1308.4136
1311.6485
1408.xxxx
DECONSTRUCTING DIMENSIONS
AND MASSIVE GRAVITY
COSMO 2014
Andrew Matas, Case Western Reserve University
Co-Authors: Claudia de Rham, Andrew Tolley
Why modify gravity (in the IR)?
Dark Matter and/or Dark Energy
Test Gravity on Large Scales
Cosmological Constant Problem(s)
⇤
Psaltis 2008
Massive Gravity
2
MPl
S=
2
Z
4
d x
✓
R
m2
U (g, f )
4
Modify gravity by adding a mass term
External “reference metric” needed to make
nontrivial potential. Breaks diff invariance.
Can address the Cosmological Constant Problem.
m can be small in a natural way.
⇤!m
◆
fµ⌫
Degrees of Freedom of a Massive Graviton
2
ds =
Tensors
Scalars
Vectors
Will 2006
2
dt + (
ij
i
+ hij ) dx dx
j
Degrees of Freedom of a Massive Graviton
2
ds =
Tensors
2
dt + (
ij
i
+ hij ) dx dx
j
A healthy massive spin
2 particle has 5
polarizations
Scalars
2s + 1 = 5
when
Vectors
Will 2006
s=2
Degrees of Freedom of a Massive Graviton
2
ds =
2
dt + (
ij
i
+ hij ) dx dx
j
Two possible scalar modes:
Tensors
Scalars
Vectors
Will 2006
- One is healthy, but must be
screened via the Vainshtein
Mechanism
Vainshtein, 1972
- The other is a ghost
(negative kinetic energy) if
it appears
Fierz and Pauli, 1939
Boulware and Deser 1972
dRGT Massive Gravity
de Rham et al, 2010
2
MPl
S=
2
4
U (g) =
X
n=2
Z
4
d x
↵n Un (g)
2
U2 = [K]
U3 = [K]3
U4 = [K]4
2
p
✓
g R
[K ]
3[K2 ][K] + 2[K3 ]
m2
U (g, f )
4
K⌫µ
=
6[K]2 [K2 ] + 3[K2 ]2 + 8[K][K3 ]
µ
⌫
⇣p
g
6[K4 ]
◆
1f
⌘µ
⌫
dRGT Massive Gravity
de Rham et al, 2010
2
MPl
S=
2
4
Z
4
d x
p
✓
g R
m2
U (g, f )
4
◆
Very special
structure chosen
to
↵ U (g)
⇣p
⌘
K =
g f
U
=
[K]
[K
]
avoid
Boulware-Deser
ghost
U = [K]
3[K ][K] + 2[K ]
U (g) =
2
3
X
n=2
n n
2
3
U4 = [K]4
µ
⌫
2
2
µ
⌫
1
3
6[K]2 [K2 ] + 3[K2 ]2 + 8[K][K3 ]
6[K4 ]
Can we get a physical picture of why dRGT works?
de Rham, AM, Tolley, 1308.4136
µ
⌫
Multi-gravity
Extension of massive gravity: Reference Metric becomes dynamical
S=
Z
2p
2
M
M
p
g
f
d4 x
gR[g] +
f R[f ]
2
2
m2 Mg2
U (g, f )
4
Hassan and Rosen, 2011
Hinterbichler and Rosen, 2012
Can have arbitrary number of metrics
S=
Z
d4 x
X M2 p
I
I
2
gI R[gI ]
m2 M12
U [gI ]
4
Kaluza-Klein Compactification
y
Consistent theories of massive gravitons arise
in extra dimensional setups: Kaluza-Klein,
DGP.
5D General Relativity has 5 propagating
degrees of freedom.
Coordinates:
µ
x
y
4 large dimensions we observe
1 compact extra dimension
Kaluza-Klein Compactification
f
x
(@y )2 !
2⇡n
kn =
L
kn2
2
n
We interpret the wavenumber along the y
2⇡n
mn =
direction as a mass in 4d
L
Infinite tower of massive modes
..
.
(5)
GAB
Consistent theory of
an infinite number
of massive gravitons
2⇡
m=2
L
(2)
Hµ⌫
2⇡
m=
L
(1)
Hµ⌫
m=0
gµ⌫
Aµ
Dimensional Deconstruction
y
Arkhani-Hamed et al, 2001, 2002
Dimensional Deconstruction: Intuition
f
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Introduce new scale:
Lattice spacing m-1
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
⇣n⌘
Arkhani-Hamed et al, 2001, 2002
N
N = mL
Ê
Ê
mn = m sin
x
Ê
Tower is truncated
0  n <1308.
N
4136
Deconstruction and Gravity
N Sites:
1 Massless Mode
N-1 Massive Modes
(2)
gµ⌫
(3)
gµ⌫
x(2)
x(3)
(1) (1)
gµ⌫ x
x(i)
(i)
gµ⌫
Can we do this while preserving the number of degrees of freedom?
Discretizing
Integrals become
sums
Z
Derivatives need to
be handled with care
@y f (x, y) ! m (fi+1 (x)
Fix y diffs before
discretizing
Ny
1 X
fi (x)
dyf (x, y) !
m i
fi (x))
Linear Level
Straightforward at the linear level
Fix gauge
hµy = 0, hyy = 1
Discretize
@y hµ⌫ (x, y) ! m2 h(i+1)
(x)
µ⌫
5d
SGR
=
Z
4
d xdy
1
hµ⌫ E µ⌫⇢ h⇢
4
Linearized GR
⇣
h(i)
µ⌫ (x)
1
[(@y h)2 ]
8
UF.P. [h] = m2 [h]2
⌘
[@y h]2
[h2 ]
However, need a non-linear theory for Vainshtein mechanism
Implementing Deconstruction
SGR,5d =
M53
Z
4
d x dy
p
g Ny R + [K]2
[K 2 ]
Implementing Deconstruction
SGR,5d =
M53
Z
4
d x dy
p
g Ny R + [K]2
[K 2 ]
The potential will come from
here after discretizing
Implementing Deconstruction
SGR,5d =
M53
Z
Fix the gauge
4
d x dy
p
g Ny R + [K]2
[K 2 ]
The potential will come from
here after discretizing
N5 = 1
Kµ⌫
Nµ = 0
1
= @y gµ⌫
2
Implementing Deconstruction
SGR,5d =
M53
Z
4
d x dy
Fix the gauge
p
g Ny R + [K]2
The potential will come from
here after discretizing
N5 = 1
Kµ⌫
Discretization prescription?
[K 2 ]
Nµ = 0
1
= @y gµ⌫
2
Kµ⌫ !?
Most procedures will introduce new
DOFs
Discretizing the metric directly will introduce the BD ghost
⇣
(2)
@y gµ⌫ ! m gµ⌫
(1)
gµ⌫
⌘
The potential is a function of the Fierz-Pauli
combination: this is known to lead to a BD ghost
Discretizing using the Vielbein
de Rham, AM, Tolley,
1308.4136
gµ⌫ = eaµ eb⌫ ⌘ab
First write derivatives of the metric
in terms of the vielbeins
Then discretize
the derivative
a
@ y eµ
We recover the ⇣dRGT
structure
⌘µ
p
K⌫µ =
µ
⌫
g
1f
⌫
!m
@y g ⇠ e@y e
⇣
(2),a
eµ
(1),a
eµ
Kµ⌫ ! Kµ⌫
⌘
Discretizing using the Vielbein
Many features of multi-gravity theories arise naturally in this construction
Symmetric Vielbein Condition
Strong Coupling Scale
Stuckleberg Fields
Discretizing using the Vielbein
Different discretizations
of the derivative lead to
different dRGT type
interactions
(2)
gµ⌫
(3)
gµ⌫
x(2)
(1)
gµ⌫
x(3)
x(1)
x(i)
Example:
⇣
@y eaµ (x, y) ! m e(2),a
(x)
µ
ce(1),a
(x)
µ
(i)
gµ⌫
⌘
Scale of interactions
⇤s ⇠
✓
M5
L
◆1/2
⇠
2 1/3
M 4 m1
The scale that arises is the usual strong
coupling scale for the lightest KK mode
Extensions
@y [f (y)g(y)] = @y f (y)g(y) + f (y)@y g(y)
We can now apply this method in other cases
We find generically that other cases are more sensitive to
discretization
Structures that are total derivatives in 5d (and
thus safe) can become physical upon
discretization
Gauss-Bonnet
We can apply Deconstruction to Gauss-Bonnet
SGB = ↵GB
Z
d5 x
p
G R2µ⌫⇢
4R2µ⌫ + R2
5d
4d
2
R⇠R+K
2
2
2
R ⇠R +K R+K
4
New Kinetic Interaction?
Hinterbichler 2013
de Rham, AM, Tolley,
1311.6485
Gauss-Bonnet
We can apply Deconstruction to Gauss-Bonnet
SGB = ↵GB
Z
d5 x
p
G R2µ⌫⇢
5d
4d
2
R⇠R+K
2
2
2
R ⇠R +K R+K
4
New Kinetic Interaction?
Hinterbichler 2013
4R2µ⌫ + R2
The deconstructed
theory has a ghost,
can be seen in
minisuperspace
de Rham, AM, Tolley,
1311.6485
Gauss-Bonnet
We can apply Deconstruction to Gauss-Bonnet
SGB = ↵GB
Z
d5 x
p
G R2µ⌫⇢
5d
4d
2
R⇠R+K
2
2
2
R ⇠R +K R+K
4
New Kinetic Interaction?
Hinterbichler 2013
4R2µ⌫ + R2
More general result:
Only consistent
kinetic term for
Massive Gravity
with flat reference
metric in 4d is
Einstein Hilbert
de Rham, AM, Tolley,
1311.6485
Charged Spin-2
Can try to keep the vector zero modes
@µ h̃n,µ⌫ ! Dµ h̃n,µ⌫ = (@µ
U (1)
Zn
de Rham, AM, Ondo, Tolley, 1408.xxxx
iqnAµ ) h̃n,µ⌫
U(1) can be restored
but this introduces new
degrees of freedom
Summary
Special structure of dRGT can
be understood from a higher
dimensional perspective
Higher derivative extensions
are challenging because total
derivative combinations can
be broken
dRGT Massive Gravity: Vielbein
Special Structure avoids BD ghost
U (e, f ) =
U1 (e, f ) =
U2 (e, f ) =
U3 (e, f ) =
Hinterbichler and Rosen, 2012
3
X
n=1
✏abcd ea
a
✏abcd e
✏abcd ea
n Un (e, f )
^ eb ^ ec ^ f d
b
c
d
^e ^f ^f
^ fb ^ fc ^ fd
Vainshtein Mechanism
Non-linear interactions screen the helicity-0 mode
Hµ⌫
gµ⌫ = ⌘µ⌫ +
MPl
1
Hµ⌫ = hµ⌫ + 2 @µ @⌫ ⇡
m
Screening when
2
@ ⇡
2
m MPl
Boulware-Deser ghost
Screening when
2
@ ⇡
2
m MPl
2
U (H) ! U(@ ⇡)
Generically, 2nd derivative operators in the action will lead to EOMs that are higher
order. This leads to a ghost, with mass
2
mghost
MPl m4
⇠
@2⇡
The mass of the ghost becomes small in the regime where the
Vainshtein mechanism is active.
Wilson Lines
Can discretize in a gauge with the shifts, just get
the Stuckleberg fields
Nµ 6= 0
WIJ
J
I
LDy f ! m (WIJ fJ
fI )
Symmetric Vielbein Condition
Metric and Vielbein formalisms equivalent only when
a,µ b
e(1) e(2),µ
b,µ a
e(1) e(2),µ
=0
In Deconstruction this condition emerges in a natural way as a gauge choice
ab
⌦y
1 a,µ
b
=
e @ y eµ
2
e
µb
a
@ y eµ
=0
Origin of Low Strong Coupling Scale
[K 2 ]
2
m
[K]2 =
[(@y h)2 ]
4
hµ⌫ ! hµ⌫ + 2@µ @⌫ ⇡
Canonically Normalize
[@y h]2 + O(MPl1 )
Kinetic term for π
(@y2 @µ ⇡)@ µ ⇡
⇡
⇡!
@y
Leads to low strong coupling scale since the y derivatives can be small
Scale of interactions
⇤ = (m2 MPl )1/3
LGR,5d
Discretized Version
X
I
Z
Diagonalize mass matrix (fourier transform)
(@ 2 ⇡I )2
N
hI+1
⇠ 3
3
⇤
⇤
Canonically Normalize
1
h̃n ! p h̃n
N
1 N
⇡
˜n ! p
⇡
˜n
N n
dy @y h @ 2 ⇡
X
h̃n1 @ 2 ⇡
˜ n2 @ 2 ⇡
˜ n3
n1 ,n2 ,n3
Identify the strong coupling scale
1
⇤s ⇠ p ⇤
N