Variation of Fundamental Constants

Scrollable formula. Use Left and Right arrow keys to reveal hidden content.

Scrollable table. Use Left and Right arrow keys to reveal additional columns.

Scrollable figure. Use Left and Right arrow keys to inspect hidden content.

Only variation of dimensionless combinations of constants is operationally meaningful. Clock tests therefore compare ratios whose transition sensitivities differ, then invert those responses for changes in quantities such as the fine-structure constant $\alpha$ , the proton-to-electron mass ratio $\mu$ , and a dimensionless light-quark parameter. Results available through 2026-08-19 are consistent with no drift or potential-linked variation in the tested directions, while scalar couplings outside the measured sensitivity subspace remain unresolved.

Required background. Equivalence Principles and Gravitational Measurement supplies the connection between position invariance and clock outcomes. Probability and Statistical Inference supplies likelihood, covariance, and identifiability concepts.

Helpful background. Weak-Field Gravity supplies the potential template used for gravitationally coupled variation.

Transition sensitivities form an inverse problem

Write $x_p=\ln X_p$ for dimensionless constants and choose a common reference frequency $\nu_*$ when defining individual transition sensitivities. For clock transition $A$ ,

\delta\ln(\nu_A/\nu_*)=\sum_p K_{Ap}\,\delta x_p, \qquad K_{Ap}=\frac{\partial\ln(\nu_A/\nu_*)}{\partial\ln X_p}.

A ratio $i=A/B$ measures the row vector $k_{ip}=K_{Ap}-K_{Bp}$ , so stacked ratio residuals obey $\boldsymbol y=K\,\delta\boldsymbol x$ . Optical electronic transitions are primarily sensitive to $\alpha$ through relativistic structure; hyperfine transitions also depend on mass ratios and nuclear magnetic structure. The $^{171}\mathrm{Yb}^{+}$ E3/E2 and E3/Cs mapping is given explicitly in Lange et al. 2021, Eqs. (1)–(3), Figs. 2–3, and Table II.

If a dimensionless scalar template $\phi(t)$ changes the constants through $\delta x_p=d_p\phi(t)$ , the observable is $y_i(t)=\sum_p k_{ip}d_p\phi(t)$ . The template may represent linear time drift, the solar-potential variation $\Delta U/c^2$ , an oscillation, or a transient. Its normalization must be declared: a dark-matter coupling inferred from an oscillation additionally depends on the assumed field density and coherence. No data can separate $d_p$ from an arbitrarily rescaled $\phi$ .

Identifiability is controlled by the rank and singular values of $K$ . One ratio constrains one combination, not every $d_p$ separately. Even several ratios can be nearly degenerate if they use similar transitions. A quoted “limit on $\alpha$ variation” may therefore assume that all other constants are fixed; that assumption is a model choice, not an experimental fact. Papers also use both $m_p/m_e$ and its inverse for the symbol $\mu$ , so signs must be normalized before datasets are stacked.

Pinned inputs, calibration, and selection

A defensible analysis pins each primary ratio release and supplement by DOI, calibration version, retrieval date, and file checksum. Examples include the 14-year Rb/Cs fountain comparison of Guéna et al. 2012, Eqs. (1)–(4) and Figs. 1–2, the direct Yb $^{+}$ optical ratio of Godun et al. 2014, Eqs. (1)–(4) and Table I, and the multiyear Yb $^{+}$ records above. A 2026 Al $^{+}$ /Yb/Sr campaign reports highly precise ratios together with cross-campaign discrepancies, a useful warning that a static ratio disagreement is not by itself temporal variation Aeppli et al. 2026, official abstract.

The analysis inputs are ratio or phase points, averaging intervals, uptime and dead-time records, frequency-comb and transfer data, systematic-shift budgets, shared reference standards, and calculated $K_{Ap}$ with uncertainty. Calibration parameters shared by epochs or ratios—blackbody coefficients, atomic polarizabilities, gravitational corrections, flywheel behavior, and link response—belong in one covariance model rather than in independent error bars.

Selections include transition state, epoch, averaging time, valid servo state, environmental bounds, outlier rule, and the set of searched frequencies. Their order and thresholds should be fixed before unblinding. Uptime is the relevant completeness measure for a continuous signal: seasonal gaps change sensitivity to an annual template, while irregular sampling creates oscillation aliases. The inference applies only to the retained transitions, timescales, and coupling basis.

Likelihood, priors, and degeneracies

For template design matrix $M$ and nuisance matrix $A$ , use

\boldsymbol y=M\boldsymbol d+A\boldsymbol\eta+\boldsymbol\epsilon, \qquad -2\ln L=\boldsymbol r^{\mathsf T}C^{-1}\boldsymbol r+\ln\det C+\text{constant},

where $\boldsymbol r=\boldsymbol y-M\boldsymbol d-A\boldsymbol\eta$ . The covariance $C$ includes clock noise, common references, systematic calibration, and sensitivity-coefficient uncertainty; when the last is appreciable, it can instead be modeled with nuisance parameters in $K$ . Priors on $\boldsymbol d$ , drifts, oscillation phase, and correlated environmental terms must be reported. A normalized null comparison uses $\boldsymbol d=0$ with the same nuisance treatment. Posterior-predictive residual spectra, coverage simulations, chain convergence or optimizer diagnostics, and stability under alternative colored-noise models are essential.

Results and limitations through 2026-08-19

Evidence or position	Result relevant to variation	Limitation or uncertainty	Discriminator
Long-baseline optical and microwave ratios	Linear-drift fits yield negative or null results in measured combinations of $\alpha$ , $\mu$ , and nuclear parameters.	Shared standards and sensitivity degeneracies prevent a universal statement about all constants.	Add ratios with linearly independent sensitivity rows.
Annual solar-potential fits	No established potential-correlated change is seen in the cited clock records.	Annual environment, ephemeris convention, and sparse seasonal sampling can covary with the template.	Fix phase from the ephemeris and compare co-located dissimilar transitions.
Oscillatory scalar searches	Clock data constrain selected frequency bands under a declared scalar-field model.	Aliases, coherence assumptions, look-elsewhere corrections, and field normalization are model-dependent.	Inject signals across frequency and phase into the actual sampling.
Static high-precision ratio discrepancies	Repeated interspecies comparisons expose calibration tension that merits resolution.	A difference between campaigns supplies no time law or scalar-sensitivity scaling.	Repeat independently and demand a coherent trend across at least three ratios.

The conservative interpretation is that no reproducible variation has been established in these data. A viable alternative is a scalar whose couplings project weakly onto existing ratios or whose timescale lies outside their coverage. That possibility is genuine, but it does not give isolated discrepancies equal evidential weight with the null: a variation claim must reproduce the same coupling vector across transitions and laboratories.

Adversarial recovery and reproducibility

A useful independent check uses at least three measured ratios whose sensitivity rows span the two coupling directions under test. Three rows in a two-column sensitivity matrix cannot all be linearly independent; the extra row instead supplies a consistency condition. Inject simultaneous $\alpha$ and $\mu$ variations into realistically sampled, correlated records; run the documented selection and likelihood; and verify calibrated interval coverage. Then change ratio coordinates with a nonsingular matrix $Q$ , transforming $\boldsymbol y'=Q\boldsymbol y$ , $K'=QK$ , and $C'=QCQ^{\mathsf T}$ . The quadratic residual and inferred physical couplings must be unchanged; the normalized data density changes only by its parameter-independent Jacobian. A combination in the left null space of $K$ should contain no injected coupling signal.

Reproduction requires the pinned measurements, masks, calibration and sensitivity inputs, covariance construction, priors, template code, injection seeds, and all searched frequencies. If only published summary points are available, the exact original likelihood cannot be claimed as reproduced; one can test only the stated reduced model. Statistical, calibration, sensitivity-theory, and template-model uncertainties should be reported separately. This conclusion should be reviewed after a superseding ratio or calibration release, a corrected atomic or nuclear sensitivity, expanded frequency coverage, or a signal that passes multi-ratio recovery, basis rotation, null-combination, and independent-laboratory tests.

Exercise

Three ratio drifts in units of $10^{-18}\,\mathrm{yr}^{-1}$ have sensitivity rows $(1,0)$ , $(0,1)$ , and $(1,1)$ for $(\dot\alpha/\alpha,\dot\mu/\mu)$ and measured central values $2$ , $-1$ , and $1$ . Find the coupling estimates and explain the role of the third ratio.

Solution

The first two independent rows give $\dot\alpha/\alpha=2\times10^{-18}\,\mathrm{yr}^{-1}$ and $\dot\mu/\mu=-1\times10^{-18}\,\mathrm{yr}^{-1}$ . Their prediction for the third ratio is the sum, $1\times10^{-18}\,\mathrm{yr}^{-1}$ , which matches its central value. Therefore the third measurement checks consistency and can reduce uncertainty, but it adds no new rank because its row is the sum of the first two. A discrepant third value would flag noise, calibration, or an incomplete coupling basis rather than uniquely identify a new constant.

References

Aeppli, Alexander, et al. “Atomic Clock Frequency Ratios with Fractional Uncertainty $\leq 3.2\times10^{-18}$ .” Physical Review Letters 137 (2026): 033201. DOI.
Godun, R. M., et al. “Frequency Ratio of Two Optical Clock Transitions in $^{171}\mathrm{Yb}^{+}$ and Constraints on the Time Variation of Fundamental Constants.” Physical Review Letters 113 (2014): 210801. DOI.
Guéna, J., et al. “Improved Tests of Local Position Invariance Using $^{87}\mathrm{Rb}$ and $^{133}\mathrm{Cs}$ Fountains.” Physical Review Letters 109 (2012): 080801. DOI.
Lange, R., N. Huntemann, J. M. Rahm, C. Sanner, H. Shao, B. Lipphardt, Chr. Tamm, S. Weyers, and E. Peik. “Improved Limits for Violations of Local Position Invariance from Atomic Clock Comparisons.” Physical Review Letters 126 (2021): 011102. DOI.