.7) Then noting aH ≈ −1/τ , eq. (2.4) can be approximated as where A * = H 2 f ,ααα /M 2 p is dimensionless, which reflects the effect of parity-violating correction of gravity.
Without loss of generality, we assume that before τ = τ mat , A * = constant, and hereafter A * = 0, where τ mat is the comoving time at the matching surface and depends on φ ini and α ini in the model. Thus when τ < τ mat , the solution is where M and W are the Whittaker functions. Initially, the perturbations are deep inside the horizon, i.e., −kτ ≫ 1. Using for |z| ≫ 1 and M (x, y, z) = Γ(2y + 1) Γ 1 2 + x + y e −iπ( 1 2 +y−x) W (x, y, z) + Γ(2y + 1) so that the initial state satisfies u (s) where H
T,inf = H 2 /π 2 and τ mat corresponds to the match surface. Then, the chirality parameter ∆χ(k, τ mat , A * ) = g(k, τ mat , A * , −1) − g(k, τ mat , A * , 1) g(k, τ mat , A * , −1) + g(k, τ mat , A * , 1) . (3.16) By requiring the continuities of u  U (a, b, x), we obtain We plot ∆χ(k, τ mat , A * ) in figure 5. The amplitude of the oscillation is determined by A * , while the position of the oscillation is determined by τ mat . What we are interested in is the oscillation of ∆χ when k > −1/τ mat . Though the amplitude of the oscillation in figure 5 is slightly larger than that in figure 3(c), due to the oversimplified assumption we made, we may use the first or the second peak of ∆χ(k, τ mat , A * ) in figure 5, as the estimation of the overall oscillating amplitude in figure 3(c).

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difference of the amplitudes of both modes, which leads to the nearly opposite oscillating shapes in the left-and right-handed GWs spectrum. The chirality oscillation of primordial GWs may bring obvious wiggles in the CMB TB/EB-mode spectrum, which is the unique fingerprint of chiral gravity. Thus highprecision CMB B-mode polarization experiments could offer us richer information on the UV-complete gravity theory than expected, though the detecting is still a challenging issue [28,33,34].