The accuracy of a redshift-space bispectrum model based on perturbation theory

CosKASI-ICG-NAOC-YITP 2017

Yukawa institute for theoretical physics

Ichihiko HashimotoCollaborators： Atsushi Taruya (Kyoto U), Yann Rasera (Paris Observatory)

Phys. Rev. D 96, 043526 (2017)

Test of gravity has opportunity to distinguish

Small scale：Free fall Large scale：Galaxies

Origin of accelerated expansionDark energy　or　Modified gravity 　？？

・Coherent motion 　→make clustering stronger

・Large scale( ~100Mpc)　

・Random motion with large velocity 　→make clustering weaker

・Around halo ( <10Mpc)　

・Finger-of-God effect (FoG)

・Strength of distortion is proportional to velocity

Correlation functionVVDS,

s?[Mpc/h]

s k[M

pc/h]

Guzzo et al. 2008

line of sight

0.6 < z < 1.2, 4deg2

v / f ⌘ d lnD+

d ln af(z)

Redshift-space distortions s = r +

vz(r)

aH(z)z red-/blue- shift due to peculiar velocity

redshift space real spaces = r +

vz(r)

aH(z)z : line of site

Current constraints

・Consistent with ΛCDM at ~10% level ・Mostly from the two point correlation or power spectrum

Okumura et al. (2015)z

f(z)�

8(z)

We want to add redshift-space bispectrum

Constraints from bispectrum

・Anisotropic information of bispectrum is not included

Hector et al. (2016)CMASS sample

・Bispectrum improves the constraint of for ~15%f�8

Combining bispectrum in future survey

z0

5

10f Power spectrumBispectrum

Power + Bispectrumerror of (%)

1�

kmax

= 0.1 [h/Mpc]

Song, Taruya and Oka (‘15)

Impact of future survey

Combining bispectrum improves the constraint by a factor of two for DESI

Aim of this work・Constructing precise model of bispectrum which can be applied for future survey・Including the information of bispectrum up to quadrupole moment

Theoretical models Previous works

・Including non-linear effects up to 1-loop levelNon-linear effects: RSD & gravitational growth

・Investigating the validity of theoretical model by comparing N-body up to quadrupole of bispectrum

・Scoccimarro, Couchman & Frieman (’98)・Smith et al. (’07), Yamamoto et al. (’16)・Gil-Marin et al. (’16,’15)

Our workBased on a formula of tree level calculation

Theoretical model of bispectrum

e.g. Taruya et al. (‘10)

This model breaks down at large scales even in the case of power spectrum

Ai = �(ri) + frzuz(ri) (i = 1, 2, 3) uz = (v · z)/(aH)

Damping effect

j4 = �ik1zf, j5 = �ik2zf

Exponential term is expanded as power seriesZdr12dr23heX · · · i =

Zdr12dr23h(1 +X +X2/2 · · · ) · · · i

Zdr12dr23heX · · · i =

Zdr12dr23h(1 +X +X2/2 · · · ) · · · i

O((�1)4) O((�1)

6)

B(s) ' Btree

+B1�loop

+ · · ·B(s) ' Btree

+B1�loop

+ · · ·B(s) ' Btree

+B1�loop

+ · · ·B(s) ' Btree

+B1�loop

+ · · ·B0PT(k1,k2,k3) B0

PT(k1,k2,k3)

Expand naively by → standard perturbation theory�1

Exact formula of redshift-space bispectrum

B(s)(k1,k2,k3) ' heX · · · i= exp{heXic}{h· · · ic · · · } Taruya et al. (‘10)

B(s)(k1

,k2

,k3

) = DFoG

(k1

,k2

,k3

)(Btree

+B01�loop

)B0PT(k1,k2,k3)B(s)(k1,k2,k3)

Calculate up to 1-loop levelOur model

Ai = �(ri) + frzuz(ri) (i = 1, 2, 3) uz = (v · z)/(aH)

Damping effect

j4 = �ik1zf, j5 = �ik2zf

Do not expand damping part → TNS modelExponential term is divided by cumulant expansion

Exact formula of redshift-space bispectrum

Theoretical model of bispectrum

Comparing with N-body

Box size:1312.5 Mpc/h, # of particles:512^3, # of realizations :512

Dark Energy Universe Simulation (DEUS)

Comparing bispectrum expanded by Legendre polynomial B(s)

` (k1, k2, ✓12) =

Z 1

�1dµ

Z 2⇡

0

d�

2⇡B(s)(k1, k2, ✓12, µ,�)P`(µ)

µ = cos!

!r2z

r1

✓12k1

k2k3

�

!

z

✓12

k1

k2k3

�

Scoccimarro et al ('98) Our definition: line of sight : line of sight

Blot et al. (‘14)

Hashimoto et al. (‘17)

Bispectrum of matter field

kmax

|k3B

(s)

`(k)|⇥10

�4[(Mpc)

3] monopole quadrupole

Simulation1-looptree

Scale dependance of bispectrum (k1 = k2 = k3)

Consistent with N-body in wide range for both case, monopole and quadrupole

✓12k1

k2 k1 = k2 = 0.096 h/Mpc

B(s)2B(s)

0

|k3B

(s)

`(k)|⇥10

�4[(Mpc)

3] monopole quadrupole

Consistent with N-body in wide range for both case, monopole and quadrupole

Shape dependance of bispectrum

Bispectrum of matter field

Consistency for power spectrum

The result of fitting bellow kmax

In our model, is consistent with power spectrum in matter field

�v

　 　 dependance of . �vkmax

Our model tree+damping

Recovery of f(z)

2 parameter fit to N-body data: f and σv

✓ streaming model• seams OK only at kmax<0.1h/Mpc• typically ~5% underestimate of f

✓ TNS model• gives unbiased estimate of f• up to kmax ~ 0.2 h/Mpc

Recovery of f(z)

2 parameter fit to N-body data: f and σv

✓ streaming model• seams OK only at kmax<0.1h/Mpc• typically ~5% underestimate of f

✓ TNS model• gives unbiased estimate of f• up to kmax ~ 0.2 h/Mpc

Recovery of f(z)

2 parameter fit to N-body data: f and σv

✓ streaming model• seams OK only at kmax<0.1h/Mpc• typically ~5% underestimate of f

✓ TNS model• gives unbiased estimate of f• up to kmax ~ 0.2 h/Mpc

Next step・Quantifying systematic error when we estimate 　cosmological parameters by using our model

・Matter density Halo number density

In the case of power spectrumFitting parameters: f,�v, b

Viewgraph by T. Nishimichi

We want to do similar test in the case of bispectrum

Method1. Construct halo number density field from N-body

2. Estimate parameters by our theoretical model

3. Compare fiducial and estimated parameters

Box size : 1 [Gpc^3/h], # of particles:1024^3, # of realizations : 300

Gadget2 :

Springel et al. (2001)

Identifying halos ( ) by using subfind code

Mh > 10�12Msun

f,�v, bFitting parameters:

For preliminary test, we employed linear bias model :�h = b�m

Statistics of halo field

In halo number density field, FoG effect becomes weaker than in matter density field

Power spectrum Bispectrum

Matter

Halo

Matter

Halo

redshift space

real space

⇥k3/210�2 ⇥k310�4

Even linear bias model, our theoretical model regenerate simulation results in high accuracy

|k3B

(s)

`(k)|⇥10

�4[(Mpc)

3]

f,�v, bFitting parameters:

Preliminary results

monopole quadrupole

Scale dependance of bispectrum (k1 = k2 = k3)

Mh > 10�12Msun

Preliminary results

・Best fit values are well consistent with fiducial

Growth rate

・ of our model is even at large�2red O(1)

kmax

・We should add error bars

Preliminary resultsVelocity dispersion

bin1 bin9

heavylight

z=0.35

Nishimichi et al. (2011)

Mh > 10�12Msun

Bispectrum Power spectrum

SummaryConstructing the model of redshift-space bispectrum taking account of nonlinear effects on gravity and RSD

What we did

Result

Future work ・Including proper bias model ・Investigating the error of estimated parameters

・Our model produce bispectrum in quasi non-linear regime in matter and halo number density field ・Best fit value of is well consistent with fiducial value ・ of bispectrum is consistent with that of power spectrum �v

f