Imperfect Dark Matter Alexander Vikman 17.04.15 Friday, April 17, - - PowerPoint PPT Presentation
Workshop on Off-the-Beaten-Track Dark Matter and Astrophysical Probes of Fundamental Physics Imperfect Dark Matter Alexander Vikman 17.04.15 Friday, April 17, 15 This talk is mostly based on e-Print: arXiv: 1403.3961 , JCAP 1406 (2014) 017 with
Alexander Vikman
17.04.15
Workshop on Off-the-Beaten-Track Dark Matter and Astrophysical Probes of Fundamental Physics
Friday, April 17, 15
e-Print: arXiv: 1403.3961, JCAP 1406 (2014) 017 with A. H. Chamseddine and V. Mukhanov and e-Print: arXiv: 1412.7136 with L. Mirzagholi
Friday, April 17, 15
Friday, April 17, 15
SM
Friday, April 17, 15
SM
Friday, April 17, 15
SM
Friday, April 17, 15
SM
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no vorticity
SM
Friday, April 17, 15
no vorticity
SM
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Friday, April 17, 15
normalized velocity
Friday, April 17, 15
normalized velocity
maµ =?λ
µ rλm Newton law
Friday, April 17, 15
normalized velocity
maµ =?λ
µ rλm Newton law
⊥µν= gµν − uµuν
with projector
Friday, April 17, 15
the dynamical part of dark sector moves along timelike geodesics
normalized velocity
maµ =?λ
µ rλm Newton law
⊥µν= gµν − uµuν
with projector
Friday, April 17, 15
ϕ → φ
dφ = dϕ/m (ϕ)
uµ = ∂µφ
the dynamical part of dark sector moves along timelike geodesics
normalized velocity
maµ =?λ
µ rλm Newton law
⊥µν= gµν − uµuν
with projector
Friday, April 17, 15
ϕ → φ
dφ = dϕ/m (ϕ)
uµ = ∂µφ
the dynamical part of dark sector moves along timelike geodesics
Constraint or the Hamilton-Jacobi equation
normalized velocity
maµ =?λ
µ rλm Newton law
⊥µν= gµν − uµuν
with projector
Friday, April 17, 15
Friday, April 17, 15
Chamseddine, Mukhanov (2013)
Friday, April 17, 15
Chamseddine, Mukhanov (2013)
One can encode the conformal / scalar part of the physical metric in a scalar field :
gµν
φ
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
Friday, April 17, 15
Chamseddine, Mukhanov (2013)
auxiliary metric
One can encode the conformal / scalar part of the physical metric in a scalar field :
gµν
φ
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
Friday, April 17, 15
Chamseddine, Mukhanov (2013)
auxiliary metric
One can encode the conformal / scalar part of the physical metric in a scalar field :
gµν
φ
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
S [˜ gµν, φ, Φm] = Z d4x √−g ✓ −1 2R (g) + L (g, Φm) ◆
gµν=gµν(˜ g,φ)
Friday, April 17, 15
Chamseddine, Mukhanov (2013)
auxiliary metric
The theory becomes invariant with respect to Weyl transformations:
˜ gµν → Ω2 (x) ˜ gµν
One can encode the conformal / scalar part of the physical metric in a scalar field :
gµν
φ
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
S [˜ gµν, φ, Φm] = Z d4x √−g ✓ −1 2R (g) + L (g, Φm) ◆
gµν=gµν(˜ g,φ)
Friday, April 17, 15
Chamseddine, Mukhanov (2013)
auxiliary metric
The theory becomes invariant with respect to Weyl transformations:
˜ gµν → Ω2 (x) ˜ gµν
The scalar field obeys a constraint (Hamilton-Jacobi equation):
gµν ∂µφ ∂νφ = 1
One can encode the conformal / scalar part of the physical metric in a scalar field :
gµν
φ
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
S [˜ gµν, φ, Φm] = Z d4x √−g ✓ −1 2R (g) + L (g, Φm) ◆
gµν=gµν(˜ g,φ)
Friday, April 17, 15
S [˜ gµν, φ, Φm] = Z d4x √−g ✓ −1 2R (g) + L (g, Φm) ◆
gµν=gµν(˜ g,φ)
with
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
Friday, April 17, 15
S [˜ gµν, φ, Φm] = Z d4x √−g ✓ −1 2R (g) + L (g, Φm) ◆
gµν=gµν(˜ g,φ)
with
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion
Friday, April 17, 15
S [˜ gµν, φ, Φm] = Z d4x √−g ✓ −1 2R (g) + L (g, Φm) ◆
gµν=gµν(˜ g,φ)
with
gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ
is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion
But it is still a system with one degree of freedom + standard two polarizations for the graviton!
Friday, April 17, 15
Nathalie Deruelle and Josephine Rua (2014)
One obtains the same dynamics (the same Einstein equations), if instead of varying the Einstein-Hilbert action with respect to the metric
gµν = F (Ψ, w) `µν + H (Ψ, w) @µΨ@νΨ
gµν
w = `µν@µΨ@νΨ `µν, Ψ
and varies with respect to
w2F ∂ ∂w ✓ H + F w ◆ 6= 0
with and
Friday, April 17, 15
Nathalie Deruelle and Josephine Rua (2014)
One obtains the same dynamics (the same Einstein equations), if instead of varying the Einstein-Hilbert action with respect to the metric
gµν = F (Ψ, w) `µν + H (Ψ, w) @µΨ@νΨ
gµν
w = `µν@µΨ@νΨ `µν, Ψ
and varies with respect to
w2F ∂ ∂w ✓ H + F w ◆ 6= 0
with and
Mimetic gravity is an exception! And it does provide new dynamics!
Friday, April 17, 15
Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)
Friday, April 17, 15
Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)
Weyl-invariance allows one to fix
gµν = ˜ gµν
Friday, April 17, 15
Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)
Weyl-invariance allows one to fix
gµν = ˜ gµν
λ (gµν∂µφ∂νφ − 1)
S [gµν, φ, λ, Φm] = Z d4x√−g ✓ −1 2R (g) + L (g, Φm) + λ (gµν∂µφ∂νφ − 1) ◆
Friday, April 17, 15
Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)
Weyl-invariance allows one to fix
gµν = ˜ gµν
λ (gµν∂µφ∂νφ − 1)
S [gµν, φ, λ, Φm] = Z d4x√−g ✓ −1 2R (g) + L (g, Φm) + λ (gµν∂µφ∂νφ − 1) ◆
The system is equivalent to standard GR + irrotational dust, moving along timelike geodesics with the velocity and energy density
uµ = ∂µφ
ρ = 2λ
Friday, April 17, 15
Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)
Dark Matter
Weyl-invariance allows one to fix
gµν = ˜ gµν
λ (gµν∂µφ∂νφ − 1)
S [gµν, φ, λ, Φm] = Z d4x√−g ✓ −1 2R (g) + L (g, Φm) + λ (gµν∂µφ∂νφ − 1) ◆
The system is equivalent to standard GR + irrotational dust, moving along timelike geodesics with the velocity and energy density
uµ = ∂µφ
ρ = 2λ
Friday, April 17, 15
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Friday, April 17, 15
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !
V (φ)
Friday, April 17, 15
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !
V (φ)
gµν ∂µφ ∂νφ = 1
Convenient to take as time
φ
Friday, April 17, 15
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !
V (φ)
gµν ∂µφ ∂νφ = 1
Convenient to take as time
φ
Adding a potential = adding a function of time in the equation
2 ˙ H + 3H2 = V (t)
Friday, April 17, 15
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !
V (φ)
gµν ∂µφ ∂νφ = 1
Convenient to take as time
φ
Enough freedom to obtain any cosmological evolution! Adding a potential = adding a function of time in the equation
2 ˙ H + 3H2 = V (t)
Friday, April 17, 15
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !
V (φ)
gµν ∂µφ ∂νφ = 1
Convenient to take as time
φ
V (φ) = 1 3 m4φ2 eφ + 1
In particular gives the same cosmological inflation as potential in the standard case
1 2m2φ2
Enough freedom to obtain any cosmological evolution! Adding a potential = adding a function of time in the equation
2 ˙ H + 3H2 = V (t)
Friday, April 17, 15
Even with potential, the energy still moves along the timelike geodesics
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Friday, April 17, 15
Even with potential, the energy still moves along the timelike geodesics
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Friday, April 17, 15
Even with potential, the energy still moves along the timelike geodesics
Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)
Φ = C1 (x) ✓ 1 − H a Z adt ◆ + H a C2 (x)
Here on all scales but in the usual cosmology it is an approximation for superhorizon scales Newtonian potential:
Friday, April 17, 15
Next term in the gradient expansion
Chamseddine, Mukhanov, Vikman (2014)
Friday, April 17, 15
Next term in the gradient expansion
Chamseddine, Mukhanov, Vikman (2014)
“unique” quadratic term
Friday, April 17, 15
Next term in the gradient expansion
Chamseddine, Mukhanov, Vikman (2014)
rµrνφrµrνφ Z d4x √−g φ;µ;νφ;µ;ν = Z d4x √−g ⇣ (⇤φ)2 − Rµνφ;µφ;ν ⌘
is not that useful:
“unique” quadratic term
Friday, April 17, 15
Next term in the gradient expansion
Chamseddine, Mukhanov, Vikman (2014)
rµrνφrµrνφ Z d4x √−g φ;µ;νφ;µ;ν = Z d4x √−g ⇣ (⇤φ)2 − Rµνφ;µφ;ν ⌘
is not that useful:
“unique” quadratic term
Friday, April 17, 15
The scalar field still obeys a constraint (Hamilton-Jacobi equation)
gµν ∂µφ ∂νφ = 1
Friday, April 17, 15
The scalar field still obeys a constraint (Hamilton-Jacobi equation)
gµν ∂µφ ∂νφ = 1
Higher time derivatives can be eliminated just by the differentiation of this Hamilton-Jacobi equation
Friday, April 17, 15
The scalar field still obeys a constraint (Hamilton-Jacobi equation)
gµν ∂µφ ∂νφ = 1
Higher time derivatives can be eliminated just by the differentiation of this Hamilton-Jacobi equation There are only minor changes (rescaling) in the background evolution equations e.g.
2 ˙ H + 3H2 = 2 2 − 3γ V (t)
Friday, April 17, 15
Chamseddine, Mukhanov, Vikman (2014)
Friday, April 17, 15
Chamseddine, Mukhanov, Vikman (2014)
δ ¨ φ + Hδ ˙ φ − c2
s
a2 ∆δφ + ˙ H δφ = 0
s =
with the sound speed
Friday, April 17, 15
Chamseddine, Mukhanov, Vikman (2014)
δ ¨ φ + Hδ ˙ φ − c2
s
a2 ∆δφ + ˙ H δφ = 0
s =
with the sound speed
Newtonian potential:
Friday, April 17, 15
Chamseddine, Mukhanov, Vikman (2014)
δ ¨ φ + Hδ ˙ φ − c2
s
a2 ∆δφ + ˙ H δφ = 0
s =
with the sound speed
Newtonian potential:
Ramazanov,Capela (2014)
cS ∼ 10−5
Friday, April 17, 15
Mirzagholi, Vikman (2014)
Friday, April 17, 15
Mirzagholi, Vikman (2014)
Tµν = εuµuν − p ⊥µν +qµuν + qνuµ
(no potential)
Friday, April 17, 15
Mirzagholi, Vikman (2014)
⊥µν= gµν − uµuν θ = rµuµ
expansion
Tµν = εuµuν − p ⊥µν +qµuν + qνuµ
(no potential)
Friday, April 17, 15
Mirzagholi, Vikman (2014)
⊥µν= gµν − uµuν θ = rµuµ
expansion energy flow
qµ = γ?λ
µrλθ
Tµν = εuµuν − p ⊥µν +qµuν + qνuµ
(no potential)
Friday, April 17, 15
Mirzagholi, Vikman (2014)
⊥µν= gµν − uµuν θ = rµuµ
expansion energy flow
qµ = γ?λ
µrλθ
energy density
ε = 2λ − γ ✓ ˙ θ − 1 2θ2 ◆
Tµν = εuµuν − p ⊥µν +qµuν + qνuµ
(no potential)
Friday, April 17, 15
Mirzagholi, Vikman (2014)
⊥µν= gµν − uµuν θ = rµuµ
expansion energy flow
qµ = γ?λ
µrλθ
energy density
ε = 2λ − γ ✓ ˙ θ − 1 2θ2 ◆
Tµν = εuµuν − p ⊥µν +qµuν + qνuµ
(no potential) pressure
p = −γ ✓ ˙ θ + 1 2θ2 ◆
Friday, April 17, 15
Mirzagholi, Vikman (2014)
Friday, April 17, 15
Mirzagholi, Vikman (2014)
no potential
φ → φ + c
symmetry
Friday, April 17, 15
Mirzagholi, Vikman (2014)
no potential
φ → φ + c
symmetry
Friday, April 17, 15
Mirzagholi, Vikman (2014)
no potential
φ → φ + c
symmetry
Noether current:
Friday, April 17, 15
Mirzagholi, Vikman (2014)
no potential
φ → φ + c
symmetry
Noether current:
charge density
Friday, April 17, 15
Mirzagholi, Vikman (2014)
no potential
φ → φ + c
symmetry
Noether current:
charge density
Friday, April 17, 15
Friday, April 17, 15
Ωµ
E = 1
2"αβγµVγ?λ
αVβ;λ '
rα ~ rβ✓ uγ
in the frame moving with the charges (Eckart frame)
Friday, April 17, 15
Ωµ
E = 1
2"αβγµVγ?λ
αVβ;λ '
rα ~ rβ✓ uγ
in the frame moving with the charges (Eckart frame) in the gradient expansion (without gravity)
Tµν ' (ε + p) UµUν pgµν + O
p ' c2
Sε + O
Friday, April 17, 15
Raychaudhuri at work!
˙ θ = −1 3θ2 − σ2 − Rµνuµuν
Friday, April 17, 15
Raychaudhuri at work!
˙ θ = −1 3θ2 − σ2 − Rµνuµuν
ε = 2 2 − 3γ n + 3c2
S ρext
SPext
Friday, April 17, 15
Raychaudhuri at work!
˙ θ = −1 3θ2 − σ2 − Rµνuµuν
ε = 2 2 − 3γ n + 3c2
S ρext
SPext
DM
Friday, April 17, 15
Raychaudhuri at work!
˙ θ = −1 3θ2 − σ2 − Rµνuµuν
ε = 2 2 − 3γ n + 3c2
S ρext
SPext
DM
S
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
rµJµ = 1 2γ0 (ϕ) θ2
:
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
easy to generate charge during radiation domination époque :
na3 = 9 2 Z t
(tcr∆t)
dt0 a3 ˙ γH2 ' 3 2a3ρrad (tcr) ∆γ
rµJµ = 1 2γ0 (ϕ) θ2
:
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
easy to generate charge during radiation domination époque :
na3 = 9 2 Z t
(tcr∆t)
dt0 a3 ˙ γH2 ' 3 2a3ρrad (tcr) ∆γ
rµJµ = 1 2γ0 (ϕ) θ2
:
∆γ = 2 3 ✓ acr aeq ◆ ' 2 3 zeq zcr Tcr ' Teq ∆γ ' eV ∆γ
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
easy to generate charge during radiation domination époque :
na3 = 9 2 Z t
(tcr∆t)
dt0 a3 ˙ γH2 ' 3 2a3ρrad (tcr) ∆γ
rµJµ = 1 2γ0 (ϕ) θ2
:
∆γ = 2 3 ✓ acr aeq ◆ ' 2 3 zeq zcr Tcr ' Teq ∆γ ' eV ∆γ
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
easy to generate charge during radiation domination époque :
na3 = 9 2 Z t
(tcr∆t)
dt0 a3 ˙ γH2 ' 3 2a3ρrad (tcr) ∆γ
rµJµ = 1 2γ0 (ϕ) θ2
:
∆γ = 2 3 ✓ acr aeq ◆ ' 2 3 zeq zcr Tcr ' Teq ∆γ ' eV ∆γ c2
S
Rappaport, Schwab, Burles, Steigman (2007)
Friday, April 17, 15
Mimetic construction and inflation shift-symmetry breaking needed for DM
easy to generate charge during radiation domination époque :
na3 = 9 2 Z t
(tcr∆t)
dt0 a3 ˙ γH2 ' 3 2a3ρrad (tcr) ∆γ
rµJµ = 1 2γ0 (ϕ) θ2
:
∆γ = 2 3 ✓ acr aeq ◆ ' 2 3 zeq zcr Tcr ' Teq ∆γ ' eV ∆γ 3
S
S
Narimani, Scott, Afshordi(2014)
c2
S
Rappaport, Schwab, Burles, Steigman (2007)
Friday, April 17, 15
Friday, April 17, 15
New large class of Weyl-invariant scalar-tensor theories beyond (but not in contradiction with) Horndeski
Friday, April 17, 15
New large class of Weyl-invariant scalar-tensor theories beyond (but not in contradiction with) Horndeski Can unite part of DM with DE, strongly decouples equation of state from the sound speed
Friday, April 17, 15
New large class of Weyl-invariant scalar-tensor theories beyond (but not in contradiction with) Horndeski Can unite part of DM with DE, strongly decouples equation of state from the sound speed Imperfect DM, with a small sound speed, transport of energy, vorticity and a perfect tracking of the external
Universe.
Friday, April 17, 15
Tanks a lot for atention! New large class of Weyl-invariant scalar-tensor theories beyond (but not in contradiction with) Horndeski Can unite part of DM with DE, strongly decouples equation of state from the sound speed Imperfect DM, with a small sound speed, transport of energy, vorticity and a perfect tracking of the external
Universe.
Friday, April 17, 15