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**Unformatted text preview: **oni constant, and the length
scale r0 is
r0 terms, so we should have < 1. Additionally, there is the
constraint that the spectral function (9.53) should be positive
deﬁnite, so that there are no ghosts. This puts a lower bound ! 0 (Dvali, 2006). It turns out that degravitation can be
made to work only for < 1=2 (Dvali, Hofmann, and
Khoury, 2007). DGP corresponds to ¼ 1=2, and so it just
barely fails to degravitate, but by extending the DGP idea to
higher codimension (Kiritsis, Tetradis, and Tomaras, 2001; de
Rham, 2008; Hassan, Hofmann, and von Strauss, 2011) or to
multibrane cascading DGP models (de Rham et al., 2008; de
Rham, 2009; de Rham, Khoury, and Tolley, 2010), < 1=2
can be achieved and degravitation made to work (de Rham
et al., 2008). Some N -body simulations of degravitation and
DGP have been done by Chan and Scoccimarro (2009),
Khoury and Wyman (2009), and Schmidt (2009). 1
M2
¼ 43 :
m 2M 5 (9.57) The potential interpolates between 4d $ 1=r and 5d $ 1=r2
behavior at the scale r0 ,
8
> 2 2M þ M 2 r1 ½ À 1 þ lnðrr Þ þ OðrÞ; r ( r0 ;
< 3 M 4r 32 M 0
0
4
4
V ð rÞ ¼ 2 M
> 3 2 2 þ Oð 13 Þ;
: 3 M 4 r
r ) r0 :
r
5 (9.58)
Physically, we think of gravity as being conﬁned to the brane
out to a distance $r0 , at which point it starts to weaken and
leak off the brane, becoming ﬁve dimensional. This is the
behavior that is morally responsible for the self-accelerated
solutions seen in DGP (Deffayet, Dvali, and Gabadadze,
2002). It has been suggested that corrections to the
Newtonian potential for r ( r0 may be observable in lunar
laser ranging experiments (Dvali, Gruzinov, and Zaldarriaga,
2003; Lue and Starkman, 2003).
The resonance massive graviton can also be generalized
away from DGP, by replacing the mass term with an arbitrary
function of the Laplacian (Gabadadze and Shifman, 2004;
Dvali, 2006; Dvali, Hofmann, and Khoury, 2007),
m2 ! m2 ðhÞ: (9.59) [See Dvali, Pujolas, and Redi (2008) for even further generalizations.] At large distances, where we want modiﬁcations to
occur, the mass term has a leading Taylor expansion
m2 ðhÞ ¼ L2ðÀ1Þ h ; (9.60) with L being a length scale and being a constant. In order to
modify Newtonian dynamics at large scales, @ ( 1=L, the
mass term should dominate over the two derivative kinetic
Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Massive gravity remains an active research area, one which
may provide a viable solution to the cosmological constant
naturalness problem. As seen, many interesting effects arise
from the naive addition of a hard mass term to Einstein
gravity. There is a well-deﬁned effective ﬁeld theory with a
protected hierarchy between the cutoff and the graviton mass,
and a screening mechanism which nonlinearly hides the new
degrees of freedom and restores continuity with GR in the
massless limit.
A massive graviton can screen a large cosmological constant, and a stable theory of massive gravity with a small
protected mass offers a s...

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