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Obviously some things, such as the speed of light in a vacuum, are defined to be a precise value. The kilogram was recently defined to have a specific value by fixing Plank's constant to $6.62607015\cdot 10^{−34}\frac{m^2 kg}{s}$.

In particular, in the case of the latter, we held off on defining this value until the two competing approaches for measuring the kilogram agreed with each other within the error bounds of their respective measurements.

Which leads me to wonder, what is the most precisely measured (not defined) value that the scientific community has measured. I am thinking in terms of relative error (uncertainty / value). Before we defined it, Plank's constant was measured to a relative error of $10^-9$. Have we measured anything with a lower relative error?

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    $\begingroup$ @Qmechanic Mind explaining why you felt this needed to be put on hold via a mod hammer, when there were clearly good answers coming forth, and everyone seems to agree upon what is meant? In fact, with 6 upvotes and 33 views, it seems people seem to feel this is a good question. $\endgroup$ – Cort Ammon Aug 16 at 5:54
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    $\begingroup$ Shouldn't the title refer to precision rather than accuracy? $\endgroup$ – Gremlin Aug 16 at 13:02
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    $\begingroup$ Discussion about this question on meta. $\endgroup$ – Emilio Pisanty Aug 16 at 13:41
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    $\begingroup$ @Gremlin Updated. You're right, I was lazy in my wording. "precision" is a better word choice $\endgroup$ – Cort Ammon Aug 16 at 15:35
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The magnetic moment of the electron has been measured to a few parts in $10^{13}$. (Source) This provides an exquisite test of quantum electrodynamics, and calculating the relevant Feynman diagrams has been a Herculean effort over decades.

Note that the more precise tests cited in other answers are basically null results: no difference between gravitational and inertial mass; no difference in magnitude of charge between proton and electron; no mass of photon. So I believe the magnetic moment of the electron is the most precise measurement that is non-null and thus “interesting”.

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    $\begingroup$ Yeah that’s the answer I was looking for but somehow only found $\alpha$. $\endgroup$ – ZeroTheHero Aug 16 at 5:35
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    $\begingroup$ While I'm not going to change the question to invalidate answers, your note is a good one that I'd edit in if I had a chance. While those null results are indeed interesting, I have the intuitive sense that it's easier to measure them with arbitrarily high precision. It is interesting, however, how few orders of magnitude separate this result from some of the equivalency tests listed in other answers. $\endgroup$ – Cort Ammon Aug 16 at 16:18
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    $\begingroup$ For full clarity, this isn't the most accurate or precise measurement known, and the fact that @CortAmmon has accepted this answer (when there are tighter uncertainties reported in other answers) is a good example of why this thread is problematic. If it comes to "non-null" results, though, this paper presents a measurement with considerably more precision than the one in this answer (but I'm not going to take the (pretty arrogant) tack that others here have taken in asserting that the first thing I found is "the" most precise experiment). $\endgroup$ – Emilio Pisanty Aug 21 at 10:18
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Carrying the fame of being one of the most precisely verified propositions in physics, the ratio of the gravitational to inertial mass was verified to be unity within $1$ in $10^{15}$ by the MICROSCOPE satellite in $2017$. The earlier best precision was $5\times10^{-14}$, obtained by Baessler, et al. in $1999$.

References:

  1. https://en.wikipedia.org/wiki/MICROSCOPE_(satellite)
  2. https://en.wikipedia.org/wiki/Equivalence_principle#Tests_of_the_weak_equivalence_principle
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A few more candidates:

  • The quantized Hall resistance is a great and surprising example, since it's an emergent property of rather complicated, "dirty" systems. As stated here (2013) one can measure the resistance to one part in $3 \times 10^{10}$. For this reason this effect is now used to define the Ohm.
  • Equivalence principle tests using torsion balances reached precisions of about one part in $10^{11}$ in $1964$, see here. Another existing answer gives a more precise result from a more modern experiment. These can be thought of as either verifying the equality of gravitational and inertial mass, or placing bounds on the strength of long-range fifth forces.
  • The electrical neutrality of bulk matter follows because the electron and proton have exactly opposite charges. Treating the charge of the electron as given, tests of neutrality effectively measure the charge of the proton, with one experiment achieving a sensitivity of one part in $10^{21}$ in $1973$. Arguments from cosmology can be used to set much larger bounds, though they require more assumptions.
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  • $\begingroup$ Can you suggest me some reference where I can look up how the Equivalence Principle tests can be interpreted as putting bounds on a long-range fifth force? $\endgroup$ – Dvij Mankad Aug 16 at 4:59
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    $\begingroup$ Another example: Earth-Moon distance measured with a laser. Distance about 3e10 cm, measured with a few cm accuracy I think. Should investigate more to be sure. $\endgroup$ – thermomagnetic condensed boson Aug 16 at 8:47
  • $\begingroup$ Electrical neutrality of bulk matter: That only proves the equality of the charges if the number of electrons and protons in bulk matter is the same - but if the charge on the electron was 1.1 times that on the proton, I would expect bulk matter to have about 10% more electrons and still be neutral. $\endgroup$ – Martin Bonner supports Monica Aug 16 at 13:03
  • $\begingroup$ Update: the ohm is now defined in terms of the elementary charge $\endgroup$ – gen-z ready to perish Aug 21 at 6:11
  • $\begingroup$ @gen-z Not particularly - in actual metrological practice, it's rather more accurate to say that the coulomb is defined in terms of the ohm. $\endgroup$ – Emilio Pisanty Aug 21 at 10:20
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A good candidate is the measurement of the fine structure constant $\alpha$. This wiki article on precision tests of QED states that:

The agreement found this way is to within ten parts in a billion ($10^{−8}$), based on the comparison of the electron anomalous magnetic dipole moment and the Rydberg constant from atom recoil measurements...

There is also an upper bound on the mass of the photon, which is in the range of $10^{-27}eV/c^2$ although that's an upper bound rather than a measurement since the expected value is $0$.

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    $\begingroup$ The ratio of the gravitational to inertial mass was verified to be unity within $1$ in $10^{15}$ in $2017$ by the MICROSCOPE satellite: en.wikipedia.org/wiki/MICROSCOPE_(satellite) $\endgroup$ – Dvij Mankad Aug 16 at 4:04
  • $\begingroup$ @DvijMankad this ought to be an answer then... $\endgroup$ – ZeroTheHero Aug 16 at 4:07
  • $\begingroup$ I think it could be added as a possible answer but I am not sure if there exist more precise measurements in particle physics, maybe some electroweak precision measurements? $\endgroup$ – Dvij Mankad Aug 16 at 4:08
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    $\begingroup$ @DvijMankad I do believe the only way a question like this can be answered on StackExchange is to put forth the best answer an individual may know, and the best answer will get upvoted. (And I highly doubt a good answer will be downvoted just because somebody finds a better answer) $\endgroup$ – Cort Ammon Aug 16 at 4:10
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    $\begingroup$ @CortAmmon Makes sense. Done! :-) $\endgroup$ – Dvij Mankad Aug 16 at 4:13
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To answer the question from a different angle, LIGO has measured gravitational waves several times over the last few years. To do so they have to observe a distortion in a 4km arm, in the order of $10^{-18}$m. In other words, it has to detect a fractional imprecision in its length of ~$2.5 * 10^{-23}$. That's pretty accurate.

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    $\begingroup$ LIGO is extremely sensitive, but pretty inaccurate, its calibration has an uncertainty of a few percent. $\endgroup$ – Bas Swinckels Aug 16 at 13:05
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    $\begingroup$ Thank you for the answer! What Bas Swinckles mentions is something I was trying to avoid when crafting the criteria of the question. The 4km long arm is not what LIGO actually measures, but rather the 10^-18m number is what is being measured. LIGO was one heck of a feat, but I was trying to avoid high-sensitivity/low-accuracy measurements because we can always come up with a "higher" value by comparing it against something bigger. $\endgroup$ – Cort Ammon Aug 16 at 15:22
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Another quantity that is well known is the frequency of the 21 cm line of Hydrogen hyperfine splitting that has been measured to about 13 significant figures, 1420405751.7667±0.0009 Hz. Unlike the electron magnetic moment, mentioned in another answer, the QED calculation for hyperfine splitting is not as accurate. This calculation also depends on other fundamental constants, including electron/proton mass ratio and proton magnetic moment. which are not known to equivalent precision.

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