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**Unformatted text preview: **t 3 degrees of freedom beyond the 2 of the massless graviton.
With regard to the cosmological constant problem, there is
the possibility of a technically natural explanation. The deformation parameter is m, the graviton mass, and GR should
be restored as m ! 0. The force mediated by a massive
graviton has a Yukawa proﬁle $rÀ1 eÀmr , which drops off
from that of a massless graviton at distances r * 1=m, so one 674 Kurt Hinterbichler: Theoretical aspects of massive gravity hopes to explain the acceleration of the Universe without dark
energy by choosing the graviton mass to be of the order of the
Hubble constant m $ H . Of course, this does not eliminate
the small cosmological constant, which reappears as the ratio
m=MP . But there is now hope that this is a technically natural
choice, because deformation by a mass term breaks the gauge
symmetry of GR, which is restored in the limit m ! 0. As we
will see, a small m is indeed protected from quantum corrections (although there are other issues that prevent this, at our
current stage of understanding, from being a completely
satisfactory realization of a technically natural cosmological
constant).
There are also interesting lessons to be learned regarding the
continuity of physical predictions. The addition of a mass term
is a brutality upon the structure of GR and does not go
unpunished. Various pathologies appear, which are representative of common pathologies associated with any infrared
modiﬁcation of gravity. These include strong classical nonlinearities, ghostlike instabilities, and a very low cutoff, or
region of trustability, for the resulting quantum effective theory. In short, modifying the infrared often messes up the UV.
New mechanisms also come into play, because the extra
degrees of freedom carried by the massive graviton must
somehow decouple themselves as m ! 0 to restore the physics
of GR.
The study of the massless limit leads to the discovery of the
Vainshtein mechanism, by which these extra degrees of freedom hide themselves at short distances using nonlinearities.
This mechanism has already proven useful for model builders
who have long-range scalars, such as moduli from the extra
dimensions of string theory, that they want to shield from
local experiments that would otherwise rule them out.
C. History and outline The possibility of a graviton mass has been studied off and
on since 1939, when Fierz and Pauli (1939) ﬁrst wrote the
action describing a free massive graviton. Following this, not
much occurred until the early 1970s, when there was a ﬂurry of
renewed interest in quantum ﬁeld theory. The linear theory
coupled to a source was studied by van Dam and Veltman
(1970) and Zakharov (1970) (vDVZ), who discovered the
curious fact that the theory makes predictions different from
those of linear GR even in the limit as the graviton mass goes to
zero. For example, massive gravity in the m ! 0 limit gives a
prediction for light bending that is off by 25% from the GR
predictio...

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