# How would one expect a massive graviton to behave?

Typically, adding a mass $m$ to a gauge boson causes the boson to only be able to travel over a finite distance, $L\sim m^{-1}$, limiting the range of the associated force. For example, photons become massive in superconductors and hence magnetic fields cannot penetrate very deep into superconductors.

Should one expect the same behavior for a massive graviton?

In the literature there are examples of massive gravity theories, such as the de Rham-Gabadadze-Tolly model (dRGT), which can give rise to self-accelerating universes due to a condensate of the graviton field (see here, for example). How does this phenomenon mesh with the usual reasoning that a mass limits the range of a gauge field?

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Before I start I should point out that it's not yet clear whether or not massive gravity works on a theoretical level. dRGT is a special theory but it still has some fundamental problems that have not yet been resolved (such as superluminal propagation around nontrivial backgrounds).

The yukawa suppression indeed is true for massive gravitons around flat space. This just follows from the basic form of the relativistic wave equation around flat space, and doesn't require a fancy nonlinear completion. Writing $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ we have

$$\square h + m^2 h = T$$

For a static, spherically symmetric source such as $T=M\delta^{(3)}(\vec{r})$, the solution to the above equation is $h=M e^{-mr}/4\pi r$.

Now you might worry that I've been too quick because in real gravity $h$ has indices. However this doesn't change the form of the solution--the yukawa suppression still works. However, it does constrain the form of the equation. Naively you would think a mass term in the equations of motion could contain any combination of $h_{\mu\nu}$ and $h \eta_{\mu\nu}$, but actually there is only one special combination that is allowed, the Fierz-Pauli tuning $h_{\mu\nu}-h \eta_{\mu\nu}$.

If you follow your intuition about the yukawa suppression and the cc a little further, you are lead to what is called 'degravitation.' Roughly speaking, the idea is that you could have a large cosmological constant, but since gravity is yukawa suppressed on very large scales, gravity doesn't see the cosmological constant. In other words, the CC is essentially a very long wavelength source, and the hope was you could have that wavelength be in the regime where the graviton propagator was suppressed.

Degravitation hasn't been able to work in any specific examples however. For example in dRGT if you try to degravitate the CC, then you end up in conflict with solar system tests, because you end up not effectively screening an extra degree of freedom that massive gravity has over normal GR. (A massless spin 2 has 2 dofs, a massive spin 2 has 5 dofs--the point is that the helicity 0 mode likes to couple strongly to matter and you need a special 'screening mechanism' to get continuity with GR. If you try to degravitate a large CC, you end up making this screening mechanism very inefficient).

So instead people who work on massive gravity try to use a condensate of gravitons to source the acceleration. This is really a fancy way of saying that you can treat the mass term as an effective source in einstein's equations $$G_{\mu\nu} = T_{\mu\nu} + m^2 T^{eff}_{\mu\nu}$$ In cosmology in particular, the $m^2 T^{eff}_{\mu\nu}$ can act like a cosmological constant term with a cosmological constant set by $m$ (recall that a real CC would like like $\Lambda g_{\mu\nu}$ on the RHS). However when you think about cosmology in these terms, the yukawa suppression isn't a good way to look at things, because you are far from flat space.

There are subtleties here because there aren't any exactly homogeneous and isotropic solutions in massive gravity, but that's the basic idea.

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Thank you, excellent answer and this is pretty much how I thought it worked. As for degravitation, was the original hope that you could take the enormous cosmological constant (CC), $\Lambda\sim M_{pl}^4$, and use the mass term as a high pass filter so that the effect of the CC is greatly reduced and we'd only end up with the relatively small amount of cosmological acceleration we observe today? –  user26866 Nov 12 '13 at 16:09
Yes that's exactly right, that's actually the language used in the original degravitation paper (degravitation is a high pass filter). –  Andrew Nov 13 '13 at 3:08