Probing Bosonic Stars with Atomic Clocks

Dark Matter could potentially manifest itself in the form of asymmetric dark stars. In this paper we entertain the possibility of probing such asymmetric bosonic dark matter stars by the use of microwave atomic clocks. If the dark sector connects to the bright sector via a Higgs portal, the interior of boson stars that are in a Bose-Einstein condensate state can change the values of physical constants that control the timing of atomic clock devices. Dilute asymmetric dark matter boson stars passing through the Earth can induce frequency shifts that can be observed in separated Earth based microwave atomic clocks. This gives the opportunity to probe a class of dark matter candidates that for the moment cannot be detected with any different conventional method.


I. INTRODUCTION
Currently cosmological observations leave little doubt about the presence of dark matter (DM) in our Universe [1]. Although its existence is very well motivated, the nature of DM still remains a complete mystery. The masses of possible DM candidates span several orders of magnitude ranging from ultralight particles of ∼ 10 −22 eV [2][3][4], to massive black holes of tens of solar masses [5][6][7]. Furthermore it is possible that DM consists of several different components. Currently the so-called Collisionless Cold Dark Matter (CCDM) paradigm is consistent with observations of the large scale structure, suggesting that DM selfinteractions are absent or very small. On the other hand, observations of the small scale structure seem to be at odds with CCDM. The core-cusp problem of dwarf galaxies, the diversity problem, and the "too big to fail" raise doubt about the validity of the CCDM paradigm (see [8] and references therein). Although these issues can be attributed to different factors, self-interacting DM (SIDM) can alleviate these problems, reconciling theory with small scale structure observations. Several studies of SIDM have been undertaken [9][10][11][12] providing constraints and an optimal range of cross sections, 0.1 − 10 cm 2 /g, for DM self-interactions that can solve the CCDM problems. Stringent constraints are imposed, for example, for the case where the self-interactions are mediated by a particle φ which is coupled to the Standard Model (SM) via a Higgs portal [13][14][15][16][17][18]. In such a case, φ must decay before the start of the Big Bang Nucleosynthesis (BBN) in order to avoid energy injection to the plasma during BBN. These constraints can be evaded if φ is also coupled to sterile or active neutrinos [19]. Also, SIDM might be needed in order to provide seeds for the existing supermassive black holes we observe in the Universe [20]. Finally, SIDM is motivated if one embeds the DM sector in Grand Unified Theories (GUTs). The punchline is that SIDM might be welcome as it can alleviate problems in the CCDM paradigm and/or explain unresolved astrophysical issues.
One particular class of SIDM theories is that of asymmetric DM (ADM). ADM is an alternative paradigm to thermally produced dark matter such as Weakly Interacting Massive Particles (WIMPs). In the usual WIMP paradigm, DM annihilates to SM particles. It turns out that for an annihilation cross section on the order of the weak interactions, DM annihilations reduce the DM relic density to the value observed today. This is the so-called WIMP miracle. However, this is not the only theoretically motivated production mechanism. Another interesting one is that of ADM. In this case, an asymmetry between the number of DM particles and antiparticles is created in the early Universe. Strong DM annihilations deplete the population of the particles in lack, leaving only DM particles of the species in excess. This is also a very well motivated paradigm. For example one can imagine a common asymmetry mechanism for baryogenesis and DM genesis. If for any baryon unit asymmetry, a DM unit is also created, then a DM particle of a mass ∼5 GeV provides the correct relic abundance of DM in the Universe. For a review on ADM see [21]. An ADM component that exhibits self-interactions among DM particles can cause the collapse of a DM cloud either by a gravothermal collapse mechanism such as in [20] or by effectively evacuating the energy of the system via "dark"-Bremsstrahlung radiation [22], leading to black hole or asymmetric dark star formation. The latter are stable compact objects where their hydrodynamic stability is caused by DM self-interactions, Fermi pressure or the uncertainty principle depending on the underlying model and the nature of the DM particle (i.e. if it is a fermion or a boson). The possibility of forming compact stable objects consisting of fermionic ADM with self-interactions was studied in [23]. The Tolman-Oppenheimer-Volkoff equation was solved and the mass-radius relation was found for these objects. It was assumed that the self-interactions were Yukawa-type and could either be attractive mediated by a scalar field φ or repulsive mediated by a vector boson field φ µ . The case of bosonic SIDM forming compact star-like objects was studied in [24], where the density profile, the mass-radius relations, and the maximum mass that these objects can withstand were derived. We should stress that asymmetric dark stars are distinctly different from dark stars that might have formed in the past if DM is of symmetric nature [25][26][27]. In the latter case, the hydrodynamic stability of the star is achieved by radiation pressure from the DM annihilation. These stars, if they ever existed, should have annihilated by now. On the contrary, due to the particle-antiparticle asymmetry, no annihilation takes place inside the asymmetric dark star. The species in excess have already annihilated away the minority component early on. Therefore, once formed asymmetric dark stars can be stable. We should also add that dark stars can exist in the form of hybrid compact stars made of baryonic and DM [28][29][30][31] as well as in the context of mirror DM [32][33][34][35].
In the case of "bosonic" stars, if matter is sufficiently cold it stays in the ground state which is a Bose-Einstein condensate (BEC) state. Several boson particles can form bosonic stars, e.g., axions, or the scalars which drive the expansion of the Universe in quintessence models [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. Recently, the authors of [56][57][58] were able to place constraints on scalar DM models based on the variation of fundamental and physical constants. One should keep in mind that, in general, dark stars can contribute to the overall DM abundance of the Universe. Gravitational lensing experiments such as MACHO [59] and EROS [60] constrain the abundance of compact objects in the mass range 10 −7 M M 10M , (M being the solar mass) to be less than 20% of the total DM density of the Universe.
Star-like objects composed of ADM can be probed both by the aforementioned gravitational lensing studies but also by gravitational wave signals produced in the coalescence of such dark objects with black holes, other compact stars such as neutron stars, or among themselves [61][62][63][64][65][66][67][68]. Additional light signals can be also produced in particular scenarios where there is a portal that couples the dark sector to the SM one [69]. However, if the dark star is sufficiently diffuse, as is the case for boson stars composed of ADM, gravitational waves produced in mergers of such objects are weak and alternative detection methods should be developed. One such method is via the use of high precision atomic clocks. The idea is simple. Atomic clocks measure time by using specific atomic transitions. For microwave atomic clocks, the ticking of the clock is sensitively dependent on the fine structure constant as well as physical constants such as the masses of the electron and the quarks. Whereas for optical atomic clocks, the ticking of the clock is only dependent on the fine structure constant [70,71]. The passing of an atomic clock through a dilute ADM boson star could, under some conditions, change these parameters and cause the atomic clock to tick at a different rate than an atomic clock not covered by the boson star. Therefore, small de-synchronizations of atomic clocks located at different places on the Earth could indicate the passing of such a ADM boson star. Clearly, once two clocks located at different places are covered by the star, they again measure time with the same rate. The de-synchronization takes place only in the case where one clock is inside and another one outside the ADM boson star.
Previously, optical atomic clocks were able to reach a precision of 10 −18 for the fractional frequency shift δω/ω [72,73], while more recently, a record precision of 10 −19 [74] has been reached. Microwave atomic clocks tend to be less sensitive, on the order of 10 −16 [75], but it has been suggested that this precision can be improved to 10 −17 (T (K)/300) 2 for certain alkali atoms [76]. These tools provide a new means to probe the existence of cosmological topological defects [71] or dilute ADM boson stars. In fact using data from the satellite born clocks of the Global Positioning System, the authors of [77] managed to set new constraints on models where DM is in the form of topological defects. Previous analyses [78,79] have also shown how atomic clocks are affected if the DM is axionic in nature.
In this paper we present potential constraints that can be set on ADM boson stars (if these objects contribute to the DM relic abundance) using microwave atomic clocks. In particular, we investigate models where the bosons couple to the SM via a Higgs portal providing a means to affect change in physical constants which determine the ticking of atomic clocks, such as the electron mass. In this study, we focus on the effect of ADM boson stars on microwave atomic clocks since, for the Higgs portal that we assume, optical atomic clocks are not sensitive probes as they are only affected by shifts in the fine structure constant. We point out that there is another DM model, relaxion DM [56,57], that connects the dark sector to the electromagnetic field. In this case, optical atomic clocks can be used as probes of boson stars composed of this type of DM. The paper is organized as follows: In Sec. II we derive the density profile and mass-radius relation of ADM boson stars and we estimate the rate of events i.e., the frequency with which these objects pass through the Earth. In Sec. III, we present the Higgs portal that is responsible for shifting the timing of atomic clocks and we present updated constraints on the couplings involved in the Higgs portal. In Sec. IV we identify the parameter space of ADM boson stars that can potentially be probed by future atomic clocks and finally we conclude in Sec. V.

II. BOSON STARS
As mentioned in the introduction star-like objects can be formed from bosonic DM which, at low temperatures, is in a BEC state. We analyze a φ 4 theory for complex scalar fields, where the self-interaction potential is given by, where λ is the self-coupling constant between bosons. Here, the positive(negative) sign denotes repulsive(attractive) self-interactions. One can find gravitationally bound systems composed of ADM subject to the self-interaction given by Eq. (1) and analyze the collision rate of such systems with Earth based atomic clocks.

A. Density Profile
In the case of repulsive self-interactions, one can solve the Einstein-Klein-Gordon (EKG) equation in order to derive the density and mass-radius profile of these objects (see e.g. [24,39]). In the case of attractive selfinteractions, the relativistic effects are suppressed and it suffices to solve the Gross-Pitaevskii-Poisson (GPP) equations [52,53]. In this paper, we focus on attractive self-interactions because they give objects that are more easily probed by atomic clocks. Namely, objects with smaller compactness (ratio of mass over radius), such as those composed of ADM with attractive self-interactions, have an increased probability of passing by the Earth, thus creating a de-synchronization in atomic clocks that are apart from each other. On the contrary, repulsive self-interactions tend to create systems with higher compactness and, therefore, lower chances of passing by the Earth. Instead of exactly solving the GPP equations, an alternative variational method can be used [51,52]. One can choose some variational ansatz for the wavefunction that characterizes matter in the boson star and minimizes the energy of the system. Taking attractive self-interactions corresponding to the negative sign in Eq. (1) and assuming a non-relativistic expansion of the complex scalar field, where m is the mass of the boson, the energy functional of the system is, where V g satisfies the Poisson equation, Here, M P is the Planck mass, and the wavefunction ψ is normalized to the particle number N , We choose an ansatz of the form [80], where σ d is the dilute minimum energy solution to be found by minimizing the energy of the system. In [51], this ansatz was found to be an excellent approximation of numerical solutions for boson stars in the dilute region. The minimization of the energy, with N fixed, results in the dilute minimum energy solution, where N max is the maximum particle number beyond which no bound state solutions exist. Therefore, for ADM boson stars subject to attractive self-interactions, the possible particle numbers are bounded from above by, The choice of ansatz given by Eq. (6) with the value of σ d that minimizes the energy, Eq. (7), provides a good approximation for the wavefunction, ψ d , of a gravitationally bound dilute ADM boson star. The central density of the ADM boson star is given by, For the parameter space shown in Fig. 2, typical values for the maximum particle number given by Eq. (8) and the central density given by Eq. (9) are on the order of 10 55 N max 10 59 and 0.3 GeV cm −3 ρ(0) 50 GeV cm −3 . We take the radius of the ADM boson star to be the radius inside which 99% percent of the mass is contained, R 99 , found via for a given particle number (or total mass). For the ansatz chosen, R 99 is approximately equal to, If all DM is in the form of such boson stars, it behaves as CCDM. If, however, only a fraction of DM bosons is in the form of boson stars, the self-interactions of the bosons have to obey well established limits from the bullet cluster and the ellipticity of galaxies (see [24,81] and the references therein). For ADM, we assume 2 → 2 scattering between like charges subject to the interaction potential given by Eq. (1). In this case, the matrix element is M = iλ and the resulting cross-section is, Using the cross-section constraint obtained in [82], we get an upper limit on the self-coupling |λ|, In this study, we choose all of the local DM density to be composed of ADM boson stars. In this case, the above constraint does not necessarily hold. However, we find that in the possible parameter space obtained, the ADM self-coupling constants are well below the maximum value given by Eq. (13), and so we choose to keep the constraint when we search for the possible parameter space in Sec. IV.

B. Collision Rate
We are interested in objects that can pass through the Earth with a rate of at least one event per year. As mentioned earlier, larger rates are achieved by objects that are relatively large and not massive. If these objects compose a component of DM, smaller masses correspond to larger number densities. Similarly, larger size increases the probability of passing through the Earth. The scattering cross-section for collisions between either Earth or a detector on Earth and a boson star is (assuming non-relativistic speeds), where R target is the radius of the target and R 99 is given by Eq. (11). For the possible parameter space analyzed in Sec. IV, the ADM boson stars have a much larger size than the Earth. Hence, the radius of the target in Eq. (14) is taken to be the radius of the Earth R E . The mean free path for collisions is, where n is the local number density of ADM boson stars which, assuming all DM is in the form of boson stars, is given by, where ρ DM 0.3 GeV/cm 3 is the Earth's local DM density. The frequency of collisions is then, where v E = 2.3 × 10 2 km s −1 is the relative velocity between the Earth and the ADM boson star. Therefore, the collision frequency will not only depend on the number density of ADM boson stars, but also the coupling constant and mass of the boson. For the parameter space shown in Fig. 2, typical values for the number density of boson stars given by Eq. (16) are on the order of 10 −44 cm −3 n 10 −42 cm −3 and typical values for the collision frequency given by Eq. (17) are shown in Fig. 3.

A. Higgs Portal and its Effect on Measured Parameters
We are interested in ADM boson stars that can potentially be detected by atomic clocks. In this case, a portal that connects the dark sector and the SM one is needed. In particular, we assume that the DM sector communicates with the SM sector via a Higgs portal (see [83][84][85]), i.e., there is a term in the Lagrangian of the form where β is a constant. Notice that this term adds to the mass term of the φ field, so that after the Higgs acquires a vacuum expectation value (VEV), the effective mass of φ is given by, where v ew is the VEV of the electroweak interaction. It should be noted that for the parameter space obtained in Fig. 2, the bare mass of φ squared must be fine tuned to be close to βv 2 ew in order to get effective masses on the order of 10 −8 eV − 10 −4 eV.
We note that one can also include a term of the form φ|H| 2 for real scalar fields, however in the case of a complex scalar field, this term is not invariant under the U(1) transformation φ → e iα φ. Such a portal can open decay channels of the field φ to SM particles as long as φ is heavier than they are. If it is assumed that the complex scalar field φ is bound in a dilute boson star, it is related to the dilute wavefunction and central density of the boson star as given by Eq. (9). Therefore, the presence of the star induces an effective change in the mass of the electron through the Higgs portal of the following form where m bare e is by definition the electron mass in the absence of any medium, y e ≡ m bare e /v ew is the Yukawa coefficient for the electron, m h is the Higgs mass, and v φ is the nonzero expecation value of φ. Note that, depending on the sign of β, the effective mass can be larger or smaller than the bare mass of the electron m bare e . In this study, we take v φ to be the central value of φ inside the boson star given by, For atomic clocks in general, the change in the counting of the clock follows [71] δω where and α, m q , m e and m p are the the fine structure constant, the quark mass, the electron mass, and the proton mass, respectively. Λ QCD is the scale of QCD and the K i are appropriate constants for the corresponding quantities i depending on the particular atom used in the atomic clock. For a typical microwave atomic clock [71], K α 2, K q −0.09 and K e/p = 1. Given the portal of Eq. (18), α remains unchanged. The change in the mass of the quarks makes very little contribution to δω due to the small factor 0.09 and since most of the mass of the proton does not come from the mass of the quarks, this is also a tiny contribution. Therefore, in our setup the overwhelming contribution to δω comes from the change of electron mass. As an ADM boson star passes through the Earth, the mass of the electron is shifted due to the nonzero value of v φ via Eq. (20). In this case, the shift in frequency is given by, It is apparent that larger boson star densities correspond to larger values of v φ , which consequently create larger shifts in the mass of the electron and therefore larger δω shifts in the clock timing. The reader should recall however that usually larger densities are achieved in heavier stars which have smaller collision rates with the Earth. Therefore, the class of boson stars that can be probed are those that have large enough δω so that the change in timing is detectable while at the same time the collision rate remains relatively high. We explore different values for the DM self-coupling constant as well as for the coefficient β. As discussed earlier, we assume that all DM is composed of ADM boson stars, and hence the DM self-coupling constant is not necessarily constrained by Eq. (13). However, in scanning the parameter space, we find that the possible self-coupling constants do, in fact, satisfy this constraint, and so we choose to search the parameter space with this constraint satisfied. The Higgs coupling constant β is also subjected to different types of constraints as demonstrated in the next subsection.

B. Bounds on Higgs Couplings
Constraints can be placed on the Higgs coupling constant, β, in Eq. (18) from fifth-force experiments if a nonzero expectation value of φ exists at the location of the experiment. One way φ can obtain an expectation value is if φ gets its mass from the Higgs and the Higgs coupling constant β is negative [86]. In this case, the expectation value is different from that defined in Eq. (21), and we leave the search for the possible parameter space corresponding to these systems for future studies. Another way fifth-force experiments can constrain β, is if the field φ is assumed to form a condensate around or inside the Earth, Sun, etc. [57,87,88]. In this case, fifth-force experiments on the Earth will always be affected by the φ expectation value given by Eq. (21). In this study, we show the constraints on β that arise given that the φ field has a nonzero expectation value that effects the fifth-force experiments. However, we do not take these constraints into account for our calculations as we assume that the ADM boson stars are not bound to the Earth as a halo and refer to [57] for such discussions.
From [83,89,90], the presence of φ, with a mass m, induces an interaction between two massive bodies with a potential, where m i is the mass of the i-th body and α is a coupling constant given by, Here, m N is the nucleon mass, is the mixing parameter which we discuss below (see Eq. (34)), and g hN N is the coupling of the Higgs to nucleons given by Here, f N T q are the nucleon parameters [89,[91][92][93] and it has been used that, For protons and neutrons, the couplings are g hpp ≈ 0.3776 m p /v ew and g hnn ≈ 0.3755 m n /v ew , respectively. Taking an average of these two couplings, Eq. (26) becomes, The value of α 2 is constrained by the aforementioned fifth-force experiments [83,[94][95][96] and from Eq. (29) one can draw constraints on β v φ as depicted in Fig. 1. An upper bound on the Higgs coupling constant β in Eq. (18) can be found from the observed constraint on the branching fraction of invisible Higgs decays. From [19], the rate for the invisible Higgs decay is given by, Recent measurements from the CMS collaboration [97] give an upper constraint on the branching fraction of invisible Higgs decays of 19% at 95% CL. For Γ(h → SM) = 4.1 MeV and taking m m h , an upper constraint on β is found to be, If the U(1) symmetry of the Lagrangian is unbroken, then φ is protected from decays into two photons. However, if the U(1) charge of φ is not conserved, one can check that the decay process φ → γγ has a lifetime  [83,[94][95][96]. The reference labels in the left panel correspond to those described in [94], while the reference labels in the right panel correspond to those described in [96].
that is several orders of magnitude larger than the age of the universe. From [86], the decay rate of a virtual Higgs to two photons is given by, where F 11/3 includes all loop contributions from charged fermions and the W bosons. Due to the interactions between φ and the Higgs, there is some mixing, , that will suppress the decay rate of φ → γγ Γ φ→γγ = 2 Γ H * →γγ (33) where is proportional to the Higgs coupling constant β in Eq. (18), The lifetime for φ is then, If we assume that all of the DM in the galaxy is in the form of boson stars, the nonzero expectation value of φ can be taken to be given by Eq. (21) inside the boson star and to be equal to zero outside the boson star. In this case, it can be shown that for β 10 −2 and a given v φ that can create a recordable clock shift, this lifetime is much larger than the age of the universe.

IV. RESULTS
In this section we would like to identify the ADM boson star parameter space that can be probed by Earth based microwave atomic clocks. The constraint on β from inverse Higgs decays, Eq. (31), the constraint on the condensate particle number for boson stars with attractive self-interactions, Eq. (8), and the constraint on the boson self-coupling constant, Eq. (13), provide boundaries for an available parameter space to scan. Again, we stress that this last constraint is not necessary since we assume that all DM is in the form of ADM boson stars. However, in searching the available parameter space without satisfying this constraint, we find that all possible solutions do, in fact, keep this constraint satisfied. We scan the available parameter space to find solutions for which the rate of collisions between boson stars and the detector, given by Eq. (17) is f ≥ 1 yr −1 and the size of the boson star is R 99 < 10 10 km. This last constraint arises due to the fact that solutions with R 99 ≥ 10 10 km will take a year or more to completely pass through a detector on Earth. We also take the constraint that the frequency shift from Eq. (24) is δω/ω ≥ (δω/ω) min where we search solutions with (δω/ω) min = 10 −16 , 10 −17 , 10 −18 . Finally, the solutions found satisfy the condition that the ADM boson stars found locally do not significantly overlap i.e., where ρ(0) is the central density of the ADM boson star given by Eq. (9). The rate of collisions depends on λ, m, and N (the number of particles composing the star), while the induced fractional frequency shift δω/ω depends on λ, m, N , and β. We take the constraints on λ, N , and β as described previously. We scan the parameter space by varying the relevant parameters within the ranges 10 −22 eV ≤ m ≤ 10 6 eV, 10 −100 λ max < λ < λ max , 0.01N max < N < N max , and 10 −50 β max < β < β max . From this scan, we identify the parameter space that can provide a rate of collision higher than one event per year with an induced δω/ω ≥ 10 −18 provided that all DM is in the form of these ADM boson stars, and that the radius and central density of the boson stars satisfy the constraints as described above.
In Fig. 2 we show the mass and radius of the boson star as a function of the DM mass where δω/ω ≥ 10 −18 with a rate of events larger than one per year, after having chosen three different values of λ and having fixed β = 0.01. In Fig. 3 we show how the aforementioned parameter space is distributed in terms of δω/ω and rate of events, while in Fig. 4 we show with different colors which part of the parameter space requires atomic clock sensitivity 10 −16 , 10 −17 or 10 −18 in order to detect the passing of such a dilute boson star from the Earth.
Several comments are in order here. First, it is obvious that further improvements to microwave atomic clock sensitivities will lead to an extension of the parameter space probed within this class of models as is apparent from Fig. 4. For all of the parameter space shown, we have assumed that these dilute boson stars make up 100% of the DM relic abundance in our galaxy. More parameter space can be probed by atomic clocks if one relaxes this condition. If boson stars compose a smaller fraction of DM, part of the parameter space can still be probed as long as the rate of events remains sufficient. This can happen for example in cases where the dilute boson star is large yet it makes up a small fraction of DM because the probability of passing through the Earth can still remain high. In addition, the clocks of the GPS network have been collecting data for more than 10 years and therefore, the same technique used in [77] can be used to probe boson stars that could have a rate of events of ∼ 0.1/year. , collision frequency f (top right), and frequency shift δω/ω (bottom right) of the ADM boson star vs. particle mass m for which β = 10 −2 , λ = 10 −45 , and the frequency of collision between the boson star and detector is f ≥ 1 yr −1 . Blue dots corresponds to δω/ω ≥ 10 −16 , red dots to 10 −16 > δω/ω ≥ 10 −17 , and black dots to 10 −17 > δω/ω ≥ 10 −18 .

V. CONCLUSIONS
In this paper, we entertain the possibility that dark matter is entirely composed of light asymmetric dark matter with attractive self-interactions that has collapsed in dilute formations. If the dark sector communicates with the visible sector via a Higgs portal, the passing of such a dilute object through the Earth can induce a small yet detectable change in physical constants like the mass of the electron. Due to the fact that dark matter in boson stars is in a BEC state, the nonzero expectation value of the boson field creates an extra contribution to the mass of the electron. Since the timing frequency of atomic clocks depends on that mass, a clock that finds itself embedded in the boson star as the latter crosses the Earth, measures time at a different rate compared to a clock that remains, at that time, outside the star. We demonstrate that current and future microwave atomic clock sensitivities can allow such clocks to probe a class of dilute boson stars for a parameter space where conventional techniques such as gravitational waves from mergers, gravitational lensing and direct dark matter detection fail. In particular, we assume that the complex scalar field composing the asymmetric dark matter boson stars has a quadratic coupling to the Higgs. We discuss the constraints that the dark matter self-coupling and Higgs coupling constant must satisfy. We then scan the available parameter space subject to these constraints as well as the additional constraints that the frequency of collisions between a boson star and the Earth is f ≥ 1 yr −1 , the induced fractional frequency shift due to the shift in the electron mass is δω/ω ≥ 10 −18 , that the boson stars do not overlap, and that the radius of the boson stars are small enough to pass the Earth within one year. We find a range of asymmetric dark matter boson star parameter space that could be detected in the near future by microwave atomic clocks.