Is there a relation between the gravitational constant G and the Hubble constant H? I know there is a whole collection of question about the relation
between constant X and constant Y in physics. So I should explain
why this is not just a gratuitous question on some of the remaining
constant combinatorics.
My original motivation was to understand what could be the effect of
universe expansion, or of inflation, on the various force fields.
This is clearly too general a question, and I had to chose a  force
that still matters at the scales where expansion makes sense (at least
to me). Gravity is the obvious candidate,
The problem is that any question about gravitation runs the risk of
getting into specifics I am trying to avoid, such as remarks on the
curvature of space (I read carefully answers on other questions about
gravitation, questions that had more or less my level of naiveness).
And the best I could com up with, hoping it makes sense, is

Is there a relation between the gravitational constant G and the
  Hubble constant H ?

i.e.  would G be affected, and how, if there was a variation of H for
whatever reason ?
And of course: what about a very fast expansion like inflation,
possibly considering other forces? But this may already be too much
for one question.
 A: The answer is given by the Friedmann's equation:
$$H^2=\dfrac{8\pi G}{3}\rho-\dfrac{kc^2}{a^2}=\dfrac{\dot{a}^2}{a^2}$$
and
$$\dot{H}+H^2=\dfrac{\ddot{a}}{a}=-\dfrac{4\pi G}{3}\left(\rho+\dfrac{3p}{c^2}\right)$$
Remarks:


*

*The Hubble "constant" is not really "constant", it varies along the cosmic history since it is really the rate $H=\dot{a}/a$, where $a(t)$ is the so called scale factor of the Universe in a Robertson-Walker metric.

*The Hubble parameter squared is related with $G=G_N$, the Newton gravitational constant, throught the relations above. It can also depend on the global "geometry" of the Universe. If the Universe is "flat", $k=0$ and you see that the Hubble parameter squared is proportional to the density of matter-content of the Universe and the proportionality constant is really "constant" if $G$ is constant, as we believe it is.

*The time variation of $H$ plus $H$ squared gives also a relationship with $\rho$ and the pressure $p$. You need some equation of state for $p$, $p=p(\rho)$, in general. 

