the dark matter bispectrum in galileon cosmologiescosmo/talks/bellini.pdf · introduction...

The Dark Matter Bispectrum in Galileoncosmologies

Emilio Bellini

Institut fur Theoretische Physik - Universitat Heidelberg

September 13, 2013

Based on:

N. Bartolo, EB, D. Bertacca, S. Matarrese. JCAP 1303 (2013) 034

Table of contents

1 IntroductionDark EnergyGalileon theoryVainshtein mechanism

2 Background evolutionCoupled GalileonCubic Galileon

3 Linear perturbation theory

IntroductionCubic Galileon

4 The weakly non-linear regimeWhy?KernelBispectrumCompensation effect

5 Conclusions

E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 2 / 20

Introduction

Table of contents






5 Conclusions


Introduction Dark Energy

A problem in Cosmology: the Dark Energy

Several observations (such as Supernovae Ia, BAO, CMB) stress thatthe universe is undergoing a phase of accelerated expansion today

Perfect fluid

p = wρ

−2.0 −1.6 −1.2 −0.8 −0.4w

0.0

0.2

0.4

0.6

0.8

1.0

P/P

ma

x

Planck+WP+BAO

Planck+WP+Union2.1

Planck+WP+SNLS

Planck+WP

[Planck Collaboration, XVI (2013)]

Acceleration

w < −1/3


Introduction Dark Energy

A problem in Cosmology: the Dark Energy

Cosmological constant (ΛCDM)w = −1

it does not affect perturbation theory (no new d.o.f.)problems:

* why is Λ so unnaturaly small?* cosmic coincidence problem

New degrees of freedom

w = w(t)⇒ background evolution modifiedit affects the perturbation theory⇒ the growth of structuresexamples:

Modified matter models (Quintessence, k-essence, UDM,. . . )Modified gravity models (f(R), scalar-tensor, Galileon, . . . )


Introduction Galileon theory

Galileon theory

Ingredients and properties:

A scalar field, π

The action preserves the Galilean-shift symmetry (∂µπ → ∂µπ + bµ)

Ostrogradski instabilities are avoided (no more than second derivativesin the equations of motion) [Ostrogradski (1850)]

“Self-screening” effect via the Vainshtein mechanism [Vainshtein (1972)]

Result

The most general Scalar-Tensor theory that contains second-orderderivatives in the action (i.e. π;µπ;µ�π) and respects the Galilean-shiftsymmetry in a flat space-time

It is a generalization of the decoupling limit of Dvali-Gabadadze-Porratitheory (DGP) [Dvali, et al. (2000)]


Introduction Galileon theory

Galileon theory

Coupled Galileon:

S =

∫d4x√−g

[(1− c0L0)

M2pl

2R+

1

2

5∑i=1

ciLi − cGLG − Lm

]where:L1 =M3π L2 = (∇π)2 L3 = (�π)(∇π)2/M3

L4 =(∇π)2[2(�π)2 − 2π;µνπ

;µν −R(∇π)2/2]/M6

L5 =(∇π)2[(�π)3 − 3(�π)π;µνπ;µν + 2π;µ

νπ;νρπ;ρ

µ+

− 6π;µπ;µνπ;ρGνρ]/M

9 [Nicolis, et al. (2009) & Deffayet, et al. (2009)]

LG =MplGµνπ;µπ;ν/M

3 L0 = 2π/Mpl [Appleby, Linder (2012)]

c0 = cG = 0 (Uncoupled Galileon) c0 = cG = c5 = c4 = 0 (Cubic Galileon)


Introduction Vainshtein mechanism

Vainshtein mechanism

Given a source of mass MS , there is acrucial scale RV (Vainshtein radius)

If R� RV ⇒ In the E.o.M. thelinear terms dominate.If R� RV ⇒ In the E.o.M. thenon-linear terms dominate.

π ' πLinπ ' πNL

RV

MS

GR

Observable effects* πLin cause the modifications of gravity;* πNL screens the effects of πLin, satisfying solar system

constraints.


Background evolution

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5 Conclusions


Background evolution Coupled Galileon

Coupled Galileon[De Felice, Tsujikawa (2010); Appleby,Linder (2012)]

FLRW flat metric

ds2 = a2(τ)[−dτ2 + δijdx

idxj]

Equations of motion

3M2plH2 = ρπ + ρm

M2pl

(3H2 + 2H′

)= −pπ

ρm′ + 3Hρm = 0

ρπ ≡c1M

3

2π +

c2π′2

2a2− 3c3π

′3HM3a4

+45c4π

′4H2

2M6a6− 21c5π

′5H3

M9a8− 9cGMplπ

′2H2

M3a4

+6c0MplH (π′ +Hπ)

a2

pπ ≡c1M

3

2π − c2π

′2

2a2− c3π

′2

M3a4(π′′ −Hπ′

)+

12c4Hπ′3

M6a6

(π′′ − 7Hπ′

8+H′π′

4H

)− 15c5H2π′

4

M9a8

(π′′ −Hπ′ + 2H′π′

5H

)− 4cGMplHπ′

M3a4

(π′′ − 3

4Hπ′ + H

′π′

2H

)+

2c0Mpl

a2(π′′ +H2π +Hπ′ + 2H′π

)E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 8 / 20

Background evolution Cubic Galileon

Cubic Galileon

Cubic Galileon

c4 = c5 = c0 = cG = 0

c1 6= 0

c1 is a linear potential:

Cosmological constant if π′ → 0

No stable de Sitter solution

0.0 0.2 0.4 0.6 0.8 1.0

0.94

0.96

0.98

1.00

1.02

a

HΠ

HaL�

HL

HaL

0.2 0.4 0.6 0.8 1.0

-1.6

-1.4

-1.2

-1.0

- 0.8

a

wΠ

HaL

[Bartolo, Bellini, Bertacca, Matarrese (2013)]

Deviations w.r.t. the ΛCDM model . 10%


Linear perturbation theory

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5 Conclusions


Linear perturbation theory Introduction

Perturbations

Metric Perturbations

ds2 = a(τ)2 [−(1 + 2ψ)dτ2 + 2ωidxidτ + [(1− 2φ)δij + χij ] dx

idxj]

DM perturbations

ρ(~x, τ) ≡ ρ(0)(τ) [1 + δ(~x, τ)]

uµ(~x, τ) ≡ 1

a

[δµ0 + vµ(~x, τ)

]ωi ≡ωi + ∂iω

χij ≡χij + ∂iχj + ∂jχi +

(∂i∂j −

1

3δij∇2

)χ

Vector perturbations transverse (∂iωi = 0)Tensor perturbations transverse and trace-free (χii = ∂iχij = 0)First-order vectors and tensors are negligible

* Sub-horizon (k2ψ � H2ψ) and Quasi-static (k2ψ � ψ) Approximations


Linear perturbation theory Cubic Galileon

Linear evolution

Equations

Einstein equations

Galileon field equation

DM stress-energy tensor conservation

* We obtain coupled equations for the scalar perturbations

* No anisotropic stress in the gravitational potentials (i.e. φ = ψ)

* Equations decoupled without choosing a gauge

Evolution of DM perturbations

δ′′ +Hδ′ = 4πG

(1− c3

2π′4

2c2M6M2pla

4α

)a2ρmδ

Modifications

Newton’s constant

friction term


Linear perturbation theory Cubic Galileon

Linear evolution[Bartolo, Bellini, Bertacca, Matarrese (2013)]

0.2 0.4 0.6 0.8 1.0

- 0.08

- 0.06

- 0.04

- 0.02

0.00

a

DHa

L�D

LHa

L-1 Growing mode

Deviations . 10%

Growth rateDeviations . 100%

f(a) =d ln δ

d ln a 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

a

fHa

L�f L

HaL-

1


The weakly non-linear regime

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5 Conclusions


The weakly non-linear regime Why?

Why?

Second-order perturbations allow to study non-Gaussianity

Non-Gaussianity comes from:

Inflation: Quantum primordial fluctuations

Gravitational instability: when perturbations enter non-linear regime.

Both effects are imprinted on the LSS. We are interested in the second.

Dark Matter Bispectrum

Useful statistic to study the DM distribution of the universe

possibility to measure the signature of modifications from standardgravity⇒ can be used to lift degeneracies among different models givingrise to the same observed power spectrum and the same backgroundcosmology


The weakly non-linear regime Kernel

Kernel[Bartolo, Bellini, Bertacca, Matarrese (2013)]

Equations (ψ(2), ψ(1)φ(1), . . .)

Einstein equations

Galileon field equation

DM stress-energy tensor conservation

In the EOM the structure of thesecond-order perturbations(ψ(2)) is the same as in thefirst-order case

ψ(1)φ(1) are source terms

Second-order DM perturbations

δ(2)′′ +Hδ(2)′ − 4πG

(1− c3

2π′4

2c2M6M2pla

4α

)a2ρmδ

(2) = S(δ)(δ(1)δ(1)

)

δ(2)~k

(a) =

∫d3q1

∫d3q2δ

(3)(~k − ~q1 − ~q2)F (a, ~q1, ~q2)δ(1)~q1

(a)δ(1)~q2

(a)

F (a, ~q1, ~q2) = A(a) +B(a)

(~k1 · ~k2

) (k1

2 + k22)

k12k2

2+ C(a)

(~k1 · ~k2

)2k1

2k22

In EdS the coefficients are A(a) = 5/7, B(a) = 1/2 and C(a) = 2/7


The weakly non-linear regime Bispectrum

Bispectrum

Definition:⟨δ(a,~k1)δ(a,~k2)δ(a,~k3)

⟩≡ (2π)

3δ(3)(~k1 + ~k2 + ~k3)B(a,~k1,~k2)

Gaussian initial conditions to remove the contribution given by the primordialfluctuations;

Sub-horizon scales (k � 10−4 h Mpc−1);

We have to exclude scales at which highly non-linear effects becomenon-negligibles (k . 10−1 h Mpc−1);

We work with the reduced bispectrum

Q(a,~k1,~k2) =B(a, k1, k2, k3)

P (a,~k1)P (a,~k2) + cyc.


The weakly non-linear regime Bispectrum

Bispectrum[Bartolo, Bellini, Bertacca, Matarrese (2013)]

0.0 0.2 0.4 0.6 0.8 1.0

- 0.035

- 0.030

- 0.025

- 0.020

- 0.015

- 0.010

- 0.005

0.000

Θ � Π

QHΘ

�ΠL�

QL

HΘ�Π

L-1

k1 =k2 =0.001 h Mpc-1

0.0 0.2 0.4 0.6 0.8 1.0

- 0.03

- 0.02

- 0.01

0.00

Θ � Π

QHΘ

�ΠL�

QL

HΘ�Π

L-1

k1 =k2 =0.05 h Mpc-1


The weakly non-linear regime Compensation effect

Compensation effectWhy this suppression?

Background . 10%

First-Order . 100%

Second-Order . 1%

F (a = 1) =

∫ a=1

am

G(a = 1, a′)da′

Equilateral configuration (removes thepower spectrum contribution)

0.2 0.4 0.6 0.8 1.0

-1

0

1

2

a'

GHa

',a

=1

L

a =1

[Bartolo, Bellini, Bertacca, Matarrese (2013)]

Galileon lines lie below theΛCDM line for a′ . 0.4

Galileon lines lie above theΛCDM line for a′ & 0.4

Red and blue lines show aminimum near a′ ' 1 thatincreases the deviations

* Other works find similar results:Borisov et al. (2008), Tatekawa etal. (2008), Gil-Marin et al. (2011)

* Vainshtein mechanism?


The weakly non-linear regime Compensation effect

Other works

Borisov and Jain (2008)f(R) gravity with second-order perturbation theory“However the reduced bispectrum, which is independent of thelinear growth factor in perturbation theory for GR, remains within afew percent of the regular gravity prediction”.

Tatekawa and Tsujikawa (2008)Brans-Dicke action with second-order perturbation theorySkewness S3 = 〈δ3〉/〈δ2〉2“..find that the difference from the ΛCDM model is only less than afew percent even if the growth rate of first-order perturbations issignificantly different from that in the ΛCDM model.”

Gil-Marın et al. (2011)N-Body simulations on f(R) gravity models.“..the effect of deviations from GR gravity on the reducedbispectrum are weak compared to those on the power spectrum (atleast for the cases considered here), opening up the possibility ofbreaking the galaxy-bias degeneracy”.


Conclusions

Table of contents






5 Conclusions


Conclusions

Conclusions

At the bispectrum level we have found a suppression effect that reducesthe deviations w.r.t. the bispectrum of the ΛCDM model. We think that agood candidate for this effect can be the Vainshtein mechanism

Due to this suppression effect, the DM bispectrum can not be usedalone to distinguish Galileon from ΛCDM

The deviations in the bispectrum are mostly given by the linear growthrate

B ∼ P 2


Thank you!

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