Stability properties in mimetic gravity theories Alexander Ganz - PowerPoint PPT Presentation

Stability properties in mimetic gravity theories Alexander Ganz Dipartimento di Fisica e Astronomia "Galileo Galilei" Università di Padova INFN, Sezione di Padova Paris-Saclay AstroParticle Symposioum 2019 16th October 2019 A. Ganz, N. Bartolo, S. Matarrese, arXiv:1907.10301 A. Ganz, P. Karmakar, S. Matarrese, D. Sorokin, arXiv:1812.02667

Mimetic Matter Starting from the Einstein-Hilbert action Chamseddine & Mukhanov (2013) � d 4 x √− gR ( g µν ) + S m ( g µν , ψ ) S = 1 2 Non-invertible conformal transformation � � g µν ≡ ˜ g αβ ∂ α ϕ∂ β ϕ g µν = − ˜ ˜ X ˜ g µν ⇒ Conformal invariant theory X → Ω − 1 ( x )˜ ˜ ˜ g µν → Ω( x )˜ g µν , X Transforming the action � S = 1 � R +3 � � d 4 x X ˜ ˜ X − 1 ˜ ˜ g µν ∂ µ ˜ X ∂ ν ˜ X − 3 � ˜ g µν , ˜ − ˜ g X + S m (˜ X , ψ ) 2 2 Alexander Ganz alexander.ganz@phd.unipd.it 2

Mimetic Matter Fixing the conformal gauge degree of freedom ˜ X = 1 Barvinsky (2014), Golovnev (2013) � � 1 � � ˜ d 4 x g µν ∂ µ ϕ∂ ν ϕ + 1) S = − ˜ g R − λ (˜ + S m (˜ g µν , ψ ) 2 Equations of Motion g µν ∂ µ ϕ∂ ν ϕ + 1 = 0 G µν − T µν = ( − G + T ) ∂ µ ϕ∂ ν ϕ � �� � 2 λ � √− g ( − G + T ) g µν ∂ ν ϕ � ∂ µ = 0 The trace λ mimics the CDM density T dm µν = ρ dm u µ u ν ≡ ( − G + T ) ∂ µ ϕ∂ ν ϕ Alexander Ganz alexander.ganz@phd.unipd.it 3

Hamiltonian analysis Performing the full Hamiltonian analysis → 3 dof Chaichian et al. (2014), Takahashi & Kobayashi (2017) Hamiltonian constraint for mimetic matter � � √ � 2 π ij π ij − 1 − 1 h ¯ 2 π 2 √ 1 + h ij ∂ i ϕ∂ j ϕ H = R + p ϕ 2 h ⇒ A necessary but not sufficient stability condition p ϕ λ = √ > 0 � 1 + h ij ∂ i ϕ∂ j ϕ h Physical interpretation: positive dark matter density Sign of λ is conserved due to the shift symmetry ϕ → ϕ + c Alexander Ganz alexander.ganz@phd.unipd.it 4

Linear stability analysis Ghost instabilities for mimetic gravity in presence of external matter Takahashi & Kobayashi (2017), Langlois et al. (2018) Second order action for pure mimetic matter � ξ 2 + 2∆˜ ξ + ( ∂ i ξ ) 2 d 3 x d t a 3 � B � − 3 ˙ a 2 ˙ S = a 2 Curvature perturbation is conserved ∆ ˙ ξ = 0 Second order Hamiltonian H = − Hp Σ Σ = H � ( p Σ − Σ) 2 − ( p Σ + Σ) 2 � 4 Ostrogradski term ⇒ equivalent to tachyon or ghost instability 1 1 p = √ ( p Σ ∓ Σ ) , q = √ (Σ ± p Σ ) 2 2 Growing mode → standard Jeans instability of dust Alexander Ganz alexander.ganz@phd.unipd.it 5

Linear stability analysis � � α − 1 2 g µν ∂ µ η∂ ν η Additional external matter fluid modeled by L = Second order Hamiltonian in presence of external matter p 2 a 3 p ξ χ + c 2 η 2 P ′ 2 χ 2 − ( ∂ i ξ ) 2 2 P ′ ( ∂ i χ ) 2 H = a 3 � − 1 η P ′ ˙ a 6 − 3 + 1 � χ m 4 ˙ a 2 a 2 2 2 P ′ Analyzing the dispersion relation in the UV-limit k → ∞ Two damped propagating modes 1 k 3 α (2 α − 1) + O ( k − 1 ) √ 2 α − 1 ω 1 , 2 = ± a + ı H Two purely damped non-propagating modes η, α ) + O ( k − 1 ) ω 3 , 4 = B ( H , ˙ Non-propagating dust modes are ghost-like Alexander Ganz alexander.ganz@phd.unipd.it 6

Higher order derivatives Introducing higher order derivatives to make the scalar field propagate Chamseddine et al. (2014) � d 4 x √− g � 1 � 2 R − γ ( � ϕ ) 2 − λ ( X + 1) S = Ghost or gradient instabilities in the scalar sector � � 2 γ − 3 � ξ 2 + 1 S (2) = ˙ d 4 x a 3 a 2 ( ∂ k ξ ) 2 γ Generalize for any higher order derivatives of the scalar field ϕ µν ϕ µν , ϕ µν ϕ µα ϕ ν α etc. Takahashi & Kobayashi (2017), Langlois et al. (2018) Alexander Ganz alexander.ganz@phd.unipd.it 7

Solving the instability problem Adding couplings of higher derivatives of the mimetic field to the curvature as � ϕ R ∼ ¯ RK Zheng et al. (2017), Hirano et al. (2017) Extra Ostrogradski modes for an inhomogeneous scalar field configuration Zheng (2018) L = R � ϕ = ( K ij K ij − K 2 + ¯ R ) � ϕ − 2 ∇ µ ( a µ − n µ K ) � ϕ � �� � T T | ϕ = t , N 2 =1 = − 2 K ∇ µ ( n µ K ) = − K 3 − ∇ µ ( n µ K 2 ) For other operators as R ϕ αβ ϕ αβ , R αβ ϕ αβ � ϕ additional degeneracy conditions are needed Alexander Ganz alexander.ganz@phd.unipd.it 8

Singular Dirac matrix Analyzing non-linear behavior by performing the Hamiltonian analysis d d t A = { A , H } D = { A , H } − { A , C I } (Ω − 1 ) IJ { C J , H } For a homogeneous scalar field ∂ i ϕ ≈ 0 the Dirac matrix Ω becomes singular Need of a case distinction: i) ∂ i ϕ �≈ 0 ii) ∂ i ϕ ≈ 0 Gomes & Guariento (2017) The theory in the singular point (case ii) is itself well defined In general more than one degree of freedom vanishes for ∂ i ϕ ≈ 0 Alexander Ganz alexander.ganz@phd.unipd.it 9

Toy model � d 4 x √− g [ � ϕ R − λ ( X + 1)] S = Linear perturbations around Minkowski and ϕ = t � S (2) = 4 d 4 x ∆Ψ∆ δϕ √ 1 + α i α i t + α i x i Inhomogeneous scalar field profile ϕ = � 1 � � � 1 + ( ∂ i ϕ ) 2 u + + 2 S (2) = (∆ u + ) 2 d 4 x u + ˜ u + ∂ i ϕ ∆ ∂ i u + + ϕ 2 ˙ � ˙ ϕ ˙ ϕ 2 2 ˙ ˙ 2 ˙ 1 � ϕ 2 ∂ i ϕ∂ j ϕ∂ i ∂ j u + ∆ u + − ( u + ⇐ − ⇒ u − ) 2 ˙ Two decoupled propagating degrees of freedom from which at least one is a ghost Alexander Ganz alexander.ganz@phd.unipd.it 10

Summary Mimetic gravity offers a unified description for CDM and DE Stable solutions require λ > 0 Mimetic matter has a tachyon/ghost instability causing the standard Jeans instability External matter decouples from the dust (non-propagating ghost modes) in the UV-limit Coupling between the curvature and higher derivatives of the scalar field solves the instability problem around FLRW For a non-homogeneous scalar field additional Ostrogradski ghost modes are revived Alexander Ganz alexander.ganz@phd.unipd.it 11