Higuchi Bound in Massive
Gravity and Bigravity
Andrew J Tolley
Case Western Reserve University
Based on work with Matteo Fasiello
1206.3852 (Massive Gravity)
1207.???? (Bigravity)
Overview
•
Why Massive Gravity?
•
Problems with FRW
•
FRW on de Sitter Massive gravity/bigravity
•
Higuchi and Vainshtein
•
Resolution - Inhomogenities? Bigravity
Why Massive Gravity?
Massive Gravity Theories are a remarkably a
constrained modification of general relativity
at large distance scales - graviton is assumed to
acquire a mass
In present talk I shall only be concerned with
models where this occurs without breaking
Lorentz or de Sitter symmetries
They are interesting in that as in GR, there are
a finite number of consistent allowed terms in
the Lagrangian that do not give rise to ghosts
By Massive Gravity we mean a nonlinear completion of Fierz-Pauli
coupled to matter
Markus Fierz and Wolfgang Pauli,
1939
hµ + · · · = m (hµ
2
1
µ
5 = 2s + 1
Fierz-Pauli mass term
guarantees 5 rather than 6
propagating degrees of
freedom
Massless spin-two in Minkowski makes sense!
h)
Why Massive Gravity?
Adding a mass to gravity weakens the strength
e mr
VY ukawa ⇠
of gravity at large (cosmological) distances
r
But thats not all!
Screening mechanism
Self-acceleration?
Degravitation mechanism?
Why Massive Gravity?
Self-acceleration?
Gravitons can condense to form a condensate
whose energy density sources self-acceleration
⇢matter ⇠ 0
H ⇠ m 6= 0
Analogous to well-known mechanism in DvaliGabadadze-Porrati model (DGP), however here it
seems possible to remove the DGP ghost??
Deffayet 2000
Koyama 2005
Charmousis 2006
Why Massive Gravity?
Gravitons can condense to form a condensate whose
energy density compensates the cosmological constant
Screening mechanism - The Cosmological Constant can be
LARGE with the cosmic acceleration SMALL
In a Massive Theory - the c.c. is a `redundant’
operator
Why Massive Gravity?
mass term
Gµ⌫
@L
M
2
+m
=
@gµ⌫
⇤gµ⌫
Graviton condensate:
Spacetime is Minkowski in presence of an arbitrary large
gµ⌫ =
✓
1+f
✓
⇤
m2
◆◆
⌘µ⌫
Gµ⌫ = 0
2 @LM
m
@gµ⌫
=
⇤
⇤gµ⌫
Equivalent Statement: The cosmological constant can be reabsorbed into a
redefinition of the metric and coupling constants - and is hence a
redundant operator
Why Massive Gravity?
Screening
Degravitation
One strong motivation for considering Massive Gravity is as a
toy model of higher dimensional gravity models (eg Cascading
Gravity) that potentially exhibit degravitation
de Rham et al 2007
Degravitation = Dynamical Evolution to a
Screened Solution from generic initial
conditions
Dvali, Hofmann, Khoury 2007
so far it is safe to say that this idea has not YET been fully realized
Why Massive Gravity?
Departure from GR is governed by essentially
a single parameter - Graviton Mass
Vainshtein Screening mechanism ensures
recovery of GR in limit m ! 0
This ensures massive gravity can be easily
made to be consistent with most tests of GR
(effectively placing an upper bound on m)
without spoiling its role as an IR modification
Why Massive Gravity?
Massive Gravity is a natural Infrared Completion of
Galileon Theories
Galileon: Nicolis, Rattazzi,
Trincherini 2010
Decoupling limit of Massive Gravity on Minkowski is a
Galileon Theory
de Rham and Gabadadze 2010
Decoupling limit of Massive Gravity on de Sitter is a
Galileon Theory (with slightly different coefficients)
de Rham and Renaux-Petel 2012
The allowed Galileon Interactions are in direct
correspondence with the allowed MG interactions
Why Massive Gravity?
Massive Gravity models share many nice features in common with
extra dimensional models such as DGP and Cascading Gravity .....
e.g. Vainshtein mechanism, Galileon limit, self-acceleration,
possible screening
.... however without the difficulty of having to solve fundamentally
higher dimensional equations
nomials The
of the
eigenvalues
ofpotential
the tensorU that has no ghosts [20] is bu
most
general
generalization
of
the
theory
proposed
in
[20].
The
Lagrangia
!
"
#
!
2
(4)
2 the µeigenvalues
polynomials
of
of
the
tensor
(4)
µ
µα
M
− gravity
g
R
+ 2mmatter
U(g,
)plus
LaM
.− Gravity
nstein
with
a scalar fun
Ghost-free
Massive
g fανthat is(2.1)
Kν f(g,
f+
)=
δνpotential
Pl
!
sl U that has no!ghosts "[20] is build Kout
µ
µ
µα f
of
characteristic
g
(g,
f
)
=
δ
−
#
αν
ν
ν
2
(4)
2
hat
(4)g
L
=
M
−
R + 2m U(g, f ) + LM .
Pl
ues of the tensor
U(g, H) = U2 + α3 U3 + α4 U4 ,
so that
!
ral
potential
U
that
has
no
ghosts
[20]
is
build
out
of
chara
µ
µ
µα
re the
αn f
are
U(g, H) = U2 + α3 U
(2.2)
Kν (g,
) =free
δν parameters,
− g fαν and
3 + α4 U4 ,
the eigenvalues "of the tensor
#
2
2
where
αn are
free
and
U2 the
= [K]
− [K
] ,parameters,
!
"µ 3
#
µ2
µα
3
"
#
g
f
f
)
=
δ
−
αν
[K]
−
3[K][K
]
+
2[K
]
,
ν3(g,
ν
U(g, H) =UU32=+K
α
U
+
α
U
,
(2.3)
2
2
3
4
4
− [K3 ] ,
" 4 U2 =2 [K]
#
2
2
2
4
" +
U4 = [K] − 6[K ][K]
8[K ][K] +
3[K ] 3−#6[K ] ,
3
2
meters, and
U3 = [K] − 3[K][K ] + 2[K ] ,
" 4 PRL,2 106,2 2311013 (2011)
de
Rham,
Gabadadze,
Tolley,
2 Th
2
#
re [. . .]2 represents
the H)
trace
of
a
tensor
with
respect
to
the
metric
g
.
U(g,
=
U
+
α
U
+
α
U
,
U4 =2 [K]3 −3 6[K ][K]
µν ] −
4 4 + 8[K ][K] + 3[K
−
[K
]
,
(2.4)
Proven
fully
ghost
free
in
ADM
formalism:
Hassan
and
Rosen
host for this theory for# a Minkowski background metric was shown in the d
2
3
2011 ofthe
re
free
parameters,
+ 2[K
,and
[. fully
. .] ]represents
the beyond
trace
a tensor
with (2.5)
respect
to the
t−in3[K][K
[17, where
18,] 20],
non-linearly
decoupling
limit
in [21,
22
#
Result reconfirmed
in
Stueckelberg
decomposition:
Result
reconfirmed
in helicity decomposition:
2 of
2
3this
2for
2 a Minkowski
4 [23,
n−the
Stückelberg
and
helicity
languages
in
24].
"
#
ghost
for
theory
background
metric
wa
6[K de
][K]
+
8[K
][K]
+
3[K
]
−
6[K
]
,
(2.6)
2 Gabadadze,
2 Tolley 2011
Rham,
de Rham, Gabadadze, Tolley 2011
U
=
[K]
−
[K
]
,
2 Hassan,
Varying
with
respect
to Strauss
the metric
gµν we find the
equations
of motion
limit
in
[17,
18,
20],
fully
beyond
the
decoupling
Schmidt-May,
von
2012 non-linearly
" 3 Kluson 2012 2
#
3 metric g . The absence
ace
of
a
tensor
with
respect
to
the
in the
Stückelberg
and2]helicity
languages
in [23, 24].
U3 = as[K]
− 3[K][K
] + 2[K
,
µν
−2
G
+
m
X
=
M
T
"
#
Now
several
other
proofs:
Mehrdad
Mirbaryi
2011,
AJT
to
appear
µν
µν
µν
pl
Minkowski background
metric
was
in the
4Varying2with
2 respect
3toshown
2 2 gdecoupling
4find the equa
the
metric
we
µν
U = [K] − 6[K ][K] + 8[K ][K] + 3[K ] − 6[K ] ,
dRGT model: allowed mass terms
de Rham, Gabadadze, Tolley 2011
Build out of
unique
combination
Mass terms are
characteristic
polynomials
K
µ
=
gµ f
µ
U (g, f ) =
i Ui (K)
i
n=d
det(
µ
+ K
µ
)=
n
Un (K)
n=0
Finite number of allowed
interactions in any dimension
Interactions protected by a
Nonrenormalization theorem
Generalized to arbitrary (dynamical - bigravity)
reference metrics by Hassan, Rosen 2011
3]. Away from the decoupling limit this is related to the constraint that was
en we
see
the
constraint
for
the
FRW
metric
to
all
orders,
by
Refs. [1, 4, 5]. Here we see the constraint for the FRW metric to all orde
4)
w.r.t.
f : of (14) w.r.t. f :
taking
variation
A No-Go
2
3
2
2
3
2
m
∂
(a
−
a
) = 0 . Gravity in Minkowski)
m ∂0 (a
− a(dRGT
) = 0model
. 0 - Massive
(15)
The simplest
model
not support
spatially
flat (or
closed)
solutions
Thisdoes
constraint
makes time
evolution
of the
scale cosmological
factor impossible.
As we have
time evolution
of
the
scale
factor
impossible.
As
we
have
noted
3
4
whichthe
areKFRW
meaning
homogeneous
and isotropic
above, keeping
and
K
terms
in
(12)
can
only
modify
the polynomial fu
4
and
Kwhich
terms
in
(12)
can
only
modify
the
polynomial
function
of
a on
∂
acts
in
(15).
Therefore,
there
are
no
nontrivial
0
where
overdot denotes the time derivative ∂homogeneou
0 . We em
in
(15).
Therefore,
no
homogeneous
Argument
isinsimple:
as inare
GR
wenontrivial
have
equation
sotropic
solutions
thethere
theory
of
massive
GR,Friedman
defined
by
(12). andand
appears in the Lagrangian only linearly. The same rem
Ittheory
is also instructive
to show
absence
FRW 1st
solutions
the unitary
equation
- thethe
2nd
follows
from
by diffininvariance
theRaychaudhuri
of massive
GR,
defined
byof(12).
α4no- fitfield
is just
theinspecial
structure
of theInterm
a α3a and
µ
for
which
φ
=
δ
x
,
and
appears
the
action
to
begin
with.
this
µ
e to show the
absence
of
FRW
solutions
in
the
unitary
gauge,
˙
that
ensures
that
f
enters
only
linearly!
This
is
a
conse
theBut
most
general
homogeneous
and
isotropic
ansatz
involves
the
lapse
function
in MG diff invariance is broken and so 2nd does not follow from
nd no f fielddecoupling
appears inlimit
the action
to
begin
with.
In
this
gauge,
the
equations
of
motion
of
this
theory
1st - consistencyds
of2 two
imposes
condition
on
scale
factor
2
2
2
2
=
−N
(t)dt
+
a
(t)d%
x
,
ogeneous and
isotropic
ansatz
involves
the
lapse
function
N(t),
derivatives acting on the helicity-0 field in particular
[3].
from
and the2 Lagrangian
(12)2 with
= αdecoupling
2 Away
2α3the
24 = 0 reads limit this is related to t
ds = −N (t)dt + a (t)d%x ,
(16)
! [1,2 4, 5]. Here we see the constraint" for the
in Refs.
aȧ
2
2 3
2
2
3
2
L
=
3M
−
−
m
(a
−
a
)
+
m
N(2a
−
3a
+
a)
.
taking
variation
of
(14)
w.r.t.
f
:
Pl
12) with α3 = α4 = 0Nreads
"
3
2
2
m
∂
(a
−
a
)
=
0
.
As a
can
verified,
the
condition
(15)
in
this
case
arises
D’Amico
et
al
2011
0
ȧ be straightforwardly
2 3
2
2
3
2
−
− mof (a
− a ) +ofmtheN(2a
− 3a
+ a) for
. the two fields,
(17)a a
requirement
consistency
equations
of motion
!
2
eeping the K3 and K4 terms in (12) can only modify the polynom
which ∂0 acts in (15). Therefore, there are no nontrivial homo
c solutions in the theory of massive GR, defined by (12).
also instructive to show the absence of FRW solutions in the un
a
a µ
h φIt
=
δ
x
,
and
no
f
field
appears
in
the
action
to
begin
with.
I
µ
is possible to find exact solutions in which the metric takes
t general homogeneous andthe
isotropic
ansatz
involves
the
lapse
fu
form ...
A No-Go?
2
2
2
2
2
ds = −N (t)dt + a (t)d%x ,
Lagrangianin(12)
with
0 reads type equation
which
a(t)αsatisfies
a Friedman
3 = α4 =
!
"
D’Amico
et
al
2011
2
a
ȧ
2
2 3
2
2
3
2
Volkov
2011
L = 3MPl −
− m (a − a ) + m
N(2a
−
3a
+
a)
.
Koyama et al 2011
N
Gratia et al 2012
Kobayashi
2012case a
be straightforwardly verified, the condition
(15)etinalthis
But
is achievedofbythe
introducing
fields
carry
ment
ofthis
consistency
equationsStuckelberg
of motion
for which
the two
fiel
the specifically,
inhomogeneities
meaning
that(15)
these
are difference
not truly
More
one can
obtain
bysolutions
taking the
i.e. the for
perturbations
inhomogeneous
ivative of FRW,
the e.o.m.
N and thearee.o.m.
for a. Technicall
Two paths
Accept inhomogeneities:
D’Amico, de Rham, Dubovsky, Gabadadze, Pirtskhalava, Tolley
`Massive Cosmologies’ 2011
Not as bad as it sounds! Vainshtein
mechanism should guarantee
inhomogeneities unobservable before late
times
Inhomogenities only appear on scale set by inverse graviton mass
Two paths
Or modify assumptions to allow FRW:
Open Universe solutions: Gumrukcuogli et al 2011
Anisotropic solutions: Gumrukcuogli et al 2012
Felice et al 2012
* Make reference metric de Sitter - AJT and Fasiello - 1206.3852
(for decoupling limit see de Rham, Renaux-Petel 2012)
* Make reference metric dynamical - Bigravity/Bimetric
von Strauss et al 2011
Comelli et al 2011
Crisostomi et al 2012
de Sitter MG and bigravity
Qualitatively for the present discussion there is no great
distinction between bigravity and de Sitter Massive gravity
This is because the second metric may not directly couple to
our observable matter (absence of ghosts) other than having
its own cosmological constant
Thus for suitably low energies bigravity looks like
MG on de Sitter (or Minkowski/AdS)
However, we will see later that quantitatively there is a
different for the Higuchi bound
Crux of problem
Although we can obtain FRW like solutions, number of
issues ...
The `mass’ of a graviton gets dressed by the background
Generically the mass grows with increasing H
Thus the Vainshtein mechanism is more subtle!! We must
send m ! 0 in away that compensates growth with H
Generalized Higuchi bound implies
2
mdressed (H)
> 2H
Successful Vainshtein mechanism (recovery of GR at large H) and
Higuchi bound are incompatible for FRW solutions
Tolley and Fasiello (to appear tomorrow)
2
Generalized Higuchi bound
Fasiello and AJT - 1206.3852
Previous Work:
2
Higuchi 1987, Deser and Waldron 2001 (de Sitter) m
Grisa, Sorbo 2009 Generalized to FRW
Berkhahn et al 2010 (Similar results to above)
Grisa and Sorbo obtain:
m2 > 2(H 2 + Ḣ)
seemingly no problem in deccelerating universe ?!?!
However! These authors assumed the equivalence of the
background FRW metric and reference metric - this is
inconsistent with known behaviour of dRGT and de Sitter/
Bigravity generalization
Necessary to use correct nonlinear theory to obtain result!
2H
2
dRGT
trics Massive Gravity
!
" (4)
#
2
2
(4)g
L
=
M
−
Rreference
+reference
2m U(g,
f )fµν +
L
.
heory
of massive
gravity
defined
on an arbitrary
metric
[53]
isMjust
Pl
avity
defined
on
an
arbitrary
metric
f
is
just
a
µν
fghtforward
massive
gravity
defined
on
an
arbitrary
reference
metric
fµν
generalization of the theory proposed in [20]. The Lagrangian takes
ation
of potential
thegravity
theory
proposed
in
Lagrangian
takes
general
Uof
that
has
no [20].
ghosts
is function
build
ofLagrang
charact
m ofgeneralization
Einstein
with
matter
a potential
thatThe
is [20]
a scalar
d
theplus
theory
proposed
in
[20]. ofout
The
o metrics
with
matter
plus
a
potential
that
is
a
scalar
function
of
sity
of
the
eigenvalues
of
the
tensor
2 !
" (4)
# a potential that is a scalar fu
MPl with
insteinStarting
gravity
matter
plus
2
(4)
point
L=
− g
R + 2m U(g, f ) + LM .
(2.1)
2
!
!
"
#
cs
µ 2 "
µ
2general potential
(4)U that
!
µα
(4)
#
ost
has
no
ghosts
is
build
out
of
characteristic
polyno- (2.1)
g
f
K
(g,
f
)
=
δ
−
MPl − g
R
+
2m
U(g,
f
)
+
L
.
αν
ν
ν
M
2
(4)
2
(4)
Sketch of argument
of the eigenvalues
L = ofMthePltensor
−
g
R + 2m U(g, f ) + LM .
!
ial U that has no
µ ghosts
µ [20]µαis build out of characteristic
(2.2)
Kν (g, f ) = δν − g fαν ,
neral
potential
U that has no ghosts [20] is build out of char
alues of
the tensor
U(g, H) = U2 + α3 U3 + α4 U4 ,
tf the eigenvalues of the tensor
!
U(g,
H)
=
U2 + α3 U3 + α4 U4 ,
(2.3)
µ
µ
µα
αn areKνfree
parameters,
and
g
f
(2.2)
(g,
f
)
=
δ
−
!
αν
ν
de Sitter spacetime metric
the αn are free parameters, and
µ
µ fµ⌫
µα
" 2 Kν 2(g,
# f ) = δν − g fαν
# ] ,
2 − [K
2
UU22 =
= 1 "[K]
[K] − [K ] ,
(2.4)
"
#
2!
3
2 #
3
"[K]
1=
U(g,
U
+
α
U
+
α
U
,
(2.3)
UU3 H)
=
−
3[K][K
]
+
2[K
]
,
3
32
2 3
34 4
(2.5) #
3 = " [K] − 3[K][K ] + 2[K ] ,
3!
4 U(g, H)
2 =2U + α3 U + α U2 ,2
4
"
#
U
=
[K]
−
6[K
][K]
+
8[K
][K]
+
3[K
]
−
6[K
] ,
2
3
3
4
4
1
4 and
ameters,
4
2
2
3
2 2
4
U4 =
[K] − 6[K ][K] + 8[K ][K] + 3[K ] − 6[K ] ,
(2.6)
4!
#parameters, and
are
free
2
2
represents
the trace of a tensor with respect to the metric(2.4)
gµν . The ab
−
[K
]
,
For experts U1 is removed
by tadpole condition and U0 is a c.c.
#
"
#
this
theory
for
a
Minkowski
background
metric
was
shown
in
the
deco
3
2
3
2
2
which
can
be
absorbed
–, 3 – into definition of matter
−U3[K][K
]
+
2[K
]
,
(2.5)
=
[K]
−
[K
]
2
, 18, 20], fully non-linearly beyond the decoupling
limit in [21, 22], a
#
4 consider FRW solut
Let us
de
Sitter.
For
simplicity
I willasassume k
=Sitter.
0. We
denote
theI d
de
For
simplicity
he
a
scale
factor
is
properly
calculated
e Gravity
"
!#
on
✓ ◆
2
2 2
2
2
example in two dimensions
Deriving Friedman equation
˙0 1 d reference
d ȧon an arbitrary
ḃ
ds
= fµνNis just
dt +
x
ravity 1defined
metric
a a(t) d~
=
(1.5)
˙0
ation of
the theory
proposed
The Lagrangian takes
N dt
N
N ˙0 dt in [20].
and
theofreference metric as
W and
solutions
in reference
massive
on a fixed
FRW
such as
vity
withthe
matter
plusgravity
a potential
that
is a geometry
scalar function
metric
as
Nice
approach
is
with
Stuckelberg
fields
icity
I
will
assume
k
=
0.
We
denote
the
dynamical
metric
as
brackets
is
the
acceleration
of
the
reference
metric
which for de
!
"
#
2
(4)
2
2 (2.1)
be
positive.
Thus
if
the
dynamics
the
a
MPl
− (4)
g
R
+
2m
U(g,
L
.
2metric switched
2 from
2
0
2
2
2 2
2 f)
2 + of
M
0
˙
ds = N dt + a(t) d~x
(1.1)
ds
=
dt
+
b
(
)d~
x
0
˙
ration, this must be accompanied by a change of sign of
which
ial U that hasthere
no ghosts
[20] is build
out
ofwecharacteristic
0
is
an
ambiguity
in
how
take
the
square
root.
Fortuna
where
is the
time-Stuce
etric
as the lapse of the reference metric must vanish.
galues
that
0
0
of
the
tensor
p
0
H
b
µ
µ
isisresolved
in
theegvierbein
formulation
of
massive
gravity,
which
wherebecause
the
time-Stuceklberg
field.
If
the
fixed
metric
is
µ!
b(
)
=
e
with
H
cons
in
de
Sitter
problem
when
we
evaluate
the
tensor
K
=
g
⌘
2
b
!⌫
⌫
⌫
2
2! 2 0
2
0
˙
0
p
ds
=
dt
+
b
(
)d~
x
(1.2)
0µ
µH
Stückelberg
fields.
This
tells
us
that
the
correct
way to takethat
the squ
b
µ!
µ
µα
b( Kν)(g,=Q
e
with
H
constant.
o evaluate
=
g
⌘
.
Now
since
The
constraint
co
!⌫
b
(2.2)
f )⌫ = δν − g fαν
˙ is
of
the
sign
of
!
e-Stuceklberg
If the
fixedthat
metric
is de Sitter
thenthe
we
have
a
branch
(the
normal
bran
! comes
Thefield.
constraint
from
variation
of
the
actio
2
˙
0
˙0
p
0j
Hb constant. 1
0
1
N
j
2
N
g
f
=
0)
a
branch
(the
normal
branch)
of
solutions
for
which
g
f
=
(1.6)
0 has
˙
0
2
b(
that
comes
from
the
variation
of
the
action
with
respect
to
b(
)
U(g, H) = U2 + α30U3 + α4 U4 ,
(2.3)
0i a ij
i
ij
2
a
al branch) of solutions for which
ameters, and With this choice the action for FRWaH
solutions
of
massive
gravity
i
=
bH
b
or (1.3)
aH
=
bH
b
We2 #mustsince
choose
sign of square
root to correlate
it is constructed
out of characteristic
polynomials of K it is li
2
] − [K ] ,
(2.4)
0
˙
˙
0
problem
in having
pass though zero. Implicity what we
#1 arises
with
sign
of
–
–
3 or
2
3
p ] ḃ,
p
] − 3[K][K ] +a 2[K
(2.5)
0
˙
˙0 <
ȧ
+ =for the zeroth eigenvalue
for
>
0
and
the
for
#
(1.4) ḃ
ȧ
4
2
2
3 ˙0
2 2
4
N ][K] +this
] − 6[K ][K] formulation
+ 8[K
3[K choice
] − 6[K
] , have The
(2.6)
acceleration
of now
the aam
s
would
been=
automatic and
0
˙
N
arisen.
the a scale factor is properly calculated as
Deriving Friedman equation
Since mass terms are characteristic
polynomials of K - linear in ˙0
⇣
Lmass = a3 N A(
0
, a) + ˙0 B(
0
, a)
⌘
It is clear than we can integrate by parts to
remove ˙0 dependence to give a nondynamical equation for 0 in terms of a and H
Constraint equation
Consistency of Friedman and Raychauduri equation
(or equation for zero Stuckelberg field) impies
1 + 2(1 + ↵3 ) + (↵3 + ↵4 )
2
✓
b
a
H
H0
◆
Normal branch of solutions is b
b
=
a
H
=
a
H0
1
Equation fixes dynamics of
Stuckelberg field
Perturbations subtlety
If metric transits from acceleration to
decceleration we need ˙0 to change sign
At thisp
point one of the eigenvalues of
g 1 f vanishes
How do we define this perturbatively?
Vierbein formulation
a
The vierbein formulation is analytic in the
Mass term is
Det
⇥
a
eµ
+
a b
⇤b f c @ µ
reference
vierbein
⇤
c
As long as it is possible to solve the equation for the
a
T
⇤
⇤⌘⇤ = ⌘
Lorentz Stuckelberg fields b
e
µa
b c
⇤c f d @ µ d
e
µb
b c
⇤c fd @ µ d
=0
6 equations for 6 unknown Lorentz transformations
Even when
˙0 = 0 we can solve for
a
⇤b
= ...@
c
H is the Hubble rate of the dynamical metric and H0 that of the refer
cal) metric, and α3 and α4 are the two free parameters in the dR
his should be compared with the associated Friedmann equation
Friedman equation
2
m
1
2 H
2
ρ − (6 + 4α3 + α4 )
+ (3 + 3α3 + α4 )m
−
H =
2
3MPl
3
H0
2
2
3
m H
2H
(1 + 2α3 + α4 )m 2 + (α3 + α4 )
.
3
H0
3 H0
H
=
H0
1
⇢dark energy
–2–
H0 is Hubble parameter of reference metric
(e.g. dRGT has 2 free parameters) and even if we employ at full the freedom
reference metric, once observations are taken into account the bound remain
stringent. We hint to a possible resolution in the Conclusions. Our final bound
by
!
"
2
H
H
2
2
2 H
(3 + 3α3 + α4 ) − 2(1 + 2α3 + α4 )
+ (α3 + α4 ) 2 ≥ 2
m̃ (H)
(H) = m
mdressed
H0
H0
H0
Dressed Mass and Higuchi
where H is the Hubble rate of the dynamical metric and H0 that of the referenc
2 parameters in the
2 dRGT
dynamical) metric,
and αbound
are the twomfree
3 and α4 is
Generalized
Higuchi
(H)
>
2H
dressed
[20]. This should be compared with the associated
Friedmann equation
H
=
H0
arises
2
m
1
2 H
21
ρ − (6 + 4α3 + α4 )
+ (3 + 3α3 + α4 )m
−
H =
2
3MPl
3
H0
2
2
3
H
m
H
from coefficient
of
kinetic
term
for
helicity
zero
mode
2
(1 + 2α3 + α4 )m 2 + (α3 + α4 )
.
3
H0
3 H0
Lhelicity zero /
2
2
mdressed (mdressed
–2–
2
2H )(@⇡)
This is a similar polynomial to what arises in the
Friedman equation
2
Partially Massless Gravity
Coefficient of kinetic term in general is
proportional
to
✓
m2dressed
H
2H = m
H0
2
2
(3 + 3↵3 + ↵4 )
H
H
2(1 + 2↵3 + ↵4 )
+ (↵3 + ↵4 ) 2
H0
H0
If we make the special choice
↵3 =
3/2
↵4 = 3/2
2
mdressed
and so if we choose
2
m =
2
2H0
2
◆
2H 2
Precisely Claudia’s
values!! (N.B. my conventions are different)
de Rham and Renaux-Petel 2012
2
H
2
2
2H = 2 (m
H0
2
2H0 )
m2dressed = 2H 2
Kinetic term vanishes regardless of matter source!!!
Higuchi versus Vainshtein
2
mdressed (H)
2
= m (1 + ) (1
2
⇢dark energy = 3m (
Higuchi
2
(2 + ↵3 (
2
) + m ↵3 (3
2
2)
3
H
=
H0
↵4 ))
) + ↵4 m
2 3
Vainshtein
d
d
2
2
2
⇢
⌧
H
dark energy
mdressed (H) > 2H
dt
dt
2
m 3
2
mdressed ⇠ ⇢dark energy ⇠ 3 H
If ↵3 + ↵4 6= 0
H0
2
m
2
2
↵3 + ↵4 = 0 (generic) mdressed ⇠ ⇢dark energy ⇠ 2 H
H0
2
m
↵4 = ↵3 = 1
m2dressed ⇠ ⇢dark energy ⇠
H
H0
1
an FRW reference metric, which has revealed several, interesting things. First of all
once stability and observations are taken into account, the Higuchi bound is still quite
stringent on the value of m, the parameter deforming GR. In summary the bound
states that
!
"
2
H
H
2
2 H
(3 + 3α3 + α4 ) − 2(1 + 2α3 + α4 )
+ (α3 + α4 ) 2 ≥ 2H 2 .
m̃ (H) = m
H0
H0
H0
Higuchi versus Vainshtein
where H is the Hubble rate of the dynamical metric and H0 that of the reference
metric.
Remarkably Ḣ drops our of generalized bound!!!!
This bound differs significantly from previous conditions, in particular [39] (see
also [41–43]) in that it is independent of Ḣ and the precise form of matter. Indeed
A direct consequence of the ghost-free form
This is a different, better theory of MG, where the freedom on fµν is also fully employed. W
onlybranch
firstwhich
derivatives
coefficient
have(action
indeed seenexpressible
how there is now with
an additional
does relax the- Higuchi
bound. of
helicity zero mode kinetic term is just a function of first
derivatives of metric in Stueckelberg analysis)
10
– 17 –
so the window found by Grisa and Sorbo for deccelerating
solutions m2 > 2(H 2 + Ḣ) is not present
an FRW reference metric, which has revealed several, interesting things. First of all
once stability and observations are taken into account, the Higuchi bound is still quite
stringent on the value of m, the parameter deforming GR. In summary the bound
states that
!
"
2
H
H
2
2 H
(3 + 3α3 + α4 ) − 2(1 + 2α3 + α4 )
+ (α3 + α4 ) 2 ≥ 2H 2 .
m̃ (H) = m
H0
H0
H0
Higuchi versus Vainshtein
where H is the Hubble rate of the dynamical metric and H0 that of the reference
the
qualitative
form
of
these
results
goes
through
in
metric.
H0 conditions,
case
ofsignificantly
bigravity from
where
is dynamical
- but [39] (see
This the
bound
differs
previous
in particular
also [41–43]) in that it is independent
Ḣ and
the precise form of matter. Indeed
with a oftwist
(later)
10
This is a different, better theory of MG, where the freedom on fµν is also fully employed. W
have indeed seen how there is now an additional branch which does relax the Higuchi bound.
Their is no regime for the de Sitter MG spatially flat
cosmologies which is simultaneously
obervationally
– 17 –
acceptable and ghost-free as long as the helicity zero
mode is present
Partia$y massless case is not included in this statement
Resolution?
One resolution to realise something like out universe in Massive
Gravity models is to return to the inhomogenous solutions
D’Amico et al 2011
Higuchi constraint is implied by representation theory of de Sitter
group. Introducing inhomogenity in the metric breaks this relation
Known exact solutions are self-accelerating type and sit in different
branches than the generic solution - as yet the general solution - the
one with all 5 degrees of freedom propagating which is continuously
connected with the normal Minkowski vacuum is not known
tion, it leads to the usual ΛCDM - like cosmological expansion of the background
in the sub-horizon approximation used here. This is clear form the fact that the
stress-tensor (24) gives rise to a de Sitter background as was shown in the previous
subsection. Hence, in the comoving coordinate system – which differs from the one
used above – the invariant de Sitter space will be the self-accelerating solution.
All this can be reiterated by performing an explicit coordinate transformation
see the
presence
of will
thebe
FRW
toWe
the can
comoving
coordinates.
This
done solutions
in two steps.inInthe
thefamous
so-called Fermi
3
2
M
!
1
⇤3be=locally
m Mwritten
normal
coordinates,
space and for all
decoupling
limitthe FRW
P metric can
P heldin fixed
times, as a small perturbation over Minkowski space-time:
de
Rham
et
al
2010
!
"
#
$ µ ν
1 2 2
2
2
2
2
2
FRW
ds = −[1 − (Ḣ + H )x ]dt + 1 − H x dx = ηµν + hµν
dx dx , (25)
2
Reasons to be hopeful?
where
the
corrections
to theform
abovefor
expression
are suppressed
by higher
powers
of
The
generic
solution
the
helicity
zero
mode
near
x=0
H 2 x2 . The Fermi normal coordinates, on the other hand, are related to those used in
which
is isisotropic
in this deformation
limit is of Minkowski space(14) (in which the FRW
metric
a small conformal
2
time), by an infinitesimal
gauge
transformation
[15]. The latter does not change the
⇡ ⇠ A(t)
+
B(t)x
π
expression (24), since Tµν
is invariant under infinitesimal gauge transformations in
the decoupling
limit. fix
OnA the
the Fermi
normal
coordinates
can be
Equations
of motion
and other
B - forhand,
example
for pure
cc source
B=constant
transformed into the standard comoving coordinates (tc , xc2) as follows [15]
A
=
Bt
!
"
1
1 2
x
2
tc = t − H(t)x , xc =
1 + H (t)x2 .
(26)
2
a(t)
4
direct calculations to the 6 order show that the helicity-1 fluctua
degravitating branch of positive
solutions,
and this
being
by
any
instability
kinetic
termwithout
as long as
κ/(qplagued
−
1)
>
0.
This
suggests the p
0
at least in the decoupling
limit. attractor solution for degravitation. The special case a3
late-time
separately below.
Reasons to be hopeful?
4.2
Degravitation in ghostless massive gravity
4.2.2
de Sitter branch
Forde
convenience
we start by recalling
the decoupling
limit Lagrangian
(9) coupled
Rham, Gabadadze,
Heisenberg,
Pirtzkhalava
2010 - ofdecoupling
Incloser
the presence
a cosmological
constant,
the fieldand
equations
(4
This solution
to the of
usual
GR
deonly
Sitter
configuration
only exis
to
usual
GR
de is
Sitter
configuration
and
exists
if
to the
an external
source
limit
2
admit
a second
branch
of solutions;
these
connect
with the self-acc
3this
≥ 3a1 a3 .can
The
stability
of#
solution
can by
be analyzed
as previously
by loo
ity of thisa2solution
be
analyzed
as
previously
looking
1 µν αβ
1 them
a3,
n and(n)
µν
µν as the de Sitter solutio
presented
in
section
we
refer
to
at
fluctuations
around
this
background
configuration,
L
=
−
X
[Π]
+
h
E
h
+
h
h
Tµν .
(35)
αβ
µν
this background 2configuration,
3(n−1) µν
MPl
eters for these
solutions
should satisfy
Λ3
n=1
1
3 2
2
1
q
Λ
+
φ
,
π
=
a1 +
3a
q
dS2a23qx
3 2
dS
3 dS = 0 ,
qdS Λ3 x + φ ,
(51)
π =
13 2
3 !
"
λ
2Λ
2
3
21
2
3
2
2 +
H
=dS
a q + a2 qdS + a3 qdS .
h
=
−
H
x
dS
2ηµν + χµν , 1 dS
µν
1 2 2
3MPl MPl
2
hµν = − HdS x ηµν + χµν ,
(52)
Tµν = −ληµν + τµν .
2
background
plus
15
(53)
To second order in fluctuations, the resulting
action is thensplit
of the form
perturbations
Tµν = −ληµν + τµν .
tuations, the resulting action
is then6H
of2the
form
1
M
(2)
µν αβ
dS Pl
αβ
ν χαβ
L
+
= − χ Eµν χαβ +
2
2
6HdS
MPl
(a2 +
3
It is
Λ3interesting
3a3 qdS )φ!φ +
3
Λ
13
1 µν
(a2 + 3a3 qdS )φ!φ +
χ τµν .
MPl
χµν τµν .
(54)
to point out again
that the helicity-0 fluctuation φ then decou
MPl
coefficient
helicity
zero order
simple
from matterof
sources
at quadratic
(however the coupling reappears at the c
nt out function
again
thatStability
the helicity-0
φ then
decouples
order).
is therefore
ensured
if the parameters satisfy o
↵of3 this↵solution
of
4fluctuation
˜
hµν
Tµν
1 2 2
= − HdS x ηµν + χµν ,
2
= −ληµν + τµν .
(52)
(53)
Reasons to be hopeful?
To second order in fluctuations, the resulting action is then of the form
L(2)
2
1 µν
1 µν αβ
6HdS
MPl
= − χ Eµν χαβ +
(a2 + 3a3 qdS )φ!φ +
χ τµν .
3
2
Λ3
MPl
(54)
ItDecoupling
is interesting tolimit
point implies
out again that
the helicity-0
fluctuation φ then decouples
existence
of inhomogenous
from matter sources at quadratic order (however the coupling reappears at the cubic
cosmological
solutions
for
massive
gravity
in
Minkowski
order). Stability of this solution is therefore ensured if the parameters satisfy one of
(dRGT)
for suitable
of parameters
of free
the
followingwhich
three constrains,
(settingrange
a1 = −1/2
and λ̃ > 0)
from Higuchi bound
a2 < 0 and
Remarkable
or
1 − 3a2 λ̃ − (1 − 2a2 λ̃)3/2
2a22
≤ a3 <
−
,
2
3
3λ̃
(55)
helicity zero does not couple to matter perts no vDVZ discontinuity
1 − 3a λ̃ + (1 − 2a λ̃)3/2
1
a2 <
Absence
or
2λ̃
and a3 >
2
2
3λ̃2
,
of Higuchi bound frees up possibility for
background Vainshtein1effect - consistency
with known
2 2
a2 ≥
and a3 > − a2 .
3
cosmology
2λ̃
(56)
(57)
ce metric, once observations are taken into account the bound rem
nt. We hint to a possible resolution in the Conclusions. Our final bo
Or .... Bigravity
!
"
2
H
H
2 H
+ 3αmetrics
2(1
+
2α
+
α
)
+
(α
+
α
)
H) Now
= m make(3both
3 + α4 ) −
3
4
3
4
dynamical,
H0
H0
H02
meaning add EH term for f metric
2p
2p
M
H is the Hubble
rate
of
the
dynamical
metric
and
H
that
of
the
refer
MP
0
f
2
L
=
g
R(g)
+
2m
U
(g,
f
)
+
f
R(f
)
ical) metric, 2and α3 and α4 are the two free parameters
in the dR
2
hisFriedman
should beunchanged
compared with the associated Friedmann equation
2
m
1
2 H
ρ − (6 + 4α3 + α4 )
+ (3 + 3α3 + α4 )m
−
H =
2
3MPl
3
H0
2
2
3
H
m
H
(1 + 2α3 + α4 )m2 2 + (α3 + α4 )
.
3
H0
3 H0
2
H
=
H0
1
we still obtain
b
H
=
a
H0
s the Hubble rate of the dynamical metric and H0 that of the referenc
) metric, and α3 and α4 are the two free parameters in the dRGT
Bigravity
HiguchiFriedmann
bound equation
should be compared
with the- associated
2
m
1
2 H
2
ρ − (6 + 4α3 + α4 )
+ (3 + 3α3 + α4 )m
−
H =
2
3MPl
3
H0
2
2
3
H
m
H
2
(1 + 2α3 + α4 )m 2 + (α3 + α4 )
.
3
H0
3 H0
b
H
=
a
H0
H
=
H0
1
–
2
–
Higuchi bound is now
2
mdressed (H)
2
H +
2
2
H 0 MP
Mf2
!
2H
4
Bigravity - Higuchi bound
Higuchi bound is now
2
mdressed (H)
suppose
2
H +
2
2
H 0 MP
Mf2
!
2H
4
H ⌧ H0
then the f-metric Friedman equation gives
suppose 3H02 Mf2 ⇡ (3 + 3↵3 +
2
mdressed
3
2
2 H0
↵4 )m MP 3
H
2
⇡ (3 + 3↵3 + ↵4 )m
2
p
H Mf
H0 ⇠ 3
mdressed MP
2
MP
H
H0
Bigravity - Higuchi bound
Higuchi bound is now
2
mdressed (H)
H0 ⇠
p
2
H +
2
2
H 0 MP
Mf2
!
2H
4
H 2 Mf
3
mdressed MP
Higuchi bound is approximately
4
4
⇠ 3H
2H
but since 3 > 2 bound is automatically satisfied!!!!!
(as long as H ⌧ H0 )
Note that in the MG decoupling limit Mf ! 1
we recover a problem
Summary
•
FRW (fully homogeneous and isotropic) solutions
are a problem in Massive Gravity
•
For Partially Massless Gravity - Higuchi bound is
automatica$y satisfied for any choice of matter
•
For Massive Gravity on a fixed reference metric,
bound is in conflict with Vainshtein mechanism
•
For Bigravity, bound is almost always satisfied
regardless of the choice of matter as long as H ⌧ H0
•
Generalized Higuchi bound is insensitive to
equation of state for matter i.e. Ḣ making it more
stringent than previously expected