Hunting for topological dark matter with atomic clocks

A. DereviankoDepartment of Physics, University of Nevada, Reno, NV 89557, USA

M. PospelovDepartment of Physics and Astronomy, University of Victoria, Victoria, BC V8P 1A1, Canada and

Perimeter Institute for Theoretical Physics, Waterloo, ON N2J 2W9, Canada(Dated: September 11, 2021)

The cosmological applications of atomic clocks so far have been limited to searches of the uniform-in-time drift of fundamental constants. In this paper, we point out that a transient in time changeof fundamental constants can be induced by dark matter objects that have large spatial extent,and are built from light non-Standard Model fields. The stability of this type of dark matter canbe dictated by the topological reasons. We point out that correlated networks of atomic clocks,some of them already in existence, can be used as a powerful tool to search for the topologicaldefect dark matter, thus providing another important fundamental physics application to the ever-improving accuracy of atomic clocks. During the encounter with a topological defect, as it sweepsthrough the network, initially synchronized clocks will become desynchronized. Time discrepanciesbetween spatially-separated clocks are expected to exhibit a distinct signature, encoding defect’sspace structure and its interaction strength with the Standard Model fields.

PACS numbers: 06.30.Ft,95.35.+d, 06.20.Jr

Despite solid evidence for the existence of dark matter(∼ 25% of the global energy budget in the Universe, andρDM ' 0.3 GeV/cm3 in Solar system neighborhood [1]),its relation to particles and fields of the Standard Model(SM) remains a mystery. A large and ambitious researchprogram in particle physics assumes that dark matter(DM) is “microscopic”, i.e. composed of heavy-particle-like matter, and searches for the results of its scatteringoff individual nuclei [2]. This assumption may not holdtrue, and significant interest exists to alternatives, amongwhich the DM composed from very light fields is of spe-cial interest. Depending on the initial field configurationat early cosmological times, such light fields could leadto dark matter via coherent oscillations around the min-imum of their potential, and/or form non-trivial stablefield configurations in physical 3D space if their poten-tial allows for such possibility. This latter option, that wewill generically refer to as the topological defects (TD), isthe main interest of our paper. The light masses of fieldsforming the TDs could lead to a large, indeed macro-scopic, size for a defect. Their encounters with the Earth,combined together with the DM-SM coupling, can lead tonovel signatures of dark matter expressed generically interms of the “transient effects”. These effects, coherenton the scale of individual detectors, are temporary shiftsin frequencies and phases of measuring devices, ratherthan large energy depositions as is the case for micro-scopic DM. In this paper we suggest the possibility ofa new search technique for the topological defect darkmatter (TDM), based on a network of atomic clocks.

Atomic clocks are arguably the most accurate scien-tific instruments ever build. Modern clocks approach the10−18 fractional inaccuracy [3, 4], which translates intoastonishing timepieces guaranteed to keep time within

a second over the age of the Universe. Attaining thisaccuracy requires that the quantum oscillator be wellprotected from environmental noise and perturbationswell controlled and characterized. This opens intrigu-ing prospects of using clocks to study subtle effects, andit is natural to ask if such accuracy can be harnessed fordark matter searches.

To put our discussion on concrete grounds, we intro-duce a collection of light fields beyond the SM, that canform TDs of different dimensionality: monopoles (0d),strings (1d), and domain walls (2d). Exact nature ofsuch defects depends on the composition of the dark sec-tor, and on self-interaction potential [5]. We provide rel-evant details in the supplementary material. For thispaper we take a simplified approach, and call φ a genericlight field from the dark sector, would it be scalar orvector, that forms a network of TD at some early stageof cosmological history. The transverse size of the de-fect is determined by the field Compton wavelength d,that is in inverse relation to the typical mass scale ofthe light fields, d ∼ ~/(mφc). The fields we are inter-ested in are ultralight: for an Earth-sized defect, themass scale is 10−14 eV. In our simplified approach wecapture only gross features of TDs [5], and call A theamplitude of the field change inside and outside a TD,A = φinside − φoutside, also choosing the outside value ofthe field to be zero.

The energy density of TDM averaged over large num-ber of defects is controlled by the energy density insidethe defect, ρinside ∼ A2/d2, and the average distance be-tween the defects, L, through natural scaling relation:

ρTDM ∼ ρinsided3−nLn−3(~c)−1 ∼ A2d1−nLn−3(~c)−1,

(1)

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1244

v2 [

phys

ics.

atom

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14

Aug

201

4

2

where n = 0, 1, 2 for a monopole, string or a domainwall, and we measure A in units of energy. This relationemerges by integrating across the TD first, which givesa monopole energy, and string(wall) linear(surface) en-ergy density, with subsequent averaging over the entirevolume.

Right combination of parameters can give a significantcontribution to or even saturate ρDM. The average timebetween “close encounters” with TD, r ≤ d, is set by thegalactic velocity of such objects vg,

T ' 1

vg× L3−n

d2−n =1

vg× A2

ρTDMd× 1

~c. (2)

Velocity of galactic objects around the Solar system isan input parameter that is relatively well known, and forthe purpose of estimates one can take vg ' 10−3 × c ≈300 km/s. If the parameter T is on the order of fewyears or less, then it is reasonable to think of a detectionscheme for TD crossing events. A typical duration of onecrossing event is τ ' d/vg.

The most crucial question is how the fields forming thedefect interact with the SM. All possible types of inter-action between TD and SM fields can be classified usingthe so-called “portals”, the collection of gauge-invariantoperators of the SM coupled with the operators from thedark sector [6]. Throughout the rest of this paper, weare going to be interested in a more general form of theSM-TD interaction in the form of the quadratic scalarportal,

− Lint = φ2

(meψ̄eψe

Λ2e

+mpψ̄pψp

Λ2p

− 1

4Λ2γ

F 2µν + ...

)(3)

→ meffe,p = me,p

(1 +

φ2

Λ2e,p

); αeff =

α

1− φ2/Λ2γ

Since inside the TD, by assumption, φ2 → A2 and outsideφ2 → 0 this portal renormalizes masses and couplingsonly when the TD core overlaps with the quantum de-vice. Here me,p and ψe,p are electron and proton massesand fields, and Fµν are electromagnetic tensor compo-nents. The appearance of high-energy scales ΛX in thedenominators of (3) signifies the effective nature of theseoperators, implying that at these scales the scalar portalswill be replaced by some unspecified fundamental theory(the same way as electroweak theory of the SM replaceseffective four-fermion weak interaction at the electroweakscale). The SM field dependence in (3) replicates corre-sponding pieces from the SM sector Lagrangian density,and this leads to the identification (the second line ofEq. (3)) of how masses and the fine-structure constantα are modulated by the TD. Thus, for every couplingconstant and SM particle mass scale X one has to firstorder in φ2

δX

X=

φ2

Λ2X

. (4)

Quadratic (as opposed to linear) dependence on φ leadsto the r−3 behaviour of correction to the gravitationalpotential, which allows escaping very strong constraintsimposed by the nil results of searches for the fifth forceand the violation of the equivalence principle [7]. Bothdirect laboratory and astrophysical constraints on ΛX donot exceed ∼ 10 TeV. Additional background informa-tion on TDM, the types of interaction with the SM, andplausible scenarios for its abundances are provided in thesupplementary material. In particular, we present an ex-plicit and ultraviolet-complete example of the so-calledAbrikosov-Neilsen-Olessen string defect [8, 9], composedof the dark gauge boson and dark Higgs field, that in-creases the value of α inside its core due to the mixing ofthe dark gauge boson and the photon [10].

Main phenomenological consequence of the interaction(3) is a temporary shift of all masses and frequencies in-side the TD. In this sense, the signature we are proposingto search for is a transient variation of fundamental con-stants. In the limit of large τ , when the size of a TD ison astronomical scales, the effect of (3) becomes identi-cal to variations of couplings and masses over time withα̇ ' const, in which case all the existing terrestrial con-straints immediately apply [11]. We also comment thatduring the TD crossing there is a new force acting onmassive bodies, giving a transient signature that can beexplored with sensitive graviometers. Also, other ways ofcoupling TD to SM are known. For example, one coulduse the so-called axionic portals, ∂µφ/fa × Jµ, where Jµis the axial-vector current. Such interactions will lead toa transient “loss” of rotational/Lorentz invariance, andcan be searched for with sensitive atomic magnetome-ters [12, 13]. By design, atomic clocks are insensitive tomagnetic fields, and therefore may also have reduced sen-sitivity to the coupling to spin, and for that reason weconcentrate on (3).

Clocks tell time by counting number of oscillationsand multiplying them by the predefined period of os-cillations 1/(2πω0), where ω0 is the fixed unperturbedclock frequency. Experimentally relevant quantity isthe total phase accumulated by the quantum oscilla-tor, φ0(t) =

∫ t0ω0dt

′; then apparently the device timereading is φ0(t)/ω0. TD would shift the oscillator fre-quency and thereby affect the phase or the time reading,φ(t) =

∫ t0(ω0 + δω(t′))dt′, where δω(t′) is the quantum

oscillator frequency variation caused by TD. We param-eterize δω(t) = gf(t), where g ∝ A2/Λ2 is the couplingstrength and f(t) ∝ |φ(r−vgt)|2 is time-dependent enve-lope (r is the clock position), so that

∫∞−∞ δω(t′)dt′ = gτ .

Suppose we compare phases of two identical clocksseparated by a distance l (see Fig.1) that encountera domain-wall-type TD. Because the TD propagatesthrough the network with a speed vg, the second clockwould be affected by TD at a later time, with a timedelay l/vg. Formally, the phase difference (or apparent

3

time

diff

eren

ce in

clo

ck re

adin

gs1

2

3

4567

8

9

1011 121

2

3

4567

8

9

1011 12

vg

l /vg

FIG. 1. By monitoring time discrepancy between twospatially-separated clocks one could search for passage oftopological defects, such as domain wall pictured here.

time discrepancy ∆t) between the clocks reads

∆ϕ(t) = g

∫ t

−∞(f(t′ − l/vg)− f(t′))dt′ ≡ ω0∆t(t) .

By monitoring correlated time difference ∆t(t) betweenthe two clocks, one could search for TDM. Before theTD arrival at the first clock, the phase difference is zero,as the clocks are synchronized. As the TD passes thefirst clock, it runs faster (or slower, depending on thesign of g), with the clock phase difference reaching themaximum value of |∆ϕ|max = |g|d/vg. ∆ϕ(t) stays atthat level while the TD travels between the two clocks.Finally, as the TD sweeps through the second clock, thephase difference vanishes. In this illustration we assumedthat d� l � L. In the limit of d . l frequency (insteadof time) comparison can be more accurate.

We may further relate the TD-induced frequency shiftto the transient variation of fundamental constants. Theinstantaneous clock frequency shift may be parameter-ized as

δω(t)

ω0=∑X

KXδX(t)

X, (5)

where X runs over fundamental constants. The di-mensionless sensitivity coefficients KX are known fromatomic and nuclear structure calculations [14]. The en-ergy density stored in the TD and various couplings en-ter implicitly through time varying deviation, δX(t) ∝|φ(r− vgt)|2, of the fundamental constant from its nom-inal value. Then the two clocks will be desynchronizedby

|∆t|max =∑X

KX

∫ ∞−∞

δX(t)

Xdt ∼

∑X

KXA2

Λ2X

τ

∼∑X

KX~cρTDMT

Λ2X

d2 . (6)

Here we used Eq. (2) and the fact that contributions tothe Lagrangian (3) factorize into the SM and TDM parts.Notice that this result does not depend on a specific classof TDs.

In practice, one needs to dissect the TD-induced desyn-chronization (6) from various noise sources present inquantum devices and the link connecting the two clocks.We neglect link noise. We assume that the TD thicknessd is much smaller than the distance between the clocks,as in Fig. 1. One would need to resolve the “hump”in the presence of background noise. Suppose we com-pare the clock readings every T seconds; then the totalnumber of measurements of non-zero phase difference isNm = l/(vgT ). For a terrestrial network with an armlength of l ∼ 10, 000 km, the TD sweep takes 30 seconds,so one could make 30 measurements sampled every sec-ond.

Because the clocks are identical and statistically inde-pendent, the variance 〈∆ϕ(t)2〉 − 〈∆ϕ(t)〉2 = 2Rϕ(T ),where Rϕ(T ) is the phase auto-covariance function [15].It can be estimated from the commonly reported Allanvariance σy(T ), which characterizes fractional instabil-ity of the clock frequency [16]: Rϕ(T ) ≈ (ω0T )2 σ2

y(T ).Thereby the uncertainty due to a single clock comparisonis√

2(ω0T )σy(T ). As we carry out Nm = l/(vgT ) mea-surements, the statistical uncertainty is reduced furtherby√Nm.

The above argument leads to the signal-to-noise ratio

S/N =c~ρTDMT d2

Tσy(T )√

2Tvg/l

∑X

KXΛ−2X . (7)

This ratio scales up with the TD size d, the sensitivitycoefficients KX , and the distance between the clocks.

The TD detection confidence would improve with bothincreasing the number of network nodes and populat-ing nodes with several clocks of different types. Clearlywhen the TD sweep is detected, all the clock pairs shouldexhibit time correlated desynchronization signature as-sociated with the sweep. Different clocks have distinctsensitivity to the variation of fundamental constants andthis could help in disentangling various couplings in (3,5).Moreover, large number of clocks in a network in an ide-alized situation will help to determine the direction ofthe TD arrival, its velocity and spatial extent.

The presented analysis can be generalized to the caseof point-like TD (monopoles), which under gravitationalforce will behave identically to the regular cold dark mat-ter. We illustrate such a case in Fig. 2. Here we as-sume that TD is an Earth-scale Gaussian-profile cloudsweeping through a clock network. Individual clocks areperturbed at different times with different amplitudes,depending on the distance to the monopole center. This

4

12

3

4567

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9

1011 12

12

3

4567

8

9

1011 12

12

3

4567

8

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1011 12

1

2

3

1

2

3

time

cloc

k ph

ase

vg

FIG. 2. Simulated response of an Earth-scale constellationof atomic clocks to a 0D Gaussian-profiled topological defect(monopole) of effective radius 0.75R⊕. The monopole centeris displaced from the collision axis by 0.2R⊕. The Earth cen-ter and the clocks lie in the collision plane. Polar angles ofthree clocks are π/2, π,−π/4 in the reference frame centeredat the Earth center.

leads to a TD-induced phase accumulation,

ϕi(t) = g

∫ t

−∞exp

{−(R(t)− ri)

2/d2}dt′ (8)

= gi

∫ t

−∞exp

{−(Z0 + vgt

′ − zi)2/d2}dt′ ,

where R(t) = {X0, Y0, Z0 + vgt} and ri = {xi, yi, zi}are the TD center and ith clock positions and d is theTD effective radius. Here we assumed that the TDpropagates along the z-axis. The coupling is rescaleddepending on the clock position gi ≡ g exp

{−ρ2

i /d2}

,

ρ = ((X0−xi)2 + (Y0−yi)2)1/2 being the impact param-eter. This translates into a differential phase accumula-tion between the clocks, similar to our “wall” example ofFig. 1, but with the step-on and step-off heights depend-ing on the difference of clock impact parameters.

A detailed discussion of atomic clock errors and sensi-tivity coefficients is presented in the supplementary ma-terial. Generally, one distinguishes between two broadclasses of atomic clocks: microwave and optical clocks.Microwave clocks operate on hyperfine transitions, thefrequency of such transitions being determined by thecoupling of atomic electrons to nuclear magnetic mo-ments and thereby these depend on the ratios of quarkand electron masses and α. Optical clocks utilize elec-tronic transitions and their clock frequency depends pri-marily on α.

Several networks of atomic clocks are already opera-tional. Perhaps the most well known are Rb and Cs mi-crowave atomic clocks on-board satellites of the Global

Positioning System (GPS) and other satellite navigationsystems. Currently there are 32 satellites in the GPSconstellation orbiting the Earth with an orbital radiusof 26,600 km with a half of a sidereal day period. Onecan envision using the GPS as a 50,000 km-arpeture darkmatter detector. As TDs sweep through the GPS con-stellation, satellite clock readings are affected. Since ac-curate ephemeris data of individual satellites are known,one could easily cross-correlate clock readings in the net-work. For two diametrically-opposed satellites the max-imum time delay between clock perturbations would be∼ 200 s, assuming the TD sweep with typical velocity of300 km/s. Different types of topological defects (e.g., do-main walls versus monopoles) would yield distinct cross-correlation signatures. While the GPS is affected by amultitude of systematic effects, e.g., solar flares, temper-ature and clock frequency modulations as the satellitescome in out of the Earth shadow, none of conventionaleffects would propagate with 300 km/s through the net-work. Additional constraints can come from analyzingextensive terrestrial network of atomic clocks on GPStracking stations.

The performance of GPS on-board clocks [17] is cer-tainly lagging behind state-of-the art laboratory clocks[3, 4]. Focusing on laboratory clocks, one could carryout a dark matter search employing the vast network ofatomic clocks at national standards laboratories used forevaluating the TAI timescale [18]. Moreover, several ele-ments of high-quality optical links for clock comparisonshave been already demonstrated in Europe, with 920 kmlink connecting two laboratories in Germany [19]. In ad-dition, Cs fountain clock and H-maser are planned to beinstalled on the international space station in the nearfuture, providing high-quality time and frequency link toseveral national laboratories around the world [20]. Re-cently proposed quantum clock network [21] would en-hance sensitivity further.

As an illustration of sensitivity to energy scales ΛX ofTDM-SM coupling (3), we consider a terrestrial network(l ∼ 10, 000 km) of Sr optical lattice clocks which aresensitive to the variation of α with Kα = 6 × 10−2. Forthese clocks one may anticipate reaching σy(1 s) ∼ 10−18

at T = 1 s measurement intervals. Requiring S/N ∼ 1in Eq.(7), substituting fiducial values for ρTDM and vg,and choosing T ∼ 1 yr, we draw sensitivity curve tothe energy scale Λα as a function of the defect size inFig. 3. Here we also show sensitivity of GPS constellation(l ∼ 50, 000 km, T = 30 s, σy(30 s) ∼ 10−11) assumingthat the TDM-SM coupling is dominated by the transientvariation of α (Kα = 2). Limits derived from both Srand GPS networks would greatly exceed the Λ < 10 TeVregion excluded by direct laboratory and astrophysicalconstraints, such as from fifth-force and the violation ofthe equivalence principle searches [7].

To summarize, we have argued that the unknown iden-tity of dark matter may be regarded as an opportunity

5

1 10 100 1000 104 105

100

105

108

1011

Excluded by terrrestial experiments and astrophysical bounds

defect size d, km

Ene

rgy

scal

e, T

eV

Trans-continental network of Sr optical lattice clocks

GPS constellation

m = 10 10 eV m = 10 14 eV

FIG. 3. Terrestrial and space networks of atomic clocks canimpose powerful constraints on characteristic energy scalesof dark-matter interaction with baryonic matter (3). Herewe show bounds on Λα that may be derived from a terres-trial network of optical lattice clocks and GPS clocks. Thehorizontal axis is the topological defect size in km and alsoincludes two characteristic TD field rest mass scale values.

for atomic physics for a new fundamental application.In particular, we have shown that dark matter in formof stable configurations of light fields (topological de-fects) may lead to occasional transient changes of parti-cle masses and coupling constants, thus giving a distinctsignature that can be searched for with the network ofsensitive atomic clocks.

Acknowledgment – We would like to thank N. Fortson,P. Graham, J. Hall, M. Murphy, J. Sherman, J. Wein-

stein, and I. Yavin for discussions. This work was sup-ported by the National Science Foundation.

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