Gravitational Waves in Axion Dark Matter

Axion dark matter is interesting as it allows a natural coupling to the gravitational Chern-Simons term. In the presence of an axion background, the gravitational Chern-Simons term produces parity violating effects in the gravitational sector, in particular on the propagation of gravitational waves. Previously, it has been shown that the coherent oscillation of the axion field leads to a parametric amplification of gravitational waves with a specific frequency. In this paper we focus on the parity violating effects of the Chern-Simon coupling and show the occurrence of gravitational birefringence. We also find deviation from the speed of light of the velocity of the gravitational waves. We give constraints on the axion-Chern-Simons coupling constant and the abundance of the axion dark matter from the observation of GW170817 and GRB170817A.


I. INTRODUCTION
The direct detection of gravitational waves in 2015 has widen the frontiers in research in fundamental physics [1]. Indeed, we are now in the era of gravitational wave astrophysics and multi-messenger astronomy and it is now possible to use gravitational wave signals to discriminate the various models beyond the standard model of particle physics as well as of cosmology. Moreover, the discovery of gravitational waves GW170817 [2] from a neutron star binary has sparked new interests in the study of nuclear physics. Remarkably, the observation of the optical counterpart of GW170817, GRB170817A [3], has given a constraint on the velocity of gravitational waves, which killed many modified theories of gravity [4][5][6].
In gravitational physics, there are three processes to be studied, namely, the production, propagation and detection of gravitational waves. The production process and detection process have been well studied. On the other hand, the propagation process has been mostly regarded as a trivial problem. In fact, in a Minkowski background, the gravitational wave equation is merely a conventional scalar wave equation. Even in the presence of the conventional matter, it is easy to solve propagation problem of gravitational waves in the curved background. However, things can get more interesting with axions. As a pseudoscalar, coupling to a gravitational Chern-Simons term is allowed [7][8][9]. As a result, parity symmetry is broken in the presence of an axion background. A peculiar feature of the axion dark matter is coherent oscillations of the axion field, which may affect the propagation of electromagnetic waves [10] and gravitational waves [11,12].
According to string theory, axions are ubiquitous in the universe [13]. Remarkably, the mass of string axions can take values in the broad range from 10 −33 eV to 10 18 GeV. In fact, it has been known that the axion is a natural candidate of an inflaton and induces a circularly polarized gravitational waves [14,15]. Recently, an axion has been intensively studied as a candidate for the dark matter. As the dark matter, we can consider the axion with mass from 10 −23 eV to 10 3 eV. The lower bound comes from observations of cosmic background radiations and the upper bound comes from observations of X-ray backgrounds [16].
In this paper, we comprehensively investigate gravitational waves propagating in an axion background. First of all, we review the results of the previous work on the parametric resonance of gravitational waves [11,12]. Then, we focus on the gravitational birefringence and the velocity modulation of gravitational waves. In particular, we discuss constraints on the Chern-Simons coupling and the abundance of the axion dark matter from observations of the velocity of gravitational waves.
The organization of the paper is as follows. In section II, we introduce basic equations for gravitational waves in axion-Chern-Simons gravity. We also discuss a potential ghost mode and a cutoff scale that is needed in order to avoid the occurrence of this un-physical feature. In section III, we analysis the propagation of gravitational waves in a background of coherently oscillating axions. In section IV, we review the parametric amplification of gravitational waves and present consistency checks. In section V, we consider the gravitational birefringence, which can be regarded as a gravitational Faraday rotation. In section VI, we derive the velocity of gravitational waves. We discuss constraints on the Chern-Simons coupling constant and the abundance of the axion dark matter using the observation of the velocity of gravitational waves. The final section is devoted to the conclusion.

II. GRAVITATIONAL WAVES IN DYNAMICAL CHERN-SIMONS GRAVITY
Let us consider the action of dynamical Chern-Simons gravity where the first term of action is the Einstein-Hilbert term, g is the determinant of the metric g µν and κ = 1/(16πG). The second term describes an action of an axion field Φ.
In this paper we will take Φ as a dark matter candidate and exploit existing observational constraints on its mass. The last term in (1) is the dynamical Chern-Simons action with ǫ αβγδ being the Levi-Civita tensor density. We have allowed a nontrivial coupling F (Φ) of the axion field to the RR Chern-Simons term.
Otherwise, the dynamical Chern-Simons term is topological and won't affect the equation of motion.
We are interested in the effect of the dark matter Chern-Simons coupling on the propagation of gravitational wave. Before we start, we need to fix the background. Let us consider a background spacetime with spatial isotropy and homogeneity Due to its structure, the Chern-Simons term does not contribute to the equation of motion of the isotropic and homogeneous universe. As a result, we have the equations of motion, where a dot denotes a time derivative and H =ȧ/a is the Hubble parameter. Generally the dark matter has a mass We will ignore self interaction as it is not relevant for our analysis.

A. Action for gravitational waves
Let us now derive the quadratic action for the gravitational waves from the action (1).
The tensor perturbation reads where h ij satisfies the transverse-traceless conditions h ij,j = h ii = 0. Substituting the metric into (1), we obtain the quadratic action where the stroke | denotes a covariant derivative with respect to the spatial coordinates.
Here, we have used the convention ǫ 0ijk = −ǫ ijk with ǫ 123 ≡ 1. It is convenient to expand h ij in terms of circular polarization basis where p A ij are the circular polarization tensor defined by and the polarization tensors p + ij and p × ij are the plus and cross modes respectively. The polarization tensors are normalized as and satisfies the helicity condition wherek i := k i /k is the unit vector in the direction of propagation of the gravitational wave.
The circular polarization modes are special as they diagonalize the dynamical Chern-Simons action term and we obtain the quadratic action (9) where the function B A is defined by In order for the perturbation to be stable, it is required that for both polarizations A. One can derive from this a natural upper bound to the energy scale of the gravitational wave in the theory [17]. This is in contrast to the parity violating gravitational waves in Lorentz violating gravity [18].

III. GRAVITATIONAL WAVES PROPAGATING IN AXION DARK MATTER
In this section, we suppose that a source in our cosmological horizon emits gravitational waves propagating in the axion dark matter background. In this situation, we can neglect the effect of cosmic expansion of the Universe in the dynamical equation (6) Since an upper bound of the dark matter density is given by the condition (17) is always satisfied for the axion dark matter with a mass m > 10 −30 eV.
For simplicity, let us consider a linear dynamical Chern-Simon coupling Following [19], we express the Chern-Simon coupling constant α in terms of a length ℓ as where M pl = √ 2κ is the reduced Planck mass. The coupling constant ℓ is experimentally constrained by the Gravity Probe B as [21] ℓ ≤ 10 8 km.
Now, we can write down the condition (16) in the present context. The axion satisfies the equation of motionΦ This can be solved as where, without loss of generality, we have made a choice of time so that the phase in (23) is zero. In this case, the no-ghost condition gives is the energy density of the dark matter field. Once we used the observed energy density, the amplitude can be determined as Φ 0 ≃ 2.1 × 10 7 eV 10 −10 eV m ρ 0.3GeV/cm 3 .
Above this scale, we cannot use the Chern-Simons gravity to describe the propagation of gravitational waves in the axion dark matter.
On energy scales below this, the equation of motion of gravitational waves in the Chern-

Simons gravity is given byḧ
where the function D A is defined by The first term is due to cosmological expansion, which we can ignore. The second term is due to the dark matter background (23). It is convenient to introduce the dimensionless parameter δ := m 2 αΦ 0 /κ in terms of which we have The parameter can be estimated as To see this, let us introduce a new variable Ψ A (t, k) defined by Then the equation of motion for the gravitational wave becomes where To the leading order of δ, the angular frequency is given by and eq.(34) takes the form of the Mathieu equation This describes an oscillator with a frequency k pumped by the polarization dependent periodic force with a magnitude f 0 and a frequency m. As is well known, the resonance occurs when k ∼ m/2 .
In this case, we obtain The amplitude of gravitational waves h A grows exponentially |h A | ∼ e Γt with the growth rate given by We can estimate the length R ×10 which the gravitational wave grows ten times bigger from the growth rate as follows, The range of the wave number for the resonance is given by This corresponds to a resonance width ∆k res : The phenomenological consequence of this result has been discussed in a previous paper [11].
Recently, more serious comparison with gravitational observation are made in [20] A. Coherence length In the above analysis, we have ignored the coherence issue of the dark matter background.
In principle, if the length scale of the gravitational perturbation becomes comparable to the Jeans length scale, the dark matter cloud can no longer remains as homogeneous and gravitational collapse will occurs. In other words, the coherence can be sustained only within the Jeans scale. As is known, the Jeans length r J of the axion dark matter can be deduced as r J = 6.7 × 10 20 eV −1 m 10 −10 eV If the fluctuations are larger than the band width ∆k res of the parametric resonance, the amplitude cannot grow efficiently. This is characterized by the ratio γ := ∆k vir ∆k res .
If γ satisfies the amplitude of gravitational waves grows. On the other hand, if γ satisfies then the frequency of the axion dark matter easily escape from the resonance band and the gravitational wave never grows. Since it is calculated as we see that the amplitude of gravitational waves grows.

V. GRAVITATIONAL FARADAY ROTATION
The fact that the angular frequency (36) is different for different circular polarization modes implies that the phase velocity v (p) is different for different circular polarization modes. It also implies a phase shift between the R and L polarization arises as the wave propagates: This is characterized by a period of phase oscillation The amplitude of the gravitational Faraday rotation is given by Hence, the gravitational Faraday rotation is sizable for axion with m 10 −5 eV.

VI. VELOCITY OF GRAVITATIONAL WAVES
The propagation of gravitational waves is characterized by the group velocity v (g) A := ∂ω A /∂k. To the leading order of δ, we obtain v (g) where we assumed δ ≪ 1 and mδ/k ≪ 1. Note that it is independent of polarization up to the second order of δ.
Once we obtain this constraint, by observing at a lower frequency, say 10 −4 Hz, we can further constrain the Chern-Simons coupling constant as For the extreme case of f gw = 10 −9 Hz and m = 10 3 eV, we obtain the stringent constraint on the coupling constant On the other hand, assuming the coupling constant ℓ = 10 2 km and f gw = 10 −9 Hz, we obtain the constraint on the abundance of the axion with m = 10 −10 eV as Ω axion < 3.4 × 10 −3 .
Thus, we see the gravitational waves can provide useful constraints to the Chern-Simons coupling constant and the abundance of the axion dark matter.

VII. CONCLUSION
We studied gravitational waves propagating in the axion dark matter. In the presence of the axion, it is natural to consider the coupling of the axion to the gravitational Chern-Simons term. Since the axion condensation violates the parity symmetry, there is a chance to observe parity violation effects in the gravity sector using gravitational waves. We found that the coherent oscillation of the axion field leads to the parametric amplification of gravitational waves with a specific frequency. We investigated the gravitational birefringence induced by the difference in the phase velocity of the different polarization modes. We also derived the group velocity of gravitational waves which is independent on the polarization at the leading order. In particular, we have given a constraint on the Chern-Simons coupling constant and the abundance of the axion dark matter from the observation of GW170817 and GRB170817A.
There are various ways to proceed. It is important to perform a comparison of our result with real data. It is interesting to study gravitational wave propagating in the ultralight vector dark matter [22,23]. It is also possible to extend the analysis to other higher spin dark matter.