Probing the matter power spectrum with the clustering ratio of - - PowerPoint PPT Presentation

Probing the matter power spectrum with the clustering ratio

• f galaxies

Julien BEL

Osservatorio Astronomico di Brera (with L. Guzzo)

Collaborators: P. BRAX, C. MARINONI and P. VALAGEAS

Frontiers of Fundamental Physics (Marseille, July 2014) Bel & Marinoni 2014, A&A, 563, 36 Bel, Marinoni, Granett, Guzzo, Peacock et al. (The VIPERS Team) 2014, , A&A, 563, 37 Bel, Brax, Marinoni & Valageas 2014, submitted, arXiv: 1406.3347B

2

Outline

• Goal: fixing the matter power spectrum
• Tool: a new clustering statistic, the clustering ratio
• Test: simulations
• Results: -Omega_m from SDSS DR7 and VIPERS PDR1
• fR0 from SDSS DR7 and DR10

3

Perturbation Theory

The Matter power spectrum

3

Perturbation Theory

The Matter power spectrum

i R N i i R g

i b δ δ

∑

=

=

,

!

Fry & Gaztañaga (1993)

3

Perturbation Theory

The Matter power spectrum

i R N i i R g

i b δ δ

∑

=

=

,

!

Fry & Gaztañaga (1993)

δg,R

z = δg,R + µk 2 f 2δR

Kaiser (1987)

4

Variance of fluctuations in spheres:

σ g,R

2

= δg,R

2(

x ) R

Statistical properties of smoothed over-densities

4

2-point correlation function:

ξg,R(r) = δg,R( x )δg,R( x +  r )

Variance of fluctuations in spheres:

σ g,R

2

= δg,R

2(

x ) R r

Statistical properties of smoothed over-densities

5

Perturbation Theory

Statistical properties of smoothed over-densities

Survey description in Guzzo et al. (arXiv:1303.2623G) and data access to PDR-1 in Garilli et al. (arXiv:1310.1008G)

DEC DEC RA

5

Perturbation Theory

Statistical properties of smoothed over-densities

DEC DEC RA

Survey description in Guzzo et al. (arXiv:1303.2623G) and data access to PDR-1 in Garilli et al. (arXiv:1310.1008G)

5

Perturbation Theory

Statistical properties of smoothed over-densities

DEC DEC RA

Survey description in Guzzo et al. (arXiv:1303.2623G) and data access to PDR-1 in Garilli et al. (arXiv:1310.1008G)

6

Homogeneous Background

Perturbation Theory

) ( ) ( ) , ( ) , ( t t x t x t ρ ρ ρ δ − =  

ρ ρ ρ x  x  x 

Deviations on smaller scales

Dynamics of the inhomogeneous universe

Linear evolution of fluctuations:

d 2δ dt2 + 2H dδ dt − 3 2 ΩmH 2δ = 0

(In Newtonian approximation)

7

Homogeneous Background

S = d 4x

∫

−g R+ f (R) 16πG + Lm # $ % & ' (

Power law function of the Ricci scalar :

Perturbation Theory

) ( ) ( ) , ( ) , ( t t x t x t ρ ρ ρ δ − =  

ρ ρ ρ x  x  x 

Deviations on smaller scales

Dynamics of the inhomogeneous universe

Modification of Einstein-Hilbert action:

f (R) = −2Λ − c2 fR0 N R0

N+1

RN

Linear evolution of fluctuations:

d 2δ dt2 + 2H dδ dt − 3 2 ΩmH 2δ = 0

(In Newtonian approximation) (Starobinsky 1980) (Carroll et al. 2007, Sotiriou 2010)

7

Homogeneous Background

m−2 ≡ 3 d3 f d3R = −3(N +1) fR0c2 R0

N+1

RN+2

Ricci scalar:

Perturbation Theory

) ( ) ( ) , ( ) , ( t t x t x t ρ ρ ρ δ − =  

ρ ρ ρ x  x  x 

Deviations on smaller scales

Dynamics of the inhomogeneous universe

Scale dependency : Linear evolution of fluctuations:

d 2δ dt2 + 2H dδ dt − 3 2 ΩmH 2δgeff (k) = 0

(In Newtonian approximation)

R(a) = 6 2H 2 + dH dt ! " # $ % & geff (k) =1 + k2 / 3 a2m2 + k2

8

2-point correlation function of smoothed matter field: ξR(r, z) = D2(z).σ 8

2(0).GR(r)

variance of smoothed matter field:

Statistical properties of smoothed over-densities

σ R

2(z) = D2(z).σ 8 2(0).F R

8

2-point correlation function of smoothed matter field:

ξR(r, z) =σ 8

2(0).GR(r, z)

σ R

2(z) =σ 8 2(0).F R(z)

variance of smoothed matter field:

Statistical properties of smoothed over-densities

8

2-point correlation function of smoothed matter field:

ξR(r, z) =σ 8

2(0).GR(r, z)

σ R

2(z) =σ 8 2(0).F R(z)

variance of smoothed matter field:

( ) ( )

k d kr W k d kR W F

TH k TH k R

ln ln

8 2 2

∫ ∫

∞ + +∞

Δ Δ =

( ) ( ) ( )

k d kr W z k d kr j kR W z r G

TH k TH k R

ln ) ( ln ) ( ) (

8 2 2

∫ ∫

∞ + +∞

Δ Δ =

and

is the dimensionless power spectrum

) ( 4

3

k P k

k

π = Δ

and where

[ ]

) cos( ) sin( ) ( 3 ) (

3

kR kR kR kR kR WTH − =

Bel & Marinoni 2012 (MNRAS)

Statistical properties of smoothed over-densities

8

( ) ( )

k d kr W k d kR W F

TH k TH k R

ln ln

8 2 2

∫ ∫

∞ + +∞

Δ Δ =

( ) ( ) ( )

k d kr W z k d kr j kR W z r G

TH k TH k R

ln ) ( ln ) ( ) (

8 2 2

∫ ∫

∞ + +∞

Δ Δ =

and

• Redshift evolution
• Non linear bias
• Redshift distortions

2-point correlation function of smoothed matter field:

ξR(r, z) =σ 8

2(0).GR(r, z)

σ R

2(z) =σ 8 2(0).F R(z)

variance of smoothed matter field:

Statistical properties of smoothed over-densities

9

The matter clustering ratio:

ηR(r) = GR(r) FR δg = b0 + b

1δ + b2

2 δ 2 +... ηg,R(r) ≡ ξg,R(r) σ g,R

2

The galaxy clustering ratio:

The clustering ratio

If the second order bias coefficient satisfies to

| b2 / b

1 |<1

Bias between galaxy and matter fluctuations:

ηg,R

z (r) =ηR(r)

9

The matter clustering ratio:

ηR(r) = GR(r) FR ηg,R

z (r) =ηR(r)

ηg,R(r) ≡ ξg,R(r) σ g,R

2

The galaxy clustering ratio:

The clustering ratio

What you see (clustering of galaxies) is what you get (clustering of matter)

δg = b0 + b

1δ + b2

2 δ 2 +...

Bias between galaxy and matter fluctuations: If the second order bias coefficient satisfies to

| b2 / b

1 |<1

10

Impact of scale dependent bias:

The clustering ratio Vs Galaxy bias

Analysis performed on HOD galaxy mock catalogues described in de la Torre et al. (2013)

11

The matter clustering ratio:

ηR(r) = GR(r) FR ηg,R

z (r) =ηR(r)

Power spectrum Alcock-Paczynski

ηg,R(r) ≡ ξg,R(r) σ g,R

2

The galaxy clustering ratio:

The clustering ratio as a cosmological test

12

The clustering ratio as a cosmological test

ηg,R

z (r,

 Ω )/ηg,R

z (r,

 Ω

true) −1

ηR(r,  Ω )/ηR(r,  Ω

true) −1

z ~1

13

SDSS DR7 + VIPERS PDR1 Planck

14

The matter clustering ratio:

ηR(r, z) = GR(r, z) F

R(z)

ηg,R

z (r) =ηR(r)

Power spectrum Fixed by the expansion rate

ηg,R(r) ≡ ξg,R(r) σ g,R

2

The galaxy clustering ratio:

Measuring the fR0 parameter of modified gravity

(Planck) d 2δ dt2 + 2H dδ dt − 3 2 ΩmH 2δgeff (k) = 0

15

The matter clustering ratio:

ηR(r) = GR(r) FR ηg,R(r) =ηR(r)

Power spectrum Alcock-Paczynski

ηg,R(r) ≡ ξg,R(r) σ g,R

2

The galaxy clustering ratio:

Sloan Digital Sky Survey Data Release 7 + 10

16

ηg,R(r) =ηR(r)

Power spectrum Alcock-Paczynski

χ 2 = (ηg,R(zi)−ηR

th(zi))2

σ i

2 i=1 3

∑

Likelihood analysis:

Constraints of deviation from Einstein’s gravity

f (R) = −2Λ − c2 fR0 N R0

N+1

RN

17

Conclusions

• A new clustering statistic: the galaxy clustering ratio ηg,R = ξg,R

σ 2

g,R

• Its amplitude is the same for galaxies and matter
• It is weakly affected by redshift-space distortions
• The estimator is simple (count-in-cell) and robust (tests on simulations)
• Assuming a flat LambdaCDM universe and combining VIPERS and SDSS

measurements

Ωm = 0.274 ± 0.017

• Next: include massive neutrinos and constrain their mass
• Adapted to cosmological tests of gravitation (quasi-linear scale)
• Assuming Planck LCDM background cosmology

fR0 < 3×10−6

10

1 h-3 Gpc3 comoving

• utput at z=1 of

MultiDark simulation

(Prada et al. 2012)

ηg,R(r) ≡ ξg,R(r) σ g,R

2

The galaxy clustering ratio:

The clustering ratio in the weakly non linear regime

14 millions of Haloes with masses between 1011.5 h-1 and 1014.5 h-1 solar masses

n = r R

Smith et al. (2003) Linear

11

Redshift space (light cone):

ηR(r) = GR(r) FR ηg,R(r) ≡ ξg,R(r) σ g,R

2

Real space (comoving output):

The clustering ratio Vs Halo bias

What you see (clustering of galaxies) is what you get (clustering of matter)

n = r R = 3

11

Luminosity dependence in SDSS

11

Linear Perturbation Theory Distance- Redshift Relation ESTIMATOR Likelihood

Ω

R g ,

η

PRIORS:

• On Ho from HST
• On baryonic

density from BigBang Nucleosynthesis

• On the spectral

index from CMB

Application of the strategy (SDSS)

R

η

PRIORS: 1 . 1 9 . 03 . . 1 4 . 5 . 5 . 1 1 1

2

≤ ≤ ≤ Ω ≤ ≤ ≤ − ≤ ≤ − ≤ Ω ≤ ≤ Ω ≤

s b X X m

n h h w

(weak) (strong)

11

Linear Perturbation Theory Distance- Redshift Relation ESTIMATOR Likelihood

Ω

R g ,

η

PRIORS:

• On Ho from HST
• On pair-wise

velocity dispersion from VVDS

• On baryonic

density from BigBang Nucleosynthesis

• On the spectral

index and norm from CMB

Application of the strategy (VIPERS)

R

η

PRIORS: 0 ≤ Ωm ≤1 0.5 ≤ h ≤ 0.9 100 ≤ σ12 ≤ 500 0.015 ≤ Ωbh2 ≤ 0.03 0.9 ≤ ns ≤1. 0.02 ≤ Δ R

2 ≤ 0.09

(weak) (strong)

Non Linear Power Spectrum

40

where

ηR(r) = GR(r) FR

RSD parameter

Non Linear redshift space distortions

Dispersion model: (Gaussian) Angle average of the power spectrum in redshift space

(Peacock & Dodds 1994)

40

where

Non Linear redshift space distortions

Simple theoretical prediction: (Gaussian)

13

“Blind test” of N-body simulations: Figuring out the hidden cosmology

0 0.2 0.4 0.6 0 .8 1.0 1.2 1.4 0 0.2 0.4 0.6 0 .8 1.0

Λ

Ω

m

Ω

m

Ω

h = 0.21 ΩΛ = Ωm = 1 Ωb = $ % & & ' & & h = 0.70 ΩΛ = 0.75 Ωm = 0.25 Ωb = 0.04 $ % & & ' & &

V ≈1h−3Gpc 3 V ≈ 40h−3Gpc 3 Ngal ≈ 4M Nhalo ≈ 200000

40

Scale dependent galaxy bias

From de la Torre & Peacock (2013):

6

Homogeneous Background

a d d f ln lnδ ≡

Growth rate:

Perturbation Theory

) ( ) ( ) , ( ) , ( t t x t x t ρ ρ ρ δ − =  

ρ ρ ρ x  x  x 

Deviations on smaller scales

3) Dynamics of the inhomogeneous universe

Velocity fluctuations:

) , ( x t Hf v p

r

   δ − = ⋅ ∇

Linear evolution of fluctuations:

2 3 2

2 2 2

= Ω − + δ δ δ H dt d H dt d

m

(In Newtonian approximation)