Supersymmetrizing Massive Gravity and Mimetic
Dark Matter
Ola Malaeb
Department of Physics
American University of Beirut
November 27, 2014
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
1 / 32
Plan
Part I: Supersymmetrizing Massive Gravity
Supersymmetry
Whats and Whys of massive Gravity
Motivation
Approach to solve the problem
Some details and final results
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
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Plan
Part II: Hamiltonian Formulation of Mimetic Dark Matter
Introducing Mimetic Dark Matter
Formulating the Hamiltonian. Why?
Results
→ End up with a conclusion and Future Work
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
3 / 32
Plan
Part II: Hamiltonian Formulation of Mimetic Dark Matter
Introducing Mimetic Dark Matter
Formulating the Hamiltonian. Why?
Results
→ End up with a conclusion and Future Work
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
3 / 32
Supersymmetry
Symmetry extension of Poicare - mixing bosons with fermions
Generalizes the Poincare algebra to a superalgebra by including a
spinor generator Q
Q turns a bosonic state into a fermionic one and vice versa
Q | boson >=| fermion >, Q | fermion >=| boson >
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
4 / 32
Supersymmetry
Symmetry extension of Poicare - mixing bosons with fermions
Generalizes the Poincare algebra to a superalgebra by including a
spinor generator Q
Q turns a bosonic state into a fermionic one and vice versa
Q | boson >=| fermion >, Q | fermion >=| boson >
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
4 / 32
Supersymmetry
Symmetry extension of Poicare - mixing bosons with fermions
Generalizes the Poincare algebra to a superalgebra by including a
spinor generator Q
Q turns a bosonic state into a fermionic one and vice versa
Q | boson >=| fermion >, Q | fermion >=| boson >
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
4 / 32
Supersymmetry
Why should we try to search for such a symmetry?
Until early 1980’s, more of a mathematical curiosity than a serious
possibility for the realistic theory of nature.
I
I
Unification of forces
Hierarchy problem
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
5 / 32
Supersymmetry
Why should we try to search for such a symmetry?
Until early 1980’s, more of a mathematical curiosity than a serious
possibility for the realistic theory of nature.
I
I
Unification of forces
Hierarchy problem
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
5 / 32
Supersymmetry
Why should we try to search for such a symmetry?
Until early 1980’s, more of a mathematical curiosity than a serious
possibility for the realistic theory of nature.
I
I
Unification of forces
Hierarchy problem
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
5 / 32
Supersymmetry
Why should we try to search for such a symmetry?
Until early 1980’s, more of a mathematical curiosity than a serious
possibility for the realistic theory of nature.
I
I
Unification of forces
Hierarchy problem
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
5 / 32
Why Massive gravity?
Our universe is accelerating now
⇓
We can either: introduce Dark energy or modify general relativity
⇓
Massive gravity is one of modified gravity theories
⇒ Interpretation of dark energy by changing the dynamics of gravity
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
6 / 32
Why Massive gravity?
Our universe is accelerating now
⇓
We can either: introduce Dark energy or modify general relativity
⇓
Massive gravity is one of modified gravity theories
⇒ Interpretation of dark energy by changing the dynamics of gravity
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
6 / 32
Why Massive gravity?
Our universe is accelerating now
⇓
We can either: introduce Dark energy or modify general relativity
⇓
Massive gravity is one of modified gravity theories
⇒ Interpretation of dark energy by changing the dynamics of gravity
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
6 / 32
Why Massive gravity?
Our universe is accelerating now
⇓
We can either: introduce Dark energy or modify general relativity
⇓
Massive gravity is one of modified gravity theories
⇒ Interpretation of dark energy by changing the dynamics of gravity
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
6 / 32
Massive gravity
M. Fierz and W. Pauli. Proc. Roy. Soc. Lond. A, 173: 211-232, 1939.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
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Massive gravity
V. I. Zakharov. JETP Lett., 12:312, 1970.
H. van Dam and M.J.G. Veltman. Nucl. Phys. B, 22:397, 1970.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
8 / 32
Massive gravity
A. I. Vainshtein. Phys. Lett. B, 39:393, 1972.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
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Massive gravity
D. G. Boulware and S. Deser. Phys. Rev. D, 6:3368, 1972.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
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Higgs Massive gravity
Einstein gravity interacting with four scalar Higgs fields with global
Lorentz symmetry φA with A = 0, 1, 2, 3
Expanding around the vacuum solution - diffeomorphism
invariance broken
Three scalars are absorbed to give the graviton mass
Forth scalar decouples at the linear level
−→ Action reduces to Fierz-Pauli mass term at the linear level
A. Chamseddine and V. Mukhanov. Journal of High Energy Physics, 08:11, 2010.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
11 / 32
Higgs Massive gravity
Einstein gravity interacting with four scalar Higgs fields with global
Lorentz symmetry φA with A = 0, 1, 2, 3
Expanding around the vacuum solution - diffeomorphism
invariance broken
Three scalars are absorbed to give the graviton mass
Forth scalar decouples at the linear level
−→ Action reduces to Fierz-Pauli mass term at the linear level
A. Chamseddine and V. Mukhanov. Journal of High Energy Physics, 08:11, 2010.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
11 / 32
Motivation For Our Work
Generalize this to the supersymmetric case
The question of the existence of a supersymmetric theory
propagating a massive graviton is a very interesting one
In supergravity, the gravitino gains mass by the super-Higgs effect
This is a way to give the graviton and the gravitino mass
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
12 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
GOING TO SUPERSYMMETRY
Consider four N = 1 chiral superfields with global Lorentz
symmetry instead of four scalars
Component fields:
I
I
I
ϕ: a complex scalar field
ψ: left-handed spinor field
F : complex scalar auxiliary field
Matter multiplets are formed out of these superfield (D-type and
F-type terms)
Couple to the Supergravity Lagrangian using the rules of tensor
calculus
Lagrangian field content:
I
I
I
spin-2 field, eaµ
spin-3/2 field, φµ
auxiliary fields u, Aµ
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
13 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
WHAT WE WANT?
Expanding the fields around the vacuum solution
ϕA = x A + χA ,
eAµ = δAµ + ēAµ ,
(1)
An action with the following required conditions
I
I
I
I
A Fierz-Pauli term for the vierbeins (ēAµ ēµA − ē2 )
No linear vierbein term
Maxwell form for the χA fields
l ∂µ χA ∂ µ χA∗ − ∂A χA ∂B χB∗
where l is a constant
Ghost free: no terms like
∂µ χA ∂ µ χA ,
I
(2)
or
∂A χA ∂B χB
(3)
Massive gravitino
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
14 / 32
MAIN RESULTS
Global supersymmetry promoted to a local one using the rules of
tensor calculus
Scalar components of the chiral multiplets ϕA acquired a vacuum
expectation value
−→ Diffeomorphism invariance and local supersymmetry broken
spontaneously
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
15 / 32
MAIN RESULTS
Global supersymmetry promoted to a local one using the rules of
tensor calculus
Scalar components of the chiral multiplets ϕA acquired a vacuum
expectation value
−→ Diffeomorphism invariance and local supersymmetry broken
spontaneously
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
15 / 32
MAIN RESULTS
Global supersymmetry promoted to a local one using the rules of
tensor calculus
Scalar components of the chiral multiplets ϕA acquired a vacuum
expectation value
−→ Diffeomorphism invariance and local supersymmetry broken
spontaneously
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
15 / 32
MAIN RESULTS
Global supersymmetry promoted to a local one using the rules of
tensor calculus
Scalar components of the chiral multiplets ϕA acquired a vacuum
expectation value
−→ Diffeomorphism invariance and local supersymmetry broken
spontaneously
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
15 / 32
MAIN RESULTS
Global Lorentz index A identified with the space-time Lorentz
index
I
I
Scalar fields χA → vectors
Chiral spinors ψA → spin-3/2 Rarita-Schwinger fields
Spectrum of the model in the broken phase:
I
I
I
A massive spin-2 field
Two massive spin-3/2 fields with different masses
A massive vector
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
16 / 32
MAIN RESULTS
Global Lorentz index A identified with the space-time Lorentz
index
I
I
Scalar fields χA → vectors
Chiral spinors ψA → spin-3/2 Rarita-Schwinger fields
Spectrum of the model in the broken phase:
I
I
I
A massive spin-2 field
Two massive spin-3/2 fields with different masses
A massive vector
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
16 / 32
DEGREES OF FREEDOM: BEFORE COUPLING
Counting degrees of freedom: supergravity coupled to a N = 1
supersymmetry model similar to the Wess-Zumino model
Before the coupling, supergravity contains
I
I
one massless spin-2 graviton (two bosonic dof)
one massless spin-3/2 gravitino (two fermionic dof)
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
17 / 32
DEGREES OF FREEDOM: BEFORE COUPLING
Counting degrees of freedom: supergravity coupled to a N = 1
supersymmetry model similar to the Wess-Zumino model
Before the coupling, supergravity contains
I
I
one massless spin-2 graviton (two bosonic dof)
one massless spin-3/2 gravitino (two fermionic dof)
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
17 / 32
DEGREES OF FREEDOM: BEFORE COUPLING
N = 1 supersymmetry model:
I
I
Four spin-0 particles, ϕA , with only six degrees of freedom (3 times
2) since ϕ0 decouples due to Fierz-Pauli choice
Six fermionic degrees of freedom forming a multiplet
→ Overall eight fermionic degrees of freedom and eight bosonic
degrees of freedom
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
18 / 32
DEGREES OF FREEDOM: BEFORE COUPLING
N = 1 supersymmetry model:
I
I
Four spin-0 particles, ϕA , with only six degrees of freedom (3 times
2) since ϕ0 decouples due to Fierz-Pauli choice
Six fermionic degrees of freedom forming a multiplet
→ Overall eight fermionic degrees of freedom and eight bosonic
degrees of freedom
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
18 / 32
DEGREES OF FREEDOM: AFTER COUPLING
After coupling to supergravity:
Bosonic degrees of freedom
I
I
A massive spin-2 particle, with five degrees of freedom
A massive vector field (spin-1 particle), three degrees of freedom
−→ Overall eight bosonic degrees of freedom
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
19 / 32
DEGREES OF FREEDOM: AFTER COUPLING
After coupling to supergravity:
Bosonic degrees of freedom
I
I
A massive spin-2 particle, with five degrees of freedom
A massive vector field (spin-1 particle), three degrees of freedom
−→ Overall eight bosonic degrees of freedom
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
19 / 32
DEGREES OF FREEDOM: AFTER COUPLING
Fermionic degrees of freedom
I
Two massive spin-3/2 particles, φµ and ψA , with four degrees of
freedom each
Overall eight fermionic degrees of freedom
−→ Same number of degrees of freedom as before coupling
O. Malaeb. Phys.Rev. D, 88:025002, 2013.
O. Malaeb. Eur. Phys. J. C, 73:2549, 2013.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
20 / 32
DEGREES OF FREEDOM: AFTER COUPLING
Fermionic degrees of freedom
I
Two massive spin-3/2 particles, φµ and ψA , with four degrees of
freedom each
Overall eight fermionic degrees of freedom
−→ Same number of degrees of freedom as before coupling
O. Malaeb. Phys.Rev. D, 88:025002, 2013.
O. Malaeb. Eur. Phys. J. C, 73:2549, 2013.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
20 / 32
Mimetic Dark matter
Chamseddine and Mukhanov modified Einstein’s theory of gravity
to propose a theory of mimetic dark matter.
Their theory is a conformal extension of Einstein’s general
relativity.
A physical metric, gµν , in terms of an auxiliary metric, g̃µν , and a
scalar field φ considered
gµν = g̃ αβ ∂α φ∂β φ g̃µν ,
(4)
Physical metric invariant with respect to the conformal
transformation of the auxiliary metric, g̃µν → Ω2 g̃µν .
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
21 / 32
Mimetic Dark matter
Chamseddine and Mukhanov modified Einstein’s theory of gravity
to propose a theory of mimetic dark matter.
Their theory is a conformal extension of Einstein’s general
relativity.
A physical metric, gµν , in terms of an auxiliary metric, g̃µν , and a
scalar field φ considered
gµν = g̃ αβ ∂α φ∂β φ g̃µν ,
(4)
Physical metric invariant with respect to the conformal
transformation of the auxiliary metric, g̃µν → Ω2 g̃µν .
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
21 / 32
Mimetic Dark matter
Chamseddine and Mukhanov modified Einstein’s theory of gravity
to propose a theory of mimetic dark matter.
Their theory is a conformal extension of Einstein’s general
relativity.
A physical metric, gµν , in terms of an auxiliary metric, g̃µν , and a
scalar field φ considered
gµν = g̃ αβ ∂α φ∂β φ g̃µν ,
(4)
Physical metric invariant with respect to the conformal
transformation of the auxiliary metric, g̃µν → Ω2 g̃µν .
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
21 / 32
Mimetic Dark matter
Chamseddine and Mukhanov modified Einstein’s theory of gravity
to propose a theory of mimetic dark matter.
Their theory is a conformal extension of Einstein’s general
relativity.
A physical metric, gµν , in terms of an auxiliary metric, g̃µν , and a
scalar field φ considered
gµν = g̃ αβ ∂α φ∂β φ g̃µν ,
(4)
Physical metric invariant with respect to the conformal
transformation of the auxiliary metric, g̃µν → Ω2 g̃µν .
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
21 / 32
Mimetic Dark matter
The scalar field satisfies the constraint equation
g µν ∂µ φ∂ν φ = 1.
(5)
Action of the extended mimetic dark matter theory
Z
1
4
1/2 1
µν
S = − d x(−g)
R + λ (1 − g ∂µ φ∂ν φ) + V (φ)
(6)
2
2
Even in the absence of matter, Tµν = 0, the conformal degree of
freedom can mimic Dark matter.
A. H. Chamseddine and V. Mukhanov. JHEP, 1311:135, 2013.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
22 / 32
Mimetic Dark matter
The scalar field satisfies the constraint equation
g µν ∂µ φ∂ν φ = 1.
(5)
Action of the extended mimetic dark matter theory
Z
1
4
1/2 1
µν
S = − d x(−g)
R + λ (1 − g ∂µ φ∂ν φ) + V (φ)
(6)
2
2
Even in the absence of matter, Tµν = 0, the conformal degree of
freedom can mimic Dark matter.
A. H. Chamseddine and V. Mukhanov. JHEP, 1311:135, 2013.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
22 / 32
Mimetic Dark matter
The scalar field satisfies the constraint equation
g µν ∂µ φ∂ν φ = 1.
(5)
Action of the extended mimetic dark matter theory
Z
1
4
1/2 1
µν
S = − d x(−g)
R + λ (1 − g ∂µ φ∂ν φ) + V (φ)
(6)
2
2
Even in the absence of matter, Tµν = 0, the conformal degree of
freedom can mimic Dark matter.
A. H. Chamseddine and V. Mukhanov. JHEP, 1311:135, 2013.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
22 / 32
Why Hamiltonian?
The Hamiltonian formulations provide insights mainly for
I
I
I
counting degrees of freedom
canonical quantization purposes
numerical investigation of solutions
The first step in constructing the canonical formalism is to write
the action in a 3 + 1 dimensional form by splitting spacetime into
space and time.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
23 / 32
Why Hamiltonian?
The Hamiltonian formulations provide insights mainly for
I
I
I
counting degrees of freedom
canonical quantization purposes
numerical investigation of solutions
The first step in constructing the canonical formalism is to write
the action in a 3 + 1 dimensional form by splitting spacetime into
space and time.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
23 / 32
Hamiltonian
The action is given by
q
Z
4
0
˙
ij
ij
St = d x h πij + pφ̇ − N p h ∂i φ∂j φ − 1 + R
√
− N i (p∂i φ + Ri ) − N hV (φ)
(7)
N and N i still appear linearly
q
Constraints: H = p hij ∂i φ∂j φ − 1 + R 0
Hi = p∂i φ + Ri
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
24 / 32
Equations of motion
Equations of motion of hij and πij are those of general relativity
plus an extra contribution from φ.
Two more equations are present by varying φ and its conjugated
momentum pφ .
The equation for the scalar
field φ is exactly the constraint
equation g µν ∂µ φ∂ν φ = 1.
That for the momentum is
the conservation of the
energy-momentum tensor.
→ Same number of equations as in GR.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
25 / 32
Equations of motion
Equations of motion of hij and πij are those of general relativity
plus an extra contribution from φ.
Two more equations are present by varying φ and its conjugated
momentum pφ .
The equation for the scalar
field φ is exactly the constraint
equation g µν ∂µ φ∂ν φ = 1.
That for the momentum is
the conservation of the
energy-momentum tensor.
→ Same number of equations as in GR.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
25 / 32
Poisson brackets
Constraints
H = Hgrav + Hφ ;
Hi = Hi grav + Hi φ .
(8)
The constraints contain extra pieces, but their algebra remains
unchanged.
O. Malaeb. arXiv:1404.4195, 2014.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
26 / 32
SUMMARY
Constructed a supersymmetric theory of massive gravity
Coupled a pure Supergravity multiplet to a matter multiplet
Once the scalar fields of the chiral multiplets acquire a vacuum
expectation value
−→ local supersymmetry is broken exactly at the same scale as
the diffeomorphism breaking
Left with a massive vector, two massive spin-3/2 fields with
different masses, and a massive spin-2 particle.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
27 / 32
SUMMARY
Constructed a supersymmetric theory of massive gravity
Coupled a pure Supergravity multiplet to a matter multiplet
Once the scalar fields of the chiral multiplets acquire a vacuum
expectation value
−→ local supersymmetry is broken exactly at the same scale as
the diffeomorphism breaking
Left with a massive vector, two massive spin-3/2 fields with
different masses, and a massive spin-2 particle.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
27 / 32
SUMMARY
Constructed a supersymmetric theory of massive gravity
Coupled a pure Supergravity multiplet to a matter multiplet
Once the scalar fields of the chiral multiplets acquire a vacuum
expectation value
−→ local supersymmetry is broken exactly at the same scale as
the diffeomorphism breaking
Left with a massive vector, two massive spin-3/2 fields with
different masses, and a massive spin-2 particle.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
27 / 32
SUMMARY
Constructed a supersymmetric theory of massive gravity
Coupled a pure Supergravity multiplet to a matter multiplet
Once the scalar fields of the chiral multiplets acquire a vacuum
expectation value
−→ local supersymmetry is broken exactly at the same scale as
the diffeomorphism breaking
Left with a massive vector, two massive spin-3/2 fields with
different masses, and a massive spin-2 particle.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
27 / 32
SUMMARY
Similar to the N = 2 supersymmetry in which we have two
gravitinos, but there they have the same mass.
Different masses since there is no N=2 supersymmetry to start
with
Before diffeomorphism breaking:
spin-1/2 and not spin-3/2 → two gravitinos with different masses
I
I
One is a genuine gravitino
Second identified with a gravitino after the breaking
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
28 / 32
SUMMARY
Similar to the N = 2 supersymmetry in which we have two
gravitinos, but there they have the same mass.
Different masses since there is no N=2 supersymmetry to start
with
Before diffeomorphism breaking:
spin-1/2 and not spin-3/2 → two gravitinos with different masses
I
I
One is a genuine gravitino
Second identified with a gravitino after the breaking
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
28 / 32
SUMMARY
Similar to the N = 2 supersymmetry in which we have two
gravitinos, but there they have the same mass.
Different masses since there is no N=2 supersymmetry to start
with
Before diffeomorphism breaking:
spin-1/2 and not spin-3/2 → two gravitinos with different masses
I
I
One is a genuine gravitino
Second identified with a gravitino after the breaking
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
28 / 32
SUMMARY
Similar to the N = 2 supersymmetry in which we have two
gravitinos, but there they have the same mass.
Different masses since there is no N=2 supersymmetry to start
with
Before diffeomorphism breaking:
spin-1/2 and not spin-3/2 → two gravitinos with different masses
I
I
One is a genuine gravitino
Second identified with a gravitino after the breaking
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
28 / 32
SUMMARY
Similar to the N = 2 supersymmetry in which we have two
gravitinos, but there they have the same mass.
Different masses since there is no N=2 supersymmetry to start
with
Before diffeomorphism breaking:
spin-1/2 and not spin-3/2 → two gravitinos with different masses
I
I
One is a genuine gravitino
Second identified with a gravitino after the breaking
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
28 / 32
Summary
Mimetic dark matter was studied in its canonical form.
Hamiltonian constructed and equations of motion analyzed
Number of equations is equal to that in general relativity plus two
more equations coming from the variation with respect to the two
extra phase variables
Equation for the scalar field was reinterpreted as the conservation
of the energy-momentum tensor and the other is just the
constraint equation.
An extra field, φ, modifying Einstein’s equations while keeping the
same number of equations. This is how dark matter is
represented.
Poisson brackets are computed and closure is proved.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
29 / 32
Summary
Mimetic dark matter was studied in its canonical form.
Hamiltonian constructed and equations of motion analyzed
Number of equations is equal to that in general relativity plus two
more equations coming from the variation with respect to the two
extra phase variables
Equation for the scalar field was reinterpreted as the conservation
of the energy-momentum tensor and the other is just the
constraint equation.
An extra field, φ, modifying Einstein’s equations while keeping the
same number of equations. This is how dark matter is
represented.
Poisson brackets are computed and closure is proved.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
29 / 32
Summary
Mimetic dark matter was studied in its canonical form.
Hamiltonian constructed and equations of motion analyzed
Number of equations is equal to that in general relativity plus two
more equations coming from the variation with respect to the two
extra phase variables
Equation for the scalar field was reinterpreted as the conservation
of the energy-momentum tensor and the other is just the
constraint equation.
An extra field, φ, modifying Einstein’s equations while keeping the
same number of equations. This is how dark matter is
represented.
Poisson brackets are computed and closure is proved.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
29 / 32
Summary
Mimetic dark matter was studied in its canonical form.
Hamiltonian constructed and equations of motion analyzed
Number of equations is equal to that in general relativity plus two
more equations coming from the variation with respect to the two
extra phase variables
Equation for the scalar field was reinterpreted as the conservation
of the energy-momentum tensor and the other is just the
constraint equation.
An extra field, φ, modifying Einstein’s equations while keeping the
same number of equations. This is how dark matter is
represented.
Poisson brackets are computed and closure is proved.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
29 / 32
Summary
Mimetic dark matter was studied in its canonical form.
Hamiltonian constructed and equations of motion analyzed
Number of equations is equal to that in general relativity plus two
more equations coming from the variation with respect to the two
extra phase variables
Equation for the scalar field was reinterpreted as the conservation
of the energy-momentum tensor and the other is just the
constraint equation.
An extra field, φ, modifying Einstein’s equations while keeping the
same number of equations. This is how dark matter is
represented.
Poisson brackets are computed and closure is proved.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
29 / 32
Summary
Mimetic dark matter was studied in its canonical form.
Hamiltonian constructed and equations of motion analyzed
Number of equations is equal to that in general relativity plus two
more equations coming from the variation with respect to the two
extra phase variables
Equation for the scalar field was reinterpreted as the conservation
of the energy-momentum tensor and the other is just the
constraint equation.
An extra field, φ, modifying Einstein’s equations while keeping the
same number of equations. This is how dark matter is
represented.
Poisson brackets are computed and closure is proved.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
29 / 32
Further work
Supersymmetric massive gravity using the quadratic formulation requires a superspace formulation of supergravity.
Structure at the non-linear level - analysis at higher orders may
reveal ghosts.
Further work could be done on applications of the Hamiltonian
quantization such as to solve the Wheeler-DeWitt equations.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
30 / 32
References
M. Fierz and W. Pauli. On relativistic wave equations for particles of arbitrary spin in an
electromagnetic field. Proc. Roy. Soc. Lond. A, 173: 211-232, 1939.
V. I. Zakharov. Linearized gravitation theory and the graviton mass. JETP Lett., 12:312,
1970.
H. van Dam and M.J.G. Veltman. Massive and massless yang-mills and gravitational fields.
Nucl. Phys. B, 22:397, 1970.
A. I. Vainshtein. To the problem of nonvanishing graviton mass. Phys. Lett. B, 39:393,
1972.
D. G. Boulware and S. Deser. Can gravity have a finite range? Phys. Rev. D, 6:3368, 1972.
A. Chamseddine and V. Mukhanov. Higgs for graviton: Simple and elegant solution.
Journal of High Energy Physics, 08:11, 2010.
O. Malaeb. Supersymmetrizing massive gravity. Phys.Rev. D, 88:025002, 2013.
O. Malaeb. Massive gravity with n=1 local supersymmetry. Eur. Phys. J. C, 73:2549, 2013.
A. H. Chamseddine and V. Mukhanov. Mimetic dark matter. JHEP, 1311:135, 2013.
O. Malaeb. Hamiltonian formulation of mimetic gravity. arXiv:1404.4195, 2014.
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
31 / 32
Acknowledgments
I would like to thank
My supervisor Ali Chamseddine for his guidance and support
All my committee members
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Khalil Bitar
Jihad Touma
Lars brink
Viatcheslav Mukhanov
All the professors in the Physics Department and the Chairman for
helping in all the arrangements
Thank you to all my friends who made my stay at AUB pleasant.
Family and husband
Ola Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity and Mimetic Dark Matter
November 27, 2014
32 / 32