# Time-variation of fundamental constants

The term physical constant expresses the notion of a physical quantity subject to experimental measurement which is independent of the time or location of the experiment. The constancy (immutability) of any "physical constant" is thus subject to experimental verification.

Paul Dirac in 1937 speculated that physical constants such as the gravitational constant or the fine-structure constant might be subject to change over time in proportion of the age of the universe.[1] Experiments conducted since then have put upper bounds on their time-dependence. This concerns the fine structure constant, the gravitational constant and the proton-to-electron mass ratio specifically, for all of which there are ongoing efforts to improve tests on their time-dependence.[2]

The immutability of these fundamental constants is an important cornerstone of the laws of physics as currently known; the postulate of the time-independence of physical laws is tied to that of the conservation of energy (Noether theorem), so that the discovery of any variation would imply the discovery of a previously unknown law of force.[3]

In a more philosophical context, the conclusion that these quantities are constant raises the question of why they have the specific value they do in what appears to be a "fine-tuned universe", while their being variable would mean that their known values are merely an accident of the current time at which we happen to measure them.[4]

## Dimensionality

It is problematic to discuss the proposed rate of change (or lack thereof) of a single dimensional physical constant in isolation. The reason for this is that the choice of a system of units may arbitrarily select any physical constant as its basis, making the question of which constant is undergoing change an artefact of the choice of units.[5][6][7]

For example, in SI units, the speed of light has been given a defined value in 1983. Thus, it was meaningful to experimentally measure the speed of light in SI units prior to 1983, but it is not so now. Tests on the immutability of physical constants look at dimensionless quantities, i.e. ratios between quantities of like dimensions, in order to escape this problem. Changes in physical constants are not meaningful if they result in an observationally indistinguishable universe. For example, a "change" in the speed of light c would be meaningless if accompanied by a corresponding "change" in the elementary charge e so that the ratio e2:c (the fine-structure constant) remained unchanged.[8]

Natural units are systems of units entirely based in fundamental constants. In such systems, it is meaningful to measure any specific quantity which is not used in the definition of units. For example, in Stoney units, the elementary charge is set to e = 1 while the reduced Planck constant is subject to measurement, ħ ≈ 137.03, and in Planck units, the reduced Planck constant is set to ħ = 1, while the elementary charge is subject to measurement, e ≈ (137.03)1/2. The 2019 redefinition of SI base units expresses all SI base units in terms of fundamental physical constants, effectively transforming the SI system into a system of natural units.

## Fine-structure constant

In 1999, evidence for time variability of the fine-structure constant based on observation of quasars was announced[9] but a much more precise study based on CH molecules did not find any variation.[10][11] An upper bound of 10−17 per year for the time variation, based on laboratory measurements, was published in 2008.[12] Observations of a quasar of the universe at only 0.8 billion years old with AI analysis method employed on the Very Large Telescope (VLT) found a spatial variation preferred over a no-variation model at the ${\displaystyle 3.9\sigma }$ level.[13]

The time-variation of fine-structure constant is equivalent to the time-variation of one or more of: speed of light, Planck constant, vacuum permittivity, and elementary charge, since ${\displaystyle \alpha ={\frac {e^{2}}{2\epsilon _{0}ch}}}$.

## Gravitational constant

The gravitational constant G is difficult to measure with precision, and conflicting measurements in the 2000s have inspired the controversial suggestions of a periodic variation of its value in a 2015 paper.[14] However, while its value is not known to great precision, the possibility of observing type Ia supernovae which happened in the universe's remote past, paired with the assumption that the physics involved in these events is universal, allows for an upper bound of less than 10−10 per year for the gravitational constant over the last nine billion years.[15]

As a dimensional quantity, the value of the gravitational constant and its possible variation will depend on the choice of units; in Planck units, for example, its value is fixed at G = 1 by definition. A meaningful test on the time-variation of G would require comparison with a non-gravitational force to obtain a dimensionless quantity, e.g. through the ratio of the gravitational force to the electrostatic force between two electrons, which in turn is related to the dimensionless fine-structure constant.

## Proton-to-electron mass ratio

An upper bound of the change in the proton-to-electron mass ratio has been placed at 10−7 over a period of 7 billion years (or 10−16 per year) in a 2012 study based on the observation of methanol in a distant galaxy.[16][17]

## Cosmological constant

The cosmological constant is a measure of the energy density of the vacuum. It was first measured, and found to have a positive value, in the 1990s. It is currently (as of 2015) estimated at 10−122 in Planck units.[18] Possible variations of the cosmological constant over time or space are not amenable to observation, but it has been noted that, in Planck units, its measured value is suggestively close to the reciprocal of the age of the universe squared, Λ ≈ T−2. Barrow and Shaw proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T−2 throughout the history of the universe.[19]

## References

1. ^ P.A.M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society A. 165 (921): 199–208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
2. ^ CODATA Recommended Values of the Fundamental Physical Constants: 2010" (March 15, 2012): "Although the possible time variation of the constants continues to be an active field of both experimental and theoretical research, there is no observed variation relevant to the data on which the 2010 recommended values are based; see, for example, the recent reviews by Uzan (2011) and Chiba (2011). Other references may be found in the FCDC bibliographic database at physics.nist.gov/constantsbib using, for example, the keywords 'time variation' or 'constants.'".
3. ^ "Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to matter. This will induce a violation of the universality of free fall. Thus, it is of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy." Uzan (2011)
4. ^ Uzan (2011), chapter 7: "Why Are The Constants Just So?": "The numerical values of the fundamental constants are not determined by the laws of nature in which they appear. One can wonder why they have the values we observe. In particular, as pointed out by many authors (see below), the constants of nature seem to be fine-tuned [Leslie (1989)]. Many physicists take this fine-tuning to be an explanandum that cries for an explanans, hence following Hoyle [(1965)] who wrote that 'one must at least have a modicum of curiosity about the strange dimensionless numbers that appear in physics.'"
5. ^ Duff, M. J. (2014). "How fundamental are fundamental constants?". Contemporary Physics. 56 (1): 35–47. arXiv:1412.2040. doi:10.1080/00107514.2014.980093 (inactive 31 May 2021).CS1 maint: DOI inactive as of May 2021 (link)
6. ^ Duff, M. J. (13 August 2002). "Comment on time-variation of fundamental constants". arXiv:hep-th/0208093.
7. ^ Duff, M. J.; Okun, L. B.; Veneziano, G. (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 2002 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023. S2CID 15806354.
8. ^ Barrow, John D. (2002), The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, ISBN 0-375-42221-8 "[An] important lesson we learn from the way that pure numbers like α define the World is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first, we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our World. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value you cannot tell, because all the pure numbers defined by the ratios of any pair of masses are unchanged."
9. ^ Webb, J. K.; et al. (2001). "Further evidence for cosmological evolution of the fine structure constant". Physical Review Letters. 87 (9): 091301. arXiv:astro-ph/0012539v3. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558. S2CID 40461557.
10. ^
11. ^ "Distant Quasars Show That Fundamental Constants Never Change". 5 January 2017.
12. ^ Rosenband, T.; et al. (2008). "Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place". Science. 319 (5871): 1808–12. Bibcode:2008Sci...319.1808R. doi:10.1126/science.1154622. PMID 18323415. S2CID 206511320.
13. ^ Wilczynska, Michael R.; Webb, John K.; et al. (2020). "Four direct measurements of the fine-structure constant 13 billion years ago". Science Advances. 6 (17): 9672. arXiv:2003.07627. Bibcode:2020SciA....6.9672W. doi:10.1126/sciadv.aay9672. PMC 7182409. PMID 32426462.
14. ^ Anderson, J. D.; Schubert, G.; Trimble, V.; Feldman, M. R. (April 2015), "Measurements of Newton's gravitational constant and the length of day", EPL, 110 (1): 10002, arXiv:1504.06604, Bibcode:2015EL....11010002A, doi:10.1209/0295-5075/110/10002, S2CID 119293843
15. ^ Mould, J.; Uddin, S. A. (10 April 2014), "Constraining a Possible Variation of G with Type Ia Supernovae", Publications of the Astronomical Society of Australia, 31: e015, arXiv:1402.1534, Bibcode:2014PASA...31...15M, doi:10.1017/pasa.2014.9, S2CID 119292899
16. ^ Bagdonaite, Julija; Jansen, Paul; Henkel, Christian; Bethlem, Hendrick L.; Menten, Karl M.; Ubachs, Wim (December 13, 2012). "A Stringent Limit on a Drifting Proton-to-Electron Mass Ratio from Alcohol in the Early Universe". Science. 339 (6115): 46–48. Bibcode:2013Sci...339...46B. doi:10.1126/science.1224898. PMID 23239626. S2CID 716087. Retrieved December 14, 2012.
17. ^ Moskowitz, Clara (December 13, 2012). "Phew! Universe's Constant Has Stayed Constant". Space.com. Retrieved December 14, 2012.
18. ^ Collaboration, Planck, PAR Ade, N Aghanim, C Armitage-Caplan, M Arnaud, et al., Planck 2015 results. XIII. Cosmological parameters. arXiv preprint 1502.1589v2 [1], 6 Feb 2015. John D. Barrow The Value of the Cosmological Constant
19. ^ Barrow, John D.; Shaw, Douglas J. (2011), "The Value of the Cosmological Constant", General Relativity and Gravitation, 43 (10): 2555–2560, arXiv:1105.3105, Bibcode:2011GReGr..43.2555B, doi:10.1007/s10714-011-1199-1, S2CID 55125081