Derivative couplings in massive bigravity

0.22 MB·

by  Xian Gao

Prepared for submission to JCAP
Derivative couplings in massive
bigravity
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0
2

r
a
M

3 Xian Gaoa and Lavinia Heisenbergb
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] aDepartment of Physics, Tokyo Institute of Technology,
h
2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan
t
p- bInstitute for Theoretical Studies, ETH Zurich,
e Clausiusstrasse 47, 8092 Zurich, Switzerland
h
[
E-mail: [email protected], [email protected]

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v Abstract. In this work we study the cosmological perturbations in massive bigravity in
0
the presence of non-minimal derivative couplings. For this purpose we consider a specific
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1 subclass of Horndeski scalar-tensor interactions that live on the unique composite effective
2
metric. For the viability of the model both metrics have to be dynamical. Nevertheless, the
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. number of allowed kinetic terms is crucial. We adapt to the restriction of having one single
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kinetic term. After deriving the full set of equations of motion for flat Friedmann-Lemaitre-
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6 Robertson-Walker background,westudylinearperturbationsontopofit. Weshowexplicitly
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that only four tensor, two vector and two scalar degrees of freedom propagate, one of which
:
v being the Horndeski scalar, while the Boulware-Deser ghost can be integrated out.
i
X
r
a
Contents
1 Introduction 1
2 Dynamical composite metric 3
3 Cosmological parametrization 4
4 Cosmological perturbations 7
5 Conclusion 9
A Details in quadratic action for the scalar modes 9
1 Introduction
Cosmology became an empirical scientific discipline thanks to the high precision achieved
by the cosmological observations. Owing to the increasing number of accurate cosmological
data we are in a position to test fundamental physics. Based on the two fundamental pillars,
the Cosmological Principle and General Relativity, we established the concordance standard
modelof theuniverseto beΛCDM, whichrequires theexistence of cold dark matter together
with dark energy in form of a cosmological constant. The model serves as an effective
cosmological model providing the best fit to current cosmological data [1–3]. In spite of this
excellent agreement the discovery of some reported anomalies like the lack of power in the
regionoflowmultipolesintheCosmicMicrowave Background, thehemisphericalasymmetry,
thedetection oflargescalebulkflowsandtheunexpectedlargescalecorrelationsmightsignal
the failure of our fundamental assumptions [4]. On the other hand, the observed unnatural
smallness of the cosmological constant is troublesome to reconcile with standard techniques
of quantum field theory [5]. Its instability under quantum corrections renders the model
technically unnatural.
The above mentioned observational and theoretical drawbacks has motivated attempts
to explain the late time cosmic acceleration by resorting to dynamical fields or modified
gravity. Modifications in form of an additional scalar field are the most extensively explored
ones, since the construction of interactions is less burdensome and it naturally provides
isotropic accelerated expansion [6–10]. The inclusion of non-trivial self-interactions for the
scalarfieldhasimportantimplicationsinthecosmologicalapplications[11–19]. Modifications
in form of vector fields is less explored because of the difficulty of generating large scale
anisotropic expansion. Nevertheless, the consideration of non-trivial self-interactions for the
vector field might yield interesting cosmological consequences [20–30].
Another interesting research line within modified gravity theories is massive gravity.
Since the carrier of the gravitational force would be massive in this case, the mediated force
would be Yukawa suppressed on large scales and this could yield a natural explanation of
the recent cosmological acceleration. The unique mass term at the linear level was already
constructed in the 1940’s by Fierz and Pauli [31]. This constitutes the unique linear theory
at the classical level without introducing any additional ghost degree of freedom. Even
theoretically being viable, this simple linear theory gives rise to the vDVZ discontinuity
– 1 –
[32,33],whichreflectsthefactthatthemasslesslimitofthetheoryyieldsadiscretedifference
to General Relativity. However, this discontinuity can be avoided by restoring non-linear
interactions that become appreciable on small scales to freeze out the field fluctuations [34].
Theinclusionofthesenon-linearinteractions hastobeperformedinawaythatmaintainsthe
Boulware-Deser ghost absent [35], which was finally accomplished in 2010 [36–40]. Besides
being the unique ghost-free non-linear theory of massive gravity, it is technically natural
in the sense that it is not subject to strong renormalization by quantum loops [41, 42].
The potential interactions have to be tuned in a very specific way to ensure the absence
of the Boulware-Deser ghost, however the one-loop contributions from the gravitons usually
destabilize this special structure. Notwithstanding this detuningof the potential interactions
is irrelevant below the Planck scale.
Still within the same line of quantum stability of the theory, new consistent matter
couplings have been proposed in [43–45]. In order to maintain the right structure of the
potentialinteractions atthequantumlevel, onlyveryrestricted mattercouplingsareallowed.
Inthiscontext,anewnon-minimalmattercouplingthroughaveryspecificeffectivecomposite
metric built out of the two metrics in massive (bi-)gravity was introduced. Even if this
couplingreintroducestheBoulware-Deserghost[43,46],itcanbeusedasaconsistenteffective
fieldtheory. Infactitwasarguedthatthereisonlythisuniqueeffectivemetricthatmaintains
thetheoryghostfreeuptothestrongcouplingscaleinthemetriclanguage[47,48],whichcan
be extended further in the unconstrained vielbein formulation of the theory [49]. Within the
same formulation, it was pointed out that the ghost freedom might be preserved fully non-
linearly [50]. However, that this was not the case was very soon proven in [51]. Nevertheless,
in the partially constrained vielbein formulation this can be indeed achieved at the price
of losing local Lorentz symmetry [52]. Moreover, the matter coupling through a composite
effective metric has also important implications in cosmology [43, 53–58]. Similarly, for
the dark matter phenomenology in form of dipolar dark matter it plays a crucial role [59–
61]. The cosmological implications of extensions of massive gravity by scalars were already
investigated in [62–69]. However, in these constructions the scalar field was living on the
dynamical metric. The generalization of this to the case of doubly coupled scalar field may
offer richer phenomenology and avoid some of the instabilities and fine tunings considerer in
the previous models [52, 70].
The theoretical implications of this effective composite metric along the line of non-
minimally coupled scalar-tensor theories were investigated in [56, 71]. The specific model
considered in [71] is motivated by the proxy theory to massive gravity [18], which is con-
structed by covariantizing the decoupling limit of massive gravity. The non-minimal deriva-
tive couplings are this time between the scalar field and terms that depend on the Riemann
tensorandRiccitensoroftheeffectivecompositemetric. In[71]itwasinvestigatedwhetheror
not these non-minimalderivative couplings maintain the ghost absent in thedecoupling limit
and on maximally symmetric backgrounds. A detailed Hamiltonian analysis of maximally
symmetric space-times showed that the only consistent non-minimal derivative coupling is
through a dynamical fiducial metric but without the inclusion of an additional kinetic term.
If the fiducial metric is fixed, then the Hamiltonian of maximally symmetric space-times
becomes highly non-linear in the lapse, which is consistent with the perturbation analysis
on top of FLRW background in [56]. This is also the case for dynamical fiducial metric in
the presence of an additional kinetic term. In this work we shall study the cosmological
perturbations of these non-minimal derivative couplings in the scenario where both metrics
are dynamical but without including the kinetic term of the fiducial metric.
– 2 –
2 Dynamical composite metric
Massive gravity theories usually involve two metrics g and f . A consistent coupling of
µν µν
some extra scalar field φ to both metrics simultaneously was introduced in [43] through a
composite metric g˜
µν
g˜ α2g +2αβg Xλ +β2f , (2.1)
µν µν µλ ν µν
≡
µ
with X defined by
ν
Xµ Xλ gµλf . (2.2)
λ ν ≡ λν
Already in the original consideration of this coupling with the effective composite metric
without derivatives acting on it, a perturbed ADM analysis divulged the existence of the
Boulware-Deser ghost originated from an operator that involved spatial derivatives [43, 46].
However, the analysis performed in the decoupling limit revealed the ghost absence at the
strong coupling scale Λ . Hence, the original massive gravity theory with this non-minimal
3
matter coupling can be considered as an effective field theory at the very least till the strong
couplingscaleandsubsequentlyitsphenomenologicalaspectscanbededuceduptothisscale.
In addition, the absence of the Boulware-Deser ghost was proven in the mini-superspace
approximation and also around exact FLRW solution since the ghostly operators are absent
intheFLRWcase. Thesimplestcouplingbetweenthescalarfieldandthecompositemetricis
throughthekinetictermforthescalar fieldg˜µν∂ φ∂ φ. Itisnaturaltoconsidermoregeneral
µ ν
couplingswhich involve derivatives ofthecompositemetric, forinstance, galileon/Horndeski-
like couplings. This possibility was investigated in [56], in which it was found that the BD
ghost would arise when the metric f is considered to be fixed. In this paper, we relax this
µν
restriction and consider bimetric theory with both dynamical metrics g and f [71]. We
µν µν
consider the following action
S = SEH+Spot+Skin+Sder, (2.3)
where SEH is the Einstein-Hilbert term for g
µν
M2
SEH = pl d4x√ gR[g] . (2.4)
2 −
Z
Note that the Einstein-Hilbert term only contains a Ricci scalar for the g metric. For the
viability of the model we are not allowed to consider an additional kinetic term for the f
metric. The non-derivative potential interactions Spot of the two metrics are given by
4
Spot = M2 dtd3x√ gm2 c e (X), (2.5)
pl − n n
Z n=0
X
with X standing for Xµ and for a matrix Mµ , e (M) are the elementary symmetric
ν ν n
polynomials defined by
e (M) n!Mµ1Mµ2 Mµn, (2.6)
n ≡ [µ1 µ2 ··· µn]
wheretheantisymmetrizationisunnormalized. Wefurthershallconsiderageneralizedkinetic
term Skin for the scalar field φ with the composite metric
Skin = d4x g˜P X˜,φ , (2.7)
−
Z
p (cid:0) (cid:1)
– 3 –
with X˜ denoting the kinetic term of φ in terms of the composite metric
1
X˜ g˜µν∂ φ∂ φ. (2.8)
µ ν
≡ −2
The most important ingredient of our action is the non-minimal derivative coupling of the
scalar field with the composite metric in Sder. Motivated by the covariantization of the
decouplinglimitofmassivegravity weshallconsidertheproxytheorylivingonthecomposite
metric [71],
Sder = d4x g˜ λ (φ)R[g˜]+λ (φ)Gµν [g˜]∂ φ∂ φ+λ (φ)Lµρνσ[g˜]∂ φ∂ φ ˜ ∂ φ ,
1 2 µ ν 3 µ ν ρ σ
− ∇
Z p (cid:16) (cid:17)(2.9)
which can be also thought of as the derivative interactions for the two metrics, where R[g˜],
Gµν[g˜] and Lµρνσ[g˜] are the Ricci scaslar, Einstein tensor and the double dual Riemann
tensor associated with g˜ , respectively. ˜ is the covariant derivative adapted to g˜ . The
µν µ µν
∇
“double dual Riemann tensor” for a metric g is defined as
µν
1
Lµρνσ[g] Rµρνσ+(Rµσgνρ+Rνρgµσ Rµνgρσ Rρσgµν)+ R(gµνgρσ gµσgνρ). (2.10)
≡ − − 2 −
InthefollowingwewillstudythisactiononFLRWbackgroundandestablishourparametriza-
tion for linear perturbations.
3 Cosmological parametrization
In this section we shall determine the full set of equations of motion that dictates the evo-
lution of spatially flat FLRW background in the presence of an additional scalar field that
couples via derivative couplings to the composite effective metric. After introducing our cos-
mological parametrization inthissection, weshallanalyzethelineartensor,vector andscalar
perturbations on top of this background in the next section. We shall particularly show, that
the nonminimal coupling between the Horndeski field and massive gravity through the com-
posite metric does not reintroduce the BD ghost. We parametrize the two metrics g and
µν
f to be
µν
g dxµdxν = N2 e2A e−H ijB B dt2+2NaB dtdxi+a2 eH dxidxj, (3.1)
µν − − i j i ij
f dxµdxν = N2(cid:16)e2ϕ (cid:0)e−Γ (cid:1)ijΩ Ω (cid:17)dt2+2N a Ω dtdxi+a2(cid:0) eΓ(cid:1) dxidxj, (3.2)
µν − f − i j f f i f ij
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
where N, a, N and a are functions of time only, and the matrix exponentials are defined
f f
perturbativelyas eH δ +H +1H kH + H3 and e−H ij = δij Hij+1Hi Hkj+
ij ≡ ij ij 2 i kj O − 2 k
H3 , etc. Throughout this paper, spatial indices are raised and lowered by δ and δij.
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) ij
O
We further decompose (with ∂2 δij∂ ∂ )
i j
(cid:0) (cid:1) ≡
B ∂ B+S , (3.3)
i i i
≡
1
H 2ζδ + ∂ ∂ δ ∂2 E+∂ F +h , (3.4)
ij ij i j ij (i j) ij
≡ − 3
(cid:18) (cid:19)
Ω ∂ ω+σ , (3.5)
i i i
≡
1
Γ 2ψδ + ∂ ∂ δ ∂2 χ+∂ ξ +γ , (3.6)
ij ij i j ij (i j) ij
≡ − 3
(cid:18) (cid:19)
– 4 –
with ∂ F 1 (∂ F +∂ F ), etc, and
(i j) ≡ 2 i j j i
∂iS = ∂iF = ∂iσ = ∂iξ = 0, hi =γi = 0, ∂ih = ∂iγ = 0. (3.7)
i i i i i i ij ij
Accordingly, it is convenient to parametrize the composite metric to be
g˜ dxµdxν = N˜2 e2A˜ (e−H˜)ijB˜ B˜ dt2+2N˜a˜B˜ dtdxi+a2(eH˜) dxidxj, (3.8)
µν i j i ij
− −
(cid:16) (cid:17)
where
N˜ αN +βN , a˜ αa+βa . (3.9)
f f
≡ ≡
Similar to (3.3)-(3.6), we may also decompose
1
B˜ ∂ B˜ +S˜, H˜ 2ζ˜δ + ∂ ∂ δ ∂2 E˜ +∂ F˜ +h˜ , (3.10)
i ≡ i i ij ≡ ij i j − 3 ij (i j) ij
(cid:18) (cid:19)
with∂iS˜ = ∂iF˜ = ∂ih˜ =δijh˜ = 0. NoteA˜etc. areexpressedintermsof A,B ,H ,ϕ,Ω ,Γ
i i ij ij i ij i ij
{ }
as
A˜= A˜(n)(A,B ,H ,ϕ,Ω ,Γ ) (3.11)
i ij i ij
n=1
X
etc., where n denotes the order in A,B ,H ,ϕ,Ω ,Γ . At the linear order, we have, for
i ij i ij
{ }
the scalar modes,
N N
A˜(1) = α A+β fϕ, (3.12)
N˜ N˜
B˜(1) = αr B +βr ω, (3.13)
1 2
a a
ζ˜(1) = α ζ +β fψ, (3.14)
a˜ a˜
a a
E˜(1) = α E+β fχ, (3.15)
a˜ a˜
with
aN N a˜+a N˜ a N Na˜+aN˜
f f f f
r , r , (3.16)
1 ≡ (Na(cid:16) +aN )a˜N˜(cid:17) 2 ≡ (Na (cid:16)+aN )a˜N˜(cid:17)
f f f f
for the vector modes,
a a
S˜(1) = αr S +βr σ , F˜(1) = α F +β fξ , (3.17)
i 1 i 2 i i a˜ i a˜ i
and for the tensor modes
a a
h˜(1) = α h +β fγ . (3.18)
ij a˜ ij a˜ ij
For later convenience, for any quantity q, we denote
1 dq 1 dq
′
q˙ = , q (3.19)
N dt ≡ N˜ dt
for short.
The background equations of motion can be determined by requiring the vanishing of
the first order action of A, ζ, ϕ, ψ and δφ, which is given by
N˜a˜3
S = dtd3xNa3 A+ 3ζ + ϕ+ 3ψ+ δφ . (3.20)
1 EA Eζ Eϕ Eψ Na3Eδφ
!
Z
– 5 –
The set of equations of motion are thus given by
a˜3
3M2H2+ pot +α com = 0, (3.21)
EA ≡ pl EA a3EA˜
N˜a˜2
M2 3H2+2H˙ + pot+α com = 0, (3.22)
Eζ ≡ pl Eζ Na2Eζ˜
(cid:16) N a˜3 (cid:17)
pot +β f com = 0, (3.23)
Eϕ ≡ Eϕ N a3EA˜
a N˜a˜2
pot +β f com =0, (3.24)
Eψ ≡ Eψ a Na2Eζ˜
where the Hubble parameter H is defined to be H a˙/a 1 da. In the above,
≡ ≡ Na dt
a a2 a3
pot = M2m2 c +3 fc +6 fc +6 fc , (3.25)
EA pl 0 a 1 a2 2 a3 3
!
aN
pot f
= b + b , (3.26)
Eζ 1 Na 2
f
N a a2 a3
pot = M2m2 f c +6 fc +18 fc +24 fc , (3.27)
Eϕ pl N 1 a 2 a2 3 a3 4
!
pot
= b +b . (3.28)
Eψ 2 3
where we have introduced
a a2
b M2m2 c +2 fc +2 fc , (3.29)
1 ≡ pl 0 a 1 a2 2
!
a a a2
b M2m2 f c +4 fc +6 fc , (3.30)
2 ≡ pl a 1 a 2 a2 3
!
N a a a2
b 2M2m2 f f c +6 fc +12 fc , (3.31)
3 ≡ pl Na 2 a 3 a2 4
!
for later convenience, and
com = 30H˜3X˜φ¯′λ 18H˜2X˜λ +6H˜2λ +6H˜φ¯′λ 2X˜P +P, (3.32)
EA˜ − 3− 2 1 1,φ− ,X˜
and
com = P +λ 6H˜2+4H˜′ +2λ (2H˜φ¯′+φ¯′′)+4X˜λ
Eζ˜ 1 1,φ 1,φφ
2λ 3(cid:16)H˜2X˜ +2H˜φ(cid:17)¯′φ¯′′+2H˜′X˜ 4H˜X˜φ¯′λ
2 2,φ
− −
6H˜X(cid:16)˜λ 2φ¯′ H˜2+H˜′ +3H˜φ¯(cid:17)′′ 12H˜2X˜2λ . (3.33)
3 3,φ
− −
(cid:16) (cid:16) (cid:17) (cid:17)
The equation of motion for the scalar field is given by
1 d
com = ¯ a˜3 , (3.34)
Eδφ L,φ− N˜a˜3dt J
(cid:0) (cid:1)
– 6 –
with
¯ = P +6λ 2H˜2+H˜′
,φ ,φ 1,φ
L
+6λ 4H˜(cid:16)2X˜ +H˜φ¯′φ¯(cid:17)′′+H˜′X˜ +3X˜λ (3H˜φ¯′+φ¯′′)+2X˜2λ
2,φ 2,φφ 2,φφφ
+3H˜X˜λ(cid:16) 11H˜2φ¯′+7H˜φ¯′′+6(cid:17)H˜′φ¯′ +6X˜λ 6H˜2X˜ +2H˜φ¯′φ¯′′+H˜′X˜
3,φ 3,φφ
5 (cid:16) (cid:17) (cid:16) (cid:17)
+ X˜2λ (3H˜φ¯′+φ¯′′)+X˜3λ , (3.35)
3,φφφ 3,φφφφ
2
and
= φ¯′P +6H˜2φ¯′λ +6H˜X˜λ +X˜φ¯′λ
J ,X˜ 2 2,φ 2,φφ
1
+18H˜3X˜λ +9H˜2X˜φ¯′λ +6H˜X˜2λ + X˜2φ¯′λ . (3.36)
3 3,φ 3,φφ 3,φφφ
2
4 Cosmological perturbations
Using the irreducible representation of the perturbation on top of FLRW background intro-
duced in the previous section, we are now at the position to study the stability analysis of
cosmological perturbations. We shall investigate the tensor, vector and scalar perturbations
separately. We shall start with the transverse traceless part of the metric fluctuations. The
quadratic action for the two tensor perturbations h and γ is given by
ij ij
1 d3k k2 k2
Stensor = dt Na3M2 h˙2 h2 +N˜a˜3 g h˜′2 w h˜2
2 8 (2π)3 pl ij − a2 ij h˜h˜ ij − h˜˜ha˜2 ij
Z (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
+Na3 2(h γ ) hij γij , (4.1)
ij ij
M − −
(cid:21)
(cid:0) (cid:1)
where h˜ is a shorthand for h˜(1) in (3.18), i.e., the linear combination of h and γ
ij ij ij ij
a a
h˜ α h +β fγ , (4.2)
ij ij ij
≡ a˜ a˜
and furthermore we have introduced the following short-cut notations for convenience,
g = 2λ 2λ X˜ 6λ H˜X˜φ¯′, (4.3)
h˜˜h 1− 2 − 3
w = 2λ +2λ X˜ 6λ X˜φ¯′′ 4λ X˜2, (4.4)
h˜˜h 1 2 − 3 − 3,φ
together with
a a N a N˜a˜
2 f M2m2 c +2 fc +2 f c +3 fc +αβ com . (4.5)
M ≡ a pl 1 a 2 N 2 a 3 NaEζ˜
" #
(cid:18) (cid:19)
(cid:16) (cid:17)
In difference to the standard equation for gravitational waves in General Relativity, we have
an additional massive tensor mode. The propagation speed of the tensor modes can be
different from the speed of light, even though observations are quite restrictive. One has to
impose the absence of ghost and gradient instabilities. In our case this will be equivalent to
imposing g > 0 and w > 0.
h˜h˜ h˜h˜
– 7 –
Similarly, we can study the stability conditions for the vector perturbations after inte-
grating out the non-dynamical ones. The quadratic action for the four vector modes S , F ,
i i
σ and ξ is given by
i i
d3k 1 1 1 2 1 1 1 2
Svector = dt Na3M2k2 S F˙ + N˜a˜3g k2 S˜ F˜′
2 (2π)3 4 pl a i − 2 i 4 h˜h˜ a˜ i − 2 i
Z (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
1 aN 2 Na3
Na3 S f σ + 2k2(F ξ )2 , (4.6)
i i i i
− 2 C − Na 16 M −
(cid:18) f (cid:19) (cid:21)
where similar to h˜ in (4.1), S˜ and F˜ should be understood as the linear combinations of
ij i i
S and σ , F and ξ through (3.17), respectively. In (4.6), 2 is given in (4.5) and we also
i i i i
M
introduce
1 αβ N˜a˜a a˜N Na˜
b + f 1+ f + com com (4.7)
C ≡ 1+ NaNaff 2 1+ NaNaff 2 Na2 (cid:16) N˜af N˜a(cid:17)EA˜ −Eζ˜ !
(cid:16) (cid:17)
with b given in (3.30) for short. Sincethe vector modes S and σ have nodynamics in (4.6),
2 i i
we may solve them in terms of F and ξ and arrive at the reduced action for F and ξ , which
i i i i
is given by
1 d3k 1 d a 2
Svector = dt Na3k2 f (F ξ ) + 2(F ξ )2 , (4.8)
2 16 (2π)3 (Gv N dt a˜ i − i M i − i )
Z (cid:20) (cid:16) (cid:17)(cid:21)
where
β2M2
pl
. (4.9)
v
G ≡ 1 N˜M2 β2 NN˜a2fr22 k2 a3
− N pl 2CNf2a˜2a2 − gh˜h˜a˜3
(cid:18) (cid:19)
From (4.8)it is transparentthat therearetwo vectorial degrees of freedom giving that β = 0,
6
which can be identified as F ξ . For the stability condition we have to impose > 0.
i i v
− G
Last but not least we study now the linear stability of the scalar modes in our model.
Initially we have 9 scalar modes, of which four (A, B, ζ and E) are from g , four (ϕ, ω, ψ
µν
and χ) are from f , and one is the perturbation of the scalar field δφ. In order to simplify
µν
the calculation, we choose a gauge in which δφ = E = 0. In the residual 7 modes, only 2
modes are dynamical, which can be conveniently chosen to be ζ˜(1) as in (3.14) and E˜(1) with
E = 0 as in (3.15), respectively. After some manipulations, the final quadratic action for
these two scalar modes takes the following general structure
d3k
Sscalar = dt (∂ ζ˜)2+2 ∂ ζ˜∂ E˜ + (∂ E˜)2
2 (2π)3 G11 t G12 t t G22 t
Z (cid:20)
+Ξ ∂ ζ˜E˜ ζ˜∂ E˜ + ζ˜2+2 ζ˜E˜ + E˜2 . (4.10)
t t 11 12 22
− W W W
(cid:21)
(cid:16) (cid:17)
The explicit expressions for the coefficients etc. are too involved to be presented in
11
G
the main text. In Appendix A we describe the details in deriving (4.10) as well as the full
expressions for the coefficients. As one can see, the would-be Boulware-Deser ghost does not
propagate and could be integrated out. Only two scalar fields are dynamical, one of which
comes from the matter field itself. This is in complete agreement with the findings in [71].
– 8 –
The hamiltonian analysis in the mini-superspace in [71] revealed that non-minimal derivate
couplings in massive bigravity with both metrics being dynamical and without the presence
of the kinetic term for the f metric does not reintroduce the Boulware-Deser ghost below
the strong coupling scale. Our cosmological perturbations analysis here affirmed the same
conclusion and is supplementary to the analysis in [71].
5 Conclusion
This paper was dedicated to the detail analysis of cosmological perturbations in massive
bigravityinthepresenceofnon-minimalderivativecouplingsofthecompositeeffectivemetric
to a Horndeski scalar field. After working out the governing background equations of motion
we payed special attention to the stability analysis of tensor, vector and scalar perturbations.
The tensor perturbations showed the presence of four propagating tensor modes and we
have seen which dynamical quantities have to be constrained in order to avoid ghost and
gradient instabilities. Similarly, the analysis of vector perturbations disclosed that out of
naively counted four vector modes only two of them are actually dynamical. The absence of
ghost and gradient instabilities required this time the condition > 0 on the background
v
G
dynamics. Finally, the analysis of scalar perturbations resulted in two propagating scalar
modes and hence reinforcing the fact that the Boulware-Deser ghost is not excited.
In this work, we considered a uniform composite metric g˜ in (2.7) and (2.9) for sim-
µν
plicity. It would be interesting to investigate the case with different composite metrics for
each non-minimal derivative coupling terms, such as those in [56]. A special case is just
the bigravity with composite metric appearing only in the kinetic term of the scalar field X˜
[57], which yields a viable theory. On the other hand, in light of the analysis in [71], the
healthiness of theories with different composite metrics for each non-minimal coupling terms
still remains an open question.
Acknowledgments
X.G.wassupportedbyJSPSGrant-in-AidforScientificResearchNo. 25287054and26610062.
LH acknowledges financial support from Dr. Max Ro¨ssler, the Walter Haefner Foundation
and the ETH Zurich Foundation.
A Details in quadratic action for the scalar modes
Inthis appendixweprovidethedetails inderiving(4.10). After fixingthegauge δφ = E = 0,
in the quadratic action for the residual 7 scalar modes S [A,B,ϕ,ω,ζ,ψ,χ], four modes A,
2
B, ϕ and ω have no explicit time derivatives and thus can be integrated out. We thus get
the quadratic action for 3 variables ζ,ψ,χ or more conveniently ζ,ζ˜,E˜ , where a˜ and E˜
{ } { }
are given in (3.14) and (3.15) after setting E = 0. The quadratic action is given by
d3k 1
S [ζ,ζ˜,E˜] = dt ζ2+ ζζ˜+ ζE˜ + ζ∂ ζ˜+ ζ∂ E˜ + ζ˜2
2 (2π)3 ∆ A1 A2 A3 A4 t A5 t A6
Z h
+ ζ˜E˜ + ζ˜∂ ζ˜+ ζ˜(∂ E˜)+ E˜2+ E˜∂ ζ˜
7 8 t 9 t 10 11 t
A A A A A
+ E˜∂ E˜ + (∂ ζ˜)2+ ∂ ζ˜∂ E˜ + (∂ E˜)2 , (A.1)
12 t 13 t 14 t t 15 t
A A A A
i
– 9 –