Cosmology with Mimetic Matter Alexander Vikman 18.07.14 Monday, - PowerPoint PPT Presentation

Cosmology with Mimetic Matter Alexander Vikman 18.07.14 Monday, July 21, 14

Mimetic Matter Chamseddine, Mukhanov (2013) Monday, July 21, 14

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 ˜ Monday, July 21, 14

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 Monday, July 21, 14

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 ) Monday, July 21, 14

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 Monday, July 21, 14

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 Monday, July 21, 14

 √− 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 ˜ Monday, July 21, 14

 √− 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 construction, see talks of Deffayet, Sivanesan, Vernizzi, Piazza Monday, July 21, 14

 √− 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 construction, see talks of Deffayet, Sivanesan, Vernizzi, Piazza But it is still a system with one degree of freedom! Monday, July 21, 14

Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) Monday, July 21, 14

Mimetic Dark Matter Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010) g µ ν = ˜ Weyl-invariance allows one to fix g µ ν Monday, July 21, 14

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) Monday, July 21, 14

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 and energy density ρ = λ u µ = ∂ µ φ Monday, July 21, 14

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 and energy density ρ = λ u µ = ∂ µ φ “Cold Dark Matter” ? Monday, July 21, 14

Mimicking any cosmological evolution, But always with zero sound speed Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014) Monday, July 21, 14

Mimicking any cosmological evolution, But always with zero sound speed Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014) Just add a potential ! V ( φ ) Monday, July 21, 14

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 Monday, July 21, 14

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 ˙ Monday, July 21, 14

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! Monday, July 21, 14

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! m 4 φ 2 V ( φ ) = 1 In particular gives the same cosmological e φ + 1 3 1 inflation as potential in the standard case 2 m 2 φ 2 Monday, July 21, 14

Perturbations I Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014) Even with potential, the field still moves along the timelike geodesics Monday, July 21, 14

Perturbations I Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014) Even with potential, the field still moves along the timelike geodesics c S = 0 Monday, July 21, 14

Perturbations I Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014) Even with potential, the field still moves along the timelike geodesics c S = 0 ✓ ◆ Z 1 − H + H Φ = C 1 ( x ) a C 2 ( x ) adt a Here on all scales but in the usual cosmology it is an approximation for superhorizon scales Monday, July 21, 14

How to give a sound speed to Mimetic Matter? Chamseddine, Mukhanov, Vikman (2014) Monday, July 21, 14

How to give a sound speed to Mimetic Matter? Chamseddine, Mukhanov, Vikman (2014) 1 2 γ ( ⇤ φ ) 2 Just add higher derivatives ! Monday, July 21, 14

How to give a sound speed to Mimetic Matter? Chamseddine, Mukhanov, Vikman (2014) 1 2 γ ( ⇤ φ ) 2 Just add higher derivatives ! γ The sound speed is c 2 s = 2 − 3 γ Monday, July 21, 14

How to give a sound speed to Mimetic Matter? Chamseddine, Mukhanov, Vikman (2014) 1 2 γ ( ⇤ φ ) 2 Just add higher derivatives ! γ The sound speed is c 2 s = 2 − 3 γ Back to waves, oscillators and normal quantum fluctuations! Monday, July 21, 14

The scalar field still obeys a constraint (Hamilton-Jacobi g µ ν ∂ µ φ ∂ ν φ = 1 equation) Monday, July 21, 14

The scalar field still obeys a constraint (Hamilton-Jacobi g µ ν ∂ µ φ ∂ ν φ = 1 equation) Higher time derivatives can be eliminated just by the differentiation of this equation Monday, July 21, 14

The scalar field still obeys a constraint (Hamilton-Jacobi g µ ν ∂ µ φ ∂ ν φ = 1 equation) Higher time derivatives can be eliminated just by the differentiation of this equation There are only minor changes (rescaling) in the background evolution equations e.g. 2 H + 3 H 2 = 2 ˙ 2 − 3 γ V ( t ) Monday, July 21, 14

Perturbations HD Chamseddine, Mukhanov, Vikman (2014) Monday, July 21, 14

Perturbations HD Chamseddine, Mukhanov, Vikman (2014) φ − c 2 δ ¨ φ + H δ ˙ a 2 ∆ δφ + ˙ s H δφ = 0 γ c 2 with s = 2 − 3 γ Monday, July 21, 14

Perturbations HD Chamseddine, Mukhanov, Vikman (2014) φ − c 2 δ ¨ φ + H δ ˙ a 2 ∆ δφ + ˙ s H δφ = 0 γ c 2 with s = 2 − 3 γ Φ = δ ˙ φ Monday, July 21, 14

Quantization Monday, July 21, 14