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Dark Matter with Massive Bigravity Theory
Luc Blanchet
Gravitation et Cosmologie (GRεCO)
Institut d’Astrophysique de Paris
27 mars 2017
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Mass discrepancy versus acceleration
[Milgrom 1983]
A critical acceleration scale a0 ' 1.2 × 10−10 m/s2 is present in the data
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The critical acceleration scale
The measured value of a0 is very close (mysteriously enough) to typical
cosmological values
a0 ' 1.3 aΛ
with
aΛ =
c2
2π
r
Λ
3
The mass discrepancy is given by
Mdyn
'
Mb
V
Vb
2
r
∼
a0
g
The scale a0 gives the transition between galaxies with high and low central
surface brightness

a0


 Σ & G for HSB galaxies (baryon dominated)

a

 Σ . 0 for LSB galaxies (DM dominated)
G
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HSB versus LSB galaxies
High Surface Brightness
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Low Surface Brightness
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The MOND formula
[Milgrom 1983, Bekenstein & Milgrom 1984]
Rotation curves of galaxies are extremely well reproduced by the MOND formula
∇·
g
g = −4π G ρbaryon
µ
a0
| {z }
avec
g = ∇U
MOND function
The Newtonian regime is recovered when g a0
In the MOND regime g a0 we have µ ' g/a0

Newtonian regime g≫ a 

MOND regime g≪ a

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
g
a
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Baryonic Tully-Fisher relation
[Tully & Fisher 1977, McGaugh 2011]
The asymptotic flat rotation velocity is approximately Vf ' G Mb a0
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DM with massive bigravity
1/4
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BTF relation fitted with Λ-CDM
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DM with massive bigravity
[Silk & Mamon 2012]
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Modified gravity versus modified dark matter
Extra fields are postulated in addition to the spin-2 metric field of GR
1
Modified gravity: The stress-energy tensor of these fields is negligible with
respect to the baryonic matter density
µν
µν
µ
ν
G
= 8π Tb + |∇ Φ∇
{z Φ} + · · ·
Φ
2
=
..
.
8π Tb
negligible w.r.t.
baryonic density
Modified dark matter: The stress-energy tensor of the extra fields is
comparable to the baryonic matter density
µν
µν
(µ ν)
G
= 8π Tb + ρ u Ω + · · ·
| {z }
ν
µ
same order as
u ∇ν Ω
= ···
baryonic density
..
.
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Pure modified gravity: TeVeS
1
[Bekenstein & Sanders 1994; Bekenstein 2004]
A disformal coupling between the physical (Jordan) and
Einstein metrics is postulated in the matter action
g̃µν = e2φ (gµν + Uµ Uν ) − e−2φ Uµ Uν
2
The vector field is a dynamical field
Z
SU = C
√
d4 x −g g µρ g νσ Uµν Uρσ
|
{z
}
U (1) like kinetic term
3
The scalar part is a k-essence theory
Z
Sφ = K
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d4 x
√
−g f (X) with X = ` (g µν − U µ U ν ) ∂µ φ∂ν φ
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Tentative conclusions on modified gravity
1
Complicated theories modifying GR with ad-hoc extra fields
2
No physical explanation for the origin of the MOND effect
3
Non-standard kinetic terms depending on an arbitrary function which is
linked in fine to the MOND function
4
Stability problems associated with the fact that the Hamiltonian is not
bounded from below [Clayton 2001, Bruneton & Esposito-Farèse 2007]
5
Generic problems to recover the cosmological model Λ-CDM at large scales
and in particular the spectrum of CMB anisotropies
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TeVeS and the CMB
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[Skordis, Mota, Ferreira & Boehm 2006]
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Dipolar dark matter?
1
[Blanchet 2006; Blanchet & Le Tiec 2008, 2009]
In electrostratics the Gauss equation is modified by the polarization of the
dielectric (dipolar) material
h
i ρ
e
∇ · (1 + χe )E =
| {z }
ε0
⇐⇒
∇·E =
ρe + ρpolar
e
ε0
D field
2
Similarly MOND can be viewed as a modification of the Poisson equation by
the polarization of some dipolar medium
g
∇· µ
g = −4π G ρb
⇐⇒
∇ · g = −4π G ρb +ρpolar
| {z }
a0
dark matter
3
The MOND function can be written µ = 1 + χ where χ appears as a
susceptibility coefficient of some “dipolar dark matter” medium
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DM with massive bigravity
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Microscopic description of DDM?
1
The DM medium by individual dipole moments p and a polarization field P
P = np
2
p = mξ
The polarization is induced by the gravitational field of ordinary masses
P =−
3
with
χ
g
4π G
ρDDM = −∇ · P
Because like masses attract and unlike ones repel we have anti-screening of
ordinary masses by polarization masses in agreement with DM and MOND
χ<0
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Anti-screening by polarization masses
+
-
+
-
pe
+
-
lines of
-
+
E
+
+
+
+
p
- +
g
+ -
charge Q
mass M
Anti-screening by polarization masses
χe > 0
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-
+ -
- +
Screening by polarization charges
lines of
χ<0
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Need of a non-gravitational internal force
1
The constituents of the dipole will repel each other so we need a
non-gravitational force
dv 0
= −∇ U + φ
dt
and look for an equilibrium when ∇ U + φ = 0
dv
=∇ U +φ
dt
2
The internal force is generated by the gravitational charge i.e. the mass
∆φ = −
3
4πG
ρ − ρ0
χ
The DM medium appears as a polarizable plasma (similar to e+ e− plasma)
oscillating at the natural plasma frequency
d2 ξ
+ ω 2 ξ = 2g
dt2
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s
with
DM with massive bigravity
ω=
−
8π G ρ0
χ
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DDM via a bimetric extension of GR
1
[Bernard & Blanchet 2014]
To describe relativistically some microscopic DM particles with positive or
negative gravitational masses one needs two metrics
gµν obeyed by ordinary particles (including baryons)
fµν obeyed by “dark” particles
2
In addition the DM particles forming the dipole moment
should interact via a non-gravitational force field,
e.g. a (spin-1) “graviphoton” vector field Aµ [Scherk 1968]
3
One needs to introduce into the action the kinetic terms for all these fields,
and one must be careful to properly define the interaction between the two
metrics gµν and fµν
4
A model was proposed but the interaction between the two metrics implies
the presence of ghosts in the gravitational sector [Blanchet & Heisenberg 2015]
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Nonlinear ghost-free massive bigravity theory
[de Rham, Gabadadze & Tolley 2011; Hassan & Rosen 2012]
The theory (called dRGT) is defined non-perturbatively as
ghost-free interaction mass term
Z
S=
z
}|
{
4
2
X
Mf2 p
M
√
√
g
2
4
−gRg + m −g
αn en (X) +
−f Rf
d x
2
2
n=0
where the interaction between the two metrics is algebraic and defined from
p the
elementary symmetric polynomials en (X) of the square root matrix X = g −1 f
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DDM via massive bigravity theory
[Blanchet & Heisenberg 2015]
1
The two species of DM particles ρg and ρf are coupled
to the two metrics gµν and fµν of bigravity
2
The ghost-free potential interactions take the form of
the square root of the determinant of the effective metric
[de Rham, Heisenberg & Ribeiro 2014]
eff
gµν
= α2 gµν + 2αβ gµρ Xνρ + β 2 fµν
3
eff
The graviphoton Aµ is coupled to the effective metric gµν
and has a
µρ νσ
non-standard kinetic term (with X = geff geff Fµν Fρσ )
Z
S=
Luc Blanchet (IAP)
2
p
Mf
Mg2
−g
d x
Rg −ρbar − ρg + −f
Rf −ρf
2
2
2
√
m
α µ
a20
µ
+ −geff
+ Aµ jg − jf +
W X
4π
β
8π
4
√
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General structure of the model

bar
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GR g 
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General structure of the model

bar
g
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GR g 
geff

A
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f

f
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General structure of the model
m
G

bar
g
GR g 
geff

A
f

f
a0
Luc Blanchet (IAP)
DM with massive bigravity
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General structure of the model
m
G

bar
g
GR g 
geff

A
f

f
a0 m  
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DM with massive bigravity
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Gravitational polarization & MOND
1
Equations of motion of DM particles in the non-relativistic limit c → ∞
dvg
= ∇ Ug + φ
dt
2
α dvf
= ∇ Uf − φ
dt
β
With massive bigravity the two g and f sectors are linked together by a
constraint equation coming from the Bianchi identities
∇ αUg + βUf = 0
3
showing that α/β is the ratio between gravitational and inertial masses of f
particles with respect to g metric
The DM medium is at equilibrium when the Coulomb force annihilates the
gravitational force, ∇Ug + ∇φ = 0, at which point the polarization is aligned
with the gravitational field
P =
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1
W 0 ∇Ug
4π
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Gravitational polarization & MOND
1
From the massless combination of the two metrics combined with the Bianchi
identity we get a Poisson equation for the ordinary Newtonian potential Ug
α ∆Ug = −4π ρbar + ρg − ρf
β
| {z }
DDM
2
With the plasma-like solution for the internal force and the mechanism of
gravitational polarization this yields the MOND equation
∇·
h
i
1 − W 0 ∇Ug = −4πρbar
| {z }
MOND function
3
Finally the DM medium undergoes stable plasma-like oscillations in linear
perturbations around the equilibrium
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Status of the model regarding ghost instabilities
1
By construction the model is safe in the gravitational sector (no ghosts)
2
The particles ρbar , ρg and ρf are separately coupled to one and only one of
the metrics of bigravity which is also safe
3
The vector field Aµ is coupled to the effective composite metric gµeffν and it
has been shown that the decoupling limit is safe i.e. the mass of possible
guosts is above the strong coupling scale [de Rham, Heisenberg & Ribeiro 2014]
Λ3 = m2 MPlanck
1/3
4
However in our model the DM particles ρg , ρf are also coupled to the
internal vector Aµ and this introduces an extra indirect interaction between
the two metrics of bigravity
5
Thus we have to check the decoupling limit of the model consistently
including all the interactions with matter fields
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Decoupling limit of the theory
1
2
[Blanchet & Heisenberg 2017]
We decouple the interactions below the strong coupling scale and concentrate
on the pure interactions of the helicity-0 mode π of the massive graviton
Using the Stückelberg trick, we restore the broken gauge invariance in the
fµν metric by replacing it by
fµν −→ fαβ ∂µ φα ∂ν φβ
where
3
φα = x α −
f αβ ∂β π
Λ33
The two g and f metrics are given by (posing Πµν = ∂µ ∂ν π)
helicity-2 mode
gµν =
ηµν
fµν =
ηµν
z}|{ 2
hµν
+
Mg
2
Πµν
−
Λ33
|{z}
helicity-0 mode
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Decoupling limit of the theory
1
2
[Blanchet & Heisenberg 2017]
We send MPl → ∞ and m → 0 while keeping Λ3 = (m2 MPl )1/3 constant
In the DL limit the gravitational part of the action takes the ghost-free form
Galileon terms
LDL
grav
3
µν
= −ĥ
ρσ
ĥρσ
Eµν
z
}|
{
3
X
bn
a3
2
(3)
+
(∂π)
(Π)
e
(Π)
+ 6 ĥµν Pµν
(n)
3n
Λ
Λ
3
n=0 3
We must specify how the vector field Aµ will behave in the DL limit. Since it
is a graviphoton, with scale given by the Planck mass, we must rescale it as
Aµ =
4
µ
MPl
When this is done we find that the contribution of the matter to the equation
of motion of the helicity-0 mode is perfectly safe
a2
δLDL
a1
µν
mat
=−
T̂bar + T̂g + 3 T̂bar
+ T̂gµν ∂µ ∂ν π
δπ
2
Λ3
|
{z
}
second-order field equation
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Conclusions
1
An important challenge is to reproduce within a single relativistic framework
The concordance cosmological model Λ-CDM and its tremendous successes at
cosmological scales and notably the fit of the CMB
The phenomenology of MOND which is a basic set of phenomena relevant to
galaxy dynamics and DM distribution at galactic scales
2
While Λ-CDM meets severe problems when extrapolated at the scale of
galaxies, MOND fails on galaxy cluster and larger scales
3
Pure modified gravity theories (extending GR with extra fields) do not seem
to be able to reproduce Λ-CDM and the CMB anisotropies at large scales
4
An hybrid approach (DM à la MOND) called DDM, defined within the
powerful framework of the massive bigravity theory, could meet the
challenges posed by MOND versus Λ-CDM
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