Mimetic Dark Matter

A Solution to the Small Scale Crisis

Fabio Capela1 Sabir Ramazanov2

1University of Cambridge, DAMTP, UK2Universite Libre de Bruxelles, Service de Physique Theorique, Belgique

Third CosPa meeting, Liege, 2014

Cold Dark Matter Particle Paradigm

Cold Dark Matter

CMB anisotropies 3

Matter power spectrum 3

Linear growth of structures 3

Galaxy rotation curves 3

Bullet Cluster 3

Missing satellite 7 (?)Core-cusp 7 (?)

Too-big-to-fail 7 (?)

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Cold Dark Matter Particle Paradigm

Cold Dark Matter Mimetic Dark Matter

CMB anisotropies 3

Matter power spectrum 3

Linear growth of structures 3

Galaxy rotation curves 3

Bullet Cluster 3

Missing satellite 7 (?)Core-cusp 7 (?)

Too-big-to-fail 7 (?)

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Mimetic Dark Matter

[Mukhanov, Chamseddine, (2013)]

gµν(g , φ) = gµν gαβ∂αφ∂βφ

where

S[gµν , φ,Ψm] =

∫d4x√−g

[−1

2R(gµν) + Lm(gµν ,Ψm)

]

1. The theory is invariant under Weyl transformations:

gµν → Ω2(x)gµν

2. The scalar field is constrained :

gµν∂µφ∂νφ = 1

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Mimetic Dark Matter rewritten

[Golonev, Barvinsky, (2013)]

S[gµν , φ, λ,Ψm] =

∫d4x√−g

(−1

2R(g) + λ(gµν∂µφ∂νφ− 1)

)

MDM is equivalent to CDM

It is equivalent to GR with an irrot. pressureless perfect fluid

Tµν = ρuµuν = 2λ∂µφ∂νφ

uµ = ∂µφ ρ = 2λ

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Mimetic Dark Matter: curious...

S[gµν , φ, λ,Ψm] =

∫d4x√−g

(−1

2R(g) + λ(gµν∂µφ∂νφ− 1)

)

1. Solutions around FRW background

λ(xµ) =λ0(x i )

a3(τ), φ(xµ) = τ

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Mimetic Dark Matter: higher-order terms

Pure pressureless perfect fluid fails at non-linear level.

Question:

Does the term γ(φ)2 spoil the dust solution (λ = λ0(x i )/a3) ?

NO!!

General higher-order terms preserving DM solution

K(∂µφ) =γ1

2(φ)2 +

γ2

2∇µ∇νφ∇µ∇νφ

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Mimetic Dark Matter: relation with other models

Mimetic Dark Matter :

LMDM(gµν , φ) = −1

2R(g) + λ(gµν∂µφ∂νφ− 1) +K(∂µφ)

whereK(∂µφ) =

γ1

2(φ)2 +

γ2

2∇µ∇νφ∇µ∇νφ

Einstein- Æther

LÆ(gµν , uµ) = −1

2R(g) + λ(gabu

aub − 1) + K abmn∇au

m∇bun

whereK abmn = c1g

abmn + c2δ

amδ

bn + c3δ

anδ

bm + c4u

aubgmn

when uµ = ∂µφ, up to the term c4uaubgmn that gives NO contributions

on-shell.

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Mimetic Dark Matter: IR limit of quantum gravity?

[D. Blas, O. Pujolas, S.Sibiryakov (2009)]

[T. Jacobson, A. Speranza (2014)]

Einstein-Aether theory is equivalent to the IR limit of the projectableversion of Horava-Lifshitz (quantum gravity) for uµ = ∂µφ

Mimetic Dark Matter is the infrared limit of acandidate theory of quantum gravity!

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Mimetic Dark Matter: Linear perturbation theory

Perturbations

1. The metric (Φ: grav. potential)

ds2 = a2(η)[(1 + 2Φ)dη2 − (1− 2Φ)δijdx

idx j]

2. ∂µφ∂µφ = 1:

δφ′

a= Φ

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Mimetic Dark Matter: Matter Dominated Stage

δφ′′ +(c2s k

2 − 3

2H2

)δφ = 0, c2

s =8πGγ

2− 24πGγ

Solution

1. long wavelength perturbations (c2s k

2 H2):

δφ ∝ η3 ⇒ Φ = const (CDM)

2. short wavelength perturbations (c2s k

2 H2):

δφ ∝ e±icskη ⇒ Φ ∝ e±icskη/η2

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Mimetic Dark Matter: Gravitational potential

10−22

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−5 10−4 10−3 10−2 10−1 100 101 102

grav

itat

iona

lpot

enti

al

η/ηeq

k = 2× 103 h Mpc−1

1/η2

mimMIMETICDARK MATTER

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Mimetic Dark Matter: Matter Power Spectrum

−k2Φ = 4πGa2δλ

Solution

1. long wavelength perturbations:

δdm ∝ a

2. short wavelength perturbations:

δdm ∝ Aδe±icskη

Solution to the missing satellite problem?

If γ ∼ 10−10M2pl ⇒ suppression for wavelength ≤ 100 kpc

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Mimetic Dark Matter: Matter Power Spectrum

10−12

10−10

10−8

10−6

10−4

10−2

100

102

104

1 10 100

P(k

)[h

−3

Mpc

3]

k [h Mpc−1]

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Mimetic Dark Matter: Radiation Dominated Stage

δdm = A + B(k) ln(η)︸ ︷︷ ︸CDM

+C (k)a(η)︸ ︷︷ ︸MDM

+decaying modes

1 the linear growth is only relevant for

λ 300 pc

2 Perturbations with these wavelengths are expected to be amplifiedby the end of the RD stage.

3 Linear growth stops when the DM perturbations (δdm) become thedominant source of the gravitational potential. Then, we get backto matter dominated stage

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Mimetic Dark Matter: δdm during RD stage

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

10−10 10−8 10−6 10−4 10−2 100

δ dm

η/ηeq

wavelength λ = 10 pc

Aδ

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Discussion

Cold Dark Matter Mimetic Dark Matter

CMB anisotropies 3 3

Matter power spectrum 3 3

Linear growth of structures 3 3

Galaxy rotation curves 3 ?Bullet Cluster 3 3 (?)

Missing satellite 7 (?) 3

Core-cusp 7 (?) 3 (?)Too-big-to-fail 7 (?) ?

First stars 3 7

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Thank You

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