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)

S = 1 2

• d4x √−gR(gµν) + Sm(gµν, ψ)

Non-invertible conformal transformation gµν = −

• ˜

gαβ∂αϕ∂βϕ

• ˜

gµν ≡ ˜ X ˜ gµν ⇒ Conformal invariant theory ˜ gµν → Ω(x)˜ gµν, ˜ X → Ω−1(x)˜ X Transforming the action S = 1 2

• d4x
• −˜

g

• ˜

X ˜ R+3 2 ˜ X −1˜ gµν∂µ ˜ X∂ν ˜ X−3˜ X

• +Sm(˜

gµν, ˜ X, ψ)

Alexander Ganz alexander.ganz@phd.unipd.it 2

Mimetic Matter

Fixing the conformal gauge degree of freedom ˜ X = 1 Barvinsky (2014),

Golovnev (2013)

S =

• d4x
• −˜

g 1 2 ˜ R − λ (˜ gµν∂µϕ∂νϕ + 1)

• + Sm(˜

gµν, ψ)

Equations of Motion

gµν∂µϕ∂νϕ + 1 = 0 Gµν − Tµν = (−G + T)

• 2λ

∂µϕ∂νϕ ∂µ √−g (−G + T) gµν∂νϕ

• = 0

The trace λ mimics the CDM density T dm

µν = ρdmuµuν ≡ (−G + T) ∂µϕ∂νϕ

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 H = 2 √ h

• πijπij − 1

2π2

• − 1

2 √ h¯ R + pϕ

• 1 + hij∂iϕ∂jϕ

⇒ A necessary but not sufficient stability condition λ = pϕ √ h

• 1 + hij∂iϕ∂jϕ

> 0 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 S =

• d3x dt a3

− 3 ˙ ξ2 + 2∆˜ B a2 ˙ ξ + (∂iξ)2 a2

• Curvature perturbation is conserved ∆ ˙

ξ = 0 Second order Hamiltonian H = −HpΣΣ = H 4

• (pΣ − Σ)2 − (pΣ + Σ)2

Ostrogradski term ⇒ equivalent to tachyon or ghost instability p = 1 √ 2 (pΣ ∓ Σ ) , q = 1 √ 2 (Σ ± pΣ) Growing mode → standard Jeans instability of dust

Alexander Ganz alexander.ganz@phd.unipd.it 5

Linear stability analysis

Additional external matter fluid modeled by L =

• − 1

2gµν∂µη∂νη

α Second order Hamiltonian in presence of external matter H = a3 − 1 2 ˙ ηP′ a3 pξχ + c2

m

2P′ p2

χ

a6 − 3 4 ˙ η2P′2χ2 − (∂iξ)2 a2 + 1 2P′ (∂iχ)2 a2

• Analyzing the dispersion relation in the UV-limit k → ∞

Two damped propagating modes ω1,2 = ± 1 √2α − 1 k a + ıH 3α (2α − 1) + O(k−1) Two purely damped non-propagating modes ω3,4 = B(H, ˙ η, α) + O(k−1) 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) S =

• d4x √−g

1 2R − γ(ϕ)2 − λ(X + 1)

• Ghost or gradient instabilities in the scalar sector

S(2) =

• d4x a3

2γ − 3 γ ˙ ξ2 + 1 a2 (∂kξ)2

• Generalize for any higher order derivatives of the scalar field ϕµνϕµν,

ϕµνϕµαϕν

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 ϕR ∼ ¯ RK Zheng et al. (2017), Hirano et al. (2017) Extra Ostrogradski modes for an inhomogeneous scalar field configuration Zheng (2018) L = R ϕ = (KijK ij − K 2 + ¯ R)ϕ − 2∇µ(aµ − nµK)ϕ

• T

T|ϕ=t,N2=1 = −2K∇µ(nµK) = −K 3 − ∇µ(nµK 2) For other operators as R ϕαβϕαβ, Rαβϕαβ 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 dt A = {A, H}D = {A, H} − {A, CI}(Ω−1)IJ{CJ, H} For a homogeneous scalar field ∂iϕ ≈ 0 the Dirac matrix Ω becomes singular Need of a case distinction: i) ∂iϕ ≈ 0 ii) ∂iϕ ≈ 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 ∂iϕ ≈ 0

Alexander Ganz alexander.ganz@phd.unipd.it 9

Toy model

S =

• d4x √−g [ϕR − λ(X + 1)]

Linear perturbations around Minkowski and ϕ = t S(2) = 4

• d4x ∆Ψ∆δϕ

Inhomogeneous scalar field profile ϕ = √ 1 + αiαi t + αixi S(2) =

• d4x

1 2 ˙ ϕ2 ˙ u+ ˜ ˙ u+ + 2 ˙ ϕ ˙ u+∂iϕ∆∂iu+ +

• 1 + (∂iϕ)2

2 ˙ ϕ2

• (∆u+)2

− 1 2 ˙ ϕ2 ∂iϕ∂jϕ∂i∂ju+∆u+ − (u+ ⇐ ⇒ u−)

• Two decoupled propagating degrees of freedom from which at least
• ne 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