How well do we know the value of G?

In article , Steven Carlip
writes:

On 3/10/21 2:09 AM, Phillip Helbig (undress to reply) wrote:
How well do we know the value of G?

G is the constant (well, as far as we know) of nature whose value is
known with the least precision. How well do we know it? Presumably
only Cavendish-type experiments can measure it directly. Other
measurements of G, particularly astronomical ones, probably actually
measure GM, or GMm. In some cases, those quantities might be known to
more precision than G itself.

Suppose G were to vary with time, or place, or (thinking of something
like MOND here) with the acceleration in question. Could that be
detected, or would it be masked by wrong assumptions about the mass(es)
involved?

The idea that G may vary in time goes back to Dirac's "large
numbers hypothesis" in the 1930s. There's been a huge amount of
experimental and observational investigation. A classic review
article is Uzan, arXiv:hep-ph/0205340; a more recent version is
arXiv:1009.5514. There are quite strong constraints on time
variation, and some weaker constraints on spatial variation,
coming from everything from Lunar laser ranging to binary
pulsar timing to Big Bang Nucleosynthesis.

I suppose that there are relatively strong constraints on variation with
time; those were used to rule out theories like Dirac's and so on: the
temperature of the Sun would change, the structure of the Earth, and so
on, and as you note some weaker constraints on spatial variation.

More interesting is how well we know it and whether different
measurements are statistically compatible. (My guess is that they are
since the precision is not very good, compared to measurements of other
constants.)

My main point is that G is rarely measured, but rather GM, and one often
has no handle on M other than by assuming G. So perhaps it could vary
from place to place within, say, the Galaxy or the Local Group. I don't
have any reason to think that it does, but, as discussed in another
thread here recently, are there actually any useful constraints?
Obviously it doesn't vary by very much, as stellar populations in
different galaxies look broadly similar and so on.

Probably most difficult to rule out is something like MOND (which
actually has a lot of evidence in support of it, at least at the
phenomenological level) where the (effective) value of G varies. In
MOND, for small accelerations, the value is higher than the Newtonian
(or GR) value.

Suppose that in the case of very strong fields, the effective value is
less than the G we measure directly. To some extent, that could be
compensated for via larger masses (as often the product GM is relevant).
To take a concrete example, in the LIGO black-hole--merger events, could
one decrease G by, say, 1 per cent, and increase the masses accordingly,
and still fit the data?