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 A. H. Chamseddine and V. Mukhanov and e-Print: arXiv: 1412.7136 with L. Mirzagholi Friday, April 17, 15
Friday, April 17, 15
SM 5% Friday, April 17, 15
SM DM 5% 27% Friday, April 17, 15
SM DE DM 5% 27% Friday, April 17, 15
SM DE DM 5% Inflation 27% Friday, April 17, 15
SM DE DM 5% Inflation 27% no vorticity on large scales Friday, April 17, 15
SM DE DM 5% Inflation 27% no vorticity on large scales u µ ∝ ∂ µ ϕ Friday, April 17, 15
Friday, April 17, 15
u µ = ∂ µ ϕ /m normalized velocity Friday, April 17, 15
u µ = ∂ µ ϕ /m normalized velocity ma µ = ? λ µ r λ m Newton law Friday, April 17, 15
u µ = ∂ µ ϕ /m normalized velocity ma µ = ? λ µ r λ m Newton law ⊥ µ ν = g µ ν − u µ u ν with projector Friday, April 17, 15
u µ = ∂ µ ϕ /m normalized velocity ma µ = ? λ µ r λ m Newton law ⊥ µ ν = g µ ν − u µ u ν with projector the dynamical part of m = m ( ϕ ) dark sector moves along timelike geodesics Friday, April 17, 15
u µ = ∂ µ ϕ /m normalized velocity ma µ = ? λ µ r λ m Newton law ⊥ µ ν = g µ ν − u µ u ν with projector the dynamical part of m = m ( ϕ ) dark sector moves along timelike geodesics u µ = ∂ µ φ ϕ → φ d φ = d ϕ /m ( ϕ ) Friday, April 17, 15
u µ = ∂ µ ϕ /m normalized velocity ma µ = ? λ µ r λ m Newton law ⊥ µ ν = g µ ν − u µ u ν with projector the dynamical part of m = m ( ϕ ) dark sector moves along timelike geodesics u µ = ∂ µ φ ϕ → φ d φ = d ϕ /m ( ϕ ) g µ ν ∂ µ φ ∂ ν φ = 1 Constraint or the Hamilton-Jacobi equation Friday, April 17, 15
How to implement this constraint? Friday, April 17, 15
Mimetic Matter Chamseddine, Mukhanov (2013) Friday, April 17, 15
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal / scalar part of the physical metric in a scalar field : φ g µ ν g αβ ∂ α φ ∂ β φ g µ ν (˜ g, φ ) = ˜ g µ ν ˜ Friday, April 17, 15
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal / scalar part of the physical metric in a scalar field : φ g µ ν g αβ ∂ α φ ∂ β φ g µ ν (˜ g, φ ) = ˜ g µ ν ˜ auxiliary metric Friday, April 17, 15
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal / scalar part of the physical metric in a scalar field : φ g µ ν g αβ ∂ α φ ∂ β φ g µ ν (˜ g, φ ) = ˜ g µ ν ˜ auxiliary metric √− g ✓ ◆� − 1 Z d 4 x S [˜ g µ ν , φ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) g µ ν = g µ ν (˜ g, φ ) Friday, April 17, 15
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal / scalar part of the physical metric in a scalar field : φ g µ ν g αβ ∂ α φ ∂ β φ g µ ν (˜ g, φ ) = ˜ g µ ν ˜ auxiliary metric √− g ✓ ◆� − 1 Z d 4 x S [˜ g µ ν , φ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) g µ ν = g µ ν (˜ g, φ ) The theory becomes invariant with respect to Weyl transformations: g µ ν → Ω 2 ( x ) ˜ ˜ g µ ν Friday, April 17, 15
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal / scalar part of the physical metric in a scalar field : φ g µ ν g αβ ∂ α φ ∂ β φ g µ ν (˜ g, φ ) = ˜ g µ ν ˜ auxiliary metric √− g ✓ ◆� − 1 Z d 4 x S [˜ g µ ν , φ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) g µ ν = g µ ν (˜ g, φ ) The theory becomes The scalar field obeys a invariant with respect to constraint (Hamilton-Jacobi Weyl transformations: equation): g µ ν → Ω 2 ( x ) ˜ g µ ν ∂ µ φ ∂ ν φ = 1 ˜ g µ ν Friday, April 17, 15
√− g ✓ ◆� − 1 Z d 4 x S [˜ g µ ν , φ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) g µ ν = g µ ν (˜ g, φ ) g αβ ∂ α φ ∂ β φ with g µ ν (˜ g, φ ) = ˜ g µ ν ˜ Friday, April 17, 15
√− g ✓ ◆� − 1 Z d 4 x S [˜ g µ ν , φ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) g µ ν = g µ ν (˜ g, φ ) g αβ ∂ α φ ∂ β φ with g µ ν (˜ g, φ ) = ˜ g µ ν ˜ is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion Friday, April 17, 15
√− g ✓ ◆� − 1 Z d 4 x S [˜ g µ ν , φ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) g µ ν = g µ ν (˜ g, φ ) g αβ ∂ α φ ∂ β φ with g µ ν (˜ g, φ ) = ˜ g µ ν ˜ 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
Dissformal Transformation 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 g µ ν one plugs in a dissformal transformation g µ ν = F ( Ψ , w ) ` µ ν + H ( Ψ , w ) @ µ Ψ @ ν Ψ ✓ ◆ w 2 F ∂ H + F with and w = ` µ ν @ µ Ψ @ ν Ψ 6 = 0 ∂ w w and varies with respect to ` µ ν , Ψ Friday, April 17, 15
Dissformal Transformation 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 g µ ν one plugs in a dissformal transformation g µ ν = F ( Ψ , w ) ` µ ν + H ( Ψ , w ) @ µ Ψ @ ν Ψ ✓ ◆ w 2 F ∂ H + F with and w = ` µ ν @ µ Ψ @ ν Ψ 6 = 0 ∂ w w and varies with respect to ` µ ν , Ψ Mimetic gravity is an exception! And it does provide new dynamics! Friday, April 17, 15
Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) Friday, April 17, 15
Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) g µ ν = ˜ Weyl-invariance allows one to fix g µ ν Friday, April 17, 15
Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) g µ ν = ˜ Weyl-invariance allows one to fix g µ ν one implements constraint through λ ( g µ ν ∂ µ φ∂ ν φ − 1) ✓ ◆ Z − 1 d 4 x √− g S [ g µ ν , φ , λ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) + λ ( g µ ν ∂ µ φ∂ ν φ − 1) Friday, April 17, 15
Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) g µ ν = ˜ Weyl-invariance allows one to fix g µ ν one implements constraint through λ ( g µ ν ∂ µ φ∂ ν φ − 1) ✓ ◆ Z − 1 d 4 x √− g S [ g µ ν , φ , λ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) + λ ( g µ ν ∂ µ φ∂ ν φ − 1) The system is equivalent to standard GR + irrotational dust, moving along timelike geodesics with the velocity u µ = ∂ µ φ and energy density ρ = 2 λ Friday, April 17, 15
Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) g µ ν = ˜ Weyl-invariance allows one to fix g µ ν one implements constraint through λ ( g µ ν ∂ µ φ∂ ν φ − 1) ✓ ◆ Z − 1 d 4 x √− g S [ g µ ν , φ , λ , Φ m ] = 2 R ( g ) + L ( g, Φ m ) + λ ( g µ ν ∂ µ φ∂ ν φ − 1) The system is equivalent to standard GR + irrotational dust, moving along timelike geodesics with the velocity u µ = ∂ µ φ and energy density ρ = 2 λ Dark Matter 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 ) g µ ν ∂ µ φ ∂ ν φ = 1 φ Convenient to take as time 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 ) g µ ν ∂ µ φ ∂ ν φ = 1 φ Convenient to take as time Adding a potential = adding a function of time in the equation H + 3 H 2 = V ( t ) 2 ˙ 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 ) g µ ν ∂ µ φ ∂ ν φ = 1 φ Convenient to take as time Adding a potential = adding a function of time in the equation H + 3 H 2 = V ( t ) 2 ˙ Enough freedom to obtain any cosmological evolution! Friday, April 17, 15
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