On the detectability of ultralight scalar field dark matter with gravitational-wave detectors

An ultralight scalar field is one of the dark matter candidates. If it couples with Standard Model particles, it oscillates mirrors in gravitational-wave detectors and generates detectable signals. We study the spectra of the signals taking into account the motion of the detectors due to the Earth’s rotation/the detectors’ orbital motion around the Sun and formulate a suitable data-analysis method to detect it. We find that our method can improve the existing constraints given by fifth-force experiments on one of the scalar field’s coupling constants by a factor of ∼ 30, ∼ 100 and ∼ 350 for mφ = 2× 10−17 eV, 10−14 eV and 10−12 eV respectively, where mφ is the scalar field’s mass. Our study demonstrates that experiments with gravitational-wave detectors play a complementary role to that Equivalence Principle tests do.


I. INTRODUCTION
Although Weakly Interacting Massive Particles (WIMPs) are promising candidates of dark matter, null results from various experiments [1][2][3][4] cast doubt on WIMPs. Therefore, it is worth searching for other candidates. Ultralight scalar field is one of the other dark matter candidates. This type of dark matter can have mass down to ∼ 10 −22 eV [5] and we especially consider a mass range of 10 −19 eV m φ 10 −10 eV. The existence of such light scalar fields is motivated by string theory [6]. Ultralight scalar field dark matter starts oscillating when H m φ , where m φ is its mass [7]. Since the occupation number is large for our mass range, it can be treated as a classical scalar field in the current universe.
An oscillating classical scalar field causes various peculiar phenomena from which we can probe its existence. For instance, the energy-momemtum tensor of the oscillating scalar field generates metric perturbations that oscillate with half the period of the oscillation of the scalar field. [8] showed that such metric perturbations can be detected in Pulsar Timing Array observations if the mass of the scalar field is around 10 −22 eV, and null results have already put constraints on its abundance [9,10]. If the scalar field couples with the Standard Model (SM) particles, it mediates additional force [11]. Such force is searched for in fifth-force experiments [12] and Equivalence Principle (EP) tests such as Lunar Laser Ranging (LLR) experiments [13,14], experiments from Eöt-Wash group [15,16] and the MICROSCOPE experiment [17]. The oscillation of the scalar field that couples with the SM particles causes time variations of physical constants such as the proton mass and the fine-structure constant. They produce the time variations of the atomic transitions as well as oscillatory force on bodies from which the constraint on the coupling to the SM particles can be obtained [18][19][20][21][22][23][24][25][26][27]. For example, hyperfine frequency comparison of 87 Rb and 133 Cs atoms has been used for this purpose [22].
In this paper, under the assumption that an ultralight scalar field comprises all dark matter and it weakly couples to the SM particles, we investigate the possibility of detecting such a field with gravitational wave (GW) detectors. As far as we know, the idea of detecting such a field by the GW detectors was firstly argued by [18]. Notice that the same idea was discussed in [28] to detect a light vector field. The main idea is the following. An ultralight scalar field as dark matter varies spatially on the scale of the de Broglie wavelength k −1 ∼ 1/(m φ v DM ) in the Galaxy, where v DM ∼ 10 −3 is the velocity dispersion of dark matter in the Galaxy. Since the values of the physical constants depend on the value of the scalar field, the masses of optical equipments such as mirrors also depend on them. Then, the spatial variations of the scalar field exerts position-dependent oscillatory force on every optical equipment. As a result, the optical equipments undergo position-dependent oscillatory motions, which ends up with non-vanishing signals in the GW detectors' outputs. Since the frequency of the motion is f φ m φ /2π and the frequency dispersion is tiny, ∆f φ ∼ f φ v 2 DM , detectors' signals are nearly monochromatic. It is worth mentioning that using the GW detectors to probe the ultralight scalar field dark matter was also investigated in [29]. In this reference, not the effects caused by the scalar force but the effects of metric perturbations sourced by the scalar field were considered, and it was found that such effects are far from the detectable level even with the future experiments. *1 .
To extract information on ultralight scalar field dark matter from real data as much as possible, we need to understand its signal's characteristics and develop a suitable data-analysis method. In this paper, we formulate a data-analysis method toward a detection of ultralight scalar field dark matter with GW detectors. First, we derive the expression of the signal in an output from a GW detector. Especially, we take into account detectors' motion due to the Earth's rotation/the detectors' orbital motion around the Sun, which is not taken into account in the previous studies [18,28]. Next, we develop a data-analysis method suitable for the signal. Taking into account the possibility that we have only one detector, we develop a method which is applicable even with one detector. Given the sophisticated analysis method, the constraints can be much tighter than the previous estimate. Finally, we update the previous estimate of the constraints.
The organization of the paper is as follows. In Section II, we explain the model we consider and how ultralight scalar field dark matter behaves in our Galaxy. In Section III, we obtain a formula for the signal in an output from a gravitational-wave detector. In Section IV, we discuss the signal's characteristics and formulate a suitable dataanalysis method to detect this signal. Then, we update the previous estimate of the constraints with our analysis method. Section V is devoted to the conclusion.

II. BASICS
We briefly review the model we consider and how the scalar field dark matter behaves around the gravitational-wave detectors.

A. Model
We consider a scalar field, φ, which couples with particles in the Standard Model and accounts for all the amount of dark matter. We assume that φ linearly couples with the particles. Then the low-energy (∼ 1 GeV) effective lagrangian is given by [11] where L SM is the Lagrangian of the Standard Model. L φ and L φ−SM are given by where κ ≡ √ 4π/M pl , β 3 is the QCD beta functions and γ mi are the anomalous dimensions of the electrons, the u and d quarks. With respect to the coupling between φ and quarks, it is more convenient to use instead of d m d and d mu in the context of equivalence principle tests [11]. Since the amplitude of φ attenuates as the universe expands, we assume that the amplitude is so tiny in the current universe and we can neglect higher-order terms in V (φ) and approximate it as B. The scalar field around a detector We let (t, x) be the rest frame of the Solar system and discuss how φ behaves in this frame. Dark matter has velocity dispersion of v DM ∼ 10 −3 in our Galaxy and the Solar system is also moving with the velocity around v DM . Therefore, the wavenumber of φ, k, satisfies Then φ can be written as where |φ k | is negligibly small for | k| m φ v DM . The angular frequency ω k ∼ m φ + k 2 /2m φ and its dispersion ∆ω k are In addition, for φ to account for all the amount of dark matter, the following condition must be satisfied, where < · · · > is an average over space with the volume of V (m φ v DM ) −3 and time of T m −1 φ and we used a result from [30] as a value of ρ DM .

III. RESPONSE OF GRAVITATIONAL-WAVE DETECTORS TO SCALAR WAVES
As we explained in the previous section, there is a scalar-waves background in the Galaxy if an ultralight scalar field accounts for dark matter. In this section, we obtain the expression for the signal, h(t), caused by the scalar waves in a GW detector's output. There are multiple sources of the signal in the output of the detector. One possible source is a time variation of the laser frequency due to the oscillation of the scalar field. This mimics common motion of the arms of the detector. Since interferometric GW detectors are mainly sensitive for their differential motion, this effect is subdominant. The second possible source is modification of laser light's propagation, which is also negligible as we prove in Appendix B. Then, we conclude that the signal is mainly sourced by the motion of optical equipments caused by the scalar waves. In what follows, we first calculate the response of the detectors to monochromatic waves, Then we generalize the expression to that for the superposition of the waves. In our work, we consider Laser Interferometer Gravitational-Wave Observatory (LIGO) [31], Einstein Telescope (ET) [32] and Cosmic Explorer (CE) [33] as representative ground-based detectors and DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) [34] and Laser Interferometer Space Antenna (LISA) [35] as representative space-based gravitational-wave detectors. The sensitivity curves for ground-based detectors are taken from [33]. For the DECIGO's sensitivity curve, we use an analytical function in [36]. For LISA, we apply configuration parameters' values summarized in [35] to calculate the sensitivity.

A. Motion of optical equipments
Due to the interaction with φ given by Eq. (3), atomic mass depends on the value of φ. Therefore, the action of an optical equipment in this theory is given by, where we neglect gravitational fields. Then the equation of motion of the optical equipment in the non-relativistic limit becomes Assuming that the equipment is at rest in the absence of φ, the second term on the right hand side is second order in the amplitude of the scalar field and is subdominant compared to the first term. Thus, the equation of motion becomes Since the masses of atoms are mostly determined by the QCD energy scale, α(φ) is approximately given by [11] α(φ) d * Substituting Eq. (10) and Eq. (14) into Eq. (13), we obtain where x 0 is the position on which the equipment is in the absence of φ. Solving this leads to

B. Response of the interferometric gravitational-wave detectors
Let us next derive the expression of the signal caused by the motion of the optical equipments. First, we calculate the response of a Michelson-interferometer with arms along the unit vectors, n and m. Following the method used in [37], we calculate the time necessary for photon to make a round trip through the arms. The oscillation of the mirror at x due to the scalar field is given by Then the position of the front mirror, x 1 , and that of the end mirror, x 2 , in the arm along n are where L is the arm length in the absence of the scalar field. Therefore the perturbation of the round-trip time due to the oscillation, δt(t; L, n), is where t is the time where photon returns back to the front mirror and higher order terms in δx have been neglected. This can be separated into two parts, The first part is and the second part is Since | k · n| ∼ m φ v DM , the round-trip time can be approximated as follows, From this, we can calculate the phase delay of the laser light after a round-trip as follows, where f laser is the frequency of the laser light. The derivative expansion in Eq. (21) breaks down for L > 1/(m φ v DM ). This situation never realizes for any detectors considered in this paper. Phase delay for another arm is obtained by replacing n by m. Since the signal in the gravitational-wave channel is the difference of the phases of the laser light traveling in the two arms, it is where Generalizing this expression into that for the superposition of the non-relativistic waves, we obtain Taking into account the time variations of the detectors' orientation, we must replace n and m by time-varying vectors, n(t) and m(t). For ground-based detectors, the time variation is due to the Earth's rotation. Then the expression for the signal is where Ω d = 2π/(sidereal day). For the space-based detector, DECIGO, the expression is where Ω y = 2π/year. The functions, g 1,i (t), g 2,i (t), c 1,i (t) and c 2,i (t), have a frequency range of . Their expressions are summarized in Appendix. C.
Next we calculate the response of LISA. There are several methods to cancel out laser-frequency noise in the LISA experiment and a measured variable depends on which method we choose. In this paper, we apply the secondgeneration TDI variable X 1 (t) [38,39]. Note that the Doppler shift caused by scalar waves can be calculated by (35) and the contribution from scalar waves to X 1 (t) is approximately where n and m are unit vectors along two different arms of LISA and L is the arm length of LISA in this case. Then it is straightforward to calculate X SW 1 (t) and the signal is where Taking into account the time variations of the detector's orientation, the expression of the signal becomes The functions, x 1,i (t) and x 2,i (t), have a frequency range of . Their expressions are also summarized in Appendix C.

IV. DETECTION METHOD AND FUTURE CONSTRAINTS
In this section, we discuss an appropriate detection method for the signal derived in the previous section and the future constraints obtained by this method. In this section, we use f φ = m φ /2π instead of m φ .

A. Characterization of the signal
Before moving to the detection method, we briefly summarize the characterization of the signal. Looking at Eq. (31), (32), (33), (34), (40) and (41), we find that the signal is a sum of 2N det (f φ ) + 1 spectra with width of for ground-based detectors and for space-based detectors. Each spectrum has frequency band of Therefore, if f φ v 2 DM f det the signal has one broadened spectrum whose frequency band is The schematic spectrum is drawn in Fig. 1.

B. Detection method
Next we discuss an appropriate method to detect this signal. The data from a GW detector is a sum of the signal, h(t), and the noise, n(t), Here we assume that the noise is stationary Gaussian. The Fourier components of the data arẽ where T obs is the observation time. From the discussions in IV A, the signal's power is in the following frequency bins, where R(x) is the integer closest to x. In what follows, we present two efficient ways to extract this signal from the noise.
1. Single detector: Incoherent sum of spectra over F (f φ ) One simple way is to take a sum of spectra over the signal's frequency range. The detection statistic in this method is where S(f ) is the one-sided power spectrum of the noise. Each term in the sum is normalized so that its expectation value is 1 if the data is pure noise. The threshold of ρ(f φ ) to claim detection is determined by its statistical properties. If the data is pure noise, the cumulative distribution function for ρ(f φ ) is where N (f φ ) denotes the number of elements in F (f φ ). Given a false alarm probability, F , the threshold ρ c is determined by Since the search is performed for various values of f φ , the number of the trials, N t , should be taken into account when F is determined. Assuming that the search is performed over the detector's frequency range at intervals of ∼ 1/T obs , the trial number is (f max − f min )T obs + 1, where f min and f max are minimum and maximum frequency of the frequency range over which the search is taken over. For example, to reject the existence of the signal with 2 sigma level, F should be The threshold of the signal's amplitude which satisfies

Two detectors: Narrow band stochastic gravitational-wave background search
Another way is to take correlation between the outputs from two detectors, which is used to search for stochastic GW background [40,41] and considered in the previous study on search for a light vector field [28]. The detection statistic is wheres 1 (f k ) ands 2 (f k ) are Fourier components of the data from two detectors.Q(f k ; f φ ) should be determined so that the search is optimal. Since the wavelength of the scalar field, 1/m φ v DM , is much longer than that of GWs with the same frequency, the overlap reduction function [40] is approximately constant. In addition, we only assume the signal's bandwidth and do not make any assumptions on its spectral shape. Under these considerations, we apply the following filter function,Q where Having explained the two detection methods, let us estimate the sensitivities of these methods. For the former method, the deviation of the expectation value of ρ(f φ ) due to the presence of the signal is where On the other hand, its variance when the data is pure noise is Therefore, the threshold of h 2 for detection is Next, we estimate the sensitivity of the latter method assuming that two detectors are parallel and their sensitivity curves are the same. The expectation value of the detection statistic when the signal is present is while its variance when data is pure noise is Therefore, the threshold amplitude for detection is which is different from Eq. (60) only by a O(1) factor. Since the former method requires only one detector, we focus on that method and estimate the constraints given by experiments with gravitational-wave detectors in this paper. Especially in the case where Note that in addition to improvement by a factor of (1/f φ v 2 DM ) 1/2 mentioned in [18], there is another improvement by a factor of N 1/4 (f φ ), which is ∼ 7 for f φ = 100 Hz and T obs = 1 year. The latter factor is overloooked in the literature, and our estimate improves the previous estimate due to this factor.

C. Future constraints
Next we estimate the future constraints on this dark matter model with the former detection method described in the previous subsection. For simplicity, we consider a plane wave satisfying (9), Averaging the square of the signal over the directions of k and time, we obtain The future constraints we can obtain with gravitational-wave detectors and the constraints given by the fifth-force experiments (Fifth-force) and the tests of equivalence principle (EP test) are shown. The shaded region is or will be excluded. We consider the case where the coupling constants other than dg are zero here. We assume that T obs = 1 year and all of them are 2σ limits. The constraints of EP tests shown here are given by an experiment from Eöt-Wash group [16] and MICROSCOPE [17]. The constraints given by LLR experiments [13,14] and a hyperfine frequency comparison experiment [22] are less stringent in this frequency range. Note that the constraints from EP tests depend on the values of the other coupling constants and they give no constraints if dg = dm e = dm u = dm d and de = 0.
for detectors other than LISA and for LISA. Then, the criteria, h 2 (f φ ) > h th (f φ ), leads to the parameter region of d g we will be able to probe with gravitational-wave detectors. Fig. 2 compares this parameter region and constraints given from the fifthforce experiments [12] and tests of equivalence principle (EP tests) [16,17]. All the detectors will provide more stringent constraints than the fifth-force experiments do in a mass band. Especially, the constraints at m φ = 2 × 10 −17 eV, 10 −14 eV and 10 −12 eV are improved by a factor of ∼ 30, ∼ 100 and ∼ 350 respectively.
On the other hand, the future constraints are weaker than those from the EP tests in most of the band. However, it is the case only if the coupling constants other than d g are zero. The EP tests constrain the following parameters, while we can constrain d * g with gravitational-wave detectors and fifth-force experiments. Especially, EP tests give no constraints if d g = d me = d mu = d m d and d e = 0. Fig. 3 shows the parameter regions which pass the MICROSCOPE experiment and experiments with gravitational-wave detectors in the case where d g and dm are non-zero. We can see that the latter experiments can exclude the parameter region along the line d g = dm, which can not be excluded by MICROSCOPE. In that sense, this search is complementary to the EP tests.

V. CONCLUSION
We formulated a suitable data-analysis method to detect signals in outputs from gravitational-wave detectors and studied their detectability. We first derived an analytical expression of the signal taking into account the motion of detectors and investigated its spectra. Then, we formulated a suitable data-analysis method to detect this signal. Taking into account the possibility that we have only one detector, we formulated a method which is applicable even with one detector. Finally, we estimated the future constraints on the coupling constants between scalar field dark matter and Standard Model particles with our method.
As a result, we found that we can improve the previous estimate of the detectability given in [18] by a factor of ∼ 7 at f φ = 100 Hz. We also found that all of the detectors we consider will provide more stringent constraints than the fifth-force experiments do in a mass band. Especially, the constraints on d g at m φ = 2 × 10 −17 eV, 10 −14 eV and 10 −12 eV are improved by a factor of ∼ 30, ∼ 100 and ∼ 350 respectively. Since experiments with gravitational-wave detectors constrain a different combination of the coupling constants from that EP tests do, they play complementary roles.
Since we formulated a data-analysis method, the next thing to do is to apply the method to real data. One of the issues to resolve for the application is line noise, which is sinusoidal noise and can mimic the signal we consider. Such noises also degrade the sensitivity of continuous-wave searches [42]. One characteristic of the signal to distinguish them can be expected bandwidth of ∼ f φ v 2 DM . Another issue is non-stationarity of the detectors caused by environmental disturbances such as earthquakes [43]. We need to generalize our method so that it can be applied to data, a part of which is unavailable due to increased noise level. Improving our analysis method to resolve these issues and analyzing real data are left for the future works.
• The previous study neglects sub-leading terms with respect to m φ L (h 1 (t) in Eq. (A4) and Eq. (A7)). However, since the leading term is suppressed by v DM ∼ 10 −3 , the sub-leading terms are actually not negligible. We derive improved signal's formula by taking into account these sub-leading terms.
• Since the signal is nearly monochromatic, we can improve signal-to-noise ratio by integrating the signal. We re-estimate the detectability with the analysis method we proposed in this paper.
Following calculations in [29], we can easily find that a metric perturbation caused by monochromatic scalar waves, Eq. (65), is where the oscillating part of Φ is The motion of optical equipments caused by the metric perturbation is Following the calculations in Sec. III, we can find that the signal caused by the metric perturbation is for detectors except for LISA and for LISA. While h 1 (t) is suppressed by m φ L compared to h 2 (t) in the regime, m φ L 1, h 2 (t) is also suppressed by v DM ∼ 10 −3 with respect to h 1 (t). Therefore, over most of the frequency range we consider, h 1 (t) is more significant than h 2 (t). Fig. 4 compares amplitudes of h 2 (t) and h 1 (t) + h 2 (t) averaged over directions of k for LIGO. We can see that the h 1 (t) becomes much larger as the signal's frequency increases.
Finally, we re-estimate the detectability of the signal with our analysis method. Fig. 5 shows the constraints on the energy density of an ultralight scalar field we can obtain with GW detectors. It also shows the constraints by LISA obtained by the same method as that of the previous study (only with h 2 (t) contributions and an integration time of 1 second). Although the constraints become much tighter, they are still far from ρ DM .  Then the equation of motion for the vector potential is In this paper, we consider the case where the scalar field behaves as the following non-relativistic matter waves in the Galaxy, where | v| 1. We obtain A µ by solving Eq. (B2) under the following assumption, The first condition means that the amplitude of the scalar field is tiny enough, which is valid in our case. The second condition means the frequency of the laser light is much larger than the frequency of the oscillation we search for, which is valid in usual gravitational-wave experiments.
We separate the vector potential into two parts, where the second part is the correction due to ψ. To simplify the calculation, we apply a gauge transformation to move to a suitable gauge. To obtainĀ µ , we apply the Lorentz gauge, Then the equation of motion forĀ µ is and its solution isĀ To obtain δA µ , we apply a following gauge, To move to this gauge, we choose Λ such that it satisfies Since the right-hand side is the sum of two modes with four momenta of k µ ± m φ u µ and it is easy to show that Eq. (B11) is solvable and In this gauge, the equation of motion for δA µ is Since the third term is negligible due to the second condition in Eq. (B4), this equation can be approximated as The solution of this equation is Therefore, the solution for Eq. (B2) is which means the amplitude of the vector potential modulates. Next we estimate the effect of this modulation on the output of the gravitational-wave detectors. Due to the second condition in Eq. (B11), the electromagnetic tensor can be approximated as Therefore laser power, I, modulates and the amplitude of this modulation is where φ amp is the amplitude of the scalar field's oscillation. Assuming the scalar field accounts for all the amount of dark matter and φ amp ∼ √ 2ρ DM /m φ , it can be estimated as where f φ = m φ /2π. On the other hand, the power spectrum density of shot noise in I is S I (f ) = 2 ωI, where ω is the angular frequency of laser light. Then where λ laser is the wavelength of laser light. Since the time for which this signal keeps the coherence is t coh ∼ 1/f φ v 2 DM , the detectable amplitude of this modulation is improved by a factor of 1 Therefore, for the modulation due to the scalar field's oscillation to be detected, the laser power must satisfy In the experiments we consider, λ laser O(1000) nm. On the other hand, the Equivalence Principle tests [15][16][17] have already provided the constraints, d e 10 −2 for the ground-based detectors' band and d e 10 −4 for the spacebased detectors' band, if it is assumed that miraculous cancellation between the coupling constants does not occur. Therefore, the required laser power (B23) is much higher than laser power in the gravitational-wave experiments we consider. In reality, we can not integrate the signal for longer time than the observational time. It makes the requirement for laser power more severe. On the other hand, if 1/f φ v 2 DM is shorter than observational time, we can improve S/N with the analysis method we proposed by a factor of N 1 4 (f φ ) as mentioned below Eq. (64). However, the difference is O(1) and it does not change the conclusion. Therefore, the modification of laser light propagation does not cause detectable signals.

Appendix C: Time variation of the detectors' orientations
In this appendix, we obtain the expressions of the functions in the right-hand sides of Eq. (31), (32), (33), (34), (40) and (41). Here we refer to Appendix C of [44]. The detectors' configurations are shown in Fig. 6.