Probing the matter power spectrum with the clustering ...ffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/BEL_Julien.pdf · Julien BEL Osservatorio Astronomico di Brera (with L. Guzzo) Collaborators:

Probing the matter power spectrum with the clustering ratio

of 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


iR

N

i

iRg i

bδδ ∑

=

=0

, !Fry & Gaztañaga (1993)

3


iR

N

i

iRg i

bδδ ∑

=

=0

, !Fry & Gaztañaga (1993)

€

δg,Rz = δg,R + µk

2 f 2δR

Kaiser (1987)

4

Variance of fluctuations in spheres:

€

σ g,R2 = 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,R2 = δg,R

2( x )

R

r


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)

DE

C

DE

C

RA

5

Perturbation Theory Statistical properties of smoothed over-densities D

EC

D

EC

RA


5

Perturbation Theory Statistical properties of smoothed over-densities D

EC

D

EC

RA


6

Homogeneous Background

Perturbation Theory

)()(),(),(

ttxtxt

ρρρ

δ−

=

ρ ρ ρ

x xx

Deviations on smaller scales

Dynamics of the inhomogeneous universe

Linear evolution of fluctuations: d 2δdt2

+ 2H dδdt−32ΩmH

2δ = 0(In Newtonian approximation)

7


S = d 4x∫ −g R+ f (R)16πG

+ Lm#

$%&

'(

Power law function of the Ricci scalar :

Perturbation Theory

)()(),(),(

ttxtxt

ρρρ

δ−

=

ρ ρ ρ

x xx



Modification of Einstein-Hilbert action:

f (R) = −2Λ−c2 fR0N

R0N+1

RN


+ 2H dδdt−32ΩmH

2δ = 0(In Newtonian approximation)

(Starobinsky 1980)

(Carroll et al. 2007, Sotiriou 2010)

7


m−2 ≡ 3 d3 fd3R

= −3(N +1) fR0c2 R0

N+1

RN+2

Ricci scalar:

Perturbation Theory

)()(),(),(

ttxtxt

ρρρ

δ−

=

ρ ρ ρ

x xx



Scale dependency :


+ 2H dδdt−32ΩmH

2δgeff (k) = 0(In Newtonian approximation)

R(a) = 6 2H 2 +dHdt

!

"#

$

%&

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:


σ R2 (z) = D2 (z).σ 8

2 (0).FR

8

2-point correlation function of smoothed matter field: ξR (r, z) =σ 82 (0).GR (r, z)

σ R2 (z) =σ 8

2 (0).FR (z)variance of smoothed matter field:


8


σ R2 (z) =σ 8


( )

( ) kdkrW

kdkRWF

THk

THk

R

ln

ln

08

2

0

2

∫

∫∞+

+∞

Δ

Δ

=

( ) ( )

( ) kdkrWz

kdkrjkRWzrG

THk

THk

R

ln)(

ln)()(

08

2

00

2

∫

∫∞+

+∞

Δ

Δ

=and

is the dimensionless power spectrum )(4 3 kPkk π=Δ

and

where

[ ])cos()sin()(3)( 3 kRkRkRkR

kRWTH −=

Bel & Marinoni 2012 (MNRAS)


8

( )

( ) kdkrW

kdkRWF

THk

THk

R

ln

ln

08

2

0

2

∫

∫∞+

+∞

Δ

Δ

=

( ) ( )

( ) kdkrWz

kdkrjkRWzrG

THk

THk

R

ln)(

ln)()(

08

2

00

2

∫

∫∞+

+∞

Δ

Δ

=and

•  Redshift evolution

•  Non linear bias

•  Redshift distortions


σ R2 (z) =σ 8



9

The matter clustering ratio:

€

ηR (r) =GR (r)FR

δg = b0 + b1δ +b22δ 2 +...

€

ηg,R (r) ≡ξg,R (r)σ g,R2

The galaxy clustering ratio:

The clustering ratio

If the second order bias coefficient satisfies to | b2 / b1 |<1

Bias between galaxy and matter fluctuations:

ηg,Rz (r) =ηR (r)

9


€

ηR (r) =GR (r)FR

ηg,Rz (r) =ηR (r)

€



The clustering ratio

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

δg = b0 + b1δ +b22δ 2 +...Bias between galaxy and matter fluctuations:

If the second order bias coefficient satisfies to | b2 / b1 |<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


€

ηR (r) =GR (r)FR

ηg,Rz (r) =ηR (r)

Power spectrum Alcock-Paczynski

€



The clustering ratio as a cosmological test

12

The clustering ratio as a cosmological test

€

ηg,Rz (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


ηR (r, z) =GR (r, z)FR (z)

ηg,Rz (r) =ηR (r)

Power spectrum Fixed by the expansion rate

€



Measuring the fR0 parameter of modified gravity

(Planck)

d 2δdt2

+ 2H dδdt−32ΩmH

2δgeff (k) = 0

15


€

ηR (r) =GR (r)FR

ηg,R (r) =ηR (r)


€



Sloan Digital Sky Survey Data Release 7 + 10

16

ηg,R (r) =ηR (r)


χ 2 =(ηg,R (zi )−ηR

th (zi ))2

σ i2

i=1

3

∑Likelihood analysis:

Constraints of deviation from Einstein’s gravity

f (R) = −2Λ−c2 fR0N

R0N+1

RN

17

Conclusions

•  A new clustering statistic: the galaxy clustering ratio ηg,R =ξg,Rσ 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 output at z=1 of MultiDark simulation (Prada et al. 2012)

€



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 = rR

Smith et al. (2003) Linear

11

Redshift space (light cone):

€

ηR (r) =GR (r)FR

€


Real space (comoving output):

The clustering ratio Vs Halo bias

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

n = rR= 3

11

Luminosity dependence in SDSS

11

Linear Perturbation Theory

Distance-Redshift Relation

ESTIMATOR

Likelihood Ω

Rg ,η

PRIORS: • On Ho from HST

• On baryonic density from BigBang Nucleosynthesis

•  On the spectral index from CMB

Application of the strategy (SDSS)

Rη

PRIORS:

1.19.003.0.014.05.05.1

1010

2

≤≤

≤Ω≤

≤≤

−≤≤−

≤Ω≤

≤Ω≤

s

b

X

X

m

nhh

w

(weak) (strong)

11

Linear Perturbation Theory

Distance-Redshift Relation

ESTIMATOR

Likelihood Ω

Rg ,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)

Rη

PRIORS:

€

0 ≤ Ωm ≤10.5 ≤ h ≤ 0.9100 ≤σ12 ≤ 500

0.015 ≤ Ωbh2 ≤ 0.03

0.9 ≤ ns ≤1.0.02 ≤ ΔR

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

ΛΩ

mΩ mΩ

€

h = 0.21ΩΛ = 0Ωm = 1Ωb = 0

$

% & &

' & &

€

h = 0.70ΩΛ = 0.75Ωm = 0.25Ωb = 0.04

$

% & &

' & &

€

V ≈1h−3Gpc 3

€

V ≈ 40h−3Gpc 3

Ngal ≈ 4MNhalo ≈ 200000

40

Scale dependent galaxy bias

From de la Torre & Peacock (2013):

6


addflnlnδ

≡Growth rate:

Perturbation Theory

)()(),(),(

ttxtxt

ρρρ

δ−

=

ρ ρ ρ

x xx


3) Dynamics of the inhomogeneous universe

Velocity fluctuations: ),( xtHfvpr

δ−=⋅∇

Linear evolution of fluctuations: 0232 2

2

2

=Ω−+ δδδ HdtdH

dtd

m(In Newtonian approximation)

Documents

Probing the matter power spectrum with the clustering ...ffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/BEL_Julien.pdf · Julien BEL Osservatorio Astronomico di Brera (with L. Guzzo) Collaborators: