Electromagnetic waves propagating in the string axiverse

It is widely believed that axions are ubiquitous in string theory and could be the dark matter. The peculiar features of the axion dark matter are coherent oscillations and a coupling to the electromagnetic field through the Chern-Simons term. In this paper, we study consequences of these two features of the axion with the mass in a range from $10^{-13}\,{\rm eV}$ to $10^{3}\,{\rm eV}$. First, we study the parametric resonance of electromagnetic waves induced by the coherent oscillation of the axion. As a result of the resonance, the amplitude of the electromagnetic waves is enhanced and the circularly polarized monochromatic waves will be generated. Second, we study the velocity of light in the background of the axion dark matter. In the presence of the Chern-Simons term, the dispersion relation is modified and the speed of light will oscillate in time. It turns out that the change of speed of light would be difficult to observe. We argue that the future radio wave observations of the resonance can give rise to a stronger constraint on the coupling constant and/or the density of the axion dark matter.


I. INTRODUCTION
According to string theory, axions are ubiquitous in the universe, dubbed the string axiverse [1][2][3][4][5].Remarkably, the axions could be a dark component of the universe and might be a dominant piece of the dark matter [6][7][8][9][10][11][12].In fact, it is difficult to discriminate between the axion dark matter and the cold dark matter on large scales.Therefore, it is important to find a method for proving the existence of the axions.
The key feature of the axion dark matter is its coherent oscillation.In particular, if the axion has the mass 10 −23 eV, the time scale of the oscillation is a few years and the oscillation produces the oscillation in the gravitational potential.Hence, one can use pulsar timing arrays to observe oscillating gravitational potential [13][14][15][16].There are other methods proposed for detecting the axion dark matter, for example, the super-radiance instability of the axion field in the rotating black holes constraining the mass range 10 −20 ∼ 10 −10 eV [2, [17][18][19], gravitational wave interferometers for probing the axion with mass 10 −22 ∼ 10 −20 eV [20], the dynamical resonance of the binary pulsars probing the mass range 10 −23 ∼ 10 −21 eV [21], and cosmological axion oscillations for exploring a wide mass range [22,23].
Recently, we have studied the gravitational waves in dynamical Chern-Simons gravity in the axion dark matter background [24].Then, we found that there occurs the parametric resonance of gravitational waves with parity-violation, that is, circularly polarized gravitational waves which allows us to probe the axions with the mass range 10 −14 ∼ 10 −10 eV.
Apparently, we can expect the same phenomena for electromagnetic waves.Since electromagnetic waves are often used to explore the universe, it is worth studying the phenomena in detail.The electrodynamics in the presence of the axion is called the axion electrodynamics [25] which has the Chern-Simons coupling between the axion and the gauge field.We see this interaction induces the parametric resonance of electromagnetic waves and also yields to the oscillation of the speed of light in time.In this paper, we study these two effects to gives rise to a new way to explore the axion dark matter in a mass range 10 −13 ∼ 10 3 eV corresponding to the observable frequency range of electromagnetic waves 10Hz ∼ 10 5 THz.Note that the axions with the mass above 10 3 eV are unstable against decaying into photons [9][10][11][12].
The organization of the paper is as follows.In Sec.II, we introduce the axion electrodynamics.In Sec.III, we derive wave equations in the oscillating axion background.In Sec.IV, we study the parametric resonance in the axion background.In Sec.V, we investigate the speed of light.The final section is devoted to conclusion.

II. AXION ELECTRODYNAMICS
The action of the axion electrodynamics is given by where each part of this action reads Here λ is a coupling constant, U (Φ) is a potential function for an axion field Φ, and A µ = (A 0 , A) is a gauge field with the field strength The dual of the field strength F µν is defined by where the anti-symmetrical epsilon tensor ǫ µνρσ is given by Here, ǫµνρσ is the Levi-Civita symbol.
From the above action, we get the equations of motion for the electromagnetic waves and the equation for the axion field Now, we can study electromagnetic wave propagation in the axion background.

III. WAVE EQUATIONS IN THE AXIVERSE
We assume the background spacetime is the Minkowski spacetime, because the dynamics of the cosmic expansion can be neglected on inter-galactic scales [26].Then, the metric reads Now, the covariant derivative is simply reduced to a partial derivative ∂ µ .We are interested in the time-evolution of the gauge field in the axion background.The gauge field is considered as the perturbed field A µ = δA µ .Next, we consider a homogeneous axion background Then, the equation of motion of axion is given by Here, we assumed the potential of the axion as It is easy to obtain the solution where Φ 0 is determined by the density of the dark matter ρ and the mass of the axion m as The equations of motion of the axion electrodynamics can be deduced as Here, the epsilon tensor in this coordinate system is defined as The time-component of the modified Maxwell equation is the same as the conventional Maxwell equation.This modified Maxwell theory is invariant under the gauge transformation, So, we can adopt the radiation gauge for the electromagnetic field, and we get the wave equations of the axion electrodynamics, where we defined the derivative operators We can diagonalize the wave equations with the circular polarization basis.In Fourier space, the vector field δA is expressed by where k is the wave number vector.The transverse gauge condition can be written as We can take polarization basis vectors, e (1) , e (2) , satisfying the following conditions Here, we defined k = |k|.Thus, the Fourier coefficient a(t) is expanded as Alternatively, we can use the circular polarization basis Now, the Fourier coefficient a(t) is expanded as Note that the components are related as This basis is useful for studying the parity violation.Using the relation we can diagonalize the wave equations as where This equation is nothing but the Mathieu equation describing the parametric resonance.Therefore, the growth rate is given by Since the axion has a non-trivial profile, the parity symmetry is violated in the equation of motion.Thus, the circular polarization should be generated.To be more precise, it is useful to define the polarization-rate of the electromagnetic field Due to the parametric amplification, the growth of one of the modes is larger that the other mode.In that case, we should have parity(t) ≃ ±1.Moreover, since the dispersion relation is modified by the axion, the speed of light is oscillating.We study the effects of these phenomena on electromagnetic waves in the following.

IV. PARAMETRIC RESONANCE
We assume that a lot of clumps whose sizes are about the Jeans length L a exist in the core of Galaxy and the axion is coherently oscillating there.These fuzzy object have the interaction with the electromagnetic fields through the Chern-Simons coupling.Thus, the coherent oscillations of the axion induce the parametric resonance of electromagnetic waves.
From the general theory of the parametric resonance, the resonance wave number k r is given by It is convenient to convert k r into the resonance frequency f r of the waves as This frequency corresponds to VLF (very low frequency) band, 3 ∼ 30 kHz.The existing FAST (Five-hundredmeter Aperture Spherical radio Telescope) has the frequency band from 70 MHz to 3 GHz in [27].Hence, this detector can survey the mass range from 10 −7 eV to 10 −5 eV.The SKA (Square Kilometre Array) has the frequency from 50 MHz to 350 MHz (SKA-low) and from 350 MHz to 14 GHz [28].Now, this detector will survey the mass range from 10 −7 eV to 10 −4 eV.If we consider the heavier axion with mass m ∼ 1 eV, the resonance frequency is that of the visible light around 10 2 THz.
On halo scales of the Galaxy, the energy density of the axion dark matter is about 0.3 GeV/cm 3 .Hence, the growth rate can be estimated as (34) Notice that this quantity is independent of the mass of the axion.In fact, the growth rate is determined by the coupling constant and the energy density of the axion dark matter.From this growth rate, we can estimate the time scale, t ×10 , for the amplitude to become ten times, as (35) Note that the time corresponding to 1pc is given by t 1pc ≃ 1.6 × 10 23 eV −1 .Thus, after the 10 Mpc propagation, the amplitude will be enhanced by 10 10 2 times.Therefore, we can obtain a stringent constrain on the coupling constant and/or the fraction of the axion dark matter in the universe.
(37) Since the band is very narrow, the circularly polarized monochromatic wave grows sharply at the resonance frequency.
If the electromagnetic waves go through near the core of the Galaxy, the energy density of dark matter gets enhanced In this situation, t ×10 becomes GeV/cm 3 ρ (39) From this estimation, the amplitude of waves going through the Galaxy core is further amplified by about 10 2 times.At the resonance frequency, when the amplitudes of waves are highly amplified, the electromagnetic wave should be fully polarized, namely, parity(t) ≃ ±1.
If we detected the resonance signal, we would argue that the axion dark matter exist.If we did not detect the resonance signal, we would be able to give the constraint on the energy density or the coupling constant.Therefore, we can say that the future very long wavelength radio wave observations of this effect can give rise to stronger constraints on the coupling constant and/or the density of the axion dark matter.

V. THE SPEED OF LIGHT
In axion electrodynamics, the dispersion relation in the axion background reads The phase velocity v p is given by Then, the deviation from the speed of light δc p is given by For example, if we observe the visible light which is in the wavelength range 380 ∼ 750 nm, we find the relative deviation of the speed of light: (42) Here, l em is the wave length of the visible light.
In fact, the group velocity is more relevant to observations.The group velocity v g is given by The deviation from the speed of light δc g is given by Notice that the linear term is canceled out in the above formula [29] and the deviation of the group velocity is given by the square of that of the phase velocity Thus, we can estimate δc g as The relative deviation from the speed of light δc is constrained by observations of gamma-ray bursts [30] as δc 10 −21 . (44) Since δc g is much smaller than the current observational constraint, we can say that there is no constraint on the energy density of axion field or the coupling constant from the speed of light.

VI. CONCLUSION
Since the axion is one of the candidates for the dark matter, it is worth seeking a method for detecting axion.In this paper, we considered the axion with the mass range from 10 −13 eV to 10 3 eV.We focused on two consequences of the coherent oscillation of the axion dark matter and a coupling to the electromagnetic field through the Chern-Simons term.First, we studied the parametric resonance of the gauge field induced by the coherently oscillating axion.It turned out that, as a result of the resonance, the amplitude of the electromagnetic waves is enhanced and the circularly polarized monochromatic waves are generated.We found that the future very long wavelength radio wave observations of this effect can give rise to stronger constraints on the coupling constant and/or the density of the axion dark matter.Second, we studied the velocity of light in the background of the axion dark matter.We found that the dispersion relation is modified and the speed of light shows oscillations in time, but this modification is too tiny to be observed.
In this paper, we have discussed the modification of the dispersion relations which leads to the change of the speed of light.However, this effect was very small in axion electrodynamics.This would also happen to gravitational waves.We report the detailed analysis in a future work.