0904/0904.0554.md

Chiral Primordial Gravitational Waves from a Lifshitz Point

Tomohiro Takahashi and Jiro Soda

Department of Physics, Kyoto University, Kyoto, 606-8501, Japan

(Dated: October 22, 2018)

We study primordial gravitational waves produced during inflation in quantum gravity at a Lifshitz point proposed by Hořava. Assuming power-counting renormalizability, foliation preserving diffeomorphism invariance, and the condition of detailed balance, we show that primordial gravitational waves are circularly polarized due to parity violation. The chirality of primordial gravitational waves is a quite robust prediction of quantum gravity at a Lifshitz point which can be tested through observations of cosmic microwave background radiation and stochastic gravitational waves.

PACS numbers: 98.80.Cq

I. INTRODUCTION

For a long time, the inflationary scenario has been regarded as an academic theory which provides an elegant solution to conceptual problems in cosmology such as the flatness problem, the horizon problem, and the origin of structure of the universe. However, the fact that all of recent cosmological observations strongly support the inflationary scenario with high precision encourages us to take the inflation more seriously. Then, taking into account that the inflation magnifies microscopic scales to macroscopic ones, it is reasonable to regard the inflation as a probe to investigate physics at the Planck scale, namely, quantum gravity.

It is widely believed that string theory is a promising candidate for quantum gravity. However, it is premature to discuss a Planckian regime of the universe using string theory. Therefore, so far, study of trans-Planckian effect on inflationary predictions has been phenomenological [1, 2]. In fact, there are various phenomenological models which can mimic trans-Planckian physics and lead to a modification of the power spectrum of curvature perturbations [3]. It is known that these quantitative trans-Planckian corrections are suffered from severe constraints due to the backreaction problem [4, 5]. However, there may be more qualitative effects due to trans-Planckian physics. For example, polarization of primordial gravitational waves could be an important smoking gun of trans-Planckian physics [6, 7, 8]. In fact, a parity violating gravitational Chern-Simons term which is ubiquitous in string theory can generate circular polarization in primordial gravitational waves [9, 10, 11]. However, it has been shown that the effect of parity violation is negligibly small for the slow roll inflation [12]. Recently, it is argued that sizable circular polarization could be generated [13, 14] by resorting to a peculiar feature due to the Gauss-Bonnet term [15]. One defect in these models is the appearance of divergence in one of circular polarization modes. This divergence suggests necessity of a consistent quantum theory of gravity.

Recently, quantum gravity at a Lifshitz point which is power-counting renormalizable is proposed by Hořava [16, 17]. In contrast to string theory, the theory is not intended to be a unified theory but just quantum

gravity in 4-dimensions. In this “small” framework, one can discuss trans-Planckian effect on cosmology in a self-consistent manner. In Hořava’s formulation of quantum gravity, the action necessarily contains a Cotton tensor, which violates parity invariance. Hence, we can expect circular polarization of primordial gravitational waves. Moreover, there exists no divergence in this model. Then, the purpose of this paper is to calculate degree of circular polarization during inflation and show observability of chiral primordial gravitational waves which is a robust prediction of quantum gravity at a Lifshitz point.

II. QG AT A LIFSHITZ POINT

The quantum gravity proposed by Hořava can be characterized by anisotropic scaling at an ultraviolet fixed point $\mathbf{x} \rightarrow b\mathbf{x}$ , $t \rightarrow b^3t$ , where $b$ , $\mathbf{x}$ and $t$ are a scaling factor, spacial coordinates and a time coordinate, respectively. This scaling guarantees the renormalizability of the theory [17]. Because of the anisotropic scaling, the time direction plays a privileged role. In other words, the spacetime has a codimension-one foliation structure in which leaves of the foliation are hypersurfaces of constant time. Since the spacetime has the anisotropic scaling and the foliation structure, the theory is not diffeomorphism invariant but invariant under the foliation-preserving diffeomorphism defined by $\tilde{x}^i = \tilde{x}^i(x^j, t)$ , $\tilde{t} = \tilde{t}(t)$ . Here, indices $i, j, k, \dots$ represent spacial coordinates.

To describe the foliation, it is convenient to use ADM decomposition of the metric $ds^2 = -N^2dt^2 + g_{ij}(dx^i + N^i dt)(dx^j + N^j dt)$ , where $N$ , $N_i$ , and $g_{ij}$ are the lapse function, the shift function and the 3-dimensional induced metric, respectively. In order for the theory to be unitary, the number of time derivative should be at most two in the action. The renormalizable kinetic part is then given by

$$S_K=\frac{2}{{\kappa }^2}\int dt{d}^{3}x\sqrt{g}N(K^{ij}{K}_{ij}-\lambda K^2),(1)S\_K\; =\; \backslash frac\{2\}\{\backslash kappa^2\}\; \backslash int\; dtd^3x\; \backslash sqrt\{g\}\; N\; (K^\{ij\}\; K\_\{ij\}\; -\; \backslash lambda\; K^2)\; ,\; \backslash quad\; (1)$$

where $K_{ij}$ is the extrinsic curvature of the constant time hypersurface defined by $K_{ij} = (\dot{g}{ij} - N{i|j} - N_{j|i})/2N$ , and $K$ is the trace part of $K_{ij}$ . Note that $\kappa$ and $\lambda$ are dimensionless coupling constants which run according tothe renormalization group flow. Hereafter, we assume $\lambda$ has already settled down to the infrared fixed point $\lambda = 1$ at the beginning of the inflation although we keep $\lambda$ in subsequent formulas.

The most crucial assumption of Hořava's theory is the detailed balance condition

$$S_V=\frac{{\kappa }^2}{8}\int dt{d}^3\mathbf{x}\sqrt{g}N{E}^{ij}{\mathcal{G}}_{ijkl}{E}^{kl},(2)S\_V\; =\; \backslash frac\{\backslash kappa^2\}\{8\}\; \backslash int\; dt\; d^3\; \backslash mathbf\{x\}\; \backslash sqrt\{g\}\; N\; E^\{ij\}\; \backslash mathcal\{G\}\_\{ijkl\}\; E^\{kl\}\; ,\; \backslash quad\; (2)$$

where we have defined $\sqrt{g} E^{ij} = \delta W[g_{ij}]/\delta g_{ij}$ with some functional $W$ . Here, we introduced the inverse of De Witt metric $\mathcal{G}^{ijkl} = (g^{ik} g^{jl} + g^{il} g^{jk})/2 - \lambda g^{ij} g^{kl}$ . The renormalizability of the theory requires $E^{ij}$ must be third order in spatial derivatives. The requirement uniquely selects the Cotton tensor

$$C^{ij}={\epsilon }^{ikl}{\mathrm{\nabla }}_k(R_l^j-\frac{1}{4}R{\delta }_l^j),(3)C^\{ij\}\; =\; \backslash varepsilon^\{ikl\}\; \backslash nabla\_k\; \backslash left(\; R\_l^j\; -\; \backslash frac\{1\}\{4\}\; R\; \backslash delta\_l^j\; \backslash right)\; ,\; \backslash quad\; (3)$$

where $\varepsilon^{ijk}$ denotes the totally antisymmetric tensor and $R_{ij}$ and $R$ are the 3-dimensional Ricci tensor and Ricci scalar. Including relevant deformations, we have

$$W=\frac{1}{w^2}\int d^3\mathbf{x}\sqrt{g}{\epsilon }^{ijk}({\mathrm{\Gamma }}_{il}^m{\mathrm{\partial }}_j{\mathrm{\Gamma }}_{km}^l+\frac{2}{3}{\mathrm{\Gamma }}_{il}^n{\mathrm{\Gamma }}_{jm}^l{\mathrm{\Gamma }}_{kn}^m)+\mu \int d^3\mathbf{x}\sqrt{g}(R-2{\mathrm{\Lambda }}_w),(4)W\; =\; \backslash frac\{1\}\{w^2\}\; \backslash int\; d^3\; \backslash mathbf\{x\}\; \backslash sqrt\{g\}\; \backslash varepsilon^\{ijk\}\; \backslash left(\; \backslash Gamma\_\{il\}^m\; \backslash partial\_j\; \backslash Gamma\_\{km\}^l\; +\; \backslash frac\{2\}\{3\}\; \backslash Gamma\_\{il\}^n\; \backslash Gamma\_\{jm\}^l\; \backslash Gamma\_\{kn\}^m\; \backslash right)\; +\; \backslash mu\; \backslash int\; d^3\; \backslash mathbf\{x\}\; \backslash sqrt\{g\}\; (R\; -\; 2\backslash Lambda\_w)\; ,\; \backslash quad\; (4)$$

where $\Gamma_{jk}^i$ and $\Lambda_w$ are Christoffel symbols and the 3-dimensional ‘‘cosmological constant’’, respectively. Here, we have introduced new coupling constants $w$ and $\mu$ . Note that the first two terms lead to the Cotton tensor. Thus, we obtain the potential part of the 4-dimensional action [17]

$$S_V=\int dt{d}^3\mathbf{x}\sqrt{g}N\times [-\frac{{\kappa }^2}{2w^4}{C}^{ij}{C}_{ij}+\frac{{\kappa }^2\mu }{2w^2}{\epsilon }^{ijk}{R}_{il}{R}_{k\mathrm{\mid }j}^l-\frac{{\kappa }^2{\mu }^2}{8}{R}_{ij}{R}^{ij}+\frac{{\kappa }^2{\mu }^2}{8(1-3\lambda )}(\frac{1-4\lambda }{4}{R}^2+{\mathrm{\Lambda }}_{w}R-3{\mathrm{\Lambda }}_w^2)],(5)S\_V\; =\; \backslash int\; dt\; d^3\; \backslash mathbf\{x\}\; \backslash sqrt\{g\}\; N\; \backslash times\; \backslash left[\; -\backslash frac\{\backslash kappa^2\}\{2w^4\}\; C^\{ij\}\; C\_\{ij\}\; +\; \backslash frac\{\backslash kappa^2\; \backslash mu\}\{2w^2\}\; \backslash varepsilon^\{ijk\}\; R\_\{il\}\; R\_\{k|j\}^l\; -\; \backslash frac\{\backslash kappa^2\; \backslash mu^2\}\{8\}\; R\_\{ij\}\; R^\{ij\}\; +\; \backslash frac\{\backslash kappa^2\; \backslash mu^2\}\{8(1-3\backslash lambda)\}\; \backslash left(\; \backslash frac\{1-4\backslash lambda\}\{4\}\; R^2\; +\; \backslash Lambda\_w\; R\; -\; 3\backslash Lambda\_w^2\; \backslash right)\; \backslash right]\; ,\; \backslash quad\; (5)$$

where a stroke $|$ denotes a covariant derivative with respect to spacial coordinates. In the above action (5), the coefficient of scalar curvature $R$ is $\kappa^2 \mu^2 \Lambda_w / 8(1-3\lambda)$ , then the gravitational constant become negative in the low energy limit unless $\Lambda_w / (1-3\lambda) > 0$ .

In addition to this gravity sector, we consider the action for an inflaton $\phi$

$$S_M=\int dt{d}^3\mathbf{x}\sqrt{g}N[-\frac{1}{2}{\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi -V(\varphi )]\mathrm{.}(6)S\_M\; =\; \backslash int\; dt\; d^3\; \backslash mathbf\{x\}\; \backslash sqrt\{g\}\; N\; \backslash left[\; -\backslash frac\{1\}\{2\}\; \backslash partial^\backslash mu\; \backslash phi\; \backslash partial\_\backslash mu\; \backslash phi\; -\; V(\backslash phi)\; \backslash right]\; .\; \backslash quad\; (6)$$

Let us assume the slow roll inflation and take the slow roll limit. Then, we can replace the action (6) with the effective cosmological constant $\bar{\Lambda}$ , namely, we have $S_M = -\int dt d^3 \mathbf{x} \sqrt{g} N \bar{\Lambda}$ .

Thus, the total action is given by $S = S_K + S_V + S_M$ . The total action $S$ breaks the detailed balance condition softly [17]. It should be emphasized that the total action $S$ reduces to the conventional Einstein theory at low energy.

III. PRIMORDIAL GRAVITATIONAL WAVES

Let us consider the background spacetime with spatial isotropy and homogeneity $ds^2 = -dt^2 + a(t)^2 \delta_{ij} dx^i dx^j$ , where $a$ is the scale factor. Using this metric ansatz, we can get the Friedmann equation with $\lambda = 1$

$$\frac{{\dot{a}}^2}{a^2}=\frac{{\kappa }^2}{12}(\bar{\mathrm{\Lambda }}-\frac{3{\kappa }^2{\mu }^2{\mathrm{\Lambda }}_w^2}{16})\equiv H^2\mathrm{.}(7)\backslash frac\{\backslash dot\{a\}^2\}\{a^2\}\; =\; \backslash frac\{\backslash kappa^2\}\{12\}\; \backslash left(\; \backslash bar\{\backslash Lambda\}\; -\; \backslash frac\{3\backslash kappa^2\; \backslash mu^2\; \backslash Lambda\_w^2\}\{16\}\; \backslash right)\; \backslash equiv\; H^2\; .\; \backslash quad\; (7)$$

The above equation leads to de Sitter spacetime, $a(t) \propto e^{Ht}$ . Here, we assumed $\bar{\Lambda} > 3\kappa^2 \mu^2 \Lambda_w^2 / 16$ . Note that if we chose $\bar{\Lambda} = 0$ , there is no Minkowski solution. Hence, there must exist residual vacuum energy in the matter sector at the end of the day. This is related to the issue of the cosmological constant, which is beyond the scope of this paper.

Now, we consider tensor perturbations $ds^2 = -dt^2 + a(t)^2 (\delta_{ij} + h_{ij}(t, \mathbf{x})) dx^i dx^j$ , where $h_{ij}$ satisfies the transverse-traceless conditions. Substituting this metric into the total action, we obtain the quadratic action

$${\delta }^{2}S=\int dt{d}^3\mathbf{x}{a}^3[\frac{1}{2{\kappa }^2}{\dot{h}}_j^i{\dot{h}}_i^j+\frac{{\kappa }^2}{8w^4{a}^6}{\mathrm{\Delta }}^2{h}_j^i\mathrm{\Delta }{h}_i^j+\frac{{\kappa }^2\mu }{8w^2{a}^5}{\epsilon }^{ijk}\mathrm{\Delta }{h}_{il}\mathrm{\Delta }{h}_{k\mathrm{\mid }j}^l-\frac{{\kappa }^2{\mu }^2}{32a^4}\mathrm{\Delta }{h}_j^i\mathrm{\Delta }{h}_i^j+\frac{{\kappa }^2{\mu }^2{\mathrm{\Lambda }}_w}{32(1-3\lambda )a^2}{h}_j^i\mathrm{\Delta }{h}_i^j],(8)\backslash delta^2\; S\; =\; \backslash int\; dt\; d^3\; \backslash mathbf\{x\}\; a^3\; \backslash left[\; \backslash frac\{1\}\{2\backslash kappa^2\}\; \backslash dot\{h\}\_j^i\; \backslash dot\{h\}\_i^j\; +\; \backslash frac\{\backslash kappa^2\}\{8w^4\; a^6\}\; \backslash Delta^2\; h\_j^i\; \backslash Delta\; h\_i^j\; +\; \backslash frac\{\backslash kappa^2\; \backslash mu\}\{8w^2\; a^5\}\; \backslash varepsilon^\{ijk\}\; \backslash Delta\; h\_\{il\}\; \backslash Delta\; h\_\{k|j\}^l\; -\; \backslash frac\{\backslash kappa^2\; \backslash mu^2\}\{32a^4\}\; \backslash Delta\; h\_j^i\; \backslash Delta\; h\_i^j\; +\; \backslash frac\{\backslash kappa^2\; \backslash mu^2\; \backslash Lambda\_w\}\{32(1-3\backslash lambda)a^2\}\; h\_j^i\; \backslash Delta\; h\_i^j\; \backslash right]\; ,\; \backslash quad\; (8)$$

where $\Delta$ represents the Laplace operator. The transverse traceless tensor $h_{ij}$ can be expanded in terms of plane waves with wavenumber $\mathbf{k}$ as

$$h_{ij}(t,\mathbf{x})=\sum _{A=R,L}\int \frac{d^3\mathbf{k}}{(2\pi {)}^3}{\psi }_{\mathbf{k}}^A(t)e^{i\mathbf{k}\cdot \mathbf{x}}{p}_{ij}^A,(9)h\_\{ij\}(t,\; \backslash mathbf\{x\})\; =\; \backslash sum\_\{A=R,L\}\; \backslash int\; \backslash frac\{d^3\; \backslash mathbf\{k\}\}\{(2\backslash pi)^3\}\; \backslash psi\_\{\backslash mathbf\{k\}\}^A(t)\; e^\{i\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\}\; p\_\{ij\}^A\; ,\; \backslash quad\; (9)$$

where $p_{ij}^A$ are circular polarization tensors which are defined by $ik_s \varepsilon^{rsj} p_{ij}^A = k \rho^A p_r^r p_s^s$ [13]. Here, $\rho^R = 1$ and $\rho^L = -1$ modes are called the right handed mode and the left handed mode, respectively. We also impose normalization conditions $p^{*i}{}_j{}^A p^j{}_i{}^B = \delta^{AB}$ , where $p^{*i}{}_j{}^A$ is the complex conjugate of $p^i{}_j{}^A$ . Substituting the expansion (9) into the gravitational action (8), we obtain

$${\delta }^{2}S=\sum _{A=R,L}\int dt\frac{d^3\mathbf{k}}{(2\pi {)}^3}{a}^3[\frac{1}{2{\kappa }^2}\mathrm{\mid }{\dot{\psi }}_{\mathbf{k}}^A{\mathrm{\mid }}^2-\{\frac{{\kappa }^2{k}^6}{8w^4{a}^6}-{\rho }^A\frac{{\kappa }^2\mu k^5}{8w^2{a}^5}+\frac{{\kappa }^2{\mu }^2{k}^4}{32a^4}+\frac{{\kappa }^2{\mu }^2{\mathrm{\Lambda }}_w{k}^2}{32(1-3\lambda )a^2}\}\mathrm{\mid }{\psi }_{\mathbf{k}}^A{\mathrm{\mid }}^2]\mathrm{.}(10)\backslash delta^2\; S\; =\; \backslash sum\_\{A=R,L\}\; \backslash int\; dt\; \backslash frac\{d^3\; \backslash mathbf\{k\}\}\{(2\backslash pi)^3\}\; a^3\; \backslash left[\; \backslash frac\{1\}\{2\backslash kappa^2\}\; |\backslash dot\{\backslash psi\}\_\{\backslash mathbf\{k\}\}^A|^2\; -\; \backslash left\backslash \{\; \backslash frac\{\backslash kappa^2\; k^6\}\{8w^4\; a^6\}\; -\; \backslash rho^A\; \backslash frac\{\backslash kappa^2\; \backslash mu\; k^5\}\{8w^2\; a^5\}\; +\; \backslash frac\{\backslash kappa^2\; \backslash mu^2\; k^4\}\{32a^4\}\; +\; \backslash frac\{\backslash kappa^2\; \backslash mu^2\; \backslash Lambda\_w\; k^2\}\{32(1-3\backslash lambda)a^2\}\; \backslash right\backslash \}\; |\backslash psi\_\{\backslash mathbf\{k\}\}^A|^2\; \backslash right]\; .\; \backslash quad\; (10)$$

Using the variable $v_{\mathbf{k}}^A \equiv a \psi_{\mathbf{k}}^A$ and conformal time $\eta$ defined by $d\eta/dt = 1/a$ , we obtain the equations of motion

$$\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\eta }^2}{v}_{\mathbf{k}}^A+(k_{eff}^2-\frac{2}{{\eta }^2})v_{\mathbf{k}}^A=0,(11)\backslash frac\{\backslash partial^2\}\{\backslash partial\; \backslash eta^2\}\; v\_\{\backslash mathbf\{k\}\}^A\; +\; \backslash left(\; k\_\{eff\}^2\; -\; \backslash frac\{2\}\{\backslash eta^2\}\; \backslash right)\; v\_\{\backslash mathbf\{k\}\}^A\; =\; 0\; ,\; \backslash quad\; (11)$$where we used $a = -1/H\eta$ and defined $k_{eff}^A{}^2 = \alpha^2 k^2 {1 + \beta(\alpha k\eta)^2(1 + \rho^A \gamma \alpha k\eta)^2}$ . We have also defined

$${\alpha }^2=\frac{{\kappa }^4{\mu }^2{\mathrm{\Lambda }}_w}{16(1-3\lambda )},\beta =H^2\frac{1-3\lambda }{{\mathrm{\Lambda }}_w{\alpha }^2},\gamma =H\frac{2}{w^2\mu \alpha }\mathrm{.}(12)\backslash alpha^2\; =\; \backslash frac\{\backslash kappa^4\; \backslash mu^2\; \backslash Lambda\_w\}\{16(1\; -\; 3\backslash lambda)\},\; \backslash quad\; \backslash beta\; =\; H^2\; \backslash frac\{1\; -\; 3\backslash lambda\}\{\backslash Lambda\_w\; \backslash alpha^2\},\; \backslash quad\; \backslash gamma\; =\; H\; \backslash frac\{2\}\{w^2\; \backslash mu\; \backslash alpha\}.\; \backslash quad\; (12)$$

Here, $\alpha$ is "the emergent speed of light" [17] and $\beta$ and $\gamma$ are dimensionless parameters.

Since there appears $\rho^A$ in Eq.(11), the evolution of right handed mode is different from that of left handed mode. Hence, the dimensionless parameter $\gamma$ characterizes "the parity violation". If $\beta = 0$ and $\alpha$ is exactly the speed of light, Eq.(11) becomes the equation for gravitational waves in pure de Sitter background in Einstein theory. Then $\beta$ measures the "deviation from Einstein theory".

IV. CIRCULAR POLARIZATION

Now, we calculate the power spectrum $|\psi_{\mathbf{k}}^A|^2$ numerically and evaluate degree of circular polarization of primordial gravitational waves.

For the numerical analysis, it is convenient to introduce dimensionless variables $k' \equiv \alpha k/H$ and $y \equiv k' H\eta$ . Using these variables and the transformation $\zeta^A \equiv \sqrt{k'H} v_{\mathbf{k}}^A/\kappa$ , we can write down the basic equation

$$\frac{d^2}{d{y}^2}{\zeta }^A+{\omega }^2(y){\zeta }^A=0,(13)\backslash frac\{d^2\}\{dy^2\}\; \backslash zeta^A\; +\; \backslash omega^2(y)\; \backslash zeta^A\; =\; 0,\; \backslash quad\; (13)$$

where $\omega^2(y) = 1 + \beta y^2(1 + \rho^A \gamma y)^2 - 2/y^2$ . Since WKB approximation is pretty good in the asymptotic past $y \rightarrow -\infty$ , we can choose the adiabatic vacuum as the initial condition. More precisely, we set the positive frequency modes as

$${\zeta }^A=\frac{1}{\sqrt{2\omega (y)}}\exp \{-i{\int }_{y_i}^y\omega (y^{\mathrm{\prime }})d{y}^{\mathrm{\prime }}\}\mathrm{.}(14)\backslash zeta^A\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash omega(y)\}\}\; \backslash exp\; \backslash left\backslash \{\; -i\; \backslash int\_\{y\_i\}^y\; \backslash omega(y\text{'})\; dy\text{'}\; \backslash right\backslash \}.\; \backslash quad\; (14)$$

On superhorizon scales $y \rightarrow 0$ , Eq.(13) has asymptotic solution $\zeta^A = C^A/y + D^A y^2$ with constants of integration $C^A$ and $D^A$ . Hence, the power spectrum defined by $k^3 |\psi_{\mathbf{k}}^A|^2 = k^3 |v_{\mathbf{k}}^A/a|^2 = \kappa^2 H^2 |y \zeta^A|^2 / \alpha^3$ reduces to

$$k^3\mathrm{\mid }{\psi }_{\mathbf{k}}^A{\mathrm{\mid }}^2=\frac{{\kappa }^2{H}^2}{{\alpha }^3}\mathrm{\mid }{C}^A{\mathrm{\mid }}^2(15)k^3\; |\backslash psi\_\{\backslash mathbf\{k\}\}^A|^2\; =\; \backslash frac\{\backslash kappa^2\; H^2\}\{\backslash alpha^3\}\; |C^A|^2\; \backslash quad\; (15)$$

on superhorizon scales $y \rightarrow 0$ . So, we only need to calculate $C^A$ using Eq.(13) with the initial condition (14). Notice that the power spectrum $P(k) = k^3 |\psi_{\mathbf{k}}^A|^2$ is scale free. From the mode function (14), we see the vacuum depends on the chirality. In the WKB regime, the amplitude of the right handed mode grows, while that of the left handed mode decays. These two effects make the difference. In Fig.1, we plotted the time evolution of the power for a right handed mode and a left handed mode and also displayed the case of Einstein theory for comparison. Clearly, one can see that the differences of initial

FIG. 1: The time evolution of the power is depicted. The thick solid line represents the evolution for the conventional Einstein gravity. The thin solid line and the dotted line show the time evolution of right handed mode and left handed mode, respectively.

amplitude and the growth rate during WKB regime lead to circular polarization.

Now, we are in a position to discuss observability of circular polarization. For this aim, we need to quantify polarization by defining degree of circular polarization

$$\mathrm{\Pi }=\frac{\mathrm{\mid }{\psi }_{\mathbf{k}}^R{\mathrm{\mid }}^2-\mathrm{\mid }{\psi }_{\mathbf{k}}^L{\mathrm{\mid }}^2}{\mathrm{\mid }{\psi }_{\mathbf{k}}^R{\mathrm{\mid }}^2+\mathrm{\mid }{\psi }_{\mathbf{k}}^L{\mathrm{\mid }}^2}=\frac{\mathrm{\mid }{C}^R{\mathrm{\mid }}^2-\mathrm{\mid }{C}^L{\mathrm{\mid }}^2}{\mathrm{\mid }{C}^R{\mathrm{\mid }}^2+\mathrm{\mid }{C}^L{\mathrm{\mid }}^2}\mathrm{.}(16)\backslash Pi\; =\; \backslash frac\{|\backslash psi\_\{\backslash mathbf\{k\}\}^R|^2\; -\; |\backslash psi\_\{\backslash mathbf\{k\}\}^L|^2\}\{|\backslash psi\_\{\backslash mathbf\{k\}\}^R|^2\; +\; |\backslash psi\_\{\backslash mathbf\{k\}\}^L|^2\}\; =\; \backslash frac\{|C^R|^2\; -\; |C^L|^2\}\{|C^R|^2\; +\; |C^L|^2\}.\; \backslash quad\; (16)$$

Numerical results are plotted in Fig.2. There are two possible channels to observe circular polarization of primordial gravitational waves. One is the in-direct detection of circular polarization through the cosmic microwave background radiation, the required degree of circular polarization has been obtained as $|\Pi| \gtrsim 0.35(r/0.05)^{-0.6}$ in [18], where $r$ is the tensor-to-scalar ratio. The relevant frequency of gravitational waves in this case is around $f \sim 10^{-17}$ Hz. From Fig.2, supposing $r = 0.05$ and $\gamma = 1$ , we see we can detect the circular polarization through the temperature and B-mode polarization correlation if $\beta > 0.2$ . The other is the direct detection of circular polarization, the required degree of circular polarization has been estimated as $\Pi \sim 0.08(\Omega_{\text{GW}}/10^{-15})^{-1}(\text{SNR}/5)$ around the frequency $f \sim 1$ Hz [19], where $\Omega_{\text{GW}}$ is the density parameter of the stochastic gravitational waves and SNR is the signal to the noise ratio [19, 20, 21]. Here, 10 years observational time is assumed. Taking look at Fig.2, one can see it is easy to get the circular polarization of the order of 0.08 in the present model. Hence, we can prove or disprove quantum gravity at a Lifshitz point by these observations.

V. CONCLUSION

We have considered the inflationary scenario in the context of quantum gravity at a Lifshitz point which isFIG. 2: The degree of circular polarization $\Pi$ for various $\gamma$ as a function of $\beta$ are shown. As can be seen from the figure, $\Pi$ grows as the $\beta$ becomes large. The dependence on $\gamma$ is not monotonic rather there is a value which gives the maximum polarization for fixed $\beta$ .

supposed to be a power-counting renormalizable theory. Because of the detailed balance condition, the action necessarily contains Cotton tensor which violates the parity invariance. We have calculated degree of circular polarization of primordial gravitational waves. As a consequence, we find that chiral primordial gravitational waves exist for generic parameters. It should be emphasized that the existence of circular polarization is a robust prediction of the theory.

In the usual discussions on the trans-Planckian effects, phenomenological approaches have been adopted

and mostly a modification of the spectrum has been discussed. While, we have used a candidate of quantum gravity and discussed chirality of primordial gravitational waves. The point is that we have found a modification of nature of gravitational waves rather than a modification of shape of the spectrum for gravitational waves. It is also apparent that the power spectrum for curvature perturbations is scale free. Thus, quantum gravity at a Lifshitz point is consistent with all of current observations. Moreover, we have a testable smoking gun of quantum gravity.

There are many issues to be pursued. One of those is to find exact solutions which represent black holes. When black hole solutions are found, it is very interesting to examine their spacetime structures, thermodynamics and Hawking radiation. It would be also intriguing to apply the idea of the anisotropic scaling in gravity to braneworld cosmology [22]. Furthermore, the possibility to generalize the anisotropic scaling in gravity to the cases where spatial isotropy is broken would be interesting from the point of anisotropic inflationary scenarios [23, 24].

After submitting our paper, we found a related work in the archive where inflation caused by a Lifshitz scalar is considered [25].

Acknowledgments

JS is supported by the Japan-U.K. Research Cooperative Program and Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science and Culture of Japan No.18540262.

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