Variation of Constants Searches

A variation-of-constants search asks whether a dimensionless parameter in the laws used to predict atomic or molecular frequencies depends on time, position, gravitational environment, or a new field. The basic observable is not an isolated frequency. It is a comparison:

$$R_{AB}(t) = \frac{\nu_A(t)}{\nu_B(t)}.$$

For small changes,

$$\delta\ln R_{AB} = \sum_X \Delta K_X^{AB} \delta\ln X, \qquad \Delta K_X^{AB} = K_{X,A}-K_{X,B}.$$

The $X$ are dimensionless constants or dimensionless combinations, and the $K$ are calculated sensitivity coefficients. A measured ratio residual is therefore a projection of possible parameter changes through atomic, molecular, or nuclear structure.

A ratio measurement first constrains a specified time-dependent signal. Interpreting it as a change of constants requires calculated sensitivities; interpreting it as a particle coupling adds a field model. The historical experiments discussed below found no significant signal in their tested models. Precision AMO research discusses dated results and open questions.

Helpful background. Fundamental constants explains correlated parameter inference. Optical clocks and frequency comparisons supply the clock response, stability statistics, and link corrections used here. The focus is laboratory comparisons; astrophysical spectra and early-universe constraints require additional source and cosmological models.

Dimensionless Changes and Frequency Ratios

Dimensionless observables first

A numerical change in a dimensional constant depends on the unit realization. For example, asking whether $m_e$ in kilograms changed while the kilogram itself is defined through exact $h$ is not a convention-independent statement. Clock comparisons instead constrain dimensionless quantities such as

$$\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c}, \qquad \mu \equiv \frac{m_p}{m_e},$$

or combinations involving light-quark masses, the QCD scale, and nuclear $g$ factors.

This page uses

$$\mu=\frac{m_p}{m_e}.$$

Some molecular and astronomical literature uses the inverse convention $m_e/m_p$. Reversing the definition reverses every quoted $\mu$-sensitivity coefficient and drift sign. A paper that writes only “mass ratio $\mu$” has not fully specified its result.

The preference for dimensionless observables does not mean dimensional measurements are useless. It means that their physical interpretation must be reduced to a ratio or otherwise tied to an operational unit realization.

A constant in an effective theory

At ordinary laboratory energies, parameters such as $\alpha$ and fermion mass ratios appear fixed. A broader theory may promote them to effective functions of a scalar field $\phi$:

$$\delta\ln X = d_X\,\varphi,$$

where $\varphi$ is a dimensionless normalized field in a declared convention and $d_X$ is a coupling coefficient. Different papers normalize $\varphi$ with the Planck scale, gravitational coupling, or another energy scale. Numerical coupling limits cannot be compared until those normalizations are translated.

The phenomenological search can be performed before adopting a microscopic model:

$$\delta\ln X(t) = x_X(t).$$

Only after limiting $x_X(t)$ should one map it to $d_X$, a dark-matter mass, a local density, or an equivalence-principle parameter.

Averaging can hide a real oscillation. A: one complete cycle inside a rectangular gate has equal positive and negative areas, giving zero average. B: the gate’s signed response is $H_{\rm gate}=\sin(\pi fT_g)/(\pi fT_g)$; zeros occur at nonzero integer $fT_g$. A negative value means a phase reversal, while $|H_{\rm gate}|$ gives amplitude attenuation. Interrogation and servo response can introduce additional reductions.

Comparing Clocks and Spectra

Why a ratio is the observable

Any frequency measurement compares phase accumulation against another oscillator. Even an “absolute” optical frequency measured in hertz is a chain of comparisons to a realization of the SI second, presently based on the caesium-133 hyperfine transition.

For two simultaneously compared references,

$$y_{AB}(t) = \ln \left[ \frac{R_{AB}(t)}{R_{AB,0}} \right] \simeq \frac{R_{AB}(t)-R_{AB,0}}{R_{AB,0}}.$$

The logarithm makes small fractional changes additive and makes inversion simple:

$$y_{BA}(t) = -y_{AB}(t).$$

A sign error in the ratio order reverses a reported drift or coupling. The numerator, denominator, beat-note convention, comb mode numbers, and frequency offsets must therefore be frozen before analysis.

Three comparison geometries

Co-located optical ratio.
Two transitions are compared through one comb and often share an optical laboratory. Gravitational-potential differences and long links are small, while common environmental and comb errors can correlate the data.

Remote optical ratio.
Clock signals are transferred by stabilized fibre, free-space optical link, or satellite link. This enables networks and independent environments, but adds link phase, reference-plane, time-transfer, and relativistic corrections.

Optical-to-microwave absolute frequency.
An optical transition is compared with a caesium primary standard or time scale. This introduces sensitivity to $\mu$ and nuclear magnetic parameters that a purely electronic optical ratio may largely cancel. It also introduces dead time, flywheel oscillators, time-scale correlations, and the caesium systematic evaluation.

Molecular spectroscopy may compare a rovibrational transition with an atomic clock through a comb. Atom–cavity and molecule–cavity comparisons are also useful, but the cavity is a material sensor with its own frequency-dependent mechanical and thermal response, not a featureless reference.

The corrected time series

A raw ratio record can be represented as

$$y_i^{\mathrm{raw}} = s(t_i;\boldsymbol\vartheta) + \sum_a c_a z_a(t_i) + b_{r(i)} + n_i.$$

Here:

• $s(t;\boldsymbol\vartheta)$ is the proposed physical signal;
• $z_a(t)$ are monitored environmental or operational regressors;
• $c_a$ are nuisance couplings;
• $b_{r(i)}$ is a run-, configuration-, or reference-plane offset; and
• $\mathbf n$ has covariance $C$ that need not be diagonal or white.

The corrected record is not obtained by subtracting every correlated monitor. A regressor should be included because a physical or diagnostic model supports it, with uncertainty in its calibration propagated. Adding regressors after inspecting a candidate peak can absorb real signals or tune away noise.

Common references and closure

Suppose three simultaneous log ratios satisfy

$$y_{AB}+y_{BC}+y_{CA}=0.$$

The closure residual

$$\Delta_{\mathrm{cl}} = y_{AB}+y_{BC}+y_{CA}$$

tests consistency of counters, comb transfer, ratio signs, and reference planes. Its uncertainty includes covariance because the three ratios share clocks and transfer oscillators.

Closure cancels every consistent change of the individual clock frequencies, including changes with different sensitivity coefficients: $(\delta\ln\nu_A-\delta\ln\nu_B)+(\delta\ln\nu_B-\delta\ln\nu_C) +(\delta\ln\nu_C-\delta\ln\nu_A)=0$. It can reveal inconsistent links or sampling, but it cannot by itself test physical variation or assign a bias to one clock.

Sampling and averaging are part of the detector

A reported point is usually an average over a gate of duration $T_g$:

$$\bar y(t) = \frac{1}{T_g} \int_{t-T_g/2}^{t+T_g/2} y(t')\,dt'.$$

For a sinusoidal input timestamped at the gate center, rectangular averaging gives the signed response

$$H_{\mathrm{gate}}(f) = \frac{ \sin(\pi fT_g) }{ \pi fT_g }.$$

The amplitude attenuation is $|H_{\mathrm{gate}}|$; its sign records a $\pi$ phase reversal. End-of-gate timestamps instead add the delay factor $e^{-i\pi fT_g}$ for an $e^{i2\pi ft}$ input. Regular sampling also aliases frequencies separated by the sampling rate, independently of gate attenuation. Interrogation, servo, dead time, counter filtering, link processing, and subsequent binning add further transfer effects. A null result above the effective bandwidth does not constrain the unattenuated physical amplitude.

Sensitivity Coefficients

Definition

After dividing frequencies by a common electronic frequency scale $\nu_{\mathrm{at}}$, and holding the other independent parameters fixed, define for transition $i$

$$K_{X,i} = \frac{\partial\ln(\nu_i/\nu_{\mathrm{at}})}{\partial\ln X}.$$

For a ratio,

$$\Delta K_X^{AB} = K_{X,A}-K_{X,B}.$$

Then

$$\delta\ln R_{AB} = \Delta K_\alpha^{AB} \frac{\delta\alpha}{\alpha} + \Delta K_\mu^{AB} \frac{\delta\mu}{\mu} + \Delta K_q^{AB} \frac{\delta X_q}{X_q} + \cdots.$$

The common dimensional frequency scale cancels. What remains is the differential dependence of the two physical systems.

Atomic electronic transitions

An electronic transition can be written schematically as

$$\nu_i = cR_\infty F_i(\alpha),$$

so

$$K_{\alpha,i} = \frac{\alpha}{F_i} \frac{\partial F_i}{\partial\alpha}$$

after the common $cR_\infty$ scale is factored out. Relativistic shifts grow roughly with $(Z\alpha)^2$, but level crossings and cancellations can produce much larger or sign-changing fractional sensitivities.

For two ordinary optical transitions, direct $\mu$ sensitivity is usually small because the leading electronic mass scale cancels. Isotope shifts, nuclear size, recoil, and hyperfine admixture can reintroduce nuclear-mass dependence.

Hyperfine transitions

A ground-state hyperfine frequency has the schematic scaling

$$\nu_{\mathrm{hfs}} \propto cR_\infty \alpha^2 F_{\mathrm{rel}}(\alpha) \frac{m_e}{m_p} g_I F_{\mathrm{nuc}},$$

where $g_I$ is a nuclear magnetic factor and $F_{\mathrm{nuc}}$ represents finite-size and nuclear-structure corrections. With $\mu=m_p/m_e$,

$$\delta\ln\nu_{\mathrm{hfs}} = \left( 2+K_{\mathrm{rel}} \right) \delta\ln\alpha - \delta\ln\mu + \delta\ln g_I + \delta\ln F_{\mathrm{nuc}} + \delta\ln(cR_\infty).$$

An optical-to-caesium comparison therefore does not constrain $\mu$ alone without assumptions or calculations for nuclear magnetic dependence. Calling it a “model-free proton-mass test” overstates the inference.

Molecular transitions

In a simple Born–Oppenheimer scaling picture,

$$\begin{aligned} \nu_{\mathrm{rot}} &\propto \mu^{-1}, \\ \nu_{\mathrm{vib}} &\propto \mu^{-1/2}, \\ \nu_{\mathrm{el}} &\propto \mu^0, \end{aligned}$$

The proportionalities refer to a fixed electronic atomic scale. All nuclear masses are assumed to scale with $\mu=m_p/m_e$, with nuclear mass ratios held fixed. A model changing those ratios needs separate sensitivities. Hence the representative sensitivities are

$$K_{\mu,\mathrm{rot}} \simeq -1, \qquad K_{\mu,\mathrm{vib}} \simeq -\frac12.$$

Inversion, tunnelling, spin–orbit, and near-degenerate transitions can have enhanced coefficients. Enhancement often comes from cancellation between larger energy contributions, so theory uncertainty and field sensitivity must be assessed at the enhanced fractional level.

Sensitivities are calculated quantities

A reliable $K$ value should state:

• the definition and sign of every varied parameter;
• which other parameters were held fixed;
• the atomic, molecular, or nuclear Hamiltonian used;
• whether the derivative was analytic or obtained by finite differences;
• numerical convergence and many-body uncertainty;
• nuclear-size, polarizability, and magnetic-structure assumptions; and
• correlations among coefficients for related transitions.

At present clock precision, treating $K$ as an exact integer can be unjustified, especially for molecular enhancement or nuclear transitions.

Multiparameter identifiability

With $M$ measured ratios and $P$ varying parameters,

$$\mathbf y(t) = K\,\mathbf x(t) + \mathbf n(t),$$

where

$$K_{aX} = \Delta K_X^{(a)}.$$

If $\operatorname{rank}K<P$, the data constrain only combinations of parameters. Even at full rank, nearly parallel sensitivity rows make the inverse problem ill-conditioned.

For covariance $C$, the local information matrix is

$$\mathcal I = K^{\mathsf T}C^{-1}K.$$

Its small eigenvalues identify weakly measured combinations. Adding a transition is valuable when its sensitivity vector points in a new direction after its actual noise and covariance are included, not merely because its individual clock uncertainty is small.

One-coupling-at-a-time limits

Many exclusion plots set all but one coupling to zero. If

$$y = K_\alpha x_\alpha + K_\mu x_\mu,$$

then an alpha-only interpretation uses

$$x_\alpha = \frac{y}{K_\alpha} \quad \text{under the assumption} \quad x_\mu=0.$$

That result is useful, but it is not the marginalized bound in a two-parameter model. Papers and plots should label one-coupling-at-a-time, profiled, marginalized, and model-correlated limits distinctly.

Signal Models

Linear drift

The simplest phenomenological model is

$$x_X(t) = x_{X,0} + \dot x_X(t-t_0).$$

For one ratio,

$$y(t) = b + D(t-t_0) + n(t),$$

with

$$D = \sum_X \Delta K_X\dot x_X.$$

Choosing $t_0$ near the weighted mean observation time reduces covariance between intercept and slope. It does not change the slope.

A linear fit is meaningful only over its stated interval. Instrument upgrades, relocks, transport, reference changes, software changes, and step-like systematic shifts can imitate or obscure a slope. A mature analysis compares:

• one global intercept against documented run-dependent offsets;
• white-noise against colored-noise covariance;
• a slope fitted before and after each major intervention;
• raw and correction-applied records;
• leave-one-run-out and leave-one-season-out results; and
• fixed analysis choices against blinded or preregistered choices.

The notation

$$\frac{\dot\alpha}{\alpha}$$

means the fitted local fractional drift rate under this model. It does not claim that the same linear law held over geological or cosmological time.

Gravitational-potential modulation

A phenomenological local-position-invariance test can write

$$\delta\ln X = k_X \frac{\Delta U}{c^2},$$

Here use the positive solar potential $U_\odot=GM_\odot/r$, as in the annual template below. It is the negative of the Newtonian potential $\Phi_N=-GM_\odot/r$ used in many mechanics and clock-redshift formulas. Changing to $\Phi_N$ reverses the fitted coupling’s sign, not the measured modulation. The $k_X$ are dimensionless. A clock ratio then has

$$\delta\ln R_{AB} = \left( \sum_X \Delta K_X^{AB}k_X \right) \frac{\Delta U}{c^2}.$$

Earth’s orbital eccentricity modulates the solar potential. To first order,

$$\frac{\Delta U_\odot(t)}{c^2} \simeq \frac{ e_{\oplus}GM_\odot }{ a_{\oplus}c^2 } \cos \left[ \Omega_\oplus(t-t_p) \right].$$

The sinusoidal amplitude is about

$$1.65\times10^{-10},$$

while the perihelion-to-aphelion difference is twice that value. A report must say whether it quotes amplitude or peak-to-peak change.

General relativity predicts a universal gravitational redshift. A co-located ratio is sensitive to species-dependent anomalous response, not to the common universal term. Remote clocks additionally require the ordinary relativistic potential and velocity corrections before a nonuniversal residual is interpreted.

An annual fit is vulnerable to annual laboratory systematics: temperature, humidity, magnetic fields, blackbody environments, grounding, air-conditioning operation, link availability, and maintenance. The solar phase is fixed by the ephemeris. Fitting an arbitrary annual phase after looking at the data changes the hypothesis and its degrees of freedom.

Deterministic oscillation

At fixed angular frequency $\omega$,

$$y(t) = a_\omega\cos\omega t + b_\omega\sin\omega t + \mathbf z(t)^{\mathsf T}\boldsymbol\gamma + n(t).$$

The quadrature amplitude and phase are

$$A_\omega = \sqrt{a_\omega^2+b_\omega^2}, \qquad \varphi_\omega = \operatorname{atan2}(-b_\omega,a_\omega),$$

for the convention

$$A_\omega\cos(\omega t+\varphi_\omega).$$

The sign in the phase definition changes if the sine convention changes. Amplitude is nonnegative, so its null distribution is not a centred Gaussian even when the fitted quadratures are Gaussian.

Stochastic narrowband signal

An ultralight field drawn from a virialized halo need not preserve one phase over an arbitrarily long dataset. It is more accurately a narrowband stochastic process with finite coherence time. If the observation time is short compared with that time, a coherent sinusoid can describe the waveform, but its amplitude and phase are random realizations of the halo field. Replacing the unknown realization amplitude by its ensemble mean can produce overly strong coupling limits; the amplitude distribution must enter the field-model likelihood (Centers and colleagues, 2021). If it is much longer, the likelihood should account for phase decorrelation and the expected line shape.

The deterministic and stochastic analyses answer different questions. Using a single coherent sinusoid across many coherence times can lose sensitivity or give incorrect coverage.

Transient signal

A transient model may predict a localized pulse, step, dispersive feature, or a sequence of arrival times across a network:

$$y_a(t) = A_a g \left[ \frac{ t-t_0-\tau_a(\widehat{\mathbf v}) }{ \tau } \right] + n_a(t).$$

Here $a$ labels the sensor, $g$ is a declared waveform, $\tau$ is a duration, and $\tau_a$ is the geometric arrival delay. A credible network candidate must exhibit:

• the predicted relative signs and amplitudes from the sensitivity coefficients;
• arrival times compatible with one propagation direction and speed;
• no corresponding link, time-transfer, power, magnetic, or environmental disturbance;
• calibrated response at the candidate duration; and
• a global significance that includes the event-time, duration, and direction scan.

A simultaneous clock glitch is not automatically a dark-matter event, especially when the clocks share a time scale, transfer oscillator, comb, software pipeline, or environmental disturbance.

Fine-Structure Constant Variation

Differential relativistic sensitivity

A useful pair of alpha-sensitive transitions has a large differential response

$$\left| \Delta K_\alpha \right|.$$

Its statistical value depends on $|\Delta K_\alpha|$ relative to the actual ratio noise and usable observation time. Relativistic level shifts in heavy atoms and ions can increase this response. Transitions near an accidental cancellation or level crossing can enhance fractional sensitivity further, but may bring difficult polarizability, quadrupole, magnetic, or many-body corrections.

The electric-octupole E3 and electric-quadrupole E2 transitions in $^{171}\mathrm{Yb}^+$ are a particularly useful pair. Their alpha sensitivities differ by approximately

$$\Delta K_\alpha^{\mathrm{E3/E2}} \simeq -6.95.$$

Both transitions can be interrogated in the same trapped ion. This removes species transport and greatly reduces some link uncertainties, while interleaved operation still leaves time-offset sampling and transition-specific systematic effects.

Converting ratio drift

For an alpha-only model,

$$\frac{d}{dt} \ln \left( \frac{\nu_{\mathrm{E3}}}{\nu_{\mathrm{E2}}} \right) = \Delta K_\alpha^{\mathrm{E3/E2}} \frac{\dot\alpha}{\alpha}.$$

Thus

$$\frac{\dot\alpha}{\alpha} = \frac{ d\ln(\nu_{\mathrm{E3}}/\nu_{\mathrm{E2}})/dt }{ \Delta K_\alpha^{\mathrm{E3/E2}} }.$$

The negative $\Delta K_\alpha$ reverses the sign between ratio drift and alpha drift.

Using long-term E3/E2 data through 2022, Filzinger and collaborators reported

$$\frac{\dot\alpha}{\alpha} = 1.8(2.5)\times10^{-19}\ {\rm yr^{-1}},$$

consistent with zero. In the solar-potential model they found

$$\frac{c^2}{\alpha} \frac{d\alpha}{dU_\odot} = -2.4(3.0)\times10^{-9},$$

also consistent with zero. These are results for the paper’s stated linear-drift and annual-potential templates, not limits on every possible time dependence.

The parentheses are one-standard-deviation fit uncertainties, not a detection interval. A two-sided upper limit requires a declared confidence construction rather than simply replacing the central value by zero.

Earlier independent comparisons

The $^{27}\mathrm{Al}^+/^{199}\mathrm{Hg}^+$ optical ratio established the modern direct-clock strategy. Repeated comparisons gave

$$\frac{\dot\alpha}{\alpha} = \left( -1.6\pm2.3 \right) \times10^{-17}\ {\rm yr^{-1}}$$

in 2008. Later $^{171}\mathrm{Yb}^+$, strontium, dysprosium, and atom–cavity measurements extended the time span, sensitivity basis, and frequency reach.

Older results remain valuable because independent atoms, laboratories, systematics, and sampling windows can reject an apparatus-specific periodicity that a more precise single dataset cannot diagnose alone.

Alpha-search systematics

Important false channels include:

• transition-specific blackbody, dc Stark, quadrupole, Zeeman, and probe shifts;
• clock-state-dependent servo or line-pulling effects;
• comb transfer and acousto-optic frequency-offset signs;
• time-dependent ion motion, trap fields, or lattice conditions;
• reference-cavity changes in atom–cavity comparisons;
• annual environmental cycles in solar-potential fits;
• maintenance steps mistaken for drift; and
• aliases of diurnal or operational cycles into a sparse spectral window.

Large $\Delta K_\alpha$ does not suppress these effects. It only converts a given ratio residual into a smaller inferred $\delta\alpha/\alpha$.

Proton-to-Electron Mass-Ratio Variation

Optical and microwave complementarity

With

$$\mu=\frac{m_p}{m_e},$$

purely electronic optical ratios have weak leading sensitivity to $\mu$. Comparing an optical transition with a hyperfine standard introduces the factor $m_e/m_p=\mu^{-1}$ and therefore stronger mass-ratio leverage.

For optical reference $o$ and hyperfine reference $h$,

$$\delta\ln \left( \frac{\nu_o}{\nu_h} \right) = \Delta K_\alpha\,\delta\ln\alpha + \Delta K_\mu\,\delta\ln\mu + \Delta K_g\,\delta\ln g_I + \cdots.$$

Multiple ratios are required to separate alpha, mass-ratio, and nuclear magnetic terms without one-at-a-time assumptions.

Lange and collaborators combined the $^{171}\mathrm{Yb}^+$ E3/E2 optical ratio with E3-to-caesium data and an external Rb/Cs constraint on nuclear magnetic dependence. Their analysis obtained

$$\frac{\dot\mu}{\mu} = -8(36)\times10^{-18}\ {\rm yr^{-1}}$$

and a solar-potential coupling

$$\frac{c^2}{\mu} \frac{d\mu}{dU_\odot} = 7(45)\times10^{-8}$$

in 2021. Both results are consistent with zero. They rely on the declared $\mu=m_p/m_e$ convention and on the sensitivity model for the caesium hyperfine reference.

Molecular leverage

Rotational, vibrational, inversion, and tunnelling transitions can probe $\mu$ with a different structure-theory basis. Useful strategies include:

• comparing vibrational overtones over several years;
• comparing rotational and vibrational lines in one molecule;
• comparing molecular and atomic references through a comb;
• using near-degenerate transitions with enhanced differential sensitivity; and
• combining isotopologues to change nuclear-mass dependence.

The tradeoff is that a molecule may have larger Stark, Zeeman, blackbody, collision, motion, and line-shape sensitivities than a clock transition. Enhanced $K_\mu$ is useful only after the long-term systematic reproducibility is demonstrated.

What does a changing proton mass mean?

Most of the proton mass arises from QCD dynamics, not the sum of bare valence-quark masses. A microscopic scalar model may change

$$\Lambda_{\mathrm{QCD}}, \qquad m_u,\ m_d,\ m_s, \qquad m_e, \qquad \alpha$$

with different couplings. The observable

$$\delta\ln\mu = \delta\ln m_p - \delta\ln m_e$$

is model-independent, but translating it into quark, gluon, or electron-sector couplings requires hadronic and nuclear response coefficients. The shorthand “proton-mass coupling” should not conceal that model layer.

Ultralight Scalar Dark Matter Overview

Classical-field regime

A sufficiently occupied ultralight bosonic mode can be treated as a classical field. In natural units $\hbar=c=1$, a simple local model is

$$\phi(t,\mathbf x) \simeq \phi_0 \cos \left( m_\phi t-\mathbf k_\phi\mathbin{\cdot}\mathbf x+\theta \right),$$

with

$$\phi_0 \simeq \frac{\sqrt{2\rho_\phi}}{m_\phi}.$$

Restoring ordinary frequency units,

$$f_\phi = \frac{m_\phi c^2}{h}.$$

The field density $\rho_\phi$ may be the full adopted local dark-matter density or only a fraction of it. Every coupling exclusion must state that choice.

For a virial speed

$$v\sim10^{-3}c,$$

the fractional linewidth is of order

$$\frac{\Delta f}{f_\phi} \sim \frac{v^2}{c^2} \sim 10^{-6}.$$

Equivalently, the coherence quality factor is roughly

$$Q_\phi \sim \frac{f_\phi}{\Delta f} \sim 10^6.$$

Order-unity factors depend on the halo velocity distribution and coherence definition.

Coupling to effective constants

One common linear-coupling convention writes

$$\begin{aligned} \delta\ln\alpha &= d_e\kappa\phi, \\ \delta\ln\mu &= d_\mu\kappa\phi, \end{aligned}$$

Here $\kappa$ is an inverse-energy normalization, often built from Newton’s constant. The dimensionless $d_e$ denotes a photon-sector scalar coupling in this convention; it is unrelated to the dimensionful electron EDM also commonly denoted $d_e$. Then a ratio has oscillation amplitude

$$A_{AB} = \left| \Delta K_\alpha d_e + \Delta K_\mu d_\mu + \cdots \right| \kappa \frac{\sqrt{2\rho_\phi}}{m_\phi}.$$

This equation makes the conditional structure visible:

• the same ratio-amplitude limit gives a coupling limit proportional to $m_\phi$ under fixed density;
• different parameter bases rotate the coupling combination;
• the inferred limit weakens if the field is only a fraction of local dark matter;
• coherence and detector response determine the applicable likelihood; and
• one-coupling-at-a-time curves set the other $d_X$ to zero.

Numerical values of $d_e$ from different papers must not be compared until their $\kappa$, field normalization, density, and confidence convention match.

Linear and quadratic portals

For a linear coupling, the dominant variation is at $f_\phi$. For a quadratic coupling,

$$\delta\ln X = d_X^{(2)} \left( \kappa\phi \right)^2,$$

and

$$\phi^2(t) = \frac{\phi_0^2}{2} \left[ 1+\cos(2\omega_\phi t+2\theta) \right].$$

This single coherent mode contains a constant offset and an oscillation at $2f_\phi$. For a virialized sum of modes, the quadratic signal also has slow difference-frequency fluctuations as well as a band near $2f_\phi$. A search at only $f_\phi$ does not constrain either model in the same way.

A Published Oscillation Search

Clock, atom–cavity, atomic-spectroscopy, and molecular-spectroscopy records have searched broad but finite mass ranges. For example, the 2023 $^{171}\mathrm{Yb}^+$ E3/E2 and E3/strontium analysis improved photon-coupling limits over much of approximately

$$10^{-24} \lesssim \frac{m_\phi}{\mathrm{eV}/c^2} \lesssim 10^{-17}$$

under its scalar-field and halo assumptions. Optical-to-microwave clock records add electron, quark, and gluon sensitivity; radio-frequency atomic and molecular spectroscopy extends to higher oscillation frequencies.

The cited experiments reported no significant dark-matter signal. Their exclusion curves are complementary, not interchangeable: each covers a particular mass band, coupling basis, field model, duty cycle, and sensor response.

Transient field structures

Some models allow macroscopic domain walls or other field structures that produce transient changes rather than a stationary narrowband signal. Atomic-clock networks, including archival satellite-clock networks, have searched for propagating correlated disturbances and found no signal at their stated sensitivities.

The existence, abundance, encounter rate, profile, and coupling of such structures are additional assumptions. A transient exclusion should state whether it is conditional on those objects making all dark matter and how the non-observation is converted into an encounter-rate or coupling bound.

Statistical Analysis

Generalized least squares

For a fixed signal template, collect the observations into

$$\mathbf y = X\boldsymbol\beta + \mathbf n, \qquad \operatorname{cov}(\mathbf n)=C.$$

The design matrix may contain an intercept, drift, sine and cosine quadratures, environmental regressors, and run offsets. The generalized-least-squares estimator is

$$\widehat{\boldsymbol\beta} = \left( X^{\mathsf T}C^{-1}X \right)^{-1} X^{\mathsf T}C^{-1}\mathbf y,$$

For a positive-definite known $C$ and a full-column-rank design matrix, the estimator has covariance

$$V_\beta = \left( X^{\mathsf T}C^{-1}X \right)^{-1}$$

when $C$ is known and the linear-Gaussian model is adequate.

Estimating $C$ from the same residuals adds uncertainty. Red noise, random-walk frequency noise, servo transients, and run-correlated systematics should be tested rather than hidden by rescaling all point errors to make reduced $\chi^2$ equal to one.

Irregular sampling and the spectral window

For sample times $t_i$, the window

$$W(\omega) = \sum_i w_i e^{-i\omega t_i}$$

determines aliases and correlations among scanned frequencies. Daily operation, weekdays, seasonal uptime, and long gaps can move power from one frequency to another.

A Lomb–Scargle periodogram or equivalent likelihood scan handles irregular times more appropriately than a naive discrete Fourier transform, but it does not by itself solve:

• colored or heteroscedastic noise;
• time averaging and servo attenuation;
• run-dependent offsets;
• shared systematics among ratios;
• frequency-dependent theory response; or
• the global trials factor.

The safest procedure is to define a generative model, simulate or inject signals through the actual timestamps and processing, and verify recovery.

Local and global significance

At one prespecified frequency, a test statistic has a local significance. Scanning $N_{\mathrm{eff}}$ effectively independent frequencies raises the chance of a noise maximum. In the simple independent approximation,

$$p_{\mathrm{global}} = 1- \left( 1-p_{\mathrm{local}} \right)^{N_{\mathrm{eff}}} \simeq N_{\mathrm{eff}}p_{\mathrm{local}}$$

when $N_{\mathrm{eff}}p_{\mathrm{local}}\ll1$. Real frequency bins are correlated, so $N_{\mathrm{eff}}$ is best calibrated with noise simulations or an appropriate extreme-value model.

Event-time, duration, waveform, direction, coupling, and subset scans also create trials. Reporting only the most impressive local peak is incomplete.

Upper limits and coverage

A null search should declare:

• frequentist confidence level or Bayesian credible level;
• the likelihood and prior, if any;
• whether nuisance parameters were profiled or marginalized;
• whether the limit is pointwise or simultaneous over a band;
• treatment of the nonnegative amplitude boundary;
• power constraints or sensitivity bands;
• calibration of coverage with synthetic data; and
• conversion from ratio amplitude to parameter or coupling space.

Expected sensitivity bands distinguish an unusually strong limit caused by a downward noise fluctuation from a robust experimental improvement.

Signal injections

End-to-end injections should test:

1. timestamp and time-scale conversion;
2. phase and ratio-sign conventions;
3. gate, servo, and link transfer functions;
4. gaps and data-quality cuts;
5. covariance estimation;
6. nuisance-regressor absorption;
7. frequency and event-parameter recovery;
8. local and global significance calibration; and
9. final coupling conversion.

Software-unit tests are necessary but cannot replace injections through the actual analysis chain.

Network covariance

For sensors $a$ and $b$,

$$C_{ab}(t,t') = \left\langle n_a(t)n_b(t') \right\rangle.$$

Off-diagonal covariance can arise from:

• one shared reference clock or time scale;
• common combs, links, or flywheel oscillators;
• environmental fields over a site;
• common theory coefficients;
• shared correction models; and
• global data-selection or time-transfer processing.

A network gains discovery power through independent sensitivity, geographical timing, and redundant closure. Counting correlated channels as independent produces overconfident limits.

Separating a Signal from Environmental Drift

A seasonal temperature change can resemble an annual gravitational-potential template. Adding temperature as a regressor does not guarantee separation: if the two time series are nearly proportional, their coefficients will have large covariance. Deliberate temperature modulation at other times or an independent laboratory helps determine the nuisance coupling.

The same principle applies to power, magnetic field, trap depth, and link drift. Record physical monitors with suitable timing and bandwidth; test their coupling separately where possible. Inject simulated signals through the actual timestamps and filters to check that correction and fitting preserve the signal being sought.

Common Pitfalls

A ratio determines a sensitivity-weighted combination. Several freely varying constants require independent sensitivity directions or explicit priors. Inverting the ratio also reverses every sensitivity difference.

Closure does not test individual clock frequencies. Consistent node-frequency changes cancel around a loop even when they arise from different physical responses.

An averaging gate has blind frequencies. A null result where the detector response vanishes cannot exclude a large unfiltered signal.

A spectral maximum needs a search-wide comparison. Correlated noise, irregular sampling, frequency scans, and event-template scans affect significance and interval coverage.

A dark-matter limit adds model assumptions. Field normalization, density fraction, coherence, and stochastic amplitude are needed beyond the measured bound on frequency modulation.

Exercises

Exercise 1: Ratio order and alpha drift

A ratio $R=\nu_A/\nu_B$ has

$$\Delta K_\alpha=K_{\alpha,A}-K_{\alpha,B}=-4.$$

Its measured logarithmic drift is

$$\frac{d\ln R}{dt} = 8.0\times10^{-18}\ {\rm yr^{-1}}.$$

Assuming only $\alpha$ varies, find $\dot\alpha/\alpha$. What value would be inferred if an analyst accidentally used $R^{-1}$ but kept $\Delta K_\alpha=-4$?

Solution

The correct inference is

$$\frac{\dot\alpha}{\alpha} = \frac{1}{\Delta K_\alpha} \frac{d\ln R}{dt} = -2.0\times10^{-18}\ {\rm yr^{-1}}.$$

For the inverse ratio,

$$\frac{d\ln R^{-1}}{dt} = -8.0\times10^{-18}\ {\rm yr^{-1}},$$

and its correct sensitivity is $+4$. Keeping the old sensitivity by mistake would give

$$+2.0\times10^{-18}\ {\rm yr^{-1}},$$

the wrong sign. Ratio and sensitivity conventions must be inverted together.

Exercise 2: Two ratios and two varying parameters

Two measured ratio changes obey

$$\begin{pmatrix} y_1\\ y_2 \end{pmatrix} = \begin{pmatrix} 2 & -1\\ 5 & 1 \end{pmatrix} \begin{pmatrix} x_\alpha\\ x_\mu \end{pmatrix}.$$

For $y_1=3\times10^{-17}$ and $y_2=4\times10^{-17}$, solve for $x_\alpha$ and $x_\mu$.

Solution

The equations are

$$2x_\alpha-x_\mu=3\times10^{-17},$$

and

$$5x_\alpha+x_\mu=4\times10^{-17}.$$

Adding gives

$$7x_\alpha=7\times10^{-17},$$

so

$$x_\alpha=1.0\times10^{-17}.$$

Substitution gives

$$x_\mu = 2x_\alpha-3\times10^{-17} = -1.0\times10^{-17}.$$

One ratio alone would constrain only a line in $(x_\alpha,x_\mu)$ space. The nonparallel sensitivity rows make the two-parameter solution identifiable.

Exercise 3: Ytterbium-ion ratio conversion

Use

$$\Delta K_\alpha^{\mathrm{E3/E2}}=-6.95$$

and

$$\frac{\dot\alpha}{\alpha} = 1.8\times10^{-19}\ {\rm yr^{-1}}$$

to find the corresponding best-fit drift of $\ln(\nu_{\mathrm{E3}}/\nu_{\mathrm{E2}})$. Repeat for the $2.5\times10^{-19}\ {\rm yr^{-1}}$ standard uncertainty.

Solution

The ratio drift is

$$\begin{aligned} D_R &= \Delta K_\alpha \frac{\dot\alpha}{\alpha} \\ &= -6.95 \left( 1.8\times10^{-19} \right) \\ &= -1.25\times10^{-18}\ {\rm yr^{-1}}. \end{aligned}$$

The standard uncertainty is

$$u(D_R) = 6.95 \left( 2.5\times10^{-19} \right) = 1.74\times10^{-18}\ {\rm yr^{-1}}.$$

Thus the underlying fitted ratio drift is also consistent with zero.

Exercise 4: Solar-potential modulation

Suppose an alpha-sensitive ratio has $\Delta K_\alpha=-7$ and the coupling model has $k_\alpha=3\times10^{-9}$. Estimate the annual ratio-modulation amplitude using

$$\left| \frac{\Delta U_\odot}{c^2} \right|_{\mathrm{amp}} = 1.65\times10^{-10}.$$

Solution

The ratio amplitude is

$$\begin{aligned} A_R &= \left| \Delta K_\alpha k_\alpha \right| \left| \frac{\Delta U_\odot}{c^2} \right|_{\mathrm{amp}} \\ &= 7 \left( 3\times10^{-9} \right) \left( 1.65\times10^{-10} \right) \\ &= 3.47\times10^{-18}. \end{aligned}$$

The sign and phase carry additional information: $\Delta K_\alpha k_\alpha<0$ specifies whether the ratio is largest near perihelion or aphelion under the adopted potential sign. Reporting only the absolute amplitude discards that check.

Exercise 5: Gate attenuation

A clock ratio is averaged in nonoverlapping $T_g=1000\ {\rm s}$ gates. Find the magnitude of the rectangular-gate response for signals at

$$f_1=10^{-4}\ {\rm Hz} \quad\text{and}\quad f_2=10^{-3}\ {\rm Hz}.$$

Solution

The response is

$$H(f) = \frac{\sin(\pi fT_g)}{\pi fT_g}.$$

For $f_1$,

$$\pi f_1T_g = 0.1\pi,$$

so

$$|H(f_1)| = \frac{\sin(0.1\pi)}{0.1\pi} \simeq 0.984.$$

For $f_2$,

$$\pi f_2T_g=\pi,$$

and therefore

$$H(f_2)=0.$$

The second signal averages to zero for ideal rectangular gates. An exclusion that ignores this transfer zero would be invalid. Moreover, with one sample every 1000 s the ordinary Nyquist frequency is $5\times10^{-4}\ \mathrm{Hz}$: $f_2$ is above it. Gate cancellation and sampling aliasing are distinct limitations.

Exercise 6: Dark-matter mass, frequency, and coherence

For

$$m_\phi=10^{-18}\ {\rm eV}/c^2,$$

use

$$\frac{1\ {\rm eV}}{h} = 2.418\times10^{14}\ {\rm Hz}$$

to estimate $f_\phi$. If $Q_\phi=10^6$, estimate the coherence time

$$\tau_c \sim \frac{Q_\phi}{f_\phi}.$$

Solution

The oscillation frequency is

$$f_\phi = 10^{-18} \left( 2.418\times10^{14} \right) = 2.418\times10^{-4}\ {\rm Hz}.$$

The period is about

$$T_\phi = \frac{1}{f_\phi} \simeq 4.14\times10^3\ {\rm s},$$

or $69$ minutes. The coherence time estimate is

$$\tau_c \sim \frac{10^6}{2.418\times10^{-4}} \simeq 4.14\times10^9\ {\rm s},$$

about $131$ years. Order-unity factors depend on whether coherence is defined with angular or cyclic frequency and on the halo line shape. For a campaign much shorter than this, a coherent-sinusoid approximation is reasonable for the waveform, but the unknown coherent realization amplitude must still be included when converting a null result into a stochastic-halo coupling limit.

Exercise 7: Look-elsewhere correction

A frequency scan has a smallest local tail probability

$$p_{\mathrm{local}}=2\times10^{-5}$$

among $N_{\mathrm{eff}}=1000$ independent frequencies. Estimate the global tail probability.

Solution

Use

$$p_{\mathrm{global}} = 1- \left( 1-p_{\mathrm{local}} \right)^{N_{\mathrm{eff}}}.$$

Numerically,

$$p_{\mathrm{global}} \simeq 1-e^{-0.02} \simeq 0.0198.$$

The small-$p$ approximation gives

$$N_{\mathrm{eff}}p_{\mathrm{local}} = 0.020,$$

which agrees well. A locally striking peak is not globally compelling after the scan.

Exercise 8: Quadratic coupling

Let

$$\phi(t) = \phi_0\cos\omega_\phi t$$

and

$$\delta\ln X = d_X^{(2)} \left( \kappa\phi \right)^2.$$

Find the constant and oscillating parts of $\delta\ln X$. At what frequency should a search look?

Solution

Using

$$\cos^2\omega_\phi t = \frac12 \left( 1+\cos2\omega_\phi t \right),$$

we obtain

$$\delta\ln X = \frac{ d_X^{(2)}\kappa^2\phi_0^2 }{2} + \frac{ d_X^{(2)}\kappa^2\phi_0^2 }{2} \cos2\omega_\phi t.$$

The first term is a constant offset and the second oscillates at

$$2f_\phi.$$

A clock-ratio search is generally insensitive to an unknown static offset because it is absorbed into the fitted baseline. Its direct signature is therefore the $2f_\phi$ modulation.

Related Reading

• Precision molecular measurements explains molecular mass sensitivity and enhancement tradeoffs.
• Frequency comparisons develops link delays, time scales, and correlated clock data.
• Fisher information explains multiparameter identifiability.
• Precision AMO research discusses dated experimental results and unresolved searches.

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