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**Unformatted text preview: **can be reduced to a
boundary term at y ¼ 0,
Z
2
1
1
ðS5 þ SGF Þ ¼ d4 x À h Áh þ hÁh
3
4
8
M5
1
1
1
À nÁn À n Án þ hÁn
2
2
2
À@ n À 1 @ h þ @ h
þn
:
2
(9.46) Kurt Hinterbichler: Theoretical aspects of massive gravity 705 Now a crucial point. We have been imagining solving the
constraints (9.36) for the independent variables. But now,
consider the action (9.46) as a function of the original variables h , n , and n. Varying with respect to n and n, we
recover precisely the constraints (9.36). Thus, we can reintroduce the solved variables as auxiliary ﬁelds, since the
constraints are then implied. The action now becomes a
function of h , n , and n.
Now add in the 4d part of the action, This branch is completely unstable, which is bad news for
doing cosmology on this branch. In addition to ghosts,
there are other issues with other nontrivial branches, such as
superluminal ﬂuctuations (Hinterbichler, Nicolis, and Porrati,
2009), and uncontrolled singularities and tunneling (Gregory,
2008). 2
M4 Z 4 pﬃﬃﬃﬃﬃﬃﬃﬃ
d x ÀgRðgÞ þ SM þ 2ðS5 þ SGF Þ;
2 The operator dependent mass term in Eq. (9.50) is known
as a resonance mass, or soft mass (Dvali, Gabadadze, and
Porrati, 2000b; Dvali et al., 2001a, 2001b; Gabadadze, 2004;
Gabadadze and Shifman, 2004; Dvali, 2006; Dvali, Hofmann,
and Khoury, 2007; Lopez Nacir and Mazzitelli, 2007;
Gabadadze and Iglesias, 2008; Patil, 2010). To see the particle content of this theory, we decompose the propagator into a
¨
sum of massive gravity propagators. The linear Stuckelberg
analysis, leading to the propagators (4.37), goes through
identically, with the replacement m2 ! mÁ. The momentum
part of the propagators now reads S¼ (9.47) where SM is the 4d matter action and the factor of 2 in front of
the 5d parts results from taking into account both sides of the
bulk (through boundary conditions at inﬁnity we have thus
implicitly imposed a Z2 symmetry).
Expanded to quadratic order,
S¼ Z 2
M4 1
h E ; h 4 2
M2 m
1
1
þ4
À h Áh þ hÁh À nÁn À n Án
4
2
4
þ hÁn þ n ðÀ2@ n À @ h þ 2@ h Þ d4 x 1
þ h T ;
2 (9.48) where E ; is the massless graviton kinetic operator (2.46),
and
m 3
2M 5
2
M4 (9.49) is known as the DGP scale.
It is invariant under the gauge transformations (9.40),
¨
under which n plays the role of the vector Stuckelberg ﬁeld.
n plays the role of a gauge invariant auxiliary ﬁeld. To get this
into Fierz-Pauli form, ﬁrst eliminate n as an auxiliary ﬁeld by
using its equation of motion. Then use Eq. (9.40) to ﬁx the
gauge n ¼ 0. The resulting action is
Z
M2 1
S ¼ d4 x 4 h E ; h 42
1
1
À mðh Áh À hÁhÞ þ h T ;
(9.50)
2
2
which is of the Fierz-Pauli form, with an operator dependent
mass term mÁ.
One can go on to study interaction terms for DGP, and the
longitudinal mode turns out to be governed by interactions
which include the cubic Galileon term $ð@Þ2 h (Luty,
Porrati, and Rattazzi, 2003;...

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