The Dark Matter Bispectrum in Galileon cosmologiescosmo/Talks/bellini.pdf · Introduction...
Transcript of 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
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 3 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 3 / 20
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, . . . )
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 4 / 20
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)]
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 5 / 20
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)
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 6 / 20
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.
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 7 / 20
Background evolution
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 8 / 20
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%
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 9 / 20
Linear perturbation theory
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 10 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 10 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 11 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 12 / 20
The weakly non-linear regime
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 13 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 13 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 14 / 20
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.
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 15 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 16 / 20
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?
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 17 / 20
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”.
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 18 / 20
Conclusions
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 19 / 20
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
E. Bellini (ITP - Uni Heidelberg) DM Bispectrum in Galileon cosmologies September 13, 2013 19 / 20
Thank you!