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 :

gµν

φ

gµν (˜ g, φ) = ˜ gµν ˜ gαβ ∂αφ ∂βφ

Monday, July 21, 14

Mimetic Matter

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

Monday, July 21, 14

Mimetic Matter

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)

Monday, July 21, 14

Mimetic Matter

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)

Monday, July 21, 14

Mimetic Matter

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)

Monday, July 21, 14

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

Monday, July 21, 14

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 construction,

see talks of Deffayet, Sivanesan, Vernizzi, Piazza

Monday, July 21, 14

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 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)

Weyl-invariance allows one to fix

gµν = ˜ gµν

Monday, July 21, 14

Mimetic Dark Matter

Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)

Weyl-invariance allows one to fix

gµν = ˜ gµν

• ne implements constraint through

λ (gµν∂µφ∂νφ − 1)

S [gµν, φ, λ, Φm] = Z d4x√−g ✓ −1 2R (g) + L (g, Φm) + λ (gµν∂µφ∂νφ − 1) ◆

Monday, July 21, 14

Mimetic Dark Matter

Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)

Weyl-invariance allows one to fix

gµν = ˜ gµν

• ne implements constraint through

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 = λ

Monday, July 21, 14

Mimetic Dark Matter

Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman; (2010)

“Cold Dark Matter”?

Weyl-invariance allows one to fix

gµν = ˜ gµν

• ne implements constraint through

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 = λ

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)

gµν ∂µφ ∂νφ = 1

Convenient to take as time

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)

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)

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)

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)

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)

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)

Monday, July 21, 14

Perturbations I

Even with potential, the field still moves along the timelike geodesics

Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)

Monday, July 21, 14

Perturbations I

Even with potential, the field still moves along the timelike geodesics

cS = 0

Lim, Sawicki, Vikman; (2010) Chamseddine, Mukhanov, Vikman (2014)

Monday, July 21, 14

Perturbations I

Even with potential, the field still moves along the timelike geodesics

cS = 0

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

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)

Just add higher derivatives !

1 2γ (⇤φ)2

Monday, July 21, 14

How to give a sound speed to Mimetic Matter?

Chamseddine, Mukhanov, Vikman (2014)

The sound speed is

c2

s =

γ 2 − 3γ

Just add higher derivatives !

1 2γ (⇤φ)2

Monday, July 21, 14

How to give a sound speed to Mimetic Matter?

Chamseddine, Mukhanov, Vikman (2014)

The sound speed is

c2

s =

γ 2 − 3γ

Back to waves, oscillators and normal quantum fluctuations!

Just add higher derivatives !

1 2γ (⇤φ)2

Monday, July 21, 14

The scalar field still obeys a constraint (Hamilton-Jacobi equation)

gµν ∂µφ ∂νφ = 1

Monday, July 21, 14

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 equation

Monday, July 21, 14

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 equation There are only minor changes (rescaling) in the background evolution equations e.g.

2 ˙ H + 3H2 = 2 2 − 3γ V (t)

Monday, July 21, 14

Perturbations HD

Chamseddine, Mukhanov, Vikman (2014)

Monday, July 21, 14

Perturbations HD

Chamseddine, Mukhanov, Vikman (2014)

δ ¨ φ + Hδ ˙ φ − c2

s

a2 ∆δφ + ˙ H δφ = 0

c2

s =

γ 2 − 3γ

with

Monday, July 21, 14

Perturbations HD

Chamseddine, Mukhanov, Vikman (2014)

δ ¨ φ + Hδ ˙ φ − c2

s

a2 ∆δφ + ˙ H δφ = 0

c2

s =

γ 2 − 3γ

with

Φ = δ ˙ φ

Monday, July 21, 14

Quantization

Monday, July 21, 14

Quantization

S = −1 2 Z dηd3x ✓ γ c2

s

δφ0∆δφ0 + ... ◆

the action

Monday, July 21, 14

Quantization

S = −1 2 Z dηd3x ✓ γ c2

s

δφ0∆δφ0 + ... ◆

the action

δφk ∼ rcs γ k−3/2

short wavelength quantum fluctuations match with the long-wave-length limit

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Newtonian potential

Φλ ' p cs/γ · Hcsk'H

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Newtonian potential

Φλ ' p cs/γ · Hcsk'H

for cS ⌧ 1

Φ` ∼ c−1/2

s

Hcsk∼Ha

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Newtonian potential

Φλ ' p cs/γ · Hcsk'H

for cS ⌧ 1

Φ` ∼ c−1/2

s

Hcsk∼Ha

Φ` ∼ c1/2

s

Hcsk∼Ha

cS 1

for

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Gravitational Waves are unchanged

hλ ' Hk'H

Newtonian potential

Φλ ' p cs/γ · Hcsk'H

for cS ⌧ 1

Φ` ∼ c−1/2

s

Hcsk∼Ha

Φ` ∼ c1/2

s

Hcsk∼Ha

cS 1

for

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Gravitational Waves are unchanged

hλ ' Hk'H

nS − 1 = nT

spectral indices

Newtonian potential

Φλ ' p cs/γ · Hcsk'H

for cS ⌧ 1

Φ` ∼ c−1/2

s

Hcsk∼Ha

Φ` ∼ c1/2

s

Hcsk∼Ha

cS 1

for

Monday, July 21, 14

Perturbations in Mimetic Inflation

Chamseddine, Mukhanov, Vikman (2014)

λ ' k−1

• n scale

Gravitational Waves are unchanged

hλ ' Hk'H

nS − 1 = nT

spectral indices

Newtonian potential

Φλ ' p cs/γ · Hcsk'H

Φλ hλ

But it seems that there is no usual Non-Gaussianity!

small sound speed

for cS ⌧ 1

Φ` ∼ c−1/2

s

Hcsk∼Ha

Φ` ∼ c1/2

s

Hcsk∼Ha

cS 1

for

Monday, July 21, 14

Conclusions

Monday, July 21, 14

New large class of scalar-tensor theories beyond (but not in contradiction with) Horndeski

Conclusions

Monday, July 21, 14

New large class of 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

Conclusions

Monday, July 21, 14

New large class of 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 New class of inflationary models with suppressed gravity waves and seemingly low non-Gaussianity

Conclusions

Monday, July 21, 14

Tanks a lot for atention! New large class of 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 New class of inflationary models with suppressed gravity waves and seemingly low non-Gaussianity

Conclusions

Monday, July 21, 14