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Pictures like the one below are often used to talk about grand unification.

enter image description here

I've never heard any physics textbook really talk about the running of the gravitational coupling constant $G$, but some popsci sources seem to vaguely imply that gravity should have a line on this diagram too, and it joins up with the others at the TOE scale.

Is there a way to make this statement precise? If so, what would gravity look like on this plot?

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    $\begingroup$ Of course the gravitational coupling also runs. However, in contrast to the other coupling strengths, it is dimensionful. So the running is almost entirely given by $\mu^2/M_\mathrm{P}^2$, which unifies with the other couplings around $\mu_\mathrm{unification}\sim M_\mathrm{P}$. In theories with extra dimensions one can change this to obtain $\mu_\mathrm{unification}=M_\mathrm{GUT}$. $\endgroup$ – user178876 Jan 13 '18 at 16:54
  • $\begingroup$ Do you want to know about the running of $G_N$ according to what theory? naïve quantum gravity? supergravity? string theory? All these theories predict different runnings. More generally, the running of a certain parameter depends on the complete theory you are embedding it into. $\endgroup$ – AccidentalFourierTransform Jan 14 '18 at 0:38
  • $\begingroup$ @AccidentalFourierTransform You don't need a grand unified theory to talk about the running of couplings below the GUT scale. Neither do you need a UV complete theory of quantum gravity to talk about the running of $G$. $\endgroup$ – knzhou Jan 14 '18 at 11:27
  • $\begingroup$ @knzhou I didn't say you do. What you do need is to specify the context. There are an infinite number of theories that implement gravitation at a quantum level. They predict different runnings for $G_N$. Without the context, the question is meaningless. You could argue that any realistic theory agrees in the IR, and therefore agrees in the running of $G_N$ for small energy. But you didn't specify that you only care about the IR. Thus, the question needs some more details to be meaningful (FWIW, I do like the question anyway, don't get me wrong). $\endgroup$ – AccidentalFourierTransform Jan 14 '18 at 14:53
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Of course the gravitational coupling also runs. However, in contrast to the other coupling strengths, it is dimensionful. So the running is almost entirely given by $\mu^2/M_\mathrm{P}^2$, which unifies with the other couplings around $\mu_\mathrm{unification}\sim M_\mathrm{P}$. In theories with extra dimensions one can change this by bringing down the Planck scale to obtain $μ_\mathrm{unification}\simeq M_\mathrm{GUT}$. This requires a parametrically largish volume of the compactified dimensions.

In string theory, the analogous question is to reconcile the GUT and string scales, which has triggered some of the most important developments of the field.

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