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F (R) bigravity
Shin’ichi Nojiri
(Шинъичи Нозири)
Department of Physics &
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI),
Nagoya Univ.
– Typeset by FoilTEX –
1
Preliminary
“bigravity’’ = system of massive spin 2 field (massive graviton)
+ gravity (includes massless spin 2 field = graviton)
F (R) extension of bigravity, Application to Cosmology
Mainly based on
S. Nojiri and S. D. Odintsov,
“Ghost-free F (R) bigravity and accelerating cosmology,”
Phys. Lett. B 716, 377 (2012) [arXiv:1207.5106 [hep-th]].
S. Nojiri, S. D. Odintsov, and N. Shirai,
“Variety of cosmic acceleration models from massive F (R) bigravity,”
JCAP 1305 (2013) 020
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2
Dark energy
Universe can be regarded as isotropic and homogeneous in the scale larger
than the clusters of galaxies
⇒ Friedmann-Robertson-Walker (FRW) metric:
ds2 =
3
∑
µ,ν=0
gµν dxµdxν = −dt2 + a(t)2
3
∑
g˜ij dxidxj .
i,j=1
a(t): scale factor, g˜ij : spacial metric
˜ ij = 2K g˜ij (R
˜ ij : Ricci curvature given by g˜ij )
R
K = 1: unit sphere, K = −1: unit hyperboloid, K = 0: flat space
{
da(t)/dt > 0 : expanding universe
d2a(t)/dt2 > 0 : accelerating expansion
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3
Assume the Universe is filled with perfect fluids.
1st FRW equation: (t, t) component of the Einstein eq.
3 2
3K
0 = − 2H − 2 2 + ρ ,
κ
κ a
κ2 ≡ 8πG
2nd FRW equation: (i, j) component
(
)
1
dH
K
+ 3H 2 + 2 2 + p ,
0= 2 2
κ
dt
κ a
ρ: energy density, p: pressure, H ≡ (1/a) da(t)/dt: Hubble rate
The Hubble constant H0: the present value of H.
H0 ∼ 70 km s−1Mpc−1 ∼ 10−33 eV in the unit ℏ = c = 1.
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4
Cosmic Microwave Background Radiation (CMB) ⇒ K ∼ 0
3 2 ( −3 )4
3
ρ ∼ ρc ≡ 2 H0 ∼ 10 eV ∼ 10−29g/cm .
κ
ρc: critical density. Flat universe ⇒ ρ ∼ ρc
Planck satellite
Density of usual matter ∼ 4.9% , dark matter ∼ 26.8% of ρc
=⇒ something unknown ∼ 68.3% · · · dark energy
(
)
3
−29
critical density ∼ 10
g/cm
-
dark energy
usual matter
dark matter
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5
Type Ia Supernovae
⇒ accelerating expansion started about 5 billion years ago.
1st and 2nd FRW equations ⇒
1 d2a(t) dH
κ2
2
=
+ H = − (ρ + 3p) .
2
a dt
dt
6
accelerating expansion ⇒ p < −ρ/3
Dark energy: large negative pressure
Equation of state (EoS) parameter: w ≡ ρp
Dark energy: w ∼ −1
Radiation: w = 1/3,
Usual matter, cold dark matter (CDM): w ∼ 0 (dust),
Cosmological constant: w = −1
Dark energy = Cosmological constant??
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6
When EoS parameter w: constant ⇒ conservation law:
dρ
+ 3H (ρ + p) = 0 ,
dt
⇒ ρ = ρ0a−3(1+w) (w ̸= −1), ρ0: constant of integration
1st FRW eq. ⇒
2
3(1+w)
In case w > −1, a(t) ∝ t
2
3(1+w)
In case w < −1, a(t) ∝ (t0 − t)
When t = t0, a(t) diverges: Big Rip singularity
2
In case w = −1, a(t) ∝ a0eH0t, H0 ≡ ρ03κ , de Sitter space-time
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7
Fine-tuning problem and Coincidence problem
Fine-tuning problem, Coincidence problem:
The definitions slightly depend on persons.
A. 1st and 2nd FRW equations (K = 0)
3 2
Λ
0 = − 2 H + 2 + ρmatter ,
κ
2κ
(
)
dH
1
Λ
2
0= 2 2
+ 3H − 2 + pmatter ,
κ
dt
2κ
Λ: cosmological constant
If the dark energy is the cosmological term, the cosmological constant is
unnaturally
( small.
)2
−33
Λ ∼ 10
eV ≪ MPlanck ∼ 1/κ ∼ 1019 GeV = 1028 eV
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8
B. Anthropic principle?
Λ
∼ ρmatter (including dark matter)
2
2κ
Very accidental! if Λ is a constant
Age of the Universe: 13.7 billion years
(
−33
∼ 10
eV
)−1
1
∼ Λ− 4
Present temperature of the Universe: (3K)
−3
∼ 10
(
1/4
eV ∼ (ρmatter)
∼
Λ
2κ2
)1/4
⇒ Dark energy might be dynamical?
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9
C. Initial condition?
If the dark energy is a perfect fluid whose EoS parameter w ∼ −1,
ρDE = ρDE 0a−3(1+w) ∼ ρDE 0
Usual matter or CDM (dust with w = 0)
ρmatter = ρmatter 0a−3
Ratio of densities of the dark energy to usual matter and dark matter
ρDE/ρmatter ∼ (ρDE 0/ρmatter 0) a−3
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10
In order that ρDE 0 ∼ ρmatter 0 in the present Universe,
because the ratio is given by ρDE/ρmatter ∼ a−3,
When transparent to radiation (a ∼ 10−3), for example:
ρDE/ρmatter ∼ 10−9
We need to fine-tune the initial condition of the ratio.
There might be a model where the dark matter interacts with dark energy
and there is a transition between them?
The EoS parameter of the dark energy changes dynamically depending on
the expansion (tracker model)?
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11
D. If the dark energy is the vacuum energy,
the quantum corrections from the matter diverge ∼ Λ4cutoff .
Λcutoff : cutoff scale
If the supersymmetry is restored in the high energy,
the vacuum energy by the quantum corrections ∼ Λ2cutoff Λ2SUSY
ΛSUSY
: the scale of the supersymmetry breaking.
If
use )the counter term in order to obtain the very small vacuum energy
( we
4
10−3 eV , we need very very fine-tuning and extremely unnatural.
Maybe we do not understand quantum gravity?
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12
The Einstein equation
1
Rµν − gµν R = κ2Tµν
2
What we consider the dark energy as a perfect fluid filling the Universe
corresponds to the modification of the energy momentum tensor Tµν of
matters, which appears in the r.h.s. in the Einstein equation. On the other
hand, there are many models to consider the modification of the Einstein
tensor in the l.h.s., which are called modified gravity models.
F (R) gravity, scalar-tensor theory (Brans-Dicke type model), Gauss-Bonnet
gravity, F (G) gravity, massive gravity, bigravity...
Recently there have been remarkable progresses in the study of massive
gravity and bigravity.
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13
Massive Gravity
(theory of massive spin two field)
Fierz-Pauli action (linearized or free theory), 3/4 century ago
M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an
electromagnetic field,” Proc. Roy. Soc. Lond. A 173 (1939) 211.
The Lagrangian of the massless spin-two field (graviton) hµν is given by
1
1
µν
µ
ν
λ µν
λ
L0 = − ∂λhµν ∂ h +∂λh µ∂ν h −∂ hµν ∂ h+ ∂λh∂ λh ,
2
2
(
h≡
hµµ
)
.
Massless graviton: 2 degrees of freedom (helicity),
Massive graviton: 5 degrees of freedom (2s + 1, spin s = 2).
The Lagrangian of the massive graviton with mass m is given by
Lm
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)
m2 (
µν
2
= L0 −
hµν h − h
2
(Fierz-Pauli action) .
14
When m = 0, gauge symmetry (linearized general covariance)
δhµν = ∂µξν + ∂ν ξµ ,
ξµ(x): space-time dependent gauge parameter.
The combination hµν hµν − h2:
Fierz-Pauli tuning (not related with any symmetry)
For the combination hµν hµν − (1 − a)h2,
if a ̸= 0, there appears ghost scalar field with mass
m2g
3 − 4a 2
=
m
2a
(
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m2g
)
→ ∞ when a → 0 .
15
Hamiltonian and counting of degrees of freedom:
D(D−1)
− 1 propagating degrees of freedom in D dimensions
2
(5 degrees of freedom for D = 4).
Legendre transformation only with respect to the spatial components hij .
∂L
= h˙ ij − h˙ kk δij − 2∂(ihj)0 + 2∂k h0k δij ,
∂ h˙ ij
∫
{
D
⇒ S = d x πij h˙ ij − H + 2h0i (∂j πij ) + m2h20i
(
)}
⃗ 2hii − ∂i∂j hij − m2hii
+h00 ∇
,
πij =
1 2
1 1
1
2
H = πij −
π + ∂k hij ∂k hij − ∂ihjk ∂j hik
2
2 D − 2 ii 2
1
1
+ ∂ihij ∂j hkk − ∂ihjj ∂ihkk + m2(hij hij − h2ii) .
2
2
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16
m = 0 case: h0i, h00: Lagrange multipliers → constraints
∂j πij = 0 ,
⃗ 2hii − ∂i∂j hij = 0 .
∇
First class constraints → gauge symmetry (⇐ general covariance)
For D = 4, hij and πij each have 6 components, respectively.
→ 12 dimensional phase space.
4 constraints + 4 gauge invariances
→ 4 dimensional phase space
(two polarizations (helicities) of massless graviton)
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17
m ̸= 0: h0i are no longer Lagrange multipliers δh0i ⇒ h0i = − m12 ∂j πij ,
∫
S=
D
{
(
2
2
˙ ij − H + h00 ∇
⃗ hii − ∂i∂j hij − m hii
d x πij h
)}
,
1 2
1 1
1
2
H = πij −
π + ∂k hij ∂k hij − ∂ihjk ∂j hik
2
2 D − 2 ii 2
)
1
1 2(
1
2
2
+ ∂ihij ∂j hkk − ∂ihjj ∂ihkk + m hij hij − hii + 2 (∂j πij ) .
2
2
m
h00: Lagrange multiplier → single constraint
2
2
⃗ hii + ∂i∂j hij + m hii = 0 ,
C = −∇
Secondary constraint:
{H, C}PB
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1
2
=
m πii + ∂i∂j πij = 0 ,
D−2
∫
H =
d
d x H,
18
Two second class constraints.
For D = 4,
12 dimensional phase space − 2 constraints = 10 degrees of freedom
(5 polarizations of the massive graviton and their conjugate momenta).
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19
Boulware-Deser ghost
D. G. Boulware and S. Deser, “Classical General Relativity Derived from Quantum Gravity,” Annals Phys.
89 (1975) 193.
In non-linear (interacting) theory, 6th degree of freedom appears as a ghost.
Non-linear massive gravity action with flat metric ηµν , hµν = gµν − ηµν
1
S= 2
2κ
∫
i dt
Nt = t0 + dt
6
N dt
t = t0
dD x
[√
]
1 2 µα νβ
−gR − m η η (hµν hαβ − hµαhνβ ) .
4
ADM formalism (N : lapse function, Ni: shift function)
2
ij
g00 = −N + g NiNj ,
g0i = Ni ,
gij = gij .
x: constant
i, j, · · · = 1, 2, 3, g ij : inverse of the spatial metric gij .
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20
m = 0 case
Einstein-Hilbert action (after partial integrations)
1
2κ2
(d)
∫
[
]
√
(d)
2
ij
d x gN
R − K + K Kij ,
D
R: curvature of spatial metric gij , Kij : extrinsic curvature
Kij =
1
(g˙ ij − ∇iNj − ∇j Ni) ,
2N
∇i: covariant derivative w.r.t. the spatial metric gij .
Canonical momenta with respect to gij :
p
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ij
)
δL
1 √ ( ij
ij
=
=
g K − Kg
,
δ g˙ ij
2κ2
21
Hamiltonian:
(∫
)
∫
ddx pabg˙ ab − L =
H=
Σt
ddx N C + NiC i .
Σt
]
√ [(d)
2
ij
C= g
R + K − K Kij ,
(
)
2
2κ
1
Kij = √
pij −
phij .
g
D−2
√
(
C = 2 g∇j K − Kh
i
ij
ij
)
,
For m = 0, Hamiltonian vanishes. N , Ni: Lagrange multipliers
⇒ C = 0, Ci = 0: first class constraints ⇔ general covariance
In D = 4,
12 phase space metric components − 4 constraints − 4 gauge symmetries
= 4 phase space degrees of freedom
= degrees of freedom in linearized theory of massless spin 2 graviton
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22
m ̸= 0 case (hij ≡ gij − δij )
η µαη µβ (hµν hαβ − hµαhµβ )
(
= δ δ (hij hkl − hik hjl) + 2δ hij − 2N δ hij + 2Ni g − δ
ik jl
ij
2 ij
ij
ij
)
Ni ,
Action
1
S= 2
2κ
∫
{
dD x pabg˙ ab − N C − NiC i
m2 [ ik jl
−
δ δ (hij hkl − hik hjl) + 2δ ij hij
4
}
(
)
]
−2N 2δ ij hij + 2Ni g ij − δ ij Nj .
N 2, NiNj terms ⇒ N 2, Ni: Not Lagrange multipliers but auxiliary fields.
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23
)
1 ( ij
C
ij −1 j
N = 2 ij
, Ni = 2 g − δ
C .
m δ hij
m
No constraints nor gauge symmetries.
Hamiltonian:
∫
{
2
)
1
C
1 i ( ij
d
ij −1 j
d x
+
C g −δ
C
2m2 δ ij hij 2m2
}
2[
]
m
+
δ ik δ jl (hij hkl − hik hjl) + 2δ ij hij
.
4
1
H= 2
2κ
12 phase space degrees of freedom, or 6 real degrees of freedom.
One more degree of freedom, compared with linearized theory
⇒ ghost scalar
Boulware-Deser ghost
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24
vDVZ(van Dam, Veltman, and Zakharov) discontinuity
H. van Dam and M. J. G. Veltman, “Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys.
B 22 (1970) 397.
V. I. Zakharov, “Linearized gravitation theory and the graviton mass,” JETP Lett. 12 (1970) 312 [Pisma
Zh. Eksp. Teor. Fiz. 12 (1970) 447].
Discontinuity of m → 0 limit in the free massive gravity with the Einstein
gravity due to the extra degrees of freedom in the limit.
⇒ the Vainstein mechanism
A. I. Vainshtein, “To the problem of nonvanishing gravitation mass,” Phys. Lett. B 39 (1972) 393.
Non-linearity screens the extra degrees of freedom (non-linearity becomes
strong when m is small).
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25
Massive gravity without ghost
C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action,” Phys. Rev. D 82, 044020 (2010)
[arXiv:1007.0443 [hep-th]],
C. de Rham, G. Gabadadze and A. J. Tolley, “Resummation of Massive Gravity,” Phys. Rev. Lett. 106
(2011) 231101 [arXiv:1011.1232 [hep-th]].
S. F. Hassan and R. A. Rosen, “Resolving the Ghost Problem in non-Linear Massive Gravity,” Phys. Rev.
Lett. 108 (2012) 041101 [arXiv:1106.3344 [hep-th]].
√
√
√
−1
−1
Non-dynamical metric fµν (∼ ηµν ), g f :
g f g −1f = g µλfλν
Minimal extension of Fierz-Pauli action:
∫
[
]
√
2
4 √
2
S = Mp d x −g R − 2m (tr g −1f − 3) .
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⇒ vDVZ discontinuity ⇒
∫
S=
√
[
d x −g R + 2m
Mp2
4
2
3
∑
]
√
βn en( g −1f ) ,
n=0
e0(X) = 1 ,
e1(X) = [X] ,
e2(X) = 12 ([X]2 − [X2]) ,
e3(X) = 16 ([X]3 − 3[X][X2] + 2[X3]) ,
e4(X) =
4
1
([X]
24
− 6[X]2[X2] + 3[X2]2 + 8[X][X3] − 6[X4]) ,
ek (X) = 0 for k > 4 ,
X = (X µν ) ,
[X] ≡ X µµ ,
∼ Galileon ⇒ Vainshtein mechanism
(longitudinal scalar mode (hµν ∼ ∂µ∂ν ϕ) ∼ Galileon scalar field)
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Hamiltonian constraint: Minimal extension case
ADM formulation, fµν = ηµν ⇒
( √
)
L =π ∂tγij + N R + N Ri − 2m γ N tr g −1η − 3 .
(
)
l
1
1
N δlj
(g −1η)µν = 2
, N i = γ ij Nj .
i
2 il
i l
−N
(N γ − N N )δlj
N
ij
0
i
2√
Highly nonlinear action in Nµ ⇒ New combinations ni
N i = (δji + N Dij )nj ,
√
√
Dij : ( 1 − nT I n) D = (γ −1 − DnnT DT )I ,
I = δij ,
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I−1 = δ ij ,
28
⇒ L =π ij ∂tγij + N R0 + Ri(δji + N Dij )nj
[√
]
√
2√
1 − nT In + N tr ( γ −1I − DnnT DT I) − 3N .
− 2m γ
Linear in N .
[ 4
]−1/2
i
ji
kl
δni ⇒ n = −Rj δ 4m det γ + Rk δ Rl
: Not including N .
[√
]
√
δN ⇒ R0 + RiDij nj − 2m2 γ
1 − nr δrsns Dkk − 3 = 0 .
+ secondary constraint = 2 constraints.
12 components of γij and π ij − 2 constraints
= 10 components (massive spin 2)
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29
Bimetric gravity (bigravity)
S. F. Hassan and R. A. Rosen, “Bimetric Gravity from Ghost-free Massive Gravity,” JHEP 1202 (2012) 126
[arXiv:1109.3515 [hep-th]].
Dynamical fµν (background independent).
∫
∫
√
√
(g)
(f )
4
2
4
d x − det g R + Mf d x − det f R
S =Mg2
∫
2
+ 2m
2
Meff
4
d x
√
− det g
4
∑
√
βn en( g −1f ) ,
n=0
2
≡ 1/Mg2 + 1/Mf2 .
1/Meff
(g)
(f )
R : scalar curvature for gµν , R : scalar curvature for fµν .
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30
Spectrum of the linearized theory
Minimal case: β0 = 3, β1 = −1, β2 = 0, β3 = 0,
Linearize
gµν
∫
⇒ S=
1
= g¯µν +
hµν ,
Mg
fµν
β4 = 1.
1
= g¯µν +
lµν ,
Mf
d4x (hµν Eˆµναβ hαβ + lµν Eˆµναβ lαβ )
[(
]
)2 ( µ
µ )2
µ
2
2 ∫
µ
lµ
hµ
hν
m Meff
lν
4
d x
.
−
−
−
−
4
Mg Mf
Mg Mf
Eˆµναβ : usual Einstein-Hilbert kinetic operator.
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31
Change of variables
1
1
1
uµν =
hµν +
lµν ,
Meff
Mf
Mg
1
1
1
vµν =
hµν −
lµν .
Meff
Mg
Mf
⇒
∫
S=
d4x (uµν Eˆµναβ uαβ + vµν Eˆµναβ vαβ )
2
m
−
4
∫
(
d x v vµν −
4
µν
v µµv νν
)
.
One massless spin-2 particle uµν and one massive spin-2 particle vµν with
mass m.
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32
F (R) bigravity
Standard F (R) gravity ⇔ scalar tensor theory
∫
SF (R) =
(
√
F (R)
d4x −g
+ Lmatter
2
2κ
)
.
Introducing the auxiliary field A,
1
S= 2
2κ
∫
√
d x −g {F ′(A) (R − A) + F (A)} .
4
Variation of A ⇒ A = R : original action
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33
Rescaling of metric
gµν → eσ gµν ,
σ = − ln F ′(A) .
⇒ Einstein frame action (Einstein-Hilbert action + real scalar field σ):
(
)
1
3
SE = 2 d4x −g R − g ρσ ∂ρσ∂σ σ − V (σ) ,
2κ
2
( −σ )
( ( −σ ))
A
F (A)
σ
2σ
V (σ) =e g e
−e f g e
= ′
− ′
.
2
F (A) F (A)
∫
√
A = g (e−σ ) ⇐ σ = − ln (1 + f ′(A)) = − ln F ′(A)
Coupling of σ with matters appears by the rescaling gµν → eσ gµν .
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34
Construction of F (R) bigravity
(by the inverse process from scalar-tensor form)
Adding the following actions to the bigravity action
∫
Sφ = −
Mg2
∫
√
4
d x − det g
{
3 µν
g ∂µφ∂ν φ + V (φ)
2
}
d4xLmatter (eφgµν , Φi) ,
{
}
∫
√
3 µν
Sξ = − Mf2 d4x − det f
f ∂µξ∂ν ξ + U (ξ) .
2
+
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35
Scale transformations gµν → e−φgµν , fµν → e−ξ fµν ,
∫
SF =Mf2
4
d x
√
{
}
−ξ
J(f
)
−2ξ
− det f J e R
− e U (ξ)
∫
2
+ 2m2Meff
∫
(√
)
4
∑
√
n
n
−1
βne( 2 −2)φ− 2 ξ en
gJ f J
d4x − det g J
n=0
{
}
√
d4x − det g J e−φRJ(g) − e−2φV (φ)
+ Mg2
∫
( J
)
4
+ d xLmatter gµν , Φi .
Kinetic terms of φ and ξ vanish. (OJ : quantities in the Jordan fame)
Coupling of φ with matters also disappears.
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36
Variation of φ and ξ ⇒
2
2
0 =2m Meff
+
2
Mg
{
2
4
∑
+
{
βn
n=0
−e
2
0 = − 2m Meff
2
Mf
(
−φ
R
J(g)
−ξ
R
2
J(f )
)
n −2)φ− n ξ
(
2 en
e 2
−2φ
+ 2e
4
∑
βn n
n=0
−e
n
−2
2
V (φ) + e
n −2 φ− n ξ
e( 2 ) 2 en
+ 2e
−2ξ
(√
gJ
−2φ
−1
)
fJ
}
′
V (φ)
,
(√
)
−1
gJ f J
U (ξ) + e
−2ξ
′
}
U (ξ)
.
In principle, can be solved algebraically with respect to φ and ξ
))
(
(√
(g)
(f )
−1
g f
,
φ = φ R , R , en
– Typeset by FoilTEX –
(
ξ=ξ R
(g)
,R
(f )
, en
(√
g −1f
))
.
37
⇒ analogue of F (R) gravity:
∫
2
SF = Mf
(
(√
))
√
−1
4
(f )
J(g)
J(f )
R ,R
, en
gJ f J
d x − det f JF
∫
2
2
+ 2m Meff
2
4
d x
√
RJ(g) ,en
(√
−1 J
gJ
f
))
(√
)
−1
en
gJ f J
(√
(
))
−1
J(g)
J(g)
J(f )
− det g JF
R ,R
, en
gJ f J
∫
+
(
n=0
∫
+ Mg
4
∑
√
( n2 −2)φ
4
d x − det g
βne
4
d xLmatter
– Typeset by FoilTEX –
(
J
gµν , Φi
)
,
38
Here
(
F
J(g)
R
J(g)
,R
J(f )
(√
(
−e
−2φ RJ(g) ,RJ(f ) ,en
−1 J
f

))
(
(√

−1 J
J(g)
J(f
)

J
 −φ R
g
f
,R
,en
J(g)
≡ e
R



(√
))
−1
gJ
fJ
( (
V
(
F
gJ
, en
))
(f )
R
J(g)
,R
J(f )
(√
, en
gJ
−1 J
f
))
R
J(g)
,R
J(f )
, en
gJ
−1 J
f

)))





,

(√
))
(

−1
J(g)
J(f
)

 −ξ R
gJ
fJ
,R
,en
J(f )
≡ e
R



(
−e
φ
(√
(√
))
−1 J
J(g)
J(f
)
J
−2ξ R
,R
,en
g
f
( (
U
ξ
R
J(g)
,R
J(f )
(√
, en
gJ
−1 J
f

)))





.
It is difficult to explicitly solve equations with respect to φ and ξ and it
might be better to define the model by introducing the auxiliary scalar fields
φ and ξ.
– Typeset by FoilTEX –
39
Cosmological Reconstruction
Usually we start from a given model and investigate the development of the universe
etc. by using the given equations. Here we consider the inverse, that is, for a given
development of the universe, we construct a model which reproduces the development,
which we call reconstruction.
Minimal case:
∫
∫
√
√
2
4
(g)
2
4
Sbi =Mg d x − det g R + Mf d x − det f R(f )
∫
(
)
√
√
√
2
+ 2m2Meff
d4x − det g 3 − tr g −1f + det g −1f .
– Typeset by FoilTEX –
40
Start from the Einstein frame action. Neglect matter.
δgµν ⇒
2
0 = Mg
(
1
(g)
(g)
gµν R
− Rµν
2
{
(
√
)
(√
)−1 ρ
(√
)−1 ρ }
1
gµν 3 − tr g −1 f
g −1 f
+ fνρ
g −1 f
2
ν
µ
[ (
)
]
3
2 1 3 ρσ
g ∂ρ φ∂σ φ + V (φ) gµν − ∂µ φ∂ν φ .
+ Mg
2 2
2
2 2
+ m Meff
)
1
+ fµρ
2
δfµν ⇒
(
)
1
(f )
(f )
fµν R
− Rµν
2
{
}
√
(√
)ρ
(√
)ρ
)
(√
(
)
1
1
2 2
+ m Meff det f −1 g
− fµρ
g −1 f
g −1 f
g −1 f fµν
− fνρ
+ det
2
2
ν
µ
[ (
)
]
3
2 1 3 ρσ
+ Mf
f ∂ρ ξ∂σ ξ + U (ξ) fµν − ∂µ ξ∂ν ξ .
2 2
2
2
0 = Mf
– Typeset by FoilTEX –
41
δφ, δξ ⇒
0 = −3□g φ + V ′(φ) ,
0 = −3□f ξ + U ′(ξ) .
□g , □f : d’Alembertian w.r.t. g, f .
(
)
(g)
Bianchi identity 0 = ∇µg 12 gµν R(g) − Rµν + field equations ⇒
{
( √
)
)−1 ρ
(√
)−1 ρ }
(√
1 µ
µ
0 = − gµν ∇g tr g −1 f + ∇g fµρ
g −1 f
+ fνρ
g −1 f
.
2
ν
µ
Similarly
[√
{
(√
)−1ν
(√
)−1µ
1
1
σµ
σν
det f −1 g
−
g −1 f
g −
g −1 f
g
2
2
σ
σ
)
}]
(√
µν
g −1 f f
.
+ det
µ
0 =∇f
– Typeset by FoilTEX –
(
)
42
In case of the Einstein gravity,
conservation law ⇐ Einstein equation + Bianchi identity
or conservation laws ⇐ scalar field equations
Scalar field equations are not independent of Einstein equation.
In case of bigravity,
only conservation laws ⇐ scalar field equations
Einstein equation + Bianchi identities + scalar field equations
⇒ new equations independent of Einstein equation.
Scalar field equations are independent of Einstein equation.
– Typeset by FoilTEX –
43
Assume FRW universes by using the conformal time t
2
dsg =
3
∑
(
µ
ν
gµν dx dx = a(t)
2
2
−dt +
µ,ν=0
2
dsf
=
3
∑
3 (
∑
dx
i
)2
)
,
i=1
µ
ν
2
2
fµν dx dx = −c(t) dt + b(t)
µ,ν=0
2
3 (
∑
i
dx
)2
.
i=1
(Most general form assuming the homogeneity, isometry, and flat spacial part.)
⇒
)
(
) (3
1
2
2
2
2
φ˙ + V (φ)a(t)
Mg ,
a − ab +
4
2
(
)
(
)
2
˙ + H 2 + m2 M 2 3a2 − 2ab − ac
δgij : 0 =Mg 2H
eff
)
(
1
a˙
3 2
2
2
φ˙ − V (φ)a(t)
Mg ,
H ≡ .
+
4
2
a
2 2
2 2
δgtt : 0 = − 3Mg H − 3m Meff
H is not exactly Hubble rate ⇐ t: conformal time
– Typeset by FoilTEX –
44
(
a3
1− 3
b
)
(
3 ˙2
1
2
+
ξ − U (ξ)c(t)
4
2
(
)
3
(
)
a c
2
2
˙ + 3K 2 − 2LK + m2 M 2
−c
δfij : 0 =Mf 2K
eff
2
b
(
)
3 ˙2
1
2
2
+
ξ − U (ξ)c(t)
Mf .
4
2
2 2
2 2 2
δftt : 0 = − 3Mf K + m Meff c
˙ ,
K ≡ b/b
)
2
Mf ,
L = c/c
˙ .
Both of equations derived from Bianchi identity:
cH = bK or
ca˙
a
˙
= b.
˙ a.
If a˙ ̸= 0, we obtain c = ab/
˙
If a˙ = 0, we find b˙ = 0, that is, a, b: constant, c can be arbitrary.
– Typeset by FoilTEX –
45
Redefinition of the scalar fields: φ = φ(η), ξ = ξ(ζ).
Identify η = ζ = t ⇒
(
)
2
˙ − H − 2m2M 2 (ab − ac) ,
=−
H
eff
(
)
2
2
2
2
2
2
2
˜
˙
V (t)a(t) Mg =Mg 2H + 4H + m Meff (6a − 5ab − ac) ,
(
) 3
(
)
c
2
2
˙ − LK − 2m2M 2 − + 1 a c ,
σ(t)Mf = − 4Mf K
eff
b
b2
(
)
3
3 2
(
)
a c
a c
2
˜ (t)c(t)2M 2 =M 2 2K
˙ + 6K 2 − 2LK + m2M 2
U
−
2c
+
.
f
f
eff
2
3
b
b
2
ω(t)Mg
2
4Mg
′
2
′
2
˜ (ζ) = U (ξ (ζ)) .
ω(η) = 3φ (η) , V˜ (η) = V (φ (η)) , σ(ζ) = 3ξ (ζ) , U
˜ (t) to
For arbitrary a(t) and b(t), if we choose ω(t), V˜ (t), σ(t), and U
satisfy the above equations, a model admitting the given a(t) and b(t)
evolution can be reconstructed.
– Typeset by FoilTEX –
46
Cosmological Models
(
FRW universe:: ds2 = a
˜(t)2 −dt2 +
a
˜(t)2 =
a
˜(t)2 =
l2
:
t2
l2n
t2n
∑3
i=1
(
)
)
2
dxi
.
de Sitter universe.
with n ̸= 1 case:
ln
tn
(
n
l
Redefinition of time coordinate: dt˜ = ± dt t˜ = ± n−1
t1−n
)
2n
(
)− 1−n
3
∑
˜
( i )2
t
2
2
˜
⇒ ds = −dt + ±(n − 1)
dx
.
l
i=1
0 < n < 1: phantom universe, n > 1: quintessence universe,
n < 0: decelerating universe
– Typeset by FoilTEX –
47
Universe with a(t) = b(t) = 1
a(t) = b(t) = 1 satisfies the previous constraint.
⇒ Einstein frame metric gµν : flat Minkowski space
Physical metric: the scalar field does not directly coupled with matter.
J
⇒ the metric we observe: Jordan frame metric gµν
= eφgµν .
2
2 ˜2
2
2
ω(t)Mg =12Mg H
= 2m Meff (c − 1) ,
2
V˜ (t)Mg
=m
2
2
Meff
(1 − c) =
2 ˜2
−6Mg H
⇒ c=1+
2
2
2
2 ˜2
σ(t)Mf =2m Meff (c − 1) = 12Mg H
,
(
˜ (t)M 2 =m2M 2 c (1 − c) = −6M 2H
˜2 1 +
U
f
eff
g
˜ 2M 2
6H
g
2
m2Meff
˜2
6H
2
m2Meff
,
)
.
Note: ω(t), σ(t) > 0 (no ghost)
– Typeset by FoilTEX –
48
Big Rip, quintessence, de Sitter and decelerating universes
l2n
a
˜(t) = 2n
t
2
12n2 Mg2
2
ω(η) Mg =
,
η2
2
2
σ(ζ)Mf =
⇒
12n2 Mg2
ζ2
,
6n2 Mg2
2
V˜ (η)Mg = −
,
2
η
˜ (ζ)M 2 = −
U
f
6n2 Mg2
ζ2
(
2
1+
6n
2 ζ2
m2 Meff
)
.
n2
e = 2 ,
t
ξ
 (

)2
(
(
)
) 
n2 
6n2
2
i 2
J 2
J
µ
ν
ξ 2
dt + dx
dsf
=
fµν dx dx = e dsf = 2 − 1 +
.
2 M 2 t2

t 
m
eff
µ,ν=0
3
∑
– Typeset by FoilTEX –
49
When t ∼ 0, redefinition:
α
t˜ ∼ 2 ,
2t
⇒
2n2t˜ ( i)2
∼ −dt +
dx
.
α
t → 0 (Big Bang or Big Rip) ⇔ t˜ → +∞.
( )2
There does not occur singularity in the metric dsJf
because the scale
factor a
˜ which is proportional to t˜ corresponds to the universe filled with
radiation.
– Typeset by FoilTEX –
(
)
J 2
dsf
6n3
α≡ 2 2 2,
m Meff t
˜2
50
Super-luminal mode in bigravity
There can be a signal whose speed is larger than the speed of light.
Speed vg of the massless particle which propagates in the universe described
J
by gµν
or gµν
2
vg2 = (dx/dt) = 1 ⇔ special relativity.
J
Speed vf in fµν
or fµν
2
vf2 = (dx/dt) = c(t)2/b(t)2
If c(t)/b(t) > 1, vf > 1 speed of light in g universe.
˜ = 0: vf = 1 + 62H˜ 22 > 1.
c(t) > 1 except of H
m M
eff
vf is greater than the speed of light. (Causality is not always violated.)
– Typeset by FoilTEX –
51
Summary
• F (R) bigravity in the conventional description with two metrics g and f .
• Explicit and exact solution of FRW equations, Big (and Little) Rip, de Sitter,
quintessence and decelerating universes.
• In general, the physical g cosmological singularity is manifested as metric f cosmological
singularity. However, there are examples where cosmological singularity of physical g
universe does not occur in the universe described by reference metric f and vice-versa.
J
• The massless particle in the space-time given by the metric fµν or fµν
can be
super-luminal.
• Other models, scalar-tensor, Brans-Dicke.
– Typeset by FoilTEX –
52
Bounce Cosmology
Matter bounce scenario
1. In the initial phase of the contraction, the universe is at the matterdominated stage.
2. There happens a bounce without any singularity.
3. The primordial curvature perturbations with the observed spectrum can
also be generated.
K. Bamba, A. N. Makarenko, A. N. Myagky, S. Nojiri and S. D. Odintsov,
“Bounce cosmology from F (R) gravity and F (R) bigravity,”
JCAP01(2014)008 [arXiv:1309.3748 [hep-th]].
– Typeset by FoilTEX –
53
Standard F (R) gravity
2
a ∼ eαt ⇔ F (R) =
1 2
R − 72R + 144α .
α
This kind of model also realizes the Starobinsky inflation.
F (R) bigravity
V˜ (η) = −12α2η 2 ,
ω(η) = 12αMg2η 2 ,
σ(ζ) =
24α Mg2ζ 2
Mf2
2
2
– Typeset by FoilTEX –
,
˜ (ζ) =
U
12α Mg2ζ 2
−
Mf2
(
2
1+
12α Mg2ζ 2
2
m2Meff
)
.
54
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