Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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